Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Finding $\int \frac{\cos(2x) dx}{\cos^4x+\sin^4x}$ $\int \frac{\cos(2x) dx}{\cos^4x+\sin^4x}$
I'm looking more into simplifying this than the solution itself (which I know involves using t=tan(x/2)).
I did:
$$\int \frac{\cos(2x) dx}{\cos^4x+\sin^4x} = \int\frac{\cos(2x)dx}{(\cos^2x+\sin^2x)^2-2\sin^2x\cos^2x} = \int\fr... | We know that
$$\begin{align}
\cos(2x) &= \cos^2(x) + \sin^2(x)\\
\sin(x) &= \dfrac{\tan(x)}{\sec(x)}\\
\sec^2(x) &= 1 + \tan^2(x)
\end{align}$$
So,
$$\int\dfrac{\cos(2x)}{\sin^4(x)+\cos^4(x)}\,\mathrm dx \equiv \int\sec^2x\left(\dfrac{-(\tan(x) - 1)(\tan(x) + 1)}{\tan^4(x) + 1}\right)\,\mathrm dx$$
Let $u = \tan(x)$. S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3480289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Taylor series of $x \cdot \ln(10+x) $ with $x=9$ point of expansion Have been trying to solve this for quiet long, however still have no idea of how to.
$$x \cdot \ln(10+x) $$
with $x=9$ point of expansion.
Will appreciate any advice, thanks in advance!
| $$f(x)=x \ln (19+x-9)= x \ln 19+ x\ln\left(1+\frac{(x-9)}{19}\right)$$
Let $(x-9)/19=z$, then
$$\implies f(x)= 19 z \ln 19+ 9 \ln 19 + 19z \ln(1+z)+9 \ln (1+z)$$
Now use $$\ln(1+z)=-z-z^2/2-z^3/2--...$$
to expand $f(x)$ in the powers of $z=(x-9)/19.$ and add the co-efficients of similar powers of $z$.WE get
$$f(x)=9\ln... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3480456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Are strictly convex Banach norms Fréchet differentiable? Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff
$$\lim_{h \to 0} \frac{\|f(x + h) - f(x) - Ah\|_W}{\|h\|_V} = 0... | Here is a counterexample on $\mathbb R^2$:
$$
\|(x,y)\| := \sqrt{ \max(x^2 + 2y^2, \ 2x^2 + y^2 )}.
$$
It is the maximum of two striclty convex norms.
It is strictly convex and not differentiable along for points with $|x|=|y|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3480598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An elegant proof for a claim about orthogonal and positive-definite matrices Let $P$ be a real $n \times n$ symmetric positive-semidefinite matrix.
Suppose that $\langle O,P \rangle = \langle I,P \rangle$ for some orthogonal matrix $O$. Here $\langle , \rangle$ is the Euclidean (Frobenius) inner product. The assumption... | The following proof does diagonalize $P$, but not in matrix form.
Let $\{v_1,\ldots,v_n\}$ be an orthonormal eigenbasis of $P$ and $Pv_i=\lambda_iv_i$ for each $i$. Then
$$
\langle P,I\rangle
= \langle P,O\rangle = \sum_i\langle Pv_i,Ov_i\rangle
\le \sum_i\|Pv_i\|\|Ov_i\|
= \sum_i\lambda_i
= \operatorname{tr}(P)
= \lan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3480694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Is this a valid mathematical model (MILP)? Is it ok to calculate values in one set of constraint and than using it for another in MILP model. Here Z and Y are binary variable.
| No, in a mathematical programming model the limits of summation cannot be variables. (In a constraint programming formulation, you can sum a variable number of variables.) One workaround (assuming the $Z_{j,q}$ are nonnegative integers) is the following:
*
*introduce binary variables $x_{1,q},\dots, x_{K_q,q}$, wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3480946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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If $m$ and $n$ are integers and $mn$ is even, $m$ is even or $n$ is even. I'm looking for feedback on my proof of the following statement:
"If $m$ and $n$ are integers and $mn$ is even, then $m$ is even or $n$ is even."
I tried using a direct proof method:
(1) Since $mn$ is even, $mn = 2k$ for some integer $k$. The ... | The really serious problem in your argument is here:
(2) In order for $k$ to be an integer, $m$ or $n$ must then have a
factor of two, and the statement is proved.
That's a problem because it is just a restatement of what you have been asked to prove.
One good way to start this problem is to suppose that both facto... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3481022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Calculating the genus of Quartic model $y^2=x^4+bx^3+cx^2+dx+e$ of elliptic curves. This link says that $y^2=x^4+bx^3+cx^2+dx+e$ is an elliptic curve.
*
*How do we compute its genus (which should be 1)?
*Under what conditions is $y^2=p(x)$ an elliptic curve where $p(x)$ is polynomial in $x$ of degeree $\ge 5$?
| This is called a hyperelliptic curve.
Basically your questions are already answered in the above wiki link, so please read that first.
I just want to add several important details as complements:
*
*One usually should assume that the polynomial $f(x) = x^4+bx^3+cx^2+dx+e$ doesn't have multiple roots, i.e. that $f$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3481144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Taylor expansion about candidate point $x^*$: $f(x^∗ + h) = f(x^∗) + hf'(x^∗) + O(h^2)$? My textbook, Algorithms for Optimization, by Kochenderfer and Wheeler, says the following:
A point can also be at a local minimum if it has a zero derivative and the second derivative is merely nonnegative:
*
*$f'(x^∗) = 0$, the... | Have you tried
$$x=x^*+h, a=x^*$$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3481277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all integers $m,\ n$ such that $m^2+4n$ and $n^2+4m$ are both squares.
Find all integers $m,\ n$ such that both $m^2+4n$ and $n^2+4m$ are perfect squares.
I cannot solve this, except the cases when $m=n$.
| Hint : There is an useful technique dealing with some problems including squares. Assume $m \ge n$ and bound $m^2+4n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3481384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Example where $n^{-1/2}S_n\Rightarrow N(0,1)$ but its variance does not converge to 1 Let $\{X_n\}$ be a sequence of independent random variables such that $X_n$ takes the values $\pm n$ each with probability $1/2n^2$ and $\pm 1$ each with probability $1/2(1-1/n^2).$ Define $S_n=X_1+\cdots+X_n$ for $n\geq 1.$ Show that... | $\sum P(Y_n=1)=\sum \frac 1 {2n^{2}} <\infty$. By Borel - Cantelli Lemma it follows that $Y_n=0$ for all large $n$ with probability $1$. This implies that the second term tends to $0$ almost surely.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3481529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove that $\tanh^2(x) \leq x^2$ I'm looking for an alternative proof to prove that $\tanh^2(x)\leq x^2$. My current proof is by observing that $\tanh'(x)=1-\tanh^2(x)\leq 1$, hence integrating for positive $x$ gives that $\tanh(x)\leq x$ hence $\tanh^2(x)\leq x^2$, the proof follows by parity of both $\tanh^2$ and $x^... | Suppose that $x\geqslant 0$. Then$$\sinh x=x+\frac1{3!}x^3+\frac1{5!}x^5+\cdots+\frac1{(2n-1)!}x^{2n-1}+\cdots$$and$$x\cosh x=x+\frac1{2!}x^3+\frac1{4!}x^5+\cdots+\frac1{(2n-2)!}x^{2n-1}.$$Therefore, since we are assuming that $x\geqslant0$, $\sinh x\leqslant x\cosh x$. In other words, $\tanh x\leqslant x$. And, since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3481753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\sqrt{\langle T(x), x \rangle}$ is a norm when $T$ is positive definite
I'm having some trouble proving the second property of a norm, i.e.
$$\| x+y\| \leqslant \| x\| + \|y\|.$$
Let $T: R^n \to R^n$ be a linear operator and $\langle, \rangle: R^n\times R^n \to R$ be defined by
$$\langle x,y\rangle=\sum... | I don't buy this:
$$ \sum_{i=1}^n T(x_i)x_i \sum_{i=1}^n T(y_i)y_i = \sum_{i=1}^n T(x_i)y_i \sum_{i=1}^n T(y_i)x_i $$
For example
$$ 45 =(1 \cdot 1 + 2 \cdot 2)(2 \cdot 2 + 3 \cdot 3) \ne (1 \cdot 2 + 2 \cdot 3)(2 \cdot 1 + 3 \cdot 2) = 64.$$
Instead, you have to prove Cauchy-Schwarz for the inner product $(x,y) \ove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3481996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Rhombus and circles angle problem Let $ABCD$ be a rhombus.
The circle $(C_1)$ of center $B$ passing through $C$ and the circle $(C_2)$ of center $C$ passing through $B$.
$E$ is one of the two points of $(C_1) \cap (C_2)$.
The line $(ED)$ meets $(C_1)$ again in $F$.
