Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What Does it Really Mean to Have Different Kinds of Infinities? Can someone explain to me how there can be different kinds of infinities?
I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me.
Any help would be ap... | This is an answer to the following question marked as duplicate which redirects here: "I've known for some time that infinitary numbers can be different in order, such as the integers (countable), and the real numbers (uncountable). I read that you can always find a higher order of infinity given any order of infinity.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "191",
"answer_count": 9,
"answer_id": 0
} |
How can you prove that the square root of two is irrational? I have read a few proofs that $\sqrt{2}$ is irrational.
I have never, however, been able to really grasp what they were talking about.
Is there a simplified proof that $\sqrt{2}$ is irrational?
| There is also a proof of this theorem that uses the well ordering property of the set of positive integers, that is in a non empty set of positive integers there is always a least element. The proof follows the approach of proof by contradiction but uses the well ordering principle to find the contradiction :) -
Let u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/5",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "64",
"answer_count": 17,
"answer_id": 1
} |
Is it true that $0.999999999\ldots=1$? I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?
| \begin{align}
x &= 0.999... \\
10x &= 9.999... \\
&= 9 + 0.999... \\
&= 9 + x \\
10x - x &= (9 + x) - x \\
(10 - 1)x &= 9 + (x - x) \\
9x &= 9 \\
x &= 1
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/11",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "348",
"answer_count": 31,
"answer_id": 0
} |
How do you calculate the semi-minor axis of an ellipsoid? Given the semi-major axis and a flattening factor, is it possible to calculate the semi-minor axis?
| Where,
$a$ = transverse radius = semi-major axis (for ellipse/oblate spheroid);
$b$ = conjugate radius = semi-minor axis (" " ");
$oe$ = angular eccentricity = $\arccos(\frac{b}{a})$;
$f$ = flattening = $\frac{a-b}{a} = 1 - \frac{b}{a} = 1 - \cos(oe) = 2\sin(\frac{oe}{2})^{2}$;
then $b = a\cos(oe) = a(1-f)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/18",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
List of Interesting Math Blogs I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills.
I'll start with my entries:
*
*Division By Zero
*Tanya Khovanova’s Math Bl... | Timothy Gowers' blog is excellent. Like Terence Tao, he is both a Fields medalist and an excellent writer. Together their blogs were my first real introduction into how professional mathematicians think, and their writing has taught me a lot, both about mathematics and about mathematical writing. If you are a seriou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "195",
"answer_count": 22,
"answer_id": 10
} |
Online resources for learning Mathematics Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning.
As someone doing a non-maths degree in college I'd be interested in finding some res... | A useful one for undergraduate level maths is Mathcentre. It has useful background material for people studying maths, or who need some maths background for other courses.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/90",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 10,
"answer_id": 2
} |
How would you describe calculus in simple terms? I keep hearing about this weird type of math called calculus. I only have experience with geometry and algebra. Can you try to explain what it is to me?
| One of the greatest achievements of human civilization is Newton's laws of motions. The first law says that unless a force is acting then the velocity (not the position!) of objects stay constant, while the second law says that forces act by causing an acceleration (though heavy objects require more force to accellera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 9,
"answer_id": 0
} |
Real world uses of hyperbolic trigonometric functions I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.
Is there any good examples of their uses outside academia?
| Velocity addition in (special) relativity is not linear, but becomes linear when expressed in terms of hyperbolic tangent functions.
More precisely, if you add two motions in the same direction, such as a man walking at velocity $v_1$ on a train that moves at $v_2$ relative to the ground, the velocity $v$ of the man re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 10,
"answer_id": 7
} |
Do complex numbers really exist? Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that comple... | The argument isn't worth having, as you disagree about what it means for something to 'exist'. There are many interesting mathematical objects which don't have an obvious physical counterpart. What does it mean for the Monster group to exist?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "530",
"answer_count": 37,
"answer_id": 3
} |
Do complex numbers really exist? Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that comple... | In the en they exists as a consistent definition, you cannot be agnostic about it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "530",
"answer_count": 37,
"answer_id": 35
} |
Why is the volume of a sphere $\frac{4}{3}\pi r^3$? I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for the formula?
| I am no where near as proficient in math as any of the people who answered this before me, but nonetheless I would like to add a simplified version;
A cylinder's volume is:
$$\pi r^2h$$
A cone's volume is $\frac{1}{3}$ that of a cylinder of equal height and radius:
$$\frac{1}{3}\pi r^2h$$
A sphere's volume is two cone... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "108",
"answer_count": 19,
"answer_id": 10
} |
Is there possibly a largest prime number? Prime numbers are numbers with no factors other than one and itself.
Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "possible factors" that number might have.
So the larger the number, it seems like the... | Another proof is:
Consider the numbers $$9^{2^n} + 1, \\ \\ n = 1,2,\dots$$
Now if $$9^{2^n} + 1 = 0 \mod p$$ then we have that, for $ m > n$ that
$$9^{2^m} + 1 = (9^{2^n})^{2^{m-n}} + 1 = (-1)^{2^{m-n}} + 1 = 1+1 = 2 \mod p$$
Thus if one term of the sequence is divisible by a prime, none of the next terms are divisib... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 9,
"answer_id": 0
} |
Proof that the sum of two Gaussian variables is another Gaussian The sum of two Gaussian variables is another Gaussian.
It seems natural, but I could not find a proof using Google.
What's a short way to prove this?
Thanks!
Edit: Provided the two variables are independent.
| I posted the following in response to a question that got closed as a duplicate of this one:
It looks from your comment as if the meaning of your question is different from what I thought at first. My first answer assumed you knew that the sum of independent normals is itself normal.
You have
$$
\exp\left(-\frac12 \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 2,
"answer_id": 0
} |
Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Can someone give a simple explanation as to why the harmonic series
$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$
doesn't converge, on the other hand it grows very slowly?
I'd prefer an easily comprehensible explanation rather tha... | Let's group the terms as follows:$$A=\frac11+\frac12+\frac13+\frac14+\cdots\\ $$
$$
A=\underbrace{(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{9})}_{\color{red} {9- terms}}
+\underbrace{(\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\cdots+\frac{1}{99})}_{\color{red} {90- terms}}\\+\underbrace{(\frac{1}{101}+\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "159",
"answer_count": 25,
"answer_id": 9
} |
What is the single most influential book every mathematician should read? If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
| There are so many, and I've already seen three that I would mention. Two more of interest to lay readers:
The Man Who Knew Infinity by Robert Kanigel. Excellently written, ultimately a tragedy, but a real source of inspiration.
Goedel's Proof by Nagel & Newman. Really, a beautiful and short exposition of the nature ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "104",
"answer_count": 30,
"answer_id": 16
} |
Simple numerical methods for calculating the digits of $\pi$ Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the first few ... | The first method that I applied successfully with function calculator was approximation of circle by $2^k$-polygon with approximating sides with one point on the circle and corners outside the circle. I started with unit circle that was approximated by square and the equation $\tan(2^{-k} \pi/4) \approx 2^{-k} \pi/4$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 7,
"answer_id": 0
} |
Calculating the probability of two dice getting at least a $1$ or a $5$ So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question.
