Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find the limit of the vector function $lim_{t\to\infty} \Big(te^{-t},\frac{t^3+t}{2t^3-1},tsin(\frac{1}{t})\Big)$
a) $lim_{t\to\infty} te^{-t} = \infty \times 0$
$lim_{t\to\infty} 1e^{-t}+-e^tt = 0+(0\times\infty)$=undefined, and repeating l'hospitals rule will render the same result over and over.
b) $lim_{t\to\infty}... | (a)
$$\lim_{t\to \infty}te^{-t}=\lim_{t\to \infty}\frac{t}{e^t}=\lim_{t\to \infty}\frac{1}{e^t}=0$$
(b)
$$\lim_{t\to \infty}\frac{t^3+t}{2t^3-1}=\lim_{t\to \infty}\frac{1+1/t^2}{2-1/t^3}=\frac{1}{2}$$
(c)
$$\lim_{t\to \infty}t\sin \frac{1}{t}=\lim_{t\to \infty}\frac{\sin\frac{1}{t}}{\frac{1}{t}}=\lim_{\frac{1}{t}\to 0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1847362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding the coefficient of $x^{50}$ in $\frac{(x-3)}{(x^2-3x+2)}$ First, the given answer is: $$-2 + (\frac{1}{2})^{51}$$
I have tried solving the problem as such:
$$[x^{50}]\frac{(x-3)}{(x^2-3x+2)} = [x^{50}]\frac{2}{x-1} + [x^{50}]\frac{-1}{x-2}$$
$$ = 2[x^{50}](x-1)^{-1} - [x^{50}](x-2)^{-1}$$
$$=2\binom{-1}{50}-\... | The beginning looks good, but I do not see how you justify the last line. I would use the geometric series instead:
$$\begin{align*}\frac{x-3}{x^2-3x+2} &= \frac{2}{x-1} - \frac{1}{x-2}\\
&= -2\frac{1}{1-x} + \frac{1}{2}\frac{1}{1-\frac 12 x} \\
& = -2 \sum_{n=0}^\infty x^n + \frac{1}{2}\sum_{n=0}^\infty \frac{x^n}{2^n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1847571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Numerical Value for $\lim \limits_{n \to \infty}\frac{x^n}{1+x^n}$
Let $$f (x) := \lim \limits_{n \to \infty}\frac{x^n}{1+x^n}$$
Determine the numerical value of $f(x)$ for all real numbers $x \ne -1$. For what values of $x$ is $f$ continuous?
I honestly do not know how to find the numerical value. I don't even know... | In general you have for $f(x)=\frac{x^n+P(x)}{x^n+Q(x)}$ that $\lim \limits_{n \to \infty}f(x)=1$ if $P$ and $Q$ are polynomials both of degree less than $n$ (divide numerator and denominator by $x^n$ to see this).
With your function $f(x)=\frac{x^n}{x^n+1}$ you have no discontinuity when $n$ is even (because the denom... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
} |
Probability of getting 2 head and 2 tail If a fair coin is tossed 4 times what is the probability that two heads and two tails will result ?
My calculation is.
no. of ways of getting exactly 2 head and 2 tails .will be $6$ out of $8$.
Eg $$HHTT,THHT,TTHH,HTTH,HTHT,THTH,HHHT,TTTH$$
| There are $2^4=16$ possible outcomes:
$HHHH$ $\ \ $ $HHHT$ $\ \ $ $HHTH$ $\ \ $ $HTHH$ $\ \ $ $THHH$ $\ \ $ $\color{red}{HHTT}$ $\ \ $ $\color{red}{HTHT}$ $\ \ $ $\color{red}{THHT}$ $\ \ $ $\color{red}{HTTH}$ $\ \ $ $\color{red}{THHT}$ $\ \ $ $\color{red}{TTHH}$ $\ \ $ $HTTT$ $\ \ $ $THTT$ $\ \ $ $TTHT$ $\ \ $ $TTTH$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Suppose $\left\{ x_{n}\right\} $ is convergent. Prove that if $c\in\mathbb{R} $, then $\left\{ cx_{n}\right\} $ also converges. Good morning! I wrote a proof of the following exercise but I don't know if it is fine:
Suppose $\left\{ x_{n}\right\} $ is convergent. Prove that if $c\in\mathbb{R} $, then $\left\{ cx_{n}\ri... | That looks fine. You may want to comment that this only works for $c\neq0$ (for obvious reasons).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Proving i-th Fibonacci number by induction, can an inductive step be used for two sequential values? I am working through the beginning of Introduction to Algorithms, and came across the problem
Prove by induction that the $i$-th Fibonacci number satisfies the equality
$$ F_{i} = \frac{\phi^{i} - \hat{\phi^{i}}}{\... | For $k\ge 1$, let $A_k$ be the assertion that $F_k$ and $F_{k-1}$ both satisfy the condition.
You have shown that if $A_k$ holds, then the condition is satisfied at $k+1$, and therefore that $A_{k+1}$ holds. So you have proved that $A_n$ holds for all $n$, and therefore that $F_n$ satisfies the condition for all $n$.
F... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Different way solving limit $\lim \limits_{ x\rightarrow 0 }{ { x }^{ x } } $ I know how to solve this problem by using L'Hospital's rule
$$\lim \limits_{ x\rightarrow 0 }{ { x }^{ x } } =\lim\limits _{ x\rightarrow 0 }{ { e }^{ x\ln { x } } } =\lim\limits _{ x\rightarrow 0 }{ { e }^{ \frac { \ln { x } }{ \frac { 1 ... | As pointed by Bernard, that is the same as showing that
$$ \lim_{x\to 0^+} x\log x = 0 \tag{1}$$
or, by setting $x=e^{-t}$,
$$ \lim_{t\to +\infty} t e^{-t} = 0\tag{2} $$
that follows by squeezing: for any $t>0$, $e^t>1+t+\frac{t^2}{2}$, hence:
$$ 0\leq \lim_{t\to +\infty} te^{-t} \leq \lim_{t\to +\infty}\frac{t}{1+t+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1848004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Integral Problem help Given definite integral
$
f(t)=\int_{0}^{t} \frac{x^2+13x+36}{1+{\cos{x}}^{2}} dx
$
At what value of $t$ does the local max of $f(t)$ occur?
What I did is replace $x$ variable with $t$...
and the $f'(t)=\frac{x^2+13x+36}{1+{\cos{x}}^{2}}=0$
because looking for local extrema points is when the... | Note that the first derivative is positive up to $-9$, then negative up to $-4$, then positive. So there is a local max at $-9$ and a local min at $-4$.
I do not know of any way of evaluating the function at the local max and local min except numerically, say using Simpson's Rule.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1848183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Can I identify $X/A$ with the one-point compactification of $X-A$, where $X\supset A$ is a topological space? Intuitively, $A$ collapses to a single point which may represent the infinity point of the one point compactification of $X-A$. Definitely, we should assume $X$ is locally compact Hausdorff.
For example, if $X=... | (Partial answer)
Edit: Corrected earlier omission: it is also necessary that $A$ be closed.
As G. Sassatelli points out, the definition of an open set in $X / A$ and in the compactification of $(X \setminus A)$ are different. A partial answer to your question is to determine under what conditions the canonical bijectio... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is the range of $λ$? Suppose $a, b, c$ are the sides of a triangle and no two of them are equal.
Let $λ ∈ IR$. If the roots of the equation $x^
2 + 2(a + b + c)x + 3λ(ab +
bc + ca) = 0$ are real, then what is the range of $λ$?
I got that $$λ ≤\frac{
(a + b + c)^
2}
{3(ab + bc + ca)}$$
After that what to do?
| For a triangle with sides $a,b,c$ by triangle inequality, we have
$$|a-b|<c$$
Squaring both sides we get,
$$(a-b)^2<c^2\tag{1}$$
Similarly,
$$(b-c)^2<a^2\tag{2}$$
And
$$(c-a)^2<b^2\tag{3}$$
Adding $(1),(2)$ and $(3)$, we get
$$a^2+b^2+c^2 <2(ab+bc+ca) \Longleftrightarrow (a+b+c)^2 <4(ab+bc+ca)\tag{4}$$
From $(4)$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1848389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Justify why there can't be a strict local extremum at $\|x\|$ = $\sqrt{1 \over 2}$ for $f(x) = \|x\|^4 - \|x\|^2$
Given $f: \Bbb R^2 \to \Bbb R$ defined by $$f(x) = \|x\|^4 - \|x\|^2$$ with $x := (x_1, x_2)$, justify why there can't be a strict local extremum at $\|x\|$ = $\sqrt{1 \over 2}$.