It is asked to find the measure of angle $\angle AFB$.... | It should be clear that $\triangle BEC$ is equilateral. Denote $\angle BFD=\alpha$ and $\angle AFD=\beta$. We need to find $\alpha+\beta$. You can show that $\angle ADF=60^{\circ}-\alpha$ through some angle chasing. Sine law for $\triangle ADF$ gives you
$$\frac{AD}{AF}=\frac{\sin\beta}{\sin(60^{\circ}-\alpha)} $$
and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3482089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Finding extrema of non-linear second-order ODE I'm dealing with a non-linear ODE of the form
$$a\frac{d^{2}y}{dx^{2}}+bA(x)\frac{dy}{dx}+cy^{3}+y=[A(x)]^{2}$$
where a,b, c are positive constants and the function A(x) is a real valued and continuously differentiable (except, maybe, at x=0). The only unchangeable bounda... | Numerically computing the solution to the differential equation using most approaches would rewrite the problem as
$$\begin{cases}z'=\frac1a\left(A^2-bAz-cy^3-y\right)\\y'=z\end{cases}$$
In which case one would be computing $y$ and $y'$ simultaneously, and the extrema can be numerically deduced based on $y'$, which you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3482336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Finding an elliptic curve with Frobenius trace zero The following theorem by Waterhouse lists all values of the Frobenius trace such that there is a corresponding elliptic curve over $\mathbb{F}_q$, $q = p^n$, $p$ prime.
The thing is, i couldn't find a single curve with $n$ even, $t = 0$ and $p \ne 1 $mod $4$ at the s... | Using that $Tr(\phi_q)=0\implies \Bbb{Z}[\phi_q] \cong \Bbb{Z}[i p^{n/2}] \subset \Bbb{Z}[i]\implies $ the curve is probably a reduction of $y^2=x^3+x$, also the dual endomorphism of $\phi_q$ is $-\phi_q$ thus the curve is supersingular, and this
I obtained the Magma code
K<a>:= GF(7^2); E:=EllipticCurve([K|1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3482443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Why does $(I-A)$ has inverse when $\|A \| < 1$ I've seen a proof using the convergence of numerical series, but it is too large, could you please tell me if there is a shorter proof. Thank you!
| We need the norm to be submultiplicative, i.e. $\|XY\|\le\|X\|\|Y\|$ for every pair of matrices $X$ and $Y$, otherwise the statement isn't true. For instance, let $\epsilon>0$ and define a matrix norm $\|X\|=\epsilon\sum_{i,j}|x_{ij}|$ for real matrices. Then $\|I\|<1$ when $\epsilon$ is sufficiently small, but $I-I=0$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Does this differential equation have an elementary solution? $$\frac{dx}{dy} =x-y^2$$
My professor told the class that this differential equation has no solutions. My question is how come? can't you just separate variables shown here.
$$y^2dy =xdx$$
$$y= \sqrt[3]{ \frac{x^2}{6} } $$
| $$\frac{dx}{dy} =x-y^2$$
The solution of this ODE is :
$$x(y)=ce^y+y^2+2y+2$$
Probably your Professor didn't say "this differential equation has no solution" but he said something that you not well understood.
Possibly he was talking of the function $y(x)$ which is the inverse function of the above known function $x(y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3482598",
"timestamp": "2023-03-29T00:00:00",
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Bilinear transform and higher order differential equation The bilinear transform as I understand corresponds to the trapezoid rule. However, I have not been able to find whether the correspondence holds for higher order ODEs, or what kind of estimate the bilinear transform corresponds to in that case. As an example I u... | Take one step more, either using $y(x+2h)$ or here for symmetry $y(x-h)$, and consider that equality is only up to terms of size $O(h^3)$,
\begin{align}
&y(x+h)-2y(x)+y(x-h)
\\
&=\frac{h}2(y'(x+h)-y'(x-h))=\frac{h^2}4(y''(x+h)+2y''(x)+y''(x-h))
\\
&=-\frac{h^2}4[ry'(x+h)+ω^2y(x+h)+2ry'(x)+2ω^2y(x)+ry'(x+h)+ω^2y(x+h)]
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3482730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Uniform Continuity of a function with the simplest way I'm trying to show that
$f(x) =
\begin{cases}
x \sin\left(\frac1x\right),\quad\text{if $x \in (0,1]$ }\\[2ex]
0, \quad \quad \quad \quad \ \text{if $x=0$}
\end{cases}$
is uniformly continuous on $[0,1]$
Let $\epsilon \gt 0$ and let $x, y \in (0,1)$.
Then
$$
\lef... | Here is an ad-hoc proof. Let $\epsilon>0$ be fixed. We can and do adjust it to $1$ if it is bigger.
The function $|f'|$ is bounded and continuous on $[\epsilon/3, \; 1]$, let $M> 3$ be an upper bound.
We set $\delta = \epsilon/M<\epsilon/3$.
Let $x,y$ be two points in $[0,1]$ at distance $<\delta$, and we can and do as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3482815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Compute telescopic sum of binomial coefficients Is there a nice or simple form for a sum of the following form?
$$ 1 + \sum_{i=1}^k \binom{n-1+i}{i} - \binom{n-1+i}{i-1}$$
Motivation: Due to a computation in the formalism of Schubert calculus the above sum with $k = \lceil n/2 \rceil -1$ is equal to the number of lin... | Hint
The following method does not use telescopic series but it makes use of elementary binomial theorem along with some geometric progression to arrive at the answer.
$$\begin{aligned}S&=\sum_{i=1}^{k}{n-1+i\choose n-1}-\sum_{i=1}^{k}{n-1+i\choose n}\\&=\left(\text{coeff. of } x^{n-1} \text{ in } \sum_{i=1}^{k}(1+x)^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3483017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Finding roots using recurrence relations Everyone knows the Fibonacci sequence:
$s[ ] = 1, 1, 2, 3, 5, 8, ...$
where
$s_{n+2} = s_n + s_{n+1}$
This represents a single solution to the polynomial, $x^2 = x + 1$.
Recurrence relations can be applied to find roots of other polynomials. For example, from the relation, $x^... | I am not sure regarding its relation to the Newton-Raphson method, but consider the function $f(x)=1+\frac 1 x$, and the recursively defined sequence
$$a_1=1,\qquad a_{n+1}=f(a_n), \quad \forall n\in \mathbb N $$.
Clearly, such sequences (i.e. defined recursively by a continuous function) can converge only to a fixed p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3483122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do Sylow subgroups distinguish representations? Let $G$ be a finite group, $X$ the set of elementary subgroups of $G$, $Y$ the set of Sylow subgroups of $G$.
Propsition 29 in Serre's Linear Representations of Finite Groups implies that the set $X$ distinguish representations, in the sense that
For any finite dimension... | The set $Y$ of Sylow subgroups does not distinguish between complex representations.
Let $G$ be cyclic of order $6$, and consider the two representations $\rho_1$ and $\rho_2$ of $G$ of degree 2 that map a generator $g$ of $G$ to
$$ \left(\begin{array}{cc}\omega&0\\0&-\omega^2\end{array}\right)\ \ \ \ \mathrm{and}\ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3483247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exercise 3.4.14 Introduction to Real Analysis by Jiri Lebl
Suppose for $f: [0,1] \to R$, we have $|f(x) - f(y) | \le K |x-y|$, and $f(0)= f(1) = 0$. Prove that $|f(x)| \le K/2$. Further show by example that $K/2$ is the best possible, that is, there exists such a continuous function for which $|f(x)| = K/2$ for some $... | Hint: for any $x \in [0,1]$ either $x$ is closer to $0$ or $x$ is closer to $1$. If $y \in \{0,1\}$ is the closer endpoint then $|x - y| \le 1/2$.
Hint2: $|f(x) - f(0)| = |f(x)| \le K|x|$. To make this an equality, we need $f(x) = \pm Kx$ for any suitable $x$. Don't forget about the condition $f(1) = 0$—if you can do t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3483338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to evaluate $\int_{0}^{\infty}\frac{1}{t}\arctan\left(\frac{t}{1+2t^2}\right)\,\mathrm dt$? I entered this integral into Wolframalpha, and got $$\int_{0}^{\infty}\frac{1}{t}\arctan\left(\frac{t}{1+2t^2}\right)\,\mathrm dt=\frac{1}{2}\pi\log{2}.$$ But it doesn't provide step by step solution for this integral.
This ... | First notice that:
$$\arctan\left(\frac{x}{1+2x^2}\right)=\arctan\left(\frac{2x-x}{1+2x\cdot x}\right)=\arctan(2x)-\arctan(x)$$
So the integral can be rewritten as:
$$I=\int_0^\infty \frac{\arctan(2x)-\arctan x}{x}dx\overset{IBP}=\int_0^\infty \ln x\left(\frac{1}{1+x^2}-\frac{2}{1+4x^2}\right)dx$$
$$2\int_0^\infty \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3483514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Deducing congruence relations from given congruence relations I am trying problems from Apostol Modular functions and Dirichlet series in number theory and I could not think about this problem from chapter 2 .
