Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Finding a probability on an infinite set of numbers I have $\Omega={1,2,3,...}$ and the possibility of each number is $P(A) = 2^{-n}$, $n=1,2,3...$
I have to prove $P(\Omega) = 1$ . I can understand that from the graph, but how do I actually prove it?
| $P(Ω) = \sum_{n=1}^{\infty} P(n) = \sum_{n=1}^{\infty} 2^{-n} = \frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3} + ..... = \frac{1/2}{1-1/2} =1 $
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Let f,g be bilinear forms on a finite dimensional vector space. Show that there exist unique linear operators Let $f,g$ be bilinear forms on a finite dimensional vector space.
(a) Suppose $g$ is non-degenerate. Show tha there exist unique linear operators $T_{1}, T_{2}$ on $V$ such that $f(a,b)=g(T_{1} a,b)=g(a,T_{2} b... | Denote the $k$-vector space in question by $V$. As $V$ is finite dimensional, $V$ and $V^* := L(V,k)$ (the dual space) have the same dimension. Define a map $\Theta_g \colon V \to V^*$ by $\Theta_g a = g(a, \cdot)$. As $g$ is non-degenerated, $\Theta_g$ is one-to-one: If $a \in \ker \Theta_g$, we have $g(a,b) = \Theta_... | {
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Find number of functions such that $f(f(a))=a$ Let X ={1, 2, 3, 4}. Find the number of functions $f : X \rightarrow X$ satisfying $f(f(a)) = a$ for all $1 \le a \le 4$.
I took the $f(x) =x$ and, then there are 1 possibilities. But answer is given as 10. How is it 10?
| HINT: If $f(x)=x$, then of course we’ll have $f(f(x))=x$ as well, but there’s another possibility: if $f(x)=y$ and $f(y)=x$, then $f(f(x))=f(y)=x$ and $f(f(y))=f(x)=y$, so both $x$ and $y$ behave correctly. These are the only possibilities, however. Let’s give each such function a code describing which elements of $X$ ... | {
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How many such strings are there? How many 50($=n$) digit strings($=f(n)$) composed of only zeros and ones are there such that all "ones" should be in groups of at least 3(The grouping should be explicit), if they occur and must be separated by atleast one "zero".
Valid Examples:
$$(111)0(111),\;\;\;(1111)00(111)0(11111... | So here's a more elementary way to find a recursion:
Denote $a(n)$ be the number of such strings that end up with a 0.
Denote $b(n)$ the number of strings that end up with a 1.
Now, we find $f(n)=a(n)+b(n)$.
Also, easily we obtain $a(n+1)=f(n)=a(n)+b(n)$ since we can add a 0 to any string.
Now, we are left to find a re... | {
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Proof that every vector field on a Lie group is left-invariant I am just starting a course on Lie groups and I'm having some difficulty understanding some of the ideas to do with vector fields on Lie groups.
Here is something that I have written out, which I know is wrong, but can't understand why:
Let $X$ be any vect... | Your problem is with the equality $(X(f))(L_g(e))= X(f\circ L_g)(e).$ Note that in $(X(f))(L_g(e))$ you first get the derivative of $f$ with respect to $X$ evaluated at $g.$ But, in $X(f\circ L_g)(e)$ you modify the function by a left translation. It holds that $(f\circ L_g)(e)=f(g)$ but you cannot say anything at nea... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate the limit without using the L'Hôpital's rule $$\lim_{x\to 0}\frac{\sqrt[5]{1+\sin(x)}-1}{\ln(1+\tan(x))}$$
How to evaluate the limit of this function without using L'Hôpital's rule?
| I thought that it would be instructive to present a way forward that circumvents the use of L'Hospital's Rule, asymptotic analysis, or other derivative-based methodologies. To that end, herein, we use some basic inequalities and the squeeze theorem.
In THIS ANSWER for $x>-1$, I showed
$$\frac{x}{x+1}\le\log (1+x)\le... | {
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How to evaluate the limit $\lim_{x\to 0} \frac{1-\cos(4x)}{\sin^2(7x)}$ I am lost in trying to figure out how to evaluate the $$\lim_{x\to 0} \frac{1-\cos(4x)}{\sin^2(7x)}.$$
So far, I have tried the following:
Multiply the numerator and denominator by the numerator's conjugate $1+\cos(4x)$, which gives $\frac{\sin^2(4... | HINT:
Using $\cos2A=1-2\sin^2A,$
$$\dfrac{1-\cos4x}{\sin^27x}=2\cdot2^2\cdot\left(\dfrac{\sin2x}{2x}\right)^2\cdot\dfrac1{7^2\cdot\left(\dfrac{\sin7x}{7x}\right)^2}$$
| {
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Nat Deduction Proof - Distributive property I need to prove (P v Q) ^ (P v R) |- P v (Q ^ R) using natural deduction and propositional logic. I should be able to do it using only AND and OR rules, but I am stuck on how to assume Q and R. This is what I have:
(P v Q) ^ (P v R)________premise
P v Q________________^e1__... | Assume $\rm \neg P$, then from 2 and 3 derive $\rm Q$ and $\rm R$. From that you have $\rm Q\wedge R$, and so on...
The rule, $\rm \neg P,\, P\vee Q \,\vdash\, Q$ , is called Deductive Syllogism. Sometimes also known as disjunctive elimination ($\rm\vee e$), or historically as modus tollendo ponens.
| {
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"timestamp": "2023-03-29T00:00:00",
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existence of a linear map Let $V$ and $W$ be finite dimensional spaces. Given a positive integer $m$ and vectors $v_1, .., v_m \in V$ and $w_1, .., w_m \in W$. We assume that for every linear combination $\sum_{i=1}^m a_i v_i = 0$, we have $\sum_{i=1}^m a_i w_i = 0$. I want to show that there exists a linear map $T : ... | It is simpler if you take the basis $e_1,\ldots,e_p$ to be a subset of $v_1,\ldots,v_m$. Without loss of generality, you can take $e_i=v_i,\quad i=1,\ldots, p.$ Let $T:V'\to W$ be the unique linear transformation satisfying $T(e_i)=w_i,\quad i=1,\ldots,p$. It follows from the hypothesis that $T$ is the required transfo... | {
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Smallest n such that there's a polynomial in $\Bbb Z_n [X]$ of degree 4 that has 8 roots in $\Bbb Z_n$
I'm looking for the smallest positive integer $n$ such that there's a quartic polynomial in $\Bbb Z_n [X]$ that has 8 distinct roots in $\Bbb Z_n$.
I have n equal to 15 with roots 1, 2, 4, 7, 8, 11, 13, 14 and equat... | Every number raied to $4$ ends in the same digit than the number itself, so $X^4-X$ has 10 roots in $\Bbb Z_{10}$, so $15$ is not the answer.
| {
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"timestamp": "2023-03-29T00:00:00",
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Integration of Legendre Polynomial I need to evaluate the following integral
\begin{equation} \int_{-1}^1 \frac{d^4P_l(x)}{dx^4}P_n(x)dx\end{equation}. Of course the answer I need is in terms of $l$ and $n$. Does anyone have any idea how to proceed?
| The facts below should allow you to compute the integral in question in terms of $l$ and $n$:
*
*The Legendre polynomials $P_0, \dots, P_n$ form a basis for the space of polynomials of degree at most $n$.
*The Legendre polynomials are orthogonal:
$$
\int_{-1}^{1} P_m(x) P_n(x)\,dx = {2 \over {2n + 1}} \delta_{mn}
... | {
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A paradox between intuition and equations regarding the surface of revolution? On the surface of revolution $$\sigma(u,v)=(f(u)cosv,f(u)sinv, g(u))$$ by the geodesic equations (i.e. $(1)\ \ddot{u}=f(u) \dfrac{df}{du} \dot{v}^2$ and $(2)\ \dfrac{d}{dt}(f(u)^2\dot{v})=0 $) we can show that every meridian (i.e. v=constant... | To give you an example, consider the sphere, which is a surface of revolution given by
$$(\sin u \cos v, \sin u \sin v, \cos u),$$
that is, $f(u) = \sin u$ and $g(u) = \cos u$, $u\in (0, \pi)$. Note that $f'(u) = \cos u$ is $0$ if and only if $u = \pi/2$, which corresponds to the great circle. Note that if $u_0 \neq \... | {
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Find an increasing sequence of rationals that converges to $\pi$ I am not sure how to construct a sequence that would convey convergence to $\pi$.
Except maybe $a_n=\{\pi + 1/n\}$ but the terms would not be rational.
