Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proofs of AM-GM inequality The arithmetic - geometric mean inequality states that
$$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$
I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone knew a proof that is unexpected in some way.... | Another answer. We can prove this alternative result:
If $a_1, \ldots, a_n$ are positive reals such that $a_1 + \dots + a_n = n$, then $a_1 \dots a_n \leq 1$.
We can suppose wlog that $a_1 \leq \dots \leq a_n$.
Notice that we can suppose $a_n \not= 1$, or else we could just solve the same problem for $n-1$.
By "pigeon... | {
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Finding minimum $\alpha > 0$ so that $\det(A - \alpha B) = 0$ for positive definite $A,B$ Given two positive definite symmetric matrices $A,B$, I'd like to find the minimum $\alpha > 0$ such that $A - \alpha B$ is singular, i.e., the threshold where $A - \alpha B$ is no longer positive definite. An algorithmic approach... | A little different, there are a number of ways, including Cholesky decomposition, to write $$ B = C^T C, $$ so that, with $G = C^{-1},$ we have
$$ G^T B G = I. $$ Then solve
$$ \det \left( G^T A G - \alpha I \right) = 0. $$ As $C$ is upper triangular, finding $G$ is not difficult. If you use some non-Cholesky metho... | {
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Show that "$\Gamma \models S \Rightarrow \Gamma \vdash S$" entails "if $\Gamma \nvdash P \And \sim P$ then $\Gamma$ is satisfiable"
Show that "$\Gamma \models S \Rightarrow \Gamma \vdash S$" entails "if
$\Gamma \nvdash P \And \sim P$ then $\Gamma$ is satisfiable"
I'm primarily confused with the notation being used ... | So I think this is what we may want to do. Suppose for contradiction that $\Gamma$ is not satisfiable. This means that $\Gamma$ has no models. Now, fix some sentence $P$ and let $S \equiv P \wedge \neg P$. Now, $\Gamma \models S$ will be vacuously true since there are no models of $\Gamma$, (i.e. any model of $\Gam... | {
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Sine substitution? My book says the following:
$$\int \frac{dx}{(16-x^2)^{3/2}}$$
$$x = 4\sin\theta$$
$$(16 - x^2)^{3/2} = (4^2\cos^2\theta)^{3/2}$$
$$=(4\cos\theta)^3$$
I don't understand the last step:
Doesn't:
$$(4^2\cos^2\theta)^{3/2} = (|4\cos\theta|)^3$$
Since:
$\sqrt{x^2} = |x|$
| We are really letting $\theta=\arcsin(x/4)$. So $\theta$ ranges over the interval $(-\pi/2,\pi/2)$, and the cosine in this interval is non-negative.
| {
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What values of $a$ make this matrix not invertible? So I'm given this matrix:
$$\left(\begin{array}{c} a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & a\end{array}\right)$$
and am told to find the values of a which make it not invertible.
I know that $a = 0$ means our matrix is invertible (since the column vectors span $\mathbb{R}... | In this answer it is shown that
$$
\det(\lambda I_n-AB)=\lambda^{n-m}\det(\lambda I_m-BA)\tag{1}
$$
We can write your matrix as
$$
\begin{pmatrix}a&1&1\\1&a&1\\1&1&a\end{pmatrix}
=(a-1)I_3+\begin{pmatrix}1\\1\\1\end{pmatrix}\begin{pmatrix}1&1&1\end{pmatrix}\tag{2}
$$
Applying $(1)$ to $(2)$, with $\lambda=a-1$, yields
... | {
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The inverse is a root of the reciprocal (reverse) polynomial. Let $F$ be a field and let $a \neq 0$ be a zero of the polynomial $a_0 + a_1x + . . . +a_nx^n$ in $F[x]$.
I want to show that $\frac{1}{a}$ is a zero of the polynomial $a_n + a_{n-1}x + . . . + a_0x^n$
How can I do this?
| Hint $ $ The second reversed (aka reciprocal) polynomial is simply $\ \hat f(x)= x^n f(1/x),\,\ n = \deg f.\,$ Now verify that $\hat f(1/a) = 0\,$ since $\,f(a) = 0.$
| {
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Solving $\arcsin(1-x)-2\arcsin(x)=\pi/2$ \begin{eqnarray*}
\arcsin(1-x)-2\arcsin(x) & = & \frac{\pi}{2}\\
1-x & = & \sin\left(\frac{\pi}{2}+2\arcsin(x)\right)\\
& = & \cos\left(2\arcsin(x)\right)\\
& = & 1-2\left(\sin\left(\arcsin(x)\right)\right)^{2}\\
& = & 1-2x^{2}\\
x & = & 2x^{2}\\
x\left(x-\frac{1}{2}\right) &... | In your first step you added an extra solution.
Since $\arcsin x$ must be smaller than $\pi/2$, the first line reads:
$$\arcsin(1-x)= \frac{\pi}{2}+2\arcsin(x) \le \frac{\pi}{2}$$
Thus, $x\le 0$ as well.
Now, by taking the $\sin$ of both sides, you took a function that was only defined up to $x=1$ (e.g. $\arcsin(x-1)$... | {
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Equivalent definitions of differentiable I am trying to show:
The two statements are equivalent:
(i) $f$ is differentiable at $a$,
(ii) $f(a + h) = f(a) + ch + o(h)$,
where c is some constant (depending on $a$) and $o(h)$
denotes some function of $h$ (also depending on $a$), with the property that
$$\lim_{h\to 0} \fra... | First we see how i) implies ii).
$f$ is differentiable at $a$ so that the limit $$\lim_{h \to 0}\dfrac{f(a + h) - f(a)}{h} = f'(a)$$ exists. This means that $$\lim_{h \to 0}\dfrac{f(a + h) - f(a) - hf'(a)}{h}= 0$$ or in other words if we let $g(h) = f(a + h) - f(a) - hf'(a)$ then $g(h)/h \to 0$ as $h\to 0$. Thus $g(h) ... | {
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Prove a that a topological space is compact iff Prove that the topological space $X$ is compact $\Leftrightarrow$ whenever {$C_j:j\in J$} is a collection of closed sets with $\bigcap_{j\in J}C_j = \varnothing$, there is a finite subcollection {$C_k:k\in K$} such that $\bigcap_{k\in K}C_k=\varnothing$.
My attempt:
First... | The basic idea is correct (taking complements and using de Morgan, essentially).
As suggestions for write-up: show the directions, for left to right e.g.:
Suppose $X$ is compact. Let $\{ C_j: j \in J \}$ be a collection of closed sets with empty intersection. Then define, for each $j \in J$, $U_j = X \setminus C_j$, w... | {
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Where have I gone wrong in trying to solve this ODE? I'm trying to solve: $\frac{dy}{dx}=\frac{x+y-1}{x+4y+2}$.
Attached is a picture of my working.
Could someone please tell me where I'm going wrong?
I'm tried both Maple and Wolfram and neither of them gives me a 'nice' answer.
I know it's wrong as I've implicitly d... | To continue on from my comment (and losing the absolute value signs for the moment, since we are taking fourth roots - but note the fourth roots vanish in the calculation) your complicated expression can have fractions cleared to give:
$$(x-2y-4)^{\frac 34}(x+2y)^{\frac 14}=\frac 1C$$
Implicit differentiation then give... | {
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Injective immersion (between smooth manifolds) that is no homeomorphism onto its image Is there an injective immersion between smooth manifolds that is no homeomorphism onto its image? With smooth I mean $C^\infty$-manifolds and of course also the immersion should be $C^\infty$.
| There is an injective immersion of $\mathbb{R}$ into the plane, whose image is the figure 8. Clearly it it not an homeomorphism to its image (since this is not a manifold).
See also: http://en.wikipedia.org/wiki/Immersed_submanifold#Immersed_submanifolds
| {
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Why would $[0,1) \times \eta$ (with lexicographic order topology) not be a manifold for $\eta > \omega_1$? From Wikipedia's entry on the long line:
And if we tried to glue together more than $\omega_1$ copies of
$[0,1)$, the resulting space would no longer be locally homeomorphic
to $\mathbb{R}$.
