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13

We assume that the only information that has been recorded is whether or not the event happened or did not happen in the various $50$-trial rounds. If we have more information, such as the total number $N$ of times the event happened in the $5000$ trials, then we make the natural estimate $N/5000$. Let $p$ be the probability of the event happening in a ...


8

You should know calculus. You should know how mathematical reasoning is done: what theorems and proofs are. It might help to know some things that are often not taught in first-year calculus: epsilons and deltas, things like the difference between pointwise convergence and uniform convergence, suprema and infima, etc. Knowing some things about power series ...


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


5

The answer relies on the following incredibly general and simple result: Given a finite morphism of schemes $f:X\to Y$ with $Y$ locally noetherian, the sheaf $f_*\mathcal O_X$ is locally trivial iff $f$ is flat. The proof consists in quoting a result in commutative algebra: a module $M$ over a ring $R$ is flat of finite presentation iff it is ...


4

$RP^n$ can be obtained as the quotient of $S^n$ by the symmetry $x\cong -x$. The spheres $S^n$ are compact and connected, hence the quotient is as well.


4

Most of the integrals I have found in my version of Gradshteyn and Rhyzik have as reference [BI], which means [BI] Bierens de Haan, D., Nouvelles tables d'integrales definies. Amsterdam, 1867. which can be found here https://archive.org/details/nouvetaintegral00haanrich The PDF has 762 scanned pages, which makes it really slow for viewing. There is also ...


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 $g:B\rightarrow C$ ...


3

William Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry Rev 2nd ed. At the beginning of ch3. You can find a detailed proof that $\mathbb RP^n$ admits a differentiable manifold structure over the quotient topology induced by the natural projection $\pi:\mathbb R^{n+1}\to\mathbb RP^n$. To see it is compact and connected, this is ...


3

“Complex Variables and Applications” by Brown and Churchill is pretty nice. It's compact, and the later chapters show applications of the theory to some physics problems.


3

Michael Hardy's answer is excellent. For a specific text I would recommend Flanigan's "Complex Variables." It is extremely clear with many pictures - aiding the geometric aspects M.H. mentioned. This is a Dover publication, so in contrast to the exorbitant cost of most math texts, this is a bargain. ...


2

As far as I'm concerned, there's no better introduction to the beautiful subject of complex analysis then Theodore Gamelien's Complex Analysis. It begins at a very basic level, requiring only simple calculus and builds slowly to PHD exam level topics such as Julia sets and meremorphic functions. It also has one of the most complete and basic presentations of ...


2

I am surmising that your definition for a Pythagorean plane is the same as mine, i.e. the one in Greenberg's excellent article. That is precisely what Hilbert carried out in his book Foundations of geometry. This book is free to the public online at the Project Gutenberg link above. It is not the most modern prose, but it's readable and at a good level of ...


2

This follows from the fact that pulling back by homotopic maps produces isomorphic bundles. Let $ p:E\to B $ and $ s:B\to E $ denote the projection and zero section, respectively. These are homotopy inverses. In particular $ sp $ is homotopic to the identity, and so $(sp)^*F=p^*s^*F =E\times_B F_B $ is isomorphic to $ F$.


2

As a follow up to the comment placed by the OP after the answer of Mark Grant, I would like to add some details. I hope the OP will find them an helpful complement to the response by Mark Grant. Let $\pi:E\longrightarrow M$ be a vector bundle, and $H$ an homotopy from $N$ to $M$ $$\begin{align}H:N\times[0,1]&\longrightarrow M\\(x,t)&\longmapsto ...


2

Let $N$ be an $R$-module. We have to show that $N \to N \otimes_R S$ is injective. As $S$ is faithfully flat as an $R$-module, it suffices to prove this after tensoring with $S$. Hence it suffices to show that $N \otimes_R S \to N \otimes_R S \otimes_R S$, $n \otimes s \mapsto n \otimes 1 \otimes s$ is injective. This is true because there is a section, ...