Looking for an adequate way to show to satisfy the three conditions.
| The sequence in graydad's answer can be written in terms of the floor function as
$$n \mapsto 10^{-n} \lfloor 10^n \pi \rfloor.$$
This suggests some cute generalizations. Instead of the decimal expansion for $\pi$, we can use the expansion in any base $q$:
$$n \mapsto q^{-n} \lfloor q^n \pi \rfloor$$
converges to $\pi$... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $C_3 \times C_3$ is not isomorphic to $C_9$ I am trying to show that $C_3 \times C_3$ is not isomorphic to $C_9$. I am new to group-theory so forgive my foolish intuitions. One natural thought I had was to show that they are of differing cardinality but intuitively, why can't we map the elements $c_1,c_2,c_3 ... | Each isomorphism preserves the orders of elements of the respective groups. But $C_9$ has an element of order $9$ whereas $C_3\times C_3$ does not. Hence they are not isomorphic.
| {
"language": "en",
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Existence of a discrete family of sets Given that in a metric space $(X,d)$, there exists a sequence $(x_n)$ which does not cluster in $X$. Then how to construct a discrete family of sets $V_n$ such that each $V_n$ is a closed neighbourhood of $x_n$? By a discrete family of sets, we mean a family such that every point ... | For $n\in\Bbb N$ let $r_n=\frac13\inf\{d(x_n,x_k):k\ne n\}$; since $x_n$ is not a cluster point of the sequence, $r_n>0$. Now for $n\in\Bbb N$ let $V_n$ be the closed ball of radius $\frac{r_n}{2^n}$ centred at $x_n$. I’ll leave it to you to check the details; the choice of $r_n$ ensures that the sets $V_n$ are pairwis... | {
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Euler number zero for odd dimensional compact manifolds I need to prove that
every compact manifold of odd dimension has Euler number zero.
The Euler number of $M$ compact and oriented is
$$
e(M):=\int_Ms_0^*\phi(TM)
$$
where $s_0$ is the zero section of $TM$ and $\phi(TM)$ is its Thom class.
We also proved that
$$
e(... | Let's suppose $M$ orientable. And let $dim(M)=2n+1$.
By Poincaré duality, we get
$$
H^q(M)\cong (H^{2n+1-q}(M))^*
$$
for every $q$.
Since every compact manifold is of finite type (hence its cohomology rings are finite dimensional) and since every finite dimensional space is isomorphic to its dual, we get
$$
h^q(M)=h^{... | {
"language": "en",
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Is the function $y=xe^x$ invertible? I'm wondering if the equation $re^r=se^s$ has any answer. If there is any answer,and $r=-1+v,s=-1-v$ in which $v$ is a positive real number,what can we say about $v$?
Thank you in advance.
| Let $r=-1+v$ and $s=-1-v$. You are looking for zeros of the function:
$$
f(v) = (-1+v)e^{-1+v} - (-1-v)e^{-1-v}.
$$
Observe that one solution is $v=0$ which corresponds to $r=s=-1$.
Compute the derivative
$$
f'(v) = ve^{-1-v} (-1+e^{2 v})
$$
and notice that it is never negative and it is zero only for $v=0$. Hence $f$ ... | {
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Rank of a matrix of binomial coefficients This question arose as a side computation on error correcting codes.
Let $k$, $r$ be positive integers such that $2k-1 \leqslant r$ and let $p$ a prime number such that $r < p$. I would like to find the rank of the following $k \times k$ matrix with coefficients in the field $F... | Write $A(r,k)$ for the matrix you describe. Then
$$
\det A(r,k) =\prod_{i,j=1}^k \frac{r-k-1+i+j}{i+j-1}.
$$
This follows from equation (2.17) in Krattenthaler's Advanced determinant calculus (with $a=n=k$, $b=r-k$). For $p\geq r+k$ every factor in the product has numerator coprime to $p$, so the reduction of $A(r,k)$ ... | {
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Existence of linear functionals
Let $f$ be any skew-symmetric bilinear form on $\Bbb R^3$. Prove that there exist linear functionals $L$ and $M$ on $\Bbb R^3$ such that $$f(a,b)=L(a)M(b)-L(b)M(a)$$
I'm having a similar problem in showing existence of linear functionals in other questions as well. How do I proceed in ... | Using the standard basis $e_1, e_2, e_3$ of $\mathbb R^3$, $f$ is represented by the skew-symmetric matrix
$$ \pmatrix{0 & a & b\cr -a & 0 & c\cr -b & -c & 0\cr}$$
where $a = f(e_1, e_2)$, $b = f(e_1, e_3)$, $c = f(e_2,e_3)$.
The case $a=b=c=0$ is easy, so suppose at least one (wlog $a$) is nonzero.
Then one solution i... | {
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Height, Time of the Ball The height $h$ in feet of a baseball hit $3$ feet above the ground is given by $h=3+75t-16t²$ where $t$ is the time in seconds.
a. Find the height of the ball after two seconds
b. Find the time where the ball hits the ground in the field
c. What is the maximum height of the ball?
| Hints :
*
*a) replace t by 2 and evaluate h
*b) replace h by 0 and find the value of t
*c) find the vertex of the parabola.
| {
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Definition of derivative of real function in baby Rudin Let $f$ be defined (and real-valued) on $[a,b]$. For any $x\in [a,b]$ form a quotient $$\phi(t)=\dfrac{f(t)-f(x)}{t-x} \quad (a<t<b, t\neq x),$$ and define $$f'(x)=\lim \limits_{t\to x}\phi(t),$$ provided this limit exists in accordance with Defintion 4.1.
I have... | In short, this is an example of Rudin's sublime succinctness. His definition includes the usual two-sided definition when $x \in (a,b)$ and also the one-sided definition $x = a$ or $x = b$. To get the one-sided definition, notice that he requires $t \in (a,b)$.
Rudin was perhaps over-succinct in that he failed to ment... | {
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Matrix exponential, computation. What is $e^A$, where$$A = \begin{pmatrix} 0 & 1 \\ 4 & 0 \end{pmatrix}?$$
| You first have to find a basis in which the matrix has the form $B= D+N$, where $D$ is a diagonal matrix, $N$ is nilpotents and $DN=ND$. Then, since $D$ and $N$ commute,
$$\mathrm e^{B}=\mathrm e^{D}\mathrm e^{N}$$
The exponential of a diagonal matrix $D$ is the diagonal matrix with the exponentials of the diagonal el... | {
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What kind of graph is this? A pie chart? Besides a gross use of Comic Sans, what kind of graph is this? I've seen it used for population statistics before, but I don't know what it's called.
My first instinct was to just say "bullet chart" due to process of elimination. After researching what a bullet chart is, which ... | e) None of the previous ones.
The graph is a scatter plot of filled squares, also know as fragmented partition chart.
| {
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Show that $\{\frac{n^2}{n^2+n}\}$ is convergent
Use only the Archimedean Property of $\mathbb{R}$ to show $\{\frac{n^2}{n^2+n}\}$ is convergent
Let $a_n=\{\frac{n^2}{n^2+n}\}=\{\frac{n}{n+1}\}=\{1-\frac{1}{n+1}\}=\{1\}-\{\frac{1}{n+1}\}$.
$\{1\}$ is convergent to $1$. Show $\{\frac{1}{n+1}\}$ is convergent. So let ... | You need to know that the sum of two convergent sequences is convergent for your argument to be 100% okay. This is probably "not known" now (if it is, your proof is fine).
You are on the right track, your proof only needs to show that $\left|(1-\frac{1}{n+1})-1\right|<\epsilon$ which you basically did!
| {
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prove the sequence is increasing Im asked to show the sequence $a_{n+1}=\sqrt{3+2a_{n}}$ where $a_{1}=0, a_{2}=1$ is increasing and bounded and therefore convergent.
I don't even know how to start the proof. Im sure it increases because its essentially adding a really small number each time and the number gets very sma... | Note that $a_n \ge 0$ and
$$a_{n+1}^2 - a_n^2 = 3 + 2a_n - 3 -2a_{n-1} = 2(a_n - a_{n-1})$$
thus one can argue by induction that $a_n$ is increasing.
Although you didn't ask, you can also prove by induction that $a_n \le 3$.
| {
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Demonstrate $f(a,b)=2^a (2b+1)-1$ is surjective using induction I am trying to show that $f:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$, where $f(a,b)=2^a (2b+1)-1$, is surjective using induction (possibly strong induction). In the case $n=0$, it is easy to see that $(0,0)$ does the trick.
Suppose there exist ele... | It is more convenient to show that every positive integer $n$ can be represented in the form $2^a(2b+1)$, where $a$ and $b$ are nonl-negative integers. We use strong induction. The result is clearly true for $n=1$. Let us suppose it is true for all $k\lt n$, and show it is true for $n$.