Why?
| Every neighbourhood of $\omega_1$ contains uncountably many ordinals, and hence
$$(\alpha,\omega_1)\times [0,1)\tag{1}$$
is not homeomorphic to a subset of $\mathbb{R}$, since it is not second countable (there are uncountably many disjoint open subsets $\{\beta\}\times \left(\frac14,\frac34\right)$, $\alpha < \beta \le... | {
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Problems-solving in Equations The distance between the two cities A and B is 300 km, set off a car from the city a toward the city b by speed 90 km/h and set off from the city b bicycle toward a by speed of 10km/h. if you knew that the car and the bike were based in nine in the morning. Select the time that the car an... | Hint:
$$ v = \frac{\mathrm{d} x}{\mathrm{d} t} = 100 $$
Integrating, you get a very simple equation of x. If you set x(0) = 0, the equation simplifies to remove the constant.
Now, x = 300. Solve for t.
| {
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Use mathematical induction to prove that a function F defined by specifying F (0) and a rule for obtaining F (n+1) from F (n)is well defined. Im just not sure what the question is asking me to prove, or how to prove it with induction.
| Hint: The result is in a certain sense obvious. We know $F(0)$, the rule tells us how to find $F(1)$, then the rule tells us how to find $F(2)$, and so on.
If we want to operate very formally, there are two things to prove: (i) There is a function $F$ that satisfies the condition and (ii) There is only one such functio... | {
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Proof in calculus Prove that, if ${f(x)}$ is any function, ${f(x) + f(-x)}$ is an even function while ${f(x) - f(-x)}$ is an odd function.
Thank you!
Note: I have used this theorem a lot of time. And I can prove it by taking specific functions. But, I have no idea about how to prove it for a general function ${f(x)}$. ... | Proof:
Let $g(x) = f(x) + f(-x)$:
Then, $g(-x) = f(-x) + f(x) = g(x)$ so it is even
Let $h(x) = f(x) - f(-x)$:
Then, $-h(-x) = -f(-x) + f(x) = h(x)$ so it is odd
| {
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Find the derivative of $\frac{(2x−1)e^{−2x}}{(1−x)^2}$ I need to find the derivative of $$\frac{(2x−1)e^{−2x}}{(1−x)^2}$$
I seems very complex to me so I'm wondering if there is a rule or formula I should be using? I attempted it using the chain rule first for the numerator (since I have $ ( 2 x- 1)$ multiplied by $e^{... | $$ \dfrac{\mathrm{d}}{\mathrm{d}x}f\left(x\right) =
\dfrac{-2{\cdot}\left(2x-1\right){\cdot}{\mathrm{e}}^{-\left(2x\right)}}{{\left(1-x\right)}^{2}}+\dfrac{2{\cdot}\left(2x-1\right){\cdot}{\mathrm{e}}^{-\left(2x\right)}}{{\left(1-x\right)}^{3}}+\dfrac{2{\mathrm{e}}^{-\left(2x\right)}}{{\left(1-x\right)}^{2}}
$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving Limit Laws Using Delta Epsilon I need to prove that
$$\lim_{x\to a} f(x) = \lim_{h\to 0}f(a+h) $$
How would I start This! I need some willing to discuss this to a beginner who is lost Hints and detailed explanations wanted plssss
ALSO IS Subsitution given or does it have to be stated
| Let us write $y=a+h$. Then, $\lim_{h\to0}f(a+h)=\lim_{y-a\to0}f(y)=\lim_{y\to a}f(y)$. This is the same as $\lim_{x\to a}f(x)$, but with a change of variable.
You could also do it this way: Using a Taylor expansion gives
$$f(a+h)=\sum^\infty_{k=0}\frac{f^{(k)}(a)}{k!}h^k=f(a)+\sum^\infty_{k=1}\frac{f^{(k)}(a)}{k!}h^k$$... | {
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What problems are easier to solve in a higher dimension, i.e. 3D vs 2D? I'd be interested in knowing if there are any problems that are easier to solve in a higher dimension, i.e. using solutions in a higher dimension that don't have an equally optimal counterpart in a lower dimension, particularly common (or uncommon)... | The kissing number problem asks how many unit spheres can simultaneously touch a certain other unit sphere, in $n$ dimensions.
The $n=2$ case is easy; the $n=3$ case was a famous open problem for 300 years; the $n=4$ case was only resolved a few years ago, and the problem is still open for $n>4$… except for $n=8$ an... | {
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Probablity of three-of-a-kind or better in a roll of four dice So. I took a math competition and one of the questions seemed simple enough.
"Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value"
Hm. Easy.
Tried solving it.
I couldn't get it.
How can I sol... | Imagine the four dice are rolled one after another. There are $6\times6\times6\times6=1296$ different possible outcomes. Of these, $6$ have all four dice showing the same value and $4\times6\times5=120$ have three dice showing the same value and the other die showing a different value. So the probability that at lea... | {
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Showing convergence or divergence of a sequence I need to determine if the series with $n$th term $\ln(n)e^{-\sqrt n}$ converges or diverges. I've tried numerous identities for $ln(x)$ and $e^{x}$ and various convergence tests but I'm still very stuck.
| To prove the given series convergent, we use the following inequalities:
*
*For $ x > 1$ , $Ln(x) < x$.
*$Exp(x) > 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dfrac{x^4}{4!} + \dfrac{x^5}{5!}$ for $x > 0.$
Let $a(n) = \dfrac{ln(n)}{e^{\sqrt{n}}}$, then $a(n) < \dfrac{2ln(\sqrt{n})}{1 + \sqrt{n}+ \dfrac{\sqrt{n}^2}... | {
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Odd torsion of elliptic curves are isomorphic $C: Y^2=X(X^2+aX+b)$
$D: Y^2=X(X^2+a_1X+b_1)$
where $a,b,\in\mathbb Z a_1=-2a,b_1=a^2-4b,b(a^2-4b)\neq0$
Let $C_{oddtors}(\mathbb Q)$ denote the set of torsion elements of $C(\mathbb Q)$ which have odd order and $D_{oddtors}(\mathbb Q)$ denote the set of torsion elements ... | Let $E$ and $E'$ be elliptic curves, and let $\phi:E\to E'$ be a $p$-isogeny (i.e., $\phi$ is an isogeny of degree $p$), where $p$ is prime. In particular, $\phi$ is a group homomorphism from $E$ to $E'$ and its kernel $\ker(\phi)$ is a group of size $p$.
*
*Prove that every $P$ in $\ker(\phi)$ has order dividing $... | {
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find a 4th order linear, non-homo ODE whose general solution: How to find a fourth order, linear, not homogenous ODE with general solution:
$y=c_1+c_2 x+c_3 e^{2x}\cos x+c_4e^{2x}\sin x-x e^{-x}$?
Is there a specific method? I feel like it is guesswork to a certain degree. I can tell some parts such as the $c_1$ term w... | Looking at the solution you know that the characteristic equation of the homogeneous equation has the double root $0$ and the complex conjugate roots $2\pm i$. The characteristic equation is then
$$
r^2((r-2)^2+1)=r^4-4\,r^3+5\,r^2=0.
$$
The equation will be
$$
y''''-4\,y'''+5\,y''=f(x).
$$
You also know that $y=-x\,e^... | {
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By Lagrange Multipliers, the function $f$ has no minima or maxima under constraint $g$? Find the extrema of $f$ subject to the stated condition.
$f(x,y)=x-y$ subject to $g(x,y)=x^2-y^2=2$.
Ok, by Lagrange Multiplier method, we find the points that satisfy $\nabla f(x,y) = \lambda \nabla g(x,y)$ for some $\lambda \in \m... | The Lagrangian method brings conditionally stationary points to the fore, if there are any. In this example there are none, as you have found out.
Now $x^2-y^2=2$ defines a hyperbola $\gamma$ with apexes at $(\pm\sqrt{2},0)$ and asymptotes $y=\pm x$. The function $f(x,y):=x-y$ essentially measures the distance from the... | {
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Show that $(u_1+u_1+...u_k)^n=\sum\limits_{r_1+r_2+...r_k=n}\dfrac{n!}{r_1!r_2!...r_k!}u_1^{r_1}u_2^{r_2}...u_k^{r_k}$
Let $r_1,...r_k$ be integers sucht that, $r_1+r_2+r_3+...,r_k=n$
The number of ways in which a subpopulation of $n$ elements can be partitioned into $k$ subpopulations of which the first contains $r_1... | Think of the simple case (binomial formula): $(\sum_{i=0}^{2}{u_i})^2 = (u_0+u_1)^2$. The "$u_i$'s" are the summands that will be raised to the power of $n$, in this case $n=2$.