This question comes from the game farkle.
| The other way to visualise this would be to draw a probability tree like so:
alt text http://img.skitch.com/20100721-xwruwx7qnntx1pjmkjq8gxpifs.gif
(apologies for my poor standard of drawing :) )
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Applications of the Fibonacci sequence The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci sequence?
| Suppose you're writing a computer program to search a sorted array for a particular value. Usually the best method to use is a binary search. But binary search assumes it's the same cost to read from anywhere in the array. If it costs something to move from one array element to another, and the cost is proportional to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 18,
"answer_id": 6
} |
What is larger -- the set of all positive even numbers, or the set of all positive integers? We will call the set of all positive even numbers E and the set of all positive integers N.
At first glance, it seems obvious that E is smaller than N, because for E is basically N with half of its terms taken out. The size of... | They are both the same size, the size being 'countable infinity' or 'aleph-null'. The reasoning behind it is exactly that which you have already identified - you can assign each item in E to a single value in N. This is true for the Natural numbers, the Integers, the Rationals but not the Reals (see the Diagonal Slash ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 9,
"answer_id": 1
} |
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$? I know there must be something unmathematical in the following but I don't know where it is:
\begin{align}
\sqrt{-1} &= i \\\\\
\frac1{\sqrt{-1}} &= \frac1i \\\\
\frac{\sqrt1}{\sqrt{-1}} &= \frac1i \\\\
\sqrt{\frac1{-1}} &= \frac1i \\\\
\sqrt{\frac{-1}1} &= \frac1i \\\\
... | Isaac's answer is correct, but it can be hard to see if you don't have a strong knowledge of your laws. These problems are generally easy to solve if you examine it line by line and simplify both sides.
$$\begin{align*}
\sqrt{-1} &= i &
\mathrm{LHS}&=i, \mathrm{RHS}=i
\\
1/\sqrt{-1} &= 1/i &
\mathrm{LHS}&=1/i=-i, \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "187",
"answer_count": 14,
"answer_id": 3
} |
Good Physical Demonstrations of Abstract Mathematics I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use.
I'm looking for nontrivial ideas in abstract mathematics that ... | I cannot resist mentioning the waiter's trick as a physical demonstration of the fact that $SO(3)$ is not simply connected. For those who don't know it, it is the following: you can hold a dish on your hand and perform two turns (one over the elbow, one below) in the same direction and come back in the original positio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "101",
"answer_count": 19,
"answer_id": 4
} |
Conjectures that have been disproved with extremely large counterexamples? I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.
I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ if it... | Another class of examples arise from diophantine equations with huge minimal solutions. Thus the conjecture that such an equation is unsolvable in integers has only huge counterexamples. Well-known examples arise from Pell equations, e.g. the smallest solution to the classic Archimedes Cattle problem has 206545 decimal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "260",
"answer_count": 18,
"answer_id": 6
} |
Can you find a domain where $ax+by=1$ has a solution for all $a$ and $b$ relatively prime, but which is not a PID? In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theo... | If I'm not mistaken, the integral domain of holomorphic functions on a connected open set $U
\subset \mathbb{C}$ works. It is a theorem (in Chapter 15 of Rudin's Real and Complex Analysis, and essentially a corollary of the Weierstrass factorization theorem), that every finitely generated ideal in this domain is princ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 5
} |
What is a Markov Chain? What is an intuitive explanation of Markov chains, and how they work? Please provide at least one practical example.
|
A Markov chain is a discrete random
process with the property that the
next state depends only on the current
state (wikipedia)
So $P(X_n | X_1, X_2, \dots X_{n-1}) = P(X_n | X_{n-1})$. An example could be when you are modelling the weather. You then can take the assumption that the weather of today can be pred... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 5,
"answer_id": 4
} |
What is the optimum angle of projection when throwing a stone off a cliff? You are standing on a cliff at a height $h$ above the sea. You are capable of throwing a stone with velocity $v$ at any angle $a$ between horizontal and vertical. What is the value of $a$ when the horizontal distance travelled $d$ is at a maximu... | I don't have a complete solution, but I attempted to solve this problem using calculus.
$x'=v \cos a$
$y''= -g$ and (at $t=0) \quad y'= v \sin a$
So, $y'= v \sin a -gt$
$x_0=0$, so $x=vt \cos a$
$y_0=h$, so $y=vt \sin a - \frac12 gt^2+c$ (integrating with respect to $t$)
Subbing in $h, y=vt \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
"answer_id": 2
} |
Why are differentiable complex functions infinitely differentiable? When I studied complex analysis, I could never understand how once-differentiable complex functions could be possibly be infinitely differentiable. After all, this doesn't hold for functions from $\mathbb R ^2$ to $\mathbb R ^2$. Can anyone explain wha... | When one uses the complex plane to represent the set of complex numbers ${\bf C}$,
$z=x+iy$
looks so similar to the point $(x,y)$ in ${\bf R}^2$.
However, there is a difference between them which is not that obvious. The linear transformation in ${\bf R}^2$, can be represented by a $2\times 2$ matrix as long as one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "83",
"answer_count": 5,
"answer_id": 4
} |
What's an intuitive way to think about the determinant? In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ mat... | If you have a matrix
*
*$H$
then you can calculate the correlationmatrix with
*$G = H \times H^H$
(H^H denotes the complex conjugated and transposed version of $H$).
If you do a eigenvalue decomposition of $G$ you get eigenvalues $\lambda$ and eigenvectors $v$, that in combination $\lambda\times v$ describes th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "793",
"answer_count": 17,
"answer_id": 10
} |
How many circles of a given radius can be packed into a given rectangular box? I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out.
Let's say I have a rectangle of length $l$ and width $w$.
Is there a simple equation that can be used to sho... | I had an answer before, but I looked into it a bit more and my answer was incorrect so I removed it. This link may be of interest: Circle Packing in a Square (wikipedia)
It was suggested by KennyTM that there may not be an optimal solution yet to this problem in general. Further digging into this has shown me that this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 2,
"answer_id": 0
} |
Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,... | it is not absolutely convergent (that is, if you are allowed to reorder terms you may end up with whatever number you fancy).
If you consider the associated series formed by summing the terms from 1 to n of the original one, that is you fix the order of summation of the original series, that series (which is not the or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "60",
"answer_count": 12,
"answer_id": 0
} |
Online Math Degree Programs Are there any real online mathematics (applied math, statistics, ...) degree programs out there?
I'm full-time employed, thus not having the flexibility of attending an on campus program. I also already have a MSc in Computer Science. My motivation for a math degree is that I like learning a... | *
*Penn State has a Master of Applied Statistics.
*Stanford has
Computational and Math Engineering.
From what I've read, many universities will not make a distinction on the degree stating whether it was earned online or not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 10,
"answer_id": 5
} |
Least wasteful use of stamps to achieve a given postage
You have sheets of $42$-cent stamps and
$29$-cent stamps, but you need at least
$\$3.20$ to mail a package. What is the
least amount you can make with the $42$-
and $29$-cent stamps that is sufficient
to mail the package?
