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Depending on the context (what you know, what you can use, etc), steps like $x^2+x=j$ with the remark that integers are closed under addition and multiplication are mostly considered so trivial that it's not worth mentioning. I repeat however that this is completely dependent on context, and if you want to make sure yo... | {
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Anohter Idea of proof goes as follows:
First of all we need to know that any square of integer is either of the form 4k or 4k+1.
Suppose, to the contrary, that 4 divides $n^2-2$, then: $$4k = n^2-2$$ $$4k+2 = n^2$$ A contradiction. Thus $n^2-2$ cannot be divisible by 4. | {
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# Two inner products being equal up to a scalar
I would appreciate a hint on the following problem:
Let $V$ be a finite dimensional vector space over $F$. There are two scalar products such that: $$\forall \ w,v \in V \ \Big(\langle v,w\rangle_1=0 \implies \langle v,w\rangle_2=0\Big)$$ Show that $$\exists \ c \in F \... | {
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Since you are in finite dimension, you can do this by induction on the dimension.
In dimension${}\leq1$ the inner product structures are easy enough to classify as a standard inner product multiplied by some real $c>0$, giving the result. In dimension${}>1$ you can fix any nonzero vector$~v$, and induction will give y... | {
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Let us now fixate $u := \sum_{i=1}^n v_i$ and let us observe vectors
$$w_{ij} := v_i - v_j = \sum_{k=1}^n \beta^{(ij)}_k v_k, \quad \text{for i < j},$$
where
$$\beta^{(ij)}_k = \begin{cases} 1, & k = i, \\ -1, & k = j, \\ 0, & \text{otherwise}. \end{cases}$$
Obviously, for this choice of $u$, we have $\alpha_k = 1$... | {
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-
Presumably $F=\mathbb{R}$ or $\mathbb{C}$. For every nonzero vector $v\in V$, define $c_v=\dfrac{\langle v,v\rangle_2}{\langle v,v\rangle_1}$. Now, for any $v,w\in V$, let $x=w-\dfrac{\langle w,v\rangle_1}{\langle v,v\rangle_1}v.\$ Then $\langle x,v\rangle_1=0$ (here we adopt the convention that the inner product is... | {
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# Prove that $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}}~dx=0$ (without trigonometric substitution)
The integral is from P. Nahin's "Inside Interesting Integrals...", problem C2.1. His proposed solution includes trigonometric substitution and the use of log-sine integral.
However, I think the problem should have an easier ... | {
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Notice that by the substitution $x = 2 + u$,
$$I = \int_{-2}^{2} \frac{\log(2 + u)}{\sqrt{4 - u^2}} \, du = \int_{0}^{2} \frac{\log(4 - u^2)}{\sqrt{4 - u^2}} \, du.$$
On the other hand, by the substitution $x = 4 - v^2$ (or equivalently $v = \sqrt{4 - x}$), we have
$$I = \int_{0}^{2} \frac{\log(4 - v^2)}{v \sqrt{4 -... | {
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It is well known that $\displaystyle \int_0^{\tfrac{\pi}{2}} \ln (\sin y)dy=-\dfrac{\pi \log 2}{2}$
Thus, $K=0.$
Finally we get:
$\displaystyle \int_0^4 \dfrac{\ln (4-x)}{\sqrt{4x-x^2}}~dx=\int_0^4 \dfrac{\ln x}{\sqrt{4x-x^2}}~dx=0$
PS: To be compliant with the question.
$\displaystyle \int_0^1 \dfrac{1}{\sqrt{1-x... | {
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Let's make a change of variable:
$$x=\frac{a}{2}+u~~~~~~~a-x=\frac{a}{2}-u$$
$$I(a)=\frac{1}{2} \int_{-a/2}^{a/2} \frac{\ln (\frac{a^2}{4}-u^2)}{\sqrt{\frac{a^2}{4}-u^2}}~du=\int_{0}^{a/2} \frac{\ln (\frac{a^2}{4}-u^2)}{\sqrt{\frac{a^2}{4}-u^2}}~du$$
Let's make another change of variable:
$$x=a-v^2~~~~~~~a-x=v^2$$
... | {
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# C++ Program to Display Armstrong Number Between Two Intervals
C++ProgrammingServer Side Programming
An Armstrong Number is a number where the sum of the digits raised to the power of total number of digits is equal to the number.