Perhaps $n$ is odd, say $2b+1$. ... | {
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Implicit Differentiation of Two Functions in Four Variables Can someone please help with this homework problem: Given the equations
\begin{align}
x^2 - y^2 - u^3 + v^2 + 4 &= 0 \\
2xy + y^2 - 2u^2 + 3v^4 + 8 &= 0
\end{align}
find $\frac{\partial u}{\partial x}$ at $(x,y) = (2,-1)$.
In case it may be helpful, we know f... | Consider $u(x,y), v(x,y)$ as functions and $x,y$ variables. Take the $x$ derivative of both equations ($y$ is a constant). Use the chain rule to derivate $u$ and $v$. Then set $x=2, y = -1$ and solve.You have 4 equations and 4 unknowns ($u, v, \frac{\partial u}{\partial x}, \frac{\partial v}{\partial x}$) , so you shou... | {
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Compute $\lim_{n \to \infty} n(\frac{{\pi}^2}{6} -( \sum_{k=1}^{n} \frac{1}{k^2} ) ) $ Basel series
$$ \lim_{n\to \infty} \sum_{k=1}^{n} \frac{1}{k^2} = \frac{{\pi}^2}{6} $$
is well known. I'm interested in computing the limit value
$$
\lim_{n \to \infty} n\left(\frac{{\pi}^2}{6} - \sum_{k=1}^{n} \frac{1}{k^2} \right)... | By the Stolz rule, for $a_n =\frac{\pi^2}{6} - \sum\limits_{k=1}^n \frac{1}{k^2}$ and $b_n = \frac{1}{n}\downarrow 0$,
$$
\lim_{n\to\infty} \frac{a_n}{b_n} = \lim_{n\to\infty} \frac{a_{n-1}-a_n}{b_{n-1}-b_n} = \lim_{n\to\infty} \frac{\frac{1}{n^2}}{\frac{1}{n-1}-\frac{1}{n}} = 1.
$$
This also can be used to get furthe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1471610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
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A combinatoric/probability puzzle A horse race has four horses A, B, C and D. The probabilities for each horse to finish first or second are P(A)=80%, P(B)=60%, P(C)=40% and P(D)=20%. The probabilities add up to 200% because one horse will finish first with 100% probability and one horse will finish second with 100% ... | Although there is no unique solution, I think useful answers can be derived by searching for the maxiumum entropy propability distribution :-)
https://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1471702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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find the solution set of the following inequality with so many radical I get lost
$\sqrt[4]{\frac{\sqrt{x^{2}-3x-4}}{\sqrt{21}-\sqrt{x^{2}-4}}}\geqslant x-5$
Edit
I get online with wolfram
-5 < x <= -2 || x == -1 || 4 <= x < 5
| First let us check where the above formula is defined:
$ x^2-3x-4 \geq 0 \; \; \Leftrightarrow \; \; x\in ] -\infty , -1] \cup [4,+\infty[$.
$x^2-4 \geq 0 \; \; \Leftrightarrow \; \; x\in ] -\infty , -2] \cup [2,+\infty[ $.
$ \sqrt{21 } - \sqrt{ x^2-4} > 0 \; \; \Leftrightarrow \; \; x\in ]-5,-2[ \cup ]2,5[.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1471835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Required number of simulation runs I have the following problem:
One wants to estimate the expectation of a random variable X. A set of 16 data values (i.e. simulation outputs) is given, and one should determine roughly how many additional values should be generated for the standard deviation of the estimate to be les... | If $X$ is a member of the normal family and you are estimating $\mu$, then
$Q = 15S_{16}^2/\sigma^2 \sim Chisq(15).$
Thus, $$P(Q > L) = P(\sigma^2 < 15 S_{16}^2/L) = 0.96,$$
where $L \approx 7.26$ cuts 5% of the probability from the lower tail
of $Chisq(15).$
Then you have a pretty good (worst case) upper bound for $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1471959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Check if a point is inside a rectangular shaped area (3D)? I am having a hard time figuring out if a 3D point lies in a cuboid (like the one in the picture below). I found a lot of examples to check if a point lies inside a rectangle in a 2D space for example this on but none for 3D space.
I have a cuboid in 3D space... | The three important directions are $u=P_1-P_2$, $v=P_1-P_4$ and $w=P_1-P_5$. They are three perpendicular edges of the rectangular box.
A point $x$ lies within the box when the three following constraints are respected:
*
*The dot product $u.x$ is between $u.P_1$ and $u.P_2$
*The dot product $v.x$ is between $v.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1472049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 0
} |
Can you choose -1 as the multiplicative unit? And what is a positive number? If one starts with the cyclic group of integers and want to introduce multiplikation the ordinare choice of multiplicative identity is the generator 1. But since 1 and its inverse -1 is sort of the same in the cyclic group, there is an automor... | *
*There is an automorphism of $\mathbb{Z}$ the abelian group that exchanges $1$ and $-1$, but it doesn't respect multiplication: that is, it isn't an automorphism of $\mathbb{Z}$ the ring, which has no nontrivial automorphisms.
*Positivity comes from an order, which in general neither exists nor is unique. $\mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1472154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the name for the mathematical property involving addition or subtraction of fractions over a common denominator? For any number $x$ where $x\in\Bbb R$ and where $x\ne0$, what is the mathematical property which states that:
$${1-x^2\over x} = {1\over x} - {x^2\over x}$$
| Just the computation rules for fractions:
$$
\frac{a - b}{c} = \frac{a + (-b)}{c} \\
\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c} \\
\frac{(-b)}{c} = -\frac{b}{c}
$$
for $a = 1$, $b = -x^2$ and $c = x \ne 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1472265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Why do we use square in measuring a qubit with probability? A superposition is as follow:
$$ \vert\psi\vert = \alpha\vert 0\rangle + \beta\vert 1\rangle. $$
When we measure a qubit we get either the result 0, with probability $\vert \alpha\vert^{2},$ or the result 1, with probability $\vert \beta\vert^{2}.$
Blockquo... | Depending on how you are setting up your quantum states, it is usually taken as an axiom of the system that if a state $|\psi\rangle$ is a linear superposition of eigenstates $\{ |e_n\rangle \}$ of some observable where
$$|\psi\rangle = \sum_n \alpha_n |e_i\rangle \ , \ \alpha_n \in \mathbb C \text{ and the coefficie... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Logic - Quantifiers Why is this proposition always true?
$$\forall x\,\forall y\,\exists z\big((x<z)\to (x\ge y)\big)$$
And where`s the flaw in my thinking:
You can always choose a $z$ larger than $x$. So the problem can be reduced to:
For all $x$ and all $y$, it is such that $x\ge y$,
which obviously isn´t true... |
Why is this proposition always true?
$\forall x\,\forall y\,\exists z\big((x<z)\implies (x\ge y)\big)$
Suppose we have $x$, $y$ in $\mathbb{R}$
By the irreflexive property of $\lt$, we have $\neg x\lt x$
Introducing $\lor$, we have $\neg x\lt x \lor x \ge y$
Applying the definition of $\implies $, we have $\neg\neg x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1472548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Finding a Real Number using epslion Fix a real number $x$ and $\epsilon>0$. If $|x-1|\le \epsilon$ show $|2-x|\ge 1- \epsilon$
I think we were supposed to use the triangle inequality to show this.
If we use the triangle inequality then $|x-1|\le ||x|-|1||$ so then $x\le \epsilon+1$ so then $-x\ge-\epsilon-1$ adding 2 t... | In general, if $a,b\in\mathbb C$ then $$\big||a|-|b|\big|\leqslant |a|-|b|. $$
For
$$|a| \leqslant |a-b| + |b|\implies |a|-|b|\leqslant|a-b| $$
and similarly
$$|b| \leqslant |a-b| + |a|\implies |a|-|b|\leqslant|a-b|.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1472693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Swapping limit at infinity with limit at 0 I am trying to calculate:
$$\lim_{x\rightarrow\infty}x^2\sin\left(\frac{1}{x^2}\right)
$$
I am pretty sure that this is equivalent to calculating:
$$\lim_{k\rightarrow0}\frac{\sin(k)}{k}=1
$$
Since $k=\dfrac{1}{x^2}$ and $\lim\limits_{x\rightarrow\infty}k=0$.
Is there any way ... | Your way is perfect !
