Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How does a Class group measure the failure of Unique factorization? I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a hell.
The Class group is given by $\rm{Cl}(F)=$ {Fra... | h=1 means that the size of the class group is 1. That means that the group is the trivial group with only one element, the identity. The identity element of the class group is the equivalence class of principal ideals. Hence h=1 is equivalent to "all fractional ideals are principal" or equivalently "all ideals are prin... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "17",
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What is the chance of an event happening a set number of times or more after a number of trials? Assuming every trial is independent from all the others and the probability of a successful run is the same every trial, how can you determine the chance of a successful trial a set number of times or more?
For example, You... | If the probability of success on any trial is $p$, then the probability of exactly $k$ successes in $n$ trials is
$$\binom{n}{k}p^k(1-p)^{n-k}.$$
For details, look for the Binomial Distribution on Wikipedia.
So to calculate the probability of $3$ or more successes in your example, let $p=0.60$ and $n=20$. Then calculat... | {
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Longest cylinder of specified radius in a given cuboid Find the maximum height (in exact value) of a cylinder of radius $x$ so that it can completely place into a $100 cm \times 60 cm \times50 cm$ cuboid.
This question comes from http://hk.knowledge.yahoo.com/question/question?qid=7012072800395.
I know that this questi... | A beginning:
Let $a_i>0$ $\>(1\leq i\leq 3)$ be the dimensions of the box. Then we are looking for a unit vector ${\bf u}=(u_1,u_2,u_3)$ in the first octant and a length $\ell>0$ such that
$$\ell u_i+2 x\sqrt{1-u_i^2}=a_i\qquad(1\leq i\leq 3)\ .$$
When $x$ is small compared to the dimensions of the box one might begin ... | {
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When can one use logarithms to multiply matrices If $a,b \in \mathbb{Z}_{+}$, then $\exp(\log(a)+\log(b))=ab$. If $A$ and $B$ are square matrices, when can we multiply $A$ and $B$ using logarithms? If $A \neq B^{-1}$, should $A$ and $B$ be symmetric?
| When they commute, or: when they have the same eigenvectors, or if $AB=BA$
| {
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If $a^n-b^n$ is integer for all positive integral value of $n$, then $a$, $b$ must also be integers. If $a^n-b^n$ is integer for all positive integral value of n with a≠b, then a,b must also be integers.
Source: Number Theory for Mathematical Contests, Problem 201, Page 34.
Let $a=A+c$ and $b=B+d$ where A,B are int... | assuming $a \neq b$
if $a^n - b^n$ is integer for all $n$, then it is also integer for $n = 1$ and $n = 2$.
From there you should be able to prove that $a$ is integer.
| {
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Is this CRC calculation correct? I am currently studying for an exam and trying to check a message (binary) for errors using a polynomial, I would like if somebody could verify that my results below are (in)valid.
Thanks.
Message: 11110101 11110101
Polynomial: X4 + x2 + 1
Divisor (Derived from polynomial): 10101
Remain... | 1111010111110101 | 10101
+10101 | 11001011101
10111 |
+10101 |
10011 |
+10101 |
11011 |
+10101 |
11101 |
+10101 |
10000 |
+10101 |
10110 |
+10101 |
111 | <-... | {
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Is concave quadratic + linear a concave function? Basic question about convexity/concavity:
Is the difference of a concave quadratic function of a matrix $X$ given by f(X) and a linear function l(X), a concave function?
i.e, is f(X)-l(X) concave?
If so/not what are the required conditions to be checked for?
| A linear function is both concave and convex (here $-l$ is concave), and the sum of two concave functions is concave.
| {
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Function writen as two functions having IVP I heard this problem and I am a bit stuck.
Given a function $f : I \rightarrow \mathbb{R}$ where $I \subset \mathbb{R}$ is an open interval.
Then $f$ can be writen $f=g+h$ where $g,h$ are defined in the same interval and have the Intermediate Value Property. I tried to constr... | Edit: In fact, all the information I give below (and more) is provided in another question in a much more organized way. I just found it.
My original post: The intermediate Value property is also called the Darboux property. Sierpinski first proved this theorem.The problem is treated in a blog of Beni Bogosel, a member... | {
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Canonical Isomorphism Between $\mathbf{V}$ and $(\mathbf{V}^*)^*$ For the finite-dimensional case, we have a canonical isomorphism between $\mathbf{V}$, a vector space with the usual addition and scalar multiplication, and $(\mathbf{V}^*)^*$, the "dual of the dual of $\mathbf{V}$." This canonical isomorphism means that... | There are two things that can go wrong in the infinite-dimensional (normed) case.
First you could try to take the algebraic dual of $V$. Here it turns out that $V^{**}$ is much larger than $V$ for simple cardinality reasons as outlined in the Wikipedia article.
On the other hand, if $V$ is a normed linear space and yo... | {
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"url": "https://math.stackexchange.com/questions/179367",
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"source": "stackexchange",
"question_score": "12",
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Enumerate certain special configurations - combinatorics. Consider the vertices of a regular n-gon, numbered 1 through n. (Only the vertices, not the sides).
A "configuration" means some of these vertices are joined by edges.
A "good" configuration is one with the following properties:
1) There is at least one edge.
2)... | We can show that vertex degrees $k \le 2$. Suppose for contradiction that $n$ is the size of a minimal counterexample, a convex $n$-gon with some degree $k \gt 2$. By minimality (discarding any vertices not connected to the given one) all vertices in the $n$-gon have degree $k$.
But it is known that the maximum numbe... | {
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Showing the sum of orthogonal projections with orthogonal ranges is also an orthogonal projection
Show that if $P$ and $Q$ are two orthogonal projections with orthogonal ranges, then $P+Q$ is also an orthogonal projection.
First I need to show $(P+Q)^\ast = P+Q$. I am thinking that since
\begin{align*}
((P+Q)^\ast f ... | To complete your proof we need the following observations.
If $\langle f,g\rangle=0$ for all $g\in H$, then $f=0$. Indeed, take $g=f$, then you get $\langle f,g\rangle=0$. By definition of inner product this implies $f=0$.
Since $\mathrm{Im}(P)\perp\mathrm{Im}(Q)$, then for all $f,g\in H$ we have $\langle Pf,Qg\rangle... | {
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"source": "stackexchange",
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Trigonometry proof involving sum difference and product formula How would I solve the following trig problem.
$$\cos^5x = \frac{1}{16} \left( 10 \cos x + 5 \cos 3x + \cos 5x \right)$$
I am not sure what to really I know it involves the sum and difference identity but I know not what to do.
| $$\require{cancel}
\frac1{16} [ 5(\cos 3x+\cos x)+\cos 5x+5\cos x ]\\
=\frac1{16}[10\cos x \cos 2x+ \cos 5x +5 \cos x]\\
=\frac1{16} [5\cos x(2\cos 2x+1)+\cos 5x]\\
=\frac1{16} [5\cos x(2(2\cos^2 x-1)+1)+\cos 5x]\\
=\frac1{16} [5\cos x(4\cos^2 x-1)+\cos 5x]\\
=\frac1{16} [5\cos x(4\cos^2 x-1)+\cos 5x]\\
=\frac1{16} [20... | {
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A Tri-Factorable Positive integer Found this problem in my SAT book the other day and wanted to see if anyone could help me out.
