Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Constructing self-complementary graphs How does one go about systematically constructing a self-complementary graph, on say 8 vertices?
[Added: Maybe everyone else knows this already, but I had to look up my guess to be sure it was correct: a self-complementary graph is a simple graph which is isomorphic to its complem... | Here's a nice little algorithm for constructing a self-complementary graph from a self-complementary graph $H$ with $4k$ or $4k+1$ vertices, $k = 1, 2, ...$ (e.g., from a self-complementary graph with $4$ vertices, one can construct a self-complementary graph with $8$ vertices; from $5$ vertices, construct one with $9$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/40745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 2
} |
Convergence of infinite/finite 'root series' Let $S_n=a_1+a_2+a_3+...$ be a series where $ {a}_{k}\in \mathbb{R}$ and let $P = \{m\;|\;m\;is\;a\;property\;of\;S_n\}$ based on this information what can be said of the corresponding root series: $R_n=\sqrt{a_1} + \sqrt{a_2} + \sqrt{a_3} + ...$
In particular, if $S_n$ is ... | If $S_n$ is convergent you cannot say anything about $R_n$, for example if $a_n=1/n^2$ then $R_n$ diverges. If $a_n=1/2^n$ then $R_n$ converges too.
If $S_n$ diverges $R_n$ will diverge too because you have for $a < 1$ that $a < \sqrt{a}$ (This reasoning assumes that $a_k \geq 0$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/40834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
How do I find roots of a single-variate polynomials whose integers coefficients are symmetric wrt their respective powers Given a polynomial such as $X^4 + 4X^3 + 6X^2 + 4X + 1,$ where the coefficients are symmetrical, I know there's a trick to quickly find the zeros. Could someone please refresh my memory?
| Hint: This particular polynomial is very nice, and factors as $(X+1)^4$.
Take a look at Pascal's Triangle and the Binomial Theorem for more details.
Added: Overly complicated formula
The particular quartic you asked about had a nice solution, but lets find all the roots of the more general $$ax^{4}+bx^{3}+cx^{2}+bx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/40864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 4
} |
Weak limit of an $L^1$ sequence We have functions $f_n\in L^1$ such that $\int f_ng$ has a limit for every $g\in L^\infty$. Does there exist a function $f\in L^1$ such that the limit equals $\int fg$? I think this is not true in general (really? - why?), then can this be true if we also know that $f_n$ belong to a cert... | Perhaps surprisingly, the answer is yes.
More generally, given any Banach space $X$, a sequence $\{x_n\} \subset X$ is said to be weakly Cauchy if, for every $\ell \in X^*$, the sequence $\{\ell(f_n)\} \subset \mathbb{R}$ (or $\mathbb{C}$) is Cauchy. If every weakly Cauchy sequence is weakly convergent, $X$ is said to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/40920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Is the factorization problem harder than RSA factorization ($n = pq$)? Let $n \in \mathbb{N}$ be a composite number, and $n = pq$ where $p,q$ are distinct primes. Let $F : \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}$ (*) be an algorithm which takes as an input $x \in \mathbb{N}$ and returns two primes $u, v$ su... | Two vague reasons I think the answer must be "no":
If there were any inductive reason that we could factor a number with k prime factors in polynomial time given the ability to factor a number with k-1 prime factors in polynomial time, then the AKS primality test has already provided a base case. So semiprime factoriz... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/40971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Simple (even toy) examples for uses of Ordinals? I want to describe Ordinals using as much low-level mathematics as possible, but I need examples in order to explain the general idea. I want to show how certain mathematical objects are constructed using transfinite recursion, but can't think of anything simple and yet ... | Some accessible applications transfinite induction could be the following (depending on what the audience already knows):
*
*Defining the addition, multiplication (or even exponentiation) of ordinal numbers by transfinite recursion and then showing some of their basic properties. (Probably most of the claims for ad... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Conditional probability Given the events $A, B$ the conditional probability of $A$ supposing that $B$ happened is:
$$P(A | B)=\frac{P(A\cap B )}{P(B)}$$
Can we write that for the Events $A,B,C$, the following is true?
$$P(A | B\cap C)=\frac{P(A\cap B\cap C )}{P(B\cap C)}$$
I have couple of problems with the equation ab... | Yes you can. I see no fault. Because if you put $K = B \cap C$ you obtain the original result
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/41092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
Compound angle formula confusion I'm working through my book, on the section about compound angle formulae. I've been made aware of the identity $\sin(A + B) \equiv \sin A\cos B + \cos A\sin B$. Next task was to replace B with -B to show $\sin(A - B) \equiv \sin A\cos B - \cos A \sin B$ which was fairly easy. I'm strug... | Note that you can also establish:
$$\sin\left(\left(\frac{\pi}{2} - A\right) - B\right) =\sin\left(\frac{\pi}{2} - (A + B)\right) = \cos(A+B)$$ by using the second identity you figured out above, $\sin(A - B) \equiv \sin A\cos B - \cos A\sin B$, giving you:
$$\sin\left(\left(\frac{\pi}{2} - A\right) - B\right) = \sin\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
2D Epanechnikov Kernel What is the equation for the $2D$ Epanechnikov Kernel?
The following doesn't look right when I plot it.
$$K(x) = \frac{3}{4} * \left(1 - \left(\left(\frac{x}{\sigma} \right)^2 + \left(\frac{y}{\sigma}\right)^2\right) \right)$$
I get this:
| I have an equation for some p-D Epanechnikov Kernel.
Maybe you will find it useful.
$$
\begin{equation}
K(\hat{x})=\begin{cases} \frac{1}2C_p^{-1}(p
+2)(1-||\hat{x}||^2)& ||\hat{x}||<1\\\\
0& \text{otherwise}
\end{cases}
\end{equation}
$$
while $\hat{x}$ is a vector with p dimensions and $C_p$ is defined as:
$$C_1 = 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Equivalent Definitions of Positive Definite Matrix As Wikipedia tells us, a real $n \times n$ symmetric matrix $G = [g_{ij}]$ is positive definite if $v^TGv >0$ for all $0 \neq v \in \mathbb{R}^n$. By a well-known theorem of linear algebra it can be shown that $G$ is positive definite if and only if the eigenvalues of ... | Let's number the definitions:
*
*$v^T G v > 0$ for all nonzero $v$.
*$G$ has positive eigenvalues.
*$v^T G v > \gamma v^T v$ for some $\gamma > 0$.
You know that 1 and 2 are equivalent. It's not hard to see that 3 implies 1. So it remains to show that either 1 or 2 implies 3. A short proof: 2 implies 3 because w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Limit of monotonic functions at infinity I understand that if a function is monotonic then the limit at infinity is either $\infty$,a finite number or $-\infty$.
If I know the derivative is bigger than $0$ for every $x$ in $[0, \infty)$ then I know that $f$ is monotonically increasing but I don't know whether the limit... | You can also prove it directly by the Mean Value Theorem:
$$f(x)-f(0)=f'(\alpha)(x-0) \geq cx \,.$$
Thus $f(x) \geq cx + f(0)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/41290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Outer product of a vector with itself Is there a special name for an outer product of a vector with itself? Is it a special case of a Gramian? I've seen them a thousand times, but I have no idea if such product has a name.
