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Stroud would be an excellent choice, especially since it has answers to problems in the back of the book (remember: you only really learn math by doing it!). I also see that it has a section on statistics, so that may be all you need. If not, I think a good book that is tailored to your needs is something like An Introduction to Medical Statistics. Do a ...


5

According to Ian Stewart, the symbol "!" was introduced because of printability. Before 1808 $\underline{n\big|} = n \cdot (n-1) \cdots 3 \cdot 2$ was [widely?] used to denote the factorial. Because it was hard to print [in non-computer ages], the French mathematician Christian Kramp chose "!". Source: Professor Stewart's Hoard of Mathematical Treasures


2

The right way to build such a category is a philosophical question. There are different approaches in the mathematical literature. One thing is clear though: the objects should be propositions, not just theorems. The problem is to define equality of proofs in a sensible way. For example, let $\Pi$ be Pythagoras' theorem. Should each of the over 100 proofs ...


3

See for example Lambek and Scott: Introduction to higher order categorical logic, ch 0.1 (unfortunately not in the net but for sure in ur university library). There first is defined a graph, then a deductive system as a graph with For each object $A$ an identity arrow $1_A:A\rightarrow A$ For each pair of arrows $f:A\rightarrow B$ and $g:B\rightarrow C$ ...


1

Yes, currently the problem is open whether or not the Partition Principle and the Axiom of Choice are equivalent. There are two major factors for this (in my opinion): Many people become less interested in choiceless results. So while they might be very happy to hear about them, they prefer to put their research efforts towards other directions. The ...


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These subjects are not all contained in what most people would think of as abstract algebra... I don't think it's really possible to fulfil your "one book only" request. Groups and rings are definitely abstract algebra. Any introductory abstract algebra book will cover them. A Book Of Abstract Algebra is quite good for a first pass, but not as comprehensive ...


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I can't confirm that every single topic above is included, but Serge Lang's Algebra is an extremely comprehensive tome for all algebra topics up to and including first year graduate-level algebra. It's not exactly readable, but it has a ton of exercises and gives you all the logical steps you need to explore these topics.


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p-regular languages are commonly known as (regular) group languages in the literature since their syntactic monoid is a finite group. If a language is accepted by a permutation automaton, then its minimal DFA is also a permutation group, but this group is transitive (since every state is accessible from the initial state). Thus your subclass is actually ...


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Stewart's Calculus will prepare you for the calculus questions in the GRE, while Rudin's Principles of Mathematical Analysis will prepare you for the introductory real analysis questions in it.


1

Another definition of laplacian matrix M( or L) = $QQ^t$ where Q is incidence matrix. By cauchy-binet theorem, one can calculate the determinant of a rectangular matrix by considering $Q and Q^t$. So, if we take a cofactor, then it can contain twigs of a tree and since determinant of matrix containing (#nodes-1) and twigs of a tree is +/- 1. So, multiplying ...


1

You might be interested in the books Algebra and Trigonometry by Gelfand. Also, it's not dry or old, but the Precalculus textbook from artofproblemsolving.com won't spoonfeed, at least. From the book description: It includes nearly 1000 problems, ranging from routine exercises to extremely challenging problems drawn from major mathematics ...


1

Yes I hear your point. Most books released these days look to spoon-feed. But that does not apply to all new books. I mean Spivak's books, Chapman Pugh's text on Analysis are examples. Now these are the books I perused during A Levels. I only got my hands on them because the government sells them for dirt cheap prices (I mean for less than 20 cents US). ...


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See this section from Computational Number Theory by Abhijit Das, 1.7.2. "We do not know any efficient algorithm for computing ${\rm ord}_m a$ unless the complete prime factorization of $\phi(m)$ is provided." He gives a pretty easy algorithm which I used as the basis for my C code. Note that you can use the Carmichael Lambda instead of the totient which ...


0

It appears that they are one and the same. It is known that $$\eta_1 = -\frac{\pi^2}{3\omega_1} \frac{\theta_1{'''}(0)}{\theta_1{'}(0)} = \frac{\pi^2}{3\omega_1}\left(1 - 24\sum_{n = 1}^\infty \frac{q^{2n}}{(1 - q^{2n})^2}\right),$$ where $\eta_1 = Z(\omega_1)$ is a quasi-period of $Z(z)$, the Weierstrass zeta function and $\omega_1$ is a period of The ...


0

'Mathematical Foundation of Elasticity' written by Marsdan for any person who hopes to become an expert in strength of materials, explicitly states on its back cover that it will teach the functional analysis from the start. First edited in 1983 then reprinted by Dover publication at a low price. A 1.5-inch thick, yet not isoteric textbook.


