Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Inverse rotation transformations I'm taking the 2-degree gibmle system and position its alignment point in a arbitrary position (denoted by the axes angles phi for the first degree, and theta for the second). How can I reverse the transformations I did (first rotation of the first axis by phi, then rotation of the seco... | If your alignment point happens to lie on either of the axes, you cannot undo the operations, because rotation about that axis will leave the alignment point in the same place, so a single "final position" leads to multiple possible input-rotations.
Assuming that the axes start out perpendicular, with the first aligne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Select a subsequence to obtain a convergent series. Does there exists strictly increasing sequence $\{a_k\}_{k\in\mathbb N}\subset\mathbb N$,
such that
$$
\sum_{k=1}^{\infty}\frac{1}{(\log a_k)^{1+\delta}}\lt \infty,
$$
where $\delta>0$ given and $$\lim_{k\to \infty }\frac{a_{k+1}}{a_k}=1.$$
| Answer. Try
$$
a_k=\left\lfloor 2^{k^{\color{red}{1/(1+\delta/2)}}}\right\rfloor.
$$
Then $\dfrac{a_{k+1}}{a_k}\to 1$, and
$$
\frac{1}{(\log a_k)^{1+\delta}}\approx\frac{c}{n^{1+\delta/2}}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/940272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$ If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$. [Hint: Given $g$ mapping $B$ onto $A$, let $f(X)=g^{-1}(X)$ for all $X \subseteq A$]
I follow the hint and obtain the function $f$. If $f$ is injective, then the statement is proven.
Qu... | If there exists a function $f:B \longrightarrow A$ such that $f$ is onto then $\lvert B \rvert \geq \lvert A \rvert$. And this means that there exists an injective function $g: A \longrightarrow B$. Now as we want to see that $2^{\lvert A \rvert} \leq 2^{\lvert B \rvert}$ it's enough to define an injective function $h:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Diagonalizing $xyz$ The quadratic form $g(x,y) = xy$ can be diagonalized
by the change of variables $x = (u + v)$ and $y = (u - v)$ .
However, it seems unlikely that the cubic form $f(x,y,z) = xyz$,
can be diagonalized by a linear change of variables.
Is there a short computational or theoretical proof of this... | If I understand correctly your question, you are asking if it is possible to write $xyz = \ell_1^3 + \ell_2^3 + \ell_3^3$ for some linear forms $\ell_1=\ell_1(x,y,z)$, $\ell_2,\ell_3$.
We can prove that $xyz \neq \ell_1^3 + \ell_2^3 + \ell_3^3$ in a few different ways. Here is a short theoretical proof that uses a litt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Applying the Law of Large Numbers? $X_k$, $k \geq 1$ are iid random variables such that
$$\limsup_{n\rightarrow\infty} \frac{X_n}{n} < \infty$$
with probability 1. We want to show that
$$\limsup_{n\rightarrow\infty} \frac{\sum_{i=1}^n X_i}{n} < \infty$$
with probability 1.
The hint says to apply the law of large number... | Consider $X_k^+ := \max(X_k,0)$. Then,
\begin{align*}
P\left(\limsup \frac{X_n}{n} < \infty\right)=1 &\Rightarrow P\left(\limsup \frac{X_n^+}{n} < \infty\right)=1 \\
&\Rightarrow \exists A: P\left(\frac{X_n^+}{n} > A \text{ i.o.}\right)=0 \text{ a.s.}\\
&\Rightarrow \sum_{i=1}^n P\left(\frac{X_i^+}{i} > A\right) < \inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Why do we use dummy variables in integrals? I want to know why we use dummy variables in integrals?
| I'll interpret the question as
Why are we not using the orignal one instead of a dummy variable?
Why use $\int_0^xf(x')dx'$ and not $\int_0^xf(x)dx$?
(If this is the case, please clarify you question with an marked(!) edit, others interpreted in differently.)
Because it has a different meaning and there is a prob... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 0
} |
How to simplify $(\sin\theta-\cos\theta)^2+(\sin\theta+\cos\theta)^2$? Simplify: $(\sin \theta − \cos \theta)^2 + (\sin \theta + \cos \theta)^2$
Answer choices:
*
*1
*2
*$ \sin^2 \theta$
*$ \cos^2 \theta$
I am lost on how to do this. Help would be much appreciated.
| $$\begin{align}
&\phantom{=}\left(\sin x-\cos x\right)^2+\left(\sin x+\cos x\right)^2\\
&=\sin^2x-2\sin x\cos x+\cos^2x+\sin^2x+2\sin x\cos x+\cos^2x\\
&=2\sin^2x+2\cos^2x\\
&=2\left(\sin^2x+\cos^2x\right)\\
&=2
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/940738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Find the Basis and Dimension of a Solution Space for homogeneous systems I have the following system of equations:
$$\left\{\begin{array}{c}
x+2y-2z+2s-t=0\\
x+2y-z+3s-2t=0\\
2x+4y-7z+s+t=0
\end{array}\right.$$
Which forms the following matrix
$$\left[\begin{array}{ccccc|c}
1 & 2 & -2 & 2 & -1 & 0\\
1 & 2 & -1 & 3 & -2... | First solve the system, assigning parameters to the variables which correspond to non-leading (non-pivot) columns:
$$\eqalign{
&t=\alpha\cr
&s=\beta\cr
z+s-t=0\quad\Rightarrow\quad &z=\alpha-\beta\cr
&y=\gamma\cr
x+2y+4s-3t=0\quad\Rightarrow\quad &x=3\alpha-4\beta-2\gamma\ .\cr}$$
So the solution set is
$$\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Probability of at least 3 red balls given 4 choices in a bag of 4 red balls and 4 black balls? Let's say there are 8 balls in a bag, where 4 are red and 4 are black.
If I choose four balls from the bag without replacement, what is the probability that I will choose at least 3 red balls?
My thinking was to use the idea ... | Here we want to find the probability of only 3 Red balls OR 4 Red balls being drawn without replacement. This means that we need to add the probability of either event together.
Another way to think of it is finding the total number of ways there are to draw 3 Red balls/4 Red balls, and then divide that by the total nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/940974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Relationship between $\int_a^b f(x) dx$ and $\sum_{i= \lceil a\rceil}^{\lfloor b\rfloor} f(i)$ Let we have a continuous function $f(x)$ in the interval $ [ a,b ] $
Does there exist any relationship between its integral and summation of function-values defined at the integers between $a$ and $b$.
i-e Relationship betwe... | Yes. Euler-McLaurin's formulas completely describe this kind of relations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/941162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding all solutions of an expression and expressing them in the form $a+bi$ $$6x^2+12x+7=0$$
Steps I took:
$$\frac { -12\pm \sqrt { 12^{ 2 }-4\cdot6\cdot7 } }{ 12 } $$
$$\frac { -12\pm \sqrt { -24 } }{ 12 } $$
$$\frac { -12\pm i\sqrt { 24 } }{ 12 } $$
$$\frac { -12\pm 2i\sqrt { 6 } }{ 12 } $$
I don't know where t... | Hint:
$$\frac{a+b}{c} = \frac ac + \frac bc$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/941253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
divisibility on prime and expression This site is amazing and got good answer.
This is my last one.
If $4|(p-3)$ for some prime $p$, then $p|(x^2-2x+4)$.
can you justify my statement?
High regards to one and all.
| If $x \geq 0$, then $x^2 - 2x + 4 = (x - 1)^2 + 3$.
So then you want to see if $-3$ is a quadratic residue modulo $p$, and that's what you use the Legendre symbol $(\frac{-3}{p})$ for, which gives you a yes ($1$) or no ($-1$) answer.
But $p \equiv 3 \mod 4$ does not guarantee $(\frac{-3}{p}) = 1$, as André's example of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/941393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solving Coin Toss Problem If a coin is tossed 3 times,there are possible 8 outcomes.
HHH HHT HTH HTT THH THT TTH TTT
In the above experiment we see 1 sequnce has 3 consecutive H, 3 sequence has 2 consecutive H and 7 sequence has at least single H.
Suppose a coin is tossed n times.How many sequence we will get which con... | Let $x^n(i,j)$ be the number of sequences of length $n$ with exactly $i$ as the length of the longest sequence of H's and ending in exactly $j$ H's. Then $x^n(i,j)=0$ if $i<j$ or $i\gt n$ or $j\gt n.$ We can fill in the table for $x^2(i,j)$ row $i,$ column $j:$
$$
\begin{array}{c|ccc}
& 0 & 1 & 2 \\
\hline
0 & 1 & 0 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/941482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$ How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$)
$$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$
Integration by parts doesn't seem to make it any simpler.