A positive integer is said to be "tri-factorable" if it is the product of three consecutive integers. How many positive integers less than 1,000 are tri-factorable?
| HINT:
$1,000= 10*10*10<10*11*12$ so in the product $n*(n+1)*(n+2)$, $n$ must be less then $10.$
| {
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Find the domain of $f(x)=\frac{3x+1}{\sqrt{x^2+x-2}}$
Find the domain of $f(x)=\dfrac{3x+1}{\sqrt{x^2+x-2}}$
This is my work so far:
$$\dfrac{3x+1}{\sqrt{x^2+x-2}}\cdot \sqrt{\dfrac{x^2+x-2}{x^2+x-2}}$$
$$\dfrac{(3x+1)(\sqrt{x^2+x-2})}{x^2+x-2}$$
$(3x+1)(\sqrt{x^2+x-2})$ = $\alpha$ (Just because it's too much to type... | Note that $x^2+x-2=(x-1)(x+2)$. There is a problem only if $(x-1)(x+2)$ is $0$ or negative. (If it is $0$, we have a division by $0$ issue, and if it is negative we have a square root of a negative issue.)
Can you find where $(x-1)(x+2)$ is $0$? Can you find where it is negative? Together, these are the numbers which ... | {
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Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an epsilon number Firstly, I give the definition of the epsilon number:
$\alpha$ is called an epsilon number iff $\omega^\alpha=\alpha$.
Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an epsilon number and there are $\kappa$ epsil... | The following is intended as a half-outline/half-solution.
We will prove by induction that every uncountable cardinal $\kappa$ is an $\epsilon$-number, and that the family $E_\kappa = \{ \alpha < \kappa : \omega^\alpha = \alpha \}$ has cardinality $\kappa$.
Suppose that $\kappa$ is an uncountable cardinal such that the... | {
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Two sums with Fibonacci numbers
*
*Find closed form formula for sum: $\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{n} \frac{F_{2k}F_{n-k}}{10^n}$
*Find closed form formula for sum: $\displaystyle\sum_{k=0}^{n}\frac{F_k}{2^k}$ and its limit with $n\to +\infty$.
First association with both problems: generating... | For (2) you have $F_k = \dfrac{\varphi^k}{\sqrt 5}-\dfrac{\psi^k}{\sqrt 5}$ where $\varphi = \frac{1 + \sqrt{5}}{2} $ and $\psi = \frac{1 - \sqrt{5}}{2}$ so the problem becomes the difference between two geometric series.
For (1) I think you can turn this into something like $\displaystyle \sum_{n=0}^{\infty} \frac{F_... | {
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Goldbach's conjecture and number of ways in which an even number can be expressed as a sum of two primes Is there a functon that counts the number of ways in which an even number can be expressed as a sum of two primes?
| See Goldbach's comet at Wikipedia.
EDIT: To expand on this a little, let $g(n)$ be the number of ways of expressing the even number $n$ as a sum of two primes. Wikipedia gives a heuristic argument for $g(n)$ to be approximately $2n/(\log n)^2$ for large $n$. Then it points out a flaw with the heuristic, and explains h... | {
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What is the name of the logical puzzle, where one always lies and another always tells the truth? So i was solving exercises in propositional logic lately and stumbled upon a puzzle, that goes like this:
Each inhabitant of a remote village always tells the truth or always lies. A villager will only give a "Yes" or a "... | Knights and Knaves?
How: read about it.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Surfaces of constant projected area Generalizing the well-known variety of plane curves of constant width, I'm wondering about three-dimensional surfaces of constant projected area.
Question: If $A$ is a (bounded) subset of $\mathbb R^3$, homeomorphic to a closed ball, such that the orthogonal projection of $A$ onto a ... | These are called bodies of constant brightness. A convex body that has both constant width and constant brightness is a Euclidean ball. But non-spherical convex bodies of constant brightness do exist; the first was found by Blaschke in 1916. See: Google and related MSE thread.
| {
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Approximating $\pi$ with least digits Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as penalty.
For example the well known $\frac{355}{113}$ is an approximation, but it gives... | Let me throw in Clive's suggestion to look at the wikipedia site. If we allow for logarithm (while not using complex numbers), we can get 30 digits of $\pi$ with
$\frac{\operatorname{ln}(640320^3+744)}{\sqrt{163}}$
which is 13 digits and 5 operation, giving a ratio of about 18/30=0.6.
EDIT: Here is another one I found ... | {
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Alternating sum of squares of binomial coefficients I know that the sum of squares of binomial coefficients is just ${2n}\choose{n}$ but what is the closed expression for the sum ${n\choose 0}^2 - {n\choose 1}^2 + {n\choose 2}^2 + \cdots + (-1)^n {n\choose n}^2$?
| Here's a combinatorial proof.
Since $\binom{n}{k} = \binom{n}{n-k}$, we can rewrite the sum as $\sum_{k=0}^n \binom{n}{k} \binom{n}{n-k} (-1)^k$. Then $\binom{n}{k} \binom{n}{n-k}$ can be thought of as counting ordered pairs $(A,B)$, each of which is a subset of $\{1, 2, \ldots, n\}$, such that $|A| = k$ and $|B| = n... | {
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Periodic solution of differential equation let be the ODE $ -y''(x)+f(x)y(x)=0 $
if the function $ f(x+T)=f(x) $ is PERIODIC does it mean that the ODE has only periodic solutions ?
if all the solutions are periodic , then can all be determined by Fourier series ??
| No, it doesn't mean that. For instance, $f(x)=0$ is periodic with any period, but $y''(x)=0$ has non-periodic solutions $y(x)=ax+b$.
| {
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Transforming an inhomogeneous Markov chain to a homogeneous one I fail to understand Cinlar's transformation of an inhomogeneous Markov chain to a homogeneous one. It appears to me that $\hat{P}$ is not fully specified. Generally speaking, given a $\sigma$-algebra $\mathcal A$, a measure can be specified either explici... | We give the condition that $\widehat P$ is a Markow kernel, and we have that
$$\widehat P((n,x),\{n+1\}\times E)=P_{n+1}(x,E)=1,$$
hence the measure $\widehat P((n,x),\cdot)$ is concentrated on $\{n+1\}\times E\}$? Therefore, we have $\widehat P((n,x),I\times A)=0$ for any $A\subset E$ and $I\subset \Bbb N$ which does... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving linear system of equations when one variable cancels I have the following linear system of equations with two unknown variables $x$ and $y$. There are two equations and two unknowns. However, when the second equation is solved for $y$ and substituted into the first equation, the $x$ cancels. Is there a way o... | $$\begin{equation*}
\left\{
\begin{array}{c}
2.6513=\frac{3}{2}y+\frac{x}{2} \\
1.7675=y+\frac{x}{3}
\end{array}
\right.
\end{equation*}$$
If we multiply the first equation by $2$ and the second by $3$ we get
$$\begin{equation*}
\left\{
\begin{array}{c}
5.3026=3y+x \\
5.3025=3y+x
\end{array}
\right.
\end{equation... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculate Rotation Matrix to align Vector $A$ to Vector $B$ in $3D$? I have one triangle in $3D$ space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current and previous $3D$ vertex positions of the triang... | General solution for n dimensions in matlab / octave:
%% Build input data
n = 4;
a = randn(n,1);
b = randn(n,1);
%% Compute Q = rotation matrix
A = a*b';
[V,D] = eig(A'+A);
[~,idx] = min(diag(D));
v = V(:,idx);
Q = eye(n) - 2*(v*v');
%% Validate Q is correct
b_hat = Q'*a*norm(b)/norm(a);
disp(['norm of error = ' nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "181",
"answer_count": 19,
"answer_id": 11
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Logical question problem A boy is half as old as the girl will be when the boy’s age is twice the sum of their ages when the boy was the girl’s age.
How many times older than the girl is the boy at their present age?
This is a logical problem sum.
| If $x$ is the boy's age and $y$ is the girl's age, then when the boy was the girl's current age, her age was $2y-x$. So "twice the sum of their ages when the boy was the girl's age" is $2(3y-x)=6y-2x$. The boy will reach this age after a further $6y-3x$ years, at which point the girl will be $7y-3x$. We are told tha... | {
"language": "en",
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"source": "stackexchange",
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Easy way to find roots of the form $qi$ of a polynomial Let $p$ be a polynomial over $\mathbb{Z}$, we know that there is an easy way to check if $p$ have rational roots (using the rational root theorem).