Problem is – Given integers $a, b, c, d\;$ with $ad-bc \equiv 1 \pmod n$, prove that there always exists in... | $(\overbrace{a\!+\!\ell n}^{\textstyle \alpha},\:\!b)=1\,$ for $\,\ell\in\Bbb Z\,$ by $\,{(a,b,n)=1,}\,$ by here. Let $\,\beta =b.\,$ We solve for $\,\delta,\gamma\in\Bbb Z$.
$\!\!\bmod n\!:\ \color{#0a0}{\alpha\equiv a}\,\Rightarrow \color{#0a0}\alpha d\!-\!b c = \color{#0a0}ad\!-\!bc = 1,\ $ so $\,\alpha d\! -\! b c ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Shortest path between two points around an obstacle? I'm trying to figure out a problem that goes like this:
A particle originally placed at the origin tries to reach the point $(12,16)$ whilst covering the shortest distance possible. But there is a circle of radius $3$, centered at the point $(6,8)$, and the point ca... | Here is one way of seeing the shortest path. If you take a rope and try to pull on either end until it is tight. The rope Will show you the shortest path. The rope wont have any angle (sharp corner) on it.
As it had been said in the comments, it will follow a tangent to the circle, then it will wrap around the circle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3483811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Given $n$ different points in a plane.
Given $n$ different points in a plane, $8$ of them are on one straight
line. The other points are in general everywhere else, so there are no
$3$ points on same one straight line. How many different triangles can
you create from these n points?
What I think is: since no three po... | As has been stated in the comments, your solution is correct; $\binom n3-\binom 83$ different non-degenerate triangles can be formed from this points.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3483916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Find $xy+yz+zx$, given quadratic form of equations. Given that $(x,y,z) \in \mathbb R^+$ and the following equations:
$$x^2 + y^2 + xy = 1,$$
$$y^2 + z^2 + yz = 2,$$
$$z^2 + x^2 + xz = 3.$$
How to find $xy + yz + zx$? Please help.
| Consider a triangle ABC.
take a point inside triangle say O. such that sides subtend angle of 2π/3 each.
let OA = x, OB = y, OC = z.
So sides will come out 1, √2, √3. ( cosine rule ).
as triangle is right angled, area = 1/2*√2
Also add up area of triangle formed by smaller ones we get,
xy + yz + zx = ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I solve $\int\limits_0^1\frac{\sqrt{x}}{(x+3)\sqrt{x+3}}dx$ without trigonometric substitution? I have the following integral to solve:
$$\int_0^1 \dfrac{\sqrt{x}}{(x+3)\sqrt{x+3}}dx$$
without using trigonometric substitution. My textbook gives me the following hint:
$$t = \sqrt{\dfrac{x}{x+3}}$$
But I don't se... | Solve for $x$:
$$t^2 = \frac{x}{x+3} = 1 - \frac{3}{x+3} \implies x = \frac{3}{1-t^2} - 3$$
then we have
$$dx = \frac{6t}{(1-t^2)^2}dt$$
and plugging in to the integral gets us
$$ \int_0^{\frac{1}{2}} \left(\frac{1-t^2}{3}\right)\cdot (t) \cdot \left(\frac{6t}{(1-t^2)^2}\right)dt = 2\int_0^{\frac{1}{2}} \frac{t^2}{1-t^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the proof that if subsets $gH$ have no common elements then $H$ must be a subgroup? In proving Lagrange's Theorem we're usually first showing that cosets have no common elements. I'm looking for the proof that if subsets $gH$ have no common elements, then $H$ must be a subgroup.
| Firstly, the claim is not true without assuming that $H$ contains the identity $e$, because any coset is a counter example.
Now let me prove:
Let $G$ be a group and $H\subseteq G$ be a subset containing the identity element $e$. Suppose that for any two elements $g, g'\in G$, either $gH = g'H$ or $gH \cap g'H = \empty... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $\forall x \in G, \exists k \in \mathbb Z^+ \backepsilon xa=a^kx $, then $\langle a \rangle$ is a normal subgroup In Pinter's A Book of Abstract Algebra, Chapter 14 Exercise E3 asks the reader to prove the following statement:
If $a$ is any element of $G$, $\langle a \rangle$ is a normal subgroup of $G$ iff $a$ has... | Hint: $ xa^nx^{-1}=a^{nk}$ can you prove this?
| {
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Population of P people, where each person knows K others, how many people mutually know each other If you have a population of $P$ people, where each person knows $K$ others within the population, (does not have to be mutual, i.e. if I know you, you don't necessarily know me), and $1 < K < P$, How many people at the le... | Here's an elementary start on investigating this interesting problem
Let $M$ be the minimum number of people in a clique. Then $M$ is non-zero if and only if $K>\frac{P-1}{2}.$
If $M=0$ then there is no pair such that each knows the other. The number of 'knowings', $KP$ , is therefore no greater than $\begin{pmatrix}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that the number of self-conjugate partitions of $n$ equals the number of partitions of $n$ into distinct odd parts First, I would love if someone can provide some clarification of this problem. Then possibly help me map out/begin a proof.
So If I were taking the number $6$ and partitioning for example (just to m... | The Wikipedia article is quite good on a proof.
You can see that $x^{2n+1}=x^n\cdot x\cdot x^n$ to form both 'legs' of a self-conjugate partition in a Ferrers diagram.
Or, if you travel along the main diagonal and read only to the right, we are looking at the number of partitions into distinct parts, $\prod 1+x^k$. We ... | {
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rank and eigenvalues Let A be a square matrix of order $14\times 14$. We know that rank(A)=12 and $\lambda=0$ is an eigenvalue with algebraic multiplicity 4. I have to decide which of the following statements is true:
*
*$\text{rank}(A^2)=12$.
*$\text{rank}(A^3)\leq11$.
*There is no matrix satisfying the given con... | Since $\dim \ker A = 2$ and $0$ has algebraic multiplicity $4$ we see that there are
two Jordan blocks corresponding to the $0$ eigenvalue. The only possible sizes are
$(1,3)$ and $(2,2)$.
In both cases, $A^2$ will drop rank by at least one, so 1. cannot hold.
Since $\operatorname{rk} A^3 \le \operatorname{rk} A^2 < 12... | {
"language": "en",
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Finding $\limsup$ of a Brownian motion function Q) Use the fact that $(1+t)^{-1/2}\exp(B_t^2/2(1+t))$ is a martingale to show that $$\limsup_{t\to \infty} \frac{B_t}{\sqrt{(1+t)\log(1+t)}}\leq 1 \text{ a.s.}$$
I can see that $(1+t)^{-1/2}\exp(B_t^2/2(1+t))$ is a non-negative martingale and hence converges to a finite l... | If $\lim \sup x_t^{2} >1$ then $(1+t)^{-1/2} e^{x_t^{2} log(1+t)} \to \infty$ along some sequence $(t_n)$ and this contradicts the fact that the martingale converges to a finite limit.
[$(1+t_n)^{a_n-\frac 1 2} \to \infty$ if $a_n >1$ for all $n$ and $t_n \to \infty$].
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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First countable space Theorem: A subspace of a first countable space is first countable.
My proof:
Let $X$ be a first countable space. So for each p $\in X$, there exists a countable neighborhood basis for $X$ at $p$. Let $A$ $\subseteq X$ be a subspace and $p\in A$. Since $p$ is a member of the space $X$, let $\mathbb... | *
*You don't have to distinguish countable from finite, it's superfluous.
*Of course $\Bbb B'_p$ is by definition the image of $\Bbb B_p$ under $U \to U \cap A$ so is also countable (an image of a countable set is countable).
The proof itself is correct. Don't try to be "hypercorrect", though.
| {
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Parity of permutation If I know that the parity of a permutation is the parity of the number of transpositions, how can prove that the parity of the permutation is the parity of
permutation decrement?
Permutation decrement is the difference between the number of truly movable elements and the number of independent cy... | I'm going to assume you're trying to prove that the parity of a $k$-cycle is equal to the parity of $k-1$. If this is the case, note that
$$(a_1~~a_2~~\cdots~~a_{k-1}~~a_k) = (a_1~~a_k)(a_1~~a_{k-1}) \cdots(a_3~~a_1)(a_2~~a_1)$$
The expression on the right-hand side is a product of $k-1$ transpositions.
| {
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Suppose $f:[0,1] \rightarrow \mathbb{R}$ is continuous. What is the value of $\int_{0}^{1} \int_{x}^{1-x} f(y) d y d x ?$ Use Fubini's theorem
*
*Suppose $f:[0,1] \rightarrow \mathbb{R}$ is continuous. What is the value of $\int_{0}^{1} \int_{x}^{1-x} f(y) d y d x ?$ Again, do not forget to justify any use of Fubini'... | Make the variable change $u=1-x$, $x=0, u=1, x=1,u=0, du=-dx$ $\int_0^1\int_x^{1-x}f(y)dydx$
$=\int_1^0\int_{1-u}^uf(y)dy(-du)=\int_1^0\int_u^{1-u}f(y)dydu=-\int_0^1\int_u^{1-u}f(y)dydu$.
| {
"language": "en",
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Ito's formula and sin(Brownian motion) I would like to compute:
$d(e^{\frac12t}sinB_t)$
using the integration by parts of Ito I come up with the following:$$\frac12e^{\frac12t}sinB_tdt +e^{\frac12t}cosB_tdB_t + 0$$ however I the solution should be only $e^{\frac12t}cosB_tdB_t$. where is the mistake?
moreover what is th... | You are computing $d(\sin B_t)$ like if it was a differentiable function.