A contest problem such as thi... | A similar problem is known in mathematics. It is called the "Postage-Stamp" problem, and usually asks which postal values can be realized and which cannot. Dynamic programming is a common but not polynomial-time solution for this. A polynomial time solution using continued fractions in the case of two stamp denomina... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Why does Benford's Law (or Zipf's Law) hold? Both Benford's Law (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and Zipf's Law (given a corpus of natural language utterances, the frequency of any word is roughly inversely proportional t... | My explanation comes from a storage perspective: Uniformly distributed high-precision numbers that range over many powers are very expensive to store. For example, storing 9.9999989 and 99999989 is much more expensive than storing 10^1 and 10^8 assuming "they" are okay with the anomaly. The "they" refers to our simula... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 10,
"answer_id": 8
} |
Separation of variables for partial differential equations What class of Partial Differential Equations can be solved using the method of separation of variables?
| There is an extremely beautiful Lie-theoretic approach to separation of variables,
e.g. see Willard Miller's book [1] (freely downloadable). I quote from his introduction:
This book is concerned with the
relationship between symmetries of a
linear second-order partial
differential equation of mathematical
physics, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Cotangent bundle This may be a poorly phrased question - please let me know of it - but what is the correct way to think of the cotangent bundle? It seems odd to think of it as the dual of the tangent bundle (I am finding it odd to reconcile the notions of "maps to the ground field" with this object).
| I'm not completely sure what you mean by this: "It seems odd to think of it as the dual of the tangent bundle (I am finding it odd to reconcile the notions of "maps to the ground field" with this object)," but maybe the following will help you see why it is natural to consider the dual space of the tangent bundle.
Give... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
Correct usage of the phrase "In the sequel"? History? Alternatives? While I feel quite confident that I've inferred the correct meaning of "In the sequel" from context, I've never heard anyone explicitly tell me, so first off, to remove my niggling doubts: What does this phrase mean?
(Someone recently argued to me that... | I would never write that (mostly because it really sounds like the definition will be in another paper...). I'm pretty sure I'd write «In what follows, ...».
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 1
} |
Probability to find connected pixels Say I have an image, with pixels that can be either $0$ or $1$. For simplicity, assume it's a $2D$ image (though I'd be interested in a $3D$ solution as well).
A pixel has $8$ neighbors (if that's too complicated, we can drop to $4$-connectedness). Two neighboring pixels with value... | This looks a bit like percolation theory to me. In the 4-neighbour case, if you look at the dual of the image, the chance that an edge is connected (runs between two pixels of the same colour) is 1-2p+2p^2.
I don't think you can get nice closed-form answer for your question, but maybe a computer can help with some Mon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
What's the difference between open and closed sets? What's the difference between open and closed sets?
Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!
| An open set is a set S for which, given any of its element A, you can find a ball centered in A and whose points are all in S.
A closed set is a set S for which, if you have a sequence of points in S who tend to a limit point B, B is also in S.
Intuitively, a closed set is a set which contains its own boundary, while a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 3
} |
Usefulness of Conic Sections Conic sections are a frequent target for dropping when attempting to make room for other topics in advanced algebra and precalculus courses. A common argument in favor of dropping them is that typical first-year calculus doesn't use conic sections at all. Do conic sections come up in typi... | If you're a physics sort of person, conic sections clearly come up when you study how Kepler figured out what the shapes of orbits are, and some of their synthetic properties give useful shortcuts to things like proving "equal area swept out in equal time" that need not involve calculus.
The other skills you typically ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Mandelbrot-like sets for functions other than $f(z)=z^2+c$? Are there any well-studied analogs to the Mandelbrot set using functions other than $f(z)= z^2+c$ in $\mathbb{C}$?
|
Here's the Mandelbrot set on the Poincaré Disk. I made it by replacing all the usual operations in the iteration
$$z_{n+1} = z_n^2+c$$
by "hyperbolic" equivalents. Adding a constant was interpreted as translating in the plane, and the hyperbolic equivalent is then
$$z \mapsto \frac{z+c}{\bar{c}z+1}$$
For the squaring ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 11,
"answer_id": 6
} |
If and only if, which direction is which? I can never figure out (because the English language is imprecise) which part of "if and only if" means which implication.
($A$ if and only if $B$) = $(A \iff B)$, but is the following correct:
($A$ only if $B$) = $(A \implies B)$
($A$ if $B$) = $(A \impliedby B)$
The trouble i... | The explanation in this link clearly and briefly differentiates the meanings and the inference direction of "if" and "only if". In summary, $A \text{ if and only if } B$ is mathematically interpreted as follows:
*
*'$A \text{ if } B$' : '$A \Leftarrow B$'
*'$A \text{ only if } B$' : '$\neg A \Leftarrow \neg B$' wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 0
} |
Proof that $n^3+2n$ is divisible by $3$ I'm trying to freshen up for school in another month, and I'm struggling with the simplest of proofs!
Problem:
For any natural number $n , n^3 + 2n$ is divisible by $3.$
This makes sense
Proof:
Basis Step: If $n = 0,$ then $n^3 + 2n = 0^3 +$
$2 \times 0 = 0.$ So it is divisi... | Given the $n$th case, you want to consider the $(n+1)$th case, which involves the number $(n+1)^3 + 2(n+1)$. If you know that $n^3+2n$ is divisible by $3$, you can prove $(n+1)^3 + 2(n+1)$ is divisible by $3$ if you can show the difference between the two is divisible by $3$. So find the difference, and then simplify... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 13,
"answer_id": 7
} |
Which average to use? (RMS vs. AM vs. GM vs. HM) The generalized mean (power mean) with exponent $p$ of $n$ numbers $x_1, x_2, \ldots, x_n$ is defined as
$$ \bar x = \left(\frac{1}{n} \sum x_i^p\right)^{1/p}. $$
This is equivalent to the harmonic mean, arithmetic mean, and root mean square for $p = -1$, $p = 1$, and $p... | One possible answer is for defining unbiased estimators of probability distributions. Often times you want some transformation of the data that gets you closer to, or exactly to, a normal distribution. For example, products of lognormal variables are again lognormal, so the geometric mean is appropriate here (or equiva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
How can I understand and prove the "sum and difference formulas" in trigonometry? The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true.
\begin{align}
\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos(\alpha \pm \beta) &= \cos \... | You might take refuge to complex numbers and use the Euler relation $\exp(i\phi)=\cos(\phi)+i\sin(\phi)$ and the fundamental property of the $\exp$ function:
$\cos(\alpha+\beta)+i\sin(\alpha+\beta)=\exp(i(\alpha+\beta))=\exp(i\alpha)\cdot\exp(i\beta)=$
$=(\cos(\alpha)+i\sin(\alpha))\cdot(\cos(\beta)+i\sin(\beta))=$
$=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "119",
"answer_count": 14,
"answer_id": 10
} |
Prove: $(a + b)^{n} \geq a^{n} + b^{n}$ Struggling with yet another proof:
Prove that, for any positive integer $n: (a + b)^n \geq a^n + b^n$ for all $a, b > 0:$
I wasted $3$ pages of notebook paper on this problem, and I'm getting nowhere slowly. So I need some hints.
