# Tag Info

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. ...

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 ...

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.

4

Usually you would say that a one-dimensional noetherian UFD is a Dedekind domain and for Dedekind domains UFD and PID is the same thing. Let us recap the proof on an elementary level: First of all we show that every prime ideal is principal: Let $0 \neq \mathfrak p$ be a prime ideal and $0 \neq f \in \mathfrak p$. Since we have an UFD, we can factorize $$f ... 4 I think you refer to a theorem of Kummer: Fermat's Last Theorem is true for an odd prime p if and only if p doe not divide the class number of the cyclotomic extension \mathbf Q(\zeta_p), i. e. the order the the group of fractionary ideals of this field modulo principal ideals. Such a prime number is called a regular prime. Kummer criterion: An odd ... 4 Short answer: The main reason is the simplification of reducing multiplication and division to addition and subtraction. Historical aspects: One application which is heavily based upon trigonometric formulas is Spherical Geometry. This realm e.g. important for astronomy and geodesy used logarithmic tables of trigonometric functions right from the ... 3 Here are a couple of books that aren't necessarily meant for mathematicians but are pretty mathematical: Foundations of Classical Electrodynamics: Charge, Flux, and Metric by Friedrich W. Hehl and Yuri N. Obukhov. This book is probably closest to what you're looking for. It is very rigorous, even starting by stating a set of axioms for classical ... 3 Numerical Partial Differential Equations I and II by Thomas have become my go-to references. They cover all the major topics (FD, FE and FV) and have lots of exercises. 3 I'm not sure you'll be able to find a text with solved exercises. My personal favorite is Folland's Real Analysis. 3 There is, I fear, not much to be done for lower values of t. But the asymptotics for t\to\infty could reasonably be estimated, depending on the values of the parameters. For instance, if b>0, write h(t)=tb^{-t}M(t) and consider the functional equation$$h(t+1)=\frac{t+1}th(t)+(t+1)b^{-t-1}\left(a+df(t)\right)+\frac cb \frac ...

3

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)

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 $|A|\geq \delta k$ ...

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

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 ...

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 ...

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.

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 ...

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 ...

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 ...

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

I have an explicit solution, which I obtained by using the method of generating functions. First, compute what I called $P_n(1)$ from a recurrence relation. Then use those to compute explicit values of $M(k)$. $P_n(1)$ is the sum of the $M(k)$ values, for $0 \le k \le n$. You don't even need to compute all $P_n(1)$ values ahead of time. You can compute the ...

2

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

2

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

1

Stevens' 1982 book is a research monograph, aimed at readers who are already experts. There's a big gap between it and Serre's book. To bridge that gap, you might like to try reading some of the articles in the Cornell--Silverman--Stevens volume on Fermat's last theorem (especially Rohrlich's article in that volume on modular curves). That and the ...

1

The more standard term for your "polygonal chain" is "piecewise linear function". Most of the interest in functional analysis is in complete function spaces; the piecewise linear functions are not complete in any common metric that I know of, so for the most part they will not be considered as a space by themselves, but rather as a subspace of a larger ...

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 ...

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

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

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 ...

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