If we apply Ito's formula with the function $f(x)= \sin x,$ we get
$$d(\sin B_t) = \cos B_t dB_t - \frac{1}{2} \sin B_t dt.$$
Now applying the integration by parts formula, you will get the result
\begin{align*}
d(e^ {\frac{1}{2}t} \sin B_t) &= ... | {
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Solving $\frac{\ln(x)\ln(y)}{\ln(1-x)\ln(1-y)}=1$ for $y$
I'm trying to solve for $y$ in terms of $x$ for the expression below.
$$\frac{\ln(x)\ln(y)}{\ln(1-x)\ln(1-y)}=1$$
First I multiplied both sides by $$ \frac{\ln(1-x)}{\ln(x)} $$
to get
$$ \frac{\ln(y)}{\ln(1-y)}=\frac{\ln(1-x)}{\ln(x)} $$
but I don't see how to... | The function $f(x)=\dfrac{\ln(1-x)}{\ln(x)}$ is monotonic in its domain $(0,1)$, hence it is invertible. So the relation between $x$ and $y$ is a bijection, and…
$$y=1-x.$$
Interestingly, the function is well approximated by $\left(\dfrac1x-1\right)^{-3/2}$, and a solution with $a$ in the RHS is approximately
$$\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3485736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How to find result of system $x+y^2-4=0$ and $y+x^2-4=0$ without using the quartic formula How do I get the values of $x$ and $y$, without using the quartic formula?
\begin{align*}
\begin{cases} x+y^2-4=0 \\ y+x^2-4=0 \end{cases}
\end{align*}
| Subtract the first equation from the second to obtain:
$$x^2-y^2+y-x=0\Leftrightarrow \\
(x-y)(x+y-1)=0 $$
| {
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"url": "https://math.stackexchange.com/questions/3485836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving the integrand $\frac{x^3}{\sqrt{x^2+10x+16}}$ I just wanted to make sure that what I did to integrate $\frac{x^3}{\sqrt{x^2+10x+16}}$ is correct.
I assumed that it is classified as a trigonometric substitution problem. And so, what I first did is to apply "completing the square":
$\int \frac{x^3 dx}{\sqrt{x^2+1... | It is correct. Here is another method without trigonometric substitution:
$$\int\frac{x^3}{\sqrt{x^2+10x+16}}dx=\int\frac{x^3+10x^2+16x-10x^2-100x-160+84x+160}{\sqrt{x^2+10x+16}}dx$$
$$ = \int\frac{(x-10)(x^2+10x+16)+84x+160}{\sqrt{x^2+10x+16}}dx$$
$$ = \int(x-10)\sqrt{x^2+10x+16}\ dx+\int\frac{84x+420}{\sqrt{x^2+10x+1... | {
"language": "en",
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Is there a non-circular explanation for computing the probability of the intersection of two dependent events? The explanation with which I'm familiar goes like this. Define the conditional probability P(A|B) as the probability of their intersections P(A and B) divided by the probability P(B). To figure out the probabi... | In practice, a lot of time you can calculate $P(A|B)$ by assuming $B$ happened and see what is the probability of $A$. For example, let's say you draw 2 cards face down, then randomly choose 1 to flip up, and let $B$ is the event that exactly 1 red card and 1 blue card was drawn, and $A$ is the event that the flipped u... | {
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An interesting contest math problem: find the maximum value of $f(a_1,a_2,...,a_n)$
Suppose the sequence $a_1,a_2,...,a_n$ is a permutation of the sequene $1+2^1,2+2^2,...,n+2^n$. Find the maximum value of $f(a_1,a_2,...,a_n)=\vert a_1-a_2\vert+\vert a_2-a_3\vert+\cdots+\vert a_{n-1}-a_n\vert$.
This is an interesting... | Inspired by John Omielan's answer,here is my attempt.
Our goal is to find the maximum value of \begin{equation}\begin{aligned}
f(a_1,a_2,...,a_n) & = \vert a_1-a_2\vert+\vert a_2-a_3\vert+...+\vert a_{n-1}-a_n\vert \\
& = \sum_{i=1}^{n-1}\vert a_{i} - a_{i+1} \vert
\end{aligned}\end{equation}
We can regard $f(a_1,a_2,.... | {
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"timestamp": "2023-03-29T00:00:00",
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difference between algebraic set and affine algebraic set in hartshorne my question is basically already in the title. I just started to read the Algebraic Geometry book by Hartshorne where he defines an algebraic set but in some propositions a bit later talks about affine algebraic sets. So does he mean the same thing... | an algebraic set (or variety) is the zero set of a polynomial whose
coefficients belong to a given
field k, considered as a subset of k^n.
if k contains R k^n may be considered affine space,
in the sense that it is closed under u,v-->au+(1-a)v for all a in R,
and then the algebraic set may be called affine.
some autho... | {
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Proving a result related to Farey Sequence This is a problem from Tom M Apostol modular functions and Dirichlet series in number theory of Chapter -5 .
I am adding image of the problem
I proved that fractions $\theta $ =$\frac{\lambda a + \mu c } { \lambda b + \mu d } $ always lie between a/ b and c/d.
But how to ... | Suppose $c/d<a'/b'<a/b$ with $d,b',$ and $b$ all positive, and with $ad-bc=1$ and $\gcd(a',b')=1.$ Consider the simultaneous equations $$La+Mc=a',$$ $$Lb+Md=b' .$$ The unique solution for $(L,M)$ is $$L=\frac {a'd-b'c}{ad-bc}=a'd-b'c,\quad M=\frac {ab'-a'b}{ad-bc}=ab'-a'b.$$ We have $L>0\iff a'd>b'c \iff a'/b'>c/d.$
... | {
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"timestamp": "2023-03-29T00:00:00",
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Absolute convergence to a rational number Let's recall the not so popular/familiar form of completeness of real numbers:
Theorem: Absolute convergence of a series implies its convergence.
Since $\mathbb{Q} $ is not complete there should exist a series $\sum_{n=1}^{\infty} u_n$ with rational terms such that $\sum_{n=1... | Every irrational number in Balanced ternary has a non-repeating expansion, and viceversa. Therefore, if we take a non-repeating sequence $(e_n)_{n\in\mathbb Z^+}$ with $e_n\in\{-1,1\}$, $$\sum_{k=1}^\infty\frac{e_k}{3^k}$$ will be irrational, while $$\sum_{k=1}^\infty\left|\frac{e_k}{3^k}\right|= \sum_{k=1}^\infty\frac... | {
"language": "en",
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Find the value of k that bisects the area Find the value of $k$ for which the line $ y= kx$ bisects the area enclosed by the curve $4y=4x-x^2$ and the $x$ - axis.
I have tried to solve this and the solution seems odd... the solution that came out was $k=1 - \sqrt [3]{2}$
The step I took was to find the interval of the ... | Hint:
The abscissa of intersections of $y=0, 4y=x^2-4x$ are the roots of $$x^2-4x=0$$
Total area $$\int_0^4\dfrac{4x-x^2}4dx=?$$
The abscissa of intersections of $y=kx, 4y=x^2-4x$ are the roots of $$x^2-4x=-4kx\iff x=0,x=4-4k$$
$$\int_0^{4-4k}\dfrac{4x-x^2}4dx=?$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Factoring $x^4-2x^3+2x^2+x+4$
I need to show that the polynomial is not irreducible and I am trying to factor the polynomial
$$x^4-2x^3+2x^2+x+4$$
I checked from a calculator that it has a factor but how do I get it by myself?
I tried grouping but It didnt work I got
$x^2(x^2-2x+2)+x+4$ And I dont know how should... | Use the Rational Root Test to check if there is any real roots. It turns out that there is no such roots of this polynomial. Hence, the conclusion is that there can only be complex roots. By the fundamental theorem of Algebra, we know any non constant single variable polynomial has at least one complex root (including ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Visualizing Conditional Gaussian I am looking at a graph that depicts a conditional Gaussian:
I understand what the titled red spheres mean - that the variables are somewhat are correlated with each other.