Update:
The case of outer product I'm talking about is $\vec{u}\vec{u}^T$ where $\vec{u}$ is a co... | In statistics, we call it the "sample autocorrelation matrix", which is like an estimation of autocorrelation matrix based on observed samples.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/41329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Symmetric and diagonalizable matrix-Jacob method: finding $p$ and $q$ Given this symmetric matrix-$A$:
$\begin{pmatrix}
14 &14 & 8 &12 \\
14 &17 &11 &14 \\
8& 11 &11 &10 \\
12 & 14 &10 & 12
\end{pmatrix}$
I need to find $p,q$ such that $p$ is the number of 1's and $q$ is the number of -1's
in the diagonal... | The characteristic polynomial of $A$ is $P(x)= x^4 - 54x^3 + 262x^2 - 192x
$. It has $0$ as a simple root, and the other three are positive. Therefore $A$ has three positive eigenvalues and one equal to zero. Since the signature can be obtained from the signs of the eigenvalues, we are done. Therefore $p=3,q=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/41364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Easy Proof Adjoint(Compact)=Compact I am looking for an easy proof that the adjoint of a compact operator on a Hilbert space is again compact. This makes the big characterization theorem for compact operators (i.e. compact iff image of unit ball is relatively compact iff image of unit ball is compact iff norm limit of ... | Here is an alternative proof, provided that you know that an operator is compact iff it is the operator-limit of a sequence of finite-rank operators.
Let $T: H \to H$ be a compact operator. Then $T= \lim_n T_n$ where the limit is w.r.t. the operatornorm and $T_n$ is a finite rank operator. Using that the $*$-involution... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 0
} |
Doubt in Discrete-Event System Simulation by Jerry Banks,4th Edition I'm new to the Math forum here, so pardon my question if it seems juvenile to some. I've googled intensively,gone through wikipedia,wolfram and after hitting dead ends everywhere have resorted to this site.
My query is this-
In chapter#8, "Random-Vari... | Here is an hypothesis. Since three coefficients only are obtained from a whole bunch of data, these could summarize some properties of the sample considered. Statisticians often use the symbol R2 for a coefficient of determination, which, roughly speaking, measures the proportion of variability in a data set.
On the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Intuitive explanation of the tower property of conditional expectation I understand how to define conditional expectation and how to prove that it exists.
Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, that is if $X$ and $Y$ are random variables (or $Y... | For simple discrete situations from which one obtains most basic intuitions, the meaning is clear.
I have a large bag of biased coins. Suppose that half of them favour heads, probability of head $0.7$. Two-fifths of them favour heads, probability of head $0.8$. And the rest favour heads, probability of head $0.9$.
P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 8,
"answer_id": 2
} |
Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings? A book on CG says:
... we can construct any affine transformation from a sequence of rotations, translations, and scalings.
But I don't know how to prove it.
Even in a particular case, I found it still hard. For ... | Perhaps using the singular value decomposition?
For the homogeneus case (linear transformation), we can always write
$y = A x = U D V^t x$
for any square matrix $A$ with positive determinant, were U and V are orthogonal and D is diagonal with positive real entries. U and V would the be the rotations and D the scaling.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Slick way to define p.c. $f$ so that $f(e) \in W_{e}$ Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W... | Perhaps the reason your solution seems ugly to you is that you appear to be excessively concerned with the formalism of representing your computable function in terms of the $\mu$ operator. The essence of computability, however, does not lie with this formalism, but rather with the idea of a computable procedure. It i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Is an integer uniquely determined by its multiplicative order mod every prime Let $x$ and $y$ be nonzero integers and $\mathrm{ord}_p(w)$ be the multiplicative order of $w$ in $ \mathbb{Z} /p \mathbb{Z} $. If $\mathrm{ord}_p(x) = \mathrm{ord}_p(y)$ for all primes (Edit: not dividing $x$ or $y$), does this imply $x=y$? ... | [This is an answer to the original form of the question. In the meantime the question has been clarified to refer to the multiplicative order; this seems like a much more interesting and potentially difficult question, though I'm pretty sure the answer must be yes.]
I may be missing something, but it seems the answer i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 1
} |
Find a first order sentence in $\mathcal{L}=\{0,+\}$ which is satisfied by exactly one of $\mathbb{Z}\oplus \mathbb{Z}$ and $\mathbb{Z}$ I'm re-reading some material and came to a question, paraphrased below:
Find a first order sentence in $\mathcal{L}=\{0,+\}$ which is satisfied by exactly one of the structures $(\mat... | Here's one:
$$
(\forall x)(\forall y)\Bigl[(\exists z)(x=z+z) \lor (\exists z)(y=z+z) \lor (\exists z)(x+y=z+z)\Bigr]
$$
This sentence is satisfied in $\mathbb{Z}$, since one of the numbers $x$, $y$, and $x+y$ must be even. It isn't satisfied in $\mathbb{Z}\oplus\mathbb{Z}$, e.g. if $x=(1,0)$, $y=(0,1)$, and $x+y=(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\lim (a + b)\;$ when $\;\lim(b)\;$ does not exist? Suppose $a$ and $b$ are functions of $x$. Is it guaranteed that
$$
\lim_{x \to +\infty} a + b\text{ does not exist}
$$
when
$$
\lim_{x \to +\infty} a = c\quad\text{and}\quad
\lim_{x \to +\infty} b\text{ does not exist ?}
$$
| Suppose, to get a contradiction, that our limit exists. That is, suppose $$\lim_{x\rightarrow \infty} a(x)+b(x)=d$$ exists. Then since $$\lim_{x\rightarrow \infty} -a(x)=-c,$$ and as limits are additive, we conclude that $$\lim_{x\rightarrow \infty} a(x)+b(x)-a(x)=d-c$$ which means $$\lim_{x\rightarrow \infty} b(x)=d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Sorting a deck of cards with Bogosort Suppose you have a standard deck of 52 cards which you would like to sort in a particular order. The notorious algorithm Bogosort works like this:
*
*Shuffle the deck
*Check if the deck is sorted. If it's not sorted, goto 1. If it's sorted, you're done.
Let B(n) be the probab... | An estimate. The probability that Bogosort doesn't sort the deck in a particular shuffle is $1 - \frac{1}{52!}$, hence $1 - B(n) = \left( 1 - \frac{1}{52}! \right)^n$. Since
$$\left( 1 - \frac{x}{n} \right)^n \approx e^{-x}$$
for large $n$, the above is is approximately equal to $e^{- \frac{n}{52!} }$, hence $B(n) \app... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that any shape 1 unit area can be placed on a tiled surface Given a surface of equal square tiles where each tile side is 1 unit long. Prove that a single area A, of any shape, but just less than 1 unit square in area can be placed on the surface without touching a vertex of any tiled area? The Shape A may have h... | Project $A$ onto a single square by "Stacking" all of the squares in the plane. Then translating $A$ on this square corresponds to moving $A$ on a torus with surface area one. As the area of $A$ is less then one, there must be some point which it does not cover. Then choose that point to be the four corners of the sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/42004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Proving an integer $3n+2$ is odd if and only if the integer $9n+5$ is even How can I prove that the integer $3n+2$ is odd if and only if the integer $9n+5$ is even, where n is an integer?
I suppose I could set $9n+5 = 2k$, to prove it's even, and then do it again as $9n+5=2k+1$
Would this work?
| HINT $\rm\ \ 3\ (3\:n+2)\ -\ (9\:n+5)\:\ =\:\ 1$
Alternatively note that their sum $\rm\:12\:n + 7\:$ is odd, so they have opposite parity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/42059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Calculate Line Of Best Fit Using Exponential Weighting? I know how to calculate a line of best fit with a set of data.
I want to be able to exponentially weight the data that is more recent so that the more recent data has a greater effect on the line.