Is there an easy way to check if $p$ have any roots of the form $qi$ where $q\in\mathbb{Q}$ (or at least $q\in\mathb... | Hint $\ f(q\,i) = a_0\! -\! a_2 q^2\! +\! a_4 q^4\! +\cdots + i\,q\,(a_1\! -\! a_3 q^2\! +\! a_5 q^4\! +\! \cdots) = g(q) + i\,q\,h(q)$
| {
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Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$ I'm having trouble computing the integral:
$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$
I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as $u=\sin(x)$ and $u=\cos(x)$, but it was not very effective.
An... | Hint: $\sqrt{2}\sin(x+\pi/4)=\sin x +\cos x$, then substitute $x+\pi/4=z$
| {
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"source": "stackexchange",
"question_score": "88",
"answer_count": 8,
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Comparing speed in stochastic processes generated from simulation? I have an agent-based simulation that generates a time series in its output for my different treatments. I am measuring performance through time, and at each time tick the performance is the mean of 30 runs (30 samples). In all of the treatments the per... | Assuming you're using a pre-canned application, then there will be an underlying distribution generating your time series. I would look in the help file of the application to find this distribution.
Once you know the underlying distribution, then "significance" is determined the usual way, namely pick a confidence lev... | {
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Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$
Possible Duplicate:
Relation of this antisymmetric matrix $r = \!\left(\begin{smallmatrix}0 &1\\-1 & 0\end{smallmatrix}\right)$ to $i$
On Wikipedia, it says that:
Matrix representation of co... | Since you put the tag quaternions, let me say a bit more about performing identifications like that:
Recall the quaternions $\mathcal{Q}$ is the group consisting of elements $\{\pm1, \pm \hat{i}, \pm \hat{j}, \pm \hat{k}\}$ equipped with multiplication that satisfies the rules according to the diagram
$$\hat{i} \right... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Is failing to admit an axiom equivalent to proof when the axiom is false? Often, mathematicians wish to develop proofs without admitting certain axioms (e.g. the axiom of choice).
If a statement can be proven without admitting that axiom, does that mean the statement is also true when the axiom is considered to be fals... | Yes. Let the axiom be P. The proof that didn't make use of P followed all the rules of logic, so it still holds when you adjoin $\neg P$ to the list of axioms. (It could also happen that the other axioms sufficed to prove P, in which case the system that included $\neg P$ would be inconsistent. In an inconsistent theor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Equivalence of norms on the space of smooth functions Let $E, F$ be Banach spaces, $A$ be an open set in $E$ and $C^2(A,F)$ be the space of all functions $f:A\to F,$ which are twice continuously differentiable and bounded with all derivatives. The question is when following two norms in $C^2(A,F)$ are equivalent:
$$
\|... | We can reduce to the case $F=\mathbb R$ by considering all compositions $\varphi\circ f$ with $\varphi$ ranging over unit-norm functionals on $F$.
Let $A$ be the open unit ball in $E$. Given $x\in A$ and direction $v\in E$ (a unit vector), we would like to estimate the directional derivative $f_v(x)$ in terms of $M=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Worst case analysis of MAX-HEAPIFY procedure . From CLRS book for MAX-HEAPIFY procedure :
The children's subtrees each have size at most 2n/3 - the worst case
occurs when the last row of the tree is exactly half full
I fail to see this intuition for the worst case scenario . Can some one explain possibly with a dia... | Start with a heap $H$ with $n$ levels with all levels full. That's $2^{i - 1}$ nodes for each level $i$ for a total of $$|H| = 2^n - 1$$ nodes in the heap. Let $L$ denote the left sub-heap of the root and $R$ denote the right sub-heap. $L$ has a total of $$|L| = 2^{n - 1} - 1$$ nodes, as does $R$. Since a binary heap ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 3
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Singular-value inequalities This is my question: Is the following statement true ?
Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators.
For every $n\in\mathbb{N}$ the following inequality holds:
$$\sum_{j=1}^n s_j(RS) \leq \sum_{j=1}^n s_j(R)s_j(S)$$
Note: $s_j(R)$ denotes the j-th singular va... | The statement is true. It is a special case of a result by Horn (On the singular values of a product of completely continuous operators, Proc. Nat.Acad. Sci. USA 36 (1950) 374-375).
The result is the following. Let $f:[0,\infty)\rightarrow \mathbb{R}$ with $f(0)=0$. If $f$ becomes convex following the substitution $x=e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Are there diagonalisable endomorphisms which are not unitarily diagonalisable? I know that normal endomorphisms are unitarily diagonalisable. Now I'm wondering, are there any diagonalisable endomorphisms which are not unitarily diagonalisable?
If so, could you provide an example?
| Another way to look at it, though no really different in essence, is to consider the operator norm on ${\rm M}_{n}(\mathbb{C})$ induced by the Euclidean norm on $\mathbb{C}^{n}$ (thought of as column vectors). Hence $\|A \| = {\rm max}_{ v : \|v \| = 1} \|Av \|.$ Since the unitary transformations are precisely the isom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Is it true, that $H^1(X,\mathcal{K}_{x_1,x_2})=0$? - The cohomology of the complex curve with a coefficient of the shaeaf of meromorphic functions... Let X be complex curve (complex manifold and $\dim X=1$).
For $x_1,x_2\in X$ we define the sheaf $\mathcal{K}_{x_1,x_2}$(in complex topology) of meromorphic functions van... | The answer is yes for a non-compact Riemann surface $H^1(X, \mathcal K_{x_1,x_2})=0$ .
The key is the exact sequence of sheaves on $X$:$$0\to \mathcal K_{x_1,x_2} \to \mathcal K \xrightarrow {truncate } \mathcal Q_1\oplus \mathcal Q_2\to 0$$ where $\mathcal Q_i$ is the sky-scraper sheaf at $x_i$ with fiber the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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For a Turing machine and input $w$- Is "Does M stop or visit the same configuration twice" a decidable question? I have the following question out of an old exam that I'm solving:
Input: a Turing machine and input w
Question: Does on running of M on w, at least one of the following things happen
-M stops of w... | We can modify a Turing machine $T$ by replacing every computation step of $T$ by a procedure in which we go to the left end of the (used portion of) the tape, and one more step left, print a special symbol $U$, and then hustle back to do the intended step. So in the modified machine, a configuration never repeats. Thus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A field without a canonical square root of $-1$ The following is a question I've been pondering for a while. I was reminded of it by a recent dicussion on the question How to tell $i$ from $-i$?
Can you find a field that is abstractly isomorphic to $\mathbb{C}$, but that does not have a canonical choice of square roo... | You can model the complex numbers by linear combinations of the $2\times 2$ unit matrix $\mathbb{I}$ and a real $2\times 2$ skew-symmetric matrix with square $-\mathbb I$, of which there are two, $\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}$ and $\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$. I see no obvious reason to pref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Prove that given any $2$ vertices $v_0,v_1$ of Graph $G$ that is a club, there is a path of length at most $2$ starting in $v_0$ and ending in $v_1$ Definition of a club: Let $G$ be a graph with $n$ vertices where $n > 2$. We call the graph $G$ a club if for all pairs of distinct vertices $u$ and $v$ not connected by... | The statement is not true. Consider a path of length 4, where $v_0,v_1$ are the endpoints of the path. There is no path of length at most two from $v_0$ to $v_1$, and the graph is a club by your definition.
Edit: After the definition of club changed
With the new definition, the proof can be made as follows:
Let $G$ be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Express Expectation and Variance in other terms. Let $X \sim N(\mu,\sigma^2)$ and
$$f_X(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$
where $-\infty < x < \infty$.