I don't understand the significance of the blue line. I know it's related to conditional gaussian but I can't qu... | Basically the bivariate normal distribution looks like this one:
Then the conditional distribution $f_{Y|X=x}(x,y)$ is here marked with the red line. We take the joint pdf and plug in $x=1.6$
$f_{X,Y}(x,y) =
\frac{1}{2 \pi \sigma_X \sigma_Y \sqrt{1-\rho^2}}
\exp\left(
-\frac{1}{2(1-\rho^2)}\left... | {
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How can I calculate the limit $\lim\limits_{n \to \infty} \frac1n\ln \left( \frac{2x^n}{x^n+1} \right)$. I have the following limit to find:
$$\lim\limits_{n \to \infty} \dfrac{1}{n} \ln \bigg ( \dfrac{2x^n}{x^n+1} \bigg)$$
Where $n \in \mathbb{N}^*$ and $x \in (0, \infty)$.
I almost got it. For $x > 1$, I observed th... | No L'hopital needed - you just have to use the fact that $\ln(xy) = \ln(x) + \ln(y)$ and break up the limits.
$\lim\limits_{n \to \infty} \dfrac{1}{n} \ln \bigg( \dfrac{2x^n}{x^n + 1} \bigg ) = $
$\lim\limits_{n \to \infty} \dfrac{\ln (2) + \ln(x^n) - \ln(x^n + 1)}{n} = $
$\lim\limits_{n \to \infty} \dfrac{\ln (2)}{n... | {
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Let $f:[0,1] \rightarrow\mathbb R$ be a continuous map such that $f(0)=f(1)$ Let $f:[0,1] \rightarrow \mathbb R$ be a continuous map such that $f(0)=f(1) .$ Let $n \geq 2$
Show that there is some $x \in[0,1]$ such that $f(x)=f\left(x+\frac{1}{n}\right) .$
My attempt. Assume $f(x)\neq f(x+1/n)$ for all $x$. Then either ... | Let $g(x)=f(x)-f(x+\frac 1 n)$. If this continuous functions is never $0$ then it is always positive or always negative. Suppose it is always positive. Write $0=f(1)-f(0)$ as $[f(\frac 1 n) -f(0)]+[f(\frac 2 n)-f(\frac 1 n)]+...+[f(\frac {n-1} n)-f(1)]$. You get a contradiction since each term is $<0$.
Similar argu... | {
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Solution of an algebraic equation having unknown within modulus The question is this:
find the number of real values of $x$ for which $|x-3|+(x-3)^2+\sqrt{x-3}+|x+3|=0.$
Here is my attempt to answer:
Since it's a real equation and $\sqrt{x-3}$ is present here, therefore we must have $x>3$. In this case this equation be... | Hint (easier direction):
None of the summands is negative.
Could they all be zero?
| {
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When can we choose a covering local basis at a point which does not contain the whole space? I was wondering if we can ensure by relatively mild seperation axioms that a local basis ,which is a cover of the space, for at a topology at a point $x_0\in X$ does not contain $X$?
I'm pretty sure that if $X$ is $T_1$ and has... | A point p in a space S can have a local base without S
iff there is an open nhood of p that is not S.
The Serpenski space ({0,1}, {empty set, {0}, {0,1}})
is a T$_0$ space where the only local base for 1 is {{0,1}}.
Exercise. Show if open U is nhood p, then
{ V : p in V, V open subset U } is a local base for p.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3487770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of multiple third roots without L'Hospital I'm not able to solve the limit of a textbook question.
The limit:
$$\lim_{x\to \infty} (\sqrt[3]{x^2}(\sqrt[3]{x+1} - \sqrt[3]{x}))$$
I've been able to simplify the limit to:
$$\lim_{x\to \infty} (\sqrt[3]{x^3+x^2} - x)$$
How do I solve this limit?
Note: no L'Hospital a... | Using
$$a-b=\frac{a^3-b^3}{a^2+ab+b^2}$$
gives
$$\sqrt[3]{x^3+x^2}-x=\frac{x^2}{(x^3+x^2)^{2/3}+x(x^3+x^2)^{1/3}+x^2}
=\frac{1}{(1+x^{-1})^{2/3}+(1+x^{-1})^{1/3}+1}\to\frac13$$
as $x\to\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3487926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $x^2(xdx+ydy)+2y(xdy-ydx)=0$ Solve $x^2(xdx+ydy)+2y(xdy-ydx)=0$
My Attempts
$$x^2(xdx+ydy)+2y(xdy-ydx)=0$$
$$\dfrac {xdx+ydy}{xdy-ydx}=-\dfrac {2y}{x}$$
Put $x=r\cos (\theta)$ and $y=r\sin (\theta)$
So, $r^2=x^2+y^2$ and $\tan (\theta)=\dfrac {y}{x}$
Now,
$$x^2+y^2=r^2$$
Differentiating both sides,
$$2xdx+2ydy=2... | $$\dfrac {xdx+ydy}{xdy-ydx}=-\dfrac {2y}{x}$$
Duvide both sides by $({x^2+y^2})$:
$$\dfrac {xdx+ydy}{x^2+y^2}=-\dfrac {2y}{x}\left ( \dfrac {xdy-ydx}{x^2+y^2} \right )$$
$$\dfrac {xdx+ydy}{x^2+y^2}=-\dfrac {2y}{x}\left ( d(\arctan (\frac {y}{x})) \right )$$
Susbtitute $\dfrac {y}{x}=z$
$$\frac 12\dfrac {d(x^2+y^2)}{x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3488027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expected number of dice rolls before rolling "1,2,3,4,5,6" QUESTION: I roll a single six-sided die repeatedly, recording the outcomes in a string of digits. I stop as soon as the string contains "$123456$". What is the expected length of the string?
My answer so far: My initial approach is to try and find the probabili... | Just to point out a simple fact for independent, identical trials with finitely many outcomes: when a string $s$ of outcomes, like "123456", has no proper initial substrings which are equal to a final substring of $s$, then the expected waiting time for $s$ is just $1$/Freq($s$) where Freq($s$) is the probability that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3488134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 2
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Problem understanding Itô's proof about product of abelian groups I am trying to understand Itô's Theorem that states the following: Let the group $G=AB$ be the product of two abelian subgroups $A$ and $B$. Then $G$ is metabelian.
I am following the book 'Product of groups' by Amberg, Franciosi and De Giovanni which is... | Sorry, my comment was wrong. These equations are derived using the definitions $[x,y] = xyx^{-1}y^{-1}$. You can check that we then have the commutator identities:
$$[x,zy] = [x,z]z[x,y]z^{-1}\ \ \ \mathrm{and}\ \ \ [xz,y] = x[z,y]x^{-1}[x,y].$$
So in your first equation we have:$$[a,b^{a_1}] = [a,a_2b_2] = [a,b_2]a_2[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3488256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A quadrilateral inscribed in a rectangle Given a rectangle $ABCD$ in which there is an inscribed quadrilateral $XYZT$, with exactly one vertex on each side of the rectangle, how could I prove that the perimeter of the inscribed quadrilateral is larger then $2|AC|$ (two diagonals)?
I tried to use the triangle inequality... | Let’s suppose that $X$, $Y$, $Z$, $T$ are in $AB$, $BC$, $CD$, $DA$, respectively. Construct the reflection of $X$ through $AD$ $X_1$, the reflection of $X$ through $BC$ $X_2$, and the reflection of $X_1$ through $CD$ $X_3$. Your diagram should look like this.
Now, we have $$XY+YZ+ZT+TX$$ $$=X_2Y+YZ+ZT+TX_1$$ $$\ge X_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3488403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Integral $\int\limits_0^a x^b (1 - c x)^d\ \cosh x\,\mathrm dx$ How does one calculate
$$\frac{2\pi^{\frac{m-1}{2}}}{\Gamma \left(\frac{m-1}{2} \right)} \left(\frac{\alpha}{\kappa} \right)^{\frac{1}{\alpha}-1}\int\limits_0^{\kappa/\alpha}x^{m+\frac{1}{\alpha}-2}\cosh(x)\left(1-\left(\frac{\alpha}{\kappa}x\right)\right)... | Doing basically the same as @Eric Towers, for the integral
$$I=\frac{2\pi^{\frac{m-1}{2}}}{\Gamma \left(\frac{m-1}{2} \right)} \left(\frac{\alpha}{\kappa} \right)^{\frac{1}{\alpha}-1}\int\limits_0^{\kappa/\alpha}x^{m+\frac{1}{\alpha}-2}\cosh(x)\left(1-\frac{\alpha}{\kappa}x\right)^{\frac{2}{\alpha}}\,dx $$
we have
$$I... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the volume of the tetrahedron with vertices $(0,0,0)$, $(2,0,0)$, $(0,2,0)$, $(0,0,2)$. I get $8$; answer is $4/3$. The following problem is from the 7th edition of the book "Calculus and Analytic Geometry Part II". It can be found in section 13.7. It is
problem number 5.