$\dfrac{n!}{r_1!r_2!...r_k!}$ ist the "multinomial coefficient" $n \choose{r_1, ..., r_k}$ and it's used to expand $(u_1+u_1+...u_k)^n=\sum\li... | {
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Centre of mass question Find the centre of mass $\overline{P}=(\overline{x},\overline{y},\overline{z}) $ of a unconstrained body $0\le z \le e^{-(x^2+y^2)}$. The density $\delta(x,y,z)$ of the body is constant.
I think we should use cylindricals.
$0\le z \le e^{-r^2}$ and then $r\in[0,+\infty]$, $\phi\in[0,2\pi]$ and $... | Obviously $\bar x=\bar y=0$. Let's call the domain $D$. Then $$\int_Dzdxdydz=\int_{\mathbb R^2} dxdy\int_0^{e^{-(x^2+y^2)}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!zdz=\int_{\mathbb R^2}\frac{e^{-2(x^2+y^2)}}{2}dz=\int_o^{2\pi}d\varphi\int_0^{+\infty}\frac{e^{-2\rho^2}}{2}\rho d\rho=-\frac{\pi}{2}[e^{-2\rho^2}]_0^... | {
"language": "en",
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Suppose that a is a group element and $a^6 = e$. What are the possibilities for $|a|$?
Suppose that $a$ is a group element and $a^6 = e$. What are the possibilities for $|a|$? (Gallian, Contemporary Abstract Algebra, Exercise 18, Chapter 3.)
I just started looking at Abstract Algebra again and I was stuck on this que... | Wouldn’t it just be the divisors of 6?
For instance, if $|a| = 2$, then $a^6$ would be $e$, in virtue of:
$a^6 = (a^2.a^2).a^2 = (e.e).e = e.e = e$
| {
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How does one find an exterior angle bisector relative to the x-axis? Let's say we're given points $A$, $B$, and $C$, which form $\Delta ABC$. Assuming $A=(0,0)$, what is the value of the exterior angle bisector formed by $\angle A$ relative to the x-axis?
(The image is simply to make what I'm asking clearer.)
| if you define two unit vectors in the direction of AB and -AC, you could use the fact that the angles between the bisector- let's call it X- and the two lines are equal, and so is the cosine. Using the dot product: $ \ u \cdot X = v \cdot X $, where u and v are the unit vectors. Once you get a subspace of vectors that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/694516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Harmonic inside with zero average Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where $|dz|$ is arc-length measure?
| We can say that $f$ is either identically $0$, or it attains both positive and negative values in $\Omega$. This follows from the maximum principle.
When $\Omega$ is a disk, we can say that $f$ is zero at the center of the disk. But for any other shapes, no such conclusion is possible: i.e., there is not a point $z_0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/694610",
"timestamp": "2023-03-29T00:00:00",
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Does the $e^z$ function always map a square to an annulus on the plane? Does the $e^z$ function always map a square to an annulus on the plane?
I was doing a few examples recently on my paper where I would take a map and then apply the $e^z$ function to it and I would always get some sort of annulus or a sector of an a... | this always occurs. e to the power of the imaginary part of z gives you an angle and the real part contributes the distance from zero. Applying this to the endpoints of a square/rectangle gives you what you observe.
edit: I'm not so sure what you mean by "take a map" mean (and I cannot comment).
edit2: Oops, I forgot a... | {
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partial Fibonacci summation Let $F_{n}$ be the n-th Fibonacci number.
How to calculate the summation like following:
$\sum_{n \geq 0} F_{3n} \cdot 2^{-3n}$
| Here's an approach via generating functions. As the Fibonacci recurrence is defined by $F_{n+2} = F_{n+1} + F_n$, we have
$$\sum_{n \ge 0} F_{n+2}z^{n+2} = \sum_{n \ge 0} F_{n+1}z^{n+1}z + \sum_{n \ge 0}F_nz^nz^2$$
which with the generating function $G(z) = \sum_{n\ge0} F_n z^n$ gives
$$G(z) - F_0 - F_1z = zG(z) - zF_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/694709",
"timestamp": "2023-03-29T00:00:00",
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Interpreting $\nabla {g}$ when $g(u,x,y)=0$. Say I have $u=f(x,y)=x^2+y^2$. Then $u-x^2-y^2=0$. We can write $g(u,x,y)=u-x^2-y^2=0$.
As a result, we have $\nabla {g}=(1,-2x,-2y)$.
How do we interpret $\nabla {g}$. If we were to plot $g$ for various values of $x$ and $y$, we'd get $0$ everywhere. But $\nabla {g}$, whi... | You are confusing yourself with poor notation.
If you have $f(x,y) = x^2+y^2$, this defines a real valued function defined everywhere. It takes values in $[0,\infty)$.
If you pick some $u \in [0,\infty)$, and consider the set of $(x,y)$ pairs that satisfy the equation $f(x,y) = u$, that is, $L_u = \{ (x,y) | f(x,y) = u... | {
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"source": "stackexchange",
"question_score": "3",
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Why is not the answer to all probability questions 1/2. Ok, I know this is wrong, but I want someone to tell me why.
Let's take a normal heads tails example of a fair coin. The probability of getting head = 1/2. And I write this is because, either it will be heads, or not. Hence two cases that makes it 1/2.
Now, I kno... | We cannot calculate a probability without using other probabilities in the calculation.
When we say that a coin has $P(H)=1/2$, or a die has $P(2)=1/6$ that is not something we learn using probability theory, it is an assumption about physics.
We assume that the die can only land with a face up, and we assume that all ... | {
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A problem on interchange of limit and integration Suppose $\lim_{h\to 0}f_n(h) = 0$ such that $g(h)=\sum_{n=1}^{\infty}f_n(h)$ converges for any $h$. Can we tell that $\lim_{h\to0}g(h) = 0$ ? i.e can we change the order of limit and sum here ? If not what is needed to make it happen ?
I don't know how to use DCT/BCT h... | $$\lim_{h\to0}g(h) = \lim_{h \to 0} \sum_{n=1}^{\infty} f_n(h)=\sum_{n=1}^{\infty} \left({\lim_{h \to 0} f_n(h)}\right) = 0 $$
as $$\sum_{n=1}^{\infty} \left({\lim_{h \to 0} f_n(h)}\right) = \sum_{n=1}^{\infty} 0 = 0 $$
| {
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Expected value of the distance between 2 uniformly distributed points on circle I have the following problem (related to Bertrand):
Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference.
Then connect these points using a line with length $b$. What is the expected length of this line? ($... | You may assume the first point $A$ at $(1,0)$ and the second point $B=(\cos\phi,\sin\phi)$ being uniformly distributed on the circle. The probability measure is then given by ${1\over2\pi}{\rm d}\phi$. The distance $D:=|AB|$ computes to $2\left|\sin{\phi\over2}\right|$, and we obtain
$${\mathbb E}(D)={1\over 2\pi}\i... | {
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What is meant with unique smallest/largest topology? I'm doing this exercise:
Let $\{T_\alpha\}$ be a family of topologies on $X$. Show that there
is a unique smallest topology on $X$ containing all the collections
$T_\alpha$, and a unique largest topology contained in all $T_\alpha$.
I have proved everything exc... | The set of all topologies contained in $\{T_{\alpha}\}$ has a partial ordering so it has a sense of maximal element. You want to show that if $\tau$ is maximal with respect to this partial ordering then $\tau \subset \tau_0$ where $\tau_0$ is the topology you mentioned. Since $\tau$ was maximal you will have $\tau=\ta... | {
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If $n \geq 6$, $G$ or $G_c$ contains a cycle of length $3$ That is the statement, if the order of $G$ is greater or equal to $6$, $G$ or its complementary contain a cycle of length $3$.
I don't really know where to start, I have drawn a lot of examples but I cant understand the essence of the proof.
Any help is welco... | Here are a few assorted hints.