Approach
Well, I would g... | The given function is
$$
f(x) = \Vert x \Vert^4 - \Vert x \Vert^2 = \left(x_1^2 + x_2^2 \right)^2 - (x_1^2 + x_2^2) = x_1^4 + 2x_1^2x_2^2 + x_2^4 -x_1^2 -x_2^2
$$
We find that
$$
\nabla f(x) = \left[ \begin{array}{c} \frac{\partial f}{\partial x_1} \\[2mm]
\frac{\partial f}{\partial x_2} \\[2mm]
\end{array} \right] = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Series with digammas (Inspired by a comment in answer https://math.stackexchange.com/a/699264/442.)
corrected
Let $\Psi(x) = \Gamma'(x)/\Gamma(x)$ be the digamma function. Show
$$
\sum_{n=1}^\infty (-1)^n\left(\Psi\left(\frac{n+1}{2}\right)
-\Psi\left(\frac{n}{2}\right)\right) = -1
$$
As noted, it agrees to many de... | Using the integral representation $$\psi(s+1) = -\gamma +\int_{0}^{1} \frac{1-x^{s}}{1-x} \, dx ,$$ we get
$$ \begin{align} \sum_{n=1}^{\infty} (-1)^{n} \left(\psi \left(\frac{n}{2} \right)- \psi \left(\frac{n+1}{2}\right) \right) &= \sum_{n=1}^{\infty} (-1)^{n} \int_{0}^{1} \frac{x^{(n+1)/2-1} - x^{n/2-1}}{1-x} \, dx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1848681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Fundamental group of a compact space with compact universal covering space I have this problem for Riemannian manifold, but think that it is just a topological problem. I know that this is probably a silly question, but it is since a while that I don't study general topology and algebraic topology..
Let $X$ be a comp... | Consider the covering map $\pi \colon \tilde{X} \rightarrow X$. Above any $p \in X$, the fiber $\pi^{-1}(p)$ is a discrete closed subset of a compact space $\tilde{X}$ and so must be finite. By the general theory of covering spaces, if we fix some $\tilde{p} \in \tilde{X}$ with $\pi(\tilde{p}) = p$ then we obtain a bij... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Question on the inequality of sequences Given two sequence $(a_n)_{n \geq 0}$ , $(b_n)_{n \geq 0}$ satisfing $a_n,b_n >0$ for all $n$ and $\sum_{n}a_n \gtrsim \sum_{n}b_n$. My question is that: For a sequence $(c_n)_{n \geq 0}$ be positive, we have the following inequality ?
$\sum_{n}a_nc_n \gtrsim \sum_{n}b_nc_n$
| The answer is negative. Consider the three sequences:
$$
(a_n)= (1,0,0\ldots) \quad (b_n)=\left( 0, \frac12, 0\ldots \right),\quad (c_n)=\left(\frac{1}{4}, 1, \ast, \ast \ldots\right).$$
(Here $\ast$ means any positive number). You have that
$$
\sum_n a_n=1>\frac12=\sum_n b_n, $$
but
$$
\sum_n c_na_n=\frac14<\frac12=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1848890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If you take the reciprocal in an inequality, would it change the $>/< $ signs? Example:$$-16<\frac{1}{x}-\frac{1}{4}<16$$
In the example above, if you take the reciprocal of $$\frac{1}{x}-\frac{1}{4} = \frac{x}{1}-\frac{4}{1}$$
would that flip the $<$ to $>$ or not?
In another words, if you take the reciprocal of $$-... | It depends if $x$ and $y$ are the same sign.
Case 1: $0 < x < y$ then $0 < x(1/y) < y(1/y)$ and $0 < x/y < 1$ and $0 < x/y(1/x) < 1 (1/x)$ so $0 < 1/y < 1/x$.
If both positive, flip.
Case 2: $x < 0 < y$ then $x/y < 0 < 1$. Then as $x < 0$ we flip when we do $x/y*(1/x) > 0 > 1*(1/x)$ so $ 1/y > 0 > 1/x$ so $1/x < 0 < 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1849081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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System of two quadratic equations in two variables with two parameters leads to quintic polynomial Actually, it's two closely related systems. Let $a,b \in \mathbb{Q}$ be the parameters.
The first system has the form:
$$(1+a y)x^2-2(a+y)x+(1+a y)=0 \\ (1-b x)y^2-2(b-x)y+(1-b x)=0$$
One of the solutions can be written ... | You're wrong in considering the two systems as consisting of quadratic equations.
The curve represented by
$$
(1+a y)x^2-2(a+y)x+(1+a y)=0
$$
is, in general, cubic (unless $a=0$). For $a\ne0$ and $b\ne0$, the first system has, properly counting multiplicities, nine solutions and it's very possible that one of the solut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1849179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to turn the reflection about $y=x$ into a rotation. If we reflect $(x,y)$ about $y=x$ then we get $(y,x)$. And because $x^2+y^2=y^2+x^2$ this can also be represented by a rotation.
Using this we get:
$$(x,y)•(y,x)=2xy=(x^2+y^2)\cos (\theta)$$
Hence
$\theta=\arccos (\frac{2xy}{x^2+y^2})$
So using complex numbers w... | If $\theta$ is the angle between the x-axis and the line from 0 to $(x,y)$, then $\theta = \arccos(\frac{y}{\sqrt{x^2+y^2}})$. We reflect about a line with angle $\frac{\pi}{4}$, so the angle between $(x,y)$ and the line $x=y$ is $\frac{\pi}{4} - \theta$. The angle between $(x,y)$ and the reflected point will be doub... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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About the existence of the diagonal set of Cantor The classic proof of the Cantor set start with the assumption that the set
$$B=\{x\in A:x\notin f(x)\}$$
exists, where $f: A\to\mathcal P(A)$ is a bijective function. I understand the proof but I dont understand the assumptions where you start to make this proof.
To be ... | Another example, from the infinite set of natural numbers:
\begin{eqnarray}
S=\mathbb{N}\\
P\left(S\right)=\{\phi, S, \forall U\neq\phi;U\subset S\}\\
f\left(x\right)=\{x\}
\end{eqnarray}
Then f is 1-1.
$B=\{x\in S;x\notin f\left(x\right)\}$
Then
$\forall x\notin B$
$B=\phi \in P\left(S\right)$
Which exists and again i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1849319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Constructing a map of degree 2 $f:T^2\rightarrow S^2$ I know the definition of degree and homology type stuff. But I don't know what a map $T^2\rightarrow S^2$ should actually look like. We never work with explicit examples in my class and I just have no idea what to write. Should I map toroidal coordinates to toroidal... | You want a map $f: S^1 \times S^1 \to S^2$ of degree two. It suffices to provide a map of degree one, since the map $z \times z^2$ is of degree two from the torus to the torus, and degree is functorial.
For a map of degree one, consider the smash product map
$$\gamma: S ^1 \times S^1 \to S^1 \wedge S^1 \cong S^2.$$
To... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is this operator called? If $x \cdot 2 = x + x$
and $x \cdot 3 = x + x + x$
and $x^2 = x \cdot x$
and $x^3 = x \cdot x \cdot x$
Is there an operator $\oplus$ such that:
$x \oplus 2 = x^x$
and $x \oplus 3 = {x^{x^x}}$?
Also, is there a name for such a set of operators ops where...
Ops(1) is addition
Ops(2) is mult... | It is known as a tetration, and it is normally written as $^na$ where n is the height of the power tower. It is the forth hyperoperation.
The zeroth hyperoperation is the successor function, and the first is the zeroth hyperoperation iterated, and so on
A more general way to define the nth hyperoperation is, using the ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate $\lim_{x \to 0^-} \left( \frac{1}{x} - \frac{1}{|x|} \right)$ (possible textbook mistake - James Stewart 7th) I was working on a few problems from James Stewart's Calculus book (seventh edition) and I found the following:
Find
$$\lim_{x \to 0^-} \left( \frac{1}{x} - \frac{1}{|x|} \right)$$
Since there's a $... | bru, I think that the problem is just about terminology. Your derivation is correct, but it is likely that what Stewart is claiming is (I guess) that a limit that goes to $-\infty$ or to $+\infty$ on only one side (as in this example, where the limit is only from the left), is "non existing". Otherwise, the statement "... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $x + y \sim x$ in a commutative monoid, does this imply $y \sim 0$? Let $(M,0,+)$ be a commutative monoid. A congruence relation is an equivalence relation, such that
$$
a \sim b, c \sim d \quad \mbox{implies} \quad a + c \sim b + d.
$$
for all $a,b,c,d \in M$.
Fix some $x,y \in M$. Does $x + y \sim x$ imply $y \... | Minimal counterexample: $(M, \oplus)$ with $M = \{0, 1, 2 \}$, $x \oplus y = \min \{ x+y , 2\}$ and $1 \sim 2$. Then $1 \sim (1 \oplus 1)$ but $1 \not\sim 0$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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why are subgroups defined based on group homomorphisms rather than on its law of composition? i am getting reacquainted with Algebra after some time. I thought i had understood it the last time, but apparently not. The question vexing me is why are subgroups (like the kernel) of a group defined on the basis of a homomo... | No, the definition of a subgroup is defined on the basis of its internal law of composition, and those subgroups which are defined on the basis of some homomorphism are defined as normal subgroups.