Express $\operatorname{E}(aX + b)$ and $\operatorname{Var}(aX +b)$ in terms of $\mu$, $\sigma$, $a$ and $b$, where $a$ and $b$ are re... | Not an answer:
Check out Wikipedia, and then learn them through comprehension and by heart.
*
*Normal Distribution (E, $\sigma$ included)
*What is Variance
*Important Properties of Variance
*Important Properties of Expected Value
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What function $f$ such that $a_1 \oplus\, \cdots\,\oplus a_n = 0$ implies $f(a_1) \oplus\, \cdots\,\oplus f(a_n) \neq 0$ For a certain algorithm, I need a function $f$ on integers such that
$a_1 \oplus a_2 \oplus \, \cdots\,\oplus a_n = 0 \implies f(a_1) \oplus f(a_2) \oplus \, \cdots\,\oplus f(a_n) \neq 0$
(where the ... | The function $f$, if it exists, must have very large outputs.
Call a set of integers "closed" if it is closed under the operation $\oplus$. A good example of a closed set of integers is the set of positive integers smaller than $2^k$ for some $k$.
Let $S$ be a closed set of integers that form the domain of $f$. Take as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Count permutations. Hi I have a compinatorial exercise:
Let $s \in S_n$. Count the permutations such that
$$s(1)=1$$ and
$$|s(i+1)-s(i)|\leq 2 \,\, \mathrm{for} \, \, i\in\{1,2, \ldots , n-1 \}$$
Thank you!
| This is OEIS A038718 at the On-Line Encyclopedia of Integer Sequences. The entry gives the generating function
$$g(x)=\frac{x^2-x+1}{x^4-x^3+x^2-2x+1}$$
and the recurrence $a(n) = a(n-1) + a(n-3) + 1$, where clearly we must have initial values $a(0)=0$ and $a(1)=a(2)=1$.
Added: I got this by calculating $a(1)$ through... | {
"language": "en",
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Proving that the number of vertices of odd degree in any graph G is even I'm having a bit of a trouble with the below question
Given $G$ is an undirected graph, the degree of a vertex $v$, denoted by $\mathrm{deg}(v)$, in graph $G$ is the number of neighbors of $v$.
Prove that the number of vertices of odd degree in... | Simply, sum of even numbers of odd number is an even number (always odd+odd=even and even+odd=odd and even+even=even). As the sum of degree of vertices needs to be even number, number of such vertices must be even. Which @Mike has presented very succinctly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/181833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 8,
"answer_id": 7
} |
A tricky but silly doubt regarding the solutions of $x^2/(y-1)^2=1$ Motivation :
I have been confused with some degree 2 equation. I suddenly came across a simple equation and couldn't get the quintessence behind that.
I have an equation $$\dfrac{x^2}{(y-1)^2}=1 \tag{1}$$ and I was looking for its solutions. It was a... | The equations $$x^2=(y-1)^2\tag{1}$$ and $$\frac{x^2}{(y-1)^2}=1\tag{2}$$ do not have the same solution set. Every solution of $(2)$ is a solution of $(1)$, but $\langle 0,1\rangle$ is a solution of $(1)$ that is not a solution of $(2)$, because $\frac00$ is undefined.
The reason is that $(1)$ does not imply $(2)$. Not... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Complete course of self-study I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study mathematics.
I request people to tell me what topics an undergraduate may/must study and the book... | I would suggest some mathematical modeling or other practical application of mathematics. Also Finite automata and graph-theory is interesting as it is further away from "pure math" as I see it, it has given me another perspective of math.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/181984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "109",
"answer_count": 23,
"answer_id": 21
} |
Simple Limit Divergence I am working with a definition of a divergent limit as follows:
A sequence $\{a_n\}$ diverges to $-\infty$ if, given any number $M$, there is an $N$ so that $n \ge N$ implies that $a_n \le M$.
The sequence I am considering is $a_n = -n^2$, which I thought would be pretty simple, but I keep ru... | If $M>0$ inequality $n^2\geq -M$ always holds. So you can take any $N$ you want.
If $M\leq 0$, then $n\geq\sqrt{-M}$, and you can take $N=\lfloor\sqrt{-M}\rfloor +1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/182060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $R$ is a ring s.t. $(R,+)$ is finitely generated and $P$ is a maximal ideal then $R/P$ is a finite field
Let $R$ be a commutative unitary ring and suppose that the abelian
group $(R,+)$ is finitely generated. Let's also $P$ be a maximal ideal
of $R$.
Then $R/P$ is a finite field.
Well, the fact that the quoti... | As abelian groups, both $\,R\,,\,P\,$ are f.g. and thus the abelian group $\,R/P\,$ is f.g....but this is also a field so if it had an element of additive infinite order then it'd contain an isomorphic copy of $\,\Bbb Z\,$ and thus also of $\,\Bbb Q\,$, which of course is impossible as the last one is not a f.g. abelia... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Frechet Differentiabilty of a Functional defined on some Sobolev Space How can I prove that the following Functional is Frechet Differentiable and that the Frechet derivative is continuous?
$$
I(u)=\int_\Omega |u|^{p+1} dx , \quad 1<p<\frac{n+2}{n-2}
$$
where $\Omega$ is a bounded open subset of $\mathbb{R}^n$... | As was given in the comments, the Gâteaux derivative is
$$
I'(u)\psi = (p+1) \int_\Omega |u|^{p-1}u\psi.
$$
It is clearly linear, and bounded on $L^{p+1}$ since
$$
|I'(u)\psi| \leq (p+1) \|u\|_{p+1}^p\|\psi\|_{p+1},
$$
by the Hölder inequality with the exponents $\frac{p+1}p$ and $p+1$.
Here $\|\cdot\|_{q}$ denotes th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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One step subgroup test help
Possible Duplicate:
Basic Subgroup Conditions
could someone please explain how the one step subgroup test works,
I know its important and everything but I do not know how to apply it as well as with the two step subgroup.
If someone could also give some examples with it it would be reall... | Rather than prove that the "one step subgroup test" and the "two step subgroup test" are equivalent (which the links in the comments do very well), I thought I would "show it in action".
Suppose we want to show that $2\Bbb Z = \{k \in \Bbb Z: k = 2m, \text{for some }m \in \Bbb Z\}$ is a subgroup of $\Bbb Z$ under addit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How is a system of axioms different from a system of beliefs? Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
| There are similarities about how people obtain beliefs on different matters. However it is hardly a blind faith.
There are rules that guide mathematicians in choosing axioms. There has always been discussions about whether an axiom is really true or not. For example, not long ago, mathematicians were discussing whether... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "101",
"answer_count": 10,
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A non-square matrix with orthonormal columns I know these 2 statements to be true:
1) An $n$ x $n$ matrix U has orthonormal columns iff. $U^TU=I=UU^T$.
2) An $m$ x $n$ matrix U has orthonormal columns iff. $U^TU=I$.
But can (2) be generalised to become "An $m$ x $n$ matrix U has orthonormal columns iff. $U^TU=I=UU^T$" ... | The $(i,j)$ entry of $U^T U$ is the dot product of the $i$'th and $j$'th columns of $U$, so
the matrix has orthonormal columns if and only if $U^T U = I$ (the $n \times n$ identity matrix, that is). If $U$ is $m \times n$, this requires $m \ge n$, because the rank of $U^T U$ is at most $\min(m,n)$. On the other hand... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Proving that $2^{2^n} + 5$ is always composite by working modulo $3$ By working modulo 3, prove that $2^{2^n} + 5$ is always composite for every positive integer n.
No need for a formal proof by induction, just the basic idea will be great.
| Obviously $2^2 \equiv 1 \pmod 3$.
If you take the above congruence to the power of $k$ you get
$$(2^2)^k=2^{2k} \equiv 1^k=1 \pmod 3$$
which means that $2$ raised to any even power is congruent to $1$ modulo $3$.
What can you say about $2^{2k}+5$ then modulo 3?