Hints please? :)
| Another non-IBP route: Consider the integral $\int_{-\infty}^\infty e^{-ax^2/2}e^{t x}\,dx$, which can be computed exactly by completing the square in the exponent. Expanding $e^{tx}$ in powers of $t$, we find that the coefficients are essentially just the desired integrals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/941570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 6
} |
Continuity of piecewise function of two variables The question looks like this.
Let $f(x, y)$ = 0 if $y\leq 0$ or $y\geq x^4$, and $f(x, y)$ = 1 if $0 < y < x^4 $.
(a) show that $f(x, y) \rightarrow 0$ as $(x, y) \rightarrow (0, 0)$ along any path through (0, 0) of the form $ y = mx^a $ with $a < 4$.
(b) Despite par... | Substitute $e^{-t}$ for $x$. In case $m=|m|$ then substitute $e^{-at+b}$ for $y$ where $b=ln(|m|)$, in case $m=-|m|$ then $y=-e^{-at+b}$. In the first case we have $t>t_0=-b/(4-a) \Rightarrow y>x^4$ so $f(e^{-t},e^{-at+b})=0,\; \forall t>t_0$. In the second case we have $y<0 \;\forall t\in \Bbb{R}$ so that $f(e^{-t},-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/941666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many functions can be constructed? How many functions $f:\left\{1, 2, 3, 4,5 \right\} \rightarrow \left\{ 1, 2, 3, 4, 5 \right\}$ satisfy the relation $f\left( x \right) =f\left( f\left( x \right) \right)$ for every $x\in \left\{ 1, 2, 3, 4, 5 \right\}$?
My book says the answer is 196.
| Hint: Let $R\subset\{1,2,3,4,5\}$ be the range of $f$.
Then $f(x)$ is completely determined for every $x\in R$, and the only choices you have about the behavior of $f$ are for $x\notin R$.
Hint 2: If $\{1,2,3,4,5\}$ is too complicated, try solving the problem for $\{1,2\}$ instead, and then see if you can apply the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/941888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Spectral Measures: Concentration Given a Hilbert space $\mathcal{H}$.
Consider spectral measures:
$$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad E(\mathbb{C})=1$$
Define its support:
$$\operatorname{supp}(E):=\bigg(\bigcup_{U=\mathring{U}:E(U)=0}U\bigg)^\complement=\bigcap_{C=\overline{C}:E(C)=1}C$$
By se... | Consider the operator $(Af)(x)=xf(x)$ on the Hilbert space $L^2([0,1])$ with Lebesgue measure. Then $E_A(M)=1_M$ is the spectral measure associated with $A$. Note that $\mbox{supp }E_A=\sigma(A)=[0,1]$, but $E_A(\{\lambda\})=0$ for any singleton $\lambda$, since $A$ does not have any eigenvalues. Now consider, say, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/942030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$ I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$, satisfying the differential equation
$$
f^{(n)}=f,
$$
and with $\,f,f',\dots,f^{(n-1)}\,$ being linearly independent.
Could you give me some hints?
Thank... | The characteristic polynomial of the equation $x^{(n)}-x=0$ is $p(\zeta)=\zeta^n-1$,
with roots $1,\lambda,\ldots,\lambda^{n-1}$, where $\lambda=\exp(i\omega)$ with
$\omega=\dfrac{2\pi}{n}$. Hence, the functions
$$
f_k(t)=\exp \big(\lambda^k t\big), \,\,\,\text{where $k\in\{0,1,2,\ldots,n-1\}$}
$$
form a basis of the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/942132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculus 2 Integral of$ \frac{1}{\sqrt{x+1} +\sqrt x}$ How would you find $$\int\frac{1}{\sqrt{x+1} + \sqrt x} dx$$
I used $u$-substitution and got this far:
$u = \sqrt{x+1}$ which means $(u^2)-1 = x$
$du = 1/(2\sqrt{x-1}) dx = 1/2u dx$ which means $dx = 2udu$
That means the new integral is $$\int \frac{2u}{u + \sqrt{... | Hint: Use that ${1 \over \sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} - \sqrt{x}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/942209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Really advanced techniques of integration (definite or indefinite) Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? Every time I search for "Advanced Techniques of Symbolic ... | You can do integration by inverting the matrix representation of the differentiation operator with respect to a clever choice of a basis and then apply the inverse of the operator to function you wish to integrate.
For example, consider the basis $\mathcal{B} = \{e^{ax}\cos bx, e^{ax}\sin bx \}$. Differentiating with r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/942263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "299",
"answer_count": 21,
"answer_id": 15
} |
Not understanding how to factor a polynomial completely $$P(x)=16x^4-81$$
I know that this factors out as:
$$P(x)=16(x-\frac { 3 }{ 2 } )^4$$
What I don't understand is the four different zeros of the polynomial...I see one zero which is $\frac { 3 }{ 2 }$ but not the three others.
| You have to solve the expression $16x^4 - 81 = 0$, and you will get
$$ 16x^4 - 81 = (4x^2 - 9)(4x^2 + 9) = 0 $$
then you will find the other three roots you didn't find.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/942368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$ If $f(x)$ is a quadratic expression such that $f(x)>0\;\forall x\in\mathbb{R},$ and if $g(x)=f(x)+f'(x)+f''(x),$
Then prove that $g(x)>0\; \forall \; x\in \mathbb{R}$.
$\bf{My\; Trial \; Solution::}$ If $f(x)>0\;\forall x\i... | Suppose $f(x)=x^2+ax+b$ with $b=f(0)>0$ and $a^2<4b$. Then,
$$
g(x)=x^2+(a+2)x+(2+a+b).
$$
We note that $g(0)=2+a+b=1+f(1)>0$ and
$$
(a+2)^2-4(2+a+b)=a^2+4a+4-8-4a-4b=(a^2-4b)-4<0
$$
so $g$ is never $0$ for real $x$. You now can infer that $g$ is always positive.
The more general case $f(x)=C(x^2+ax+b)$ is the same. It... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/942446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
General formula for $\sin\left(k\arcsin (x)\right)$ I'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig substitution. But $k=2$ is not so bad, because I can use the double an... | I would try to write $A[k](x)=\sin (k.\arcsin(x) )$ and $B[k](x)=\cos (k.\arcsin (x) )$
and then $A[k+1](x) = \sin ( \arcsin(x) + k.\arcsin (x) ) = x B[k](x) + \sqrt{1-x^2} A[k](x)$
You could then either use the same recurrence form for $B[k+1](x)$ and have a double recurrence relation, or use $B[k](x)=\sqrt{1-A[k](x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/942506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Terminology - Limit doesn't exist Take the following limit: $$ \lim_{x \to 2} \dfrac{x+2}{x-2} $$
This doesn't exist. My textbook says it doesn't because "The denominator approaches
0 (from both sides) while the numerator does not."
I don't understand what this means. I do understand that it doesn't exist. My thought ... | Consider the function $\frac{\sin x}{x}$ as x approaches 0 for a case where the limit does exist and is equal to 1. Reference if you want one for this case.
If you want another example where the limit doesn't exist consider either $\sin x$ as x tends to infinity or $(-1)^n$ as n tends to infinity, consider the cases wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/942579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Show that every proper subgroup of this group is finite. Let $G$ be the group of rational numbers in $[0,1)$ whose
denominator is a power of $2$:
\begin{align*}
G &= \{r/2^k : \text{$r \in \mathbb Z$, $0 \le r < 2^k$, $k = 0, 1,
\ldots$} \} \\
&=\{0, \frac12, \frac14, \frac34, \frac18, \frac38,... | Hint: Try to show that if a subgroup of $G$ is infinite, then for infinitely many $k$ it contains a generator of $A_k$. Then show that it is in fact true for all $k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/942709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Proving a function has real roots I am not interested in finding roots but interested in proving that the function has real roots.
Suppose a function $f(x) = x^2 - 1$
This function obviously has real roots.
$x = {-1, 1}$
How could I prove this without actually finding the roots?
