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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2answers
101 views

Prove that $\prod_{i=2}^n (1-1/i^2) = {n+1\over 2n}$

prove the $$\prod_{i=2}^n (1-1/i^2) = {n+1\over 2n}$$ for all n greater or equal to 2. $\pi$ should be a big pi from $i=2$ to $n$ for $(1-1/i^2)$. I'm really confused about the $\prod$ function. ...
5
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1answer
2k views

Meaning of mathematical operator that consists of square brackets with a plus sign as a subscript

I was reading a paper on tomographic reconstruction, and I found an operator that is not explained: $[expression]_+$ The operator was used to compute a surrogate for the log-likelihood cost ...
7
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1answer
1k views

Why use radical notation instead of rational exponents?

I'm helping my younger sister for her math class. She has recently been taught integer exponents, and has starteed studying radicals (mainly square roots). The next topic will be rational exponents, ...
0
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2answers
74 views

Problem with Math Notation

I am getting confused with this notation: $$R =\{x | x=a_i\times b_j;1\leq i\leq m \text{ and } 1\leq j\leq n\}$$ where $a$ and $b$ are vectors of length $m$ and $n$ resp. What does this mean? Does ...
-1
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2answers
96 views

For any two points, there exist a path of $n$ segments that connects them. Is there a name for this kind of set?

We consider a set $A$. $A$ is called convex if for every $x,y\in A$, we have the line segment $xy$ is also in $A$. I want to generalize this notion, such that instead of one line segment, there can ...
0
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2answers
79 views

Exponentation vs Power

What definition of $a^b$ operation is the most popular and standartized: Exponentation or Power? Is any difference between them?
0
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1answer
135 views

Why are $321,213,132$ in cyclic order

Could anyone help me understand this concept, if we have $1,2,3$ why are $$123,231,213$$ Considered in cyclic order and $$321,213,132$$ Considered in acyclic order? I am asking this in regards to the ...
1
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1answer
277 views

How can this algorithm be expressed in mathematical terms?

I have found a solution to a basic problem using an algorithm, but I'm having a bit of a hard time expressing this algorithm in terms of discrete math. ...
1
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2answers
2k views

Big O Notation and finding witnesses

I am trying to figure out some stuff here with Big O Notation. I mean I understand the concept of it and can generally be able to tell what the efficiency of something is, but I do not really ...
6
votes
5answers
842 views

Does $x/yz$ mean $x/(yz)$ or $(x/y)z$?

When people write $x/yz$, do they usually mean $x/(yz)$ or $(x/y)z$? For example, from Wikipedia If $p\geq 1/2$ , then $$ \Pr\left[ X>mp+x \right] \leq \exp(-x^2/2mp(1-p)) . $$ Thanks!
0
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2answers
232 views

Why is Fermat's spiral formula written as $r^2=a^2\theta$ instead of $r=a\sqrt{\theta}$?

I'm reading Clifford A. Pickover's Math Book, in the Fermat's spiral page, it says the Fermat's spiral formula is $r^2=a^2\theta$, why isn't it written as $r=\pm a\sqrt{\theta}$? What's the problem in ...
3
votes
2answers
258 views

When do I write $\sin(x)$ and when $\sin x$?

I sometimes see $\sin x$ and sometimes $\sin(x)$. Are the parenteses needed since the sine is a function or is it more an operator that can be premultiplied to the variable? Or are people just lazy?
1
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3answers
188 views

How to denote a function that depends on $a_1, \ldots, a_n$ without the dots?

I know that I can wrote something like $$a_1 + \cdots + a_n$$ without the dots as $$\sum_{i=1}^n a_i$$ which seems clearer to me. As a programmer, I'd rather have a rule set with variables than ...
2
votes
0answers
76 views

What's the equivalent to floor(x) and ceil(x) for real numbers?

Are there equivalents to the notation of $\lfloor x \rfloor$ and $\lceil x \rceil$ that don't round to the next integer, but to a specified digit of a real number? Examples $floorReal(2.3656, 1) = ...
3
votes
1answer
3k views

What is i (mod p)?

Is there a difference between $i$ mod $p$, and $i$ (mod $p$)? To give context, this is the original problem: if $i \geq 0$ what is $i$ (mod $p$)? edit: Forgot to add the parentheses to example
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4answers
1k views

Why not write $\sqrt{3}2$?