How can I do this?
| Most linear least squares algorithms let you set the measurement error of each point. Errors in point $i$ are then weighted by $\frac{1}{\sigma_i}$. So assign a smaller measurement error to more recent points. One algorithm is available for free in the obsolete version of Numerical Recipes, chapter 15.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/42242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist I've been trying to solve the following problem:
Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\i... | Hint: If you assume $\lim _{x \to \infty } f'(x) = L \ne 0$, the contradiction would come from the mean value theorem (consider $f(x)-f(M)$ for a fixed but arbitrary large $M$, and let $x \to \infty$).
Explained: If the limit of $f(x)$ exist, there is a horizontal asymptote. Therefore as the function approaches infini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/42277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "55",
"answer_count": 6,
"answer_id": 4
} |
Find control point on piecewise quadratic Bézier curve I need to write an OpenGL program to generate and display a piecewise quadratic Bézier curve that interpolates each set of data points:
$$(0.1, 0), (0, 0), (0, 5), (0.25, 5), (0.25, 0), (5, 0), (5, 5), (10, 5), (10, 0), (9.5, 0)$$
The curve should have continuous t... | You can see that it will be difficult to solve this satisfactorily by considering the case where the points to be interpolated are at the extrema of a sinusoidal curve. Any reasonable solution should have horizontal tangents at the points, but this is not possible with quadratic curves.
Peter has described how to achie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/42395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Is -5 bigger than -1? In everyday language people often mix up "less than" and "smaller than" and in most situations it doesn't matter but when dealing with negative numbers this can lead to confusion.
I am a mathematics teacher in the UK and there are questions in national GCSE exams phrased like this:
Put these numbe... | Like all too many test questions, the quoted question is a question not about things but about words.
Roughly speaking the same question will have appeared on these exams since before the students were born. And in their homework and quizzes, students will have seen the question repeatedly.
Let's assume that the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/42556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 4
} |
Why are the periods of these permutations often 1560? I ran across a math puzzle that went like this:
Consider the list $1,9,9,3, \cdots$ where the next entry is equal to the sum mod 10 of the prior 4. So the list begins $1,9,9,3,2,3,7,\cdots$. Will the sequence $7,3,6,7$ ever occur?
(Feel free to pause here and solve... | Your recurrence is linear in that you can add two series together term by term and still have it a series. The period of (0,0,0,1) is 1560, so all periods will be a divisor of that. To get 1560 you just have to avoid shorter cycles.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/42880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Is it possible for function $f : \mathbb{R} \to \mathbb{R}$ have a maximum at every point in a countable dense subset of its domain? Is it possible for function $f : \mathbb{R} \to \mathbb{R}$ have a maximum at every point in a countable dense subset of its domain ? The motivation for this question is I have a sequenc... | Sample paths of Brownian motion have this property (with probability $1$), see here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/42944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding double coset representatives in finite groups of Lie type Is there a standard algorithm for finding the double coset representatives of $H_1$ \ $G/H_2$, where the groups are finite of Lie type?
Specifically, I need to compute the representatives when $G=Sp_4(\mathbb{F}_q)$ (I'm using $J$ the anti diagonal with ... | Many such questions yield to using Bruhat decompositions, and often succeed over arbitrary fields (which shows how non-computational it may be). Let P be the parabolic with Levi component GL(2)xSL(2). Your second group misses being the "other" maximal proper parabolic Q only insofar as it misses the GL(1) part of the L... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/42995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Angle of a javelin at any given moment I am using the following formula to draw the trajectory of a javelin (this is very basic, I am not taking into consideration the drag, etc.).
speedX = Math.Cos(InitialAngle) * InitialSpeed;
speedY = Math.Sin(InitialAngle) * InitialSpeed;
javelin.X = speedX * timeT;
javeli... | I am making the assumption that the javelin is pointed exactly in the direction of its motion. (This seems dubious, but may be a close enough approximation for your purposes).
The speed in the X direction is constant, but the speed in the Y direction is $\text{speedY} -g\cdot \text{timeT}$. So the direction of motion... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Real-world applications of prime numbers? I am going through the problems from Project Euler and I notice a strong insistence on Primes and efficient algorithms to compute large primes efficiently.
The problems are interesting per se, but I am still wondering what the real-world applications of primes would be.
What ... | Thought I'd mention an application (or more like an explicit effect, rather than a direct application) that prime numbers have on computing fast Fourier transforms (FFTs), which are of fundamental use to many fields (e.g. signal processing, electrical engineering, computer vision).
It turns out that most algorithms for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 19,
"answer_id": 8
} |
Convex hull problem with a twist I have a 2D set and would like to determine from them the subset of points which, if joined together with lines, would result in an edge below which none of the points in the set exist.
This problem resembles the convex hull problem, but is fundamentally different in its definition.
One... | It looks like you are looking for the lower [convex] hull. Some algorithms such as the Andrew's variant of Graham Scan actually compute this and compute the upper hull and then merge these two to obtain the convex hull. Andrew's algorithm can also be seen as a sweep algorithm, so if you want a quick implementation, you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
A simple question about Iwasawa Theory There has been a lot of talk over the decades about Iwasawa Theory being a major player in number theory, and one of the most important object in said theory is the so-called Iwasawa polynomial. I have yet to see an example anywhere of such a polynomial. Is this polynomial hard/im... | Here is a function written for Pari/GP which computes Iwasawa polynomials. See in particular the note.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/43267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $\sum\limits_{k=1}^n k^2$ and $\sum\limits_{k=1}^n k(k+1)$ combinatorially
$$\text{Evaluate } \sum_{k=1}^n k^2 \text{ and } \sum_{k=1}^{n}k(k+1) \text{ combinatorially.}$$
For the first one, I was able to express $k^2$ in terms of the binomial coefficients by considering a set $X$ of cardinality $2k$ and par... | For the first one, $\displaystyle \sum_{k=1}^{n} k^2$, you can probably try this way.
$$k^2 = \binom{k}{1} + 2 \binom{k}{2}$$
This can be proved using combinatorial argument by looking at drawing $2$ balls from $k$ balls with replacement.
The total number of ways to do this is $k^2$.
The other way to count it is as fol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 3
} |
Need a hint: prove that $[0, 1]$ and $(0, 1)$ are not homeomorphic I need a hint: prove that $[0, 1]$ and $(0, 1)$ are not homeomorphic without referring to compactness. This is an exercise in a topology textbook, and it comes far earlier than compactness is discussed.
So far my only idea is to show that a homeomorphis... | There is no continuous and bijective function $f:(0,1) \rightarrow [0,1]$. In fact, if $f:(0,1) \rightarrow [0,1]$ is continuous and surjective, then $f$ is not injective, as proved in my answer in Continuous bijection from $(0,1)$ to $[0,1]$. This is a consequence of the intermediate value theorem, which is a theorem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken... | Let the considered integral be I i.e
$$I=\int_0^{\infty} \frac{1}{1+t^4}\,dt$$
Under the transformation $t\mapsto 1/t$, the integral is:
$$I=\int_0^{\infty} \frac{t^2}{1+t^4}\,dt \Rightarrow 2I=\int_0^{\infty}\frac{1+t^2}{1+t^4}\,dt=\int_0^{\infty} \frac{1+\frac{1}{t^2}}{t^2+\frac{1}{t^2}}\,dt$$
$$2I=\int_0^{\infty} \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 7,
"answer_id": 5
} |
Quick ways for approximating $\sum_{k=a_1}^{k=a_2}C_{100}^k(\frac{1}{2})^k(\frac{1}{2})^{100-k}$? Consider the following problem:
A fair coin is to be tossed 100 times, with each toss resulting in a head or a tail. Let
$$H:=\textrm{the total number of heads}$$
and
$$T:=\textrm{the total number of tails},$$
wh... | Chebyshev's inequality, combined with mixedmath's and some other observations, shows that the answer has to be D without doing the direct calculations.