An other way :
Since $$\sin\frac{1}{x^2}=\frac{1}{x^2}+\frac{1}{x^2}\varepsilon(x)$$
where $\varepsilon(x)\to 0$ when $x\to\infty $, you get
$$\lim_{x\to\infty }x^2sin\frac{1}{x^2}=\lim_{x\to \infty }\frac{\sin\frac{1}{x^2}}{\frac{1}{x^2}}=\lim_{x\to\infty }\frac{\frac{1}{x^2}+\frac{1}{x^2}\vare... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1472756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Is $\frac{4x^2 - 1}{3x^2}$ an abundancy outlaw? Let $\sigma(x)$ denote the sum of the divisors of $x$. For example, $\sigma(6) = 1 + 2 + 3 + 6 = 12$.
We call the ratio $I(x) = \sigma(x)/x$ the abundancy index of $x$. A number $y$ which fails to be in the image of the map $I$ is said to be an abundancy outlaw.
My ques... | A partial answer.
Let $I(y)=\frac{4x^2 - 1}{3x^2}$.
If $\gcd(4x^2 - 1, 3x^2) = 1$ then $3$ is a factor of $y$ contradicting the fact that $I(y)<\frac{4}{3}$.
Therefore $\gcd(4x^2 - 1, 3x^2) = 3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1472823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Where am I going wrong in solving this exponential inequality? $$(3-x)^{ 3+x }<(3-x)^{ 5x-6 }$$
Steps I took:
$$(3-x)^{ 3 }\cdot (3-x)^{ x }<(3-x)^{ -6 }\cdot (3-x)^{ 5x }$$
$$\frac { (3-x)^{ 3 }\cdot (3-x)^{ x } }{ (3-x)^{ 3 }\cdot (3-x)^{ x } } <\frac { \frac { 1 }{ (3-x)^{ 6 } } \cdot (3-x)^{ 5x } }{ (3-x)^{ 3 }\cdo... | Hint: use the fact that $a^x$ is decreasing if $0<a<1$ and increasing if $a>1$.
Now consider first $2<x <3$. In that case the inequality is satisfied iff $2<x <9/4$. If $x>3$ there's no solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1472928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Incorrect General Statement for Modulus Inequalities $$|x + 1| = x$$
This, quite evidently, has no solution.
Through solving many inequalities, I came to the conclusion that,
If,
$$|f(x)| = g(x)$$
Then,
$$f(x) = ±g(x)$$
And this was quite successful in solving many inequalities. However, applying the above to this pa... | The definition of the absolute value is
$$
\lvert x \rvert =
\begin{cases}
x & \text{if $x\geqslant 0$} \\
-x & \text{if $x < 0$},
\end{cases}
$$
so
$$
\lvert f(x) \rvert = g(x)
\iff
\begin{cases}
f(x) = g(x) \\
f(x) \geqslant 0
\end{cases}
\qquad\text{or}\qquad
\begin{cases}
-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Intuitive proof of the formula ${}_nC_r + {}_nC_{r-1} = {}_{n+1}C_r$ I came across this formula in combination— ${}_nC_r + {}_nC_{r-1} = {}_{n+1}C_r$. Even though I know its rigorous mathematical proof, I want a logical and elegant proof of this.
For example, the famous formula of combination ${}_nC_r = {}_nC_{n-r}$ sa... | Assume the $n+1$ objects are balls, with only one of them being red and the rest white.
A particular selection of $r$ balls has to either contain the red ball or not.
Those selections omitting the red ball are $^{n}C_r$. Those selections containing the red ball have choice in which $r-1$ white balls have to be chosen, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Solving another bessel equation by substitution For a problem solving class I need to find the general solution of ODE $x^2y''+(\frac{3}{16}+x)y=0$ in terms of $J_{\nu}$ and $J_{-\nu}$, if possible. $\nu$ represents the Bessel parameter.
A hint is given, namely that useful substitutions would be $y=2u \sqrt{x}$ and $\... | The previous remark answers your question, keeping track of the expansions
\begin{gather*}
\cos z = J_0(z)-2J_2(z)+2J_4(z)-\cdots;\\
\sin z= 2J_1(z)-2J_3(z)+\cdots.
\end{gather*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Permutations and Combinations Heads/Tails Out of all of the potential sequences of 73 Heads/Tails games, each being Heads or Tails, how many sequences contain 37 tails and 36 heads?
Express the output in terms of factorials.
Because there are sequences of 73 games my initial thought is, that the answer would be $\fra... | Yes that is correct.
You can basically assume that you have 73 boxes and you have to fill 36 boxes with heads and 37 boxes with tails. Thus you can fill heads first and then fill the remaining ones with tails. Thus the answer is $\dbinom{73}{36}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show that $f(0)=f(-1)$ is a subspace Let $F(\Bbb R,\Bbb R)$ is a vector-space of functions $f: \Bbb R\to \Bbb R$. The functions $f(0)=f(-1)$ are an example of subspace?
If it's possible I would like some example of functions such that $f(0)=f(-1)$.
| $$F(\mathbb{R},\mathbb{R})=\left\{\mathcal{f}:\mathbb{R}\rightarrow \mathbb{R} | \mathcal{f}(0) = \mathcal{f}(-1) \right\}$$
We have to show-
*
*It is not a empty set or null element is a part of this set.
*If $\mathcal{f},\mathcal{g} \in F$, then $\left(\mathcal{f}+\mathcal{g}\right) \in F$.
*If $\mathcal{f} \in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How discontinuous can the limit function be? While I was reading an article on Wikipedia which deals with pointwise convergence of a sequence of functions I asked myself how bad can the limit function be? When I say bad I mean how discontinuous it can be?
So I have these two questions:
1) Does there exist a sequence o... | The following is a standard application of Baire Category Theorem:
Set of continuity points of point wise limit of continuous functions from a
Baire Space to a metric space is dense $G_\delta$ and hence can not be countable.
Another result is the following:
Any monotone function on a compact interval is a pointwise li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Convergence of difference of series Let $f(x)$ be a probability density function, i.e $f(.) \geq 0$ and $\int_{\mathbb{R}}f(x) \ dx=1$. Let $I_{n,k}= \big[\frac{k}{n}, \frac{k+1}{n}\big), \ k \in \mathbb{Z}$ and $p_{n,k}$ denote the area of $f(.)$ in the interval $I_{n,k}$, i.e,
$$
p_{n,k}=\int_{\frac{k}{n}}^{\frac{k+... | This is indeed true. We compute
\begin{eqnarray*}
\sum_{m}\left|a_{n,m+1}-a_{n,m}\right| & = & \sum_{m}\left|\sum_{k}\left[p_{n,\left(m+1\right)-k}-p_{n,m-k}\right]p_{n,k}\right|\\
& \leq & \sum_{k}\left[p_{n,k}\cdot\sum_{m}\left|p_{n,\left(m+1\right)-k}-p_{n,m-k}\right|\right]\\
& \overset{\ell=m-k}{=} & \sum_{k}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The names of 8 students are listed randomly. What is the probability that Al, Bob, and Charlie are all in the top 4? My attempt at a solution: There are $C_{4,3}$ ways to arrange Al, Bob, and Charlie in the top 4, and $C_{5,4}$ ways to arrange the other 5 people. All of this is over $C_{8,4}$, giving $$\frac{C_{4,3}C_{... | $4$ slots are up for grabs, so a simple way is to just compute $\dfrac48\cdot\dfrac37\cdot\dfrac26= \dfrac{1}{14}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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checking if a number is a prime I was reading Wikipedia, and it was given that "all primes are of the form 6k ± 1" (other than 2 and 3), where k = 1,2,3,4,...
Is this statement correct? If yes, can we use this to test if a given number is a prime number? For instance, we can say that 41 is a prime number, since there ... | For $k\ge 1$ no prime can be of the form $6k$ (obvious), $6k+2=2(3k+1)$, $6k+3=3(2k+1)$, $6k+4=2(3k+2)$. There remains $6k+1$ and $6k+5=6(k+1)-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Conjugacy classes for rotations of $D_{2n}$ It says in my notes that for a dihedral group $D_{2n}$, if $n$ is even, then each conjugacy class has at most $1$ element. It says that for the conjugacy class $C_{r^k}$ where $r^k$ is some reflection, the conjugacy class is $\{r^{\frac{n}{2} = k}\}$ (or just $\{r^{k}\}$).
I ... | Something is certainly incorrect in your notes: a group is abelian if and only if every conjugacy class has size one, and the dihedral group $D_{2n}$ is not abelian for $n>2$. Possibly what is meant is that the center of $D_{2n}$ consists of at most one non-trivial element; and for $n$ even, this element is $r^{n/2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1474146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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exponential distribution involving customers
Customers arrive at a certain shop according to a poisson process at
the rate of $20$ customers per hour. what is the probability that the
shop keeper will have to wait more than $5$ minutes (after opening)
for the arrival of the first customers?
