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.
0
votes
3answers
89 views
Distance between two points
The distance between the two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is the quantity
$$\mathrm{distance}(P, Q) = \sqrt{(\Delta x)^2 + (\Delta y)^2}.$$
Is $(P, Q)$ above indicating an open ...
0
votes
1answer
28 views
Birkhoff Lattice theory notation question- probably easy to answer
In Lattice Theory p. 30 3rd edition:
Lemma 3. In any distributive lattice, every polynomial is equivalent to a join of meets, and dually:
$p(x_1,...,x_r)=\lor_{\alpha \in A}\{ \land_{S_\alpha} x_i ...
2
votes
3answers
44 views
A question about $[c_0,c_1,\ldots,c_n]$ notation for continued fractions
I try to understand why by definition
$[c_0,c_1,\ldots,c_n]=[c_0,[c_1,\ldots,c_n]]$ and also
$[c_0,c_1,\ldots,c_n]=[c_0,c_1,\ldots,c_{n-2},[c_{n-1},c_n]]$ .
Those are continued fractions, and ...
1
vote
0answers
51 views
Is $\langle f \rangle $ an “inner product”?
Let $$\langle f(x,y)\rangle = \iint_S f(x,y)\,\mathrm{d}x\,\mathrm{d}y$$
I have seen the above in multiple papers as the definition of $\langle f(x,y)\rangle$. I would normally associate angle ...
-2
votes
4answers
93 views
Derivative of product notation?
Presume $f(x,y)$ is a continuous function. How would I take the derivative of $$\prod_{x=1}^N f(x,y)$$?
Edit: derivative with respect to $x$, that is.
6
votes
2answers
138 views
Is a bra the adjoint of a ket?
The instructor in my quantum computation course sometimes uses the equivalence
$$(\left|a\right>)^\dagger\equiv\left<a\right|$$
I understand that this is true for the typical matrix ...
2
votes
1answer
97 views
Does bra-ket notation work for all inner product spaces?
My quantum computation instructor keeps referring to the vector space in which he is using Dirac's bra-ket notation as an "inner product space", but doesn't it need additional properties to use that ...
2
votes
2answers
52 views
Can minutes be used to measure the length?
I am using an Old American maths textbook and it uses US customary units. For the length, they use minutes (like $6'$ for the altitude of a triangle). How is this related to foot which is the unit of ...
1
vote
1answer
39 views
Indexed Family of Sets
Most books write a family of sets $A_i$ with index set $I$ as $\{ A_i \}_{i \in I}$. However, I've read other books that have criticized this notation; they insist that one should write $(A_i )_{i \in ...
1
vote
1answer
78 views
What does the semicolon ; mean in a function definition
Cauchy's Hypothesis or Noll's theorem states that $\vec{t}(\vec{X},t;\partial \Omega) = \vec{t}(\vec{X},t;\vec{N})$ where $\vec{N}$ is the outward unity normal to the positively oriented surface ...
0
votes
2answers
40 views
Hyperfactorial curiosity
I've been familiarizing myself with the hyperfactorial, and I'm simply curious if it has an extension/analogue into the world of rational numbers, irrational numbers, and complex numbers like the ...
6
votes
1answer
216 views
Is there a rigorous theory of context, whereby sets can gain additional structure within a context?
Consider sets $G$ and $H$ and a function $f : G \rightarrow H$. So far, it doesn't really make sense to ask whether $G$ and $H$ are groups (technically, the answer is "no, they're not groups"), and ...
1
vote
2answers
82 views
How many total order relations on a set $A$?
Let's define a set $T_A$ which is the set of all total order relations on $A$.
This set is a subset of the set of all $2$-adic relations on $A$: $$T_A \subset \mathcal P(A^2) $$
1-Which is the ...
2
votes
2answers
69 views
Explain to me what these symbols mean.
Okay this may sound stupid but I need a little help...
What does:
$\Large \frac{d}{dx}$ and $\Large \frac{dy}{dx}$ mean?
I need a through explanation. Thanks.
1
vote
2answers
124 views
What does the plus sign contained in a circle ($\oplus$) mean in this case?
The fundamental group of the torus is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$.
I know that the $\oplus$ symbol is the exclusive or symbol but I don't understand how two of the same sets are ...
0
votes
1answer
32 views
Big Oh power difference?
Can a function with higher power like $n^3$ become big oh for a lower power function let say $O(n^2)$
1
vote
1answer
38 views
Big oh and big Omega?
I have question is about big oh and big omega
if $f(n)$ is $\Omega(n^2)$ is $f(n)$ $O(n)$?
