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0

It depends on what she wants out of the book, but it sounds like "Contemporary Abstract Algebra" by Gallian would be a great fit. The book introduces all of the major concepts in a way that is both easy to grasp and motivated. There are also interesting historical insights placed throughout the book. I had a friend lend it to me when I was starting to learn ...


0

A classic text is Marcel Berger's Geometry, although it jumps right into group actions in chapter 1 and uses them to define affine spaces. I'm not sure how suitable it will be without some abstract algebra/group theory exposure, which you haven't listed. Having studied linear algebra, you're already familiar with a very special family of groups, the group ...


3

Here are a couple of books that aren't necessarily meant for mathematicians but are pretty mathematical: Foundations of Classical Electrodynamics: Charge, Flux, and Metric by Friedrich W. Hehl and Yuri N. Obukhov. This book is probably closest to what you're looking for. It is very rigorous, even starting by stating a set of axioms for classical ...


2

In the première partie, chapitre VI, $\S$1, the 1er Théorème is: Lorsque les differens [sic] termes de la série (1) sont des fonctions d'une même variable $x$, continues par rapport à cette variable dans le voisinage d'une valeur particulière pour laquelle la série est convergente, la somme $s$ de la série est aussi, dans le voisinage de cette valeur ...


0

Maybe you want to take a look to this paper: http://arxiv.org/abs/1104.4345


1

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.


3

As for an upper bound, we get into work related to Erdos' conjecture on arithmetic progressions (which has not been proven even in the case of arithmetic progression of length 3). I believe the strongest known result is Sander's proof that, if $f(k)$ is the size of the largest subset of a $k$-element progression free from arithmetic progression of length ...


3

Roth's Theorem will give you an upper bound: http://wiki.math.toronto.edu/TorontoMathWiki/images/2/2d/Expo_paper.pdf In that paper it is Theorem 1.3 (bottom of first page). It says: Theorem (Roth, 1953) For any $\delta>0$ if $k>\exp(\exp(c\delta^{-1}))$ (for some absolute constant $c>0$) and $A\subseteq \{1, 2, 3, ..., k\}$ and $|A|\geq \delta k$ ...


1

To elaborate on Thomas Andrews' comment: Addition is well-defined: For all $a,b,c,d\in\mathbb{R}$, if $a=b$ and $c=d$, then $a+c=b+d$. The well-defined nature of addition in $\mathbb{R}$ is taken as an axiom. Hence, if $a=b$ and $c=d$, we have $$ a+c=b+d\Longleftrightarrow a+c=b+c, $$ but I think there is something more interesting you could show, namely ...


1

You start with $$ a \oplus c = x \quad (*) $$ where $x$ is the resulting value, then note that $a = b$, so you replace $a$ with $b$ in equation $(*)$. This gives $$ b \oplus c = x $$ Of course this means $$ a \oplus c = x = b \oplus c $$ and thus $$ a = b \Rightarrow a \oplus c = b \oplus c $$ Let us redo this for variables $a, b, c \in \mathbb{R}$ and the ...


2

The bound should depend only on $k$, not at all on $n$. It suffices to find a lower bound on the size of the largest subset of the first $k$ positive integers with no arithmetic progression of length $3$. In other words, as user TravisJ explains it, we want the largest subset of $\{1,2,\dots, k\}$. We can achieve a very weak but easy lower bound by simply ...


1

In addition to the ones mentioned above, there are also (in no particular order): R. Bartle, Introduction to Measure theory - it has a particularly nice section on integrals of the form $$ h(x) = \int f(x,t)dt $$ G. De Barra, Measure Theory and Integration"- does a nice job of differentiation. Halmos, Measure Theory - a little dated now, but I have looked ...


1

The 3 big options are: Folland, Real Analysis and its Applications 2e Royden & Fitzpatrick, Real Analysis 4e Rudin, Real & Complex Analysis 3e I liked each of them for different reasons, but I found Folland + Royden & Fitzpatrick (To supplement chapter 3 of Folland) was a good option. Lots of solved problems is generally an undergraduate ...


0

I am keen on the Schaum's outline series. Try this: "Schaum's Outline of Theory and Problems of real Variables: Lebesgue Measure and Integration with Applications to Fourier Series" by Murray Spiegel This series of books have hundreds of solved problems on measure theory (& other topics like topology, complex analysis & differential geometry). I ...


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It seems to me that what you ask is a particular instance of Theorem 2.10.10 (page 176) of H. Federer "Geometric Measure Theory" (Springer 1969) (here the Zmath reference)


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I'm not sure you'll be able to find a text with solved exercises. My personal favorite is Folland's Real Analysis.


0

In mathematical logic, and computer programming languages like Haskell, F#, OCaml, and Scala, () represents the (single possible) value of the Unit type, which is used to denote a function that returns no value. It is not valid to add a number with this representation of ().