First, rewrite D as $48 \leq H \leq 52$. A is a subset of D, and because the binomial distribution with $n = 100$ and $p = 0.5$ is symmetric about $50$, C is less lik... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
How can I solve this infinite sum? I calculated (with the help of Maple) that the following infinite sum is equal to the fraction on the right side.
$$
\sum_{i=1}^\infty \frac{i}{\vartheta^{i}}=\frac{\vartheta}{(\vartheta-1)^2}
$$
However I don't understand how to derive it correctly. I've tried numerous approaches b... | Several good methods have been suggested. Here's one more. $$\eqalign{\sum{i\over\theta^i}&={1\over\theta}+{2\over\theta^2}+{3\over\theta^3}+{4\over\theta^4}+\cdots\cr&={1\over\theta}+{1\over\theta^2}+{1\over\theta^3}+{1\over\theta^4}+\cdots\cr&\qquad+{1\over\theta^2}+{1\over\theta^3}+{1\over\theta^4}+\cdots\cr&\qquad\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Bounding ${(2d-1)n-1\choose n-1}$ Claim: ${3n-1\choose n-1}\le 6.25^n$.
*
*Why?
*Can the proof be extended to
obtain a bound on ${(2d-1)n-1\choose
n-1}$, with the bound being $f(d)^n$
for some function $f$?
(These numbers describe the number of some $d$-dimensional combinatorial objects; claim 1 is the case $... | First, lets bound things as easily as possible. Consider the inequality $$\binom{n}{k}=\frac{(n-k)!}{k!}\leq\frac{n^{k}}{k!}\leq e^{k}\left(\frac{n}{k}\right)^{k}.$$ The $n^k$ comes from the fact that $n$ is bigger then each factor of the product in the numerator. Also, we know that $k!e^k>k^k$ by looking at the $k^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Descriptive examples for beta distribution Do you have descriptive/typical examples for processes whose results are described by a beta distribution? So far i only have one:
You have a population of constant size with N individuals and you observe a single gene (or gene locus).
The descendants in the next generation ar... | Completely elementary is the fact that for every positive integers $k\le n$, the distribution of the order statistics of rank $k$ in an i.i.d. sample of size $n$ uniform on the interval $(0,1)$ is beta $(k,n-k+1)$.
Slightly more sophisticated is the fact that, in Bayesian statistics, beta distributions provide a simple... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
equivalent definitions of orientation I know two definitions of an orientation of a smooth n-manifold $M$:
1) A continuous pointwise orientation for $M$.
2) A continuous choice of generators for the groups $H_n(M,M-\{x\})=\mathbb{Z}$.
Why are these two definitions equivalent? In other words, why is a choice of basis o... | Recall that an element of $H_n(M,M-\{x\})$ is an equivalence class of singular $n$-chains, where the boundary of any chain in the class lies entirely in $M-\{x\}$. In particular, any generator of $H_n(M,M-\{x\})$ has a representative consisting of a single singular $n$-simplex $\sigma\colon \Delta^n\to M$, whose bound... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Converting a QBFs Matrix into CNF, maintaining equisatisfiability I have a fully quantified boolean formula in Prenix Normal Form $\Phi = Q_1 x_1, \ldots Q_n x_n . f(x_1, \ldots, x_n)$. As most QBF-Solvers expect $f$ to be in CNF, I use Tseitins Tranformation (Denoted by $TT$). This does not give an equivalent, but an ... | To use Tseitin's Transformation for predicate formulas, you'll need to add new predicate symbols of the form $A(x_1, ..., x_n)$. Then the formula $Q_1 x_1, ..., Q_n x_n TT(f(x_1,...,x_n))$ will imply "something" about this new predicate symbols, so the logical equivalence (which I assume what is meant by $\equiv$) does... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Isomorphism on commutative diagrams of abelian groups Consider the following commutative diagram of homomorphisms of abelian groups $$\begin{array} 00&\stackrel{f_1}{\longrightarrow}&A& \stackrel{f_2}{\longrightarrow}&B& \stackrel{f_3}{\longrightarrow}&C&\stackrel{f_4}{\longrightarrow}& D &\stackrel{f_5}{\longrightarro... | This is wrong. Consider
\begin{array}{ccccccccccc}
0 & \to & 0 & \to & 0 & \to & A & \to & A & \to & 0\\
\downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow\\
0 & \to & 0 & \to & A & \to & A & \to & 0 & \to & 0
\end{array}
where all maps $A \to A$ are the identity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/43894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
Let $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.
I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as $g(xy) = g(x) + g(y)$
I looked up the hint and it says let $g(x) = \log f(a^x) $
T... | Both the answers above are very good and thorough, but given an assumption that the function is differentiable, the DE approach strikes me as the easiest.
$ \frac{\partial}{\partial y} f(x y) = x f'(xy) = f(x)f'(y) $
Evaluating y at 1 gives:
$ xf'(x) = f(x)f'(1) $
The above is a separable DE:
Let $ p = f'(1) $ and $ z ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 3,
"answer_id": 0
} |
Edge coloring a graph to find a monochromatic $K_{2,n}$ I am trying to prove or disprove the following statement: Let $n>1$ be a positive integer. Then there exists a graph $G$ of size 4n-1 such that if the edges of $G$ are colored red or blue, no matter in which way, $G$ definitely contains a monochromatic $K_{2,n}$. ... | The claim does not hold for $n = 2$. Consider the following observations for any graph $G$ hoping to satisfy the claim.
*
*$G$ is a $K_{2,2}$ with three edges appended.
*Without loss of generality, $G$ is connected and has no leaves.
*$G$ has at least five vertices.
Draw a $K_{2,2}$ and plus one more vertex. Sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Trouble with absolute value in limit proof As usual, I'm having trouble, not with the calculus, but the algebra. I'm using Calculus, 9th ed. by Larson and Edwards, which is somewhat known for racing through examples with little explanation of the algebra for those of us who are rusty.
I'm trying to prove $$\lim_{x \to ... | Because of the freedom in the choice of $\delta$, you can always assume $\delta < 1$, that implies you can assume $x$ belongs to the interval $(0, 2)$.
Edit: $L$ is the limit of $f(x)$ for $x$ approaching $x_0$, iff for every $\epsilon > 0$ it exists a $\delta_\epsilon > 0$ such that:
$$\left\vert f(x) - L\right\vert <... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How many points in the xy-plane do the graphs of $y=x^{12}$ and $y=2^x$ intersect? The question in the title is equivalent to find the number of the zeros of the function $$f(x)=x^{12}-2^x$$
Geometrically, it is not hard to determine that there is one intersect in the second quadrant. And when $x>0$, $x^{12}=2^x$ is eq... | If you are solving a multiple choice test like GRE you really need fast intuitive, but certain, thinking. I tried to put myself in this rushed set of mind when I read your question and thought this way: think of $x^{12}$ as something like $x^2$ but growing faster, think of $2^x$ as $e^x$ similarly, sketch both function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 1
} |
Game theory textbooks/lectures/etc I looking for good books/lecture notes/etc to learn game theory. I do not fear the math, so I'm not looking for a "non-mathematical intro" or something like that. Any suggestions are welcome. Just put here any references you've seen and some brief description and/or review. Thanks.
Ed... | Coursera.org offers an excellent game theory course by Dr.s Shoham, Leyton-Brown, and Jackson (https://www.coursera.org/course/gametheory).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/44246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 7,
"answer_id": 6
} |
How do I get the square root of a complex number? If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
| Here is a direct algebraic answer.
Suppose that $z=c+di$, and we want to find $\sqrt{z}=a+bi$ lying in the first two quadrants. So what are $a$ and $b$?