Trial and error could work, number theo... | One way is using the discriminant of the quadratic equation:
$$\sqrt{b^2-4ac}$$
If the value inside the square root is greater than 0, then there are two real roots
If it is equal to 0, there is one real root
If it is less than 0, it has imaginary roots
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/942850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Why is the polynomial $f(x)=x^3+x^2+x+1$ monotonic? I have to argue why the polynomial $f(x)=x^3+x^2+x+1$ has a reverse function $f^{-1}$ which is defined in on the whole of $\mathbb R$. I'm certain the argument would simply be that because $f(x)$ is monotonic on $\mathbb R$ it is also injective on $\mathbb R$. However... | Without calculus, you could look at $f(y)-f(x)=(y-x)(x^2+xy+y^2+x+y+1)$, and show that the right-hand side is positive whenever $y>x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/942922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Sum of products of positive operators I'm trying to answer the following question: Given two positive self adjoint operators $\mathcal{A}$ and $\mathcal{P}$ on a Hilbert space, is the following composition:
$\mathcal{AP}+\mathcal{PA}$ also positive?
One possible condition under which this is true is when $\mathcal{AP}... | Observe that
\begin{align*}
\langle (\mathcal A \mathcal P+\mathcal P\mathcal A)x,x\rangle&=\langle \mathcal A \mathcal Px,x\rangle+\langle\mathcal P\mathcal Ax,x\rangle\\
&=\langle \mathcal Px,\mathcal Ax\rangle+\langle\mathcal Ax,\mathcal Px\rangle\\
&=2\mbox{Re }\langle\mathcal Px,\mathcal Ax\rangle=2\mbox{Re }\lan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/943049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
induction for idempotent matrix : $P^n = P$
Given that $P^2 = P$ how do i prove by induction that $P^n = P$?
I have tried the following: we know that $P^k = P$ holds for $k = \{1,2\}$. If we now take $k=3$:
$$
\begin{align}
P^3 &= P^2P
\\ &=PP \tag*{($P$ is idempotent)} \\
\\&= P^2
\\&=P
\end{align}
$$
therefore $P^... | Suppose $P^{n-1}=P$.
Then
$$\begin{align*}P^n&=P(P^{n-1})\\
&=PP\\
&=P^2\\
&=P.
\end{align*}
$$
We're given $P^2=P$, so by induction on $n$, we're done.
Thinking of induction as reaching back to the previous cases instead of reaching forward to the next case can be insightful.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/943239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
finding the generating function $\phi(s) = \mathbb{E}(s^{H_0})$. i just started the course of markov chains and i'm having a few problems with one of the excercises.
Let $Y_1,Y_2, \dots$ be i.i.d random variables with:
$\mathbb{P}(Y_1 = 1) = \mathbb{P}(Y_1 = -1) = \frac{1}{2}$ and set $X_0 = 1, X_n = X_0 + Y_1+ \cdots ... | If $Y_1=-1$, then $X_1=0$ hence $H_0=1$. If $Y_1=1$, then $X_1=2$ hence $H_0=1+H'_0+H''_0$, where, in the RHS, $1$ accounts for the first step, $H'_0$ for the time to hit $1$ starting from $2$ and $H''_0$ for the time to hit $0$ starting from $1$. Thus, $H'_0$ and $H''_0$ are independent and distributed like $H_0$.
Tu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/943344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Square root for Galois fields $GF(2^m)$ Can we define a function similar to square root for $G = GF(2^m)$ (Galois field with $2^m$ elements) as $\sqrt{x} = y$ if $y^2 = y \cdot y = x$ ? For which elements $x \in G : \exists y \in G : y^2 = x$ this function would be defined?
Can I approach this question like this:
If we... | For the field $GF(p^m)$ the map
$$F: x\mapsto x^p$$ is an automorphism of order $m$, that is
$$F^m(x) =x^{p^m} = x$$
and so the the inverse automorphism is
$$F^{-1} = F^{m-1}$$
or
$$\sqrt[p]{x}= x^{p^{m-1}}$$
The observation about normal bases of @Dilip Sarwate is excellent; also see http://en.wikipedia.org/wiki/No... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/943417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Order of parameters in quantified predicates I'm studying up for my midterm in Discrete Math and I've been looking at sample questions and their solutions. There is one I don't really understand and I was hoping someone could help me out.
2. Let the domains of x and y be the set of all integers.
Compute the Boolean val... | $\forall x.P$ means that every possible value of $x$ will make $P$ true.
$\exists x.P$ means that there is a value of $x$ that will make $P$ true.
One such value is enough, but there has to be at least one.
Usually each variable is restricted to some domain. For example, since this is
a discrete math course, can we sti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/943502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How many numbers can a typical computer represent? I couldn't find this elsewhere so I thought I'd give it a try to figure out exactly how many numbers a typical desktop computer can represent in memory. I'm thinking about this in the context of numerical algorithms to solve mathematical problems.
I'm pretty sure si... | The eight byte signed integers are not a subset of the double precision numbers if by double precision you mean $64$ bits. The eight bit signed integers have $63$ bits of mantissa plus a sign bit, while the double precision floats only have $52$ bits of precision. To compute the overlap is not so easy. Let us focus ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/943589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Proving a set-theoretic identity Context: Measure theory.
Reason: Just curious.
Question: Given $\{A_k\}$ with $A_k$ not disjoint, $B_1=A_1$ and $B_n = A_n - \bigcup\limits_{k=1}^{n-1} A_k$ for $n \in \mathbb{N}-\{1\}$ and $k \in \mathbb{N}$, how can I show that $$\bigcup\limits_{n=2}^{\infty}A_n = \bigcup\limits_{n=2}... | Try showing the double inclusion. One side is easy as $B_i \subseteq A_i$ for all $i$. For the other side, think of $x \in \bigcup A_i$, and let $A_k$ the first $k$ such that $x \in A_k$, what can you say about $x$ and $B_k$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/943714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Inhomogeneous modified Bessel differential equation I'm trying to solve the following inhomogeneous modified bessel equation.
$$y^{\prime\prime}+\frac{1}{x}y^{}\prime-\frac{x^2+4}{x^2}y=x^4$$
I know the homogeneous solution for this differential equation is $y_h=c_1I_2(x)+c_2K_2(x)$
Where $I_2$ and $K_2$ are the modif... | $$y^{\prime\prime}+\frac{1}{x}y^{\prime}-\frac{x^2+4}{x^2}y=x^4$$
The solution for the associated homogeneous ODE is $y_h=c_1I_2(x)+c_2K_2(x)$
The solution for the non-homogeneous ODE can be found on the form $y=y_h+p(x)$ where $p(x)$ is a particular solution of the ODE.
The seach of a particular solution using the va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/943945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Is division axiomatizable? Consider a set $G$ with a group operation. We can define a division operation $a*(b^{-1})$ and call it $\operatorname{div}$. Is the class of division operations first order axiomatizable? And if so, is it finitely axiomatizable?
| Let $\star$ be your operator. On a group, this can be axiomatized as:
$$\forall x(1\star(1\star x)=x)\text{ (A)}\\ \forall x(x\star x = 1)\text{ (B)}\\\forall x,y,z\left((x\star y)\star z = x\star(z\star(1\star y))\right)\text{ (C)}$$
We can quickly show:
$$\begin{align}
x\star 1 &= (1\star(1\star x))\star 1 \text{ (A)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/944028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Game of dots: winning strategy? The game begins with a row of $n$ numbers, in increasing order from $1$ to $n$. For example, if $n=7$, we have a row of numbers $(1,2,3,4,5,6,7)$.
On each turn, a player must either remove 1 number, or remove 2 consecutive numbers. For example, the first player to move can remove $2$ or ... | Example strategy for $n=7$:
The first player takes $4$, and then until the last element:
*
*If the second player takes $x$, then the first player takes $8-x$.
*If the second player takes $(x,x+1)$, then the first player takes $(7-x,8-x)$.
General strategy for an odd $n$:
The first player takes $\dfrac{n+1}{2}$, an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/944097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Determining consistency of a general overdetermined linear system For $m > 2$, consider the $m \times 2$ (overdetermined) linear system
$$A \mathbf{x} = \mathbf{b}$$
with (general) coefficients in a field $\mathbb{F}$; in components we write the system as
$$\left(\begin{array}{cc}a_{11} & a_{12} \\ \vdots & \vdots \\ a... | When $m=3$ and $\mathbb F$ is infinite, there are no other obstructions besides the determinant. When $\mathbb F$ is finite, there are many others : for example
if we put $\chi_{\mathbb F}(X)=\prod_{t\in {\mathbb F}^*} (X-t)$, $\chi(t)$ is zero iff
$t$ is nonzero, so that the following $n$ polynomials are all obstruct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/944213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Can a number have infinitely many digits before the decimal point? I asked my teacher if a number can have infinitely many digits before the decimal point. He said that this isn't possible, even though there are numbers with infinitely many digits after the decimal point. I asked why and he said that if you keep adding... | What is the underlying reason for having infinitely many digits following the decimal point, but not infinitely many digits left of the decimal point?