And there are a lot of subgroups which are not normal: for example, the group $A_5$ is simple; this means that there are n... | {
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Differential Forms on the Riemann Sphere I am struggling with the following exercise of Rick Miranda's "Algebraic Curves and Riemann Surfaces" (page 111):
Let $X$ be the Riemann Sphere with local coordinate $z$ in one chart and $w=1/z$ in another chart. Let $\omega$ be a meromorphic $1$-form on $X$. Show that if $\omeg... | EDIT: By multiplying by an appropriate polynomial, we may assume that $\omega$ has poles (at most) at $0$ and $\infty$.
On $\Bbb C-\{0\}$ you now have holomorphic functions $f$ and $g$ (your $f_2$) with $$z^2f(z)=-g(1/z).$$ Since $f$ and $g$ have at worst poles at $0$, this equation tells us that each of their Laurent ... | {
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"timestamp": "2023-03-29T00:00:00",
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Is the subgroup of homotopically trivial isometries a closed subgroup of the isometry group? Let $(M,g)$ be a connected Riemannian manifold. Then according to the Steenrod-Myers-Theorem, the isometry group $\text{Isom}(M,g)$ of $(M,g)$ is a compact lie group with the compact-open topology.
Is the subgroup $G$ of isomet... | Let $I$ denote the isometry group of $(M,g)$. Being a Lie group, it is locally path connected. It follows that the subgroup $G< I$ you are interested in, is open. Now, it is a nice exercise to work out is that each open subgroup of an arbitrary topological group is also closed.
| {
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"timestamp": "2023-03-29T00:00:00",
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Infinitely Concatenated Sine and Cosine Using a graphing calculator, if one concatenates sine and cosine repeatedly, i.e.
$$y=\sin(\cos(\sin(\cos(x))))$$
the graph appears to approach a horizontal line, suggesting that at infinite concatenation, there is a single value of the function for all $x$. Is this correct? If... | This is an example of an attractive fixed point.
A fixed point of the function $x\mapsto \sin(\cos(x))$ is a number $x_0$ satisfying $x_0 = \sin(\cos(x_0))$, i.e. the number you put in is the same number that you get out. An attractive fixed point is one for which, if $x$ is sufficiently close to $x_0$, then $\sin(\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1850050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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In this proof for the rationals being countably infinite, is the enumeration strategy important?
During some set theory exercises I came across the proof that the rationals are countably infinite. It used what appears to be a common proof where all the pairs are listed and a zig-zagging path is taking through them, st... | For the traditional bijection with the positive integers you only get one free trip to infinity. The bijection you proposed requires infinitely many trips to infinity. In set theory ordered mappings beyond infinity are associated with "ordinal" numbers.
Ordinal $\omega$ maps to {1 .. $\infty$}. The sequence {2 .. $\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1850159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Mills Test Running Time Can Miller's Test be replaced with the bound below in hopes that it would make a faster general-purpose primality test (compared to ECPP).
If $n$ is an $a$-SPRP for all primes $a$ $<$ ($\log_2 n$)/$2$, then $n$ is prime.
I checked the bounds for the first few prime bases $a$, and the tests are ... | A lot of work has been done on the Miller-Rabin primality test to examine how large a number can be (deterministically) assessed with a few given bases, see wikipedia chapter here. According to that, testing with the prime bases up to $37$ is good to $ 318665857834031151167461 \approx 2^{78}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1850270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differential Geometry for General Relativity I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on Manifolds for me on the side.
I do like mathematical rigor, and I'd like a t... | Check out Barrett O'Neill's book on semi-Riemannian geometry. This book is written exactly for your purposes: it discusses manifolds with symmetric nonsingular metrics, and in particular spacetime metrics. There are even chapters on cosmology and the Schwarzchild metric.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1850381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$. I'm stuck with the proof of the following:
Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.
I've tried starting with $s_... | Since $\{s_n\}$ converges to $s$ and $\{t_n\}$ converges to $t$, $\{t_n - s_n\}$ converges to $t - s$. Since $s_n \leq t_n$ for all $n$, each term $t_n - s_n$ is nonnegative. It thus suffices to show that a sequence of nonnegative terms cannot converge to a negative limit (use proof by contradiction).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1850471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Laurent series of $\frac{1}{z^2(z-1)}$ when $0<\lvert z\rvert<1$ $\frac{1}{z^2(z-1)} = -\left(\frac{1}{z}+\frac{1}{z^2}+\frac{1}{1-z}\right)$.
I know that $\frac{1}{1-z}=\sum\limits_{n=0}^\infty z^n$, but what about the other two terms, should they be left as they are, since we can already think of $\frac{1}{z}$ and $\... | Your approach is right, at the end you just add the three series to obtain the Laurent series of $\frac{1}{z^2(z-1)}$ :
$$-(\sum\limits_{n=0}^\infty z^n+\frac{1}{z}+\frac{1}{z^2})=-\sum\limits_{n=-2}^\infty z^n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1850620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Classifying singularities of $\frac{\sin(\pi z)}{z^4+1}$ If $f(z)=\frac{\sin(\pi z)}{z^4+1}$, we have four roots of unity, which are isolated singularities of $f$: $$z=-(-1)^{1/4},z=(-1)^{1/4}, z=-(-1)^{3/4}, z=(-1)^{3/4}.$$
Do we need to find the Laurent series about all four of the singularities in order to classify ... | No need to compute the Laurent series. All the singularities are zeroes of order $1$ of the denominator, and the numerator does not vanish at them. This implies that they are poles of order one. You can see it as follows. Call $z_i$, $1\le i\le4$ the singularities. Then
$$
f(z)=\frac{1}{z-z_1}\,\frac{\sin(\pi\,z)}{(z-z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1850731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Method for calculating integral of $e^{-2ix\pi\psi}/(1+x^2)$ I am seeking the method for calculating the following integral
$$\int_{-\infty}^\infty\frac{e^{-2ix\pi\psi}}{1+x^2} dx $$
Ideas I have are:
1) substition (however which one?)
2) integration by parts
The integral comes from the Fourier transform of $$\frac{1}... | $$\int_{-\infty}^\infty\frac{e^{-2ix\pi\psi}}{1+x^2} dx =\int_{-\infty}^\infty\frac{\cos(2\pi\psi\,x)}{1+x^2}dx-i\int_{-\infty}^\infty\frac{\sin(2\pi\psi\,x)}{1+x^2}dx$$
Please check this question
$$\color{red}{I(\lambda)=\int_{-\infty}^{\infty}{\cos(\lambda x)\over
x^2+1}dx=\frac{\pi}{e^{\lambda}}}$$
and
$$\color{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1850829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How is defined the inner product $g_p$ on $T_p \mathbb{R}^n/\Gamma$ at the point $p$? In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$, I have some questions related to the resolution of the spectrum of the tori.
The lattice acts on $\mathbb R^n$
by $$γ(x)=x+γ$$ for $x \in\mathbb R^n$, $γ ∈ ... | The metric on $R^n/\Gamma$ is induced by the scalar product of $R^n$. More generally, let $(X,g)$ be a manifold $X$ endowed with a differentiable metric $g$, $G$ a subgroup of isometries which acts properly and freely on $X$, $X/G$ is a manifold and $g$ induces a metric on $X/G$ as follows:
Let $p:X\rightarrow X/G$ the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove by induction that $3^{2n+3}+40n-27$ is divisible by 64 for all n in natural numbers I cannot complete the third step of induction for this one. The assumption is $3^{2n+3}+40n-27=64k$, and when substituting for $n+1$ I obtain $3^{2n+5}+40n+13=64k$. I've tried factoring the expression, dividing, etc. but I cannot ... | Let $A_n = 3^{2n+3}+40n-27$, then $A_n = 11A_{n-1} - 19A_{n-2} + 9A_{n-3}$.
From this it's clear that if 64 divides $A_n$ three consecutive values of $n$ then it holds for the next. So by induction it's enough to check it for $n = -1,0,1$, which is easy enough to do by hand.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Math symbols on vectors e.g item in, item not in, for all I have a vector s= <1,2,3> and I want to perform various operations on it like these ones:
check if an item x exists in s,
x not in s,
a for all i in s where i is the item.
What is the correct math symbols for doing this on a vector? This is perhaps an unconven... | Viewing a (coordinate) vector as a map from an index set to the set of reals (or whatever), "$x$ is a component of $s$" is the same as saying that $x$ is in the image of that map ...
And unless you are also doing something really vector-ish with them, I'd prefer to call $s$ a finite sequence of reals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1851354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is intermediate value property equivalent to Darboux property? I always thought that a function $f:\mathbb{R} \to \mathbb{R}$ has the intermediate value property (IVP) iff it maps every interval to an interval (Darboux property):
Proof:
Let $f$ have the Darboux property and let $a<b$ and $f(a) < f(b)$.
Then $f([a,b])$... | That is correct according to the definition of intermediate value property saying that for all $a<b$ in the domain, for all $u$ between $f(a)$ and $f(b)$, there exists $k\in(a,b)$ such that $f(k) = u$. The two properties are equivalent, and your proof of that is correct.