So i am trying to th... | Let the discrete random variable $X=$ The number of minutes the shopkeeper will have to wait for the first customers to arrive.
$X\sim Po(\lambda)$ and working in minutes $\implies 3 \space$ customers arrive per minute, such that $\lambda=3$ so $$X\sim Po\left(3\right)$$ $$P\left(X \gt 5\right)=1-P\left(X \le 5\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1474225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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How many integers between 1000 and 9999 inclusive consist of How many integers between $1000$ and $9999$ inclusive consist of
(a) Distinct odd digits,
(b) Distinct digits,
(c) From the number of integers obtained in (b), how many are odd integers?
(a) ${}^5P_4 = 120$
(b) $9 \times 9 \times 8 \times 7 = 4536$
(c) I d... | If you place $0,2,4,6,$ or $8$ in the units place, it becomes even. Hence, you will have to place $1,3,5,7,$ or $9$ in the units place to make it odd. So in the units place, you have $5$ choices. In the thousand's place we have $8$ choices ($8$ because we have left out $0$ and the number in the units place). Similarly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1474463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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$\cot(x+110^\circ)=\cot(x+60^\circ)\cot x\cot(x-60^\circ)$ How to find the solution of this trigonometric equation
$$\cot(x+110^\circ)=\cot(x+60^\circ)\cot x\cot(x-60^\circ)$$
I have used the formulae $$\cos(x+60^\circ)\cos(x-60^\circ)=\cos^2 60^\circ - \sin^2x$$
$$\sin(x+60^\circ)\sin(x-60^\circ)=\sin^2x-\sin^260^\cir... | $$\cot(x+110^\circ)=\cot(x+60^\circ)\cot x\cot(x-60^\circ)$$
$$\frac{\cot(x+110^\circ)}{\cot x} = \cot(x+60^\circ)\cdot \cot(x-60^\circ)$$
$$\frac{\cos(x+110^\circ)\cdot \sin x}{\sin (x+110^\circ)\cdot \cos x} = \frac{\cos(x+60^\circ)\cdot \cos(x-60^\circ)}{\sin(x+60^\circ)\cdot \sin(x-60^\circ)}$$
Now Using Componendo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1474586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A question about gradient. The gradient of a function of two variable $f(x,y)$ is given by
$$\left( \frac{\partial f}{\partial x} ,\frac{\partial f}{\partial y}\right). $$
It is also evident that gradient points in the direction of the greatest increase or decrease of a function at a point. My question is that whether... | The answer is that your question doesn't actually make sense: the graph of $f(x,y)$ lives in $\mathbb{R}^2 \times \mathbb{R}$, whereas $\nabla f$ is in $\mathbb{R}^2$.
What is true is that $\nabla f(a,b)$ is normal to the level curve of $f$ passing through $(a,b)$: this is because, taking a local parametrisation of a l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1474685",
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"source": "stackexchange",
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} |
homeomorphism of the closed disc on $\mathbb R ^2$ I just found a statement that for any closed disc on $\mathbb R^2$ and any two points inside it, there is a homeomorphism taking one point to another and is identity on the boundary. But I can't write down the homeomorphism myself. Can anyone help? Many thanks in advan... | For each $\mathbf v_0=(x_0,y_0)\in D$, the unit disk, I will define a specific homeomorphism $f:D\to D$ so that $f_{\mathbf v_0}(\mathbf 0)=\mathbf v_0$ which fixes the boundary $S^1$.
Now, for $\mathbf v\neq 0$, we have $\mathbf v=r\mathbf s$ for a unique $(r,\mathbf s)$ for $r\in(0,1]$ and $\mathbf s\in S^1$. This i... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is differentiating on both sides of an equation allowed? Let's say we have $x^2=25$
So we have two real roots ie $+5$ and $-5$.
But if we were to differentiate on both sides with respect to $x$ we'll have the equation $2x=0$ which gives us the only root as $x=0$.
So does differentiating on both sides of an equation alt... | mcb's answer suggests that the equal sign doesn't always have the same meaning. I disagree, the difference is in the role of $x$. When we write $(x-1)(x+1)=x^2-1$, we mean that for all $x$, $(x-1)(x+1)=x^2-1$. When we write $x^2=25$, it means for which $x$ is $x^2=25$ ? (Answer: $x=-5$ and $x=+5$).
In the first case, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
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"answer_id": 4
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What is the formula for $\int \cos(\pi x)dx$? I want to evaluate the integral $$\int \cos(\pi x)dx$$
I thought it would simply be $\sin(\pi x) +C$
However in the solutions book the answer is $\frac{1}{\pi}\sin(\pi x)$
This is an integral where it seems as if the chain rule is being performed?
| Let's take the derivative of $\frac{1}{\pi}\sin(\pi x)$
to see that we get the function being integrated.
Remember that $$\frac{d}{dx}f(g(x))= g'(x)f'(g(x))$$
$g(x)$, here, the inside function, is simply $\pi x$. The outer function, f(x), is the sine function.
$$g'(x)=\frac{d}{dx}(\pi x)=\pi$$
$$f'(g(x))=\frac{d}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to differentiate $x^{\log(x)}$? I want to differentiate $x^{\log(x)}$ with respect to $x$.
By using chain rule ($x{log(x)}\frac{1}{x}$), I got the answer $\log x$. Is it right?
| $$y=x^{\log x}$$
$$\log y=\log x \log x=\log^2 x$$
$$\frac{y'}{y}=\frac{2}{x}\log x$$
$$y'=\frac{2y}{x}\log x$$
$$y'=2\frac{x^{\log x}}{x}\log x$$
$$y'=2x^{\log x-1}\log x$$
| {
"language": "en",
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Can a unbounded sequence have a convergent sub sequence? I have been using the Bolzano-Weierstrass Theorem to show that a sequence has a convergent sub sequence by showing that it is bounded but does that mean that if a sequence is not bounded then it does not have a convergent sub sequence?
The sequence I am strugglin... | Yes, an unbounded sequence can have a convergent subsequence.
As Weierstrass theorem implies that a bounded sequence always has a convergent subsequence, but it does not stop us from assuming that there can be some cases where unbounded sequence can also lead to some convergent subsequence.
For example-
if we conside... | {
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"source": "stackexchange",
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Proving the nested interval theorem
Theorem:
Let $\{I_n\}_{n \in \mathbb N}$ be a collection of closed intervals with the following properties:
*
*$I_n$ is closed $\forall \,n$, say $I_n = [a_n,b_n]$;
*$I_{n+1} \subseteq I_n$$\forall \,n$.
Then $\displaystyle\bigcap_{n=1}^{\infty} I_n \ne \emptyset$.
Pf: Let $I_n... | Here is a non-constructive proof.
Construct a sequence by choosing an element $x_n \in I_n$ for every $n$; you can do this however you like. Since this sequence is bounded then the Bolzano-Weierstrass theorem says there exists a convergent subsequence $x_{n_k}$ converging to some real number $c$. Suppose $c \not\in \bi... | {
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"source": "stackexchange",
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Maximizing $\sin \beta \cos \beta + \sin \alpha \cos \alpha - \sin \alpha \sin \beta$ I need to maximize
$$ \sin \beta \cos \beta + \sin \alpha \cos \alpha - \sin \alpha \sin \beta \tag{1}$$
where $\alpha, \beta \in [0, \frac{\pi}{2}]$.
With numerical methods I have found that
$$ \sin \beta \cos \beta + \sin \alpha \co... | As Dan suggested, solve for the stationary points of the gradient. You'll end up solving two (identical) quartics in $\sin \alpha$, $\sin \beta$ (which are really just 2 quadratics due to the symmetry) and find that the only stationary point in the range is at $\alpha = \beta = \pi/6$. For this to not be the maximum on... | {
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"source": "stackexchange",
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Prove that $(A-B)\cup (B-A)=(A\cup B)-(A\cap B)$ Prove that:
$(A-B)\cup (B-A)=(A\cup B)-(A\cap B)$
My Attempt:
$x\in (A-B)\cup (B-A)$
so $x\in (A-B)\vee x\in (B-A)$
$(x\in A \wedge x\notin B)\vee (x\in B\wedge x\notin A)$
I suppose I should try different cases from here? I haven't been able to make progress after this ... | Looks like you're on the right track. You could make two cases now based on your "or" statement.
Case $1$: $x \in A$ and $x\notin B$. Then obviously $x\notin A\cap B$.
Case $2$: $x \in B$ and $x\notin A$. Then again $x \notin A\cap B$.