The underlying reason is that real numbers can have infinite precision, but only finite size. You can find larger and larger real numbers, but each of them has a finite... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/944284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "60",
"answer_count": 9,
"answer_id": 6
} |
Show $\sum\limits_{k=1}^n{n-1\choose k-1} =2^{n-1}$
*
*Given $$\sum\limits_{k=1}^n k{n\choose k} = n\cdot 2^{n-1}$$
*
*I know that $$k\cdot{n\choose k}=n\cdot{n-1\choose k-1}=(n-k+1)\cdot{n\choose k-1}$$
Therefore $$\sum\limits_{k=1}^n k{n\choose k} = \sum\limits_{k=1}^n n{n-1\choose k-1} = n\cdot 2^{n-1}$$
So,... | With $j=k-1$
$$\sum_{k=1}^n {n-1\choose k-1}=\sum_{j=0}^{n-1} {n-1\choose j}=2^{n-1} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/944422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Is there a name for the function that gives me the signal of a number only? I know the function that gives the absolute value of a number is called either absolute function or 'modulus' function, such as:
$$
modulus(-6) = modulus(6) = 6
$$
Now, I want to name a function that gives me a unit value with the same signal a... | That's that signum function.
Actually:
$$\text{signum}(x)=\begin{cases}\begin{align}1,\quad x>0\\0,\quad x=0\\-1,\quad x<0\end{align}\end{cases}$$
See it at wikipedia or WolframMathworld.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/944548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$\prod\left(1-p_n\right)>0$ I want to prove that if $0\le p_n<1$ and $\sum p_n<\infty$, then $\prod\left(1-p_n\right)>0$ .
There is a hint : first consider the case $\sum p_n<1$, and then show that $\prod\left(1-p_n\right)\ge1-\sum p_n$ .
How can I use this hint to show the statement above?
| Why it is sufficient to prove the hint: Suppose $\sum p_n < \infty$. Then there is an integer $N$ such that $\sum_{n \geq N} p_n < 1$. Now observe that both $\prod_{n < N} (1-p_n)$ (a finite product) and $\prod_{n \geq N} (1-p_n)$ (using the hint) are positive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/944678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
The rank after adding a rank-one matrix to a full rank matrix Suppose $A$ and $B$ are all $N$ by $N$ matrices.
$$rank(A) = N, rank(B) = 1$$
What's the lower bound of: $$rank(A+B)?$$
My guess in this specific case is $$rank(A+B) \geq N-1,$$ but I don't know if it's true, how to prove it, and under what condition we h... | Think of $A,B$ as linear transformations. $rank(A)=N$ implies $A$ is one to one, hence maps every subspace of dimension $k$ to another subspace of the same dimension. $rank(B)=1$ implies $\dim\ker(B)=N-1$. Now $$\dim(A+B)(\ker(B))=\dim(A(\ker(B)))=\dim(\ker(B))=N-1$$implies $$\dim(Im(A+B))\geq N-1,$$and in other words ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/944737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why is this set a $\sigma$-algebra?? $X$ is an uncountable set.
Why is $\mathcal{A}=\{A \subset X: A \text{ or } X \setminus A \text{ is countable } \}$ a $\sigma$-algebra ??
$$$$
A $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a collection of subsets of $X$ such that :
(1) $\varnothing \in \mathcal{A}$
(2) $A \in \ma... | Well, (1) and (2) are obvious.
Hint for (3): let $A_1,A_2,\dots\in \mathcal A$. Consider two cases:
*
*Either all $A_n$'s contain only countable points.
*Or at least one of them (w.l.o.g., say $A_1$) contains all but countable points.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/944823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can $\Theta(f_1) = \Theta(f_2)$? Does $\Theta(n^3+2n+1) = \Theta(n^3)$ hold? I'm so used to proving that a concrete function is Big-Whatever of another function, but never that Big-Whatever of a function is Big-Whatever of another function.
| The problem is that $f=\Theta(g)$ is bad notation, because the two sides aren't "equal" in the obvious sense. One way to make the notation precise is to think of $\Theta(g)$ as the collection of all functions which "are big-theta of $g$." In other words, $\Theta(g)$ consists of all the functions $f$ so that $$c\cdot g(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/944942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Infinitely Many Circles in an Equilateral Triangle
In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.
I need to find the ... | Look at the following figure carefully,
As the triangle is equilateral ($AC$ is the angle bisector). So, $\angle ACD = 30^{\circ}$
$$\tan 30^{\circ} = \frac{AD}{DC} = 2AD\ (\because DC = 1/2) $$
$$\therefore AD = \frac{1}{2\sqrt{3}}$$
This is the radius of the bigger circle, let its area be $A_1$
$$\therefore A_1 = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/945123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Example of a commutative square without a map between antidiagonal objects? In an abelian category, can there be a commutative diagram of (vertical/horizontal) exact sequences
$$
\require{AMScd}
\begin{CD}
0 @>>> N @>>> M\\
@. @VVV @VVV \\
0 @>>> X @>>> Y\\
@. @VVV \\
@. 0
\end{CD}
$$
such that the following conditions... | Consider $0 \to \mathbb{Z} \to \mathbb{Q}$, and $\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$. Then the only map in either direction between $\mathbb{Q}$ and $\mathbb{Z}/2\mathbb{Z}$ is the zero map, since $\mathbb{Q}$ is torsionfree, and every quotient of $\mathbb{Q}$ is divisible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/945221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving this summation: $\sum_{i=1}^k i\cdot2^i$ $$\sum_{i=1}^k i\cdot2^i$$
I'm working on a recurrence relation question and now I'm stuck at this point. I have no idea how to simplify this down to something I can work with. Can I seperate the terms into
$$\sum_{i=1}^k i \cdot \sum_{i=1}^k 2^i$$ and then just use the... | Consider the series
\begin{align}
\sum_{k=0}^{n} t^{k} = \frac{1-t^{n+1}}{1-t}.
\end{align}
Differentiate both sides with respect to $t$ to obtain
\begin{align}
\sum_{k=0}^{n} k t^{k-1} &= \frac{1}{(1-t)^{2}} \left( -(n+1) (1-t) t^{n}+(1-t^{n+1}) \right) \\
&= \frac{1 -(n+1) t^{n} + n t^{n+1}}{(1-t)^{2}}.
\end{align}
N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/945281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
For any integer $n\geq 1$, define $\sin_n=\sin\circ ... \circ \sin$ ($n$ times). Prove that $\lim_{x\to 0}\frac{\sin_nx}{x}=1$ for all $n\geq 1$ I got this problem:
For any integer $n\geq 1$, define $\sin_n=\sin\circ ... \circ \sin$ ($n$ times). Prove that $\lim_{x\to 0}\frac{\sin_nx}{x}=1$ for all $n\geq 1$.
Some hint... | Hint: $\sin(x)\sim x$ then, $\sin(\sin(x))\sim\sin x\sim x$ then $\sin_n(x)\sim\sin_{n-1}(x)\sim...\sim x$. I let you conclude.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/945452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
If the product $(x+2)(x+3)(x+4)\cdots(x+9)(x+10)$ expands to $a_9x^9+a_8x^8+\dots+a_1x+a_0$, then what is the value of $a_1+a_3+a_5+a_7+a_9$?
When expanded, the product $(x+2)(x+3)(x+4)\cdots(x+9)(x+10)$ can be written as $a_9x^9+a_8x^8+\dots+a_1x+a_0$. What is the value of $a_1+a_3+a_5+a_7+a_9$?
| Let $P(x) = (x+2)(x+3)..(x+10)$. Then what is $\dfrac{P(1)-P(-1)}2=?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/945590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Upper and/or lower Bound for Numbers of different topologies on the set $\{1,...n \}$ As the title says I am looking for upper and lower bound for the cardinality of different topologies on a set $\{1,....n\}$ for natural n!