Precisely we have
$$a=\sqrt{\frac{c+\sqrt{c^{2}+d^{2}}}{2}}$$ and
$$b=\frac{d}{|d|}\sqrt{\frac{-c+\sqrt{c^{2}+d^{2}}}{2}}.$$ (The factor of $\frac{d}{|d|}$ is used ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "121",
"answer_count": 12,
"answer_id": 3
} |
Paths with DFA? My teacher made an example to explain DFA, it was about paths (URL paths), the rules were as follows:
S ::= /
S ::= /O
O ::= [a-z]
O ::= [a-z]R
O ::= [a-z]S
R ::= [a-z]
R ::= [a-z]R
R ::= [a-z]S
Examples of paths could be: /foo, /foo/, foo/bar and so on.
However, I don't understand why you would need t... | You don't need them, in fact. The grammar you wrote is equivalent to the one obtained by deleting the R rules and substituting the second O rule by
O ::= [a-z]O
... No idea why your teacher wrote it that way, sorry.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/44445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Are the inverses of these matrices always tridiagonal? While putzing around with the linear algebra capabilities of my computing environment, I noticed that inverses of $n\times n$ matrices $\mathbf M$ associated with a sequence $a_i$, $i=1\dots n$ with $m_{ij}=a_{\max(i,j)}$, which take the form
$$\mathbf M=\begin{pma... | Let $B_j$ be the $n\times n$ matrix with $1$s in the upper-left hand $j\times j$ block and zeros elsewhere. The space of $L$-shaped matrices you're interested in is spanned by $B_1,B_2,\dots,B_n$. I claim that if $b_1,\dots,b_n$ are non-zero scalars, then the inverse of
$$ M=b_1B_1+b_2B_2+\dots + b_nB_n$$ is then the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?
The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif
Laczkovich gave a solut... | I have no answer to the question, but here's a picture resulting from some initial attempts to understand the constraints that exist on any solution.
$\qquad$
This image was generated by considering what seemed to be the simplest possible configuration that might produce a tiling of a rectangle. Starting with the two ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "108",
"answer_count": 2,
"answer_id": 0
} |
One divided by Infinity? Okay, I'm not much of a mathematician (I'm an 8th grader in Algebra I), but I have a question about something that's been bugging me.
I know that $0.999 \cdots$ (repeating) = $1$. So wouldn't $1 - \frac{1}{\infty} = 1$ as well? Because $\frac{1}{\infty} $ would be infinitely close to $0$, perha... | There is one issue that has not been raised in the fine answers given earlier. The issue is implicit in the OP's phrasing and it is worth making it explicit. Namely, the OP is assuming that, just as $0.9$ or $0.99$ or $0.999$ denote terminating decimals with a finite number of 9s, so also $0.999\ldots$ denotes a term... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 5,
"answer_id": 4
} |
Can a circle truly exist? Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says...
A circle is a simple shape of Euclidean geometry consisting of the set of points in a plane that are a given distance from a given point, the centr... | In the same sense as you think a circle is impossible, a square with truly perfect sides can never exist because the lines would have to have infinitesimal width, and we can never measure a perfect right angle, etc.
You say that you think a square is physically possible to represent with 4 points, though. In this case... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Independence of sums of gaussian random variables Say, I have independent gaussian random variables $t1, t2, t3, t4, t5$ and I have two new random variables $S = t1 + t2 - t3$ and $K = t3 + t4$.
Are $S$ and $K$ independent or is there any theorem about independece of random variables formed by sum of independent gaussi... | In fact, the distribution of the $t_i$ plays no significant role here, and, moreover, existence of the covariance is not necessary.
Let $S=X-Y$ and $K=Y+Z$, where $X$, $Y$, and $Z$ are independent random variables generalizing the role of $t1+t2$, $t3$, and $t4$, respectively.
Note that, by independence of $X$, $Y$, an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
bayesian networks for regression Would it be possible to use bayesian network for regression and/or prediction? I understand that it is a tool one can use to compute probabilities, but I haven't found much material about possible applications for forecasting.
| The Naive Bayes classifier is a type of classifier which is a Bayesian Network (BN). There are also extensions like Tree-Augmented Naive Bayes and more generally Augmented Naive Bayes.
So not only is it possible, but it has been done and there is lots of literature on it.
Most of the applications I see deal with classi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\tan(\frac{\pi}{2}) = \infty~$?
Evaluate $\displaystyle \int\nolimits^{\pi}_{0} \frac{dx}{5 + 4\cos{x}}$ by using the substitution $t = \tan{\frac{x}{2}}$
For the question above, by changing variables, the integral can be rewritten as $\displaystyle \int \frac{\frac{2dt}{1+t^2}}{5 + 4\cos{x}}$, ignoring the upper an... | Continuing from my comment, you have
$$\cos(t) = \cos^2(t/2) - \sin^2(t/2) = {1-t^2\over 1+ t^2}.$$
Restating the integral with the transformation gives
$$\int_0^\infty {1\over 5 + 4\left({1-t^2 \over 1 + t^2}\right)}{2\, dt\over 1 + t^2} = 2\int_0^\infty {dt\over 9 + t^2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/45099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$ How do I show that:
$$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$
This is actually problem B $4371$ given at this link. Looks like a... | Use $\sin(x) = \cos(\frac{\pi}2 - x)$, we can rewrite this as:
$$\frac{1}{\cos^2 \frac{3\pi}{7}} + \frac{1}{\cos^2 \frac{2\pi}{7}} + \frac{1}{\cos^2 \frac{\pi}{7}}$$
Let $a_k = \frac{1}{\cos \frac{k\pi}7}$.
Let $f(x) = (x-a_1)(x-a_2)(x-a_3)(x-a_4)(x-a_5)(x-a_6)$.
Now, using that $a_k = - a_{7-k}$, this can be written a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 5,
"answer_id": 0
} |
Raising a square matrix to the k'th power: From real through complex to real again - how does the last step work? I am reading Applied linear algebra: the decoupling principle by Lorenzo Adlai Sadun (btw very recommendable!)
On page 69 it gives an example where a real, square matrix $A=[(a,-b),(b,a)]$ is raised to the... | What you are using is that for a given complex number $z=a+bi$, we have $\frac{z+\overline{z}}{2}=a={\rm Re}(z)$ and $\frac{z-\overline{z}}{2}=ib=i{\rm Im}(z)$ (where $\overline{z}=a-bi$). Also check that $\overline{z^k}=\overline{z}^k$ for all $k \in \mathbb{N}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/45297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Minimal free resolution I'm studying on the book "Cohen-Macaulay rings" of Bruns-Herzog
(Here's a link and an image of the page in question for those unable to use Google Books.)
At page 17 it talks about minimal free resolution, but it doesn't give a proper definition (or I'm misunderstanding the one it gives), could... | I can't see that book online, but let me paraphrase the definition from Eisenbud's book on commutative algebra. See Chapter 19 page 473-477 for details.
Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m},$ then
Definition: A free resolution of a $R$-module $M$ is a complex
$$\mathcal{F}: ...\rightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
} |
Weierstrass Equation and K3 Surfaces Let $a_{i}(t) \in \mathbb{Z}[t]$. We shall denote these by $a_{i}$. The equation $y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}$ is the affine equation for the Weierstrass form of a family of elliptic curves. Under what conditions does this represent a K3 surface?
| A good reference for this would be Abhinav Kumar's PhD thesis, which you can find here. In particular, look at Chapter 5, and Section 5.1. If an elliptic surface $y^2+a_1(t)xy+a_3(t)y = x^3+a_2(t)x^2+a_4(t)x+a_6(t)$ is K3, then the degree of $a_i(t)$ must be $\leq 2i$.