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 ...


2

With your assumptions, $f$ is a flat morhism, and $H^1(X_y,\mathscr O_{X_y})=0$ for all $y\in Y$. Then the theory of cohomology and base change says that $f_\ast\mathscr O_X$ is locally free. It is rank $d$ because $d=h^0(X_y,\mathscr O_{X_y})$. If you want a precise statement, here it is: Suppose you have a proper morphism $f:X\to Y$, with $Y$ locally ...


2

Groupoids in the category theory can be described as partial groups. What you are asking for is called ringoids, see also here.


1

It would be helpful to know your background and what level of text you are looking for. In the absence of this information, I will recommend three that I like very much. In increasing order of difficulty: Axler's Linear Algebra Done Right Lang's Linear Algebra Roman's Advanced Linear Algebra


1

Start with $x_1 = 1$ and $x_3 = 0$, and let's find a way to make sure that players $1$ and $3$ are the only ones eliminated. Well let's assume that $x_2 = 1$. This choice would be symmetric with $x_2 = 0$ and reversing the direction of the circle. With $x_2 = 1$ we require $x_4 = 1$, which requires $x_6 = 1$, etc, up to the last even number, $x_{n-1} = 1$. ...


1

Before Eudoxus of Cnidus the only way to define a ratio of two geometric segments was to assume commensurability and to subdivide them into smaller segments of equal lengths. So all theorems that used ratios of segments, about similar triangles for example, had to rely on it and were put in doubt when Hippasus (allegedly) discovered that the side and the ...


1

Here's a reference: Shapiro, H., A survey of canonical forms and invariants for unitary similarity, Linear Algebra Appl. 147:101–168 (1991). I quote from the introduction: Many authors have studied the problem of finding a canonical form for unitary similarity and proposed methods for reducing a matrix to a canonical form under unitary ...


1

I agree with André's comment about the "incorrectness" of the reference to a Greek axiom of commensurability. In Craig Smorynski, History of mathematics : A supplement (2008) we can find some refernce to a "commensurability assumption" [page 50] and to an "axiom of commensurability" [page 51]. I strongly support the first locution; the ...


1

Combinatorial Optimization: Theory and Algorithms (Algorithms and Combinatorics) (ISBN-13: 978-3642244872) and Combinatorial Optimization: Polyhedra and Efficiency (Algorithms and Combinatorics) (ISBN-13: 978-3540443896) The first as general introduction and enough for most cases, the latter as reference material covering basically the whole topic


1

I recommend you a book that I thoroughly enjoyed when I studied complex analysis: "An Introduction to Complex Analysis", by Agarwal, Perera and Pinelas. This book is divided into 50 lectures that will take you from the very basics of complex analysis to advanced topics; it starts with the usual introductory material and then goes through the Cauchy ...


1

I am one to vouch for the calculus books by James Stewart that are alluded to above by cjferes. I teach my calc courses out of these books (7E). The books are easy to understand and have quality examples. They are in color and really explain how to do the integration methods. The book has proofs, but is light on proofs. This is most likely the preferred ...


1

I think you will probably like any of the introductory books by Rey Pastor. The issue there is that he was Spanish, so you won't probably be able to find a book by him in English. I have the three volumes of his Calculus course, and it's the most comprehensive book I've ever seen on the subject. A book I like that has a small introduction including some ...


1

Hilbert's Collected works (gesammelte abhandlungen) in three volumes (and in German) were published by Chelsea Publishing. I have the first volume on number theory. As far as I can tell, the AMS now offers only volume 1 (http://www.ams.org/bookstore/chelsealist). You can find used copies of the other volumes by searching for "hilbert gesammelte abhandlungen" ...


1

1) Yes and its available online. 2) No.


1

There is a Wikipedia Project to make Hilbert's collected works avaioable online: https://de.m.wikisource.org/wiki/David_Hilbert_Gesammelte_Abhandlungen_Erster_Band_–_Zahlentheorie It still needs some help!



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