Are there some known bounds? My teacher says that there no formula which gives the exactly numb... | Let $X=\{1,....n,\}$and $m(n)$ is the number of topologies on $X$. We want to show that $m(n)\leq 2^{n(n-1)}$
We denote $\mathcal{M}$ for the set of all topologies on $X$ and we define for every $x \in X$ a function as follows:
$f_x:\mathcal{M}\rightarrow \mathcal{P}(X)$, $f(\tau )= \bigcap_{U\in\tau,x\in U}U$
So the m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/945727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\text{If } |z_1| = |z_2|, \text{ show that } \frac{z_1 + z_2}{z_1-z_2} \text{is imaginary.} $ $\text{If } |z_1| = |z_2|, \text{ show that } \frac{z_1 + z_2}{z_1-z_2} \text{is imaginary.} $
The first thing I tried to do was to multiply both top and bottom by the conjugate of the denominator...
$$
\frac{z_1 + z_2}{z_1-z... | Multiply instead by $\dfrac{\bar{z_1} + \bar{z_2}}{\bar{z_1} + \bar{z_2}}$ to get $\displaystyle \frac{|z_1|^2 + z_1\bar{z_2} + \bar{z_1}{z_2} + |z_2|^2}{|z_1|^2 + z_1\bar{z_2} - \bar{z_1}{z_2} - |z_2|^2}$. The denominator is imaginary and the numerator is real.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/945824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Absolute continuity under the integral Let $f:[0,T]\times \Omega \to \mathbb{R}$ where $\Omega$ is some bounded compact space. Let $t \mapsto f(t,x)$ be absolutely continuous. Is then
$$t \mapsto \int_\Omega f(t,x)\;dx$$
also absolutely continuous provided the integral exists?
I think we need $|f(t,x)| \leq g(x)$ for... | Let $\epsilon > 0$ and $(t_i, t_{i+1})_{i=1}^N$ be any non-overlapping set of intervals such that $\sum_j|f(t_j,x) - f(t_{j-1},x)| \leq \epsilon$ whenever $\sum_j |t_{j+1} - t_j| < \delta_x$, where the $\delta_x$ comes from the absolute continuity of $f$.
$\sum_j |F(t_j) - F(t_{j-1})|\leq\sum_j \int_{\Omega} |f(t_j,x) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/945907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How to solve DE $y'=1/(x^2y)(y^2-1)^{3/2}$ Man, I'm having troubles with this differential equation. I just can't do any math if I'm tired...
What I have done:
$$\frac{y'\cdot y}{(y^2-1)^{3/2}}=\frac{1}{x^2}$$
Now I integrated both sides from $a$ to $x$ and I've got
$$-\frac{1}{(y^2-1)^{1/2}}+ \frac{1}{(y(a)^2-1)^{1/2... | I don't understand the $_a^x$ part. Why would you do that? You can evaluate the antiderivative directly(we may have the same solution but here it goes anyway);
$$\int \frac{y' y}{(y^2-1)^{3/2}} \mathrm{d}x =\frac12 \int \frac{1}{u^{3/2}} \mathrm{d}u=-\frac{1}{\sqrt{y^2-1}}+c$$
And $$\int \frac{1}{x^2} \mathrm{d}x = -\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/945964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Quotient Topology, why is this set "saturated"?
It says $[2,3]$ is saturated with respect to $q$, but not open in $Y$.
BUt it doesn't make sense to me because
$q(A) = q([0,1) \cup[2,3]) = [0,1) \cup [2-1,3-1] = [0,1) \cup [1,2] = [0,2] = Y$, so the image is open in $Y$ as it is $Y$.
| Note that $p$ is almost 1-1, the only exception is $p(1) = 1 = p(2)$, this is where the two closed intervals of $X$ are "glued together", giving the result $[0,2]$, the image of $p$.
$p$ induces an equivalence relation on $X$ that only identifies $1$ and $2$ and no other points.
A set $B \subset X$ is saturated under ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/946114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find solution for $A * X = B$ I have a Matrix:
$$A= \pmatrix{1 & -2 & 1 \\ -1 & 3 & 2 \\ 0 & 1 & 4}$$
My task is to find $X$ from:
$$A * X = \pmatrix{4 & 0 & -3 & 1 \\ 1 & 5 & 2 & -1 \\ 0 & 1 & -1 & 2}$$
My problem is, that i dont know how to do this. I mean i could build several equations like:
$$1 * x1,1 -2 *x1,2 + ... | If you denote by $x_1$, $x_2$, $x_3$ and $x_4$ the columns of $X$ and by $b_1$, $b_2$, $b_3$ and $b_4$ the columns of the given $3\times 4$ matrix (call it $B$), you have essentially to solve the linear systems
$$
Ax_i=b_i\qquad(i=1,2,3,4)
$$
If you consider the “multiaugmented” matrix
$$
\left[\begin{array}{c|c|c|c|c}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/946255",
"timestamp": "2023-03-29T00:00:00",
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Multivariable optimization - how to parametrize a boundary?
A metal plate has the shape of the region $x^2 + y^2 \leq 1$. The plate is heated so that the temperature at any point $(x,y)$ on it is indicated by
$$T(x,y) = 2x^2 + y^2 - y + 3$$
Find the hottest and coldest points on the plate and the temperature at each ... | We parametrize $x=\sin t, y=\cos t$ (although it works if we switch $x$ and $y$ as well). We obtain: $$2\sin^2 t+\cos^2 t-\cos t+3.$$Take the derivative and solve: $$\sin t(2\cos t+1)=0.$$
Edit
An important part I left out is that $0\leq t \leq 2\pi$. We choose this time interval because it traverses the circle exactl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/946411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Concepts in mathematics which are referred to as 'generalizations' I am curious to know some theorems usually taught in advanced math courses which are considered 'generalizations' of theorems you learn in early university or late high school (or even late university).
For example, I know that Stokes's theorem is a gen... | The Parallelogram law of inner product spaces is a generalization of a theorem of Euclidean geometry.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/946486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 21,
"answer_id": 12
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Can the partial derivative of f(x,y) at (a,b) exist if f(x,y) is not continuous at (a,b)? Suppose f(x,y) is continuous for all $(x,y) \neq (a,b)$, (not continuous at (a,b)), can the partial derivative with respect to x (or y) at (a,b) still exist?
| Another example is
$$f(x,y) = \begin{cases} \frac{xy}{x^2+y^2} & \text{ if}\ (x,y)\neq (0,0) \\ 0 & \text{ if}\ (x,y)=(0,0) \end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/946527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Product rule trig This was given as a solution to a question and I've tried working it out but can never get the same answer. Here $x=rcosϕ$ and $y=rsinϕ$
It's mostly the first 2 lines I don't understand. Wouldn't $x^2 = r^2cos^2ϕ$ and $y^2 = r^2sin^2ϕ$? And how did the first 2 terms of the second line come along? I c... | $$\ r^2d\phi=r^2\cdot1\cdot d\phi=r^2\cdot(\cos^2(\phi)+\sin^2(\phi))\cdot d\phi=$$
$$\ =r^2\cos^2(\phi)d\phi+r^2\sin^2(\phi)d\phi=r\cos(\phi) r\cos(\phi)d\phi+r\cos(\phi) r\cos(\phi)d\phi$$
Now you have that:
$$\ r\cos(\phi)=x$$
and
$$d(r\sin(\phi))=\sin(\phi)dr+rd(\sin(\phi))=\sin(\phi)dr+r\cos(\phi)d\phi$$
$$d(r\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/946660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question on Lie derivative For a Lie derivative $\mathscr{L}_{X} Y$ of $Y$ with respect to $X$, we mean that for two smooth vector fields $X$ and $Y$ on a smooth manifold $M$ such that the following holds
$$
\mathscr{L}_{X} Y = \underset{t \rightarrow 0}{\lim} \frac{(\varphi_{-t})_{\star} Y - Y}{t}, \tag{1}
$$
where ... | Since $Y$ is smooth in the variable $p$, and now $Y_{\varphi_t(p)}$ (or maybe this notation: $Y_{p(t)}$) is just the restriction of $p$ to the integral curve, and therefore also smooth. Think of it as a composition of the smooth map $t\mapsto p(t)$ with $p\mapsto Y_p$.
On the other hand, $(\varphi_{-t})_\star$ is nothi... | {
"language": "en",
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Show a sequence such that $\lim_{\ N \to \infty} \sum_{n=1}^{N} \lvert a_n-a_{n+1}\rvert< \infty$, is Cauchy
Attempt. Rewriting this we have, $$\sum_{n=1}^{\infty} \lvert a_n-a_{n+1}\rvert< \infty
\,\,\,\Longrightarrow\,\,\, \exists N \in \mathbb{N}\ \ s.t,\ \ \sum_{n \geq N}^{\infty} \lvert a_n-a_{n+1}\rvert < \inf... | I'd go like this:
Assuming $\;m>n\;$ , we have
$$|a_m-a_n|=|a_m-a_{m-1}+a_{m-1}-a_{m-2}+a_{m-2}-a_{m-3}+\ldots+a_{n+1}-a_n|\le$$
$$\le\sum_{k=0}^{m-n-1}|a_{m-k}-a_{m-k-1}|\xrightarrow[m,n\to\infty]{}0$$
The last limit is not actually a double one, but rather "make $\;n\to \infty\;$ and thus also $\;m\to\infty\;$"
| {
"language": "en",
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What's the closure of $(a,b)$ in discrete topology on the real number $\mathbb{R}$ In my opinion, by definition, the closure of $(a,b)$ in discrete topology on the real number $\mathbb{R}$ is $(a,b)$. However, I just saw the answer for this question is $[a,b]$. Now I am not sure which one.
| A different approach is the following. The discrete topology is induced by the discrete metric.