It is good to keep in mind that you can take powers of con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Equivalence of a Lebesgue Integrable function I have the following question:
Let $X$: $\mu(X)<\infty$, and let $f \geq 0$ on $X$. Prove that $f$ is Lebesgue integrable on $X$ if and only if $\sum_{n=0}^{\infty}2^n \mu(\lbrace x \in X : f(x) \geq 2^n \rbrace) < \infty $.
I have the following ideas, but am a little unsu... | $$\frac12\left(1+\sum_{n=0}^{+\infty}2^n\,\mathbf 1_{f\geqslant2^n}\right)\leqslant f\lt1+\sum_{n=0}^{+\infty}2^n\,\mathbf 1_{f\geqslant2^n}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/182527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 0
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Prime factorization, Composite integers. Describe how to find a prime factor of 1742399 using at most 441 integer divisions and one square root.
So far I have only square rooted 1742399 to get 1319.9996. I have also tried to find a prime number that divides 1742399 exactly; I have tried up to 71 but had no luck. Surely... | Note that the problem asks you do describe how you would go about factorizing 1742399 in at most 442 operations. You are not being asked to carry out all these operations yourself! I think your method of checking all primes up to the squareroot is exactly what the problem is looking for, but to be safe you should check... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is it about modern set theory that prevents us from defining the set of all sets which are not members of themselves? We can clearly define a set of sets. I feel intuitively like we ought to define sets which do contain themselves; the set of all sets which contain sets as elements, for instance. Does that set pro... | crf wrote:
"I understand that all sets are classes, but that there exist classes
which are not sets, and this apparently resolves Russell's
paradox...."
You don't need classes to resolve Russell's paradox. The key is that, for any formula P, you cannot automatically assume the existence of $\{x | P(x)\}$. If $P(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
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About the sequence satisfying $a_n=a_{n-1}a_{n+1}-1$ "Consider sequences of positive real numbers of the form x,2000,y,..., in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of x does the term 2001 appear somewhere in the sequence?
(A) 1 (B) 2 (... | (This is basically EuYu's answer with the details of periodicity added; took a while to type up.)
Suppose that $a_0 , a_1 , \ldots$ is a generalised sequence of the type described, so that $a_i = a_{i-1} a_{i+1} - 1$ for all $i > 0$. Note that this condition is equivalent to demanding that $$a_{i+1} = \frac{ a_i + 1 }... | {
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"timestamp": "2023-03-29T00:00:00",
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Find all linearly dependent subsets of this set of vectors I have vectors in such form
(1 1 1 0 1 0)
(0 0 1 0 0 0)
(1 0 0 0 0 0)
(0 0 0 1 0 0)
(1 1 0 0 1 0)
(0 0 1 1 0 0)
(1 0 1 1 0 0)
I need to find all linear dependent subsets over $Z_2$.
For example 1,2,5 and 3,6,7.
EDIT (after @rschwieb)
The answer for presen... | Let us denote with $M$ (the transpose) of your matrix,
$$M= \begin{pmatrix}
1 & 0 & 1 & 0 & 1 & 0 & 1 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}.$$
As rschwieb already noted, a vector $v$ with $Mv=0... | {
"language": "en",
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"source": "stackexchange",
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Intuition on proof of Cauchy Schwarz inequality To prove Cauchy Schwarz inequality for two vectors $x$ and $y$ we take the inner product of $w$ and $w$ where $w=y-kx$ where $k=\frac{(x,y)}{|x|^2}$ ($(x,y)$ is the inner product of $x$ and $y$) and use the fact that $(w,w) \ge0$ . I want to know the intuition behind this... | My favorite proof is inspired by Axler and uses the Pythagorean theorem (that $\|v+w\|^2 =\|v\|^2+\|w\|^2$ when $(v,w)=0$). It motivates the choice of $k$ as the component of $y$ in an orthogonal decomposition (i.e., $kx$ is the projection of $y$ onto the space spanned by $x$ using the decomposition $\langle x\rangle\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Two problems with prime numbers
Problem 1. Prove that there exists $n\in\mathbb{N}$ such that in interval $(n^2, \ (n+1)^2)$ there are at least $1000$ prime numbers.
Problem 2. Let $s_n=p_1+p_2+...+p_n$ where $p_i$ is the $i$-th prime number. Prove that for every $n$, there exists $k\in\mathbb{N}$ such that $s_n<k^2<s... | Problem 2:
For any positive real $x$, there is a square between $x$ and $x+2\sqrt{x}+2$. Therefore it will suffice to show that $p_{n+1}\geq 2\sqrt{s_n}+2$. We have $s_{n}\leq np_n$ and $p_{n+1}\geq p_n+2$, so we just need to show $p_n\geq 2\sqrt{np_n}$, i.e., $p_n\geq 4n$. That this holds for all sufficiently large ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Attaching a topological space to another I'm self-studying Mendelson's Introduction to Topology. There is an example in the identification topology section that I cannot understand:
Let $X$ and $Y$ be topological spaces and let $A$ be a non-empty closed subset of $X$. Assume that $X$ and $Y$ are disjoint and that a co... | I think it's good that you ask this question, plus one. Your intuition will eventually develop, don't worry. I had trouble understanding identification topologies too when I saw them first. It just takes some time to get used to, don't worry. The way I think about it now, is as follows:
You have two spaces $X,Y$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Deducing formula for a linear transformation The question I'm answering is as follows:
Let $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear transformation such that $ T(1,1) = (2,1) $ and $ T(0,2) = (2,8) $. Find a formula for $ T(a,b) $ where $ (a,b) \in \mathbb{R}^2 $.
Earlier we proved that $\{(1,1), (0,2)\}... | We have
$$(1,0)=(1,1)-\frac{1}{2}(0,2)\qquad\text{and} \qquad(0,1)=\frac{1}{2}(0,2).\tag{$1$}$$
Note that $(a,b)=a(1,0)+b(0,1)$. So
$$T(a,b)=aT(1,0)+bT(0,1).$$
Now use the values of $T(1,1)$ and $T(0,2)$, and Equations $(1)$, to find $T(1,0)$ and $T(0,1)$, and simplify a bit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/183077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Explicitness in numeral system
Prove that for every $a\in\mathbb{N}$ there is one and only one way to express it in the system with base $\mathbb{N}\ni s>1$.
Seems classical, but I don't have any specific argument.
| We can express a natural number $a$ to any base $s$ by writing $k = \lceil \log_s a \rceil$, the number of digits we'll need, $a_k=\max(\{n\in \mathbb{N}: ns^k\leq a\}),$ and recursively
$a_{i}=\max(\{n\in \mathbb{N}: ns^i \leq a-\sum_{j=i+1}^k a_j s^j\})$. It's immediate that each of these maxes exists, since $0s^i\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Stabilizer of a point and orbit of a point I really need help with this topic I have an exam tomorrow and am trying to get this stuff in my head. But the book is not explaining me these two topics properly.
It gives me the definition of a stabilizer at a point where $\mathrm {Stab}_G (i) = \{\phi \in G \mid \phi(i) = i... | In simple terms,
Stabilizer of a point is that permutation in the group which does not change the given point
=> for stab(1) = (1), (78)
Orbit of a point(say 1) are those points that follow given point(1) in the permutations of the group.
=>orbit(1) = 1 for (1); 3 for (132)...; 2 for (123)...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/183190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Sum of a stochastic process I have a question regarding the distribution of the sum of a discrete-time stochastic process. That is, if the stochastic process is $(X_1,X_2,X_3,X_4,\ldots)$, what is the distribution of $X_1+X_2+X_3+\ldots$? $X_i$ could be assumed from a discrete or continuous set, whatever is easier to c... |
Are there any types of stochastic processes, where the distribution of the sum can be computed numerically or even be given as a closed-form expression?
