Question on the meaning, history, and usage of mathematical symbols and notation. Please remember to mention where (book, paper, webpage, etc.) you encountered any mathematical notation you are asking about.

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0
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1answer
13 views

Limit notation where variable does not approach anything

I was reading an example in my probability textbook that states a limit as $$\lim_{n}{P\left\{X \leq 3 - \frac{1}{n} \right\}}$$ where the RV $X=k$ is defined for $ k \in \mathbb{R}$ What exactly ...
2
votes
3answers
54 views

Expected value of $g(X)$.

If $\mathrm{E}(X) = \sum_{x\in I} x\,\mathrm{P}(X=x)$, how can I deduce that $E(g(X)) = \sum_{x\in ?} g(x)\,\mathrm{P}(X=x)$? I don't see why it isn't $E(g(X)) = \sum_{g(x)\in ?} ...
0
votes
1answer
14 views

Is it correct to write $argmin(x, y) \sum_i^n |p_{x_i} - x| + |p_{y_i} - y| = argmin(x) \sum_i^n |p_{x_i} - x| + argmin(y) \sum_i^n |p_{y_i} - y| $?

$argmin(x, y) \sum_i^n |p_{x_i} - x| + |p_{y_i} - y| = argmin(x) \sum_i^n |p_{x_i} - x| + argmin(y) \sum_i^n |p_{y_i} - y| $ Is it a legit way of separating argmins to show independence of $x$ and ...
0
votes
1answer
29 views

Regarding taking powers of prime ideals in a ring

My question is simple to ask: given some prime ideal $P$ in a ring $R$, we can talk about $P^2, P^3$ etc. but can we discuss $P^0$? Is there a convention that says $P^0 = R$, or is there something ...
5
votes
1answer
326 views

What does an arrow under a sigma mean?

In sigma notation, what does it mean when an arrow is used underneath? In particular, in this equation: $$y_k = f_k\left(\alpha_k + \sum_{j \to k}w_{jk}f_j\left(\alpha_j + \sum_{i \to j}w_{ij}x_i ...
1
vote
2answers
176 views

What does $y_i=f(\sum_j w_{ij}y_j)$ mean (in an artificial neuron model)?

While trying to understand artificial neural networks, I came upon an equation for finding the net input of an artificial neuron. Can someone explain this to me and what it means? Here is the original ...
7
votes
4answers
868 views

The meaning of various equality symbols

I'm interested in knowing what is the meaning of the various equality symbols: $=,\sim, \cong,\approx,\equiv$. For example, the speed of a car $V$ in m/s: what would be the meaning of each of these ...
-1
votes
2answers
46 views

Using the definition of $f$ is $O(g)$ proof:

I'm studying for my discrete math class and I don't understand how to prove big O notation. I understand that $f$ is $O(g)$ of another if $f(x) \le c g(x)$ holds. How would I go about proving $\sin ...
1
vote
1answer
38 views

Confusing delta notation for multivariate expectation

I'm seeing this odd notation describing process error covariance in an extended Kalman filter over a time interval $(t, t')$. It says that $$ Q(t) = \left[ \begin{matrix} Q_1(t) ...
1
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0answers
53 views

On a Probability notation - $\mathbb{E}[X(.)|\mathcal{F}]_G$

What could mean this notation : $\mathbb{E}[X(.)|\mathcal{F}]_G$ ? where G : $\Omega \rightarrow \mathbb{R}$ is a random variable on a probability space $(\Omega,P, \mathcal{F})$. X could be a ...
0
votes
1answer
11 views

Notation: square brackets with a unique scalar?

my question is purely about notation. I am reading papers in computer science and I see that people use the following notation $[x]$ to denote $\{1,2,\ldots,x\}$. Is that correct? Or does it mean ...
1
vote
1answer
24 views

How to interpret scientific notation?

I'm having a problem understanding scientific notation. What is the difference between the following: $$\text{5e2, 5e-2, -5e2, -5e-2}$$
1
vote
1answer
21 views

The sum of $V=U+W$ of a vectorspace V and subspaces $U$, and $V$

I know what the sum of two subspaces is and how we notate but is it ok to write a minus to denote what I hope should be obvious is meant. So we have $V=U+W+Y$ where $V$ is a v.space and $U,W,Y$ ...
0
votes
0answers
40 views

Is the notation $\mathbb{P}(X \cap Y) = \mathbb{P}(X,Y)$ common?