The blog's author Beni Bogoşel clarified in a c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Decreasing sequence numbers with first digit $9$
Find the sum of all positive integers whose digits (in base ten) form a strictly decreasing sequence with first digit $9$.
The method I thought of for solving this was very computational and it depended on a lot of casework. Is there a nicer way to solve this question?... | Here a simple way to compute it with haskell. The idea is to take all subsequences of "876543210", prepend "9", parse that as an integer and sum them all:
Prelude> (sum $ map (read.("9"++)) $ Data.List.subsequences "876543210")::Integer
23259261861
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1851478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Complexity of LUP decomposition of tri-diagonal matrix to solve an equation? Doing LU decomposition of tri-diagonal matrix and then solving the eqn by using forward substitution followed by backward substitution is done is O(n) time.
http://www.cfm.brown.edu/people/gk/chap6/node13.html
But what are the number of operat... | LU factorization with partial pivoting for banded matrices of size $n\times n$ with bandwidth $w$ (number of subdiagonals plus number of superdiagonals) requires $O(w^2n)$ flops, triangular solvers require $O(wn)$ flops.
Thus, solving linear equations with tridiagonal matrix using LU factorization with partial pivoting... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1851549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A rectangle with perimeter of 100 has area at least of 500, within what bounds must the length of the rectangle lie? Problem
The problem states that there is a rectangle that has a perimeter of $100$ and an area of at least $500$ and it asks for the bounds of the length
which can be given in interval notation or in th... | You are correct that $x + y = 50$ and that $xy \geq 500$. We can solve the inequality by completing the square.
\begin{align*}
xy & \geq 500\\
x(50 - x) & \geq 500\\
50x - x^2 & \geq 500\\
0 & \geq x^2 - 50x + 500\\
0 & \geq (x^2 - 50x) + 500\\
0 & \geq (x^2 - 50x + 625) - 625 + 500\\
0 & \geq (x - 25)^2 - 125\\
125 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1851791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question on infinite abelian group Let $G$ be an infinite abelian group such that every proper non-trivial subgroup of $G$ is infinite cyclic ; then is $G$ cyclic ? ( The only characterization I know for infinite abelian groups to be cyclic is that every non-trivial subgroup has finite index . But I am not getting a... | The answer is yes. Here is an elementary argument.
Suppose firstly that for each $n\in\mathbb N$ we have $nG=G$. Since $G$ is clearly torsion free, it follows that $G$ is a $\mathbb Q$-vector space, immediately violating the hypothesis.
Thus for some $n\in\mathbb N$ we have $nG<G$, so by hypothesis $nG\cong\mathbb Z$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why there is no value for $x$ if $|x| = -1$? According to the definition of absolute value negative values are forbidden.
But what if I tried to solve a equation and the final result came like this:
$|x|=-1$
One can say there is no value for $x$, or this result is forbidden.
That reminds me that same thinking in the pa... | The beauty of math is that you can define everything. The question is: what properties you want this "j" to satisfy? For example, I guess that you want the absolute value $|\cdot|$ to satisfy the triangle inequality. Note that
$$
0=|0|=|j+(-j)|\leq|j|+|-j|=-1-1=-2
$$
a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Formal solution without handwaving about Jordan normal form Let $A$ be a $7\times 7$ matrix over $\mathbb C$ with minimal polynomial $(t-2)^3$. I need to prove $\dim \ker (A-2)\geq 3$.
The handwavy argument I have is that $\deg m$ is the size of the greatest Jordan block while $\dim \ker (A-2)$ is the number of blocks,... | I wouldn’t call your argument handwavy, but we can replace the use of the Jordan normal form by the underlying calculations:
Lemma: Let $V$ be a finite-dimensional vector space and $f, g \colon V \to V$ be two endomorphisms. Then $\dim \ker (fg) \leq \dim \ker f + \dim \ker g$.
Proof: We have $g( \ker(fg) ) \subseteq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $77!$ divisible by $77^7$?
Can $77!$ be divided by $77^7$?
Attempt:
Yes, because $77=11\times 7$ and $77^7=11^7\times 7^7$ so all I need is that the prime factorization of $77!$ contains $\color{green}{11^7}\times\color{blue} {7^7}$ and it does.
$$77!=77\times...\times66\times...\times55\times...\times44\times...\... | If $p$ is a prime number, the largest number $n$ such that $p^n \mid N!$ is
$\displaystyle
n = \sum_{i=1}^\infty \left \lfloor \dfrac{N}{p^i}\right \rfloor$.
Note that this is really a finite series since, from some point on, all of the $\left \lfloor \dfrac{N}{p^i}\right \rfloor$ are going to be $0$. There is also ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
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How I can find all solutions of the ODE $(y')^{2}+y^{2}=4$ I want to find all solutions of this ordinary differential equation:
$$
(y')^{2}+y^{2}=4
$$
but I don't know how. It is impossible by use of series method or Laplace transform?
| $$y'(x)^2+y(x)^2=4\Longleftrightarrow$$
$$y'(x)=\pm\sqrt{4-y(x)^2}\Longleftrightarrow$$
$$\frac{y'(x)}{\sqrt{4-y(x)^2}}=\pm1\Longleftrightarrow$$
$$\int\frac{y'(x)}{\sqrt{4-y(x)^2}}\space\text{d}x=\int\pm1\space\text{d}x\Longleftrightarrow$$
$$\int\frac{y'(x)}{\sqrt{4-y(x)^2}}\space\text{d}x=\text{C}\pm x\Longleftright... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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What is the general formula for these matrix coefficients? Question
I devised an interesting math puzzle for myself but couldn't deduce any solution:
Given:
$$AB=BA=A+B$$
$$ (AB)^n = \sum_{j=1}^n a_j A^j + b_j B^j$$
It's obvious $a_j=b_j$ but what is the general formula for any given $n$?
$$a_j=b_j = ?$$
For Example
... | Let's take a more general approach by evaluating $A^m B^n$.
Let $$A^m B^n=\sum_{i=1}^m {f_{i}(m,n)A^i}+\sum_{j=1}^n {g_{j}(m,n)B^j}$$
It can be seen that $f_{1}(1,n)=g_{i}(1,n)=1$ for $1\le i\le n$
By considering $A^{m+1}B^n$, one can show that: $$f_{1}(m+1,n)=\sum_{i=1}^n {g_{i}(m,n)}$$
$$g_{i}( m+1,n)=\sum_{j=i}^n {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Faster way to find Taylor series I'm trying to figure out if there is a better way to teach the following Taylor series problem. I can do the problem myself, but my solution doesn't seem very nice!
Let's say I want to find the first $n$ terms (small $n$ - say 3 or 4) in the Taylor series for
$$
f(z) = \frac{1}{1+z^2}... | For this particular problem, try a different substitution: $x=z^2$.
Then
$$
\frac1{1+x} = \sum (-1)^nx^n$$
so $$
\frac1{1+z^2} = \sum (-1)^nz^{2n}$$
The probelm of finding a closed form is not always easy. If you can find a closed form for the coefficient of $z^k$ in
$$
\frac{1}{(1-z)(1-z^2)(1-z^3)(1-z^4)\cdots}
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
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How are proofs formatted when the answer is a counterexample? Suppose it is asked:
Prove or find a counterexample: the sum of two integers is odd
The fact that 1 + 1 = 2 is a counterexample that disproves that statement. What is the proper format in which to write this? I will provide my attempt.
Theorem: the sum of... | Cite the counterexample. Since $1 + 1 = 2$, the sum of two arbitrary integers is not always odd.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 15,
"answer_id": 0
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Exponentiation with negative base and properties I was working on some exponentiation, mostly with rational bases and exponents. And I stuck with something looks so simple:
$(-2)^{\frac{1}{2}}$
I know this must be $\sqrt{-2}$, therfore must be imaginary number. However, when I applied some properties I have something u... | Note that $4^{\frac {1}{4}}$ has 4 values $(\sqrt {2},-\sqrt {2},i\sqrt {2},-i\sqrt {2})$
When you square a number or an equation then you are increasing solution values.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The map $f:\mathbb{Z}_3 \to \mathbb{Z}_6$ given by $f(x + 3\mathbb{Z}) = x + 6\mathbb{Z}$ is not well-defined By naming an equivalence class in the domain that is assigned at least two different values prove that the following is not a well defined function.
$$f : \Bbb Z_{3} \to \Bbb Z_{6} \;\;\;\text{ given by } f(\o... | It is easy to show that
$\mathbb Z_3=\left\{\begin{array}\{
\{...,-6,-3,0,3,6,...\},\\
\{...,-5,-2,1,4,7,...\},\\
\{...,-4,-1,2,5,8,...\}\end{array}\right\}$
and that
$\mathbb Z_6=\left\{\begin{array}\{
\{...,-12,-6,0,6,12,...\},\\
\{...,-11,-5,1,7,13,...\},\\
\{...,-10,-4,2,8,14,...\},\\
\{...,-9,-3,3,9,15,...\},\\
\{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 2
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Proof: A function is convex iff it is convex when restricted to any line ..