Hint 1: Is it clear that the $n=6$ case implies the $n>6$ case?
Hint 2: This problem is usually phrased in slightly different language. Let me present it to you, as it may help you think about things in a more clear fashion.
Consider the complete graph with six vertices: $K_6$. Take a pa... | {
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What's the "ridge" in Ridge Regression? In normal least squares, we try to find $\hat\beta$ which minimizes
$$\|y-X\beta\|^2$$
Ridge regression expands this to "penalize" certain values of $\beta$ via a matrix $\Gamma$:
$$\|y-X\beta\|^2+\|\Gamma\beta\|^2$$
I'm wondering where the term "ridge" comes from. My best guess ... | As you can see in this link (page 5, col.2), Hoerl (presumably the inventor of ridge regression)
"gave the name "ridge regression" to his procedure because of the similarity of its mathematics to methods he used earlier i.e., "ridge analysis," for graphically depicting the characteristics of second order response surfa... | {
"language": "en",
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Prove by induction that for all $n$, $8$ is a factor of $7^{2n+1} +1$ I want to prove by induction that for all $n$, 8 is a factor of $$7^{2n+1}+1$$
I have proved it true for the base case and assumed it true for $n=k$, but when I cannot figure out when to go towards the end of proving it true for $n=k+1$ assuming it i... | Your approach was right until you considered $7^{2k+1}+1=8m$
$49 \times 7^{2k+1} + 1 = 49 \times(8m-1) +1 = 8 \times 49m -48$ which is divisible by 8.
| {
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Example of a subset of $\mathbb{R}^2$ that is closed under vector addition, but not closed under scalar multiplication? I've found several examples which are closed under scalar multiplication, but not vector addition, but I can't come up with one that is closed under vector addition, but not scalar multiplication.
| The set $\{(x,y): x\ge0, y\ge0\}$ is closed under addition, but not under scalar multiplication, since $-1\cdot(1,1)=(-1,-1)$, for example.
| {
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Interesting applications of Taylor's theorem I am assistant for a real analysis course (kind of a TA, holding a couple of hours complementary to the lecture). I have to treat Taylor's theorem this Monday, and I'd like to give a few examples of where it is useful. The only thing I can think of right now is approximation... | I think that deriving an explicit formula for the $n^{th}$ fibonnaci number by using generating functions is a pretty cool "mathy" application. Check out the book by Wilf ``generatingfunctionology''
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sine defined for a triangle inscribed in a circle with a diameter of one Let a circle be drawn with a diameter of one (and thus a radius of one half). Then let a triangle with vertices A, B, and C be inscribed in the circle (i.e. points A, B, and C are arbitrary points on the circle).
Then a, the side of the triangle ... | Although lab bhattacharjee has already said, we have to use the Law of Sines. If you aren't familiar with it or its proof, see the link. I will tell you how to proceed in a detailed manner.
Here we have our $\triangle ABC$ and its circumscribed circle with center $O$. We now construct a diameter $BOD$. So, $\angle BA... | {
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Understanding the Gram-Schmidt process I would like to better understand the gram-schmidt process. The statement of the theorem in my textbook is the following:
The Gram-Schmidt sequence $[u_1, u_2,\ldots]$ has the property that $\{u_1, u_2,\ldots, u_n\}$ is an orthonormal base for the linear span of $\{x_1, x_2, \ldo... | Consider the following diagram, courtesy of mathinsight.org:
You can think of $(a \cdot u) u$ as the piece of $a$ that is in the direction of $u$. The part that is left over, $a - (a \cdot u) u$, must naturally be the missing side of the triangle, and hence is perpendicular to $u$. So at each step of the Gram-Schmidt ... | {
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Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In SAGE this is E.padic_lseries(p).series(n) where n is the precision to which the series is computed. So please give me the... | As far as I know there is no code to solve this problem distributed with Magma as standard. Your choices are: use the Sage implementation; implement the algorithms in Magma for yourself; or find someone who has appropriate Magma code and persuade them to share it with you.
EDIT. This answer is totally wrong, as ccorn'... | {
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Solving simultaneous equations in terms of variables If $x+y = m$ and $x-y=n$ then $(x^2-y^2) -2x$ is equal to
in terms of $m$ and $n$ only!
How do you solve?
| Notice that
$$(x^2 - y^2) - 2x = (x + y)(x - y) - [(x + y) + (x - y)]$$
But you know what $x + y$ and $x - y$ are ($m$ and $n$ respectively). I believe it is very, very simple to continue on from here.
This is the simplest method to solve this particular problem. Of course, you could also choose to express $x$ and $y$ ... | {
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Subnormal versus quasinormal subgroups Let $G$ be a group and $H$ a subgroup.
$H$ is subnormal if it exists a finite normal chain from $H$ to $G$.
$H$ is quasinormal if $HS=SH$ for all subgroup $S$ of $G$.
If $G$ is a finite group, then every quasinormal subgroup is subnormal.
What about the converse :
Question:... | For an explicit example, take $\;H:=\{(1)\,,\,(12)(34)\}\le S_4\;$ . Then this subgroup is subnormal since
$$H\lhd \{(1)\,,\,(12)(34)\,,\,(13)(24)\,,\,(14)(23)\}\lhd A_4\lhd S_4$$
but it is not quasinormal since if we take $\;K:=\langle(123)\rangle=\{(1)\,,\,(123)\,,\,(132)\}\;$ then
$$HK=\{(1)\,,\,(123)\,,\,(132)\,,\,... | {
"language": "en",
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Is $\infty / \infty = 1$? Lately, my friend and I were arguing about what $\infty / \infty$ equals.
My thinking was that $\infty / \infty = 1$, since no matter how high you go in the numerator, it would have to go equally as high in the denominator.
My friend pointed out that one is not the smallest it can go, and can ... | The following limits all have the indeterminate form of $\frac{\infty}{\infty}$, but they are not all $1$.
$$\lim \limits_{x \to \infty} \frac{x^2}{x}$$
$$\lim \limits_{x \to \infty} \frac{x}{x^2}$$
$$\lim \limits_{x \to \infty} \frac{x}{x}$$
However, if you are given $\frac{\infty}{\infty}$ without context, the value ... | {
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How to solve : $\lim_{n\rightarrow \infty} \frac{n!}{n\cdot 2^{n}}$ $$\lim_{n\rightarrow \infty} \frac{n!}{n\cdot 2^{n}}$$
I need to solve the limit problem above. I have no idea about what to do. What do you suggest?
Thanks in advance.
| Use Stirling's formula $n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^n(1+O(\frac{1}{n}))$. You wil get
$\frac{n!}{n2^n}=\sqrt{\frac{2\pi}{n}}\left(\frac{n}{2e}\right)^n(1+O(\frac{1}{n}))\to \infty$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Definite integral of partial fractions? So I'm to find the definite integral of a function which I'm to convert into partial fractions.
$$\int_0^1 \frac{2}{2x^2+3x+1}\,dx$$
Converting to partial fractions I get...
$\frac{A}{2x+1} + \frac{B}{x+1}$ with $A = 4$ and $B = -2$
Thus the definite integral is...
$$
\begin{alig... | You neglected the chain rule:
$$
\int \frac{4}{2x+1} \, dx = 2\ln|2x+1|+C \ne 4\ln|2x+1|+C.
$$
| {
"language": "en",
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For all $n \geq 1$, and positive integers $a,b$ show: If $\gcd (a,b)=1$, then $\gcd(a^n,b^n)=1$ For all $n \geq 1$, and positive integers $a,b$ show:
If $\gcd (a,b)=1$, then $\gcd(a^n,b^n)=1$
So, I wrote the $gcd (a,b)=1$ as a linear combination: $ax+by=1$
And, I wrote the $gcd(a^n,b^n)=1$ as a linear combination: $a^... | You can use the unique prime factorization to show this result logically . If $a$ & $b$ are relatively prime then they have no common prime factors. Therefore any power of $a$ & $b$ will just repeat each of the prime factors $n$ times. Therefore $a^n$ and $b^n$ still have no common prime factors and therefore are relat... | {
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Give an explicit ring isomorphism
I want to give an explicit isomorphism between $\mathbb{F}_7[X]/(X^2+2X+2)$ and $\mathbb{F}_7[X]/(X^2+X+3)$.