I hope this helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/45401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof that the set of incompressible strings is undecidable I would like to see a proof or a sketch of a proof that the set of incompressible strings is undecidable.
Definition: Let x be a string, we say that x is c-compressible if K(x) $\leq$ |x|-c.
If x is not c-compressible, we say that x is incompressible by c. K(x... | Roughly speaking, incompressibility is undecidable because of a version of the Berry paradox. Specifically, if incompressibility were decidable, we could specify "the lexicographically first incompressible string of length 1000" with the description in quotes, which has length less than 1000.
For a more precise proof ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How to compute the transition function in non-determinism finite accepter NFA? I'm currently teaching myself Automaton using Peter Linz book - An Introduction to Formal Languages and Automata 4th edition. While reading chapter 2 about NFA, I was stuck this example (page 51):
According to the author, the transition fu... | be careful that your machine should read 'a' to accept destination state. in your nfa, before reading 'a', 2 lambda transitions should be placed. first, to go to q2, and second, to go to q0. after that your machine can read a 'a' and places on q1. now transition to q2 and q0 are take placed by one and two lambda transi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Number of possible sets for given N How many possible valid collections are there for a given positive integer N given the following conditions:
All the sums from 1 to N should be possible to be made by selecting some of the integers. Also this has to be done in way such that if any integer from 1 to N can be made in m... | The term I would use is "multiset". Note that your multiset must contain 1 (as this is the only way to get a sum of 1). Suppose there are $r$ different values $a_1 = 1, \ldots, a_r$ in the multiset, with $k_j$ copies of $a_j$. Then we must have $a_j = (k_{j-1}+1) a_{j-1}$ for $j = 2, \ldots, r$, and $N = (k_r + 1) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Why is an empty function considered a function? A function by definition is a set of ordered pairs, and also according the Kuratowski, an ordered pair $(x,y)$ is defined to be $$\{\{x\}, \{x,y\}\}.$$
Given $A\neq \varnothing$, and $\varnothing\colon \varnothing \rightarrow A$. I know $\varnothing \subseteq \varnothing ... | The empty set is a set of ordered pairs. It contains no ordered pairs but that's fine, in the same way that $\varnothing$ is a set of real numbers though $\varnothing$ does not contain a single real number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/45625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
"answer_count": 3,
"answer_id": 2
} |
How to find a positive semidefinite matrix $Y$ such that $YB =0$ where $B$ is given $B$ is an $n\times m$ matrix, $m\leq n$.
I have to find an $n\times n$ positive semidefinite matrix $Y$ such that $YB = 0$.
Please help me figure out how can I find the matrix $Y$.
| If $X$ is any (real) matrix with the property that $XB=0$, then $Y=X^TX$ will do the trick. Such a matrix $Y$ is always positive semidefinite. To see this note that for any (column) vector $v$ we have $v^TYv=(Xv)^T(Xv)=|Xv|^2\ge0$.
How to find such a matrix $X$? If $m=n$ and $\det B\neq0$, then there is no other choic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Interesting integral related to the Omega Constant/Lambert W Function I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am sorry.
$$\displaystyle\frac{1}{\int_{-\infty}^{\infty}\frac{1}{(e^{x... | While this is by no means rigorous, but it gives the correct solution. Any corrections to this are welcome!
Let
$$f(z) := \frac{1}{(e^z-z)^2+\pi^2}$$
Let $C$ be the canonical positively-oriented semicircular contour that traverses the real line from $-R$ to $R$ and all around $Re^{i \theta}$ for $0 \le \theta \le \pi$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 3,
"answer_id": 0
} |
Calculate the area on a sphere of the intersection of two spherical caps Given a sphere of radius $r$ with two spherical caps on it defined by the radii ($a_1$ and $a_2$) of the bases of the spherical caps, given a separation of the two spherical caps by angle $\theta$, how do you calculate the surface area of that int... | Here's a simplified formula as a function of your 3 variables, $a_1$, $a_2$, and $\theta$:
$$
2\cos(a_2)\arccos \left ( \frac{-\cos(a_1) + \cos(\theta)\cos(a_2)}{\sin(\theta)\sin(a_2)} \right ) \\
-2\cos(a_1)\arccos \left ( \frac{\cos(a_2) - \cos(\theta)\cos(a_1)}{\sin(\theta)\sin(a_1)} \right ) \\
-2\arccos \left ( \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 1
} |
Can an element other than the neutral element be its own inverse? Take the following operation $*$ on the set $\{a, b\}$:
*
*$a * b = a$
*$b * a = a$
*$a * a = b$
*$b * b = b$
$b$ is the neutral element. Can $a$ also be its own inverse, even though it's not the neutral element? Or does the inverse property requ... | Your set is isomorphic to the two-element group: $b=1$, $a=-1$, $*=$multiplication. So yes, $a$ can very well be its own inverse.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/45847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 0
} |
Subspace intersecting many other subspaces V is a vector space of dimension 7. There are 5 subspaces of dimension four. I want to find a two dimensional subspace such that it intersects at least once with all the 5 subspaces.
Edit: All the 5 given subspaces are chosen randomly (with a very high probability, the interse... | Assuming your vector space is over $\mathbb R$, it looks to me like "generically" there should be a finite number of solutions, but I can't prove that this finite number is positive, nor do I have a counterexample. We can suppose your two-dimensional subspace $S$ has an orthonormal basis $\{ u, v \}$ where $u \cdot e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
bijective morphism of affine schemes The following question occurred to me while doing exercises in Hartshorne. If $A \to B$ is a homomorphism of (commutative, unital) rings and $f : \text{Spec } B \to \text{Spec } A$ is the corresponding morphism on spectra, does $f$ bijective imply that $f$ is a homeomorphism? If not... | No. Let $A$ be a DVR. Let $k$ be the residue field, $K$ the quotient field. There is a map $\mathrm{Spec} k \sqcup \mathrm{Spec} K \to \mathrm{Spec} A$ which is bijective, but not a homeomorphism (one side is discrete and the other is not). Note that $\mathrm{Spec} k \sqcup \mathrm{Spec}K = \mathrm{Spec} k \times K$, s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Why are horizontal transformations of functions reversed? While studying graph transformations I came across horizontal and vertical scale and translations of functions. I understand the ideas below.
*
*$f(x+a)$ - grouped with x, horizontal translation, inverse, x-coordinate shifts left, right for -a
*$f(ax)$ - gro... | For horizantal shift:
The logical reason for horizantal shift is that in(f)(x)=y=x the origin is (0,0)and in f(x)=(x-2)is (2,0) for this we should add 2 to get 0 becouse in parent function become 0 when we add 0 and in shifted function to make zero our function we ahould add 2
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/46053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 4
} |
Fractional part of $b \log a$ From the problem...
Find the minimal positive integer $b$ such that the first digits of $2^b$ are 2011
...I have been able to reduce the problem to the following instead:
Find minimal $b$ such that $\log_{10} (2.011) \leq \operatorname{frac}(b~\log_{10} (2)) < \log_{10} (2.012)$, where... | You are looking for integers $b$ and $p$ such that $b\log_{10}2-\log_{10}(2.011)-p$ is small and positive. The general study of such things is called "inhomogeneous diophantine approximation," which search term should get you started, if you want something more analytical than a brute force search. As 6312 indicated, c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Effect of adding a constant to both Numerator and Denominator I was reading a text book and came across the following:
If a ratio $a/b$ is given such that $a \gt b$, and given $x$ is a positive integer, then
$$\frac{a+x}{b+x} \lt\frac{a}{b}\quad\text{and}\quad \frac{a-x}{b-x}\gt \frac{a}{b}.$$
If a ratio $a/b$ is gi... | How about something along these lines: Think of a pot of money divided among the people in a room. In the beginning, there are a dollars and b persons. Initially, everyone gets a/b>1 dollars since a>b. But new people are allowed into the room at a fee of 1 dollar person. The admission fees are put into the pot. The ave... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 8,
"answer_id": 6
} |
Reference book on measure theory I post this question with some personal specifications. I hope it does not overlap with old posted questions.