3

Strictly speaking, there is no "class of functions between classes" because any member of a class is a set. That's all there is to it. However, the above does not answer the question of whether the (meta)category $\mathbf{Cls}$ of classes in NBG forms a cartesian closed category – all it shows is that the obvious candidate does not work. Instead, we make ...


1

As far as I remember GBN, you cannot, because any function whose domain is a proper class is not a set. Indeed, let $f:X\rightarrow Y$ a function (hence a subclass of $X\times Y$ satisfying certain requirements). Consider the map $\pi^X:X\times Y\rightarrow X$ that sends $(x,y)\mapsto x$. This is a function between classes. Now, suppose $f$ is a set. ...


0

2 years of the degree course in Bologna University,Italy is pretty good if you want to learn true infinitesimal calculus !! All The Best !


3

Anything is valid if you define it. Mathematicians tend to not only use different notation for the same thing, but often the same notation to denote different things. That's all right as long as they define up front what a given notation is supposed to mean. In many cases, the meaning is so commonplace that all mathematicians agree upon it. You'd find no ...


17

We usually denote $n$-tuples in the form $(a_1,\ldots,a_n)$, so for example $(x,y,z)$ is a triple, $(x,y)$ is a pair, somewhat redundantly $(x)$ could be called a one-tuple and $()$ a (in fact: the) zero-tuple.


13

Generally, no. But you could say: Let us denote something with (). ... and start using (), if you think this would convey your point. There is no need for academic reference validating this, it is the matter of basic author freedom. Let's say you want to wear a violet tie with blue dots. You don't ask if there is a law permitting you to do this. ...


4

It's certainly not conventional, and it's hard to think of any occasion when one might want to use such notation explicitly. That said, it would not be illogical to define () as zero, just as the empty set is sometimes written $\{\}$. As a precedent, an empty sum, such as $\sum_{k=1}^0x_k$, is defined to be zero by a standard convention.


11

We use parentheses to indicate the order of operations. To refer to your example: the operator $+$ takes two arguments, in the form of $a+b$. You can think of $+$ as a function that takes two variables. In this example, the $+$ is missing the second argument: there's nothing there. And $15 + $ isn't a valid mathematical statement -- it's not equal to ...


1

Notation is based on common acknowledgement. Sure you can invent any random thing, for example you can even draw a monkey instead of parenthesis. But this makes people confused about the notation. If here you mean a variable, usually people put a letter here. When the letters are not used up (or not for a specific purpose, eg. Greek letter for angle, an ...


0

Suppose $A$ is an $n\times n$ symmetric matrix and $x\in{\mathbb R}^n$. Let $f(x)=x^TAx.$ When you use Lagrange multipliers to find the minimum and maximum of $f(x)$ subject to $\|x\|^2=1$, the eigenvalues show up as feasible Lagrange multipliers, and eigenvectors show up as vectors corresponding to the feasible Lagrange multipliers (in fact, you get the ...


1

The numbers defined by $x_{n} + y_{n} \, \sqrt{2} = (1+\sqrt{2})^{n}$ are called Pell and Pell-Lucas numbers. As stated by the proposer question B should be $y_{2n+1} = y_{n+1}^{2} + y_{n}^{2}$ as will be shown. 1) From $x_{n} + y_{n} \sqrt{2} = (1+\sqrt{2})^{n}$ it is seen that \begin{align} x_{n+1} + y_{n+1} \sqrt{2} &= (1+\sqrt{2})^{n+1} = ...


0

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 ...


0

Depending the on the level of mathematical sophistication you have you might try Eduardo Sontag's book. Here is a link to it. http://www.math.rutgers.edu/~sontag/mct.html


2

As stated it is false. Let $$ U = (-1,0) \cup (10,11) \subset \mathbb{R}$$ and $$ V = (1,2) \cup (15,16) \subset \mathbb{R} $$ Take $$ K = \{ -2/3, -1/3 , 10.5\} $$ Let $\gamma$ be the map $$ \gamma(x) = \begin{cases} x + 16 & x \in (-1,0) \\ x - 9 & x \in (10,11)\end{cases} $$ which is clearly a diffeomorphism of $U$ and $V$. But ...


2

In the literature on constructive and linear logic it is the other way around, although $\to$ doesn't strictly belong to either group because it isn't monotone. Troelstra's Contructivism in Mathematics defines the almost negative formulas of Heyting arithmetic as formulas that have no $\vee$ and limited use of $\exists$. In linear logic the terminology made ...


1

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


2

Another good reference is Diamond and Shurman's "A First Course in Modular Forms", or William Stein's "Modular Forms".


1

In fact $$\eqalign{ x_n &= \dfrac{(1 + \sqrt{2})^n + (1 - \sqrt{2})^n}{2}\cr &= \sum_{j=0}^{\lfloor n/2\rfloor} {n \choose 2j}\; 2^j\cr y_n &= \dfrac{(1 + \sqrt{2})^n - (1 - \sqrt{2})^n}{2\sqrt{2}}\cr &= \sum_{j=0}^{\lfloor (n-1)/2 \rfloor} {n \choose 2j+1} \; 2^j }$$ The recursions can be derived from $$\eqalign{x_{n+1} + y_{n+1} ...