As stated, the problem is quite equivalent to compute the distribution of the sum of an arbritary set of random variables. Little can be said in general, as the fact... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Multiplicative group of integers modulo n definition issues
It is easy to verify that the set $(\mathbb{Z}/n\mathbb{Z})^\times$ is closed under multiplication in the sense that $a, b ∈ (\mathbb{Z}/n\mathbb{Z})^\times$ implies $ab ∈ (\mathbb{Z}/n\mathbb{Z})^\times$, and is closed under inverses in the sense that $a ∈ (... | Well $2\cdot 3\equiv 1\; \text{mod}\ 5$, so $2$ and $3$ are multiplicative inverses $\text{ mod } 5$.
How to find the inverse of a number modulo a prime number was the topic of one of my previous answers. Modulo a composite number, inverses don't always exist.
See Calculating the Modular Multiplicative Inverse without... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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How to find the maxium number of edge-disjoint paths using flow network Given a graph $G=(V,E)$ and $2$ vertices $s,t \in V$, how can I find the maximum number of edge-disjoint paths from $s$ to $t$ using a flow network? $2$ paths are edge disjoint if they don't have any common edge, though they may share some common v... | Hint: if each edge has a capacity of one unit, different units of stuff flowing from $s$ to $t$ must go on edge-disjoint paths.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding a basis for the solution space of a system of Diophantine equations Let $m$, $n$, and $q$ be positive integers, with $m \ge n$.
Let $\mathbf{A} \in \mathbb{Z}^{n \times m}_q$ be a matrix.
Consider the following set:
$S = \big\{ \mathbf{y} \in \mathbb{Z}^m \mid \mathbf{Ay} \equiv \mathbf{0} \pmod q \big\}$.
It ... | Even over a field, a fair amount goes into this. Here are two pages from Linear Algebra and Matrix Theory by Evar D. Nering, second edition:
From a row-echelon form for your data matrix $A,$ one can readily find the null space as a certain number of columns by placing $1$'s in certain "free" positions and back-substit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Can you construct a field with 6 elements?
Possible Duplicate:
Is there anything like GF(6)?
Could someone tell me if you can build a field with 6 elements.
| If such a field $F$ exists, then the multiplicative group $F^\times$ is cyclic of order 5. So let $a$ be a generator for this group and write
$F = \{ 0, 1, a, a^2, a^3, a^4\}$.
From $a(1 + a + a^2 + a^3 + a^4) = 1 + a + a^2 + a^3 + a^4$, it immediately follows that
$1 + a + a^2 + a^3 + a^4 = 0$. Let's call this (*).
Si... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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"answer_id": 2
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can one derive the $n^{th}$ term for the series, $u_{n+1}=2u_{n}+1$,$u_{0}=0$, $n$ is a non-negative integer derive the $n^{th}$ term for the series $0,1,3,7,15,31,63,127,255,\ldots$
observation gives, $t_{n}=2^n-1$, where $n$ is a non-negative integer
$t_{0}=0$
| The following is a semi-formal variant of induction that is particularly useful for recurrences.
Let $x_n=2^n-1$. It is easy to verify that $x_0=0$. It is also easy to verify that
$$x_{n+1}=2x_n+1,$$
since $2^{n+1}-1=2(2^n-1)+1$.
So the sequence $(x_n)$ starts in the same way as your sequence and obeys the same recurr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Is $ \ln |f| $ harmonic? I'd like to show that $\ln |f| $ is harmonic, where $f$ is holomorphic defined on a domain of the complex plane and never takes the value 0.
My idea was to use the fact that $\ln |f(z)| = \operatorname{Log} f(z) - i*\operatorname{Arg}(f(z)) $, but $Log$ is only holomorphic on some part of the c... | This is a local result; you need to show that given a $z_0$ with $f(z_0) \neq 0$ there is a neighborhood of $z_0$ on which $\ln|f(z)|$ is harmonic.
Fix $z_0$ with $f(z_0) \neq 0$. Let $\log(z)$ denote an analytic branch of the logarithm defined on a neighborhood of $f(z_0)$. Then the real part of $\log(z)$ is $\ln|z|$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Does there exist a nicer form for $\beta(x + a, y + b) / \beta(a, b)$? I have the expression
$$\displaystyle\frac{\beta(x + a, y + b)}{\beta(a, b)}$$
where $\beta(a_1,a_2) = \displaystyle\frac{\Gamma(a_1)\Gamma(a_2)}{\Gamma(a_1+a_2)}$.
I have a feeling this should have a closed-form which is intuitive and makes less he... | $$
\beta(1+a,b) = \frac{\Gamma(1+a)\Gamma(b)}{\Gamma(1+a+b)} = \frac{a\Gamma(a)\Gamma(b)}{(a+b)\Gamma(a+b)} = \frac{a}{a+b} \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} = \frac{a}{a+b} \beta(a,b).
$$
If you have, for example $\beta(5+a,8+b)$, just repeat this five times for the first argument and eight for the second:
$$
\fr... | {
"language": "en",
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Find a side of a triangle given other two sides and an angle I have a really simple-looking question, but I have no clue how I can go about solving it?
The question is
Find the exact value of $x$ in the following diagram:
Sorry for the silly/easy question, but I'm quite stuck! To use the cosine or sine rule, I'd ne... | Use the cosine rule with respect to the 60 degree angle. Then you get an equation involving $x$ as a variable, Then you solve the equation for $x$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question about applying Arzelà-Ascoli An example of an application of Arzelà-Ascoli is that we can use it to prove that the following operator is compact:
$$ T: C(X) \to C(Y), f \mapsto \int_X f(x) k(x,y)dx$$
where $f \in C(X), k \in C(X \times Y)$ and $X,Y$ are compact metric spaces.
To prove that $T$ is compact we ... | Following tb's comment:
Claim: If $\{f_n\}$ is equicontinuous and $f_n \to f$ uniformly then $\{f\} \cup \{f_n\}$ is equicontinuous.
Proof: Let $\varepsilon > 0$.
(i) Let $\delta^\prime$ be the delta that we get from equicontinuity of $\{f_n\}$ so that $d(x,y) < \delta^\prime$ implies $|f_n(x) - f_n(y)| < \varepsilon$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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direct sum of image and kernel in a infinitedimensional space Is it true that in an infinitdimensional Hilbert space the formula $$\text{im} S\oplus \ker S =H$$holds, where $S:H\rightarrow H$ ?
I know it is true for finitely many dimensions but I'm not so sure about infinitely many. Would it be true under some addition... | Consider $l_2$, and the translation operator $T:~e_n\mapsto e_{n+1}$, it's injective but not surjective. So that $ker~T=0,im~T\neq l_2,ker~T\oplus im~T\neq l_2$.
If $rank~im T$ is finite, i remember i have learned that the equality holds in some book. (but it's vague to me now, so take care)
You can try to consider the... | {
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How to find $\lim\limits_{x\to0}\frac{e^x-1-x}{x^2}$ without using l'Hopital's rule nor any series expansion? Is it possible to determine the limit
$$\lim_{x\to0}\frac{e^x-1-x}{x^2}$$
without using l'Hopital's rule nor any series expansion?
For example, suppose you are a student that has not studied derivative yet (and... | Define $f(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$. One possibility is to take $f(x)$ as the definition of $e^x$. Since the OP has suggested a different definition, I will show they agree.