Now sequences $x_n$ converge to $x$ in the discrete metric if and only if there exists $n_0\in \mathbb{N}$ such that $x_n=x$ for all $n\geq n_0$. Consequently, $(a,b)$ is the closure of itself in this topology.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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finding a function from given function here is a function for:
$f(x-\frac{\pi}{2})=\sin(x)-2f(\frac{\pi}{3})$
what is the $f(x)$?
I calculate $f(x)$ as follows:
$$\begin{align}
x-\frac{\pi}{2} &= \frac{\pi}{3} \Rightarrow x= \frac{5\pi}{6} \\
f(\frac{\pi}{3}) &=\sin\frac{5\pi}{6}-2f(\frac{\pi}{3}) \\
3f(\frac{\pi}{3}) ... | Assuming $f$ is defined for all $x\in\mathbb{R}$. First, note that for any $x$,
$$
f(x) = \sin\!\left(x+\frac{\pi}{2}\right)-2f\!\left(\frac{\pi}{3}\right) = \cos x -2f\!\left(\frac{\pi}{3}\right)
$$
so it only remains to compute $f\!\left(\frac{\pi}{3}\right)$. From the expression above
$$
f\!\left(\frac{\pi}{3}\right... | {
"language": "en",
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Show that $x \sin (\frac {1 } {x } ) $ is uniformly continuous on $(0,1) $ I want to prove that $f(x)=x \sin (\frac {1 } {x } ) $ is uniformly continuous on $0<x<1$.
If we consider the same function with the extra condition that $f $ is defined to equal zero at $x=0 $. then this new function would be continuous on $[0,... | In general if a function $f(x)$ is continuous on $(a,b)$ such that both $\displaystyle\lim_{x\to a+}f(x)$ and $\displaystyle\lim_{x\to b-}f(x)$ exists then $f$ is uniform continuous on $(a,b)$
| {
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"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Closed form of $\sum_{n=1}^{\infty}(-1)^{n+1} n (\log(n^2+1)-\log(n^2))$ How would you start computing this series?
$$\sum_{n=1}^{\infty}(-1)^{n+1} n (\log(n^2+1)-\log(n^2))$$
One of the ways to think of would be Frullani integral with the exponential function , but it's troublesome due to the power of $n$ under logar... | I'm getting the same answer as you.
$$ \sum_{n=1}^{\infty} (-1)^{n+1} n \log \left( \frac{n^{2}+1}{n^{2}}\right) = \sum_{n=1}^{\infty} (-1)^{n+1} n \int_{0}^{1} \frac{1}{n^{2}+x} \ dx$$
Then since $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{n^{2}+x}$ converges uniformly on $[0,1]$,
$$ \begin{align} &\sum_{n=... | {
"language": "en",
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Runge-Kutta methods and Butcher tableau What does the Butcher tableau of a Runge-Kutta method tell me about the method, besides the coefficients in its formulation? In particular, what requirements about it guarantee consistency and therefore convergence? I have been told something necessary is the row-sum condition, i... | The Butcher Tableau determines the stability function $R(z)$ of the corresponding method.
In particular, for the Linear Test equation due to Dahlquist
$$u'(t) = \lambda u(t) \Rightarrow u(t) = u_0 e^{\lambda (t - t_0)}$$
the stability function determines how the approximation $u_{n+1}$ follows from the previous iterate... | {
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Number of pizza topping combinations It seems there are lots of pizza questions but I'm not sure how to apply the answers to my problem. Obviously I'm not a mathematician. Essentially I'm trying to determine how many different variations of pizzas there are given the following parameters. You can choose unlimited, u... | You can choose the sauce in $\binom{6}{1} + 1$ ways (the $1$ is for no sauce). Then consider filling up $7+15+7+2$ blanks with either $0$ or $1$, $1$ if you want that topping, $0$ if you don't want it. This can be done in $2^{31}$ ways. So the total number of ways is
$$
(\binom{6}{1} + 1) \cdot 2^{31} = 7 \cdot 2\;147\... | {
"language": "en",
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Prove $\frac{n^2+2}{(2 \cdot n^2)-1} \to \frac{1}{2}$
Prove $\frac{n^2+2}{(2 \cdot n^2)-1} \to \frac{1}{2}$ for $n \to \infty$.
I've been looking at this for hours! Also, sorry I don't have the proper notation.
This is where I'm at:
$$
\left| \frac{n^2 + 2}{2 \cdot n^2 - 1} - \frac{1}{2}\right| = \left| \frac{5}{4 ... | $\left|\frac{n^2+2}{2n^2-1} - \frac {1}{2}\right| = \frac{5}{4n^2-2}$
Let $\epsilon>0$ be given. Let $n_0$ be the smallest integer such that $n\geq n_0> \sqrt{\frac{5}{4\epsilon} + \frac 12}$. Equivalently, $\epsilon>\frac{5}{4n^2-2}$.
Thus, $\left|\frac{n^2+2}{2n^2-1} - \frac {1}{2}\right|< \epsilon$, for all $n\geq ... | {
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Saving for retirement - how much? I'm working through a problem in the book "An Undergraduate Introduction to Financial Mathematics" and there is an example I can follow.
The problem is: Suppose you want to save for retirement. The savings account is compounded monthly at a rate of 10%. You are 25 years and you plan to... | Your payments-in are made at the end of the months. This information is not given explicit in the text.
The equation is
$x\cdot \frac{1-(1+\frac{0.1}{12})^{480}}{-\frac{0.1}{12}}=1500\cdot\frac{1-(1+\frac{0.1}{12})^{360}}{-\frac{0.1}{12}}\cdot \frac{1}{(1+\frac{0.1}{12})^{360}}$
This gives $x\approx 27.03$
| {
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3 Variable Diophantine Equation Find all integer solutions to $$x^4 + y^4 + z^3 = 5$$ I don't know how to proceed, since it has a p-adic and real solution for all $p$.
I think that the only one is (2, 2, -3) and the trivial ones that come from this, but I can't confirm it.
| After a careful investigation I present some results which might be helpful in the final resolution of this problem. Lets start with the original equation i.e.
$$x^4+y^4+z^3=5$$
It is easy to see that there is no solution of this equation where $x,y$ and $z$ are all positive. Second if $(x,y,z)$ is a solution then so i... | {
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What are Darboux coordinates? What are Darboux coordinates?
Are they different from coordinates on $\Bbb R^n$ or some smooth manifold?
I am familiar with Riemannian manifolds, but Darboux coordinates came up in some materials.
| A smooth manifold $M$ equipped with a closed, non-degenerate two-form $\omega$ is called a symplectic manifold. It follows almost immediately that a symplectic manifold is even dimensional.
Darboux's Theorem: Let $(M, \omega)$ be a symplectic manifold. For any $p \in M$, there is a coordinate chart $(U, (x^1, \dots, ... | {
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How to prove invariance of dot-product to rotation of coordinate system Using the definition of a dot-product as the sum of the products of the various components, how do you prove that the dot product will remain the same when the coordinate system rotates?
Preferably an intuitive proof please, explainable to a high-... | First you should show that for any two vectors $v$ and $w$ in $\mathbb{R}^n$ (taking $n=3$ if necessary) $v\cdot w = |v||w|\operatorname{cos}\theta $, where $\theta$ is the (smaller) angle between both vectors.