3
votes
2answers
74 views
Summation/Sigma notation
There are lots of variants in the notation for summation. For example, $$\sum_{k=1}^{n} f(k), \qquad \sum_{p \text{ prime}} \frac{1}{p}, \qquad \sum_{\sigma \in S_n} (\operatorname{sgn} \sigma) a_{1 , ...
5
votes
5answers
230 views
+50
If $f(x)\to f(a)$ when $x\to a$, why don't we denote it as $\displaystyle \lim_{x\to a}f(x)\to f(a)$?
If $f(x)\to f(a)$ when $x\to a$, why don't we denote it as
$$
\lim_{x\to a}f(x)\to f(a)
$$
instead of
$$
\lim_{x\to a}f(x) = f(a)?
$$
I need a comprehensible explanation for a newbie like me!
0
votes
1answer
46 views
Examples of formulas or equations with several variables/constants? [closed]
This may be a bizarre question (and probably not well-defined). However, are there any formulas or equations that utilizes many number of distinct variables?
For instance, Euler's identity uses 5:
...
2
votes
2answers
41 views
Tensor notation and rules
I have a few questions about tensors:
I appreciate that $g^{\alpha\beta}=g^{\beta\alpha}$ but when contracting say $T^{\sigma}_{\mbox{ }\;\mu\nu\rho}$ to $T_{\;\;\mu\nu}$, first of all can it be ...
0
votes
0answers
45 views
What does this notation mean?
In complex analysis, what does $z \in \overline{\mathbb{C}^-}$
Is it saying that the conjugate of all complex numbers whos imaginary part is negative?
0
votes
3answers
58 views
Question about changing a logarithm's base
I've been using the following method to derive/remember the logarithm base conversion formula: If I want to convert $\log_a(x)$ to an expression in base $b$, I say,
$$a^{\log_a(x)}=x\\
...
2
votes
1answer
75 views
A problem with lambda calculus notation and semantics for function-valued functions
I would like to understand how to use the $\lambda$-notation to write usual (set-theoretic) functions, and if it is possible at all.
Here are my naïve attempts. Assume that all variables are ...
0
votes
0answers
38 views
Kronecker delta, expansion
Can anyone help me understand how to expand the following:
$$
\delta u_{i,j}(\delta_{im} +u_{i,m})
$$
Where $\delta u_{i,j} $ is the first variation of $u$, with the comma representing the ...
6
votes
0answers
169 views
Looking for an approach to mathematical notation wherein the universe is divided into disjoint worlds.
Is there a rigorous approach to mathematical notation wherein the "universe" is divided into disjoint "worlds," and the meaning of notation is world-dependent? This would solve a few pesky problems. ...
0
votes
2answers
58 views
Leibniz notation - how to get $dx$ out of a derivative $v = \frac{dx}{dt}$
I know that velocity equals $v = \frac{dx}{dt}$ which is writen in Leibniz notation. How can i get $dx$ out of it in a proper way? I don't like it when people say that i should just multiply ...
2
votes
1answer
42 views
Does this summation index make any sense?
From my textbook, I have this summation:
$$
y_f(k) = \sum_{\tau = k_0}^{k-1}a^{k-1-\tau}g(\tau)
$$
So far so good. But then there is a "change of variable" $\tau = \theta - m$ and the summation ...
2
votes
1answer
91 views
In Logic is ⇒, →, and ⊃ basically the same symbol?
I need to create a few truth tables and I got confused by the logic symbols as some of the questions use either one or the other which is really confusing especially if they all mean the same thing.
...
10
votes
1answer
185 views
Using $\bigvee$ and $\bigwedge$ instead $\exists$ and $\forall$
My professor of Algebra use some "strange" notation for me. He uses $\bigvee$ instead $\exists$ and $\bigwedge$ instead $\forall$. For example $$\displaystyle\bigwedge_{x\in \mathbb{Z}}\bigwedge_{m\in ...
1
vote
1answer
19 views
Multiple Indexing Sets
I suspect that this is a trivial thing, but I can't seem to find an example.
Say I have some Set indexed by Set $I$:
$$\{ x_i \}_{i \in I}$$
This called a "Family" of Sets, correct?
Next, say I ...
1
vote
2answers
107 views
The meaning of $\int\limits_\mathbb{R}$
Which of
$\lim_{a\to\infty} \int_{-a}^{a}$ and $\lim_{r\to-\infty} \lim_{s \to \infty} \int_{r}^{s}$
does $\int_{\mathbb{R}}$
mean?
0
votes
2answers
66 views
Nested Set subscript notation
Suppose I have a Set $A$ with elements $P$:
$A = \{P_1,P_2,...,P_n\}$
And another Set $B$, where the elements are Set $A$:
$B = \{A_1,A_2,...,A_i\}$
How would I refer to a specific element $P$, as ...