In either case (that is, whether we start with $x\in A$ or $x\in B \equiv x \in A\cu... | {
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homeomorphic topology of quotient space of $S^1$ I got stuck on the problem about quotient space from General Topology of Stephen Williard. Here is the problem:
Let $\sim$ be the equivalence relation $x \sim y$ iff $x$ and $y$ are diametrically opposite, on $S^1$. Which topology is the quotient space $S^1/\sim$ homeo... | The space you will get is $\mathbb{RP}^1$, the one dimensional real projective space which is by definition of all lines through the origin in $\mathbb{R}^2$.
| {
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Permutation group conjugacy proof.
I tried this example in $S_3$ and saw that this held but I couldn't figure out how to prove it for all $S_n$. Any help would be appreciated.
| First, you should show that if $\sigma$ is a cycle, then $\alpha\sigma\alpha^{-1}$ is also a cycle of the same length. In fact, you can actually write out what this permutation has to be.
Take an arbitrary permutation $\sigma = \sigma_1 \sigma_2 \dots \sigma_n$, written in disjoint cycle notation. Then $\alpha\sigma\al... | {
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Computing the dimension of a vector space of matrices that commute with a given matrix B, This is part $2$ of the question that I am working on.
For part $1$, I showed that the space of $5\times 5$ matrices which commute with a given matrix $B$, with the ground field = $\mathbb R$ , is a vector space.
But how can I com... | There is a theorem by Frobenius:
Let $A\in {\rm M}(n,F)$ with $F$ a field and let $d_1(\lambda),\ldots,d_s(\lambda)$ be the invariant factors $\neq 1$ of $\lambda-A$, let $N_i=\deg d_i(\lambda)$. Then the dimension of the vector space of matrices that commute with $A$ is $$N=\sum_{j=1}^s (2s-2j+1)n_j$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Do uniformly grey sets of positive density exist? Let us call a set $A\subset \mathbb{R}^2$ uniformly grey if the measure of its sections is constant, but not full. (There may be a standard name for this; I would be glad if someone tells me.)
Formal definition: There are intervals $[a_1,b_1]$ and $[a_2,b_2]$ and consta... | Take a fat Cantor set $C$ (such as the Smith-Volterra-Cantor set) in $[0,1]$ of measure $\frac{1}{2}$. Then $C \times C$ will satisfy your section condition.
Does it have positive density? The points in the fat cantor set have positive density on $\mathbb{R}$, and so given any ball around our point we can find a square... | {
"language": "en",
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"source": "stackexchange",
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How many roots are rational?
If $P(x) = x^3 + x^2 + x + \frac{1}{3}$, how many roots are rational?
EDIT:
$3x^3 + 3x^2 + 3x + 1 = 0$, if any rat roots then,
$x = \pm \frac{1}{1, 3} = \frac{-1}{3}, \frac{1}{3}$, and none of these work. Complete?
| The polynomial $3P(x) = 3x^3 + 3x^2 + 3x + 1$ has integer coefficients. So, if $p/q$ with $\operatorname{gcd}(p,q) = 1$ is a rational root, necessary $p$ divides the constant term $1$, and $q$ divides the dominant coefficient $3$. So we have four candidates $-1$, $-1/3$, $1/3$ and $1/3$. Clearly, the roots of $P(x)$ ar... | {
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$\lim_{x\to 0} \frac{1-\cos (1-\cos (1-\cos x))}{x^{a}}$ is finite then the max value of $a$ is? I know the formula $\frac{1-\cos x}{x^{2}}=\dfrac{1}2$
But how do i use it here and tried l hospital and no use and doing l hospital twice will make it very lengthy how do i approach ?
| Hint.
$$ \frac{1 - \cos\bigl(1 - \cos(1-\cos x)\bigr)}{x^a}
= \frac{1 - \cos\bigl(1 - \cos(1-\cos x)\bigr)}{\bigl(1 - \cos(1 - \cos x)\bigr)^2} \cdot \left(\frac{1 - \cos(1 - \cos x)}{(1 - \cos x)^2}\right)^2 \cdot \frac{(1 - \cos x)^4}{x^a} $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differential Equation (First order with separable variable) Given that $\frac{dy}{dx}=xy^2$. Find the general solution of the differential equation.
My attempt,
$\frac{\frac{dy}{dx}}{y^2}=x$
$\int \frac{\frac{dy}{dx}}{y^2} dx=\int x dx $
$-\frac{1}{y}=\frac{x^2}{2}+c$
$y=-\frac{2}{x^2+2c}$
Why the given answer is $y=... | The constant is arbitrary. You can equally well have $3c,-c$, or even a different letter, like $A$; they are all equivalent. The given answer has just chosen a simpler-looking form.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1476540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Laplace's Method for Integral asymptotes when g(c) = 0 I am using these notes as my reference, but I am running into some questions.
Say I am trying to find, for large $\lambda$
$$I(\lambda)=\int_0^{\pi/2}dxe^{-\lambda\sin^2(x)}$$
This has our maximum at $c=0$, where g(c)=0 and g'(c)$\neq$0. So when I pull out the $... | Note that you have an endpoint maximum, and hence the integral should only be one-sided; this is the sort of case where Watson's lemma applies.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many (3x3) square arrangements?
In how many ways can we place 9 identical squares - 3 red, 3 white and 3 blue, in a 3x3 square in a way that each row and each column has squares of each three colours?
Let the colors be $RGB$, let the white=green for simplicity, then the top row has the following cases:
$RGB, RBG,... | "In total, there are 2+1=32+1=3 possible arrangements here"
No, you don't add them. You multiply them. 2 choices for the second row and each choice yields 1 choice for the third.
"In total, there are 2x1=2 possible arrangements here"
=======
There are six ways you can do the first row.
In the second row there are 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1476797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Moves to P-positions in Nim Let $A$ be an N-position in Nim such that all moves to P-positions reqire exactly $k$ tokens to be removed. What can we say about $A$?
| As you may know, a winning move is one that changes the ginary XOR of theheap sizes to $0$, which means to XOR one of the existing heap sizes with the current XOR sum $s$. By assumption, for each heap size $n_i$ in $A$, we have either $n_i\operatorname{XOR} s>n_i$ or $n_i\operatorname{XOR} s=n_i-k$. The first case occu... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the Matrix Diagonalizable if $A^2=4I$ I have two question:
Let $A$ be a non-scalar matrix, $A_{k \times k} \in \Bbb R$, and $A^2=4I$. Is the matrix A, always diagonalizable in $\Bbb R$?
Answer
I know that the answer is yes:
$$A^2=4I \rightarrow A^2-4I=0$$
Then by the Cayley Hamilton theorem I know that the matrix ... | The matrix $A$ is a root of the polynomial $t^2-4$, hence the minimal polynomial of $A$, a divisor of $t^2-4$, has only simple roots. This is is equivalent to $A$ being diagonalisable.
Answer to the second question: a matrix over a field $K$ is diagonalisable over $K$ if and only if its minimal polynomial splits over ... | {
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Is my proof of $-(-a)=a$ correct? I'm trying to prove theorem $1.4$ with the following:
$$a=a$$
$$a-a=\stackrel{0}{\overline{a-a}}\tag{Ax.5}$$
$$a-a=0\tag{Inverse def.}$$
$$a+(-a)=0\tag{Thm 1.3}$$
Here I thought about packing the $a$ with a minus sign and then it would yield a new minus sign on Its inverse due to th... | Better wording:
By theorem 1.2. For every a there is a unique -a such that a + (-a) = 0. Likewise for (-a) there is a unique -(-a) such that (-a) + (-(-a)) = 0.
Therefore:
(-a) + (-(-a)) = 0
a + (-a) + (-(-a)) = a+ 0 = a
By associativity:
[a + (-a)] + (-(-a)) = a so
0 + (-(-a)) = (-(-a)) = a.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do you factorize quadratics when the coefficient of $x^2 \gt 1$? So I've figured out how to factor quadratics with just $x^2$, but now I'm kind of stuck again at this problem:
$2x^2-x-3$
Can anyone help me?
| Depending on your level, you may not have seen the quadratic formula that JMoravitz and BLAZE posted. That formula is often the way to find the zeroes of and factor a quadratic.
But what Adam suggests may be more up your alley, and it's what I'll discuss as well: You can make educated guesses and then check to see if i... | {
"language": "en",
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Distribution of $Y= \frac{X_1}{|X_2|}$? If $X_1$ and $X_2$ are independent and identically distributed Gaussian random variables with parameters $0$ and $\sigma^2$, how do I find the distribution of $Y= \frac{X_1}{|X_2|}$?