Let $f\colon \Bbb R^n\to \Bbb R$ be a function. Then, $f$ is convex if
and only if the function $g\colon\mathbb{R} \to \mathbb{R}$ defined as $g(t) \triangleq
f(x+tv)$, with domain
$$
\operatorname{dom}(g)=\{t\mid x+tv \in \operatorname{dom}(f... | The "$\Rightarrow$" part is easy.
The other direction can be proven by contradiction:
Assume that $f$ is not convex. Then,
$\operatorname{dom}(f)$ is not convex or there exist $x,y \in \operatorname{dom}(f)$ and $\lambda \in (0,1)$ with $f( \lambda \, x + (1-\lambda) \, y ) > \lambda \, f(x) + (1-\lambda) \, f(y)$.
*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Show that $[L:K]=1 \Leftrightarrow L=K$ Let $L/K$ be a field extension.
I want to show that $$[L:K]=1 \Leftrightarrow L=K$$
$$$$
I have done the following:
For the direction $\Rightarrow \ : $
Since $[L:K]=1=\text{dim}_KL$ we have that there exist $a\in L$ with $\langle a\rangle$ a $K$-basis of $L$.
So, let $\el... |
To get the desired result, can we just take $a = 1$?
Yes, you are using that in a one-dimensional vector space, any non-zero vector gives a basis.
Can you give me a hint for the other direction?
You have to show that $K$ is one-dimensional as a vector space over itself.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1852968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determine number of real roots on an incomplete polynomial Let's say that I have an incomplete quartic equation with real coefficients, which is $$x^4 - 3x^3 + ... - 10 = 0$$
And also given 2 complex roots, $a + 2i$ and $1 + bi$ where $a$ and $b$ are real numbers.
The problem asks the sum of the real roots, but firstl... | Edit : The OP changed $-3x^2$ to $-3x^3$.
Let $\alpha,\beta,\gamma,\omega$ be the four roots.
Then, by Vieta's formulas,
$$\alpha+\beta+\gamma+\omega=3$$
$$\alpha\beta\gamma\omega=-10$$
Case 1 : If $(a,b)=(1,-2)$, then we may suppose that $\alpha=1+2i,\beta=1-2i$, so
$$\gamma+\omega=1,\quad \gamma\omega=-2$$
So, $\gamm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Where is the fault in my proof? I had some spare time, so I was just doing random equations, then accidentally came up with a proof that showed that i was -1. I know this is wrong, but I can't find where I went wrong. Could someone point out where a mathematical error was made?
$$(-1)^{2.5}=-1\\
(-1)^{5/2}=-1\\
(\sqrt{... | Your mistake is that you have "$(-1)^{5/2} = -1$". It actually holds that $(-1)^{5/2} = i$ since you get by euler identity that
$$(-1)^{5/2} = {e^{i\pi}}^{5/2} = e^{5/2 i\pi} = i.$$
Furthermore you shouldn't write $\sqrt{-1} = i$ because the root isn't defined for negative values and you can get all sorts of wrong pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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quadratic variation of brownian motion doesn't converge almost surely I just came across the following remark:
If $(B_t)_{t\geq0}$ is a one dimensional Brownian motion and if we have a subdivison $0=t_0^n<...<t_{k_n}^n=t$ such that $\sup_{1\leq i\leq k_n}(t_i^n-t_{i-1}^n)$ converges to $0$ is $n$ converges to $\infty$ ... | Consider the sequence of partitions $$\pi(n) = \bigcup_{i=0}^n \left\{\frac in t \right\},\ n\geqslant1 $$ of $[0,t]$, that is, $t_i^n = \frac int$ for $0\leqslant i\leqslant n$. Then $$\sup_{1\leqslant i\leqslant n}\left(t_i^n-t_{i-1}^n\right) = \frac1n\stackrel{n\to\infty}\longrightarrow0, $$ but $$\sum_{n=1}^\infty\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Sensitivity analysis in linear programming Could someone please explain in detailed steps how to apply a sensitivity analysis to such problem:
$$maximize \ \ 2x_1 + 3x_2 \\
s.t. \ \ 4x_1+3x_2≤600 \\ 2x_1+2x_2≤320 \\ 3x_1+7x_2≤840 \\ x_i≥0$$
The goal is it to determine the boundaries of $x_2$.
| I am not particularly keen to do your homework for you "in detail". But here is a starting point. The red line is a contour of $2x+3y$. The blue line is $4x+3y=600$; the orange line is $x+y=160$; the green line is $3x+7y=840$. The other lines are $x\ge0,y\ge0$.
The first question to settle is obviously where the allowe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Trigonometric identities: $ \frac{1+\cos(a)}{1-\cos(a)} + \frac{1-\cos(a)}{1+\cos(a)} = 2+4\cot^2(a)$ I don't really know how to begin, so if I'm missing some information please let me know what it is and I'll fill you guys in :).
This is the question I can't solve:
$$
\frac{1+\cos(a)}{1-\cos(a)} + \frac{1-\cos(a)}{1+\... | Put $t=\tan (\frac a2)$.You have $$\frac{1+\cos(a)}{1-\cos(a)}=\frac{1+\frac{1-t^2}{1+t^2}}{1-\frac{1-t^2}{1+t^2}}=\frac{1}{t^2}$$
Besides $\cot(a)=\frac{1-t^2}{2t}$ so we have to prove
$$\frac{1}{t^2}+t^2=2+4\left(\frac{1-t^2}{2t}\right)^2$$
Immediate calculation gives $$\frac{1+t^4}{t^2}=\frac{1+t^4}{t^2}$$ which is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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$x(a^{1/x}-1)$ is decreasing
Prove that $f(x)=x(a^{1/x}-1)$ is decreasing on the positive $x$ axis for $a\geq 0$.
My Try:
I wanted to prove the first derivative is negative.
$\displaystyle f'(x)=-\frac{1}{x}a^{1/x}\ln a+a^{1/x}-1$. But it was very difficult to show this is negative. Any suggestion please.
| Using simple algebra it is possible to prove that $g(x) = f(1/x) = \dfrac{a^{x} - 1}{x}$ is strictly increasing for $x > 0, a > 0, a \neq 1$ and $x$ being rational. The extension to irrational values of $x$ is easily done by considering sequences of rationals converging to $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Hypergeometric Random Variable Expectation In a binomial experiment we know that every trial is is independent and that the probability of success, $p$ is the same in every trial. This also means that the expected value of any individual trial is $p$. So if we have a sample of size $n$, by the linearity property of the... | As others have pointed out, the probability of a red ball at each of your $n$ draw actually is $R/N$. They are just correlated.
You can also compute this expectation directly from the identity
$$
\sum_{r=0}^n r\binom{R}{r}\binom{N-R}{n-r} = R\binom{N-1}{n-1}
$$
To see this, the rhs counts the number of ways to pick a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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How to solve system of differential equations? I would like to solve a system of differential equations
\begin{align*}
&x''(t) = -a_0(a_1 - bz'(t))\cos(wt), &&x(t_o)= 0, &&x'(t_o)=0\\
&z''(t)= -a_0 bx'(t)\cos(wt), &&z(t_o) =0, &&z'(t_o)= 0
\end{align*}
It reduces to a third order equation
$z'''(t) = a(1-cz'(t))\cos^2... | $$x''(t) = -a_0(a_1 - bz'(t))\cos(wt),\ \ \ \ x(t_o)= 0,\ \ \ \ \ x'(t_o)=0$$
let $x'(t)=y(t) ,z'(t)=\beta(t)$
than we get
$$y'(t)=-a_0(a_1 - b\beta(t))\cos(wt)$$
the second equation converts to $$\beta'(t)= -a_0b y(t)\cos(wt) $$
i think it can be solved now
on solving and referring results from work of @okrzysik So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1853782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$x^2-x+1$ has a root $\!\bmod p\,$ for infinitely many primes $p$ Prove that the equation $$x^2 - x + 1 = p(x+y)$$ has integral solutions for infinitely many primes $p$.
First, we prove that there is a solution for at least one prime, $p$. Now, $x(x-1) + 1$ is always odd so there is no solution for $p=2$. We prove ther... | This answer uses Quadratic Reciprocity (QR).
$$x^2 - x + 1 = p(x+y)$$
$$\iff x^2-x+1-px=py$$
for a fixed $x\in\mathbb Z$ and prime $p$ has a solution $y\in\mathbb Z$ if and only if $p\mid x^2-x+1-px$, i.e. if and only if $p\mid x^2-x+1$.
Your problem is equivalent to proving that $p\mid x^2-x+1$ has a solution $x\in\ma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding percentage of a dollar amount I'm working with a website that can be used to pay contractors on my behalf, instead of requiring them to submit to me their W9 for taxes. The website takes $2.75\%$ in processing fees.
If I'm paying someone $\$22$ per hour, and the website requires $2.75\%$, I believe that would ... | If the processing fee is $2.75\%$ of the amount processed, and you want to have $\$22$ after the fee is taken out, then you have the following equation:
$$x-x\times2.75\%=22,$$
where $x$ is the initial amount (i.e. before the processing fee is taken).
