Intereting Posts

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I
Circle Packing Algorithm
Proof: Tangent space of the general linear group is the set of all squared matrices
Fly and Two Trains Riddle
On the possible values of $\sum\varepsilon_na_n$, where $\varepsilon_n=\pm1$ (i.e., changing signs of the original series)
Maximize :: $A = B \times C$
The Aleph numbers and infinity in calculus.
Direct product of two normal subgroups
Is the function $ f(x,y)=xy/(x^{2}+y^{2})$ where f(0,0) is defined to be 0 continuous?
What exactly are fractals
A simple inequality for floor function
Proving $\int_a^b \frac {x dx}{\sqrt{(x^2-a^2)(b^2-x^2)}} = \frac {\pi}{2} $
Uniform convergence of $\sum_{n=0}^{\infty} \frac{\int_{\sin nx}^{\sin(n+1)x}\sin t^2dt \int_{nx}^{\infty}\frac{dt}{\sqrt{t^4+1}}}{1+n^3x^2}$
Why is the shortest distance between two circles along the segment connecting their centers?
If $r : X \to A$ is a deformation retract and $i : A \to X$ is inclusion, then $i(r)$ is homotopic to $id$ on $X$

I am reading a paper (to be able to implement the Baum-Welch algorithm in it) and the following notation is defined:

$$

[ a_k ]_{k=i}^j ≡ (a_i, a_{i+1}, \ldots , a_j)

$$

$$

[a(k)]_{k=i}^j ≡ (a(i), a(i+ 1), \ldots , a( j))

$$

- Sole minimal element: Why not also the minimum?
- Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?
- Size of a union of two sets
- Strange set notation (a set as a power of 2)?
- How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
- Cardinality of the set of all pairs of integers

I (think) the first is shorthand of a n-tuple. I guess the second is something like that, but I don’t understand the difference between the first and the second. Is this common notation something odd, or what? I am pretty rusty on my math, so if I am missing something obvious please don’t hesitate to point that out.

- Prove or find a counter example to the claim that for all sets A,B,C if A ∩ B = B ∩ C = A ∩ C = Ø then A∩B∩C ≠ Ø
- Upper bound on cardinality of a field
- Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.
- Alternative notation for exponents, logs and roots?
- Prove that two sets are the same
- If $A,B,$ and $C$ are sets, then $A\times(B-C)$ = $(A \times B)$ $-$ $(A \times C)$.
- Proof of $(A - B) - C = A - (B \cup C)$
- Notation for intervals
- Size of a union of two sets
- Is the function $f:\mathbb{R}^2\to\mathbb{R}^2$, where $f(x,y)=(x+y,x)$, one-to-one, onto, both?

As the comments state, there is no essential difference between the two. They simply both mean:

“The part of the sequence $a$ starting at index $i$ and ending at $j$.”

As Brian M. Scott points out, there are several justifications for both of these notations for a sequence occurring:

Sometimes you want to emphasize the fact that a is a function of $k$. Sometimes the choice is made on the basis of readability: one version combines badly with other notation in use, resulting in cluttered expressions that are hard to read. Sometimes you just happened to use one instead of the other.

- what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by 47?
- Don't we need the axiom of choice to choose from a non-empty set?
- Asymptotic inverses of asymptotic functions
- Why do disks on planes grow more quickly with radius than disks on spheres?
- A closed form for $\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx$
- Proper subgroups of non-cyclic p-group cannot be all cyclic?
- Linear independence of images by $A$ of vectors whose span trivially intersects $\ker(A)$
- What are the differences between rings, groups, and fields?
- Why is this Inequality True for all Positive Real Numbers?
- Finding the sum of the series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$
- Is it possible to use mathematical induction to prove a statement concerning all real numbers, not necessarily just the integers?
- $C^{1}$ function such that $f(0) = 0$, $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$ and $\int_{0}^{1}f(x)\, dx = 1$
- Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$
- Is there a purely algebraic proof of the Fundamental Theorem of Algebra?
- Counting number of moves on a grid