$1.$ What technique would you use to prove this ... | Induction.
For $n=1$ it is trivially true
Assume true for $n=k$
i.e. $$(a + b)^k \ge a^k + b^k$$
Consider case $n=k+1$
\begin{align}&(a+b)^{k+1}
=(a+b)(a+b)^k\\
&\ge(a+b)(a^k+b^k)\\
&=a^{k+1}+b^{k+1}+ab^k+ba^k\\
&\ge a^{k+1}+b^{k+1}\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
"answer_id": 2
} |
Why $PSL_3(\mathbb F_2)\cong PSL_2(\mathbb F_7)$? Why are groups $PSL_3(\mathbb{F}_2)$ and $PSL_2(\mathbb{F}_7)$ isomorphic?
Update. There is a group-theoretic proof (see answer). But is there any geometric proof? Or some proof using octonions, maybe?
| Can't leave comments yet, but the details of there being only one simple group of order 168, and why PSL(2,7) and PSL(3,2) are order 168 and simple, are spelled out on pages 141-147 in Smith and Tabachnikova's "Topics in Group Theory".
Steve
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 7,
"answer_id": 4
} |
Finding an addition formula without trigonometry I'm trying to understand better the following addition formula: $$\int_0^a \frac{\mathrm{d}x}{\sqrt{1-x^2}} + \int_0^b \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \int_0^{a\sqrt{1-b^2}+b\sqrt{1-a^2}} \frac{\mathrm{d}x}{\sqrt{1-x^2}}$$
The term $a\sqrt{1-b^2}+b\sqrt{1-a^2}$ can be... | Replace the first integral by the same thing from $-a$ to $0$, and consider the points W,X,Y,Z on the unit circle above $-a,0,b$ and $c = a\sqrt{1-b^2} + b \sqrt{1-a^2}$. Draw the family of lines parallel to XY (and WZ). This family sets up a map from the circle to itself; through each point, draw a parallel and take ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Is this a counter example to Stromquist's Theorem? Stromquist's Theorem: If the simple closed curve J is "nice enough" then it has an inscribed square.
"Nice enough" includes polygons.
Read more about it here: www.webpages.uidaho.edu/~markn/squares
An "inscribed square" means that the corners of a square overlap with... | Regarding your edit: (0.2, 0) — (1, 0.2) — (0.8, 1) — (0, 0.8) (and many others)
http://www.imgftw.net/img/326639277.png
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Non-completeness of the space of bounded linear operators If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. $B(X,Y)$ is not complete?
| Let $X = \mathbb{R}$ with the Euclidean norm and let $Y$ be a normed space which is not complete. You should find that $B(X, Y) \simeq Y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Indefinite summation of polynomials I've been experimenting with the summation of polynomials. My line of attack is to treat the subject the way I would for calculus, but not using limits.
By way of a very simple example, suppose I wish to add the all numbers between $10$ and $20$ inclusive, and find a polynomial which... | You seem to be reaching for the
calculus of finite differences, once a well-known topic but rather
unfashionable these days. The answer to your question is yes: given a polynomial
$f(x)$ there is a polynomial $g(x)$ (of degree one greater than $f$) such that
$$f(x)=g(x)-g(x-1).$$
This polynomial $g$ (like the integral ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Is the function $(x, a) \mapsto (F(x,a), a)$ continuous whenever $F$ is? Let $A$, $X$, and $Y$ be arbitrary topological spaces. Let $F:X\times A\rightarrow Y$ be a continuous function. Let $P$ be the map from $X\times A$ to $Y\times A$ taking $(x,a)$ to $(F(x,a),a)$. Does it follow from continuity of $F$ that $P$ is co... | Yes. This follows from the fact that a function $U \to V \times W$ is continuous if and only if its component functions $U \to V, U \to W$ are, and from the fact that the projection maps $V \times W \to V$ and $V \times W \to W$ are continuous. Both of these facts in turn follow from the universal property of the pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why is Gimbal Lock an issue? I understand what the problem with Gimbal Lock is, such that at the North Pole, all directions are south, there's no concept of east and west. But what I don't understand is why this is such an issue for navigation systems? Surely if you find you're in Gimbal Lock, you can simply move a sma... | I don't imagine that this is a practical issue for navigation any longer, given the advent of GPS technology. However, it is of practical concern in 3-d animation and robotics. To get back to your navigation example, suppose that I have a mechanical gyro mounted in an airplane flying over the North Pole.
If the gyro i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 1
} |
How to find eigenvectors/eigenvalues of a matrix where each diagonal entry is scalar $d$ and all other entries are $1$ How would you find eigenvalues/eigenvectors of a $n\times n$ matrix where each diagonal entry is scalar $d$ and all other entries are $1$ ? I am looking for a decomposition but cannot find anything for... | The matrix is $(d-1)I + J$ where $I$ is the identity matrix and $J$ is the all-ones matrix, so once you have the eigenvectors and eigenvalues of $J$ the eigenvectors of $(d-1)I + J$ are the same and the eigenvalues are each $d-1$ greater. (Convince yourself that this works.)
But $J$ has rank $1$, so it has eigenvalue ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Methods to see if a polynomial is irreducible Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now. First is Eisenstein criterion. Another is that if a polynomial is irreducible mod p then it is irreducible. Are there any others?
| One method for polynomials over $\mathbb{Z}$ is to use complex analysis to say something about the location of the roots. Often Rouche's theorem is useful; this is how Perron's criterion is proven, which says that a monic polynomial $x^n + a_{n-1} x^{n-1} + ... + a_0$ with integer coefficients is irreducible if $|a_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 7,
"answer_id": 1
} |
How to get inverse of this matrix using least amount of space? I'm working on a problem from a past exam and I'm stuck, so I'm asking for help.
Here it is: $A = \frac12 \left[\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1
\end{array}\right]$ find $\mathbf A^{-1}$.
My ... | Gauss/Jordan Elimination will do it. It'll let you find |A|^1 with out the bother of finding the determinant. Just augment your original matrix with the identity and let her rip.
On an aside, you can still deduce the determinant from the inverse.
{ |A|^1= (1/det)[adj|A|]
therefore the determinant is equal to the lowes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges? We know that $\displaystyle\zeta(2)=\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ and it converges.
*
*Does there exists a bijective map $f:\mathbb{N} \to \mathbb{N}$ such that the sum $$\sum\limits_{n=1... | We show that the for any $f: \mathbb{N} \rightarrow \mathbb{N}$ bijective this is not cauchy. Suppose it is for given $\epsilon > 0$ there exists $N$ such that $\sum_{n=N}^{2N} \frac{f(n)}{n^2} < \epsilon$. We have $\sum_{n=N}^{2N} \frac{f(n)}{n^2} \geq \frac{1}{(2N)^2}\sum_{n=N}^{2N}f(n)\geq \frac{1}{(2N)^2} \frac{N(N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 1
} |
Division of Factorials [binomal coefficients are integers] I have a partition of a positive integer $(p)$. How can I prove that the factorial of $p$ can always be divided by the product of the factorials of the parts?
