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4

I am not entirely sure what your question is, but here is my interpretation of it: is every $n$-manifold $X$ (without boundary) the boundary of some $(n+1)$-manifold with boundary $Y$? The answer is yes: just take $Y=X\times [0,\infty)$, identifying $X$ with $\partial Y=X\times\{0\}$. (Mike Miller gave an answer to the contrary in the comments; however, ...


1

No reference here, but I can at least answer your direct question of why the two conditions are equivalent. In particular, we need to figure out what the "first partial derivatives of the total derivative" look like. Assume, for the time being, that we're talking about a function $f:\Bbb R^n \to \Bbb R$. The total derivative is given by $$ Df = ...


1

$$\def\Ind{\mathop{\mathrm{Ind}}\nolimits} \def\ind{\mathop{\mathrm{ind}}\nolimits} \def\Hom{\mathop{\mathrm{Hom}}\nolimits}$$ Let me talk about these two functors for a bit. The first thing I should mention is that while the representations $\Ind_{H}^{G} \sigma$ and $\ind_{H}^{G} \sigma$ are isomorphic (for finite groups $G$ and $H$), they are not ...


0

For an interesting book, why not try 50 challenging problems in probability by Mosteller? That might be the book you're looking for. Somebody has already mentioned Feller, but that is too much for bedtime reading.


0

My favorite undergraduate statistics textbook is the textbook "All of statistics" by Larry Wasserman. In particular it is great for acquiring statistical intuition. The flow throughout the text is very coherent, i.e. you really get the sense of how all the different methods come together and form a statistician's toolbox. In contrast, I find that in many ...


0

Many people recommend the book All of Statistics: A Concise Course in Statistical Inference by Wasserman.


1

You won't get any farther than regulated functions. In fact, assuming that you mean $\mathcal{P}$ to be an operator that takes a class of functions to functions that are piecewise the restriction of elements of that class. $$\mathcal{P(U(P(}C))) = \mathcal{U(P(}C))$$ That is, a piecewise regulated function is regulated.This is fairly immediate if you take ...


0

There are two books that most colleges use currently: Understandable Statistics by Brase & Brase, and the other is Triola's Elementary Statistics. I recommend the first one since the author does every thing for the students.


0

A very good introduction to the subject is Combinatorics: an introduction By Faticoni


2

To answer your question: yes, it still can be beneficial to read old textbooks or publications, especially from a history of math point of view it is even necessary to read the original works. You get an idea on how the original ideas have been developed. Very interesting - no question about it. However, you have to take into account that these old books ...


0

Go to archive.org and look up Gauss' Werke, Band 1. This is in German and it includes the unfinished notes that would have become part of Section 8. But I would strongly recommend reading Mathews book on number theory first because it attempts to go over the content of Gauss' DA in a more up-to-date and accessible fashion.


0

I’m sure you are aware of the fact that the norm of $a$ is also, up to sign, the constant in the characteristic polynomial of $a$ in $K$ (not the minimal polynomial). It’s also the determinant of $a$ in the regular representation. That is, $z\mapsto za$ is an $F$-linear transformation of the $F$-vector space $K$, and has a determinant, which is independent ...


3

This happens to have been the topic of an article in the May 2015 American Mathematical Monthly. You can get access here. Edit: instead of me sumerizing the article, you can find it here for free (thanks to Raymond Manzoni for the link).


5

There is a nice proof in Remmert's book "Funktionentheorie" using residue calculus. Also, Tom M. Apostol has given a short elementary proof in The American Mathematical Monthly 1973, based on quadratic cotangent identities. The article also surveys other proofs of this famous identity, and gives many useful references.


3

As per the comments, here is an analytic proof (due to Euler) that can be generalised to the ring of integers of a number field (a number field is a finite extension $K$ of $\mathbb Q$, and its integer ring is the ring of algebraic integers contained in it). This can then be used to show that $\mathbb Z[\sqrt d]$ has infinitely many primes. (If $d \not\equiv ...


1

Derivatives on fractals sets was considered by several authors: Jiang, H. & Su, W. Some fundamental results of calculus on fractal sets Communications in Nonlinear Science and Numerical Simulation , 1998, 3, 22 - 26; Parvate, A. & Gangal, A. Fractal differential equations and fractal-time dynamical systems Pramana J Phys, Springer India, 2005, 64, ...