Find the volume of the tetrahedron who... | Note that the given volume is a cone with the height 2 and a right isosceles triangle of side 2 as the base. Thus, its volume can be calculated as
$$\frac13 Area_{base} \cdot Height = \frac13 (\frac12 \cdot 2\cdot 2)2=\frac43$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3488590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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Discrete subspace in upper limit space Let $A$ be $\mathbb R$ under the upper limit topology (having as basis $\{(a,b] : a < b\}$), then the subspace $\{(x, -x)\mid x
\text{ is irrational}\}$ of $A\times A$ is closed, uncountable and discrete.
I think the subspace is closed since singleton is closed in upper limit sp... | In fact in $A \times A$ it's the antidiagonal that is closed and discrete, and so are all its subsets, the antidiagonal being
$$C=\{(x,-x): x \in A\}$$ while the diagonal $$\Delta=\{((x,x):x \in A\}$$
is just homeomorphic to $A$ (this holds in any space), and as $A$ is far from discrete and the irrationals in $A$ too, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Seeking the result: $\lim_{j,M \to \infty}\frac{1}{M}\prod_{k=1}^{M}\left[\prod_{n=j}^{2j}\left(1+\frac{1}{kn}\right)\right]^{\frac{1}{ln 2}}$ I am seeking the result of this limit
$$\lim_{j,M \to \infty}\frac{1}{M}\prod_{k=1}^{M}\left[\prod_{n=j}^{2j}\left(1+\frac{1}{kn}\right)\right]^{\frac{1}{ln 2}}=X$$
$X=1.78107..... | \begin{align}
\log\left(\frac{1}{M}\prod_{k=1}^{M}\left[\prod_{n=j}^{2j}\left(1+\frac{1}{kn}\right)\right]^{\frac{1}{\ln 2}}\right)
&=-\log(M)+\frac{1}{\log 2}\sum_{k=1}^{M}\sum_{n=j}^{2j}\log\left(1+\frac{1}{kn}\right)\\
&=-\log(M)+\frac{1}{\log 2}\sum_{n=j}^{2j}\sum_{k=1}^{M}\left(\frac{1}{kn}+O\Bigl(\frac{1}{k^2n^2}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do we deal with modulus signs when finding the solution of a differential equation? Consider: $\quad y'=\frac{y}{2x} \quad $ where we're required to find the general solution in the form $y=y(x)$
$\quad y'=\frac{y}{2x} \quad \rightarrow \quad \int \frac{1}{y} dy=\int \frac{1}{2x} dx$
$\hspace{2.5cm} \rightarrow \qu... | You can always write
$$
\frac{y(x)}{y_0}=\sqrt{\frac{x}{x_0}},
$$
as in any solution, there can be no sign change in neither $x$ nor $y$. Thus the fractions are always positive. Along the way this also takes care of the integration constant by directly expressing it in the initial conditions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3488934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Understanding Austin's two moving knives procedure for cutting a cake fairly From wikipedia description, the Austin procedure goes as follows
*
*Alice places one knife on the left of the cake and a second parallel to it on the right where she judges it splits the cake in two.
*Alice moves both knives to the right i... | Assume George thinks there is less than half the cake between the knives at the start. At the end the other piece is between the knives, so George thinks there is more than half between the knives. As George's estimate of the value of the cake between the knives is a continuous function of the position of the knives,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing that $(\mathbb{R},+,\cdot)$ and $(\mathcal{M}_2(\mathbb{R}),+,\cdot)$ are not isomorphic rings Prove that $(\mathbb{R},+,\cdot)$ and $(\mathcal{M}_2(\mathbb{R}),+,\cdot)$ are not isomorphic rings.
I came up with the following argument, but I am not sure it works : the equation $x^2=1$ has only two solutions in ... | One is commutative; the other isn't.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3489116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Hidden random walk in shallow sums? Prove that $\sum_{k=0}^n k \cdot \binom{2n-k}n 2^k = (2n+1) \cdot \binom{2n}n - 4^n $ Motivation: I was perusing through Laurent's solution to a recent online puzzle here.
Basically he managed to prove that, from the expected value of the probability distribution (in that question), ... | In evaluating
$$\sum_{k=0}^n {2n-k\choose n} k 2^k$$
we write
$$\sum_{k=0}^n {2n-k\choose n-k} k 2^k
= \sum_{k=0}^n k 2^k [z^{n-k}] (1+z)^{2n-k}
\\ = [z^n] (1+z)^{2n} \sum_{k=0}^n k 2^k z^k (1+z)^{-k}.$$
Now we may extend the sum in $k$ beyond $n$ because there is no
contribution to the coefficient extractor $[z... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Characterizing smooth, square-integrable functions on $(0,1]$ Is there a simple way to characterize the functions in $C^\infty((0,1])\cap L^2((0,1])$?
That is, given a function $f(t)\in C^\infty((0,1])$, is there a necessary/sufficient condition I can check to see if it's square integrable? An example of such a functi... | Define
$$f(t)= \frac{1}{\sqrt t \sqrt{1+(\ln t)^2}}.$$
Then $\int_0^1 f(t)^2\,dt = \pi/2,$ but
$$\lim_{t\to 0+} \frac{f(t)^2}{t^p}=\infty$$
for all $p>-1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3489347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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If all second partial derivatives exist and are continuous then all first partial derivatives are also continuous I want to prove that for $A \subseteq \mathbb R^n$ open set and $f:A\to \mathbb R$
if for all $i,j = 1,\dots, n$ $$\frac{\partial ^2f}{\partial x_i\partial x_j}(x)$$ is continous on $A$, then $$\frac{\parti... | From the definition $\frac{\partial^2 f}{\partial x_i\partial x_j}:=\frac{\partial }{\partial x_i}\left[\frac{\partial f}{\partial x_j}\right]$, you are saying that $\frac{\partial f}{\partial x_j}$ has continuous partial derivatives. Therefore it is a differentiable function.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $p$ and $q$ such that $x^2+px+q
Find $p$ and $q$ such that $$x^2+px+q<x$$ iff $$x \in (1,5)$$
I tried the following:
$$x^2+px+q = (x+\frac{p}{2})^2+q-\frac{p^2}{4}$$
where the global minimum is $$q-\frac{p^2}{4}$$ if $$x = \frac{-p}{2}$$
However this doesn't seem to help with the problem at hand...
| First rearrange as follows: $$x^2+px+q<x\iff x^2+(p-1)x+q<0$$ Recall from the properties of parabolas that $$ax^2+bx+c<0$$ between the (real) roots of the equation if $a>0$. So since in our case $a=1>0$, the parabola $$x^2+(p-1)x+q<0$$ between its zeros. So since we are given that $x\in(1,5)$ is the solution set, we su... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof verification in Hatcher Algebraic Topology, Proposition 3.25 First, this is the link of the book, for convenience: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf#page=244
Proposition 3.25 If $M$ is connected, then $M$ is orietable iff $\tilde M$ has two components.
(Here $M$ is an $n$-manifold. The definition of ... | Fix the two global orientations and call them $+1$ and $-1$. This gives you a map $\tilde M \to \{+1, -1\}$ which is continuous and surjective. So $\tilde M$ is disconnected. Conversely, the inverse images of $+1$ and $-1$ are each homeomorphic to $M$ so those are connected.
To see why the map is continuous you use the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solving integral of $\frac{\sin^2x}{x^2}$ with distributions I tried to evaluate $\int_{\mathbb{R}}\frac{\sin^2x}{x^2}dx$, using the theory of distribuions.
I guess I should evaluate it by thinking of a distribution generated by $T=\frac{\sin^2x}{x^2}$ and testing it on a $\phi \in \mathcal{D}(\mathbb{R})$:
\begin{equa... | Take the function $$ f(x) = \left\{\begin{array}{ll} 1 & \text{for } |x|<1\\
\frac12 & \text{for } |x|=1 \\ 0 & \text{for } |x|>1 \end{array} \right.$$
The Fourier transform of this function is
$$ \tilde f(k) = \int_{-\infty}^\infty f(x) e^{-ikx} dx = \int_{-1}^1 e^{-ikx} dx = \frac{2\sin k}{k} $$
The Parseval's t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Understanding how contrapositive work I want to understand contrapositive clearly. I'll start by saying: "If it is sunny, then there is light". The statement is true. But now consider the contrapositive: " If there is no light then it is not sunny". The contrapositive is false because you could create light with a fla... | Read your correct contrapositive again:
If there is no light then it is not sunny.
That's true - it's equivalent to the first statement. You could create light with a flashlght, but then there would be light.
What
you could create light with a flashlight.
tells you is that the original true statement is not "if and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3490056",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "5",
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Prove length relationship of median and sides in triangle using triangle inequality AD is the median in the triangle ∆ABC, from the corner A.