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One cannot be realistic, honest to himself, in approaching Folland's manual on functional anlysis, with merely three or four light-weight calculus courses, two college-level linear algebra course and one ordinary differential equations course for biologist. At which profession do you focus? and at which university level? Folland's book is too deep (and too ...


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I'm not an expert, but I think there's a "baby step giant step" method, when $n$ is too large for brute force to be effective. If you know the factorization of $\phi(n)$ already, though, then you can find the order by calculating $a^k$ modulo $n$ for increasingly small divisors $k$ of $\phi(n)$. For example, take $a=342$ and $n=803=73\cdot11$, so that ...


1

Constructive proof: Consider some nonzero substochastic matrix $P$ with equal-row-sums. Call $P_{is(i)}$ one of the minimal nonzero entries of row $i$, and consider the deterministic matrix $D$ such that $D_{is(i)}=1$ for every $i$ and $D_{ij}=0$ otherwise. Finally, define $m(P)=\min\{P_{ij}\mid P_{ij}\ne0\}$. Then $P-m(P)D$ is a substochastic matrix with ...


1

I found this a valuable resource: A Survey of Arithmetical Definability by Alexis Bès


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Euclid's "The Elements". Greek. Old. It doesn't get much more classic than that. As well as his other writings. http://en.wikipedia.org/wiki/Euclid#Other_works Also, the Principia Mathematica by Newton. As well as his other writings. http://en.wikipedia.org/wiki/Isaac_Newton#Mathematics Archimedes probably deserves a mention as well. You know, pi ...


3

The founder of the Institute for Advanced Study, A. Flexner, once published a paper called "The usefulness of useless knowledge" http://library.ias.edu/files/UsefulnessHarpers.pdf. This paper justifies all theoretical sciences. I get a kick out of reading the article. I think about another one, "The Spirit and the Uses of the Mathematical Sciences", ...


2

If you want to do applied math without theory, then respectfully, you shouldn't go into applied math. Even applied mathematicians care about where things come from and how to justify them, so you won't be able to avoid proofs and theorems. With that said, a few of my favorite resources are as follows: Finite Difference Methods for Ordinary and Partial ...


2

Not exactly what you were looking for (because these are "secondary sources"), but maybe interesting nevertheless: Mathematics and Its History by Stillwell walks you through the history of mathematics showing original problems in modern notation with many good exercises at an undergraduate level and with lots of pointers to the original sources. Euler - ...


1

Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. Although not strictly speaking a purely mathematical book, surely I would put it among the classics.


1

If you want to learn algebraic geometry, a classical paper is Jean-Pierre Serre's FAC. See here for the French original, and here for the Englisch translation. See here for some advertisement by Georges Elencwajg.


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Three books: Euler's Introduction to the Analysis of the Infinite and Foundations of the Differential Calculus both translated by JD Blanton and published by Springer, also the very informative Analysis by its History by Hairer and Wanner. There are always the original papers by the biggies which are more often than not very interesting, illuminating and ...


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There are several Source Books that have made nice selections for you to pick from, e.g., Smith, Struik, Fauvel and Gray, Stedall. But for an extended read, you can do nothing wrong by immersing yourself in Gauss' Disquisitiones.


5

For introductory Number Theory, you could go with Gauss, Disquisitiones Arithmeticae. Don't worry, you don't have to read Latin, it is available in English and other living languages.


2

Today was the first day of class in my complex analysis course. I sometimes attempt new derivations in real time to keep it fresh. Today, we got to the point of asking what was the reciprocal of $z=x+iy$. We said, let $w=a+ib$ and seek solutions of $wz=1$. This gives: $$ wz = (a+ib)(x+iy) = ax-by+i(bx+ay) = 1+i(0).$$ Equating real and imaginary parts ...


5

This paper, called Physics, Topology, Logic and Computation: A Rosetta Stone does just that in section 3.2. If you have time and interest, I would suggest reading the entire paper (since the whole thing is pretty cool).


1

The first component of $\mathcal{A}$ corresponds to the trivial representation, the second component, to the sign representation, and the third, to the standard representation (see here). By choosing a specific matrix basis, we obtain: identity $\to$ $(1) \oplus (1) \oplus \begin{pmatrix} 1& 0 \newline 0& 1 \end{pmatrix}$, $(123)\to$ $(1) ...


0

I have found the following published reference: Elementary proof of Zsigmondy's theorem by M. Teleuca, which also cleans up the proof of Birkhoff & Vandiver.


