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2

I don't think You can find a specific notation for it, but you can write it as set: $$\big\{ \bigcup \limits_{T \in U} T : \; U \in \mathcal{P} (S) \big\}$$

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too long for a comment: Greek letters are precariously overloaded. There is both the Dirichlet eta $\eta(s)$ and the Dedekind eta function $\eta(\tau)$. "$\pi$" can occasionally mean a permutation or a prime in a field of characteristic $k$ as well as the ratio of the circumference to the diameter of a circle and the prime counting function $\pi(n)$ ...

1

The Cyrillic Л, which is analogous to L, is the first letter in the name Lobachevsky and has been used in hyperbolic geometry for the Lobachevsky function $$Л(\theta) = -\int_0^\theta \log|2\sin t|\,dt.$$ This notation was introduced by Milnor. See 1) chapter 7 of Thurston's "Geometry and Topology of 3-manifolds" (written by Milnor), 2) the appendix ...

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For anything you write that you want other people to read, do not use Cyrillic letters if you don't know what you're doing. If there were any trend to use Cyrillic in math then the Russians would use them, and they don't. They write almost everything with Latin and Greek letters like everyone else. And they write "sin" for the sine function, "lim" for limit, ...

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Independence is denoted $\perp \!\!\! \perp$ not orthogonal $\perp$. Use "\perp \ ! \ !\ ! \perp" in Tex (remove space between \ and !). A and B will be assumed to be not independent unless shown otherwise, but I know of no symbol for it.