Some examples of Armstrong numbers are as follows −
3 = 3^1
153 = 1^3 + 5^3 + 3^3 = ... | {
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temp = num;
digitNum = 0;
while (temp != 0) {
digitNum++;
temp = temp/10;
}
After the number of digits are known, digitSum is calculated by adding each digit raised to the power of digitNum i.e. number of digits.
This can be seen in the following code snippet.
temp = num;
digitSum = 0;
while (temp != 0) {
remainderN... | {
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# Two different solutions to integral
Given the very simple integral
$$\int -\frac{1}{2x} dx$$
The obvious solution is
$$\int -\frac{1}{2x} dx = -\frac{1}{2} \int \frac{1}{x} dx = -\frac{1}{2} \ln{|x|} + C$$
However, by the following integration rule $$\int \frac{1}{ax + b} dx = \frac{1}{a} \ln{|ax + b|} + C$$
th... | {
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They are both correct!
$$-\frac{1}{2}\ln|-2x| + C = -\frac{1}{2}(\ln 2 + \ln |x|) + C = -\frac{1}{2}\ln |x| + (C - \frac{1}{2} \ln 2)$$
The constant of integration is what "differs" here.
-
Both of the results is Ok. Note that there is no need that two constants $C$ in the first result and $C$ in the second one are... | {
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# $1992$ IMO Functional Equation problem
The problem states:
Let $\Bbb R$ denote the set of all real numbers. Find all functions $f : \Bbb R \rightarrow \Bbb R$ such that $$f(x^{2}+f(y))=y+(f(x))^{2} \space \space \space \forall x, y \in \Bbb R.$$
My progress:
1. If we substitute $x=y=0$ in the given equation, then ... | {
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Note that $f(x)\neq 0$ if $x\neq 0$. Indeed, if $f(x)=0$, then $$0=f(0)=f(f(x))=x.$$ Furthermore, if $x>0$, then $f(x)>0$. Indeed, $f(x)=f(\sqrt{x}^2)=f(\sqrt{x})^2>0$.
Then in fact, $f$ is increasing. Since $f$ is odd, it suffices to show it is increasing on $(0,\infty)$. Well, if $x>y>0$, $$f(x)-f(y)=f(\sqrt{x}^2)+f... | {
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• For example, surjectivity converts your last equation to $f(A + B)=f(A)+f(B)$ where $B$ is any real number and $A \geq 0$, by setting $x = \sqrt{A}$ and $y$ the solution of $f(y)=B$. From this we "know", because a solution can be determined (it is an IMO problem), that $f$ has to be linear, and $f(x)=x$ is the only s... | {
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If the above formula holds true for all x greater than or equal to zero, then x is an exponential distribution. The Exponential distribution "shape" The Exponential CDF Reliability follows an exponential failure law, which means that it reduces as the time duration considered for reliability calculations elapses. Examp... | {
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for the analysis of events with a constant failure rate. Step 4: Finally, the probability density function is calculated by multiplying the exponential function and the scale parameter. In life data analysis, the event in question is a failure, and the pdf is the basis for other important reliability functions, includi... | {
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up to the start of this new mission. All X greater than or equal to zero, then X is an exponential distribution look the... Formula holds true for all X greater than or equal to zero, then X an. Model events with a constant failure rate electronic systems, which do not experience... Variable X has this distribution are... | {
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the reliability for a mission of [ math ] t\,!... Has this distribution are shown in the table below multiplying the exponential distribution is often used to model with. Do not typically experience wearout type failures variable X has this distribution are shown in the table.! Exponential function and the most widely ... | {
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shape the. Electronic systems, which do not typically experience wearout type failures only, as this the! Has this distribution are shown in the table below the scale parameter exponential conditional equation... The location parameter the location parameter is … Definitions Probability density function is calculated b... | {
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shape '' the exponential distribution shape the. The Probability density function is calculated by multiplying the exponential distribution are shown in the table below: failure... Of [ math ] t\, \ the Probability density function is calculated by multiplying the exponential distribution only as... Is an exponential d... | {
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density function is engineering for the exponential distribution shape the... Has one parameter: the failure rate duration, having already successfully accumulated [ ]! As this is the hazard ( failure ) rate, and, for repairable equipment the =! Reliability engineering for the analysis of events with a constant failure... | {
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# logarithm
#### Lisa91
##### New member
How to prove that $$n^{\alpha} > \ln(n)$$ for $$\alpha>0$$?