3
votes
0answers
52 views
Why is the Euclidean metric called the prime at infinity?
I've been studying p-adic analysis recently and after a bit of searching on the web, I haven't found an answer as to why the Euclidean metric is referred to as the 'prime at infinity', and given the ...
4
votes
2answers
46 views
Naming variables with same number of subscripts
I have a very stupid question.
I am writing a LaTeX document with a complex mathematical model and I'm running out of letters and symbols. I also don't want to keep adding more and more \hat{}'s and ...
3
votes
0answers
49 views
What is the name of the function that indexes Grothendieck universes?
Assume Tarski-Grothendieck set theory. Then Grothendieck universes form a well-ordered proper class, so we can let $U_\alpha$ denote the $\alpha$'th Grothendieck universe, where $\alpha$ is an ...
0
votes
2answers
97 views
Evaluating using BRA KET Notation?
Evaluate $\langle 0 \mid x^3 \mid 1\rangle$, assuming that all the wave functions you encounter are normalized eigenfunctions of the harmonic oscillator Hamiltonian, without Mathematica, Maple, or ...
3
votes
2answers
62 views
two notation: semi-metric and pesudometric
There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different?
Thanks ahead.
0
votes
1answer
37 views
What does $C^{\text{1,stw}}(0,b)$ mean
I know that this C with the one means continoulsy differentiable functions but what does this stw stand for? Does anyone know this?
1
vote
2answers
126 views
what is $C^{-\infty}(\mathbb{R})$
Thanks in advance.
what is $C^{-\infty}(\mathbb{R})$?
Is that the same as the "distribution" defined in differential geometry?
It would be helpful if someone can describe it in another way ...
0
votes
2answers
70 views
Proper Use Of Notation? (Determine whether the function is surjective)
Determine whether the function $f: Z×Z→Z$ is onto if
a) $f(m,n)=m+n$.
b) $f(m,n)=m^2+n^2$
c) $f(m,n)=m$.
d) $f(m,n)=|n|$.
e) $f(m,n)=m−n$.
I was able to answer these ...
2
votes
3answers
138 views
Is adding a plus before a positive number a mistake?
I had a simple problem to find out the function equation from a sinusoidal graph.
$$F(x)= -2\sin(2x)$$
Would it be considered a "mistake" to write it down the following way?
$$F(x)= -2\sin(+2x)$$
...
3
votes
2answers
149 views
Difference between formula and algorithm
What is the difference between the terms formula and algorithm in mathematics? I haven't seen the definition of formula anywhere. I know that algorithm means that Turing machine halts for every input. ...
0
votes
0answers
110 views
Using the Cauchy integral formula to evaluate $\int_{\gamma=(a,a)} \frac{z}{z^4-1} dz$.
I'm trying to understand how to use the Cauchy integral formula, but a bit confused as to how to use it as I cant seem to get the right answer!
$$\int_{\gamma=(a,a)} \frac{z}{z^4-1} dz$$ where ...
1
vote
1answer
37 views
Problem converting matlab formula for relative frequency vector to mathematical formula
I have the following matlab code:
...
7
votes
0answers
77 views
If $H$ is a subgroup of $G$ and $x,y\in G$, what is $xHy$ called?
For a group $G$, its subgroup $H$ and $x,y\in G,$ we call $xH$ a left coset of $H,$ and we call $Hy$ a right coset of $H.$ Is there a special name for sets of the form $xHy$? Is there a name or ...
0
votes
1answer
32 views
How could I write the intersection of all sets in $S$ containing $E$ in symbols?
How could I write the intersection of all subsets in $S$ containing $E$ in symbols?
Something like? $$\bigcap_{E\subset C}C$$
0
votes
1answer
31 views
Mathematical description of an identifier which consists of text and numbers [closed]
in my code I use an artificial identifier of the form UIxxx where xxx is in the range of 1 to 999. In my assay, where I describe this unique identifier I am not sure how to mathematically describe it. ...
2
votes
2answers
143 views
What does $\propto$ mean?
From this article:
$\ldots$a maximum-a-postiori "a maximum-a-posteriori
$(MAP_{x,k}^{\,\,\,\,\,1})$ estimation, seeking a pair $(\hat{x}, \hat{k})$ maximizing:
$$p(x, k\mid y) \propto p(y|x, ...
4
votes
1answer
43 views
Abuse of notation in declaring a variable is a function of another?
The standard way to write $ \text{ y is a function of x} $ is
$ y = f(x) $
This is taken to mean that $y$ is the value of function $f$ evaluated at $x$. For simplicity let's take $f$ to be some ...