I'm not supposed to use the method using the Jacobian but I'm not sure what the "easier" way to g... | If $X_1$ and $X_2$ are independent and identically distributed Gaussian random variables then their common distribution is circularly symmetrical around the origin. Let this distribution be denoted by $N(u,v)$. Then the cdf of Y can be calculated as follows
$$F_Y(y)=P\left(\frac{X_1}{|X_2|}<y\right)=P(X_1<y\ |X_2|)=\ii... | {
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Understaning a problem involving measurable sets of a bounded sequence of measures Hi guys I am trying to understand my problem,
We have a sequence of measureable sets $\{A_n\}$ Where $\sum _{n=1} ^{\infty}m(A_n) < \infty$
Define a set $B= \{x \in \mathbb R: \# \{ n:x \in A_n \}=\infty \}$
We want to show m(B)=0
My qu... | Let $f_i = \chi_{A_i}$ be the characteristic function of $A_i$ and $f = \sum_{i=1}^\infty f_i $. Then
$$\int f d\mu = \sum_{i=1}^\infty m(A_n) < \infty $$
(The equality can checked using, e.g., monotonce convergence theorem) Thus $f$ is integable. Thus it is finite a.e.. This is what we want as $B = \{ f= \infty\}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Let $G$ be a graph with $n$ vertices and $e$ edges. Let $m$ be the smallest positive integer such that $m \ge 2e/n$. Let $G$ be a graph with $n$ vertices and $e$ edges. Let $m$ be the smallest positive integer such that $m \ge 2e/n$.
Prove that $G$ has a vertex of degree at least $m$.
My approach is sum of all the degr... | Suppose that there does not exist a vertex $v$ such that $\deg(v) \geq m$. Then the sum of the degrees of all vertices in $G$, $\sum_{i=1}^{n}v_i \leq (m-1)n$. Since $\sum_{i=1}^{n}v_i=2e$, we have $2e\leq(m-1)n$. Divide both sides by $n$ to get $\frac{2e}{n} \leq m-1$. So, we have $\frac{2e}{n}+1 \leq m$. This is a co... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve this limit: $\lim_{n \to \infty} \frac{(2n+2) (2n+1) }{ (n+1)^2}$ $$
\lim_{n\to\infty}\frac{(2n+2)(2n+1)}{(n+1)^{2}}
$$
When I expand it gives:
$$
\lim_{n\to\infty} \dfrac{4n^{2} + 6n + 2}{n^{2} + 2n + 1}
$$
How can this equal $4$? Because if I replace $n$ with infinity it goes $\dfrac{\infty}{\infty}$ on... | HINT: rewrite it in the form
$$\frac{4+\frac{6}{n}+\frac{2}{n^2}}{1+\frac{2}{n}+\frac{1}{n^2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1477946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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What are polyhedrons? Polyhedrons or three dimensional analogues of polygons were studied by Euler who observed that if one lets $f$ to be the number of faces of a polyhedron, $n$ to be the number of solid angles and $e$ to be the number of joints where two faces come together side by side $n-e+f=2$.
It was later seen ... | The polyhedra that Euler studied was not the same as we call polyhedra today. Today a polyhedra is a body with (only) flat polygonal faces.
Those that Euler studied was a subset of these. Basically those whose nodes and vertices form a planar graph.
There are some complications that can arise in the generic definition ... | {
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Simple Proof of FT of Algebra Which proof of the Fundamental Theorem of Algebra requires minimum mathematical maturity and has the best chances to be understood by an amateur with knowledge of complex numbers and polynomials?
| Suppose the polynomial has no zeros. Then every value of $f(z)$ has a direction.
Draw a large circle, where $z^n$ dominates over the other terms in the polynomial. As you travel around the circle, the direction of $f(z)$ is near the direction of $z^n$, so it rotates $n$ times as $z$ goes once around the circle.
Now g... | {
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How to construct an n-gon by ruler and compass? Since $\cos[\frac{2\pi}{15}] $ is algebraic and equal to $\frac{1}{8}(1+\sqrt{5}+\sqrt{30-6\sqrt{5}})$ we know that the regular 15-gon is constructible by ruler and compass.
Although I know how to construct a hexagon by ruler and compass and have seen the construction of... | In order to construct an angle equal to $\frac{2\pi}{15}$, you just need to construct an equilateral triangle and a regular pentagon, since:
$$ \frac{2\pi}{15} =\frac{1}{2}\left(\frac{2\pi}{3}-\frac{2\pi}{5}\right).$$
Have a look at the Wikipedia page about contructible regular polygons.
| {
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Find the max, min, sup and inf of a sequence
Let $\{a_n\}=\{x\mid x\in\mathbb {Q},x^2 <2\}$, find the max, min, sup and inf of a sequence
Clearly, sup is $\sqrt {2} $ and inf is $-\sqrt {2} $, so we have $-\sqrt {2}<a_n <\sqrt {2} $. Since $\mathbb {Q} $ is dense of $\mathbb {R} $, max and min both don't exist cause... | Okay, I guess my first answer wasn't clear.
If you have a bounded set A one of three things can happen.
1) A has a maximum element x. If so then x is also the sup of A and the sup of A is a member of the set A. (sup means least upper bound and if x is maximal it is a least upper bound.) So max A = sup A = x; x $\in$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1478370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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How many ways are there of rolling 6 different coloured 6-sided dice so that exactly 3 different values are showing? Any help with this problem would be greatly appreciated. I have tried solving this problem like this:
1st die: 6 choices,
2nd die: 5 choices,
3rd die: 4 choices,
4th die: 3 choices,
5th die: 3 choices,
6... | Basically, you need to choose 3 distinct #s from 6,
choose patterns of 4-1-1 of a kind, 3-2-1 of a kind and 2-2-2 of a kind and permute them.
$4-1-1$ of a kind: $\binom31\binom63\cdot\frac{6!}{4!} = 1800$
$3-2-1$ of a kind: $\binom31\binom21\binom63\cdot\frac{6!}{3!2!} =7200$
$2-2-2$ of a kind: $\binom63\cdot\frac{6!}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1478458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Prove by Induction of a Total Function I'm given a total function that spans Positive Integers ($\mathbb Z^+$ to Positive Real numbers $\mathbb R^+$)
I'm given that $S(1) = 7$
Also I'm given that
$S(2^k) = S(2^{k-1}) +5$ // equivalent to $2^{k-1} + 5$ (Right hand side)
where $k$ is an element of positive Integers
I'm t... | You need to prove it for $n+1$ assuming that it's true for $n$.
$$S(2^{n+1})=S(2^{n})+5=5n+7+5=5(n+1)+7$$
Where I have used the given condition in the first equality and the induction hypothesis in the second one. Doing this you have proved the given formula using mathematical induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1478856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Square root of two How would you find root 2? I have been told to use a number line. I have tried to visualize it on a number line using triangles. But am unsure of where to go from there.
| The square root of two can not be expressed as a fraction of whole numbers or as a decimal. It is the classic example of an irrational number. The real numbers are infinite limits of rationals but the irrationals can not be expressed completely in any finite form. Geometrically the diagonal of a square with sides of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1478996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Can I construct a bijection between (0,1) and [0,1] using sets and sup/inf? (Proof verification) This question comes from a class I'm in currently:
Find a 1-1 onto map between $(0,1)$ and $[0,1]$.
Let $r \in (0,1)$. Now, define the function $f: (0,1) \rightarrow S$ such that
$$ f(r) = s = \begin{cases}
\lbrace ... | First, note that there's a problem: if $r={1\over 2}$, then $f(r)=\emptyset$, so $g(f(r))$ is undefined.
However, that's not the only problem. Let's look at $g\circ f$ directly. If $r<{1\over 2}$, then $f(r)=[r, {1\over 2})$, which has empty intersection with $({1\over 2}, 1]$, so $g(f(r))=r$. Meanwhile, if $r>{1\over ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1479124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that if $\gcd(a, b)\mid c$, then the equation $ax + by = c$ has infinitely many integer solutions for $x$ and $y$. Show that if $\gcd(a, b)\mid c$, then the equation $ax + by = c$ has infinitely
many integer solutions for $x$ and $y$.
I understand that if there is one, solution for $ax+by =c$, then there are infin... | Euclidean algorithm says there is at least one solution.
if $ax + by = c$ then $a(x - b/gcd(a,b)) + b(y + a/gcd(a,b)) = ax + by = c$. Inductively there are infinitely many further solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1479220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Two point form of solving straight line problems in coordinate geometry If we find the slope of a line via two point form we find that when the for different points in the same straight line we have different equations. Why is it so? Say, if the points were $3,5$ and $6,10$ then the equation was $2x-y=1$ and when the p... | The two st lines you mentioned are compeltely different so of course you will have two different equations !!
For the first one passing through $ (3,5)$ and $(6,10)$, its equation is $y=5/3x$.