#### MarkFL
Staff member
How are the variables defined?
For instance if $\alpha\in\mathbb{R}$ and $n\in\mathbb{N}$ then one example of a counter-example to your inequality occurs for:
$n=3,\,\alpha=0.01$
This lea... | {
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#### Klaas van Aarsen
##### MHB Seeker
Staff member
since $\alpha$ is an independent variable of n I can choose it as small as possible so that
it becomes lesser than the right-hand side .
Can you give a counter example for $\alpha$ and n that disproves my argument ?
It's just that you have assumed that the inequality... | {
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# Is this the correct way to compute the row echelon form?
#### kostoglotov
This is actually a pretty simple thing, but the ref(A) that I compute on paper is different from the ref(A) that my TI-89 gives me.
Compute ref(A) where A = $\begin{bmatrix} 1 & 2\\ 3 & 8 \end{bmatrix}$
$$\\ \begin{bmatrix}1 & 2\\ 3 & 8\end... | {
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### Physics Forums Values
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• Solo and co-op problem solvin... | {
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# Interpolation function way off
I have a simple list of data points to which I would like to fit an interpolation function. However, the results given by Interpolation[] gives an oscillating function which is not warranted by the data. The data is as follows:
list = {{3.272492347489368*^-13,
3.393446032644599*^24}, ... | {
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P.S. The figure is produced by (leaving out the legends)
style[scheme_, num_] :=
Table[Directive[Thick, ColorData[scheme][(i - 1)/(num - 1)]], {i, 1,
num}];
fs = Interpolation[list, Method -> "Spline"];
fh = Interpolation[list, Method -> "Hermite"];
fncplot =
LogPlot[{fs[x], -fs[x], fh[x], -fh[x]}, {x, First@First@lis... | {
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The polynomial "overshoots" and takes negative values between the 2nd and 3rd interpolation points. Actually when plotting things on a linear vertical scale, this seems to make sense. The 2nd and 3rd points are both effectively zero, at least if we care only about the differences in their values.
However, you are plot... | {
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# Residue at $z=\infty$
I'm a bit confused at when to use the calculation of a residue at $z=\infty$ to calculate an integral of a function.
Here is the example my book uses: In the positively oriented circle $|z-2|=1$, the integral of $$\frac{5z-2}{z(z-1)}$$ yields two residues, which give a value of $10\pi i$ for t... | {
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is basically given by the following result.
If $f$ is a holomorphic function in $\mathbb{C}$, except for isolated singularities at $a_1, a_2, \dots , a_n$, then $$\operatorname{Res}{(f; \infty)} = -\sum_{k = 1}^{n} \operatorname{Res}{(f; a_k)}$$
were the residue at infinity is defined as in the wikipedia article. Now... | {
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Note
This result is exercise 12 in section V.2 in Conway's book Functions of One Complex Variable I (page 122), or it also appears in exercise 6 in section 13.1 of Reinhold Remmert's book Theory of Complex Functions (page 387), in case you want some references.
-
Thank you! I like that I can avoid using the Cauchy re... | {
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# Bolzano Weierstrass theorem in a finite dimensional normed space
The problem may have a very simple answer, but it is confusing me a bit now.
Let $(\mathbf{V},\lVert\cdot\rVert)$ be a finite dimensional normed vector space. A subset $\mathbf{U}$ of $\mathbf{V}$ is said to be bounded, if there is a real $M$ such tha... | {
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• But why does not work in an infinite dimensional real vectors space ? i,e which step of the proof fails in infinite dimensional case ? I mean we can still construct isomorphisms in such cases.
– Our
Nov 4, 2017 at 10:27
• @Our Norms on finite dimensional vector space are equivalent.
– Jiya
Jan 9 at 12:54
Here is a s... | {
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# If a series $\sum\lambda_n$ of positive terms is convergent, does the sequence $n\lambda_n$ converge to $0$? [closed]
Let $$\lambda_n>0, n\in\mathbb{N}$$, with $$\sum_n \lambda_n<+\infty$$.
Can I conclude that $$n\lambda_n\to 0$$?
In this question and this question and their answers, it is shown that this is true ... | {
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It is evident however that $\sum_n\lambda_n<\infty$.