Is it just for aesthetic purposes, or is there a deeper reason why we write $2\sqrt{3}$ and not $\sqrt{3}2$?
1
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0answers
1k views

Semi-Factorial Skipping Every $k^\text{th}$ Number

For an integer $n$, the semi-factorial $n!!$ can be defined as $$ n!! = n(n-2)(n-4)\cdots $$ In other words, the semi-factorial of $n$ is the familiar factorial, but with every other term omitted. For ...
5
votes
5answers
283 views

Is there a definition or standard for the symbol $\pm$

In college, I had been taught the famous formula $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ Here $\pm$ means that I choose either $+$ or $-$. But I have seen sometimes in physics that $\pm$ can mean some ...
0
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2answers
113 views

Why is this set of $2 \times 2$ matrices of real numbers a nonabelian group under multiplication?

My book wrote something like this Consider $M_{2}^*(\mathbb{R}) = \left \{ A \in M_{2}(\mathbb{R}) : \det(A) \neq 0\right \}$ $A = \bigl(\begin{smallmatrix} 1 &1 \\ 1 &0 ...
2
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1answer
5k views

Notation: Matrix with a prime

I can't find this over the Internet: what does a matrix with a prime mean? A' I found this in an exercise, this is in basic linear algebra.
1
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1answer
465 views

little o notation with natural logs

I'm having trouble with little o notation. Help me show that: $2(n^2 + 100n)\log^5n = o(n^2\sqrt{n})$. It is the last hwk on my sheet and I don't understand it, if someone can help me with ...
2
votes
3answers
683 views

big O notation with asymptotically nonnegative increasing functions

Let $f(n)$ and $g(n)$ be asymptotically nonnegative increasing functions. Show: $f(n) · g(n) = O((\max\{f(n), g(n)\})^2)$, using the definition of big-oh. I can't quite figure this out, can ...
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2answers
74 views

Is there a particular notation for a function confined in a set?

For example $f:\mathbb{R}\to\mathbb{R}$ is a function. How to simply express a correspondent function $g:\mathbb{Q}\to\mathbb{R}$ such that $g(x)=f(x)$, $\forall x\in\mathbb{Q}$?
0
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1answer
71 views

infimum - notational problem

I have divided a two-dimensional coordinate system into several regions that form a partition. Now I want to define a line that is formed by the lower bound of some of these regions, and I cannot ...
5
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4answers
305 views

What is the correct definition of the absolute value of $x$, $|x|$?

What is the correct definition of the absolute value of $x$, $|x|$? Option A $$ |x|= \begin{cases} -x&\text{if } x < 0\\ 0& \text{if } x=0\\ x&\text{if } x>0 \end{cases} $$ Option ...
4
votes
1answer
297 views

What does $c.c.$ mean in this proof?

This is a proof from Wikipedia of Moore-Penrose inverse being the optimal solution of a least squares problem, in which there is a acronym $c.c.$ occurred in some of the equations. Mind if I ask what ...
0
votes
1answer
88 views

Two $\psi$ functions

This is either a notation/history question or a point of confusion. In (for example) Ramanujan's proof of Bertrand's postulate, he uses the following notation: $\log [x]!$ means $\log ([x]!),$ in ...
0
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2answers
875 views

Is there a mathematical symbol for “the value grows”?

Is there a mathematical symbol for "the value grows?" For example: This result will be increasingly difficult as the value of n grows to infinity.
0
votes
1answer
111 views

Is this function valid in the mathematical sense?

I'm trying to find the error in a proof that yields a contradictory result, and I was wondering if the (rather silly and convoluted) function involved in the proof is "valid" in the mathematical ...
2
votes
2answers
218 views

What does $\mathbb R^S$ mean when $S$ is a set? [duplicate]

Possible Duplicate: What's the meaning of a set to the power of another set? What does $\mathbb R^S$ mean when $S$ is a set? I am reading a text and I wonder if it has a special meaning ...
4
votes
3answers
5k views

Symbol for the set of odd naturals?

Obviously the set of naturals is denoted $\mathbb{N}$, but is there a symbol for the set of odd naturals? Would $2\mathbb{N}+1$ (or $2\mathbb{N}-1$) be a standard notation?
5
votes
2answers
625 views

Resources to learn the meaning of any math symbol

There is lots of symbols and may be operators like || in this expression $$ 3^k||n$$ that I would like to be able to quickly find the meaning of. I tried Wolfram|Alpha but I think it expects the ...
6
votes
1answer
396 views

Why is base-10 the most common positional notation radix for number representation?