2

Let $A\in M_n(\mathbb{R})$. $A$ has $n$ distinct real eigenvalues iff there are $\lambda\in\mathbb{R}$ and $P,Q$ real symmetric $>0$ s.t. $(A+\lambda I)^2=PQ$ AND the dimension of the commutant of $A$ is $n$. EDIT. Proof: $(\Rightarrow)$ Clearly $A$ is diagonalizable and the dimension of its commutant is $n$. Let $\lambda\in\mathbb{R}$ s.t. $A+\lambda ...


1

[EDIT] : the following solution applies to highly irregular graphs (sometimes known as irregular graphs) https://en.wikipedia.org/wiki/Highly_irregular_graph I'd be surprised if you could find a nice lower bound. Testing for the irregular graphs on 12 vertices (110 graphs according to Brendan McKay https://cs.anu.edu.au/~bdm/data/graphs.html), most have ...


1

For forcing I recommend "Set Theory:An introduction To Independence Proofs" by Kunen, which is self-contained, and has a whole chapter (out of 8 chapters) om Martin's Axiom.And a great set of exercises and problems. Also "Lectures In Set Theory", edited by Morley, as a supplement.


-1

I'm not familiar with this book. I recommend "Set Theory: An Introduction to Independence Proofs" by Kunen. Also two books by Jech on forcing (known as the larger and the smaller Jech). An entire chapter in Kunen on Martin's Axiom precedes the forcing chapters, although Solovay and Tennenbaum needed Iterated Forcing to prove the consistency of Martin's ...


0

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 ...


1

Rice's book has worthwhile examples of various computing techniques using real data, but the main proofs are what you will find in Casella/Berger, perhaps at a slightly lower mathematical, pre-measure-theoretic level. One of several very good books introducing basic statistical methods in R (descriptive statistics, t and chi-squared tests, ANOVA models for ...


3

Once you have an algorithm for deciding whether two graphs are isomorphic, you can use it to find the isomorphism whenever one exists. Given isomorphic graphs $G$ and $H$, let $v$ be the first vertex of $G$ (meaning "first" in the ordering used to present the graph, for example as a matrix), and check, for each vertex $w$ of $H$ whether the induced ...


6

$\newcommand{\sech}{\operatorname{sech}}\newcommand{\arctanh}{\operatorname{arctanh}}\newcommand{\Res}{\operatorname*{Res}}$ $\Res\limits_{z=\frac\pi2i}\left(\frac{\sech^2(z)}{\pi^2+4z^2}\right)=-i\frac{3+\pi^2}{12\pi^3}$ and for $k\ge1$, $\Res\limits_{z=\frac{(2k+1)\pi}2i}\left(\frac{\sech^2(z)}{\pi^2+4z^2}\right)=\frac{8z}{\left(\pi^2+4z^2\right)^2}$. ...


2

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 ...


3

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 ...


1

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 ...


0

Might it have been this one? ${}\qquad{}$ http://www.amazon.com/Geometric-Algebra-Dover-Books-Mathematics/dp/0486801551


12

Approach 1: For the first integral \begin{align} 2\int^1_{0}\frac{{\rm d}x}{\pi^2+\ln^2\left(\frac{1-x}{1+x}\right)} &=-\frac{4}{\pi}\mathrm{Im}\int^1_0\frac{{\rm d}x}{(\ln{x}+\pi i)(1+x)^2}\tag1\\ &=-\frac{4}{\pi}\mathrm{Im}\int^1_{-1}\frac{{\rm d}x}{\ln{x}(1-x)^2}\tag2\\ ...


2

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.


0

I know this question is quite old, but I wanted to point out that Illusie has an article "Frobenius and Hodge Degeneration" attempting to explain the proof in the book Introduction to Hodge Theory by Bertin et al.. Demailly (one of the authors) has a copy of the book on his website so I think it's okay to link to it.


1

Though it's not a superb first introduction, I have to recommend Linear Algebra Done Right by Axler.


6

Try these books: Introduction to Linear Algebra by Strang. Linear Algebra and its Applications by Lay.


1

It appears on page 99 of Wilf's Generatingfunctionology, which is available on his website here.