Read the equation as:
*
*From the initial amount $x$
*take out $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1854111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Real roots of $z^2+\alpha z + \beta=0$ Question:-
If equation $z^2+\alpha z + \beta=0$ has a real root, prove that $$(\alpha\bar{\beta}-\beta\bar{\alpha})(\bar{\alpha}-\alpha)=(\beta-\bar{\beta})^2$$
I tried goofing around with the discriminant but was unable to come with anything good. Just a hint towards a solution,... | Let the roots be $-x,-y$ with $x$ real. Then $\alpha=x+y$ and $\beta=xy$, hence
$$
\alpha\bar{\beta}-\beta\bar{\alpha}=(x+y)x\bar{y}-xy(x+\bar{y})=x^2(\bar{y}-y) \\
(\bar{\alpha}-\alpha)=\bar{y}-y \\
(\beta-\bar{\beta})^2 =x^2 (y-\bar{y})^2 \\
$$
It is easy to see that the product of the first two is the las... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1854220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Concept of Trigonometric identities The value of the expression
$$\dfrac{\sin x}{ \cos 3x} + \dfrac{\sin 3x}{ \cos 9x} + \dfrac{\sin 9x}{ \cos 27x}$$ in terms of $\tan x$ is
My Approach
If I take L.C.M of this as $\cos 3 \cos 9x \cos 27x$ and respectively multiply the numerator then it is getting very lengthy. Even ... | $$ {\sin x \over \cos 3x}$$
$$= {\sin x \over \cos x(1 - 4\sin^2 x)}$$
$$= {\tan x \over (1 - 4\sin^2 x)}$$
Now you can get $$\sin^2 x = {\tan^2 x \over 1 + \tan^2 x}$$ from $$ \cot^2 x + 1= { 1\over \sin^2 x}$$
$$\therefore {\sin x \over \cos 3x} = {\tan x \over (1 - 4\sin^2 x)} = {\tan x (1 + \tan^2 x) \over (1 - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1854289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What does it mean for a pdf to have this property?
What does it mean for a probability density function $f(x)$ to have the following property?
$$1+\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx>0$$
I have tried a lot to simplify this condition and see what it means (in terms of moments of $f(x)$... | That can be written as
$$ \int_{0}^{+\infty} x^2 \cdot\frac{d^2}{dx^2}\log(f(x))\cdot f(x)\,dx < 1 $$
that is a constraint that depends on minimizing a Kullback-Leibler divergence.
It essentially gives that your distribution has to be close to a normal distribution (in the KL sense).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1854474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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General solution for $\frac{\mathrm{d}^2 y}{\mathrm{d} x^2} = y$? Start with
$$\frac{\mathrm{d}^2 y}{\mathrm{d} x^2} = y$$
then
$$\frac{1}{\mathrm{d} x} \, \mathrm{d} \left(\frac{\mathrm{d} y }{\mathrm{d} x}\right) = y$$
$$\frac{\mathrm{d} y}{\mathrm{d} x} \, \mathrm{d} \left(\frac{\mathrm{d} y }{\mathrm{d} x}\right) =... | For linear equations with constant coefficients, the "guess-and-check" method, which amounts to assuming $y=e^{\lambda x}$ and solving for $\lambda$, actually does generalize to all possibilities (provided that you can solve the necessary equation for $\lambda$, and with appropriate adjustment for duplicate roots). One... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Writing domains: $∈$ or $⊆$? Usually when we write domains for functions (e.g. $f(x)=x^2$) in set notation, we would write something like this:
$$D=\{x∈ℝ\}$$
This means that all values of x are part of the set of real numbers. However, would it not be more appropriate to write
$$D=\{x⊆ℝ\}$$
or
$$D=\{x⊂ℝ\}$$
Because the... | When a set $D$ is specified or desribed by use of brace brackets, it means that the members of $D$ are all those and only those things that satisfy the conditions written between the brackets.
So $D=\{x\in \mathbb R\}$ means that for any $x,$ we have $ x\in D\iff x\in \mathbb R$. Of course that means $D=\mathbb R$.
And... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1854641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sorting rows then sorting columns preserves the sorting of rows From Peter Winkler's book:
Given a matrix, prove that after first sorting each row, then sorting each column, each row remains sorted.
For example: starting with
$$\begin{bmatrix}
1 & -3 & 2 \\
0 & 1 & -5 \\
4 & -1 & 1
\end{bmatrix}$$
Sorting each row in... | Let's say after sorting, the new matrix is $A$ with m rows and n columns. We have to prove that
\begin{equation}
A_{ij}\leq A_{ik},\, \forall j \leq k
\end{equation}
where $A_{xy}$ is element at $x_{th}$ row and $y_{th}$ column.
Here we know that there are at least $i$ elements (including $A_{ik}$) in column $k$ whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1854750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$ $G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where $n=5, 11, 71$ and order of $G$ is $n^2-1$.
Question:
*
*Does $G$ (as defined above for $n=5, 11, 71$) exist?
*How can I compute such calculation ? Is there a online... | This is only a partial attempt at tackling a more general underlying question.
Note that if $n \ge 5$ is odd, then $n^{2} - 1$ divides $n!$.
In fact $n^{2} = (n - 1) (n +1)$. Clearly $n-1$ divides $n!$. As to $n+1$, since $n$ is odd, $n+1$ is even, so
$$
n+1 = 2 \cdot \frac{n+1}{2}.
$$
Clearly also $2$ and $\frac{n+1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1854953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find the union&intersection of two lines by their equations? I will try to be as clear as possible concerning my confusion, and I will use some examples(several ones).
Case number 1.
Assume two equations(in cartesian form) of two planes.
$2x+2y-5z+2=0$ and $x-y+z=0$
Now,we need to find their vectors.
For the fir... | First of all: $Ax+By+Cz+D=0$ is plane equation.
Case 1:
Intersection of to planes is line. To find equation of that line you have to solve system of equations:
$$
2x+2y-5z+2=0\\
x-y+z=0 \Rightarrow x=y-z \\
$$
If we substitute second equation into first we got
$$
2(y-z)+2y-5z+2=0 \Rightarrow 4y-7z+2=0 \Rightarrow y=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1855054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sierpinski triangle formula: How to take into account for $0^{th}$ power? The formula to count Sierpinski triangle is $3^{k-1}$ .It is good if you don't take the event when $k=0$.But how can you write a more precise formula that takes the $k=0$ into account which gives $3^{-1}$? Just to note, I did figure out the equat... | As I understand it, you want a formula to count the number $n$ of triangles that remain at level $k$ in the standard trema construction of the Sierpinski triangle. If we say that level one is the initial triangle, then that leads to a sequence of images that looks like so:
We can then clearly see your formula: $n=3^{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1855106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving $b(5px - 3x) = a(qx - 4)$ for $x$, and stating any restrictions on the variables I am a high school student in Algebra II and while I normally have no trouble with problems dealing with algebraic equations, I simply cannot muster the answer to this question.
Solve for $x$:
$$b(5px - 3x) = a(qx - 4)$$
State any... | In the between the second and third lines, it looks like you just made a typo. You added $3bx$ to both sides, but then you wrote $3bc$ on the other side. The equation in the third line should have been
$$ 5bpx = aqx - 4a + 3bx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1855183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Fundamental solution of linear system of ODEs I struggle to understand what the fundamental solution is supposed to be.
Specifically it's about a linear system of homogen ODEs with constant coefficents of the form: $\dot{\textbf{F}}=\textbf{AF}$ where $\textbf{F},\dot{\textbf{F}}:\mathbb{R} \to \mathbb{R^n}, \textbf{A}... | This is a subtle point of terminology. The solution to the first-order, linear, homogeneous system $\dot{f}(t) = A\,f(t)$ where $f,\dot{f} : \mathbb{R} \rightarrow\mathbb{R}^{n\times1}$ and $A\in\mathbb{R}^{n\times n}$ has the solution
$$
f(t) = \sum_{k=1}^{n} c_k f_k(t)
$$
where $c_k \in \mathbb{R}$ and $f_k: \mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1855279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Double integration over a general region $\iint x^2 +2y$ bound by $y=x$ $y=x^3$ $x \geq 0$
this is either a type I or type II since the bounds are already nicely given for a type I, I integrated it as a type I:
Finding the bounds:
$x^3=x \to x^3-x=0 \to x(x^{2}-1)= 0 \to x=0, x=\pm1$
Since $-1\lt 0$ my bounds for $x$... | Note that $x^3\lt x$ in the interval $(0,1)$. (A picture always helps in this kind of problem.)
So $y$ travels from $y=x^3$ to $y=x$.