As a quick example $\frac{9!}{(2!3!4!)} = 1260$ (no remainder), where $9=2+3+4$.
I can nearly see it b... | If you believe (:-) in the two-part Newton case, then the rest is easily obtained by induction. For instance (to motivate you to write a full proof):
$$\frac{9!}{2! \cdot 3! \cdot 4!}\ =\ \frac{9!}{5!\cdot 4!}\cdot \frac{5!}{2!\cdot 3!}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 12,
"answer_id": 6
} |
For any $n$, is there a prime factor of $2^n-1$ which is not a factor of $2^m-1$ for $m < n$? Is it guaranteed that there will be some $p$ such that $p\mid2^n-1$ but $p\nmid 2^m-1$ for any $m<n$?
In other words, does each $2^x-1$ introduce a new prime factor?
| No. $2^6-1 = 3^2 \cdot 7$. But we have that $3|2^2-1$ and $7|2^3-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$ Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
| You can also prove it by induction, which is nice and easy, although it doesn't give much intuition as to why it works.
It works for $1$ since $\frac{1\times 2}{2} = 1$.
Let it work for $n$.
$$1 + 2 + 3 + \dots + n + (n + 1) = \frac{n(n+1)}{2} + (n + 1) = \frac{n(n+1) + 2(n+1)}{2} = \frac{(n+1)(n+2)}{2}.$$
Therefore, i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "136",
"answer_count": 36,
"answer_id": 28
} |
Applications of algebraic topology What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, any subgroup of a free group is free.
The deeper the methods used, the be... | How about the ham sandwich theorem? Or is that Ham and Cheese?
There is a plane which cuts all 3 regions into equal parts.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 12,
"answer_id": 11
} |
How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition) I'm doing exercise on discrete mathematics and I'm stuck with question:
If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f \circ\ g)$ is given by $(f \circ\ g) ^{-1... | Use the definition of an inverse and associativity of composition to show that the right hand side is the inverse of $(f \circ g)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 7,
"answer_id": 5
} |
Definition of an Algebraic Objects How did the definition of Algebraic objects like group, ring and field come up? When groups were first introduced, were they given the 4 axioms as we give now. And what made Mathematicians to think of something like this.
| Let me address the question: "what made Mathematicians ... think of something like this"?
The answer is: Galois, in studying the problem of factorization of polynomials, realized that
reasoning about the symmetries of the roots could be more powerful a tool than studying explicit formulas (which, loosely speaking, had ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Formal notation for number of rows/columns in a matrix Is there a generally accepted formal notation for denoting the number of columns a matrix has (e.g to use in pseudocode/algorithm environment in LaTeX)? Something I could use in the description of an algorithm like:
if horizontaldim(V) > x then
end if
or
if size... | None that I know of, but I've seen numerical linear algebra books (e.g. Golub and Van Loan) just say something like $V\in\mathbb{R}^{m\times n}$ for a matrix V with m rows and n columns, and then use m and n in the following algorithm description.
MATLAB notation, which some other people use, just says rows(V) and colu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solving a quadratic inequality $x^2-3x-10>0$ I am solving the following inequality, please look at it and tell me whether am I correct or not. This is an example in Howard Anton's book and I solved it on my own as given below, but the book has solved it differently! I want to confirm that my solution is also valid.
| If you graph the function $y=x^2-3x-10$, you can see that the solution is $x<-2$ or $x>5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Separability of $ L_p $ spaces I would like to know if the Lebesgue spaces $L_p$ with $ 0 < p < 1 $ are separable or not.
I know that this is true for $1 \leq p < + \infty$, but I do not find any references for the
case $ 0 < p < 1 $.
Thank you
| Please refer this article. It talks about $L_{p}$ spaces for $0 < p \leq 1$. Link:
http://www.jstor.org/stable/2041603?seq=2
Look at the step functions, the ones that take rational values and whose steps have rational endpoints there should be only countably many of those. And then you can perhaps apply the same argu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
probability and statistics: Does having little correlation imply independence? Suppose there are two correlated random variable and having very small correlation coefficient (order of 10-1). Is it valid to approximate it as independent random variables?
| It depends on what else you know about the relationship between the variables.
If the correlation coefficient is the full extent of your information, then the approximation is unsafe, as Noldorin points out.
If, on the other hand, you have good external evidence that the coefficient adequately captures the level of a s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 2
} |
What is the importance of the Collatz conjecture? I have been fascinated by the Collatz problem since I first heard about it in high school.
Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process indefinitely. Th... | Aside from the mathematical answers provided by others, the computational verification of the Collatz problem is a good exercise for programmers. There are many optimization opportunities (e.g., time-space trade-off using lookup tables, parallelism), many pitfalls (e.g., integer type overflow), possibilities of exploit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "247",
"answer_count": 9,
"answer_id": 7
} |
When is $n^2+n+p$ prime?
Possible Duplicate:
Behaviour of Polynomials in a PID!
Prove: if $p$ is a prime, and if $n^2+n+p$ is prime for $0\leq n \leq \sqrt{p/3}$, then it is also prime for $0 \leq n \leq p-2$.
This appeared on reddit recently, but no proof was posted. With $p=41$, it is Euler's famous prime-generati... | This follows by employing in Rabinowitsch's proof a Gauss bound, e.g. see Theorem 9.1 here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Companions to Rudin? I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend?
| There is a set of notes and additional exercises due to George Bergman. See his web page...
http://math.berkeley.edu/~gbergman/ug.hndts/
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 3
} |
How to tell if a line segment intersects with a circle? Given a line segment, denoted by it's $2$ endpoints $(X_1, Y_1)$ and $(X_2, Y_2)$, and a circle, denoted by it's center point $(X_c, Y_c)$ and a radius $R$, how can I tell if the line segment is a tangent of or runs through this circle? I don't need to be able to ... | There are too many answers already, but since no one mentioned this, perhaps this might still be useful.
You can consider using Polar Coordinates.
Translate so that the center of the circle is the origin.
The equation of a line in polar coordinates is given by
$r = p \sec (\theta - \omega)$
See the above web page for w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 8,
"answer_id": 5
} |
Show $\sqrt 3$ is irrational using $3p^2=q^2$ implies $3|p$ and $3|q$ This is a problem from "Introduction to Mathematics - Algebra and Number Systems" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer... | Try to write out the factorization of the right and left handed sides.
Now compare the order of the 3 on the left and right side, one of them is equal, forcing the other side to become odd. Contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 5
} |
Why are $x$ and $y$ such common variables in today's equations? How did their use originate? I can understand how the Greek alphabet came to be prominent in mathematics as the Greeks had a huge influence in the math of today. Certain letters came to have certain implications about their meaning (i.e. $\theta$ is almost... | This question has been asked previously on MathOverflow, and answered (by Mariano Suárez-Alvarez). You can follow this link, and I quote his response below.
