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Since given a product $\prod_i a_i$ setting $a_i' = a_i - 1$ will transform it in a product $\prod_i (1 + a_i')$ there is nothing special about that type of product without further restrictions. That being said, the form you gave is common when considering infinite products, among others due to the inequality you mentioned that helps to establish that ...


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Real and Functional Analysis by Serge Lang. Functional Analysis by Walter Rudin.


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I think this has nothing in particular to do with the Borel fibration. Given any fiber bundle $F \to E \to B$ with connected fiber $F$, one has $H^0(F;\mathbb Z) \cong \mathbb Z$, and the monodromy action induced on this group by the fundamental group $\pi_1(B)$ of the base is trivial. Thus in the cohomological Serre spectral sequence of this bundle, ...


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From pg. 35 Complex Analysis by Serge Lang, conformal maps are differentiable (holomorphic) so this fact follows from the fact that differentiable functions are continuous and continuous functions have this property. Not sure what this means. Every translation z--> z+b is conformal. It maps C --> C. How can this be extended "onto" a nowhere dense set? This ...


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From what you write, it seems you are defining the following sequence of sets: $(A_n)_{n\ge 1}$ such that $A_1=A_2=\{1\}$ and $A_{n+1}$ is the set of all possible non-empty sums with values in $A_1,\ldots,A_n$ and such that they do not belong to $A_1 \cup \cdots \cup A_n$. Well, at this point, it is not difficult to prove by induction that $$\forall n\ge 4, ...


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Here is a guess: First, we need to define "geometry": I will assume that it means a complete simply-connected Riemannian manifold of constant curvature $0$ or $\pm 1$. (This, of course, excludes many other geometries.) Next, given a Riemannian manifold $(M,g)$ with the Levi-Civita connection $\nabla$, a diffeomorphism $f: M\to M$ is called affine if it ...


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"Problems in Algebraic Number Theory" http://www.amazon.com/Problems-Algebraic-Number-Graduate-Mathematics/dp/0387221824 And you might consider Marcus "Number Fields" http://www.amazon.com/Number-Fields-Universitext-Daniel-Marcus/dp/0387902791/ref=sr_1_1?s=books&ie=UTF8&qid=1422568022&sr=1-1&keywords=marcus+number+fields While it is an ...


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One can define hyperbolic geometry as the study of points and chords of a disk $D$ invariant under affine transformations of $D$ to itself; see Klein's disk model of non-Euclidean geometry. However, there are also others models of hyperbolic space, e.g., the Poincare model. The remark could be related to Klein's disk model of hyperbolic geometry, but I am ...


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Consider the generic transfer function $$ H(s) = \frac{Y(s)}{X(s)} = \frac{b_0s^n + \cdots + b_ns^0}{s^n + a_1s^{n-1} + \cdots + a_n} $$ Then the state space model is \begin{align} \dot{\mathbf{q}} &= \mathbf{Aq} + \mathbf{B}u\\ y &= \mathbf{Cq} + Du \end{align} where \begin{align} \mathbf{A} &= \begin{bmatrix} -a_1 & 1 & 0 & \cdots ...


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If you want a probability book that uses real analysis but not measure theory, then you want an older book back before measure theory became so central to the subject. I recommend Feller's two volume work, An Introduction to Probability and Its Applications.


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Ash's book on Probability Theory is fantastic and covers what you want, although at a bit higher level, with some measure theory: http://www.amazon.com/Probability-Measure-Theory-Second-Robert/dp/0120652021/ref=asap_bc?ie=UTF8


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Let me concisely but properly address your questions. 1) Learn about tensor products and multilinear algebra in general. (Since you mention that you read Lang's linear algebra book, there is a more advanced algebra book by Lang, called - guess what - Algebra; it is a standard reference.) These things are totally crucial in e.g. commutative algebra, ...


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Discrete just have to be separated from one another, finite means the total number must be $<\infty$. So $f(x)=x$ has no discontinuities at all, and $f(x)={1\over x}$ has one discontinuity at $0$ and again, this number is finite. On the other hand the floor function $f(x)=[x]$ (definition below if you haven't seen it) has infinitely many of them--one at ...


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I highly recommend Advanced Linear Algebra by Steve Roman. It has complete proofs for everything covered in the book. Moreover, the proofs are very well written and are a pleasure to read. It is written abstractly and assumes that you already know abstract algebra.


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How about this book? At my school, it is the text used for Advanced Linear Algebra. Where I go, Abstract Algebra is a prereq for it. I would imagine that book would be along the lines of what you're looking for. The book also has positive reviews from what I've seen.


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You are right, there are not many books on MCF in a Riemannian manifolds. There are basically two reasons: (1) Not much is known in the general case. One early result was given by G. Huisken, which says that if the curvature of the ambient space is pinched and the initial embedding is convex, then the MCF shrinks to a round point (This is a generalization ...