One can see without checking details that the answer $-\frac{4}{21}$ cannot be right. Your integrand is positive in the region, so the answer must be positive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1855395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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compute marginal I have tried to solve this exercise
Let $X$ and $Y$ be random variables with joint probability density function
given by: $f(x,y)=\frac{1}{8}(x^2-y^2)e^{-x}$ if $x>0$, $|y|<x$
Calculate $E(X\mid Y=1)$
so, the marginal $f_Y(y)$ is $\int_y^\infty \frac{1}{8}(x^2-y^2)e^{-x} dx +\int_{-y}^\infty \frac{1}{8... | $$
\text{What you need for the marginal is }
\begin{cases} \displaystyle \int_y^\infty & \text{if } y\ge 0, \\[10pt]
\displaystyle \int_{-y}^\infty & \text{if } y<0. \end{cases}
$$
Or you can just write it as $\displaystyle \int_{|y|}^\infty\!\!.~~$ At any rate in $f_{Y=1}(y)$ you'd have $\displaystyle\int_1^\infty$.... | {
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"timestamp": "2023-03-29T00:00:00",
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Why the probability is $0$ but possible We want to take a random number from natural numbers how much is the probability that,the number be $1$?
When we want to say the probability we say it is $0$ but we say zero for impossible things but that is possible.I know that every number divided by infinity is zero but maybe ... | There are two issues here:
Firstly, you haven't specified what probability distribution on the natural numbers we should assume. You probably mean one which is in some sense uniform: $Pr(0)=Pr(1)=Pr(2)=\cdots$ to infinity. However there is no such distribution, as explained here.
The other issue is that "possible" isn'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1855618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Which one is bigger $\sqrt[1023]{1024}$ or $\sqrt[1024]{1023}$ Which one is bigger $\sqrt[1023]{1024}$ or $\sqrt[1024]{1023}$
I am really stuck with this one.My friend says that it can be solved by $AM-GM$
but I didn't succes.Any hints?
| Raise both numbers to the power of $1023\cdot 1024$ to get $1024^{1024}$ and $1023^{1023}$. Which one looks bigger now?
Alternatively, pick your fravourite from among the two numbers $\sqrt[1023]{1023}$ or $\sqrt[1024]{1024}$, and compare each of the original two numbers to the one you picked.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1855716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Martingale Convergence Theorem I have a Question regarding MCT which I am stuck in, the question goes like this:
Let $X_0 = 1$ and assume that $X_n$ is distributed uniformly on $(0,X_{n-1})$.
and $Y_n = 2^nX_n$.
the questions are:
a) Show that $\left( Y_n\right)$ converges to $0$ a.s.
b) Is $Y_n$ uniformly integrable... | As you say, we can write $Y_n = U_1 \cdots U_n$ where $U_i$ are iid $U(0,2)$. That means $\ln Y_n = \sum_{i=1}^n \ln U_i$. Compute $E[\ln U_i]$ and note that it is negative. So by the strong law of large numbers, $\frac{1}{n} \ln Y_n \to E[\ln U_i] < 0$ a.s. This implies $\ln Y_n \to -\infty$ a.s. which is to say $... | {
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"answer_id": 0
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On the Liouville-Arnold theorem A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to find a transformation which leads to action-angle coordinates and, in these set of variable... | Let $M= \{ (p,q) \in \mathbb{R}^{n} \times \mathbb{R}^n \}$ ($p$ denotes the position variables and $q$ the corresponding momenta variables). Assume that $f_1, \cdots f_n$ are $n$ commuting first integrals then you get that $M_{z_1, \cdots, z_n} := \{ (p,q) \in M \; : \; f_1(p,q)=z_1, \cdots , f_n(p,q)=z_n \} $ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1855956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
} |
Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$ Can anyone give a proof of the derivative of this type of function? Specifically showing that
$\dfrac{d(x^w)}{dx} = wx^{w-1}$ for a real $w$?
I tried to use the Taylor series expansion for $(x+dx)^w$ and got the correct result. However, the proof of the T... | I wrote an answer to a question that was closed as a duplicate of this one. I thought I would add a different answer to this question.
Integer Case
For integer $n\ge0$, the Binomial Theorem says
$$
(x+h)^n=\sum_{k=0}^n\binom{n}{k}x^{n-k}h^k\tag1
$$
So
$$
\frac{(x+h)^n-x^n}h=\sum_{k=1}^n\binom{n}{k}x^{n-k}h^{k-1}\tag2
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 10,
"answer_id": 2
} |
Prove: if $f(0)=0$ and $f'(0)=0$ then $f''(0)\geq 0$
let $f$ be a nonnegative and differentiable twice in the interval $[-1,1]$
Prove: if $f(0)=0$ and $f'(0)=0$ then $f''(0)\geq 0$
*
*Are all the assumptions on $f$ necessary for the result to hold ?
*what can be said if $f''(0)= 0$ ?
Looking at the taylor polynom... | Your proof is enough when $f''$ is continuous.
Here's a way without the continuity assumption.
This Taylor expansion $f(x) = f(0) + xf'(0) + \frac{x^2}{2}f''(0) + o(x^2)$ yields here: $$f(x) = \frac{x^2}{2}f''(0) + o(x^2)$$
Either $f''(0)=0$ and we're good, otherwise the previous equality rewrites as
$\displaystyle \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
A simple question about the Hamming weight of a square Let we define the Hamming weight $H(n)$ of $n\in\mathbb{N}^*$ as the number of $1$s in the binary representation of $n$. Two questions:
*
*Is it possible that $H(n^2)<H(n)$ ?
*If so, is there an absolute upper bound for $H(n)-H(n^2)$?
It is interestin... | Let $n_k=2^{2k-1}-2^k-1$. We have
$$H(2^{2k-1}-2^k-1)=2k-2,$$
because we flip one of the $2k-1$ ones of $2^{2k-1}-1$ to a zero.
On the other hand
$$
n_k^2=2^{4k-2}-2^{3k}+2^{k+1}+1.
$$
Here the integer $m_k=2^{4k-2}-2^{3k}$ has Hamming weight $k-2$, so $H(n_k^2)=k$.
Therefore
$$H(n_k)-H(n_k^2)=k-2,$$
and the answers a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How to find the coordinate vector x with respect to the basis B for R^3?
Find the coordinate vector of $x = \begin{bmatrix}-5\\-2\\0\end{bmatrix}$ with respect to the basis $B = \{ \begin{bmatrix}1\\5\\2\end{bmatrix}, \begin{bmatrix}0\\1\\-4\end{bmatrix}, \begin{bmatrix}0\\0\\1\end{bmatrix} \}$ for $\mathbb{R}^3 $
$[x... | Basically I found that you need to just Put the Basis B matrices together with the vector x at the end and you get $[x]_B = \begin{bmatrix}-5\\23\\102\end{bmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Roulette and Discrete Distribution A roulette wheel has 38 numbers. Eighteen of the numbers are black, eighteen are red, and two are green. When the wheel is spun, the ball is equally likely to land on any of the 38 numbers. Each spin of the wheel is independent of all other spins of the wheel. One roulette bet is a be... | "What is the chance that one places exactly 9 bets before stopping?"
The last two bets have to both be wins or both be loses otherwise stopping would happen after 7 bets, not 9 bets. That makes 7 bet sequence results
possible for a final, 7 bet ahead, win and 7 bet sequence results possible for a final, 7 bet behind, l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
The product of five consecutive positive integers cannot be the square of an integer
Prove that the product of five consecutive positive integers cannot be the square of an integer.
I don't understand the book's argument below for why $24r-1$ and $24r+5$ can't be one of the five consecutive numbers. Are they saying t... | $24r-1$ and $24r+5$ are also divisible neither by $2$ nor by $3$. So they must also be coprime to the remaining four numbers, and thus must be squares.
But this is impossible, because we already know that $24r+1$ is a square, and two non-zero squares can't differ by $2$ or $4$.
For the second part: $6r+1$ is coprime to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving that $2^{2a+1}+2^a+1$ is not a perfect square given $a\ge5$ I am attempting to solve the following problem:
Prove that $2^{2a+1}+2^a+1$ is not a perfect square for every integer $a\ge5$.
I found that the expression is a perfect square for $a=0$ and $4$. But until now I cannot coherently prove that there are n... | I will assume that $a \ge 1$
and show that
the only solution to
$2^{2a+1}+2^a+1 = n^2$
is
$a=4, n=23$.
This is very non-elegant
but I think that
it is correct.
I just kept charging forward,
hoping that the cases
would terminate.
Fortunately,
it seems that they have.