If $x=\frac{p}{q}$ is rational, then
\begin{eqnarray*}
f(x)&=&\lim_{n\to\infty}\left(1+\frac{p}{qn}\right)^n\\
&=&\lim_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 7,
"answer_id": 1
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Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$? I would like to evaluate the sum
$$
\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)
$$
Where $\operatorname{Si}$ is the sine integral, defined as:
$$\operatorname{Si}(x) := \int_0^x \frac{\sin t}{t}\, dt$$... | We want (changing the sign and starting with $n=1$) :
$$\tag{1}S(0)= -\sum_{n=1}^\infty \left(\mathrm{Si}(n)-\frac{\pi}{2}\right)$$
Let's insert a 'regularization parameter' $\epsilon$ (small positive real $\epsilon$ taken at the limit $\to 0^+$ when needed) :
$$\tag{2} S(\epsilon) = \sum_{n=1}^\infty \int_n^\infty \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
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"answer_id": 2
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Trigonometry- why we need to relate to circles I'm a trigonometry teaching assistant this semester and have a perhaps basic question about the motivation for using the circle in the study of trigonometry. I certainly understand Pythagorean Theorem and all that (I would hope so if I'm a teaching assistant!) but am looki... | Suppose that instead of parametrizing the circle by arc length $\theta$, so that $(\cos\theta,\sin\theta)$ is a typical point on the circle, one parametrizes it thus:
$$
t\mapsto \left(\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2}\right)\text{ for }t\in\mathbb{R}\cup\{\infty\}. \tag{1}
$$
The parameter space is the one-point c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The base of a triangular prism is $ABC$. $A'B'C'$ is an equilateral triangle with lengths $a$... The base of a triangular prism is $ABC$. $A'B'C'$ is an equilateral triangle with lengths $a$, and the lengths of its adjacent sides also equal $a$. Let $I$ be the midpoint of $AB$ and $B'I \perp (ABC)$. Find the distance f... | Let $BB'$ be along $x$ axis and $BC$ be along $y$ axis ($B$ being the origin). Given that $B'I$ is perpendicular to $BA$, $\angle{ABC}$ will be $\pi/3$ (as $\Delta BIB'$ is a $(1,\sqrt{3},2)$ right-triangle). The co-ordinates of $A$ will then be of the form $\left(a\cos{\pi/3},a\cos{\pi/3},h\right)$. As the length of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Application of Radon Nikodym Theorem on Absolutely Continuous Measures I have the following problem:
Show $\beta \ll \eta$ if and only if for every $\epsilon > 0 $ there exists a $\delta>0$ such that $\eta(E)<\delta$ implies $\beta(E)<\epsilon$.
For the forward direction I had a proof, but it relied on the use of the ... | Assume that $\beta=h\eta$ with $h\geqslant0$ integrable with respect to $\eta$, in particular $\beta$ is a finite measure. Let $\varepsilon\gt0$.
There exists some finite $t_\varepsilon$ such that $\beta(B_\varepsilon)=\int_{B_\varepsilon} h\,\mathrm d\eta\leqslant\varepsilon$ where $B_\varepsilon=[h\geqslant t_\varep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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using central limit theorem I recently got a tute question which I don't know how to proceed with and I believe that the tutor won't provide solution... The question is
Pick a real number randomly (according to the uniform measure) in the interval $[0, 2]$. Do this one million times and let $S$ be the sum of all the ... | Let's call $S_n$ the sum of the first $n$ terms. Then for $0 \le x \le 1$ it can be shown by induction that $\Pr(S_n \le x) = \dfrac{x^n}{2^n \; n!}$
So the exact answers are
a) $1 - \dfrac{1}{2^{1000000} \times 1000000!}$
b) $1 - \dfrac{1}{2000^{1000000} \times 1000000!}$
c) $1$
The first two are extremely clo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that for this function the stated is true.
For the function
$$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$
show that
$$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$
Hey everyone, I'm very new to this kind of maths and would really appreciate any help. Hopefully I can get an idea from this and apply it to other... | Use the definition for the complex sine:
$$ \sin(z)=\frac{ e^{iz}-e^{-iz} } {2i} $$
Thus,
$$-\sqrt{2}ie^{i\frac{w}{2}}\sin\frac{w}{2} =-\sqrt{2}ie^{i\frac{w}{2}}(\frac{1}{2i}(e^{i\frac{w}{2}} - e^{-i\frac{w}{2}})) $$
Now simplify to get your result.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Safe use of generalized inverses Suppose I'm given a linear system $$Ax=b,$$ with unknown $x\in\mathbb{R}^n$, and some symmetric $A\in\mathbb{R}^{n\times n}$ and $b=\in\mathbb{R}^n$. Furthermore, it is known that $A$ is not full-rank matrix, and that its rank is $n-1$; therefore, $A$ is not invertible. However, to comp... | Let $\tilde x = A^+b$. Then obviously $A\tilde x = AA^+b$. But since $AA^+$ is an orthogonal projector, and specifically $I-AA^+$ is the projector to the null space of the Hermitian transpose of $A$, $\tilde x$ is a solution iff $b$ is orthogonal to the null space of $AA^+$, that is, orthogonal to the null space of the... | {
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Why is the last digit of $n^5$ equal to the last digit of $n$? I was wondering why the last digit of $n^5$ is that of $n$? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ideas about how I can figure out the proof myself? Thanks.
| If $\gcd(a, n) = 1$ then by Euler's theorem,
$$a^{\varphi(n)} \equiv 1 \pmod{n}$$
From the tables and as @SeanEberhard stated, $$ \varphi(10) = \varphi(5*2) = 10\left( 1 - \frac{1}{5} \right) \cdot \left(1 - \frac{1}{2} \right)$$
$$= 10\left(\frac{4}{5} \right) \cdot \left(\frac{1}{2} \right) = 4$$
Let $n=10$ and thus,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 7,
"answer_id": 1
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divisibility for numbers like 13,17 and 19 - Compartmentalization method For denominators like 13, 17 i often see my professor use a method to test whether a given number is divisible or not. The method is not the following :
Ex for 17 : subtract 5 times the last digit from the original number, the resultant number sho... | Your professor is using the fact that $100000001=10^8+1$ is divisible by $17$. Given for example your $80$-digit number, you can subtract $98765432\cdot 100000001=9876543298765432$, which will leave zeros in the last $16$ places. Slash the zeros, and repeat. After $5$ times you are left with the number $0$, which is di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Solving a literal equation containing fractions. I know this might seem very simple, but I can't seem to isolate x.
$$\frac{1}{x} = \frac{1}{a} + \frac{1}{b} $$
Please show me the steps to solving it.
| You should combine $\frac1a$ and $\frac1b$ into a single fraction using a common denominator as usual:
$$\begin{eqnarray}
\frac1x& = &\frac1a + \frac1b \\
&=&{b\over ab} + {a\over ab} \\
&=& b+a\over ab
\end{eqnarray}$$
So we get: $$x = {ab\over{b+a}}.$$
Okay?
| {
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"timestamp": "2023-03-29T00:00:00",
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Why Does Finitely Generated Mean A Different Thing For Algebras? I've always wondered why finitely generated modules are of form
$$M=Ra_1+\dots+Ra_n$$
while finitely generated algebras have form
$$R=k[a_1,\dots, a_n]$$
and finite algebras have form
$$R=ka_1+\dots +ka_n$$
It seems to me that this is an flagrant abuse... | The terminology is actually very appropriate and precise. Consider that "A is a finitely generated X" means "there exists a finite set G such that A is the smallest X containing G".
Looking at your examples, suppose $M$ is a finitely generated module, generated by $a_1,\dots,a_n$. Then $M$ contains $a_1,\dots,a_n$. Sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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relation between integral and summation What is the relation between a summation and an integral ? This question is actually based on a previous question of mine here where I got two answers (one is based on summation notation) and the other is based on integral notation and I do not know yet which one to accept . So ... | The Riemann or Lebesgue integral is in a sense an continuous sum. The symbol $\int$ is adapted from a letter looking like a somewhat elongated 's' from the word summa. In the definitions of the Riemann and the Lebesgue integrals the ordinary finite summation $\sum _{k=1}^n$ is used but the relation between the two is d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 5,
"answer_id": 3
} |
A variable for the first 8 integers? I wish to use algebra to (is the term truncate?) the set of positive integers to the first 8 and call it for example 'n'.