You'll find details on this point (and precise references) in Cajori's History of mathematical notations, ¶340. He credits Descartes in his La Géometrie for the i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
Which one result in mathematics has surprised you the most? A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox. It states that you can sep... | I was very surprised to learn about the Cantor set, and all of its amazing properties. The first one I learnt is that it is uncountable (I would never have told), and that it has measure zero.
I was shown this example as a freshman undergraduate, for an example of a function that is Riemann-integrable but whose set of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "200",
"answer_count": 91,
"answer_id": 23
} |
Which one result in mathematics has surprised you the most? A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox. It states that you can sep... | Rather basic, but it was surprising for me:
For any matrix, column rank = row rank.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "200",
"answer_count": 91,
"answer_id": 55
} |
Which one result in mathematics has surprised you the most? A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox. It states that you can sep... | The Chinese Magic Square:
816
357
492
It adds up to 15 in every direction! Awesome! And the Chinese evidently thought so too, since they incorporated it into their religious writings.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "200",
"answer_count": 91,
"answer_id": 87
} |
Could you explain why $\frac{d}{dx} e^x = e^x$ "intuitively"? As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself.
thanks.
| Suppose $\frac{d}{dx}f(x)=f(x)$.
Then for small $h$, $f(x+h)=f(x)+hf(x)=f(x)(1+h)$. If we do this for a lot of small intervals of length $h$, we see $f(x+a)=(1+h)^{a/h}f(x)$. (Does this ring a bell already?)
Setting $x=0$ in the above, and fixing $f(0)=1$, we then have $f(1)=(1+h)^{1/h}$, which in limit as $h\rightarro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "48",
"answer_count": 21,
"answer_id": 1
} |
Find the coordinates in an isosceles triangle Given:
$A = (0,0)$
$B = (0,-10)$
$AB = AC$
Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?
| edit (to match revised question): Given your revised question, there is still the issue of C being on either side of the y-axis, but you have specified that AB=AC and that you are given $\mathrm{m}\angle BAC$ (the angle between AB and AC), so as in my original answer (below), the directed (trigonometric) measure of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Einstein notation - difference between vectors and scalars From Wikipedia:
First, we can use Einstein notation in
linear algebra to distinguish easily
between vectors and covectors: upper
indices are used to label components
(coordinates) of vectors, while lower
indices are used to label components
of cov... | A vector component is always written with 1 upper index $a^i$, while a covector component is written with 1 lower index $a_i$.
In Einstein notation, if the same index variable appear in both upper and lower positions, an implicit summation is applied, i.e.
$$ a_i b^i = a_1 b^1 + a_2 b^2 + \dotsb \qquad (*) $$
Now, a ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to convert a hexadecimal number to an octal number? How can I convert a hexadecimal number, for example 0x1A03 to its octal value?
I know that one way is to convert it to decimal and then convert it to octal
0x1A03 = 6659 = 0o15003
*
*Is there a simple way to do it without the middle step (conversion to decimal... | A simpler way is to go through binary (base 2) instead of base 10.
0x1A03 = 0001 1010 0000 0011
Now group the bits in bunches of 3 starting from the right
0 001 101 000 000 011
This gives
0 1 5 0 0 3
Which is your octal representation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 2
} |
What does "only" mean? I understand the technical and logical distinction between "if" and "only if" and "if and only if". But I have always been troubled by the phrase "only if" even though I am able to parse and interpret it. Also in my posts on this and other sites I have frequently had to make edits to migrate th... | I think analogies in plain English are the way to internalize this... so here's one:
Given that you want to wear socks with your shoes, put your shoes on only if you have already put your socks on.
The idea is that there is no other way to arrive at the state of having your socks and shoes on (aside from the ridiculous... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
"answer_id": 0
} |
What is $\sqrt{i}$? If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary?
Is it used or considered often in mathematics? How is it notated?
| With a little bit of manipulation you can make use of the quadratic equation since you are really looking for the solutions of $x^2 - i = 0$, unfortunately if you apply the quadratic formula directly you gain nothing new, but...
Since $i^2 = -1$ multiply both sides of our original equation by $i$ and you will have $ix^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "88",
"answer_count": 10,
"answer_id": 0
} |
How many ways are there to define sine and cosine? Sometimes there are many ways to define a mathematical concept, for example the natural base logarithm. How about sine and cosine?
Thanks.
| An interesting construction is given by Michael Spivak in his book Calculus, chapter 15. The steps are basically the following:
$1.$ We define what a directed angle is.
$2.$ We define a unit circle by $x^2+y^2=1$, and show that every angle between the $x$-axis and a line origined from $(0,0)$ defines a point $(x,y)$ in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 0
} |
When does the product of two polynomials = $x^{k}$? Suppose $f$ and $g$ are are two polynomials with complex coefficents (i.e $f,g \in \mathbb{C}[x]$).
Let $m$ be the order of $f$ and let $n$ be the order of $g$.
Are there some general conditions where
$fg= \alpha x^{n+m}$
for some non-zero $\alpha \in \mathbb{C}$
| We don't need the strong property of UFD. If $\rm D$ is a domain $\rm D$ then $\rm x$ is prime in $\rm D[x]$ (by $\rm D[x]/x \cong D$ a domain), and products of primes factor uniquely in every domain (same simple proof as in $\Bbb Z$). In particular, the only factorizations of the prime power $\rm x^i$ are $\rm \,x^j x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 1
} |
Rules for rounding (positive and negative numbers) I'm looking for clear mathematical rules on rounding a number to $n$ decimal places.
Everything seems perfectly clear for positive numbers. Here is for example what I found on math.about.com :
Rule One Determine what your rounding digit is and look to the right side o... | "Round to nearest integer" is completely unambiguous, except when the fractional part of the number to be rounded happens to be exactly $\frac 1 2$. In that case, some kind of tie-breaking rule must be used. Wikipedia (currently) lists six deterministic tie-breaking rules in more or less common use:
*
*Round $\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 2,
"answer_id": 1
} |
$F_{\sigma}$ subsets of $\mathbb{R}$ Suppose $C \subset \mathbb{R}$ is of type $F_{\sigma}$. That is $C$ can be written as the union of $F_{n}$'s where each $F_{n}$'s are closed. Then can we prove that each point of $C$ is a point of discontinuity for some $f: \mathbb{R} \to \mathbb{R}$.
I refered this link on wiki : h... | I believe you're looking for something like the construction mentioned here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Beta function derivation How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments?
I'm pretty sure this is possible, but can't see how.
I can see that
$$\newcommand{\Beta}{\mathrm{Beta}}\sum_{a=0}^n {n \choo... | The multinomial generalization mentioned by Qiaochu is conceptually simple but getting the details right is messy. The goal is to compute $$\int_0^1 \int_0^{1-t_1} \ldots \int_0^{1-t_1-\ldots-t_{k-2}} t_1^{n_1} t_2^{n_2} \ldots t_{k-1}^{n_{k-1}} t_k^{n_k} dt_1 \ldots dt_{k-1},$$ where $t_k = 1 - t_1 - \ldots - t_{k-1},... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 2,
"answer_id": 1
} |
Why does this sum mod out to 0? In making up another problem today I came across something odd. I've been thinking it over and I can't exactly place why it's true, but after running a long Python script to check, I haven't yet found a counter example.