If
$2^{2a+1}+2^a+1 = n^2$,
then
$2^{2a+1}+2^a = n^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How do I know that this system of equations has infinitely many solutions? \begin{cases}
2x + 4y - 2z = 0\\
\\3x + 5y = 1
\end{cases}
My book is using this as an example of a system of equations that has infinitely many solutions, but I want to know how we can know that just from looking at the equations?
| One can write your system $A x = b$ as augmented matrix and bring it into row echelon form
$$
\left[
\begin{array}{rrr|r}
2 & 4 & -2 & 0 \\
3 & 5 & 0 & 1
\end{array}
\right]
\to
\left[
\begin{array}{rrr|r}
1 & 2 & -1 & 0 \\
3 & 5 & 0 & 1
\end{array}
\right]
\to
\left[
\begin{array}{rrr|r}
1 & 2 & -1 & 0 \\
0 & -1 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Show that $\inf \{ m+n\omega: m+n\omega>0,and,m,n\in{Z}\}= 0$, where $\omega>0 $ is irrational. Let $\omega\in\mathbb {R}$ be an irrational positive number. Set
$$A=\{m+n\omega: m+n\omega>0,and,m,n\in{Z}\}.$$
Show that $\inf{A}=0.$
How should I start this problem? I don't get this problem.
| Fix $\epsilon>0$. Fix an integer $N>1/\epsilon$. Consider the fractional parts of the numbers $k\omega, 0<k\le N$. These are commonly denoted
$$
\{k\omega\}:=k\omega-\lfloor k\omega\rfloor.
$$
Because $\omega$ is irrational the numbers $\{k\omega\}\in(0,1)$, $k=1,2,\ldots,N$, are all distinct.
Because there are $N$ of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1856956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Union of two vector spaces Can someone tell me how to union two vector spaces, it might be simple question but I forgot how to do that.
Lets say I was given two vector spaces $W$ and $U$:
$$W = \operatorname{span}\{ (1,2), (1,1) \}$$
$$U = \operatorname{span}\{ (3,4), (2,2) \}$$
What is $W \cup U$?
| If these are both subspaces of $\mathbb{R}^2$, then in fact $U = W = \mathbb{R}^2$, so $W \cup U = \mathbb{R}^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How do you rigorously explain the fact that $u \in L^p$ can be non defined over sets of measure 0? In all the definitions of $L^p(\Omega)$ spaces I have been given these are defined to be the set of functions $f: \Omega \to \mathbb{R}$ whose norm $||\cdot||_{L^p}$ is finite. We define is as the quotient with the equiv... | It doesn't matter - the two versions of the definition give isometrically isomorphic spaces. Allowing functions to be undefined on a set of measure zero can be convenient, for example allowing us to refer to $f(x)=|x|^{-1/2}$ as an element of $L^1([-1,1])$ without having to define $f(0)$. Or allowing us to define $f=\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Constructing $\mathbb{R}$ from $\mathbb{Z}$? I have been told that the real number line $\mathbb{R}$ can be constructed from the cartesian product $\mathbb{Z} \times [0,1)$.
How exactly is that true? Surely, the cartesian product $\mathbb{Z} \times [0,1)$ would give a set of ordered pairs of numbers? How is this equiva... |
I have been told that the real number line $\mathbb{R}$ can be constructed from the cartesian product $\mathbb{Z} \times [0,1)$.
"constructed from" is perhaps relatively vaguely defined, but I assume you mean "has the same cardinality as" or, equivalently, "a bijection exists to".
$$ f((n, r)) = n + r $$ is exactly s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
$C_c^\infty(\Omega)\subseteq L^p(\Omega)$ for any open $\Omega$? Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$. Can we show that $$C_c^\infty(\Omega)\subseteq L^p(\Omega)\tag 1$$ for all $p\in [1,\infty]$? It's clear that $(1)$ holds if $\Omega$ has finite Lebesgue measure. And it's clear that $(1)$ holds for $p... | Let $f\in C^\infty_c(\Omega)$. Then $f$ is supported in a compact set $K$ and $|f|$ attains a maximum $C$ in this $K$. Thus
$$\int_{\Omega} |f|^p dx = \int_K |f|^p dx \le \int_K C^p dx = \text{Vol}(K) C^p.$$
Thus $f\in L^p$ for all $p$. Indeed $C^\infty_c(\Omega)$ is dense in $L^p$ for all $1\le p <\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Using cross product prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$ I am asked to elaborate on the following proof:
Let $\vec{u} \neq \vec{0}$. Prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$.
My attemp... | Hint: both $\sin$ and $\cos$ can't be $0$ at the same time. Note the formula for dot and cross product in terms of angles and magnitudes.
$$a\cdot b=\|a\|\|b\|\cos \theta$$
$$\|a\times b\|=\|a\|\|b\| \sin \theta $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding the basis and dimension of a subspace of the vector space of 2 by 2 matrices I am trying to find the dimension and basis for the subspace spanned by:
$$
\begin{bmatrix}
1&-5\\
-4&2
\end{bmatrix},
\begin{bmatrix}
1&1\\
-1&5
\end{bmatrix},
\begin{bmatrix}
2&-4\\
-5&7
\end{bmatrix},
\begin{bmatrix}
1&-7\\
-5&1
\en... | Inputs
$$
\alpha =
\left(
\begin{array}{rr}
1 & -5 \\ -4 & 2
\end{array}
\right),
\qquad
\beta =
\left(
\begin{array}{rr}
1 & 1 \\ -1 & 5
\end{array}
\right),
\qquad
\gamma =
\left(
\begin{array}{rr}
2 & -4 \\ -5 & 7
\end{array}
\right),
\qquad
\delta =
\left(
\begin{array}{rr}
1 & -7 \\ -5 & 1
\end{arra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How would you work out these combinations?
*
*If there are 16 different ice-cream flavours, how many combinations are there for a two scoop?
*If there are still 16 different ice-cream flavours, how many combinations are there for a three scoop?
How would you work out the above combinations? I found it just sitting ... |
1) working out the number of combinations including duplicate scoops
(e.g. chocolate-chocolate-vanilla)
Consider the case where there is only one scoop of ice cream. There are 16 flavors (choices), and thus 16 "combinations." The next case is 2 scoops. One way to think about this problem is to consider how many cho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$A \subseteq \mathbb R^n $ s.t. for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ , is $A$ closed $\mathbb R^n$? Let $A \subseteq \mathbb R^n $ such that for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ ; then I know that $A$ is bounded ; my ques... | Assume $A$ is not closed. Then there is some $x_0 $ in the closure of $A$, but not in $A$.
This implies that the continuous function $f(x) := |x - x_0| $ assumes only positive values on $A$, but the closure of the image contains $0$, which contradicts closeness of $f(A)$.
Hence, $A$ is closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Negation of definition of continuity This should be a very easy question but it might just be that I'm confusing myself. So we have the definition of a function $f$ on $S$ being continuous at $x_0$:
For any $\epsilon$>0, there exists $\delta>0$ such that: whenever $|x-x_0|<\delta$, we have $|f(x)-f(x_0)|<\epsilon$
An... | The negation is:
there exists $\epsilon >0$ such that for any $\delta>0$ we can find an $x$ such that $|x-x_0|<\delta$ and $|f(x)-f(x_0)| > \epsilon$.
And you have just proved this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1857945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
} |
Integral over infinity of $F(x) = e^{ix}$ how to calculate integral of following function over infinity ?
$F(x) = e^{ix}$ ($i$ imaginary)
$$
\int\limits_{-\infty}^\infty e^{ix} \, dx
$$
| $$\int_0^{\infty}e^{ix}dx=\left[\frac{e^{ix}}{i}\right|_0^{\infty}=\left[-ie^{ix}\right|_0^{\infty}=\left[ie^{ix}\right|^0_{\infty}=i-e^{i\infty}$$
this result not converge, are you sure the sign in the exponential is correct?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1858043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Chain rule, time derivative and change of variables A simple calculus question. If I apply the chain rule to a composite function:
$$\frac{d}{dt}f(x(t))=\frac{\partial}{\partial x}f(x(t))\frac{dx}{dt}$$
Now, if I change variables, and define:
$$x=x_1+\lambda x_2$$
I can say:
\begin{equation}
\frac{d}{dt}f(x_1(t),x_2(t... | \begin{equation}
\frac{df(x)}{dt}=\frac{df(x)}{dx}\frac{dx}{dt}=\frac{df(x)}{dx}(\frac{dx}{dx_1}\frac{dx_1}{dt}+\frac{dx}{dx_2}\frac{dx_2}{dt})=\frac{df(x)}{dx}(\frac{dx_1}{dt}+\lambda^{-1}\frac{dx_2}{dt})
\label{eq:1}
\end{equation}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1858208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $f(n) =\displaystyle\sum_{r=1}^{n}\Biggl(r^n\Bigg(\binom{n}{r}-\binom{n}{r-1}\Bigg) + (2r+1)\binom{n}{r}\Biggr)$, then what is $f(30)$? Please give me hints on how to solve it.
I tried 2-3 methods but it doesn't go beyond two steps. I am out of ideas now.
Thank you
| We may simply deal with each piece separately:
$$ \sum_{r=1}^{n}(2r+1)\binom{n}{r}=\left.\frac{d}{dx}\sum_{r=1}^{n}\binom{n}{r}x^{2r+1}\right|_{x=1}=\left.\frac{d}{dx}\left(x\cdot\left(1+x^2\right)^n-1\right)\right|_{x=1}=2^n(n+1)-1.$$
On the other hand, by summation by parts:
$$ T_n=\sum_{r=1}^{n}\left(\binom{n}{r}-\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1858347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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