This is a very geometric fact and you can probably prove it to them if they know the cosines law. First obse... | {
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finding code of a function in GAP packages How can I find the codes related to a function in GAP? When I use "??" in front of the name of function there is no help, so I want to find the code of function among packages.
| First, GAP is an open-source project, and both the core system and GAP packages are supplied in the GAP distribution with their source code. You may use various tools available at your system to search for a string in the source code, for example
grep -R --include=*.g* BoundPositions pkg/*
Secondly, you may print a fu... | {
"language": "en",
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2nd order homogeneous ODE I am trying to solve a system of differential equations (for the full system see below) and I am stuck the following 2nd order ODE (with $a$ and $b$ being constants and $\dot x = \frac{dx}{dt}$):
$$\ddot x - \frac34 a\dot x^2 -b \dot x + 2 a b = 0$$
I tried to substitute $v := \dot x$, which l... | As I said in a comment, start defining $z=x'$; so the differential equation becomes $$\frac{dz}{dt} - \frac34 a z^2 -b z + 2 a b = 0$$ that is to say $$\frac{dz}{dt} = \frac34 a z^2 +b z - 2 a b $$ then, as Semsem suggested, it is separable; so $$\frac{dt}{dz} = \frac{1}{\frac34 a z^2 +b z - 2 a b}$$ so $$t+C=-\frac{2 ... | {
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How do I solve this geometric series I have this geometric series $2+1+ \frac{1}{2}+ \frac{1}{4}+...+ \frac{1}{128}$to solve. So I extract the number two and get $2(\frac{1}{2}^0+ \frac{1}{2}^1+...+ \frac{1}{2}^7)$
I use the following formula $S_n= \frac{x^{n+1}-1}{x-1}$ so I plug in the values in this formula and get ... | Hint: Its $$2 + (1+\frac12+ \frac14+ \cdots + \frac{1}{128})$$ not multiplied with $2$.
You can also think of it as follows: The first term is $a_1=2$ and the common ratio is $r=1/2$ and then you sum it using the formula where you last term is $a_9=1/128$.
Edit: If you do want to factor out a $2$, then you get $$2(1+\... | {
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Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module. Exercise: Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module.
I'm not sure about all different kind of modules, but this is a question of a book about associative algebras. It is not really e... | For a field $k$, $k[x]$ is a principal ideal ring. By the correspondence theorem, the only ideals of $k[x]/(x^2)$ are those generated by divisors of $x^2$. Thus $k[x]/(x^2)$ has exactly three submodules: $(x^2)/(x^2),(x)/(x^2)$ and $k[x]/(x^2)$.
So the existence of $(x)/(x^2)$ immediately tells you why $k[x]/(x^2)$ isn... | {
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Example of a non-trivial function such that $f(2x)=f(x)$ Could you give an example of a non-constant function $f$ such that
$$
f(x) = f(2x).
$$
The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the rationals.
I am wondering if there is any other such function o... | One more example
$$f(x) = \sin(2\pi\log_2x)$$
$$f(2x) = \sin(2\pi\log_2(2x)) = \sin(2\pi(1 + \log_2x)) = \sin(2\pi\log_2x) = f(x)$$
| {
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Proof of induction principle Theorem 1.1.3 (induction principle) of Dirk Van Dalen "Logic and Structure" states:
Let $A$ be a property, then $A(\phi)$ holds for all $\phi \in PROP$ if:
*
*$A(p_i)$, for all i;
*$A(\phi),A(\psi) \Rightarrow A(\phi \square \psi)$
*$A(\phi) \Rightarrow A(\neg \phi)$
I don't underst... | For the sake of avoiding confusion, I feel it should be pointed out precisely in what sense PROP is the "smallest" set of well-formed formulae. For example, $\{10,11\}$ is certainly the smallest set of consecutive integers that add up to $21$, but that does not mean that it is a subset of $\{6,7,8\}$.
However, when it ... | {
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How many non-collinear points determine an $n$-ellipse? $1$-ellipse is the circle with $1$ focus point, $2$-ellipse is an ellipse with $2$ foci. $n$-ellipse is the locus of all points of the plane whose sum of distances to the $n$ foci is a constant.
I know that $3$ non-collinear points determine a circle. $5$ non-coll... | The number of points needed to identify a $n$-ellipse is $2n+1$. This directly follows from the general equation of a $n$-ellipse
$$\sum_{i=1}^n \sqrt{(x-u_i)^2+(y-v_i)^2}=k$$
where the number of parameters is $2n+1$. So, for a $1$-ellipse (circle) we need $3$ noncollinear points to identify $3$ parameters ($u_1,v_1,k$... | {
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Books that use probabilistic/combinatorial/graph theoretical/physical/geometrical methods to solve problems from other branches of mathematics I am searching for some books that describe useful, interesting, not-so-common, (possibly) intuitive and non-standard methods (see note *) for approaching problems and interpret... | Mark Levi, The Mathematical Mechanic: Using Physical Reasoning to Solve Problems.
| {
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"answer_id": 2
} |
If $f(x)\in\mathbb{Q}[x]$ of degree $p$ and $\operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$ then $f(x)$ is irreducible. Let $f(x)\in\mathbb{Q}[x]$ , $p$ prime, $\deg f(x)=p$ and $G = \operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$, where $K$ the the splitting field of $f(x)$ over $\mathbb{Q}$.
Sh... | Assume that $\sigma$ is an automorphism of order $p$, also $\sigma$ is a permutation of the roots $a_1, \ldots , a_p$ of $f(x)$. Thus $\sigma$ must permute these roots in a cycle. This means that all the roots are in fact conjugate. Thus $f(x)$ is irreducible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/948711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Prove this equality by using Newton's Binomial Theorem Let $ n \ge 1 $ be an integer. Use newton's Binomial Theorem to argue that
$$36^n -26^n = \sum_{k=1}^{n}\binom{n}{k}10^k\cdot26^{n-k}$$
I do not know how to make the LHS = RHS. I have tried $(36^n-26^n) = 10^n $ which is $x$ in the RHS, but I don't know what to do ... | Bring the $26^n$ to the other side. You are then looking at the binomial expansion of $(26+10)^n$. The $26^n$ is the $k=0$ term that was missing in the given right-hand side.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/948779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the precedence of the limit operator? I would like to know the precedence of the $\lim$ operator.
For instance, given the following expression:
$$f(x) = \lim_{x \to a} u(x) + v(x)$$
Does the limit apply only to the term?
$$f(x)=\left(\lim_{x \to a} u(x)\right) + v(x)$$
Or does it apply to the entire expression?... | In most textbooks I've seen the limit operator has higher precedence than addition/subtraction:
$$\lim_{x \to a} u(x) + v(x) \equiv \left(\lim_{x \to a} u(x)\right) + v(x)$$
Where it gets hairy is whether the limit operator has higher precedence than multiplication/division:
$$\lim_{x \to a} u(x) v(x) \stackrel?= \left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/948861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A question on endomorphisms of an injective module This is a homework question I am to solve from TY Lam's book Lectures on Modules and Rings, Section 3, exercise 23.
Let $I$ be an injective right $R$-module where $R$ is some ring.
Let $H= \operatorname{End}(I),$ the endomorphisms on $I$.
I need to show that given $f,... | Hint. $0\to\operatorname{Im}(h)\to I$ and define $\bar f:I/\ker h\to I$ by $\bar f(\bar x)=f(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/948950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is there a proof for what I describe as the "recursive process of mathematical induction for testing divisibility". I was working on my homework for Discrete Math, and we were asked to "Prove: $6 \mid n^{3}+5n$,where $n\in \mathbb{N}$" my solution varied significantly from how I have seen it done by others. I noticed ... | Nice!
Yes that's correct, if you have a sequence $a_1,a_2,a_3,\ldots$ of terms which are all divisible by a fixed integer $m$, then their differences must be also. If you are trying to prove that each of the $a_i$ are divisible by $m$, then you could construct a new sequence:
\begin{align*}
b_1&=a_2-a_1 \\
b_2&=a_3-a_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/949062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Exercise books in analysis I'm studying Rudin's Principles of mathematical analysis and I was wondering if there are some exercise books (that is, books with solved problems and exercises) that I can use as a companion to Rudin.
