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What you could write is just $$\{(g_m \circ g_{m-1} \circ \ldots \circ g_1)(x) | m\leq k; g_i \in F\, \forall i=1,\ldots,m\}$$ This does cover repetitions and allows any sequence of $f_i$'s up to length $k$.

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Your third equation can be thought of as integrating 1 with respect to F(y) over the interval [0,x]. $$\int_0^x 1 dF(y)$$ Let g=F(y), so we have $$\int_0^x 1 dg$$ which is trivial to solve: $$\int_0^x 1 dg =(g)|_0^x= g(x)-g(0)$$ by FTC, which is then equivalent to $$F(x)-F(0)$$ by the original definition of g.

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Here it is a recursive definition of $\Gamma(x)$, the orbit of $x$: Let $\Gamma_1(x)=\{f_1(x),\ldots,f_n(x)\}$. Let $\Gamma_{m+1}(x)=\{ f_1(y),\ldots,f_n(y) / y\in \Gamma_{m}(x)\}$. Finally, $$\Gamma(x)=\bigcup_{m\geq 1}\Gamma_m(x).$$

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A possibly related concept is that of an Iterated Function System, and the Hutchinson operator: $$H(S) = \bigcup_{i=1}^n f_i(S)$$. In this setting, the collection of all iterates of your collection of functions $\{f_i\}$ applied to $x$ could be described as $$\bigcup_{j=0}^\infty H^{j}(\{x\})$$ where the exponent represents repeated applications of the ...