Recently I strongly feel that I have to review the knowledge of measure theory for the sake of starting my thesis.
I am not totally new with measure theory, since I have taken a... | Donald L. Cohn-"Measure theory". Everything is detailed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/46213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "86",
"answer_count": 16,
"answer_id": 1
} |
Book/tutorial recommendations: acquiring math-oriented reading proficiency in German I'm interested in others' suggestions/recommendations for resources to help me acquire reading proficiency (of current math literature, as well as classic math texts) in German.
I realize that German has evolved as a language, so ide... | I realize this is a bit late, but I just saw by chance that the math department of Princeton has a list of German words online, seemingly for people who want to read German math papers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/46313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 6,
"answer_id": 2
} |
Many convergent sequences imply the initial sequence zero? In connection to this question, I found a similar problem in another Miklos Schweitzer contest:
Problem 8./2007 For $A=\{a_i\}_{i=0}^\infty$ a sequence of real numbers, denote by $SA=\{a_0,a_0+a_1,a_0+a_1+a_2,...\}$ the sequence of partial sums of the series $... | I would suggest you try using the alternating harmonic series. It is conditionally convergent so you can try rearrangements that might pop out convergent to zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/46350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
Alternative to imaginary numbers? In this video, starting at 3:45 the professor says
There are some superb papers written that discount the idea that we should ever use j (imaginary unit) on the grounds that it conceals some structure that we can explain by other means.
What is the "other means" that he is referring... | Maybe he meant the following: A complex number $z$ is in the first place an element of the field ${\mathbb C}$ of complex numbers, and not an $a+bi$. There are indeed structure elements which remain hidden when thinking in terms of real and imaginary parts only, e.g., the multiplicative structure of the set of roots of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 4
} |
Reduction formula for $I_{n}=\int {\cos{nx} \over \cos{x}}\rm{d}x$ What would be a simple method to compute a reduction formula for the following?
$\displaystyle I_{n}=\int {\cos{nx} \over \cos{x}} \rm{d}x~$ where $n$ is a positive integer
I understand that it may involve splitting the numerator into $\cos(n-2+2)x~$ ... | The complex exponential approach described by Gerry Myerson is very nice, very natural. Here are a couple of first-year calculus approaches. The first is kind of complicated, but introduces some useful facts. The second one, given at the very end, is quick.
Instead of doing a reduction formula directly, we separate ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Connected planar simple Graph: number of edges a function of the number of vertices Suppose that a connected planar simple graph with $e$ edges and $v$ vertices contains no simple circuit with length greater than or equal to $4.\;$ Show that $$\frac 53 v -\frac{10}{3} \geq e$$
or, equivalently, $$5(v-2) \geq 3e$$
| As Joseph suggests, one of two formulas you'll want to use for this problem is Euler's formula, which you may know as
$$r = e - v + 2 \quad\text{(or}\quad v + r - e = 2)\qquad\qquad\quad (1)$$
where $r$ is the number of regions in a planar representation of $G$ (e: number of edges, v: number of vertices). (Note, for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix? I am trying to understand a proof concerning commuting matrices and simultaneous diagonalization of these.
It seems to be a well known result that when you take the eigenvectors of $A$ as a basis and diagonaliz... | Suppose that $A$ and $B$ are matrices that commute. Let $\lambda$ be an eigenvalue for $A$, and let $E_{\lambda}$ be the eigenspace of $A$ corresponding to $\lambda$. Let $\mathbf{v}_1,\ldots,\mathbf{v}_k$ be a basis for $E_{\lambda}$.
I claim that $B$ maps $E_{\lambda}$ to itself; in particular, $B\mathbf{v}_i$ can be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 2
} |
Proving Stewart's theorem without trig Stewart's theorem states that in the triangle shown below,
$$ b^2 m + c^2 n = a (d^2 + mn). $$
Is there any good way to prove this without using any trigonometry? Every proof I can find uses the Law of Cosines.
| Geometric equivalents of the Law of Cosines are already present in Book II of Euclid, in Propositions $12$ and $13$ (the first is the obtuse angle case, the second is the acute angle case).
Here are links to Proposition $12$, Book II, and to Proposition $13$.
There is absolutely no trigonometry in Euclid's proofs.
Thes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 4,
"answer_id": 0
} |
Atiyah-Macdonald, Exercise 8.3: Artinian iff finite k-algebra.
Atiyah Macdonald, Exercise 8.3. Let $k$ be a field and $A$ a finitely generated $k$-algebra. Prove that the following are equivalent:
(1) $A$ is Artinian.
(2) $A$ is a finite $k$-algebra.
I have a question in the proof of (1$\Rightarrow$2): By using t... | The claim also seems to follow from the Noether normalization lemma:
Let $B := k[x_1, \dotsc, x_n]$ with $k$ any field and let $I \subseteq B$ be any ideal.
Since $A$ is a finitely generated $k$-algebra you may let $A := B/I$. By the Noether normalization lemma it follows that there is a finite set of elements $y_1, \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 3,
"answer_id": 2
} |
How to prove the implicit function theorem fails Define $$F(x,y,u,v)= 3x^2-y^2+u^2+4uv+v^2$$ $$G(x,y,u,v)=x^2-y^2+2uv$$
Show that there is no open set in the $(u,v)$ plane such that $(F,G)=(0,0)$ defines $x$ and $y$ in terms of $u$ and $v$.
If (F,G) is equal to say (9,-3) you can just apply the Implicit function theore... | To say $(F,G) = (0,0)$ is to say that $y^2 - 3x^2 = u^2 + 4uv + v^2$ and $y^2 - x^2 = 2uv$. By some algebra, this is equivalent to $x^2 = -{1 \over 2}(u + v)^2$ and $y^2 = -{1 \over 2}(u - v)^2$. So you are requiring the nonnegative quantities on the left to be equal to the nonpositive quantities on the right. Hence th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Permutation/Combinations in bit Strings I have a bit string with 10 letters, which can be {a, b, c}. How many bit strings can be made that have exactly 3 a's, or exactly 4 b's?
I thought that it would be C(7,2) + C(6,2), but that's wrong (the answer is 24,600).
| Hint: By the inclusion-exclusion principle, the answer is equal to
$$\begin{align} & \text{(number of strings with exactly 3 a's)}\\ + & \text{(number of strings with exactly 4 b's)}\\ - &\text{(number of strings with exactly 3 a's and 4 b's)} \end{align}$$
Suppose I want to make a string with exactly 3 a's. First, I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a - b|$ I came across the following problem in my self-study of real analysis:
For any real numbers $a$ and $b$, show that $$\max \{a,b \} = \frac{1}{2}(a+b+|a-b|)$$ and $$\min\{a,b \} = \frac{1}{2}(a+b-|a-b|)$$
So $a \geq b$ iff $a-b \g... | I know this is a little bit late, but here another way to get into that formula.
If we want to know $\min(a,b)$ we can know which is smaller by taking the sign of $b-a$. The sign is defined as $sign(x)=\frac{x}{|x|}$ and $msign(x)=\frac{sign(x)+1}{2}$ to get the values $0$ or $1$; if $msign(a-b)$ is $1$ it means that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 4,
"answer_id": 3
} |
Group theory intricate problem This is Miklos Schweitzer 2009 Problem 6. It's a group theory problem hidden in a complicated language.