I think the way to do it would be to send a root $\alpha$ of $X^2+2X+2$ to the element $\beta$ of $\mathbb{F}_7[X]/(X^2+X+3)$ so that $\beta$ is a root of $(X^2+X+3)$.
| Hint: Note that $X^2+X+3=0$ and be rewritten as $4X^2+4X+12=0$, and then as $(2X+1)^2+4=0$.
Also, $X^2+2X+2=0$ can be rewritten as $(2X+2)^2+4=0$.
| {
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Bounding $\sum_{p\leq x} \chi(p )$ for non-principal character $\chi$ Suppose $\chi$ is a non-principal Dirichlet character mod $k$. Let $A(x)=\sum_{n\leq x} \chi(n)$. Since $\sum_{n\leq k} \chi(n)=0$, we easily get the bound $|A(x)|\leq \varphi(k)$ where $\varphi$ is the Euler totient function.
Now let's define $B(x)=... | It is known that for $x$ large and if $\chi$ is primitive with modulo $k > 2$, then
$$B(x) \ll k^{1/2} x (\log x)^{-A}$$
for any $A > 0$. The implied constant, which is ineffective, depends only on $A$.
Ref H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS 53, 2004, page 124.
| {
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Prove that if $d$ is a common divisor of $a$ & $b$, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$
Prove that if $d$ is a common divisor of $a$ & $b$, then $d=\gcd(a,b)$ if and only if $\gcd(\frac{a}{d},\frac{b}{d})=1$
I know I already posted this question, but I want to know if my proof is valid:
So for my pre... | It seems fine however this proof can be reduced massively. To prove '$\Leftarrow$',we know $gcd(\frac{a}{d},\frac{b}{d})=1$ therefore we can write this as a linear combination. $$\frac{a}{d}x+\frac{b}{d}y=1$$
Now multiply through by $d$: $$ax+by=d$$.Therefore $gcd(a,b)=d$
To prove'$\Rightarrow$' is mostly the same logi... | {
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Partial Fractions I am working on some online calculus 2 partial fraction problems and I just can not seem to do this one. The question reads: "Evaluate the integral $\int \frac{17x^2}{(x+1)(x^2+1)}\, dx$."
I approached the problem by setting $\frac{17x^2}{(x+1)(x^2+1)}= \frac{A}{x+1} +\frac{Bx+C}{x^2+1}$. I then set $... | Okay, we've found $A = \dfrac{17}2$.
We also know $$A(x^2 + 1) + (Bx + C) (x + 1) = 17x^2$$
Now, if you want to stick with real valued $x$, first let $x = 0$
Then $$\underbrace{\frac{17}2}_{A} + C = 0 \iff C = -\frac{17}{2}$$
Now, let $x = 1$:
$$\underbrace{17}_{2A}+ 2B + 2C = 17 \iff 2B + 2C = 0 \iff B = -C = \frac{1... | {
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Is there a closed form? Is there a closed form for $k$ in the expression
$$am^k + bn^k = c$$
where $a, b, c, m, n$ are fixed real numbers?
If there is no closed form, what other ways are there of finding $k$?
Motivation: It came up when trying to apply an entropy model to allele distribution in genetics. The initial po... | A closed form solution can only exist if m is a rational power of n, and/or $abc=0$. If such is not the case, let $\gamma=\dfrac1{\ln m-\ln n},\quad\alpha=\dfrac cb,\quad\beta=-\dfrac ab$ . Then $k=-x$, where x is the solution to the recursive equation $x=\gamma\ln(\alpha m^x+\beta)$, which can be computed using the fo... | {
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Integral $ \int_{-\infty}^{\infty}\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x $
*
*I need to compute the improper integral
$$
\int_{-\infty}^{\infty}\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x
$$
using contour integration and possibly principal values. Trying to approach this as I would normally approach evaluating ... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
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"answer_id": 4
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Finding a $δ$ for the limit $\lim_{x\to 2} x^4 = 16$ $$\lim_{x\to a} x^4 = L$$
for some arbitrary a
Picking $a$ to be 2, we get:
$$\lim_{x\to 2} x^4 = 16$$
To show that is the limit I tried doing the epsilon-delta definition of a limit to show how to find a $δ$ such that $|f(x) - L| < \epsilon $ for all x satisfying $0... | Let's start at d = 1 ( d = delta ) : 1 < x < 3 ==> 1 < x^2 < 9 ==> 5 < x^2 + 4 < 13 ( yours is 9 ). Next, 3 < x + 2 < 5 ==> /x + 2/ < 5. Finally: /x^2 - 16/ = /(x - 2)(x + 2)(x^2 + 4)/ < 5*13*/x - 2/ = 65*/x - 2/. We need that 65*/x - 2/ < e ( e = epsilon ) ==> /x - 2/ < e/65. We want that /x - 2/ < 1 and also that /x ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that a real number in $(0,1]$ is rational if and only if it has a repeating decimal representation. A decimal expression is said to be repeating if it ends in a repeating pattern of digits. For example, the following are repeating decimal expressions:
$$.333..., .1231333..., 123121312131213...$$
Show that a real n... | Any periodic decimal can be written as a geometric series, where the sum formula is then a rational expression.
The other way around, any rational number $\frac mn$ can be rewritten as $10^{-k}\cdot \frac pq$ with $gcd(p,q)=1$ and $q$ containing no factors $2$ or $5$. Set $d=\phi(q)$ the value of Eulers totient functio... | {
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"source": "stackexchange",
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Maximal ideal contains a zero divisor
Suppose $R$ is a commutative and unital ring. Let the ideal $I$ be maximal and $a,b$ be (nonzero) zero divisors in $R$.
Show that $ab = 0$ implies $a \in I$ or $b\in I$
We've only had a bit of exposure to ideals: we know that $I$ maximal $\to R/I$ field, a little about the Eucl... | In a commutative ring $R$, a maximal ideal $A$ is prime.
$A$ maximal $\Rightarrow R/A$ is a field $\Rightarrow R/A$ is an integral domain $\Rightarrow A$ is prime.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/697361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A family has three children. What is the probability that at least one of them is a boy? According to me there are $4$ possible outcomes:
$$GGG \ \
BBB \ \
BGG \ \
BBG $$
Out of these four outcomes, $3$ are favorable. So the probability should be $\frac{3}{4}$.
But should you take into account the order of their birt... | The possibilities are
ggg ggb gbg bgg gbb bgb bbg bbb
at least one boy... 7/8
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a name for the generalization of the concept "Abelian group" where the axiom $-x+x = 0$ is weakened to the following? Is there a name for the generalization of the concept "Abelian group" where the axiom $−x+x=0$ is replaced by the following list?
*
*$−0=0$
*$−(x+y)=−x+−y$
*$−(−x)=x$
*$x+(-x)+x = x$
In... | Apparently it's called a (commutative) inverse monoid. For further details, see Wikipedia1, 2, 3 or Lawson's Inverse Semigroups4.
(I haven't proven that the sets of axioms are equivalent. You may want to reserve the bounty for someone who does so.)
| {
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I found this odd relationship, $x^2 = \sum_\limits{k = 0}^{x-1} (2k + 1)$. I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ should equal $x^2$.
| Notice : $$\begin{align}(x + 1)^2 - x^2 &= x^2 + 2x + 1 - x^2 \\&= 2x + 1\end{align}$$
We take a summation on both sides and see that a lot of cancellation occurs on the LHS:
$$\sum_{k = 0}^{x-1}\left((x+1)^2 - x^2\right) = \sum_{k = 0}^{x-1}(2x+1)\\
(x -1 + 1)^2 - 0^2 = \sum_{k = 0}^{x-1}(2x+1)\\
x^2 = \sum_{k = 0}^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fermat's last theorem fails in $\mathbb{Z}/p\mathbb Z$ for $p$ sufficiently large
Statement
For any $n, \;x^n+y^n=z^n$ has non-trivial solutions in $\mathbb{Z}/p\mathbb Z$ for all but finitely many $p$.