Why is $\sum_{n=1}^{m}{n^m}\equiv 0\mod m$ true for all odd $m \ge 3... | If it is odd, each n can be paired with -n. So we get $n^m+(-n)^m=n^m-n^m=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the $N$-th derivative of $f(x)=\frac {x} {x^2-1}$ I'm practicing some problems from past exams and found this one:
Find the n-th derivative of this function:
$$f(x)=\frac {x} {x^2-1}$$
I have no idea how to start solving this problems. Is there any theorem for finding nth derivative?
| To add to Derek's hint: you will have to show the validity of the formula
$\frac{\mathrm{d}^k}{\mathrm{d}x^k}\frac1{x}=\frac{(-1)^k k!}{x^{k+1}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 0
} |
Finding the Heavy Coin by weighing twice Suppose you have $100$ coins. $96$ of them are heavy and $4$ of them are light. Nothing is known regarding the proportion of their weights. You want to find at least one genuine (heavy) coin. You are allowed to use a weight balance twice. How do you find it?
Assumptions:
Heavy... | I think this works.
Divide the coins into three groups: $A$ with $33$ coins, $B$ with $33$ coins and $C$ with $34$ coins.
Weigh $A$ and $B$ against each other.
Now if $A$ is heavier than $B$, then $A$ cannot have two or more light coins, as in that case, $A$ would be lighter (or equal to $B$). Now split $A$ into group... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 0
} |
Does a section that vanishes at every point vanish? Let $R$ be the coordinate ring of an affine complex variety (i.e. finitely generated, commutative, reduced $\mathbb{C}$ algebra) and $M$ be an $R$ module.
Let $s\in M$ be an element, such that $s\in \mathfrak{m}M$ for every maximal ideal $\mathfrak{m}$. Does this impl... | Not in general, no. For example, if $R = \mathbb C[T]$ and $M$ is the field of fractions
of $R$, namely $\mathbb C(T)$, then (a) every maximal ideal of $R$ is principal; (b) every
element of $M$ is divisible by every non-zero element of $R$. Putting (a) and (b) together
we find that $M = \mathfrak m M$ for every maxi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.
I do not understand anything more than the following.
*
*Elementary row operations.
*Linear dependence.
*Row reduced forms and their relations with the original matrix.
If the... | Since inverse/transpose are not allowed we start by writing
$$A = \begin{bmatrix}
a_{11} & a_{12} & a_{13} & \dots & a_{1n} \\
a_{21} & a_{22} & a_{23} & \dots & a_{2n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & a_{n3} & \dots & a_{nn}
\end{bmatrix}$$
and similarly
$$B = \begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "379",
"answer_count": 34,
"answer_id": 14
} |
Find the Frequency Components of a Time Series Graph For a periodic (and not so periodic) function, it is always possible to use Fourier series to find out the frequencies contained in the function.
But what about function that cannot be expressed in mathematical terms? For example, this graph (accelerogram):
Is there... | Quinn and Hannan's The estimation and tracking of frequency is dedicated to this topic. I can highly recommend it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
A $1-1$ function is called injective. What is an $n-1$ function called? A $1-1$ function is called injective. What is an $n-1$ function called ?
I'm thinking about homomorphisms. So perhaps homojective ?
Onto is surjective. $1-1$ and onto is bijective.
What about n-1 and onto ? Projective ? Polyjective ?
I think $n-m$ ... | I will:
*
*suggest some terminology for three related concepts, and
*suggest that $n$-to-$1$ functions probably aren't very interesting.
Terminology.
Let $f : X \rightarrow Y$ denote a function. Recall that $f$ is called a bijection iff for all $y \in Y$, the set $f^{-1}(y)$ has precisely $1$ element. So define t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
CAS with a standard language I hope this question is suitable for the site.
I recently had to work with Mathematica, and the experience was, to put it kindly, unpleasing. I do not have much experience with similar programs, but I remember not liking much Matlab or Maple either. The result is that I am a mathematician w... | Considering that Maxima is developed in Common Lisp and accepts CL sintax, maybe this system would suit your requirement.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 4
} |
Group as the Union of Subgroups We know that a group $G$ cannot be written as the set theoretic union of two of its proper subgroups. Also $G$ can be written as the union of 3 of its proper subgroups if and only if $G$ has a homomorphic image, a non-cyclic group of order 4.
In this paper http://www.jstor.org/stable/269... | One way to ensure this happens is to have every maximal subgroup be characteristic. To get every maximal subgroup normal, it is a good idea to check p-groups first. To make sure the maximal subgroups are characteristic, it makes sense to make sure they are simply not isomorphic. To make sure there are not too many m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Intuitive explanation of Cauchy's Integral Formula in Complex Analysis There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive direction, then for every $z$ in the disc we can writ... | Expanding on my comment, this result can be translated into:
"A surface in $\mathbb{R}^3$ which satisfies the Maximum-Modulus principle is uniquely determined by specifying it's boundary"
To see this, write the holomorphic function $f(z)$ in terms of its real and imaginary parts:
$$ f(z) = f(x,y) = g(x,y) + ih(x,y)$$
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "100",
"answer_count": 14,
"answer_id": 3
} |
How do you show monotonicity of the $\ell^p$ norms? I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
| For completeness I will add this as an answer (it is a slight adaptation of the argument from AD.):
For $a\in[0,1]$ and any $y_i\geq 0, i\in\mathbb N$, with at least one $y_i\neq0$ and the convention that $y^0=1$ for any $y\geq0$, \begin{equation}\label{*}\tag{*}\sum_{i=1}^\infty \frac{y_i^a}{\left(\sum_{j=1}^\infty y_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "65",
"answer_count": 5,
"answer_id": 4
} |
Beautiful identity: $\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$ Let $m,n\ge 0$ be two integers. Prove that
$$\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$$
where $\delta_{mn}$ stands for the Kronecker's delta (defined by $\delta_{mn} = \begin{cases} 1, & \text{if } m=n; \\ 0, & ... | This follows easily from the Multinomial Theorem, I believe.
$$ 1 = 1^n = (1 - x + x)^n$$
$$ = \sum_{a+b+c=n} {n \choose a,b,c} 1^a \cdot (-x)^b \cdot x^c$$
$$ = \sum_{m=0}^{n} \sum_{k=m}^{n} {n \choose m,k-m,n-k} 1^{m} \cdot (-x)^{k-m} \cdot x^{n-k} $$
$$ = \sum_{m=0}^{n} \left[ \sum_{k=m}^{n} (-1)^{k-m} {k \choos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 7,
"answer_id": 1
} |
What is the $x$ in $\log_b x$ called? In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is $x$ called here? The exponent?
| Another name (that I've only ever seen when someone else asked this question) is "logarithmand".