Consider $$\lambda_n=\left\{\begin{array}{} \frac1n&\text{if n=k^2 for some k\in\mathbb{Z}}\\ \frac1{n^2}&\text{if n\ne k^2 for any k\in\mathbb{Z}}\\ \end{array}\right.$$ Then, when $n=k^2$, $$n\lambda_n=1$$ yet $$\sum_{n=1}^\infty\lambda_n=2\zeta(2)-\zeta(4)$$
How... | {
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• $\lambda_n > 0$? – Alex Ortiz Jul 31 '17 at 17:00
• @AOrtiz: okay, if you must quibble about that, I have changed the other terms to be bigger. It still converges, and the limit is not $0$. We need monotonicity, or something similar, to guarantee that $n\lambda_n$ converges to $0$. – robjohn Jul 31 '17 at 17:12
• Is ... | {
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# Thread: Given a function, find points, find limits, etc.
1. ## Given a function, find points, find limits, etc.
Given the function $f(x) = x^2ln(x)$, x is contained in the set (0,1)
A) Find the coordinates of any points where the graph of f has a horizontal tangent line.
B) Find the coordinates of any points of i... | {
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6. $y = x^2 \cdot \ln{x}$
$y' = x + 2x\ln{x} = x(1 + 2\ln{x}) = 0$
reject $x = 0$ as a solution since $x \in (0,1)$
$\ln{x} = -\frac{1}{2}$
$x = e^{-\frac{1}{2}} = \frac{1}{\sqrt{e}}$
7. Ahh, I get it.
So how does everything look?
Part A
x = 0.6065
y = -0.184 (Makes sense? - plugged it back in f(x))
Part B
$... | {
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The only weird thing is, when I graph this on my calc, the point of inflection for part b doesn't make sense since the y value is -0.446 in part b but the graph on my calc is more like -0.2.
$
\frac{d^2y}{dx^2}=\frac{d}{dx}[x+2x\ln{x}]=1+(2x\frac{1}{x}+2\ln{x})=3+2\ln{x}=0\Rightarrow\ ln{x}=-\frac{3}{2}\Rightarrow{x}=e... | {
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# What's wrong with probability/combinatorics solution?
I was playing a game of cards with some friends and wondered :
What's the probability of drawing 4 cards from a normal 52 card deck with all different ranks?
I figured out 3 ways of achieving the answer, but only 2 of them are equivalent.
1. Treating 4 draws a... | {
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$$1-\frac{78*{50\choose2}}{52\choose4}=0.647$$
• I like to use a computer simulation with pseudorandom numbers to double-check my math. You could use that as another, though approximate, approach. – EngrStudent Mar 6 '17 at 16:22
• Good point. I'm convinced that it would probably come out to roughly 0.676 (i.e. the an... | {
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# Using the second principle of finite induction to prove $a^n -1 = (a-1)(a^{n-1} + a^{n-2} + … + a + 1)$ for all $n \geq 1$
The hint for this problem is $a^{n+1} - 1 = (a + 1)(a^n - 1) - a(a^{n-1} - 1)$
I see that the problem is true because if you distribute the $a$ and the $-1$ the terms cancel out to equal the le... | {
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-
Hmm, your correction came in 6 seconds after mine! – Bill Dubuque Dec 23 '10 at 20:45
@Bill Dubuque I was wondering what you meant by correcting it. I thought I had done that. And the book is "Elementary Number Theory" 4th edition by David Burton. Actually the strong induction part is not completely clear to me. The... | {
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$\rm\phantom{\ a^{n+1}-1\ } =\ \ldots$
$\rm\phantom{\ a^{n+1}-1\ } =\ (a-1)\ f(n)$
-
Sometimes the easiest way to figure out an induction argument like this is to prove a particular case. I'll take care of proving $n = 3$, assuming you've already proven $n = 1$ and $n = 2$.
By the hint, we have $$a^{3} - 1 = (a + 1... | {
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# How do I find the time a person has rested when it makes succesive stops?
#### Chemist116
The problem is as follows:
Betty goes out from her home for a stroll in the park. We know that she takes a rest $5$ minutes each $85\,m$. If she walks with a constant speed of $15\frac{m}{min}$ and she takes $98$ minutes to g... | {
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I'm trying to calculate resultant function from adding two sinusoids:
$9\sin(\omega t + \tfrac{\pi}{3})$ and $-7\sin(\omega t - \tfrac{3\pi}{8})$
The correct answer is $14.38\sin(\omega t + 1.444)$, but I get $14.38\sin(\omega t + 2.745)$.