The books I'm searching for should be:
*
*full of hard, non-obvious, non-common, and th... | This is what I recommend to students learning analysis as a good companion:
http://minds.wisconsin.edu/handle/1793/67009
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/949197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 0
} |
Finding cartesian equation for trigonometric parametric forms I'm trying to find the cartesian equation for these parameteric forms:
$$
x = sin\theta + 2 cos \theta \\
y = 2 sin\theta + cos\theta
$$
I tried:
$$\begin{align}
x^2 & = sin^2\theta + 4cos^2\theta \\
& = 1 - cos^2\theta + 4cos^2\theta \\
& = 1 + 3cos^2\the... | In general, $(a+b)^2\ne a^2+b^2$ unless $ab=0$
Solve for $\sin\theta,\cos\theta$ in terms of $x,y$
Then use $\sin^2\theta+\cos^2\theta=1$ to eliminate $\theta$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/949290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding Solutions to Trigonometric Equation Find all $x$ in the interval (0, $\frac{\pi}{2}$) such that $$\frac{\sqrt{3}-1}{\sin x} + \frac{\sqrt{3}+1}{\cos x} = 4\sqrt{2}.$$
| Rewrite it in the form
$$2\sqrt2\left(\frac{\sqrt3-1}{2\sqrt2}\cos x+\frac{\sqrt3+1}{2\sqrt2}\sin x\right)=2\sqrt2\sin 2x.$$
For $\phi=\arcsin\frac{\sqrt3-1}{2\sqrt2}$ it implies
$$\sin(x+\phi)=\sin 2x,$$
i.e. $x+\phi=2x+2\pi n$ or $x+\phi=\pi-2x+2\pi n$, $n\in\Bbb Z$. Therefore, the only solutions in $(0,\pi/2)$ are $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/949390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $V$ be finite dimensional v.s. and $0 \ne T\in \mathscr L(V)$ , then $\exists$ $S \in \mathscr L(V)$ such that $0 \ne T \circ S$ is idempotent If $V$ is a finite dimensional vector space and $T \ne0$ is a linear operator on $V$ , then how may we prove that there is a linear operator $S$ on $V$ such that $T\circ S$ ... | Hint: Suppose $(c,v)$ is an eigenpair of $T$. Consider $T \circ P_v$, where $P_v$ is the orthogonal projector onto the span of $v$. When $c=1$ or $c=0$, I claim this is idempotent. How do you fix it otherwise?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/949504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that if $f:[a,b]\to\Bbb{R}$ is one to one and has the intermediate value property, then $f$ is strictly monotone I got this problem:
Prove that if $f:[a,b]\to\Bbb{R}$ is one to one and has the intermediate value property, then $f$ is strictly monotone.
That is, we must show that $\forall x,y\in[a,b], x<y \to f(x... | I'll show that if $f$ is one to one and not monotone in $[a,b]$ then there exist $x_1,x_2,x_3\in[a,b]$ such that $x_1<x_2<x_3$ and $f(x_2)<f(x_1),f(x_3)$ or $f(x_1),f(x_3)<f(x_2)$:
(Note: I tried to write the proof similiar to nested if statements in computer programming for better flow)
Since $f$ is one to one we get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/949567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Onion-peeling in O(n^2) time I am working on the Onion-peeling problem, which is: given a number of points, return the amount of onion peels. For example, the one below has 5 onion peels.
At a high level pseudo-code, it is obvious that:
count = 0
while set has points:
points = find points on convex hull
set.re... | Jarvis' March (Gift Wrapping) Algorithm takes $O(nh)$ time, where $h$ is number of points on convex hull.
Hint:
Suppose algorithm takes $i$ iterations to complete, and $h_k$ be the number of points on the $k$th convex hull $(1 \le k \le i )$. Also, let $n_k$ be the number of points remaining after first $k-1$ iteration... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/949697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Differentiability question ends up in contradiction. Let $f(x)=x^3cos\frac{1}{x}$ when $x\neq0$ and $f(0)=0$.
Is $f(x)$ differentiable at $x=0$?
My first attempt
Definition: A function is differentiable at $a$ if $f'(a)$ exists.
$$f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{x+h-x}$$
$$f'(0)=\lim_{h \to 0}\frac{f(h)-f(0)}{h... | $1$) You showed correctly, from the definition of the derivative, that $f'(0)=0$.
$2$) You used the ordinary differentiation formula to find the derivative of $f(x)$ when $x\ne 0$. That is perfectly fine.
Then you decided to use the limit as $x\to 0$ of the $f'(x)$ calculated in $2$) to calculate $f'(0)$. That is in p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/949771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Prove the difference of roots is less than or equal to the root of the difference I am doing a larger proof that requires this to be true:
$|\sqrt{a} - \sqrt{b}| \leq \sqrt{|a - b|}$
I can square both sides to get
$a - 2\sqrt{a}\sqrt{b} + b \leq |a - b|$
Note that a and b are > 0.
I also know that
$|c| - |d| \leq |c - ... | $\sqrt{a}$ and $\sqrt{a}$ are positive, therefore,
$$-\sqrt{b}\leq \sqrt{b}$$ ant it follows that
$$\sqrt{a}-\sqrt{b}\leq \sqrt{a} +\sqrt{b} $$
by symmetry we have
$$|\sqrt{a}-\sqrt{b}|\leq |\sqrt{a} +\sqrt{b} |=\sqrt{a} +\sqrt{b}$$
Now muptiply both sides by $|\sqrt{a}-\sqrt{b}|$ to get
$$|\sqrt{a}-\sqrt{b}|^2\leq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/949866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Why is $\sum_{k=j}^{i+j}(j+i-k) = \sum_{k=1}^{i}(k)$ $\displaystyle\sum_{k=j}^{i+j}(j+i-k) = \displaystyle\sum_{k=1}^{i}(k)$
I know the above are equal through testing it out with arbitrary values, but I can't get an intuitive grasp as to why this is.
| \begin{align}
S &= \sum_{k=j}^{i+j} (i+j-k) \\
&= (i) + (i-1) + \cdots + ((i+j)-(i+j-1)) + ((i+j)-(i+j)) \\
&= (i) + (i-1) + \cdots + 1 + 0 \\
&= 0 + 1 + \cdots + (i-1) + i \\
&= \sum_{k=0}^{i} k.
\end{align}
Also
\begin{align}
\sum_{n=1}^{m} n = \frac{m(m+1)}{2}
\end{align}
such that
\begin{align}
\sum_{k=j}^{i+j} (i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/949996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Bibinomial coefficient integer For integers $n \ge k \ge 0$ we define the bibinomial coefficient. $\left( \binom{n}{k} \right)$ by
$$ \left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .$$
What are all pairs $(n,k)$ of integers with $n \ge k \ge 0$ such that the corresponding bibinomial coefficient is an integer?
(... | Hint:
Use that
$$
n!! = 2^kk!
$$
when $n=2k$ and that
$$
n!! = \frac{n!}{2^k k!}
$$
when $n=2k+1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/950119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Reroll 2 dice sum probability My statistics are very far in my memory and I am not a math guru so I do not understand half of fuzy symbols used in most post that could have the answer I am looking for. So I would ask for a very simple and easy to understand answer pretty please :)
I have 2 dice, numbered {0,0,1,1,2,2}.... | Consider we roll the two dice and, conditional on the sum of the face values, roll again (if the sum of face values is 0, 1, or 2) or stop (if the sum of the face values is 3 or 4). The sum of the two dice on the first roll will be 0 (zero) with probability $\dfrac{4}{36}$ as indicated. In this case, roll again and the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/950228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$\int\frac{2x+1}{x^2+2x+5}dx$ by partial fractions $$\int\frac{2x+1}{x^2+2x+5}dx$$
I know I'm supposed to make the bottom a perfect square by making it $(x+1)^2 +4$ but I don't know what to do after that. I've tried to make $x+1= \tan x$ because that's what we did in a class example but I keep getting stuck.
| $$
\int\frac{2x+1}{x^2+2x+5}dx
$$
I would first write $w=x^2+2x+5$, $dw=(2x+2)\,dx$, and then break the integral into
$$
\int\frac{2x+2}{x^2+2x+5}dx + \int\frac{-1}{x^2+2x+5}dx.
$$
For the first integral I would use that substitution. Then
$$
\overbrace{\int\frac{-dx}{x^2+2x+5} = \int\frac{-dx}{(x+1)^2 + 2^2}}^{\text{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/950354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Russell's paradox and axiom of separation I don't quite understand how the axiom of separation resolves Russell's paradox in an entirely satisfactory way (without relying on other axioms).
I see that it removes the immediate contradiction that is produced by unrestricted comprehension, but it seems that we still need f... | I don't think the axiom scheme of separation "resolves" Russell's paradox at all, but restricts the way of using predicates to determine sets.
The paradox is nothing but a proof that there is no one-to-one correspondence between predicates and classes: there are predicates that not defines a class. When writing sets as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/950428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
$\operatorname{rad}(I)=\bigcap_{I\subset P,~P\text{ prime}}P$ $R$ commutative ring with unity. $I$ $R$-ideal. Then $\operatorname{rad}(I)=\bigcap_{I\subset P,~P\text{ prime}}P$. That is, the radical of $I$ is the intersection of all prime ideals containing $I$.
There is a proof of this in my textbook, but I do not und... | The following theorem, or some equivalent, probably appears in your book.
In a ring $R$, given an ideal $I$ and a multiplicative subset $S$ such that $S \cap I = \emptyset$, there exists a maximal ideal among those containing $I$ and disjoint from $S$; any such ideal is prime.
To prove this, note that such ideals are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/950562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
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