0

The unit group of the ring of integers $\mathbb{Z}[\sqrt{2}]$ is given by $\{ \pm(1+\sqrt{2})^n\mid n\in \mathbb{N}\}$, i.e., $1+\sqrt{2}$ is a fundamental unit of the real quadratic number field $\mathbb{Q}(\sqrt{2})$. Note that $(1+\sqrt{2})(\sqrt{2}-1)=1$, hence $(1+\sqrt{2})^n(\sqrt{2}-1)^n=1$, so that all $(1+\sqrt{2})^n$ are invertible in the ring ...


1

For example, and using an inductive argument: $$x_{n+1}+y_{n+1}\sqrt2=(1+\sqrt2)^{n+1}=(1+\sqrt2)(1+\sqrt2)^n=(1+\sqrt2)(x_n+y_n\sqrt2)\implies$$ $$\implies x_{n+1}+y_{n+1}\sqrt2=(x_n+2y_n)+(x_n+y_n)\sqrt2$$ Thus, we have $$\begin{cases}y_{n+1}=x_n+y_n\\{}\\x_{n+1}=x_n+2y_n\end{cases}\implies x_{n+1}=(y_{n+1}-y_n)+2y_n=y_{n+1}+y_n$$ Try now to take it ...


0

It is not bounded sorry for the inconvenience. Here is the proof a friend gave me (thanks a lot to him): (It can also be found here) Let consider the set $W_1 = \{w = (s,w_1,\dots)| w_1 \leq n\}$. Then $\mathcal{E}^n_n$ is bigger than $\displaystyle\sum_{w\in W_1}O^n_n(w)*P(w)$. \begin{align} \sum_{w\in W_1}O^n_n(w)*P(w) &= \sum_{k=0}^n \sum_{i=0}^k ...


4

Short answer: The main reason is the simplification of reducing multiplication and division to addition and subtraction. Historical aspects: One application which is heavily based upon trigonometric formulas is Spherical Geometry. This realm e.g. important for astronomy and geodesy used logarithmic tables of trigonometric functions right from the ...


1

Most number theory textbooks I read are algebraic in nature. They are not that visual - except possibly for geometry of numbers - and are no pictures. With programs like SAGE, it's possible to use computers to feel out number theory concepts, but it's still not common to try to visualize them. Here I try to show that $\mathbb{Z}[\frac{1+ \sqrt{3}}{2}]$ is ...


3

Numerical Partial Differential Equations I and II by Thomas have become my go-to references. They cover all the major topics (FD, FE and FV) and have lots of exercises.


0

After a bit more digging, I discovered that this is simply the expression for the volume element on a Riemannian manifold. The general expression in this setting is $$ \int_Mf dV_M = \int_Uf(\Phi(x))\sqrt{\det(g_{ij}(x))}dx $$ where $\Phi:U\subset\Bbb{R}^n\rightarrow M$ is the parametrization of $M$, and $g_{ij}$ is the Riemannian metric. The Riemannian ...


2

This book is very good: Numerical Solution of Partial Differential Equations Morton and Mayers


0

There appear to be plenty of references on google (as Ryan pointed out). Here is one on Stokes theorem for Lipschitz forms on a smooth manifold: http://arxiv.org/pdf/0805.4144.pdf Does that help?


2

I'd certainly recommend all of the titles presented above; I personally worked from the Colin Adams book. Here is a YouTube channel which I've found is helpful for those studying elementary knot theory.


0

I've found an answer for my question: Theorem 3.4 of Pertti Mattila's Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability.


1

There is a long strand of literature relating hyperbolicity of groups and of group actions to second bounded cohomology of groups. The first result in this theory is Brooks' proof that free groups of positive rank have second bounded cohomology of uncountably infinite dimension. After further progress eventually Epstein and Fujiwara proved the same ...


1

I understand the perron tree construction of a Kakeya set but I do not fully understand the visualisation of the second diagram (but I have seen it before). Here is a link to a lecture by Timothy Gowers that may help from the talk he gave on the importance of mathematics: https://www.youtube.com/watch?v=gjA8S82Q810 Something that myself & one of my ...


1

As pointed out in comments, the property that characterizes functions such that $$\forall s<t<u\qquad f(t)\leq\max\{f(s),f(u)\}$$ is Quasiconvexity. In particular a function satisfying the above inequality is called quasiconvex (or quasi-convex or quasi convex) and it is called strict quasiconvex if the inequality is strict. See article on ...


0

This generative model has some interesting properties. It has a fixed average degree with each node having degree at least one. It is also my intuition that it minimizes or comes very close to minimizing clustering for a fixed number of edges. I'm not familiar with any models which behave as the one you are describing. Good for you, it's hard to come up with ...



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