3

$\dfrac{dt}{t}$ is the (well, unique up to a constant factor, so "a", strictly) Haar measure on the topological group $(0,+\infty)$ (with multiplication). That makes some transformations particularly nice when written in that style, $$\int_0^\infty f(at)\,\frac{dt}{t} = \int_0^\infty f(t)\,\frac{dt}{t}$$ for all $a > 0$, so $$\int_0^\infty t^s ... 5 Your difficulty stems from the use of the letter y for two different purposes: (a) as coordinate variable in the (x,y)-plane, and (b) as variable for (unknown) functions x\mapsto y(x) whose graphs are lying in the (x,y)-plane. When dealing with ODEs for the first time we are given a function f:\ (x,y)\mapsto f(x,y) defined in some region \Omega ... 3 The notation \dfrac{\mathrm dy(x)}{\mathrm dx} is short for \dfrac{\mathrm dy}{\mathrm dx}(x) or y'(x), if you prefer. In this context, the equality \dfrac{\mathrm{d}y(x)}{\mathrm{d}x} = f(x,y) should be read as \dfrac{\mathrm dy}{\mathrm dx}(x)=f(x,y(x)). As for your last example, you got the wrong idea, y is a function whose domain is a ... 0 "The load of edge e is f_e = \sum_{p\in P} such that e is in _pf_p" should be:$$f_e = \sum_{p\in P \text{ such that }e\text{ is in }p} f_p$$1 You could do something simple representing the "order" or degree to which you are referring. For example, instead of saying that F_n := 2 G_n, why not say G_{n}^{(2)} = 2 G_n. Now you can easily represent G_n^{(5007)} and you don't need to cycle through letters ;) 0 I‘d go with e^{2 \pi i} = 1. I believe this is at least more complicated than taking two limites. But of course the question which one is more complicate is very subjective. 1 An integral, one (or both) of limits of which is infinite, is an improper integral. For such type of integral you can't take Newton-Leibniz formula as is, you'll have to use the definition of improper integral:$$I=\int_0^\infty 2dx\equiv\lim_{t\to\infty}\int_0^t2dx.$$Now \int_0^t2dx=2t, and, following the definition,$$I=\lim_{t\to\infty} 2t=\infty,$$... 0 The expression 2 \cdot (+\infty) - 2 \cdot 0 does indeed make sense. It is rather useful to do arithmetic/analysis with \pm \infty alongside the other real numbers; e.g \lim_{x \to +\infty} has all of the same properties that \lim_{x \to a} does, and \lim_{x \to a} f(x) = +\infty behaves much more like a convergent limit than other sorts of ... 4 Just to amplify Henning Makholm's headline answer just a bit ... Some older texts use (x) [without the rotated 'A'] for the universal quantifier, and some use (\forall x) [with the rotated 'A' and brackets]. In those notations multiple universal quantifiers will look like (x)(y)\varphi or (\forall x)(\forall y)\varphi. The modern habit is to use ... 6 No, they are just typographical variants of the same mathematical meaning. 1 In a physics or mathematical physics context I've often seen \int \mathrm{d}x\; f(x) rather than \int f(x) \;\mathrm{d}x, e.g. in courses I've done on waves, diffusion, and statistical mechanics. Briefly the benefits of the former are the emphasis on integration being a linear operator, and clarity for multi-dimensional integrals. The latter is more ... 1 I don't think there is a special notation for that. You can call it H for "half", i.e. you can just say "Let H=\{(X,Y,Z)\in\mathbb{R}\mid X=Y,Z\geq X\}". Usually if P is a point on H, you just say P\in H. Do you need more specific notation? 1 A large plus symbol (or product symbol), would graphically look very bad, especially if you try to put indices on it. However large version of circled plus or times, are used: \bigoplus, \bigotimes. 0 There is Newton's notation for the derivative: \dot x instead of x' or Leibnitz's notation. I'm not sure this is a physicists notation, but I've only found it in physics and differential geometry courses, so that's saying something: given an injective function (of x) x\mapsto y(x), denote the inverse, y^{-1}, by y\mapsto x(y). 1 One problem is that "the third element of a set" makes no sense -- recall that$$\{1,2,3\} = \{3,1,2\} = \{ 3,1,1,2,1,1,3,3,1,2,3,1,2\} $$What you are referring to by the "third element" is not of the set at all, but of the specific notation you have chosen to write the set. If you wanted to refer to 3, you would just say "3". Or if you had a variable ... 5 The Landau big-O notation is extremely strange. One writes$$f(x) = O(g(x))$$which looks like f is the composition of O and g, but it is nothing of the sort. Is O() an operator that can be applied to any term? Can I write$$O(x^2) = O(x^3)$$or O(x^2) = 2x^2? Not normally. It is easily confused with a whole family of similar notations for ... 6 The single worst use of mathematical notation I have ever seen was in a set of lecture notes in which the author wanted to construct a sequence of equivalence relations, each one (\equiv_n) derived from the previous one (\equiv_{n-1}). After i_0 iterations of this procedure, the construction has no more work do do, and the sequence has converged to a ... 0 How about using pairs of letters like r,s or u,v , or m,n when writing on a blackboard? Unless you're extremely careful, the two in any pair get very easily confused with each other. Or, when you're told you have two collections of objects ( with maybe some additional propreties ) , say S,X , and then you have that a, or worse x is an element in ... 2 From a proof that convergence a.e. implies convergence in measure for \mu(\Omega)<\infty:$$\bigcup_{r\geq 1}\bigcap_{n\geq 1}\bigcup_{j\geq n}\{|{f_j-f}|>\frac{1}{r}\}=\{\omega:f_j(\omega) \not \to f(\omega)\}$$Also, labeling graphs of functions as f(x) (which I end up still doing to my undergraduates, who are bored when I mention my reservations ... 4 The usage of pi: \pi is a constant. \pi(x) is the prime counting function. \prod(x) is a product of a sequence. 1 I took a long time to get used to derivative of integrals like this$$\frac{\partial}{\partial x}\int_{x_0}^x f(x,y) \ dx$$It's just too much x's in the same formula, and each one has a different meaning. Nevertheless, its common to see people writing down this way. 0$$\large{\prod_{n = 1}^3 \mathbb{R} = \mathbb{R}^3}$$Edit: Apparently this is common notation. MJD suggests a better example:$$\large{\prod_{n = 1}^3 S \neq S^3}$$10 There is an old story about Lang and Mazur, Mazur tried to get Lang attention by using the worst notation possible. He wrote Xi conjugated over Xi, which looks like:$$\frac{\overline{\Xi}}{\Xi}$$P.S. You can read the story, narrated by Paul Vojta, in the AMS Notices issue dedicated to Lang: AMS Nottices Lang It is on pages 546-547. 0 When I first learnt trigonometry (at the age of 6\frac{1}{2}), I encountered the \sin,\cos,\tan,\sec,\csc,\cot notation. This confused me since I believed that \sin x was s\cdot i\cdot n\cdot x. (However, I got over it and now I'm doing category theory.) One more strange notation that I've encountered is \partial\Sigma, used to denote the boundary ... 5 There is already the potential for a difference in meaning without recourse to complex numbers: compare \sqrt[2]{(-1)^2} = \sqrt{1} to (-1)^{2/2} = (-1)^1. The point is that there isn't really a way to define a (singly-valued) function x^y across negative/complex values of x,y so that (x^a)^b is always equal to x^{ab}. This makes order of ... 5 Some people prefer to consider \sqrt[n]{\cdot} as the inverse of (\cdot)^n. In particular, they write freely \sqrt[3]{-1}. On the other hand they prefer not to write (-1)^{\frac{1}{3}} because this would be different than (-1)^{\frac{2}{6}}. I have seen this distinction mainly in high-school teachers, because at that level it might be hard to ... 1 There is no difference between \sqrt[p]{x^q} and x^{\frac{q}{p}}, they are just different forms of notation for the same thing. 1 First of all, there are two similar symbols: U+00D8 Ø latin capital letter o with stroke U+2205 ∅ empty set The first is a letter used in Danish, Norwegian, and Faroese languages. The second is the empty set symbol. The rendering of "\emptyset" on this site looks like the first, and ought to look like the second, in my opinion. Now, after that diversion, ... 1 Let X, Y be Banach spaces and U\subset X open. A function u\colon U\to Y is said to be differentiable at a\in U if there exists a linear operator Du_a\colon X\to Y (the differential of u at a) such that$$ u(a+h)=u(a)+Du_a(h)+o(h). $$If u is continuous, then Du_a is also continuous. If u is continuous and differentiable at very point ... 4 The notation (\mathbb C\backslash\{0\})\times\mathbb R refers to the set of points of the form (a + bi,c) where i^2 = -1; a,b and c are real numbers; and at least one of a and b is non-zero. 0 "Per degree" (perhaps better use radians if talking math?) isn't really an unit, it is a dimensionless ratio. So, to be really rigurous, there is no unit. In any case, it is better to use the general ISO guidelines for readability. 2 I have seen 5 \text{ deg}^{-1} and like that. 2 Personally, I have never heard it called that (of course, it is always possible that some in subfield of mathematics I'm not familiar with, it is an accepted name). However, non-Americans will usually call \mathbb{Z} (the integers) "zed", perhaps you misunderstood? 2 I suppose you could write this as$$ \bigcup \left(\mathbb Z\big/b\mathbb Z\right)^*, $$where the union just throws the invertible cosets in \mathbb Z\big/b\mathbb Z together. Anyway, this isn't standard in any way, nor is it more comprehensible than defining some notation like$$ \mathcal C_b = \{\,a\in\mathbb Z \mid \gcd(a,b)=1\,\} $$on your own. ... 1 I don't think there's a standard notation for the set of representatives of the invertible elements under a quotient map of rings. The set is the union of all the arithmetic progressions {\cal P}_b(a)=a+{\Bbb Z}b as a varies in a set of representatives of (\Bbb Z/b\Bbb Z)^\times. If you want a symbol for that, an option could be$$ \cal ...