I remember seeing this problem on an first year undergraduate problem sheet, but never succeeded in solving it. I ... | Not sure what you consider to be elementary but it can be solved with Schur's theorem. See:
http://math.mit.edu/~fox/MAT307-lecture05.pdf
It is theorem 4 in that paper.
| {
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Find the sum of $\binom{100}1 + 2\binom{100}2 + 4\binom{100}3 +8\binom{100}4+\dots+2^{99}\binom{100}{100}$ Find the sum of
$\binom{100}1 + 2\binom{100}2 + 4\binom{100}3 +8\binom{100}4+\dots+2^{99}\binom{100}{100}$
How you guys work on with this question? With the geometric progression? Combination? Or anyother way to ... | $$\sum_{r=1}^{100}2^{r-1}\binom{100}r=\frac12\sum_{r=1}^{100}2^r\binom{100}r=\frac12\left[(1+2)^{100}-1\right]$$
| {
"language": "en",
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How to find the sum of $k$th powers of all proper divisors of first $n$ numbers I am trying this problem but unable to come up with efficient algorithm can someone help with this problem.
I have solved the easier version of the problem
below is the problem link.
Thanks in advance
Spoj 14175. Power Factor Sum Sum (hard... | Simply count the number of times $m$ appears in the list of all the divisors of $\{1,2,...,n\}$, it is $[\frac{n}{m}]$, (where, $[a]$ is the floor of $a$). So the sum of $k$-th power of proper divisors is $\sum\limits_{m=2}^n m^k([\frac{n}{m}]-1)$.
| {
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Proof that $x^n \mod b = (x \mod b)^n$ I've been messing around with modular arithmetic recently, and stumbled across this, but couldn't find a proof for it anywhere. I hate taking things as truth without knowing why, so could anyone provide a (fairly simple) proof?
| Presumably you mean the following congruence $\ X\equiv x\,\Rightarrow\, X^n\equiv x^n\pmod b.\,$ This is simply the Congruence Power Rule, proved below.
Congruence Sum Rule $\rm\qquad\quad A\equiv a,\quad B\equiv b\ \Rightarrow\ \color{#c0f}{A+B\,\equiv\, a+b}\ \ \ (mod\ m)$
Proof $\rm\ \ m\: |\: A\!-\!a,\ B\!-\!b\ ... | {
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A hint on why if $c$ is not a square in $\mathbf{F}_p$, then $c^{(p - 1)/2} \equiv -1 \mod p$ Let $\mathbf{F}_p$ be a finite field and let $c \in (\mathbf{Z}/p)^\times$. If $x^2 = c$ does not have a solution in $\mathbf{F}_p$, then $c^\frac{p - 1}{2} \equiv -1 \mod p$.
I will try to prove the contrapositive: Suppose th... | We assume $p$ is odd, and use an argument that yields additional information.
There are two possibilities, $p$ is of the form $4k-1$, and $p$ is of the form $4k+1$.
Let $p$ be of the form $4k-1$. If $c^{(p-1)/2}\equiv 1\pmod{p}$, then $c^{(p+1)/2}\equiv c\pmod{p}$. But $\frac{p+1}{2}=2k$, and therefore
$$(c^k)^2\equ... | {
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Expected value of random variable I have this question:
What's the expected value of a random variable $X$ if $P(X=1)=1/3$, $P(X=2)=1/3$, and $P(X=6)=1/3$?
I am very confused as to how I can work this problem out. I was thinking it would be something like:
$$E[X] = P(X=1) \cdot (1/3) + P(X=2) \cdot 1/3 + P(X=6) \cdot... | Matt's answer is correct. The expected value is by definition what you expect to get. In investments in gambling you're expected value would be want you expect to run home with after many trials. Most likely the expected value is then negative. That's the concept behind it.
| {
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Linear maps using Tensor Product While I was reading some posts (Definition of a tensor for a manifold, and Tensors as matrices vs. Tensors as multi-linear maps), I encountered the following explanation:
"To give a linear map $V \rightarrow V$ is the same as to give a linear map $V^* \otimes V\rightarrow \mathbb{R}$, a... | Not so detailed as you want but I'll give a hint.
It means you have a vector space isomorphism $\mathcal{L}(V^*\otimes V, \mathbb R)\simeq \mathcal{L}(V)$. For seeing this you have to define a linear bijective map $$\Phi:\mathcal{L}(V)\longrightarrow \mathcal{L}(V^*\otimes V, \mathbb R).$$ To each $T\in \mathcal{L}(V)... | {
"language": "en",
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Solving a Second Order PDE I'm trying to solve the equation $u_t = \alpha^2 U_{yy}$ given $u(y,t)$ bounded $y \rightarrow\infty$ and $u(0,t) = U_o e^{iw_ot}$. Initial is $u(y,0) = 0$. I have gotten both separations as $Y'' - \lambda Y=0$ and $T' = \alpha^2T$, but from here I get confused what to do, I never learnded P... | I would like to point out that if $u(y,0)=0$, the solution is $u(y,t)=0$. This is a classical Possion problem which is derive from heat diffusion. So this equation is also called heat equation. I think a simple way to solve this equation is to use Fourier Transformation.
Let $\hat{u}(x,t)=F[u(y,t)]$ denote the correspo... | {
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A problem with concyclic points on $\mathbb{R}^2$ I am thinking about the following problem:
If a collection $\{P_1,P_2,\ldots,P_n\}$ of $n$ points are given on the $\mathbb{R^2}$ plane, has the property that for every $3$ points $P_i,P_j,P_k$ in the collection there is a fourth point $P_l$ in the collection such that... | This follows directly by applying Sylvester-Gallai Theorem and inversion.
Consider any collection of $n+1$ points $\{ P_1, P_2, \ldots P_{n+1}\}$. Fix $P_{n+1}$, and apply inversion (with respect to a unit circle) to the remaining $n$ points to obtain $\{Q_1, Q_2, \ldots Q_{n}\}$. Note that $Q_{n+1}$ is the point at in... | {
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Divergence of $ \sum_{n = 2}^{\infty} \frac{1}{n \ln n}$ through the comparison test? I have shown that it diverges through the integral test, but I am curious about how this would be shown using the comparison test. I can't use harmonic series because this is lesser than it. I had one idea: harmonic series can be comp... | Using Cauchy condensation, if $\displaystyle \sum_{n = 2}^{\infty} \frac{2^n}{2^n \log 2^n}$ converges or diverges, then the same must be true of my desired series.
This series is equal to $\frac{1}{\log 2} \displaystyle \sum_{n = 2}^{\infty} \frac{1}{n}$, the harmonic series, thus it diverges, and so does my desired s... | {
"language": "en",
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"source": "stackexchange",
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Closedness of the closed half-space Suppose we have a hyperplane $H(p, \alpha) = \{x \in \mathbb{R}^n
\mid p \cdot x = \alpha\}$ , then how do we prove that one of the corresponding closed half-spaces, $H^*(p, \alpha) = \{x \in \mathbb{R}^n \mid p \cdot x \leq α\}$ is indeed closed?
For every $x$ that is an element of... | Let $p$ and $a$ be given, and suppose that $p\cdot x > a$ for some $x$. The goal is to show that there exists $\delta > 0$ such that $p \cdot (x+y) > a$ for all $y$ satisfying $|y| < \delta$. If $\epsilon = p\cdot x -a$, then everything works if $|p\cdot y| < \epsilon/2$, which is definitely accomplished if $|p||y| < \... | {
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Opens in a convergence space By the book "contemporary mathematics", Beyond Topology (F.mynard , E.Pearl)
I am now studying convergence spaces on the book mentioned above. On this book (p.123) I find this definition:
A subset O of a convergence space is open if lim $\mathcal{F}$ $\cap$ O $\neq$ $\emptyset$ implies that... | Well, for a topological space $X$ a set $O \subset X$ is open iff for all $x \in O$, for any filter $\mathcal{F}$ on $X$ with $\mathcal{F} \rightarrow x$ we have $O \in \mathcal{F}$.
(Because $O$ is a neighbourhood of $X$ and a filter convergent to $x$ in a topological space means that all neighbourhoods of $x$ are in ... | {
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Sequences of Rationals and Irrationals
*
*Let $(x_n)$ be a sequence that converges to the irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational?
*Let $(y_n)$ be a sequences that converges to the rational number $y$. Must $y_1, y_2, \dots$ all be rational?