In order to define $r_n = 2n$ or similar.
This means:
$$r_0 = 0$$
$$r_1 = 2$$
$$\ldots$$
$$r_7 = 14$$
However there would not be an $r_8$.
edit: Changed "undefin... | You've tagged this abstract-algebra and group-theory but it's not entirely clear what you mean.
However, by these tags, perhaps you are referring to $\left(\Bbb Z_8, +\right)$?
In such a case, you have $r_1+r_7 = 1+_8 7 = 8 \mod 8 = 0$. So there is no $r_8$ per se; however, the re-definition of the symbols $r_0, r_1, \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Can this type of series retain the same value? Let $H$ be a Hilbert space and $\sum_k x_k$ a countable infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation only in those blocks, like this (for brevity illustrated only for the first ... | If both series converge, it doesn't change anything. This can be easily seen by considering partial sums.
Put $k_j$ as the cumulative length of the first $j$ blocks. Then clearly $\sum_{j=1}^{k_n} x_j=\sum_{j=1}^{k_n} x_j'$ for any $n$, so assuming both series converge, we have that
$$\sum_j x_j=\lim_{n\to \infty}\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Graph $f(x)=e^x$
Graph $f(x)=e^x$
I have no idea how to graph this. I looked on wolframalpha and it is just a curve. But how would I come up with this curve without the use of other resources (i.e. on a test).
| You mention in your question that "I have no idea how to graph this" and "how would I come up with this curve without the use of other resources (i.e. on a test)". I know that you have already accepted one answer, but I thought that I would add a bit.
In my opinion, then best thing is to remember (memorize if you will)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
definition of the exterior derivative I have a question concerning the definition of $d^*$. It is usually defined to be the (formally) adjoint of $d$? what is the meaning of formally?, is not just the adjoint of $d$? thanks
| I will briefly answer two questions here. First, what does the phrase "formal adjoint" mean in this context? Second, how is the adjoint $d^*$ actually defined?
Definitions: $\Omega^k(M)$ ($M$ a smooth oriented $n$-manifold with a Riemannian metric) is a pre-Hilbert space with norm $$\langle \omega,\eta\rangle_{L^2} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
The primes $p$ of the form $p = -(4a^3 + 27b^2)$ The current question is motivated by this question.
It is known that the number of imaginary quadratic fields of class number 3 is finite.
Assuming the answer to this question is affirmative, I came up with the following question.
Let $f(X) = X^3 + aX + b$ be an irreduci... | For (229, -4,-1) the polynomial factors as $(x-200)^2(x-58)$
For (1373, -8,-5) the polynomial factors as $(x-860)(x-943)^2$
For (2713, -13,-15) the polynomial factors as $(x-520)^2(x-1673)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/185261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Predicting the next vector given a known sequence I have a sequence of unit vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_k,\ldots$ with the following property: $\lim_{i\rightarrow\infty}\vec{v}_{i} = \vec{\alpha}$, i.e. the sequence converges to a finite unit vector.
As the sequence is generated by a poorly known proces... | A lot of methods effectively work by fitting a polynomial to your data, and then using that polynomial to guess a new value. The main reason for polynomials is that they are easy to work with.
Given that you know your functions have asymptotes, you may get better success by choosing a form that incorporates that fact,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Rigorous proof of the Taylor expansions of sin $x$ and cos $x$ We learn trigonometric functions in high school, but their treatment is not rigorous.
Then we learn that they can be defined by power series in a college.
I think there is a gap between the two.
I'd like to fill in the gap in the following way.
Consider the... | Since $x^2 + y^2 = 1$, $y = \sqrt{1 - x^2}$,
$y' = \frac{-x}{\sqrt{1 - x^2}}$
By the arc length formula,
$\theta = \int_{0}^{x} \sqrt{1 + y'^2} dx = \int_{0}^{x} \frac{1}{\sqrt{1 - x^2}} dx$
We consider this integral on the interval [$-1, 1$] instead of [$0, 1$].
Then $\theta$ is a monotone strictly increasing functio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
The measure of $([0,1]\cap \mathbb{Q})×([0,1]\cap\mathbb{Q})$ We know that $[0,1]\cap \mathbb{Q}$ is a dense subset of $[0,1]$ and has measure zero, but what about $([0,1]\cap \mathbb{Q})\times([0,1]\cap \mathbb{Q})$? Is it also a dense subset of $[0,1]\times[0,1]$ and has measure zero too?
Besides, what about its comp... | To give a somewhat comprehensive answer:
*
*the set in question is countable (as a product of countable sets), so it is of measure zero (because any countable set is zero with respect to any continuous measure, such as Lebesgue measure).
*it is also dense, because it is a product of dense sets.
*it has measure zer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
How to solve for $x$ in $x(x^3+\sin x \cos x)-\sin^2 x =0$?
How do I solve for $x$ in $$x\left(x^3+\sin(x)\cos(x)\right)-\big(\sin(x)\big)^2=0$$
I hate when I find something that looks simple, that I should know how to do, but it holds me up.
I could come up with an approximate answer using Taylor's, but how do I s... | Using the identity $\cos x=1-2\sin^2(x/2)$ and introduccing the function ${\rm sinc}(x):={\sin x\over x}$ we can rewrite the given function $f$ in the following way:
$$f(x)=x^2\left(x^2\left(1-{1\over2}{\rm sinc}(x){\rm sinc}^2(x/2)\right)+{\rm sinc}(x)\bigl(1-{\rm sinc}(x)\bigr)\right)\ .\qquad(*)$$
Now ${\rm sinc}(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
What is the order type of monotone functions, $f:\mathbb{N}\rightarrow\mathbb{N}$ modulo asymptotic equivalence? What about computable functions? I was reading the blog "who can name the bigger number" ( http://www.scottaaronson.com/writings/bignumbers.html ), and it made me curious. Let $f,g:\mathbb{N}\rightarrow\math... | If you want useful classes of "orders of growth" that are totally ordered, perhaps you should learn about things like Hardy fields. And even in this case, of course, Asaf's comment applies, and it should not resemble a well-order at all.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/185542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Equivalent definitions of linear function We say a transform is linear if $cf(x)=f(cx)$ and $f(x+y)=f(x)+f(y)$. I wonder if there is another definition.
If it's relevant, I'm looking for sufficient but possibly not necessary conditions.
As motivation, there are various ways of evaluating income inequality. Say the vect... | Assume that we are working over the reals. Then the condition $f(x+y)=f(x)+f(y)$, together with continuity of $f$ (or even just measurability of $f$) is enough. This can be useful, since on occasion $f(x+y)=f(x)+f(y)$ is easy to verify, and $f(cx)=cf(x)$ is not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/185651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Ellipse with non-orthogonal minor and major axes? If there's an ellipse with non-orthogonal minor and major axes, what do we call it?
For example, is the following curve a ellipse?
$x = \cos(\theta)$
$y = \sin(\theta) + \cos(\theta) $
curve $C=\vec(1,0)*\cos(\theta) + \vec(1,1)*\cos(\theta) $
The major and minor axes ... | Hint: From $y=\sin\theta+\cos\theta$, we get $y-x=\sin\theta$, and therefore $(y-x)^2=\sin^2\theta=1-x^2$. After simplifying and completing the square, can you recognize the curve?
The major and minor axes do turn out to be orthogonal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/185718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
What are affine spaces for? I'm studying affine spaces but I can't understand what they are for.
Could you explain them to me? Why are they important, and when are they used? Thanks a lot.
| The first space we are introduced in our lives are euclidean spaces, which are the classical beginning point of classical geometry. In these spaces, there is a natural movement between points that are translations, i.e., you can move in a natural way from a point $p$ to a point $q$ through the vector that joint them $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 5,
"answer_id": 3
} |
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