I understand that number can have multiple representations, and I can conceive that the positional notation system was better adapted for arithmetics than say the Roman numerals system which led to ...
5
votes
1answer
446 views

Writing “$\nabla f$” or “$\operatorname{grad} f$”

When hand-writing the gradient of $f$ as "$\nabla f$" or "grad $f$", is it necessary to indicate that it is a vector using the usual vector markings (cap, arrow, wavy line, etc.)?
1
vote
2answers
41 views

Argumentation, series and convergence

Consider $\sum_{n=1}^{\infty} \frac{1}{n^{3}}$. We know that it converges. Given $k\in\left(0,\infty\right)$. Is it then "okay" to say that there exists a $j\in\mathbb{N}$ such that ...
2
votes
1answer
128 views

Are $\mathscr{A}_\mu$ and $\mathscr{M}_{\mu^*}$ the same?

I am reading Cohn and he uses the notation $\mathscr{A}_\mu$ to mean the completion of $\mathscr{A}$ under $\mu$, and he says that a set in $\mathscr{A}_\mu$ is called $\mu$-measurable. He uses the ...
1
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2answers
61 views

Category formulas without explicit specifying of objects

Consider the following example: $C$ is a category each of whose Hom-sets is partially ordered. Let $f$, $g$, and $h$ are morphisms of this category. Consider the formula: $g\circ f \ge h$. Intuition ...
1
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3answers
130 views

Sigma question: is it legal to write something like this?

Is this mathematical syntax correct? $$\sum_{n+1}^m\sin(n-2) $$ As you see, the starting value is $n+1$ instead of being just purely one variable.
5
votes
2answers
1k views

Notation showing a set is non-empty and finite

Does there exist any mathematical notation that would indicate that a set $V$ is non-empty and finite? Or would I have to write this out in words?
-4
votes
3answers
305 views

What is the meaning of $P(A \cup B)$ in probability and statistics?

In probability, letters $A$ and $B$ are used to denote various events. Then we write $P(A)$ for the probability of event $A$ happening. Same for $P(B)$. But I often see the notation $P(A\cup B)$ as ...
4
votes
2answers
695 views

How should “7 $\log_{10}$” be interpreted?

A cookery related article I want to refer to mentions a "7 log10 relative reduction of salmonella". A few related sources suggest this evaluates to 10,000,000, although I would have imagined that 107 ...
3
votes
2answers
124 views

What letter should I use to denote an ideal?

In commutative algebra, there seem to be two rather different notational conventions for ideals: either $I,J, \dots$ or $\mathfrak{a}, \mathfrak{b}, \dots$. By itself, it is hardly surprising - after ...
0
votes
1answer
102 views

What does $f:\mathbb{R}^d \rightarrow \in \mathbb{R}^{d'}$ mean?

In this MSR Technical Report, I came across a definition of the manifold learning problem: Given a set of $k$ unlabelled observations $\{v_1,v_2,\dots,v_i,\dots,v_k\}$ with $v_i \in ...
1
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3answers
137 views

Help with odd partial derivatives in velocity $\bar v^2 = \dot x ^2+\dot y^2$

I am doing a physics -course Tfy-0.2061. My teacher claims that this is velocity squared, $\bar v^2 = \dot x ^2+\dot y^2$. I cannot understand why it is not $\bar v^2 = (\dot x +\dot y)^2$. If ...
6
votes
2answers
142 views

Is the 'arc-' notation for inverse trigonometric and hyperbolic functions discouraged?

Any books we've used throughout high school and university preferred the '^-1' notation, leading me to believe that the 'arc-' notation is archaic. It feels like we're taught it, but discouraged from ...
9
votes
5answers
1k views

How does one read aloud Vinogradov's notation $\ll$ and $\ll_{\epsilon }$?

How does one read aloud the Vinogradov's notation $\ll$ and $\ll_{\epsilon }$ as in $$f(x)\ll g(x)$$ and $$c\ll_{\epsilon }\left( \prod\limits_{p\mid abc}p\right) ^{1+\epsilon}.$$ Is the first one ...
0
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1answer
203 views

Set operations on asymptotic notations

How do you solve something like this? Find A and B from the following expressions: ...
6
votes
2answers
280 views

Symbols and terminology for distinguishing derivability from sequents

First some definitions to make it clear what I'm talking about: A deductive system is a set $J$ of judgments together with a set $R$ of inference rules each of the form $$ j_0 \leftarrow j_1, ...
0
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0answers
96 views

Parsing complicated formula

On a homework assignment, we were given $$x \lt 5y \leftrightarrow x \gt z \rightarrow x + z \geq yw \wedge -x \lt z$$ I broke it up into $$1~~~x <5y$$ $$2~~~~~x > z$$ $$3~~~x + z \geq yw ...
6
votes
1answer
253 views

History of Lie algebra notation (in Fraktur)?

Does anyone know how it has become the standard to express Lie algebras in fraktur? I'd also like to know how it's established for each era and region, not only the origin. It doesn't seem that ...