Prove that $\frac{AB+AC}{2}$$>AD>$$\frac{AB+AC-BC}{2}$.
I have that
$AB+AC>BC$,
$AB+BC>AC$,
$AC+BC>AB$
as well as
$AC+AD>CD$
$AD+CD>AC$
$AC+CD>AD$
and
$AB+AD>BD$
$AD+BD>AB$
$AB+... | This is just a simple application of the triangle inequality.
Take $A'$ so that $ABCA'$ is a parallelogram. Then $2AD=AA'< AB+BA'=AB+AC$. For the other inequality, we have $AD+DB> AB$ and $AD+DC> AC$. Adding gives $2AD> AB+AC-BC$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3490180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$
Prove without evaluating the integrals that:$$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$
Or equivalently:
$$\boxed{\int_0^\pi\... | It suffices to show the vanishing integral below
\begin{align}I=& \int^\frac{\pi}{2}_0\frac{(3x-\pi)\ln(1-\sin x)}{\sin x}dx\\
=& \int^\frac{\pi}{2}_0\int^\frac{\pi}{2}_0 \frac{(\pi-3x)\cos y}{1-\sin y \sin x}dy\>dx\\
=& \int^\frac{\pi}{2}_0\int^\frac{\pi}{2}_0 (\pi-3x)\frac{d}{dx}
\left(2\tan^{-1}\frac{\sin\frac{x-y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3490404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "65",
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Is $(2+i)^n + (2-i)^n $ a real number ($\in \Bbb R$)? The question:
$$\forall n \in \Bbb N$$
is the number
$$(2+i)^n+(2-i)^n$$
in the real numbers ($\Bbb R$)?
My try for solution using Newton binom:
$$(2+i)^n = \sum_{k=0}^{n}\binom{n}{k}2^{n-k}i^{k}$$
$$(2-i)^n = \sum_{k=0}^{n}\binom{n}{k}2^{n-k}(-i)^{k}$$
$$(2+i)^n +... | More efficient way:
A complex number plus its complex conjugate is real, so
$(2+i)^n+(2-i)^n=(2+i)^n+(\overline{2+i})^n=(2+i)^n+\overline{(2+i)^n}\in\mathbb R.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3490524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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A problem related to domain and range of real functions. I am trying to solve a problem related to real functions. For which I need properties of domain and range which a number must follow to be in domain and range of the real function.
I have found the property, for domain but I'm not able to find of range.
Since fun... | The range is characterized as the set of all real numbers $y$ such that $f(x)=y$ for at least one $x$ in the domain of $f$.
[The domain consists of all $x$ with $16-x^{2} \geq 0$ which means $-4 \leq x \leq 4$. The range consists of all non-negative real numbers less than or equal to $4$. [If $0 \leq y \leq 4$ then $x... | {
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Inequality $\frac{x^3}{x^2+y^2}+\frac{y^3}{y^2+z^2}+\frac{z^3}{z^2+x^2} \geqslant \frac{x+y+z}{2}$ Help to prove this Inequality:
If x,y,z are postive real numbers then:
$\dfrac{x^3}{x^2+y^2}+\dfrac{y^3}{y^2+z^2}+\dfrac{z^3}{z^2+x^2} \geqslant \dfrac{x+y+z}{2}$
I tied to use analytic method with convex function but no... | Hint: We have $$ \frac {x^3}{x^2 + y^2}=x-\frac{xy^2}{x^2+y^2}\ge x-\frac{y}{2}$$ because by AM-GM $x^2+y^2\geq 2xy$ so that $$\frac{xy}{x^2+y^2}\le\frac12$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3490814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Where precisely is topology required to prove the existence of non-zero eigenstates in the proof of the spectral theorem? My professor today remarked that the proof of the spectral theorem (even for the discrete spectrum case) uses not just algebra but also topology to prove the existence of eigenstates. However, I'm n... | At the very least you need to establish that the spectrum is nonempty. This usually requires the Fundamental Theorem of Algebra in the finite-dimensional case, or some form of Liouville's Theorem in the general case. Examples:
*
*Theorem VII.3.6 in Conway's A Course in Functional Analysis.
*Theorem 1.2.5 in Murphy'... | {
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Is there a lower bound to density at boundary points of a convex set? Let $X \subset \mathbb R^d$ be convex and compact. For each $x \in X$ define
$$D(x) = \lim_{r \to 0}\frac{\mu(X \cap B(x,r))}{\mu(B(x,r))}$$
where $B(r,d)$ is the ball with centre $x$ and radius $r$ and $\mu$ is the Lebesgue measure. The density m... | I'll make a suggestion (maybe I'm wrong).
Let's fix $a$ - an interior point of $X$ and $\varepsilon > 0$ s.t. $B(a,\varepsilon) \subset X$. Then for arbitrary $x \in X$ you have that $x + t(a + z - x) \in X$ for $||z|| < \varepsilon$ and $t \in [0,1]$. So, for arbitrary $t < \frac{r}{b + \varepsilon}$ (where $b$ is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3491213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Floating Point Arithmetic dealing with a Taylor expansion for e^-x Suppose we want to compute $e^{-a}$ for $a>>1$. Which of the following techniques should I use?
(a) Taylor expansion for $e^{-x}$ about $x=0$ or
(b) Taylor expansion for $e^x$ about $x=0$, then take its reciprocal.
Would a Taylor expansion about $e^{-x... | As a rule of thumb, the direct, naive evaluation of a sum $a_1+...+a_n$ will have accumulated floating point errors of a size $(|a_1|+...+|a_n|)\mu$ where $\mu$ is the floating point machine constant. In your case this means that the series evaluation of $e^x$ has a floating point error, additional to the truncation er... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3491333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Triangular numbers ($\text{mod } 2^n$) as a permutation of $\{0,1,2,\dots,2^n-1\}$ My question is based on an observation made by Vepir in their question "Grasshopper jumping on circles".
Vepir's observation was essentially that the sequence of triangle numbers $T\colon \mathbb{Z} \rightarrow \mathbb{Z}$ $$
T(n) = \f... | I'll be using $\equiv$ between non-integers to denote that the two sides differ by a multiple of the modulus $b^k$.
The map is a permutation, that is, bijective, exactly if $\frac12m(m+1)\equiv\frac12n(n+1)$ implies $m=n$ for $0\le m,n\lt b^k$. So assume $\frac12m(m+1)\equiv\frac12n(n+1)$. Adding $\frac18$ yields $\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3491464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Variant of coin toss problem Suppose you play the following game: You toss a fair coin. If you get heads, a hundred dollars are added to your reward. If you get tails, however, the game is stopped and you do not get anything at all. After each throw you can decide, whether you want to take the money or keep playing. Wh... | I would assume that the 100 dollars would be split proportionally to how many of the 3 completed flips each player guessed correctly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3491602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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How do I know if the number is divisible by $60$? I was study Babel Civilization ( Babylon ) , I see what about the number $60$ now I'm going to find a trick or fast method to know the rest of divisibility by number $60$ for example :
$6689=60^{2}+51.60+29$
$2567=42.60+47$
I know that the number $60$ divisible by $1... | Being divisible to $60$ is equivalent to being divisible simultaneously to $3,20$. So we can use divisibility test for these, which are easy in base $10$. Divisibility by $20$ mean the last $2$ digits must be and even number followed by $0$. Divisibility by $3$ mean sum of digits is divisible by $3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3491854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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(X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀ ∈ X, B(x,ε) is contained in some member of the cover.
Let (X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that for every x ∈ X, B(x,ε) is contained in some member of the cover.
My attempt:
(X... | The Wiki proof linked in the comments uses the fact that a continuous function on a compact set reaches its extrema. If you want a proof from scratch and closer to what you are trying to do, here are a few hints:
$1).\ $ Let $\mathcal A$ be an open cover of $X$. For each $x\in X$ there is an open neighborhood $B_{\epsi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3491978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
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} |
Is central projection of the imaging process a projective transformation? I am reading the book Multiple View Geometry in Computer Vision(Second Edition) and encounter some questions.
On Page 7, it says
In applying projective geometry to the imaging process, it is
customary to model the world as a 3D projective spa... | Central projection from 3D space to 2D space is not a projective transformation, as you have correctly concluded.
Central projection from 2D space to 2D space is a projective transformation as the text concludes in the last line of the same section you have highlighted. You can also see an example of the latter case o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find how many parts are in a triangle Recently, I got this interesting riddle in a math test, which I still can't solve. Here are the exact words:
Each side of an equilateral triangle was divided into 100 equal parts. Points received
connected by segments. How many parts did you get?
Here is an example for a triang... | I'll take it you want number of regions created. This is tedious to count them all, until you realize there's symmetry. Breaking the triangle into 4 equilateral triangles, 3 are just rotations of each other there are 26 in each of these for 78 in those triangles. You can do this again with the last triangle, giving 14... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
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Colored areas in a triangle that don't share a line A friend and I came up with this puzzle and I'm looking for a proof.