For two random variables $X,Y$, is the notation $\mathbb{P}(X \cap Y) = \mathbb{P}(X,Y)$ common? In a probability class last year we had always used $\mathbb{P}(X \cap Y)$. This year in a stochastic ...
2
votes
1answer
79 views

Uniform norm $ \|u\|_{C(\overline{U})}$ in PDE

Let $U\subset \Bbb{R}^n\to\Bbb{R}$ be an open set (not necessarily bounded) and $u:U\to\Bbb{R}$ be a bounded continuous function. In Evans's PDE textbook, the author defines a norm $$ ...
0
votes
1answer
13 views

Clarification of Direct sum meaning with $\geq 3$ subspaces

Let $V$ be a vectorspace and $U_i$ subspaces of V. In the definition of $\oplus_{i \in I} U_i$ it is said that is does not suffice for $U_i$ to be pairwise disjoint. Instead we must have the stronger ...
1
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1answer
17 views

Explanation Notation Union Probability

Could somebody explain how to create intuition for the probability/union notation below? I don't know how to read it. And is this a situation where events are disjoint, but dependent?
-2
votes
1answer
31 views

Big O notation $O(\epsilon)$ [on hold]

What does it mean to say that $\tau=O(\epsilon)$? Where $\epsilon$ is small, meaning that $\epsilon \ll 1$.
3
votes
2answers
88 views

Origin of the notation for statistical divergence

The unusual notation $D(P||Q)$ seems to be universally used for statistical divergences (e.g. KL divergence). What is the origin of this notation, and do the double bars (pipe symbols) have any ...
0
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0answers
18 views

Topology; difference between open subsets of $X$ containing $x$ and open neightborhood od $x$?

I see my lecture notes and some texts alternate between the two. What is the difference in saying that "an open subset of $X$ containing $x \in X$" and an "open neighborhood of $x \in X$"?
1
vote
4answers
48 views

Mathematical Notation of Sequence of Functions

Let's say I have a finite set of functions $F=\{f_1,f_2,f_3,...,f_n\}$ and I want to show a recursive function that is constructed by an arbitrary sequence of applications of functions in $F$ to input ...
0
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2answers
58 views

Notation in commutative algebra

I am doing some exercises on commutative algebra and came along the following expressions, which were not elaborated on. Is someone familiar with them? The first is for $p$ a prime number ...
2
votes
1answer
374 views

Confused about notation and derivatives inside integrals

EDIT: To make what I am asking more clear. I've simplified it and have a more direct question. Let's say I am writing out an expression, and I want to write: $$\int_0^xF'(y)\,dy$$ However, for ...
0
votes
1answer
41 views

In sequent calculus, what's going on with sequents with multiple formulae in the succedent?

The sequent proof systems I learned only allowed one formula on the right hand side of the sequent, and $\phi_1, \ldots, \phi_n \Rightarrow \psi$ (or ... $\vdash \psi$) is understood as saying that ...
0
votes
1answer
18 views

Are these two notations equivalent?

I've seen both $\forall a,b$ and $\forall a\forall b$ of these being used in various posts and wondered if they are equivalent or if there is a subtle difference between the two.
1
vote
1answer
30 views

How can I write a set of equations in summation form?

I have a system of equations as follows: \begin{align} & A_1^{11} + A_1^{12} + A_1^{13} + \cdots + A_1^{1n}=X \\[8pt] & A_1^{21} + A_1^{22} + A_1^{23} + \cdots+ A_1^{2n}=X \\[8pt] & ...
1
vote
1answer
13 views

Notation To Define A Mapping From A Set to a Relation Between Two Elements Of That Set

Say I had a set $S=\{s_1,\dots, s_n\}$, and each $s_i$ denotes an outcome. If I wanted to define a function, $f$ which takes two elements of $S$, $\{s_i, s_j\}$, and maps it to a relation, either ...
0
votes
0answers
10 views

reference request: $C^k(\overline\Omega)$ as restriction of $C^{k}$ functions on $\Omega$

Let $\Omega\subset\mathbb{R}^d$ be an open set. $C^k(\Omega)$ is defined as the space of functions $f:\Omega\to\mathbb{R}$ such that $\partial^nf$ is continuous for $0\leq|n|\leq k$. There are ...
1
vote
1answer
25 views

Symmetric Simple Random Walk - Definition Clarification

I'm finding conflicting answers everywhere, including in my own notes. In the phrase "symmetric simple random walk", which part, "symmetric" or "simple" refers to having a probability of $0.5$ to go ...
0
votes
1answer
42 views

Notation for kernel object

When $f: A \mapsto B$ is a morphism in some category with a zero object and limits, we can use $\ker(f)$ to refer to an equivalence class of morphisms to $A$ which satisfy a particular universal ...
0
votes
0answers
26 views

How to formalise a procedure involving Cartesian products of sets of vectors and transformation in matrices?

I am asking for an help to formalise with the correct notation the following procedure. Let $n\in \mathbb{N}$. Let $\{0,1\}^{n-1}$ be the set of vectors of dimension $(n-1)\times 1$ with each ...
0
votes
0answers
44 views

Why isn't the identity/unit matrix upright?