1

It is an application of Courat-Fischer Theorem (or Min-Max Theorem), and here there is a problem on this subject with it's answer


1

i have heard the name "Cauchy interlacing theorem" (as well as "Cauchy's interlace theorem" or "Interlacing eigenvalues theorem for bordered matrices")Infor that theorem. that could be a hint that cauchy did it first or was among the first. sorry i have no idea where to find an original paper of that, but i hope my post helps anyways.


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Perhaps [1] could be of use. However, see [2] if you want a very abstract approach (an approach that is related to topics in the links I gave in my comment here). [1] Handbook of Mathematical Induction by David S. Gunderson (over 900 pages) [2] Elementary Induction on Abstract Structures by Yiannis N. Moschovakis


3

The first few chapters of GH Hardy's "A Course of Pure Mathematics" may be worth a read.


0

Recently, I came across the example in an Encyclopedia of Mathematics (Finite-difference calculus). This partially answers my question in a simple case, but it has some nuances that I want to be cleared, so I edited the head post to include more questions. The following is the part from the ecyclopedia relevant to my question. $$\varPhi(x, f(x), f(x+1), ...


2

As dry, old and rigorous as it gets "Advanced Mathematics Precalculus with discrete mathematics and data analysis." It's what I had in High School, although I had a modern textbook as a suppliment. There might be newer versions out, but I assume you want the older ones.


1

Let $G$ be a bipartite graph with bipartition $(A,B)$. The idea is to apply Menger's Theorem to a new graph $G^\prime$ obtained from $G$ by adding two vertices $u$ and $v$, and joining $u$ to all vertices in $A$, and $v$ to all vertices in $B$. Now you just need to check that a matching in $G$ corresponds to a set of internally-disjoint $(u,v)$-paths in ...


3

I think Milnor's "Topology from a Differentiable Viewpoint" (accessible online) has a proof of this in the appendix. The proof is pretty simple, but uses one slightly technical lemma. I would say the Theorem is easy to see since the idea is straightforward, but requires slightly more work to prove. Here it is. See page 55.


4

We break up this answer into a few different categories which partially overlap. Non-textbook resources. Supplemental Resources, which might try to give a bird's eye view, but doesn't give a complete course. Introductory resources, or what might usually be used in a first year of calculus in a university, usually where proofs do not form a large ...


3

Well, if you're serious about applied mathematics-and serious in that you don't just want "reciepe" books,rather applications that build on the meaty theory background you have-then you should avoid such texts and try and locate books that don't avoid theory,but merely downplay it. Those are the "real" applied mathematics textbooks. You definitely need to ...


2

I recommend Gilbert Strang's books: 1) Introduction to Applied Math; 2) Computational Science and Engineering.


0

Q1: These are right singular vectors, see Singular value decomposition Q2: The geometric meaning of the product of two largest singular values is the maximal amount of area increase under the map. That is, if you have a 2-dimensional plane (or surface) transformed by $T$, the area will increase by at most $\sigma_{n-1}\sigma_n$. Q3: I can't really say, ...


1

According to my calculations in sage, there are 11 graphs (out 156) on six vertices such that $\det(I-M)=1$ and exactly two of these eleven are trees. This shows that is is going to be very difficult to determine information about cycles from the value of $\det(I-M)$. (There is nothing special about the value '1' here, for example example there are 35 graphs ...


0

Writing a chess engine that plays only moderately well, is a considerable programming challenge. Imo, very few people have the discipline, ability and patience to actually organize and execute something like that from scratch, programming-wise. There are some open sources which you can see and read, like that of GNU chess, Borland's Turbo chess and several ...


0

Start here.          


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The Knights tour is a famous chess/math problem.


0

My own version is as follows: if $k$ is the Fourier transform of a certain function from $L^1$, then $K$ belongs to the trace class. Proof. $\,$ Suppose that $k=\mathcal F u$ with $u\in L^1$. Then $$ k(x-y)=\int_{-\infty}^{+\infty} u(t)\exp(it(x-y))\,dt =\int_{-\infty}^{+\infty} u(t)\exp(itx)\exp(-ity)\,dt. $$ For $t\in\mathbb R$, denote by $A_t$ the ...


0

You might enjoy Piergiorgio Odifreddi, The Mathematical Century: The 30 Greatest Problems of the Last 100 Years, published by Princeton University Press in 2006. Also, Ben Yandell, The Honors Class: Hilbert's Problems and Their Solvers, published by AKPeters in 2001. Here's a link to a review.



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