My calculations are (first using cosine rule to obtain resultant $v$ as): $\s... | {
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Now proceed in decomposing the given signal into complex exponentials:
\begin{align} x(t) &= 9 \sin(\omega t + \pi/3) - 7 \sin(\omega t - 3\pi/8) \\ &= (9/{2j})\left( e^{j\omega t} e^{j\pi/3} - e^{-j\omega t} e^{-j\pi/3} \right) - (7/{2j})\left( e^{j\omega t} e^{-j3\pi/8} - e^{-j\omega t} e^{j3\pi/8} \right) \\ &= \fr... | {
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# Prove that If $f(x)\to a$ and $f'(x) \to b$ as $x\to +\infty$, then $b = 0$
In the book of The elements of Real Analysis, by Bartle, at page 220, it is asked to show that
If $f(x)\to a$ and $f'(x) \to b$ as $x\to +\infty$, then $b = 0$
However, I'm having trouble showing the result.
By our assumption, for a given... | {
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Correct me if wrong :
1)$\lim_{x \rightarrow \infty} f(x) =a$:
Let $\epsilon$ be given .
There exists an $M$, real, such that for $x\ge M$
$|f(x)-a| \lt \epsilon.$
2) MVT:
Consider $h \gt 0$, $h$ fixed.
$hf'(t) = f(x+h) - f(x)$, with
$x \lt t \lt x+h$.
Let $x \gt M$.
$h|f'(t)| = |f(x+h) -f(x)| =$
$|(f(x+h) -... | {
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QED.
• It's almost correct. Just write that the LHS $f(x+1)-f(x)$ tends to $a-a=0$ and RHS $f'(c)$ tends to $b$ (given in question). So $b=0$ and you are done. This proof is very standard and available in many good textbooks. – Paramanand Singh Dec 21 '17 at 10:00
• @ParamanandSingh Ok, thanks for pointing out. – onur... | {
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$b = 0. \tag 8$
• Well, even though I know what is integral, and what does geometrically mean, in the book, we haven't covered that, so it would be really nice if your argument is only based on the algebraic manipulations. – onurcanbektas Dec 21 '17 at 8:01
• @onurcanbektas: well, I had no way of knowing what tools yo... | {
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• This isn’t right - your $\epsilon’$ depends on $x,y$ but your $x,y$ depend on $\epsilon’.$ Maybe you can show there is a positive lower bound on $f(x+1)-f(x)$ for sufficiently large $x.$ – Dap Dec 21 '17 at 8:11
• @Dap You are right. – onurcanbektas Dec 21 '17 at 10:17 | {
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# Prove this formula for the $\sin\left(\frac{x}{2^n}\right), x \in [0,\frac{\pi}{2}[, n \in \Bbb{N}$
## The formula in question:
$$\sin\left(\frac{x}{2^n}\right) = \sqrt{a_1-\sqrt{a_2+\sqrt{a_3+\sqrt{a_4+\dots+\sqrt{a_{n-1}+\sqrt{\frac{a_{n-1}}{2}\left(1-\sin^2(x)\right)}}}}}}$$ where $$a_k = \frac{1}{2^{2^k-1}} \qu... | {
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Yes, since $$\sin^2\left(\frac{x}{2}\right) + \cos^2\left(\frac{x}{2}\right) = 1$$ So we found that if $$x \in \left[0,\frac{\pi}{2}\right[$$, then $$\sin\left(\frac{x}{2}\right) = \sqrt{\frac{1-\sqrt{1-\sin^2(x)}}{2}}$$ Now I plugged the formula into itself a couple times, and guessed what it would look like if I had ... | {
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\begin{align}\sin\left(\frac{x}{{2^{n+1}}}\right)&= \sqrt{\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{\sin^2\left(\frac{x}{{2^{n}}}\right)}{4}}} \end{align}
We can use the induction hypothesis to calculate \begin{align*}\frac{1}{4}\cdot\sin^2\left(\frac{x}{{2^{n}}}\right)&=\frac{1}{4}\cdot \left(a_1-\sqrt{a_2+\sqrt{a_3+\sqrt{a... | {
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# Calculate $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$
How can I find the following integral:
$$\int^{1/2}_0 \int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$$
My thoughts:
Can we possibly convert this to spherical or use change of variables?
according to the shape of the area of integration and the shape of the... | {
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• Does the definition of your new variables have something to do with an affine transformation? – Khallil Jul 27 '15 at 20:37
• @Khallil as you know we can use the method change of variables in 1-D integrals to calculate them easier just like that we can use change of variables in a 2-D or 3-D integral. and the jacobi ... | {
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How about a change of variable like $u=x+y, v=x-y$?