Given an equilateral triangle of area 1, color parts of the triangle red,
blue, and green such that
*
*Each color makes exactly one connected region strictly inside the triangle
*Ther... | Just as a first step in proving an upper bound on the area, here is a proof I found that details $\dfrac{1}{5}$ as an upper bound, although it's clear from the proof the bound is unachievable.
For this proof I only considered two of the three sides of the triangle; that is, no two colors can lie on a line parallel to a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Evaluating $\int\frac{\mathrm{d}u}{u\log u}$. I know we use $u$-substitution for $\log u$ and then we find the derivative but after that i'm confused. This is because the question is already using $u$ as a variable. Any explanation would be appreciated.
$$\int\frac{\mathrm{d}u}{u\log u}$$
| If $x=\log u$, then $dx/du=1/u$, so we have
$$\int\frac{du}{u\log u}=\int\frac{dx}{x}=\log|x|+C=\log|\log u|+C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Dimension of vector subspace Let $V$be the vectorspace of all polynomials. And $W$ is the subspace spanned by $t^2+t+2, t^2+2t+5, 5t^2+3t+4,2t^2+2t+4$.
The question asks us to find the dimension of $W$. Here is my try:
Let us call the four polynomials as $A$,$B$,$C$&$D$ in the order as they appear in the question. Th... | The dimension is at most $3$, since $W\subset P_2$. It is at least $2$, as the first two are not multiples of each other. The last is twice the first, so can be thrown out.
Let's check if the first three are independent, by computing the following determinant: $\begin{vmatrix}1&1&2\\1&2&5\\5&3&4\end{vmatrix}=1\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\binom{a_1}{2} + \binom{a_2}{2} + \cdots + \binom{a_n}{2} \ge r\binom{k+1}{2} + \left(n-r\right)\binom{k}{2}$
If $a_1,a_2,\cdots,a_n$ are positive integers, and $a_1+a_2+\cdots +a_n=nk+r$, where $k$ and $r$ are integers such that $0\le r<n$, prove that $$\dbinom{a_1}{2} + \dbinom{a_2}{2} + \cdots + \dbinom... |
We obtain with OP's problem setting and applying Jensen's inequality
\begin{align*}
\color{blue}{\sum_{j=1}^n\binom{a_j}{2}}&\geq n\binom{\frac{1}{n}\sum_{j=1}^n a_j}{2}\\
&=n\binom{\frac{1}{n}(nk+r)}{2}\\
&=\frac{n}{2}\left(\frac{nk+r}{n}\right)\left(\frac{nk+r}{n}-1\right)\tag{1}\\
&=\frac{1}{2}\left(nk+r\right)\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Where is the mistake ( in using mean value theorem)? $$ f(x)=
\begin{cases}
x^2\sin \frac1x & x \ne 0 \\
0 & x=0\\
\end{cases}
$$
$f$ is differentiable everywhere and
$$ f'(x)=
\begin{cases}
2x\sin \frac1x-\cos \frac1x & x \ne 0 \\
0 & x=0\\
\end{cases}
$$
$f$ satisf... | The mean value theorem says that there exists some $c$ in the interval $(0,x)$ such that [...]. And it is indeed the case that for any $x>0$ you can find such a $c$. That is not to say that any $c$ in $(0,x)$ satisfies the MVT, or even that the other numbers in $(0,x)$ behave nicely.
So what the MVT actually tells you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Using the numbers $0,1,2,3,4,5,6,7$ (repetition allowed) how many odd numbers can be created which will be less than $10000$?
Using the numbers $0,1,2,3,4,5,6,7$ (repetition allowed) how many odd numbers can be created which will be less than $10000$?
I have tried to solve this in the following way:
Numbers which wil... | If you have Mathematica here is a way to check your answer. As it has been pointed out the mistake is in the number of four-digit odd numbers.
a[n_] :=
If[ContainsNone[IntegerDigits[n], {8}] &&
ContainsNone[IntegerDigits[n], {9}] && OddQ[n], 1, 0];
Total[a /@ Table[i, {i, 1000, 9999}]]
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Calculting interest on investment Given an investment of 9550000 USD I want to know what the value of this investment is after one year if I have to pay 10% of interest per year.
According to my understanding, that would be 90% of the investment which equals to 8595000 USD.
According to the solutions, however, it is ... | Hint: You discount the investment. That means you divide it by $(1+i)=(1+0.1)=1.1$
$C_0=9,550,000\cdot \frac1{1.1}=...$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How can we show that $n < 2^n$ for all natural numbers $n$? I proved it using calculus or by drawing their graph but I was thinking if there is any simpler way to prove it. Please help me.
Proof by induction >
$P(n) : n < 2^n$ for all $n \in\mathbb{N}$
$P(1) : 1 < 2^1$, i.e.
$1 < 2,$ this is a true statement.
Now lets... | Induction:
If $k < 2^k$ (which is true for the first few natural $k$) and $k \ge 1$ (which is true for all $k$) then:
$k +1 \le k + k < 2^k + 2^k = 2^{k+1}$ and thus for any natural number this is true for it will be true for the next and there will be none where it isn't true.
.....
In essence, multiplying a number, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 10,
"answer_id": 7
} |
HAPPY NEW YEAR $2020$ Remainder Problem I framed a new question just now. What is the Remainder when the number $20^{20}$ is divided by $2020$
My try:
$$\frac{20^{20}}{2020}=\frac{20^{19}}{101}$$
Now Consider:
$$20^{18}=(400)^9=(404-4)^9=101k-2^{18}$$
Now i was trying to find Remainder without calculator or by manual d... | $20^{19}=100^9\cdot4^9\cdot20=100^9\cdot4^{10}\cdot5=100^9\cdot1024^2\cdot5 \equiv - 14^2\cdot5=-980 \equiv 30 \; (\mod 101)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Transformation which takes Fermat curve $x^n+y^n=1$ to a hyperelliptic curve? Motivated by this where it is possible to take certain Fermat curves like $x^3+y^3=1$ into Elliptic curves.
I was wondering if it is always possible to transform any Fermat curve $x^n+y^n=1$ birationally into some hyperelliptic curve?
| There are non-hyperelliptic Fermat curves.
*
*According to "The Group of Automorphisms of the Fermat Curve" (Tzermias 1995), the automorphism group of the Fermat curve with $n \ge 4$ in characteristic $0$ is the semidirect product $\Sigma_3 \ltimes (\Bbb{Z}/n)^2$ which has order $6n^2$.
*The genus of the Fermat c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
For a set function on a semi-ring, does additive imply finitely additive? Prove or disprove: If $\mu$ is an additive (i.e., 2-additive) set function on a semi-ring $S$, then $\mu$ is finitely additive (i.e., n-additive for every finite $n$) on $S$.
If $S$ is a ring, this is true by induction. For a semi-ring, the naiv... | Additive function on a semiring may not be finitely-additive. There is a simple example.
Consider a set $X = \{a,b,c\}$ and a semiring $S = \{X, \{a\}, \{b\}, \{c\}, \emptyset\}$ and a function $\mu:S \rightarrow \mathbb{R}$ that is defined in the following way:
$\mu(X) = 1$, $\mu(A) = 0$ for all $A \in S$ s.t. $A \ne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find last two digits of a number $A=(2016^{2015^{2014}}+2014^{2015^{2016}}+2017)^{2017}$ Find last two digits of a number $A=(2016^{2015^{2014}}+2014^{2015^{2016}}+2017)^{2017}$.
I have tried to write $A=100k+r$ and find r in $A$ but I stuck at it. Any solution will be aprreciated. Thank you.
| We work below modulo $100$. Then
$$
\begin{aligned}
A
&=\left(\ 2016^{2015^{2014}}+2014^{2015^{2016}}+2017\right)^{2017}
\\
&=\left(\ 16^{2015^{2014}}+14^{2015^{2016}}+17\right)^{2017}
\\
&=\left(\
16^{2015^{2014}\text{ taken modulo }5}
+14^{2015^{2016}\text{ taken modulo }10}
+17\right)^{2017}
\\
&\qquad\text{ since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $\frac ab= \frac bc= \frac cd$ then $(a^2+b^2)(c^2+d^2)=(ab+cd)^2$ [sic?] This is a basic algebra question. I found in class 9 math book and it is little tricky for me.
If $$\dfrac ab= \dfrac bc= \dfrac cd$$
prove that $$(a^2+b^2)(c^2+d^2)=(ab+cd)^2$$
Note (by @Blue). As observed in comments, the problem is inco... | Using complex numbers and their absolute value one can transform
$$
(a^2+b^2)(c^2+d^2)=|a-ib|^2|c+ id|^2=|(ac+bd)+i(ad-bc)|^2,$$
so with $ad=bc$ the right side is pretty fixed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3493943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
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