0

Additionally, here contains a list of Problems in Finite Model Theory, which should give you some ideas.


0

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.


0

refer to Robin hartsthorne: Geometry, Euclid and Beyond. No need to endorse this book!


1

I really like Terence Tao's An introduction to Measure Theory (which is available from his website) but it has no contents related to Hilbert spaces, weak and *-weak convergence or anything related to functional analysis. You can also consider Folland's Real Analysis for this part. For the functional analysis part I would recommend you Lax's Functional ...


14

Sub $x=\tanh{u}$, $dx = \operatorname{sech^2}{u} \, du$. Then the integral is $$\int_{-\infty}^{\infty} du \, \frac{\operatorname{sech^2}{u}}{\pi^2+4 u^2} $$ Now, use Parseval. The Fourier transforms of the pieces of the integrand are $$\int_{-\infty}^{\infty} du \, \frac{e^{i u k}}{\pi^2+4 u^2} = \frac14 \frac{\pi}{\pi/2} e^{-\pi |k|/2} $$ ...


0

Since the formula uses the sin function, it has to be there in some form. If you want to compute the area of a 2n-gon in terms of the area of a n-gon, you can use the method of Archimedes, which is described very nicely in chapter 38 of "100 Great Problems of Elementary Mathematics" by Heinrich Dorrie ($13 from Amazon, also available online from Google ...


1

Area of regular $n$-gon in terms of circumradius $b$ is $$A=\frac{1}{2} nb^2 \sin\left(\frac{2\pi}{n}\right),$$so to avoid the sine function expand in a taylor series: $$A=\frac{1}{2} nb^2\sum_{k=0}^\infty \frac{(-1)^k\left(\frac{2\pi}{n}\right)^{1+2k}}{(1+2k)!}.$$


0

I wrote about this counterexample in another answer, but I suppose it'd be useful to list the relevant articles here. In addition to the sources below, a good friend of mine wrote a nice exposition of this topic, and there is a very nice overview by Kraft: Hanspeter Kraft, Challenging problems on affine $n$-space, Astérisque (1996), no. 237, Exp. No. ...


2

Let's prove it. Suppose $k$ is in the kernel of $A$ and $x_0$ is a specific solution, so that $Ax_0=b$. We have $$A(x_0+k)=Ax_0+Ak=Ax_0=b.$$ Conversely, suppose that $x_0$ and $y_0$ are two solutions. Then $$A(x_0-y_0)=0$$ implying that $x_0-y_0=:k$ is in the kernel. Therefore any two solutions differ by an element of the kernel.


0

The best book as far as i know are these two: Discrete Mathematics By Norman L.Biggs or Discrete Mathematics and Its Applications by Kenneth H. Rosen


3

Have a look at $$\frac{dy}{dx}=\frac{y^k}{x}$$ for an example where you really can't swap orders of taking the limit. For one order the answer is $$y = \ln x$$ and for the other order it is a peculiar expression which depends on whether $x$ is less than or greater than $1$.


2

For $n\geq k\in\mathbb{N}$, you can prove that $\dfrac{n!}{k!\cdot(n-k)!}$ is an integer from a combinatorial/counting argument. Establish that the formula gives the number of ways to choose a subset of $k$ from a set of $n$, and that automatically makes it an integer. Or you can prove that $\dfrac{n!}{k!\cdot(n-k)!}$ is an integer by showing that the ...


1

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 ...


0

Suppose that you have $n$ equations for $n$ unknowns of the form $$f_i=f_i(x_1,x_2,\cdots,x_n)=0$$ I think that the fastest way to solve is to minimize function $$\Phi(x_1,x_2,\cdots,x_n)=\sum_{i=1}^n f_i^2$$ and not use Newton-Raphson which, as you wrote, can be slow. If the solution is unique, no problem. Generating good starting values is too much ...


3

It is a consequence of the co-area formula. In particular, I studied it in the books by Giaquinta and Modica, volumes 4 and 5.


0

For combinatorics, I would highly recommend the book by Yao Zhang. Its an excellent wealth of information and provides common strategies for counting that tackle pretty much all problems out there. It has both text that teaches you, example problems, and "challenge" problems. You can find it easily online.



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