This was one of my midterm que... | Lemma 1: For any real number $x$, there exists a sequence of rational numbers $(x_n)$ such that $\lim (x_n) = x$.
Proof: take e.g. $x_n$ to be the decimal expansion of $x$ truncated to $n$ digits.
Lemma 2: For any real number $x$, there exists a sequence of irrational numbers $(y_n)$ such that $\lim (y_n) = x$.
Proof: ... | {
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Subscript before a function symbol? Does anyone know what the subscript before the function means?
$$
_pf_p
$$
It's part of a definition for selfish routing in networks:
Let $N = (V,E)$ be the network, which is a directed graph. There are $k$ source-destination paris $\{{s_1}, {t_1}\}, ..., \{{s_k}, {t_k}\}$. ${P_i}$ ... | "The load of edge $e$ is $f_e = \sum_{p\in P}$ such that $e$ is in $_pf_p$" should be: $$f_e = \sum_{p\in P \text{ such that }e\text{ is in }p} f_p$$
| {
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How prove this set isn't dense? I want to prove this set $M=\{U \in X ,\ \|U\|\le 1\}$ isn't dense in $X =C[a,b]$.
Can you help me?
| The set is not dense in $C([a,b])$. This means that the closure of $M$ does not equal $C([a,b])$. To see this, let $f\in C([a,b])$ such that $\|f\|=2$. Take the open ball of radius $1/2$ centered at $f$, $B(f,1/2)$. So if $g\in B(f,1/2)$ we have that $3/2< \|g\|< 5/2$. Hence, $g\notin M$ and so $B(f,1/2)\cap M=\emptyse... | {
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Proving that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence. Let $(x_n)_{n=1}^{\infty}$ be a sequence such that $|x_{n+1} - x_n| < r^n$ for all $n \geq 1$, for some $0 < r < 1$. Prove that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.
I understand that a Cauchy sequence means that for all $\varepsilon > 0$ $\exists N$ so t... | For every $\epsilon>0$, take a natural number $N$ such that $r^N <(1-r)\epsilon$, for example by taking $N=\lfloor\frac{\ln (1-r)\epsilon}{\ln r}\rfloor+1$. Then, for all $m,n\geq N$, assume $m<n$, we have
\begin{align}
|x_n - x_m|&=|(x_n - x_{n-1}) + (x_{n-1} - x_{n-2}) + ... + (x_{m+1} - x_m)|\\
&\leq |(x_n - x_{n-1}... | {
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Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ Setting: Let $\sum a_n z^n$ have radius of convergence $R$. We have that
$$
R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}}
$$
via Hadamard's formula for the radius of convergence.
Question: What are the radii of con... | Your calculation for the second is correct. For the first one note that
$$\sum_{n=0}^\infty a_n z^{2n} = \sum_{n=0}^\infty b_n z^n$$
with $b_n=a_{k}$ if $n=2k$ for $k\in\mathbb{N}$ and $b_n=0$ otherwise.
Then
$$R_1=\frac{1}{\limsup_{n\rightarrow\infty}|b_n|^{1/n}}=\frac{1}{\limsup_{n\rightarrow\infty} |a_n|^{1/(2n)}}=... | {
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$\forall x\in\mathbb R$, $|x|\neq 1$ it is known that $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$. Find $f(x)$.
$\forall x\in\mathbb R$, $|x|\neq 1$ $$f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$$Find $f(x)$.
Now what I'm actually looking for is an explanation of a solution to th... | Since it works for all $x$ it means that $t$ has to take all the values of the domain,since t will be equal to all of those values,both $$\frac{x-3}{x+1},\frac{x+3}{1-x}$$
Will take all those values.so basically $f(t)$ in first will be equal to the $f(t)$ in other,or you can take that for example $$a=\frac{x-3}{x+1},b=... | {
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Finding the second derivative; What am I doing wrong? Original Question:
$xy+y-x=1$
Find the second derivative;
$d^2y\over{dx^2}$$(xy+y-x=1)$
We are allowed to use either notation as far as I know: ${dy\over{dx}}$ or ${y'}$. Because ${dy\over{dx}}=y'$ according to my Math 1000 prof.
$(xy)'+y'-x'-1'=0'$
$(y+xy')+y'-1=0$... | Edit: After your equation changes, your work is correct and the "supposed" answer is wrong.
| {
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"timestamp": "2023-03-29T00:00:00",
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Fermat's Last Theorem with negative exponent FLT says that the Diophantine equation $a^n+b^n=c^n$ isn't satisfied by any triplet $(a,b,c)$ where $n\in\mathbb{N}$ and $n>2$.
But what happens if $n\in\mathbb{Z}$ and thus can be negative?
$\textbf{My first thoughts:}$ If we have a negative exponent coupled with a real $a... | If you had $a^{-5}+b^{-5}=c^{-5}$
then you would also have $(\frac{d}{a})^{5}+(\frac{d}{b})^{5}=(\frac{d}{c})^{5}$ for any $d$
but if we choose $d = lcm(a,b,c)$ then we have a solution to $a^5+b^5=c^5$ in integers.
So we can only solve for $-n$, if we can solve for $n$.
| {
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Help with a simple derivative I am trying to solve $\dfrac {6} {\sqrt {x^3+6}}$
and so far I made it to $6(x^3+6)^{-\frac 1 2}$
then I continued and now I have $(x^3+6)^{- \frac 3 2} * 3x^2$
and I cannot figure out what how to find the constant that should be before the parenthesis.
| $$\frac{d}{dx}(\frac{6}{\sqrt{x^3+6}})= 6\frac{d}{dx}(\frac{1}{\sqrt{x^3+6}})\\
\implies 6\frac{d}{dx}(\frac{1}{\sqrt{x^3+6}})=6\frac{d}{d(x^2+6)}(\frac{1}{\sqrt{x^3+6}})\frac{d}{dx}(x^3+6)=\frac{-3}{(x^3+6)^{3/2}}\cdot 3x^2=-9\frac{x^2}{(x^3+6)^{3/2}}$$
| {
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Prove that the logarithmic mean is less than the power mean. Prove that the logarithmic mean is less than the power mean.
$$L(a,b)=\frac{a-b}{\ln(a)-\ln(b)} < M_p(a,b) = \left(\frac{a^p+b^p}{2}\right)^{\frac{1}{p}}$$ such that $$p\geq \frac{1}{3}$$ That is the $\frac{1}{p}$ root of the power mean.
| Here is my proof. I feel like I made a huge leap at the end. I was not sure how to embed my LaTex code, it would not work. So I took screenshots. The last two lines, I have a gut feeling that I am missing a key step that links the two.
| {
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"timestamp": "2023-03-29T00:00:00",
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Euler's Theorem to solve for X $x^{138} \bmod 77 = 25$. How can I use Euler's Theorem to solve for $x$. $77$ is not a prime number but its factors are. $7$ and $11$ are prime, so the totient function of $77$ will be $60$
| $\displaystyle x^{138}\equiv25\pmod{77}\implies x^{138}\equiv25\pmod7\equiv4$
Clearly, $(x,7)=1$ so using Fermat's Little Theorem, $\displaystyle x^6\equiv1\pmod7$
As $138\equiv0\pmod6,x^{138}\equiv1\pmod7$
or $\displaystyle\implies x^{138}=(x^6)^{23}\equiv1^{23}\pmod7\equiv1$
So, we need $\displaystyle1\equiv25\pmod7\... | {
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Non analytic numbers We know that some real numbers (actually, most of them) are not algebraic and the proof of this fact is beautiful: algebraic numbers, like polynomials with integer coefficients, are countable, contrary to $\mathbb R$, hence there should be some non algebraic numbers.
I was wondering if any real nu... | Without loss of generality, we may assume $|\alpha| < 1$.
We can construct the desired Taylor series
$$ f(x) = \sum_{n=0}^{+\infty} a_n x^n $$
as follows. Let $f_k(x)$ be the polynomial
$$ f_k(x) = \sum_{n=0}^{k} a_n x^n $$
*
*Choose $a_0$ so that $0 < a_0 < \alpha$
*Choose $a_k$ so that $|f_{k-1}(\alpha) + a_k \a... | {
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"timestamp": "2023-03-29T00:00:00",
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Question about logic implications If $X\implies Y$ and $X\implies Z$, does that mean that $Y\implies Z$?