For the second one passing through $ (4,6)$ and $(6,11)$, its equation is $ y=5/2x-4$.
Here is their plot :
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1479334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The cardinality of the set Let $\mathbb{G} =\{ a^b + \sqrt{c}: a,b,c\in \mathbb Q \}$
I guess the set $\mathbb{G}$ is countable set, but I can't show it properly.
How to start the proof?
| The proof is basically the same as for $\mathbb Q$ being countable. You start with an enumerations $a_j$, $b_k$ and $c_l$ (for $a$, $b$ and $c$) in your formula and iterate the triples by first taking those where $j+k+l=0$ then $j+k+l=1$, then $j+k+l=2$ and so on. For each level you have finitely many triples and by th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Confused about Euclidean Norm I am trying to understand that the Euclidean norm $\|x\|_2 = \left(\sum|x_i|^2\right)^{1/2}$ is in fact a norm and having trouble with the triangle inequality.
All the proofs I have referred to involve the Cauchy-Schwarz inequality. But it seems that this inequality is proved in an inner p... | Hint:
Note that the Euclidean norm is a particular case of a $p$-norm and for these norms the triangle inequality can be proved using the Minkowky inequality.
Anyway, the Euclidean norm is the only $p$-norm that satisfies the parallelogram identity ( see: Determining origin of norm), so it is coming from an inner prod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1479509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
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Chain Rule for Differentials I'm trying this problem from Lee's Smooth Manifolds, but I'm not sure where I'm making a mistake:
Problem: Let $M$ be a smooth manifold, and let $f,g\in C^{\infty}(M)$. If $J\subset\mathbb{R}$ containing the image of $f$, and $h:J\to\mathbb{R}$ is a smooth function, then $d(h\circ f)=(h'\ci... | In the beginning, it is better to track the base points of all the objects involved in order to make sure that everything "compiles" correctly. Here,
$$ d(h \circ f)|_p (X) = d(f^{*}(h))|_p (X) = (f^{*}(dh))|_p(X) = dh|_{f(p)}(df|_p(X)) = h'(f(p)) \cdot df|_p(X) = ((h' \circ f) \cdot df)|_p(X) $$
so you lost your $f$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1479640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$F(F(x)+x)^k)=(F(x)+x)^2-x$ I have no idea about this problem. But I feel we have to use chain rule of differentiation here.
The Function $F(x)$ is defined by the following identity:
$F(F(x)+x)^k)=(F(x)+x)^2-x$
The value of $F(1)$ is such that a finite number of possible values of $F'(1)$ can be determined solely from ... | $$F((F(x)+x)^k) = (F(x)+x)^2 - x
$$
differentiate both sides to get
$$F'((F(x)+x)^k)\cdot k (F(x)+x)^{k-1}\cdot (F'(x)+1) = 2(F(x)+x)\cdot(F'(x)+1) -1
$$
Let denote $u(x)=F(x)+x$ then
$$ F'(u(x)^k) \cdot k u(x)^{k-1} u'(x) = 2u(x)u'(x) -1
$$
The only appropriate value for $F(1)$ I could think is $F(1)=0$, i.e. $u(1)=1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1479774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to solve $e^x=kx + 1$ when $k > 1$? It's obvious that $x=0$ is one of the roots. According to the graphs of $e^x$ and $kx + 1$, there's another root $x_1 > 0$ when $k > 1$. Is there a way to represent it numerically.
| If $t = -x - 1/k$, we have $$t e^t = (-x - 1/k) e^{-x} e^{-1/k} = - e^{-1/k}/k $$
The solutions of this are $t = W(-e^{-1/k}/k)$, i.e. $$x = - W(-e^{-1/k}/k) - 1/k$$ where $W$ is one of the branches of the Lambert W function.
If $k > 1$, $-e^{-1/k}/k \in (-1/e,0)$, and there are two real branches: the $0$ branch (whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1479894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Using arithmetic mean>geometric mean Prove that if a,b.c are distinct positive integers that
$$a^4+b^4+c^4>abc(a+b+c)$$
My attempt:
I used the inequality A.M>G.M to get two inequalities
First inequality
$$\frac{a^4+b^4+c^4}{3} > \sqrt[3]{a^4b^4c^4}$$
or
$$\frac{a^4+b^4+c^4}{3} > abc \sqrt[3]{abc}$$ or new --
first ine... | I just want to add another way to solve the problem.
$$\begin{align} & a^4 + b^4 + c^4 = (a^2)^2+(b^2)^2+(c^2)^2 \\& \ge a^2 b^2 + b^2c^2 + c^2a^2 \text{ (From Cauchy-Schwarz)}\\&= (ab)^2 + (bc)^2 + (ca)^2 \\ & \ge abbc + bcca + caab\ (\because a^2 + b^2 + c^2 \ge ab + bc + ac) \\ &= abc(a+b+c), \end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1480000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
What did this sum become: $\left(\sum_{n=a}^{b} z_n\right)^2 $? Can I see something meaningfull about this sum? Where is it equal to?
$$\left(\sum_{n=a}^{b} z_n\right)^2$$
Is it equal to: $\sum_{n=a^2}^{b^2} z_n^2$ or something else, I've no idea how to deal with it.
Thanks for the help.
| $$\left(\sum_{n=a}^b z_n\right)^2 = \left(\sum_{j=a}^b z_j\right)\left(\sum_{k=a}^b z_k\right) = \sum_{j,k=a}^b z_jz_k$$
If you want, you can pull the squares and reduce repetitive terms into the following form:
$$\left(\sum_{n=a}^b z_n\right)^2 = \sum_{n=a}^b z_n^2 + 2\sum_{j=a+1}^b\left[\sum_{k=a}^{j-1} z_jz_k\right]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1480112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Express the following product as a single fraction: $(1+\frac{1}{3})(1+\frac{1}{9})(1+\frac{1}{81})\cdots$
I'm having difficulty with this problem:
What i did was:
I rewrote the $1$ as $\frac{3}{3}$
here is what i rewrote the whole product as:
$$\left(\frac{3}{3}+\frac{1}{3}\right)\left(\frac{3}{3}+\frac{1}{3^2}\right... | Write it in base-3 notation. The factors are $$1.1,\; 1.01,\;1.0001,\;1.00000001\,,...,\;1.(2^n-1\text{ zeros here})1,...\;.$$They are easy to multiply: you get successively$$1.1,\; 1.111,\;1.1111111\,,...,\;1.(2^n-1\text{ ones here}),...\;.$$The limit is $1.111...$, which you probably recognize as the limit of the geo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1480214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Proving Logarithims I'm trying to figure out how to go about proving this statement:
$$n^{\log(a)} = a^{\log(n)}$$
I'm told that I cannot prove from both sides. I tried to $\log$ the first side to get:
$\log(n)\log(a)$
But I'm not sure where to go from here, any ideas would be much appreciated.
| Suppose $n=a^x$, then $\log n=x\log a$ by taking logs.
Then, by substitution, $n^{\log a}=(a^x)^{\log a}=\dots$
How to find the method - well you start with $n$ to some power, and you want to end with $a$ to some power, so it is natural to express $n$ as a power of $a$, and once this is done the result drops out.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1480361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Prove that $\forall z \in \Bbb C : \lvert \Re(z) \rvert \le \lvert z \rvert \le \lvert \Re(z) \rvert + \lvert \Im(z)\rvert$ I'm having difficulties trying to prove these two complex inequalities :
$\forall z \in \Bbb C :$
$$\lvert \Re(z) \rvert \le \lvert z \rvert \le \lvert \Re(z) \rvert + \lvert \Im(z)\rvert$$
$$\lve... | You have
$$ |z| = \sqrt{(\Re z)^2 + (\Im z)^2} \geq \sqrt{(\Re z)^2} = |\Re z|$$
and
$$ |z| = \sqrt{(\Re z)^2 + (\Im z)^2} \leq \sqrt{|\Re z|^2 +2|\Re z| |\Im z|+ |\Im z|^2} = \sqrt{(|\Im z| + |\Re z|)^2} = |\Im z| + |\Re z|. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1480463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Given Column Space and Null Space construct a matrix. Construct a matrix whose column space contains (1 1 1) and (0 1 1) and whose null space contains (1 0 1) and (0 1 0), or explain why none can exist.
So I am not sure if I did this right, but after applying the null space to A I got that
$A=\begin{bmatrix}
a& 0 &-a\... | Before beginning, I just consider the case where the matrix we want to construct is a $3\times 3$ matrix (since you tried to construct such a matrix). I claim that we cannot construct such a matrix.
Indeed, since the $2$ vectors which belong to the column space are linearly independent, we have that $$2\le \text{ rank ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1480533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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