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For the math subject test, you shouldn't just focus on Calculus, but if you do, Spivak would be my suggestion since it provides more rigor. However, you need to know analysis (real and complex), linear algebra, topology, probability, abstract algebra, combinatorics, and some other fields at a decent level. Here is the link to practice book supplied by ETS. ...


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I suggest Thomas's Calculus as well as Hille's Calculus (as a secondary source).


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What follows is an attempt to motivate this beautiful and difficult (in my opinion) subject. It is just an attempt, I cannot promise it will be useful. Suppose you have a family of curves over $\mathbb A^1=\textrm{Spec }\mathbb C[t]$, like for instance the family $$\pi:\textrm{Spec }\mathbb C[x,y,t]/(xy-t)\to \mathbb A^1$$ given by $t\mapsto t$. As it is ...


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I figured it out on my own, and since none of the answers answer my question I will just post mine. The third proportion to $a$ and $b$ is defined as the number $x$ such that $a,b,x$ are in a geometric progression. If the question was instead the third proportion to $b,a$, then it would be the number $x$ such that $b,a,x$ are in geometric progression. Then ...


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A continued proportion is a pair of equations like: $$\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$$ A fourth proportional is a solution of the equation $$\frac{a}{b}=\frac{c}{d}$$ where three of the numbers are known and the other is the unknown. There are several cases. As mean proportional is a solution of either $$\frac{a}{b}=\frac{b}{c}\ \ \ \ \text{ or ...


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If you're trying to find mathematical sense in the physical world, then let me ask you this: How do you measure length? Do you count the number of molecules? atoms? quarks? Tell me what you basic unit is, and I will ask you to represent $\frac12$ with it. By the way, these numbers are called irrational because they don't make (rational) sense. In ...


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The important fact here is that you can't have EXACLTY a $1$ meter slab because there is (even if very small) an "uncertainty" with the length of the slab. In fact we could measure the length of the slab infinitely many times but it won't never be $1$ exactly (for more details see this numberphile video). So if we can't have a length $1$ we can't have ...


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Another great book is Introduction to Calculus and Classical Analysis by Omar Hijab. It contains many solved problems to show you how the reasoning is like and is well written. http://www.amazon.com/Introduction-Calculus-Classical-Undergraduate-Mathematics/dp/1441994874


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I liked A course in computational number theory by David Bressoud and Stan Wagon, since I like to use Mathematica. You probably don't want the otherwise excellent book 104 Number Theory Problems: From the Training of the USA IMO Team by Titu Andreescu, Dorin Andrica and Zuming Feng because the problems are not graduated.


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The best physical intuitive and interactive explanation of eigenvalues/eigenvectors can be found in the link below http://setosa.io/ev/eigenvectors-and-eigenvalues/


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I would consider Mathematics of Choice by Niven and Graphs and Their Uses by Ore. http://www.maa.org/publications/ebooks/anneli-lax-new-mathematical-library These books are part of a series intended for talented high schoolers, and the authors were first-rate mathematicians.


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Irving Adler in 'a new look at geometry' gives a circular chain of proofs of alternate versions of the fifth postulate (that if a line striking each of two lines at the same angle, then the two lines do not cross at any distance). The idea of such a proof shows that all of these propositions are identical in function.


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The first example that comes to my mind is the Cauchy-Schwarz inequality. Here there is a paper I found some time ago in which the authors claim (I write claim, because I did not check each proof) to be able to show twelve different proofs of the result. Actually, there is an entire book on inequalities that starts from this very basic one, with various ...


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Here I found that 20 different proofs for the Euler Formula$$V-E+F=2.$$


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A classic is Hadamard's Lessons in Geometry, recently translated into English. The original French version is free on the internet (legally). If you do choose to read Euclid directly, as proposed in a comment, you might want to supplement your reading with Hartshorne's Geometry: Euclid and Beyond to get a modern perspective. I think Pogorelov's Geometry ...


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I would absolutely recommend Artin's Algebra in your situation. Apart from the book being excellently written, one of its major advantages is that it develops linear algebra and abstract algebra in parallel. Many of the most interesting applications of groups are to geometric problems, and Artin's book is great for that. Many other books on abstract ...


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My functional analysis course's Textbook : Functional Analysis , K.Yosida. This book contains a wide range of topics in FA, and many proofs are elegant. The only shortage is that there is no exercises in the book.


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You can start out with the book Introductory Functional Analysis by B. Daya Reddy. I started reading it and was not able to stop because everything was so clear and rigorously defined. This book is unlike any other book you find off of the shelf, which are filled with indecipherable notations that are not transferable beyond what that particular book ...