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Just a suggestion.. Usually $\sigma(A)$ denotes the set of all possible permutations of elements of $A$ To simplify you could write $\sigma(n) = \sigma(\{1, \dots, n\})$ and write then $f_{\sigma(n)}$ the set of all functions you describe Or if you mean also subsequence (i.e. Permutations shorter than $n$) you could define $\sigma(n)= \cup_{i = 1}^n ... 0 They are the same. Technically, the notation$\forall a \, \forall b$is what one is supposed to write, but$\forall a,b$is a shorthand usually used in lectures, notes, proof scratch work, etc. But you should definitely remember that$\forall a \forall b$is the correct choice. I have also seen some professors write$\forall a \ni \forall b$where$\ni$... 1 The sequent calculus is based on the notation$\Gamma \Rightarrow \Delta$(or$\Gamma \vdash \Delta$), with$\Gamma, \Delta$finite (possibly empty) sequences of formulas, called a sequent. The intuitionistic sequent calculus is obtained with the restriction that$\Delta$consists of at most one formula. For the semantics for sequents, see Gaisi ... 1 You could write $$\sum_{i=1}^nA_j^{ki}=X$$ with the point being that this equation is true for all$j,k$. You have$mn$equations here, each one summing over one of the indices on$A$. 2 Let's say$\succ$is a total ordering on$S$. Therefore, either$a \succ b$or$b \succ a$for all$a, b \in S$. In this case, to do what you want to do, you need to create a function from$S^2$to$S^2$. It will take as input$(a, b)$for$a, b$in$Sand then output the ordered tuple. Here is the formal definition: $$f(a, b)=\begin{cases}(a, b) \ \ \ ... 2 Like Geoff suggests in the comments, you can use \text{Ker} for the object and \text{ker} for the morphism. But I don't think this convention is universal so you should probably say that you're using it. I sometimes use \text{ker} for both. 0 I think you can't denote the elements of your set as n,m. What is an n,m? Is it a pair? You can separate the set as a union of two subsets:$$\{n| n=2k, k\in \mathbb{N} \text{ and } 1\leq n\leq 10 \}\cup\{m|m=2k-1, k\in \mathbb{N} \text{ and } 11\leq m \leq 30\}$$-1 In my point of view "most important mathematical symbols" are neither Latin nor Greek.$$\color{Green}{0, 1, 2, 3, 4, 5, 6, 7, 8, 9,\cdots}$$0 Big O notation gives you an upper bound. Saying f(x)=x+O(\epsilon), for example, is saying that f(x)=x plus some error term which is bounded by some constant times epsilon, which is small. In your case, they are saying that \tau is bounded above by something on the order \epsilon up to a constant 1 Maybe you are looking for this: (-1)^nx^{2n}. One may recall that$$ \cos x= \sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}x^{2n} $$giving$$ f(x)=\cos x. $$1 The number of elements of a set A, called the cardinality of A is denoted |A|. Here's what I understand you are trying to say: I have a set L and a set S which is a subset of L. S is the set \{A, B, C\}, where:$$A = \{a_1, a_2, ... a_n\}B = \{b_1, b_2, ... b_n\}C = \{c_1, c_2, ... c_n\}$$for some integer n such that 1 \le n \lt ... 0 (1) and (2) are not the same in the sense that functions are defined in different domains unless U=\mathbb{R}^d. The notation C(\overline{U}) is seldom defined as in (2) in PDE books (I have never seen one), although the space C(X) (which denotes the set of continuous complex or real functions) where X is a compact Hausdorff space is well discussed ... 3 It's common to write \Delta x instead of h. It would be bad to use dx, because then dx would mean two different things. 1 I think the name "amalg" may come from the terminology free product with amalgamation, which occurs in algebraic topology, for instance in Van Kampen's theorem. 0 I am reading some notes from a hungarian friend. It seems to me like this means "independent" random variables in probability. I was wondering if this is common notation or if it's common only in Hungary. 2 Copyright is very rarely an issue in mathematics, as long as proper attribution is given. It would be absurd to try to copyright a theorem, insofar as someone else could rediscover it for themselves, and often someone else does. If you're duplicating large portions of a published textbook to the open Internet, that could in principle be a copyright ... 1 \delta_{ij}=\begin{cases}1 & \text{if } i=j \\ 0 & \text{otherwise}\end{cases} https://en.m.wikipedia.org/wiki/Kronecker_delta \epsilon_{ijk}=\begin{cases} sgn(ijk) & \text{as a permutation, if } i,j,k \text{ are different} \\ 0 & \text{otherwise}\end{cases} https://en.m.wikipedia.org/wiki/Levi-Civita_symbol And of course repeated ... 7 Complementary set of primes: all the other primes. Concretely, if \mathscr P=\{all primes\}, then$$\pi'=\mathscr P\setminus\pi,\quad p'=\mathscr P\setminus\{p\}\,.$$A p'-group consists of elements that all avoid the prime p in their orders. 4 I regard d and \rm d as the same. In general, I think of it like this: If f: \Bbb R \to \Bbb R, the derivative at x is f'(x) or \frac{{\rm d}f}{{\rm d}x}(x); If f: \Bbb R^n \to \Bbb R, the k-th partial derivative at x is \frac{\partial f}{\partial x^k}(x), or D_kf(x), or D_{x^k}f(x), or \partial_kf(x) or \partial_{x^k}f(x); If ... 1 D_x is the same thing as \frac{d}{dx}, it's just a different notation inveted by Euler. The capital D of course stands for derivative, and derivative is with respect to the "base" of D, for example D_t is derivative in respect to t. \frac{\partial}{\partial x} is a partial derivative used in multi-variable calculus. In this case, it is with ... 0 I think that it depend upon the last expression you get. If your expression can't be simplified as in x=1\pm\sqrt{2}, then use the "\pm" notation. Otherwise, as in x=1\pm\sqrt{4}, add an equivalent statement such as x=3 \vee x=-1, or even x=3,-1 which the same by convention. 0 Extending Regret's suggestion: $$\text{sign}(\frac{\min A + \max A}{2})\cdot\max\{|min A|,|max A|\}.