I think it does, but can anyone show this as a proof?
Thanks
| No.
What is true is $$[(X \rightarrow Y) \land (X \rightarrow Z)]\implies (X \rightarrow (Y \land Z))$$
In the case that you know $$(X\rightarrow Y) \land (Y\rightarrow Z),$$ then you can infer $$X \rightarrow Z.$$
| {
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Dirichlet kernel identity $\sum\limits_{k=-n}^{n}e^{ikx}=1+ 2\sum_{k=1}^{n}\cos(kx)$ My question is about Dirichlet kernel identity. Why is the following true?
$$\sum_{k=-n}^{n}e^{ikx}=1+ 2\sum_{k=1}^{n}\cos(kx)$$
| Hint: $e^{ikx}+e^{i(-k)x}=2cos(kx)$, for all $1\le k\le n$.
| {
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Do users of RTL languages adopt an LTR standard for mathematics (in the same way they often do when using LTR words or phrases in RTL text)? Non-mathematician here. There is a discussion on this forum titled "Is “applying similar operations from left to right” a convention or a rule that forces us to mark one answer wr... | In Hebrew (& in Israel) you always read equations in LTR.
There are no exceptions (not even inline equations, as one might expect).
RTL math doesn't exist here, so that's just a no, and it would be just as confusing and odd as it would in any other language or place.
So basically I'd say that no one will understand you... | {
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If $(x^y)^z = x^{y\cdot z}$, why does $(-5)^{2^{0.5}}$not equal $(-5)^1$? As shown by Wolfram Alpha, $(x^2)^{0.5}$ is equal to |x|, but if you tried to simplify it to $x^{2\times {0.5}}$, it would just be $x^1$, or $x$.
Is there some unwritten rule about that distribution law that means you can't do it with fractional ... | Whenever we have grouping symbols we work from the inside out , following PEMDAS , WIA included. If you want the exponents to cancel do it this way , $$ (x^{0.5})^2$$
Your numerical example becomes $$((-5)^{0.5})^2 = ( \sqrt{-5})^2 = (i \sqrt{5})^2 = i^2 5 = -5 $$
And in general for real numbers x , $$( \sqrt{x})^2 = x... | {
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Who proved Fundamental Theorem of algebra using Liouville's theorem? One of the most famous proofs of the Fundamental Theorem of Algebra involves Liouville's theorom stating that a bounded entire function in constant.
Who first came up to the idea of deriving FToA from Liouville's theorem? Was it Liouville himself?
... | At the bottom of page 124 of Jesper Lützen's "Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics", it is stated that Liouville was in fact the first person to use this approach to proving the Fundamental Theorem of Algebra. The date given for Liouville's theorem is 1844, but Liouville's formulation was... | {
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Every free module is a projective one I'm trying to understand this proof in Hungerford's book using the universal property of the free modules:
In the whole proof I didn't understand just this line, because we can use the uniqueness just in a function from $F$ to $A$ but $gh$ and $f$ are from $F$ to $B$, what am I mi... | We can use uniqueness in any function from $F$ to any module. Both $f$ and $gh$ are maps $F \to B$ and by design they agree on the basis of $F$. Since any map out of $F$ is uniquely determined by where it sends the basis that means these two maps are the same, because they are both the unique map $F \to B$ determined... | {
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Longest parallel chord of an ellipse I am searching for a source demonstrating that, for any set of parallel chords spanning an ellipse, the longest chord passes through the center of the ellipse. I am not referring to the major and minor axes, which I know are the longest and shortest diameters. Rather, I am referrin... | Every invertible linear transformation preserves the ratio of lengths of parallel line segments. Use a linear transformation that maps the ellipse to a circle.
| {
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Computing the lengths of the obtained trapezium $ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets $BC$ produced at $P$. My book asked me to show that the area of $APB$ and $ABCD$, are the same, which I did. But it aroused my interest. So I researched on how do we compute the sides or simply character... | Edit：
It is easy find all for $ACPD$, $AC$ is trivial to get. then you can have $F$, since $AB$ is known,so $AF,FB,FC$ can be obtained. then you can find $CP,DP$ since $AC$ \\ $DP$
| {
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"timestamp": "2023-03-29T00:00:00",
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References for elliptic curves over schemes As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic curves over fields. Could anyone suggest me any references(books, papers, lectur... | You could try reading (the relevant parts of) Qing Liu's book on Algebraic geometry or the book on Neron models by Bosch-Lutkebohmert-Raynaud to get a feeling for elliptic curves over one-dimensional schemes. You could also try reading some papers where abelian schemes are used, e.g., Szpiro's asterisque (1985) on the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Describe all ring homomorphisms from $\mathbb Z\times\mathbb Z$ to $\mathbb Z\times\mathbb Z$ Note: In this class, a ring homomorphism must map multiplicative and additive identities to multiplicative and additive identities. This is different from our textbook's requirement, and often means there are fewer situations ... | I have a manual that goes through the solutions to this problem. Although Quimey is on track, there is actually 9 possibilities, and they all describe a ring homomorphism.
Think of it this way. Let $f\colon \mathbb Z\times\mathbb Z \to \mathbb Z\times\mathbb Z$ be the function. Then suppose that $f(1,0) = (m,n)$.
Wel... | {
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Example: Irreducible component - affine varieties
Again, I know how to prove the statement. But, I cannot find any example. Please help me for finding an example. Thank you:)
| If $n=2$, take for $X$ the circle $x^2+y^2=1$, for $H$ the "hypersurface" $y=0$ ( a good old line!) and then $X\cap H$ consists of the two irreducible components (=points) $\{(-1,0)\}$ and $\{(1,0)\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/700869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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} |
Probability in monopoly
Based off of this classical monopoly board, my friend told me that it is better statistically to get 3 properties because you are more likely to land on the properties because they are close together. Because the properties are close together, it means that the probability of you landing on the... | Your real mistake here is in assuming that your chance of landing on any particular property each time around the board is $1$ in $40$, which is where you got your 10% and 7.5% chances for landing on a railroad or 3 properties, respectively. You seem to be saying, for example, that in a game where you go around the bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/701007",
"timestamp": "2023-03-29T00:00:00",
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Derivative of the nuclear norm The nuclear norm is defined in the following way
$$\|X\|_*=\mathrm{tr} \left(\sqrt{X^T X} \right)$$
I'm trying to take the derivative of the nuclear norm with respect to its argument
$$\frac{\partial \|X\|_*}{\partial X}$$
Note that $\|X\|_*$ is a norm and is convex. I'm using this for so... | Alt's answer has a fundamental error. First of all, the nuclear norm is the sum of all singular values not the absolute of the singular values.
To make it right, we need to first define the square root for matrix as $\sqrt{BB}=B$. As Alt shown,
$||x||_*=tr(\sqrt{V\Sigma^2V^T})$
But we cannot use the circularity of tra... | {
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"url": "https://math.stackexchange.com/questions/701062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that a triangle that has two congruent angles is isosceles I'm having some trouble with the following problem:
Prove that a triangle that has two congruent angles is isosceles
I tried to prove this by separating it into two triangles and use the ASA or the SAS postulate. However, I am stuck. I need some help. T... | If you bisect the vertex angle, you find that you have created two congruent triangles. The triangles are congruent because of AAS congruence. Because of CPCTC, the sides are congruent as well. It is hard to describe. See this image:
| {
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Expectation of Integrals of Brownian Motion Hello I am not a native english speaker so please let me know if something does not make sense. I am interested in computing the following:
$$E\int_0^T(B_s(\omega,t))^4dt$$
Or at least showing it is finite because I want to prove that $(B_s(\omega,t))^2\in\mathcal H[0,T]$. Th... | Hint Apply Tonelli's theorem and use that $\mathbb{E}(B_t^4)=3t^2$ as $B_t$ is Gaussian with mean $0$ and variance $t$.
Remark Please note that the expression $B_s(\omega,t)$ does not make sense at all. The Brownian motion depends on the time (usually denoted by $t$) and the "random" $\omega$. So, in your case, it sho... | {
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"url": "https://math.stackexchange.com/questions/701302",
"timestamp": "2023-03-29T00:00:00",
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