I realize this is more of a typesetting problem then a mathematical one. I've already tried the TeX stack exchange and the question got canned. In ISO 80000-2:2009, variables and running numbers are ...
1
vote
0answers
21 views

Conventions for named equations [closed]

In the document I'm writing, I find myself referring back to a small number of equations often enough that it makes sense to give them a name; perhaps some sort of initialism. (I hate when texts ...
0
votes
0answers
19 views

Slow time variable

If I have a time variable $t$ such that $t \in[0,5]$ and I introduce a new variable $\tau$ st $\tau =\epsilon t$, where $\epsilon <<1$. Why is $\tau$ called a 'slow time variable', when $\tau ...
0
votes
1answer
18 views

Basic set notation in combining different ranges of numbers

What is the proper way to specify a set which contains all even numbers between 1 and 10, and all odd numbers between 11 and 30? Would this work? $$ U = \{n, m\ |\ n \ \text{is even},\ 1 \le n \le ...
0
votes
1answer
48 views

Notation and Quantifiers

I was wondering what is a natural way to write certain formal expressions, without make them look too cumbersome. In particular, what I learned from various books is that, when we deal with the ...
0
votes
0answers
48 views

in spanish do you have to do the upside down exclamation marks before a number for factorials? [closed]

for example $4! = 4 \times 3 \times 2 \times 1 = 24$ if you were to write this in spanish would you have to write $¡4! = 24$ ? a quick google search doesn't give me anything
104
votes
22answers
11k views

Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a ...
0
votes
2answers
37 views

Is this element-of_{ij} - looking symbol the Levi-Civita symbol?

I'm reading this formula: from a page Is the symbol that looks like an element-of symbol with two indices i and j the Levi-Civita symbol? Mathematics is my weak-side so I'm not sure. Actually I ...
17
votes
7answers
2k views

Why do we (mostly) restrict ourselves to Latin and Greek symbols?

99% of variables, constants, etc. that I run into are named for either a Latin character (like $x$) or a Greek character (e.g. $\pi$). Sometimes I twitch a little when I have to keep two separate ...
1
vote
1answer
24 views

Write the series using sigma notation: $f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! +\cdots$

$$f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + \cdots$$ I don't know how to get the signs to work like negative, then positive. I have tried to make it like the following: $(-1)^{n-1} \frac{x^{2n-1}}{ ...
2
votes
1answer
44 views

Notation for sets of unordered pairs

Let $A$ be a finite set of unordered pairs, e.g., $$A = \{\{1, 2\}, \{1, 3\}, \{2, 3\}\} \enspace .$$ Which of the following is proper notation for "the element $\{1, 2\}$ belongs to $A$"? $\{1, 2\} ...
0
votes
0answers
21 views

Question about notation for a statement about conditional probability distribution

Consider the random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, P)$. Suppose $Y$ is a discrete random variable with support $\mathcal{Y}\subset \mathbb{R}$. Suppose ...
1
vote
6answers
7k views

Is there a mathematical symbol for unknown?

I'm searching for two symbols - considering they exist - (1) unknown value; (2) unknown probability. Note: I thought that $x$ was used in a temporary context, whenever I see it, it remains unknown ...
0
votes
1answer
21 views

Set builder notation: defining the number of elements

I have a set L and I have a subset S which is part of L and contains three elements A, B and C. Finally, each of these elements are subsets that consist of their own elements: $A=\{a_1...a_n\}$ ...
1
vote
0answers
27 views

Question regarding terminlogy and wording of the derivative

When doing calculus, we typically say that we "take the derivative of a function f(x)." However, rigorously, f(x) is not a function but rather the value of the function f evaluated at x. Thus, in ...
4
votes
2answers
22k views

What does ! mean in sequences?

I'm doing a sequences problem where I have to write the first five terms of a sequence. It looks normal, but there is an exclamation mark on the denominator: $$a_n = \frac{1}{(n + 1)!}$$ & ...
2
votes
2answers
108 views

What does the notation $C(\bar U)$ mean for $U\subset\Bbb{R}^d$ open?

Let $U$ be an open subset of $\Bbb{R}^d$. In Evans's PDE book, $$ C(U)=\{u: U\to\Bbb{R} \mid u\ \hbox{continuous}\} $$ and $$ C(\bar U)=\{u\in C(U)\mid u\ \hbox{ is uniformly continuous on bounded ...
2
votes
1answer
61 views

Why don't we use “dx” as the limiting variable when teaching the definition of the derivative with respect to x?

For a function $f(x)$ that is differentiable at $x$, the derivative of $f$ with respect to $x$ at the point $x$ is usually given as $$\frac{df(x)}{dx} = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}.$$ I think ...
0
votes
0answers
14 views

Notations of conditions

Assume I have the Mean Square Error: $\mathcal{L}(X, Y) = \frac{1}{\lvert X\rvert}\sum_{i=0}^{\lvert X\rvert}\left(f(\mathbf{x}_{i}) - \mathbf{y}_{i}\right)^2 \space \space ,\space \space \lvert ...