A set system $(S,L)$ is called a Steiner triple system if $L \neq \emptyset$, any pair $x,y \in S, x \neq y$ of points lie on a unique line $\ell \in L$, and every line $\ell \in L$ c... | Let $g:S\rightarrow A$ be defined as $g(x) = h(x)^{-1} f(x)$.
Now, if $\{x,y,z\}\in L$, then $g(y) = h(z)g(x)h(z)^{-1}$. This means that the image of $g$ is closed under conjugation by elements of $A$ since $A$ is generated by the image of $h.$
Also, since this formula does not depend on the order of $x,y,z$, it me... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
$F[a] \subseteq F(a)?$ I think this is probably an easy question, but I'd just like to check that I'm looking at it the right way.
Let $F$ be a field, and let $f(x) \in F[x]$ have a zero $a$ in some extension field $E$ of $F$. Define $F[a] = \left\{ f(a)\ |\ f(x) \in F[x] \right\}$. Then $F[a]\subseteq F(a)$.
The way... | (1) Yes, you are correct. Note that $F(a)=\{\frac{f(a)}{g(a)}:f,g\in F[x], g(a)\neq 0\}$; in other words, $F(a)$ is the field of fractions of $F[a]$ and therefore certainly contains $F[a]$.
(2) Yes, the notation $F[a]$ is standard for the set you described.
Exercise 1: Prove that if $a$ is algebraic over $F$, then $F[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Is there a name for the matrix $X(X^tX)^{-1}X^{t}$? In my work, I have repeatedly stumbled across the matrix (with a generic matrix $X$ of dimensions $m\times n$ with $m>n$ given) $\Lambda=X(X^tX)^{-1}X^{t}$. It can be characterized by the following:
(1) If $v$ is in the span of the column vectors of $X$, then $\Lambda... | This should be a comment, but I can't leave comments yet. As pointed out by Rahul Narain, this is the orthogonal projection onto the column space of $X$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/47093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Null Sequences and Real Analysis I came across the following problem during the course of my study of real analysis:
Prove that $(x_n)$ is a null sequence iff $(x_{n}^{2})$ is null.
For all $\epsilon>0$, $|x_{n}| \leq \epsilon$ for $n > N_1$. Let $N_2 = \text{ceiling}(\sqrt{N_1})$. Then $(x_{n}^{2}) \leq \epsilon$ f... | You could use the following fact:
If a function $f:X\to Y$ between two topological spaces is continuous and $x_n\to x$, then $f(x_n)\to f(x)$.
(In case you do not have learned it in this generality, you might at least know that this is true for real functions or for functions between metric spaces. In fact, in the case... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
(Organic) Chemistry for Mathematicians Recently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do with physical chemistry: cond-mat, fluids, QM of molecules, and analysis of spect... | Organic chemistry
S. Fujita's "Symmetry and combinatorial enumeration in chemistry" (Springer-Verlag, 1991) is one such endeavor. It mainly focuses on stereochemistry.
Molecular biology and biochemistry
A. Carbone and M. Gromov's "Mathematical slices of molecular biology" is recommended, although it is not strictly a b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 2,
"answer_id": 1
} |
Question about proof for $S_4 \cong V_4 \rtimes S_3$ In my book they give the following proof for $S_4 \cong V_4 \rtimes S_3$ :
Let $j: S_3 \rightarrow S_4: p \mapsto \left( \begin{array}{cccc}
1 & 2 & 3 & 4 \\
p(1) & p(2) & p(3) & 4 \end{array} \right)$
Clearly, $j(S_3)$ is a subgroup $S_4$ isomorphic with $S_3$, h... |
It is only used to identify the subgroup S3 of S4, and is only needed as a technicality.
If you view S3 as bijections from {1,2,3} to {1,2,3} and S4 as bijections from {1,2,3,4} to {1,2,3,4}, and you view functions as having domains and ranges (not just rules), then no element of S3 is an element of S4. The function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Qualitative interpretation of Hilbert transform the well-known Kramers-Kronig relations state that for a function satisfying certain conditions, its imaginary part is the Hilbert transform of its real part.
This often comes up in physics, where it can be used to related resonances and absorption. What one usually find... | Never heard of the Kramers-Kronig relations and so I looked it up. It relates the real and imaginary parts of an analytic function on the upper half plane that satisfies certain growth conditions. This is a big area in complex analysis and there are many results. For example, in the case of a function with compact supp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Involuted vs Idempotent What is the difference between an "involuted" and an "idempotent" matrix?
I believe that they both have to do with inverse, perhaps "self inverse" matrices.
Or do they happen to refer to the same thing?
| A matrix $A$ is an involution if it is its own inverse, ie if
$$A^2 = I$$
A matrix $B$ is idempotent if it squares to itself, ie if
$$B^2 = B$$
The only invertible idempotent matrix is the identity matrix, which can be seen by multiplying both sides of the above equation by $B^{-1}$. An idempotent matrix is also known ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Show that $f \in \Theta(g)$, where $f(n) = n$ and $g(n) = n + 1/n$ I am a total beginner with the big theta notation. I need find a way to show that $f \in \Theta(g)$, where $f(n) = n$, $g(n) = n + 1/n$, and that $f, g : Z^+ \rightarrow R$. What confuses me with this problem is that I thought that "$g$" is always suppo... | You are sort of right about thinking that "$g$" is supposed to be simpler than "$f$", but not technically right. The formal definition says nothing about simpler.
However, in practice one is essentially always comparing something somewhat messy, on the left, with something whose behaviour is sort of clear(er) to the e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
"Counting Tricks": using combination to derive a general formula for $1^2 + 2^2 + \cdots + n^2$ I was reading an online article which confused me with the following. To find out $S(n)$, where $S(n) = 1^2 + 2^2 + \cdots + n^2$, one can first write out the first few terms:
0 1 5 14 30 55 91 140 204 285
Then, get th... | The key word here is finite differences. See Newton series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/47509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 1
} |
A double integral (differentiation under the integral sign) While working on a physics problem, I got the following double integral that depends on the parameter $a$:
$$I(a)=\int_{0}^{L}\int_{0}^{L}\sqrt{a}e^{-a(x-y+b)^2}dxdy$$
where $L$ and $b$ are constants.
Now, this integral obviously has no closed form in terms of... | Nowadays many mathematicians (including me -:)) would be content to use some program to have
$$I'(a)=\frac{e^{-a (b+L)^2} \left(2 e^{a L (2 b+L)}-e^{4 a b L}-1\right)}{4 a^{3/2}}.$$
As for the proof, put $t=1/a$ and let $G(b,t)=e^{-b^2/t}/\sqrt{\pi t}\ $ be a fundamental solution of the heat equation $u_t-u_{bb}/4=0\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Completeness and Cauchy Sequences I came across the following problem on Cauchy Sequences:
Prove that every compact metric space is complete.
Suppose $X$ is a compact metric space. By definition, every sequence in $X$ has a convergent subsequence. We want to show that every Cauchy sequence in $X$ is convergent in $X$... | Let $\epsilon > 0$. Since $(x_n)$ is Cauchy, exists $\eta_1\in \mathbb N$ such that
$$ \left\vert x_n - x_m\right\vert < \frac \epsilon 2$$
for each pair $n, m > \eta_1$.
Since $x_{k_n} \to a$, exists $\eta_2 \in \mathbb N$ such that
$$ \left\vert x_{k_n} - a\right\vert < \frac \epsilon 2$$
for each $n > \eta_2$.
Let $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.