$$ This would not work if |\min A| = |\max A| \neq 0, in which case Blah(.,.) wouldn't be a function anyway (unless you arbitrarily choose one of the two possible answers). 1 Another idea is to indicate the domain and codomain. This makes it clear the object you're considering is a function. Suppose f \colon (0,1) \to \mathbb R is differentiable. Then f is continuous. 0 A function f: \>A\to B is a law that produces for each input "point" x\in A an output value y\in B according to some formula (or geometrical, verbal, algorithmical, \ldots description). The output value is denoted by f(x). Here A and B can be arbitrary nonempty sets; but in any case there is an agreed on format to identify the individual ... 2$$\left(\sum x_i\right)^2 = \sum x_i^2 + 2\sum_{i<j}x_i x_j$$2$$\sum_{i=1}^n x_i^2=x_1^2+x_2^2+\cdots+x_n^2\Big(\sum_{i=1}^n x_i\Big)^2=(x_1+x_2+\cdots+x_n)^2$$1 For example, let x_i represent i^{th} natural number, and n = 4. Then \left(\sum_{i=1}^n x_i\right)^2 = (1 + 2 + 3 + 4)^2 = (10)^2 = 100 and \left(\sum_{i=1}^n x_i^2\right) = (1^2 + 2^2 + 3^2 + 4^2) = (1 + 4 + 9 + 16) = 30. 2 The ring \mathbb{Z}_{(p)} is defined as \{ \frac{a}{b}\in \mathbb{Q}\,|\, p\nmid b\}. In general if R is an integral ring and P a prime ideal, R_P:= \{ \frac{a}{b}\in Frac(R)\,|\, b\not\in P\}. 0 \mathbb Z_{(p)} is the localization of \mathbb Z at the prime ideal p\mathbb Z. P_P stands for the maximal ideal of R_P. Other (better?) notation: PR_P. 0 The colon represents a contraction over the same tensor, hence$$ |T_{i,;}|=\sqrt{\sum_{j}T_{ij}T_{ij}} . 6 After a bit of poking around, I found the same result as @SiongthyeGoh, but not from a mathematical P.O.V: From The Census of Mathematical Notations: On the Helsinki University of Technology, Department of Mathematics website, we find that they use syt instead of \gcd, and it is called 'Suurin yhteinen tekijä' in a Finnish context. Also find syt ... 1 Possibly: 1) \frac{\partial^{|A|}}{\prod_{i\in A}\partial_{x_i}}f(x_1,x_2,x_3,x_4,x_5). Sorry I don't have any ideas for the 2 and 3. However, you could just define some notion such as a large capital iota which functions like the large sigma for sums and large pi for products. 2 I believe it is just a typo. syt is just gcd. \begin{align*} A_d &= \left\{ x: 1 \leq x \leq n\text{ and } gcd(x,n)=d \right\}\\ &= \left\{ dx': 1 \leq x' \leq n/d \text{ and } gcd(dx',n)=d \right\}\\ &= \left\{ dx': 1 \leq x' \leq n/d \text{ and } gcd(x',n/d)=1 \right\}.\\ \end{align*} We can see that the set above have the same cardinality ... 1 Since \log_an=(\lg_ab)(\log_bn), you have that \log_an is just a constant multiple of \log_bn, and we know that constant multiples are invisible to asymptotic estimates. 1 Both of these\vDash \forall x\,(\, |x|=6 \iff x^2=36)\vDash\forall x\, (\,x\in \{6,-6\} \iff x^2=36)$$are equivalent and correct (with basic assumptions as the domain being \Bbb R, etc). 0 For example, it is used to write$$x^2=36\implies |x|=6\iff x=\pm 6$$Or more "logically":$$x=\pm 6\vDash x^2=36\;,\;\;\text{or}\;\;\;\;|x|=6\vDash x^2=36$$0 f:x \mapsto y means that f is a function which takes in a value x and gives out y. But, f: \mathbb{N} \to \mathbb{N} means that f is a function which takes a natural number as domain and results in a natural number as the result. 0 It means that f is a function that takes the value x to the value y. For instance,$$f: x\mapsto x^2is an alternate way of writing f(x) = x^2. 2 The set of matrices which size is p\times m is denoted in different ways in mathematics. For instance I use M_{p,m}(\mathbb{R}). Another notation is \mathbb{R}^{p\times m}, which means vectors where every components lies in \mathbb{R}^m. 1 No there is no standard name for the bijection \begin{align} \mathbb R^2 &\to\mathbb C\\ (x,y)&\mapsto x+yi.\end{align} It is assumed that everyone understands that \mathbb C is a two-dimensional vector space over \mathbb R, so no one ever explicitely writes down this bijection. In fact, I can't even think of a situation where there would be a ... 1 First note that\int_a^b f(x)\ \mathrm dx=\int_a^b f(t)\ \mathrm dt$$Since the function and the limits are equivalent, then the variable of integration that we use does not affect the result. Now for your integral we have$$\int_a^b f(a+b-x)\ \mathrm dx$$Using u-substitution, we have$$u=a+b-x\Rightarrow -\mathrm du=\mathrm dx$$Which implies that ... 0 Adding to Tobias' comment, you can make a distinction between a group G = (S, \cdot, 1), i.e., a structure, and the set S of its elements. 0 What you are describing is effectivelly simply a mapping from \mathbb R^n to \mathbb R^n. There is plenty of theory covering such mappings, especially the fields of multivariate calculus and linear algebra. 0 You could write$$ \underset x{\operatorname{arg\,max}}\left|\{a\in A\mid x\in a\}\right|\;.\$ See Wikipedia. Strictly speaking, this denotes the set of elements that maximise the expression, but if this is known to be a single element, the notation is often abused to refer to that element itself.

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Whether you use a bracket or a chevron depends on the convention of the book you may be reading. However, it is important to consider if we are dealing with a column or a row vector. As far as I'm concerned, if we plainly say vector, that means that it is written in the form of a column. You should be careful when dealing with this, as some operations with ...

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Per request by GamrCorps, I post my original comment here. What you have written says that the truth of the expression on the left implies the truth of the expression on the right. It is correct. If you want equivalence you can use the double arrow that shows how the truth of either implies the truth of the other.

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