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There's a proof of this fact (in the category of CW-complexes, at least) in section 4.G of Hatcher's "Algebraic Topology." The basic idea is to construct an explicit map $X\to N(\mathfrak{U})$, then use Whitehead's theorem to show that it's a homotopy equivalence. The proof uses paracompactness, but it's only used to construct a partition of unity; the local ...


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Your question is quite broad, but I agree with Modded Bear that AP calc is unlikely to be of much help. The Concrete Mathematics book would be excellent, but I might throw in the book Discrete Mathematics and Its Applications by Kenneth Rosen. This book is an absolute tome with thousands of exercises (literally) that range from the very easy to the ...


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knowing AP calculus will probably not help a lot unless you want to learn generating functions. In my experience you the prerequisites you might not know in some books (although not all) may be linear algebra or group theory. Here are some books I recommend on the subject: Concrete Mathematics Problem solving methods in combinatorics Bondy and Murty GTWA ...


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I'd recommend learning about other topics in mathematics that strongly employ the language and techniques of linear algebra. The two clearest examples to me are functional analysis and differential geometry. In functional analysis you get to see the ideas of vector spaces, except these are expanded to the cases of infinite-dimensional Banach and Hilbert ...


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For any linear map $S:\mathscr{H}_1\to\mathscr{H}_2$ between two vector spaces we can define its transpose as $S^*:\mathscr{H}_2^*\to\mathscr{H}_1^*$ by letting $(S^*f)(x)=f(Sx)$ for any $x\in\mathscr{H}_1$ and $f\in\mathscr{H}_2^*$. Hilbert spaces are naturally self-dual (by the Riesz representation theorem!), so we can think of $S^*$ as a map ...


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Here are some hints about how to do this constructively. Suppose $p,q$ are points in the hyperbolic plane such that the length of the arc of the horocycle between $p,q$ is equal to $\ell$. Then we may assume that, in the upper half plane, $p=(0,1)$ and $q=(\ell,1)$. To get a formula for the length of the geodesic segment between $p,q$, construct a ...


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Proposition 2.1.13 from Elementary Number Theory: Primes, Congruences, and Secrets by William Stein (freely available on his site) is: If $\text{gcd}(a,n) = 1$, then the equation $ax ≡ b ~(\text{mod } n)$ has a solution, and that solution is unique modulo $n$. From there, it is only a small step to the required statement.


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Consider $SL(d,\mathbb{C})$ as a real algebraic group (ie, replace each matrix entry with a $2 \times 2$ matrix representing a complex number with real entries). Then the complexification has the structure of $SL(d,\mathbb{C}) \times SL(d,\mathbb{C})$ and is hence $SL(d,\mathbb{C})$, considered as a group over $\mathbb{R}$, is not absolutely simple.


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No, there is no general way to calculate a "general series" because there are just so many series, they can be very involved. There are even very simple finite sum like $\sum_{i=1}^n 1/i$ that have no simple closed form.


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It seems the following. There is no such $\alpha$. Indeed, assume the converse. Fix $y=1$ and let $x\to\infty$. Then we obtain $\alpha>1$. From the other side, put $x_n=e^{-2n}$, $y_n=e^{-n}$, for each $n$. Then $$|x_n-y_n|^\alpha=e^{-n\alpha}+o(e^{-n\alpha}).$$ $$x\ln x – x - y\ln y+y=-2ne^{-2n}-e^{-2n}+ne^{-n}+e^{-n}=ne^{-n}+o(ne^{-n}).$$ Then we ...


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Isn't this just a graphical representation of the Sieve of Eratosthenes, published "Introduction to Arithmetic" (60–120 AD) by Nicomachus...?


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here is a FREE pdf of Robert Ash's Probability and Measure theory. It has answers in the back.


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Definitely "Extending Mathematics" by T.J.Heard. This book is an old A-level maths book and comes in two volumes. It includes many challenging "STEP type" (Sixth Term Examination Paper) problems and in the Preface the author said he wrote it to cover the whole S-level mathematics syllabus. In the second volume, I was also surprised to find a few IMO problems ...


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I would recommend Hall, Knight - Elementary Trigonometry. It's not as elementary as the title suggests and is quite rigorous for a text at that level. It would fit your requirements as it uses a lot of geometric proofs where possible, and this will motivate the geometric intuition you seek. I'll give you an example of this: in order to show that 1 radian is ...


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$2 \left((\text{ay}-\text{bx})^2+(\text{az}-\text{cx})^2+(\text{bz}-\text {cy})^2\right)$ Why work hard on such a straightforward arithmetic task when Mathematica gives you the answer immediately? Is there really a benefit in doing this by hand? And if you want to make the value small, just set the individual squares small (or zero).


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HINT: For real $p,q$ we have $(p-q)^2\ge0$ When does the equality occur?



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