2

I believe you are talking about a hereditary property. From Wikipedia: “In mathematics, a hereditary property is a property of an object, that inherits to all its subobjects, where the term subobject depends on the context.”

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This is the usual notation for the unit group of the ring $R$, that is, the group of invertible elements using ring multiplication as the group operation. Note that this is usually not the same thing as $R \setminus \{0\}$, because most elements aren't invertible (if it is the same, your ring is a field, by definition). While you can talk about $R \setminus ... 0 The notation is often used on the form$\Bbb R^*$i.e. with a star and it means$\Bbb R\setminus \{0\}$and we have$(\Bbb R^*,\times)$is a multiplicative group. 3 Wikipedia cites a notation by Gauss: $$x = a_0 + \dfrac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{a_3}}}$$ would be written: $$x = a_0 + \mathop{\mathrm{K}}_{k = 1}^3 \frac{1}{a_k}$$ 3 It's a little difficult to give you any really useful help because you showed the proof of the theorem without saying what the theorem was.$\{x\}$is the set that contains the point$x$and nothing else. (Similarly,$\{x, y\}$is the set that contains$x, y, $and nothing else.)$A\times B$generally is the set of all ordered pairs$(a,b)$where$a$is ... 4 Yes,$\{x\}$is a set which has a single element (and thus we call it a singleton), and$\{x\}\times B=\{(x,b)\mid b\in B\}$. Clearly it is homeomorphic to$B$. Note that ordered pairs are not necessarily ordered pairs of real numbers. You can talk about product of two sets, or two spaces. And this is somewhat similar to the case of$\Bbb R^2$, or the real ... 1 The answer is neither, but rather$\{G_\alpha\}\cup\{X\}$. 0 You could use$H_{T1,A,B}$, or$H_{T1;A,B}$. Your idea for your function is almost right, but you've included some superfluous information. Try$H_{T1,A,B,C,D}=\cases{a, \text{if } A=1,B=1,∀C,D,\\b, \text{if }A=2,B=2, ∀ C,D,\\c, \text{if } A=3,B=3,C=1,D=2,\\d, \text{if } A=3,B=3,C=1,D=3}$or use your simplifying trick and say$H_{T1,A,B,C,D}=\cases{a, ...

0

What is the context? If you're handwriting something for calculus class, I think one clear thing to do (if a bit nonstandard) would be to write something like $$\lim_{x\to p}\begin{cases} f\left(x\right)\\ =f_{1}\left(x\right)\\ =f_{2}\left(x\right)\\ =x \end{cases}$$ $$=p$$ Alternatively, if you're doing lots of calculations with the same limit, you could ...

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Here is how Accumulate works: Accumulate[{a, b, c, d}] produces {a, a + b, a + b + c, a + b + c + d} Thus, the mth entry of the list Accumulate[Table[MoebiusMu[k], {k, 1, n}]] is $$M(m)=\sum_{k=1}^m\mu(k)$$ so that Mean[Accumulate[Table[MoebiusMu[k], {k, 1, n}]]], the average of the n entries on this list, is the just average of the first $n$ values of ...

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