From page 36 of The Spirit of Mathematical Analysis by Martin Ohm, translated from the German by Alexander John Ellis, 1843:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Alternate definition of prime number I know the definition of prime number when dealing with integers, but I can't understand why the following definition also works:
A prime is a quantity $p$ such that whenever $p$ is a factor of some product $a\cdot b$, then either $p$ is a factor of $a$ or $p$ is a factor of $b$.
... | As far as I know, your definition
A prime is an element p such that whenever p divides ab, then either p divides a or p divides b,
is the true definition of "prime". The usual one,
... an element p which cannot be expressed as a product of non-unit elements,
is the definition of an irreducible element. Now, in eve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 1
} |
Are all algebraic integers with absolute value 1 roots of unity? If we have an algebraic number $\alpha$ with (complex) absolute value $1$, it does not follow that $\alpha$ is a root of unity (i.e., that $\alpha^n = 1$ for some $n$). For example, $(3/5 + 4/5 i)$ is not a root of unity.
But if we assume that $\alpha$ i... | Let me first mention an example in Character Theory. Let $G$ be a finite group of order $n$ and assume $\rho$ is a representation with character $\chi:=\chi_\rho$ which is defined by $\chi(g)=Tr(\rho(g))$. Since $G$ is a finite group then, by invoking facts from linear algebra, one can show $\chi(g)\in\mathbb{Z}[\zeta_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "114",
"answer_count": 7,
"answer_id": 0
} |
End of preview. Expand in Data Studio
StackMathQA
StackMathQA is a meticulously curated collection of 2 million mathematical questions and answers, sourced from various Stack Exchange sites. This repository is designed to serve as a comprehensive resource for researchers, educators, and enthusiasts in the field of mathematics and AI research.
Configs
configs:
- config_name: stackmathqa1600k
data_files: data/stackmathqa1600k/all.jsonl
default: true
- config_name: stackmathqa800k
data_files: data/stackmathqa800k/all.jsonl
- config_name: stackmathqa400k
data_files: data/stackmathqa400k/all.jsonl
- config_name: stackmathqa200k
data_files: data/stackmathqa200k/all.jsonl
- config_name: stackmathqa100k
data_files: data/stackmathqa100k/all.jsonl
- config_name: stackmathqafull-1q1a
data_files: preprocessed/stackexchange-math--1q1a/*.jsonl
- config_name: stackmathqafull-qalist
data_files: preprocessed/stackexchange-math/*.jsonl
How to load data:
from datasets import load_dataset
ds = load_dataset("math-ai/StackMathQA", "stackmathqa1600k") # or any valid config_name
Preprocessed Data
In the ./preprocessed/stackexchange-math directory and ./preprocessed/stackexchange-math--1q1a directory, you will find the data structured in two formats:
- Question and List of Answers Format:
Each entry is structured as {"Q": "question", "A_List": ["answer1", "answer2", ...]}.
math.stackexchange.com.jsonl: 827,439 linesmathoverflow.net.jsonl: 90,645 linesstats.stackexchange.com.jsonl: 103,024 linesphysics.stackexchange.com.jsonl: 117,318 lines- In total: 1,138,426 questions
dataset_info:
features:
- name: Q
dtype: string
description: "The mathematical question in LaTeX encoded format."
- name: A_list
dtype: sequence
description: "The list of answers to the mathematical question, also in LaTeX encoded."
- name: meta
dtype: dict
description: "A collection of metadata for each question and its corresponding answer list."
- Question and Single Answer Format:
Each line contains a question and one corresponding answer, structured as {"Q": "question", "A": "answer"}. Multiple answers for the same question are separated into different lines.
math.stackexchange.com.jsonl: 1,407,739 linesmathoverflow.net.jsonl: 166,592 linesstats.stackexchange.com.jsonl: 156,143 linesphysics.stackexchange.com.jsonl: 226,532 lines- In total: 1,957,006 answers
dataset_info:
features:
- name: Q
dtype: string
description: "The mathematical question in LaTeX encoded format."
- name: A
dtype: string
description: "The answer to the mathematical question, also in LaTeX encoded."
- name: meta
dtype: dict
description: "A collection of metadata for each question-answer pair."
Selected Data
The dataset has been carefully curated using importance sampling. We offer selected subsets of the dataset (./preprocessed/stackexchange-math--1q1a) with different sizes to cater to varied needs:
dataset_info:
features:
- name: Q
dtype: string
description: "The mathematical question in LaTeX encoded format."
- name: A
dtype: string
description: "The answer to the mathematical question, also in LaTeX encoded."
- name: meta
dtype: dict
description: "A collection of metadata for each question-answer pair."
StackMathQA1600K
- Location:
./data/stackmathqa1600k - Contents:
all.jsonl: Containing 1.6 million entries.meta.json: Metadata and additional information.
Source: Stack Exchange (Math), Count: 1244887
Source: MathOverflow, Count: 110041
Source: Stack Exchange (Stats), Count: 99878
Source: Stack Exchange (Physics), Count: 145194
Similar structures are available for StackMathQA800K, StackMathQA400K, StackMathQA200K, and StackMathQA100K subsets.
StackMathQA800K
- Location:
./data/stackmathqa800k - Contents:
all.jsonl: Containing 800k entries.meta.json: Metadata and additional information.
Source: Stack Exchange (Math), Count: 738850
Source: MathOverflow, Count: 24276
Source: Stack Exchange (Stats), Count: 15046
Source: Stack Exchange (Physics), Count: 21828
StackMathQA400K
- Location:
./data/stackmathqa400k - Contents:
all.jsonl: Containing 400k entries.meta.json: Metadata and additional information.
Source: Stack Exchange (Math), Count: 392940
Source: MathOverflow, Count: 3963
Source: Stack Exchange (Stats), Count: 1637
Source: Stack Exchange (Physics), Count: 1460
StackMathQA200K
- Location:
./data/stackmathqa200k - Contents:
all.jsonl: Containing 200k entries.meta.json: Metadata and additional information.
Source: Stack Exchange (Math), Count: 197792
Source: MathOverflow, Count: 1367
Source: Stack Exchange (Stats), Count: 423
Source: Stack Exchange (Physics), Count: 418
StackMathQA100K
- Location:
./data/stackmathqa100k - Contents:
all.jsonl: Containing 100k entries.meta.json: Metadata and additional information.
Source: Stack Exchange (Math), Count: 99013
Source: MathOverflow, Count: 626
Source: Stack Exchange (Stats), Count: 182
Source: Stack Exchange (Physics), Count: 179
Citation
We appreciate your use of StackMathQA in your work. If you find this repository helpful, please consider citing it and star this repo. Feel free to contact zhangyif21@tsinghua.edu.cn or open an issue if you have any questions.
@misc{stackmathqa2024,
title={StackMathQA: A Curated Collection of 2 Million Mathematical Questions and Answers Sourced from Stack Exchange},
author={Zhang, Yifan},
year={2024},
}
- Downloads last month
- 222