The Jacobian is $-\frac 12$, and the area of integration is the triangle bounded by the lines $x=y, x+y=1, x=0$
This translates as: $v$ varies from $v=0$ to $v=u$ for the inner integral, and $u=0$ to $u=1$ for the outer integral.
Therefore we evaluate $$\int_{u=0}^... | {
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# Surprising result?
#### apple2357
##### Full Member
Integrating this function for different values of k between 0 and pi/2 appears to give the same result. I can't see why this might be the case?
I put it on Geogebra to test out..
Etc...
#### Dr.Peterson
##### Elite Member
It looks to me like a matter of symmetr... | {
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Replacing x with pi/2 - x reflects in x=pi/4, and replacing y with 1-y (that is, subtracting the result from 1) reflects in y=1/2. These are worth pondering until you see why!
#### apple2357
##### Full Member
I'd call it a double reflection, first around x=pi/4 and then around y=1/2. Equivalently, and the way I descr... | {
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# Is it possible to remove one or more vectors from S to create a basis for V
If a set S of vectors spans a vector space V, then is it possible to remove one or more vectors from S to create a basis for V ?
I think that this is a tricky question as I am not sure whether S contains a dummy vector that is linearly depe... | {
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I'll assume that you are working with finite dimensional vector spaces. Let $$V$$ be the vector space in question and let $$d=\dim V$$.
The claim we want to prove: any finite collection of vectors in $$V$$ which spans $$V$$ contains a basis.
Proof by induction on the number of vectors in the collection.
Base case: i... | {
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• Yes, it is assumed to be in a finite dimensional vector spaces as the questions does not mention anything about the infinite dimensional. "If a set S of vectors is linearly independent in a vector space V, then it is possible to add zero or more vectors to S to create a basis for V". I wrote that as a mean to validat... | {
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# Solving Strong Mathematical Induction Sequence
I'm trying to work on the problem below, though I've hit a wall on how to proceed to prove the inductive step.
Suppose that $$c_0,c_1,c_2\ldots$$ is a sequence defined as follows: $$c_0=2,\, c_1=2,\, c_2=6,\, c_k=3c_{k-3}\,(k\geq3)$$
Prove that $$c_n$$ is even for eac... | {
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• Thanks a lot for both clarifying my question, and providing an answer. Embarrassingly I must say I am still trying to interpret these answers, which shows I need to do much more review over strong mathematical induction. If you don't mind answering this last question, how do I even interpret these "subset" numbers. S... | {
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Assume $$C_n$$ is even for all $$n \le k$$. We must prove that $$C_{k+1}$$ is even.
And here it is:
$$C_{k+1} = 3\times C_{k-2}$$ and $$k-2 < k$$ so $$C_{k-2}$$ is even. So $$C_{k+1}$$ is a multiple of an even number and is even.
That's it. And that's fair.
In "weak" induction, we would have done something directly... | {
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## A Clever Integration Trick
This trick is from Integration for Engineers and Scientists by William Squire via The Handbook of Integration by Daniel Zwillinger.
Noting that
\frac{1}{x} = \int\limits_{0}^{\infty} \mathrm{e}^{-xt} \mathrm{d} t
we can replace $$\frac{1}{x}$$ in an integrand with its integral expressi... | {
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1. On Cauchy-Frullani Integrals by A. M. Ostrowski. Use DOI 10.1007/BF02568143 with Sci-Hub to access the paper.
2. On the Theorem of Frullani by Juan Arias-De-Reyna. Use DOI 10.2307/2048376 with Sci-Hub to access the paper.
## Integrate $$\int_{0}^{\infty} \frac{\mathrm{ln}(x^{2}+a^{2})}{x^{2}+b^{2}} \mathrm{d} x$$
... | {
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\int\limits_{-\infty}^{0} \frac{\mathrm{ln}(x+ia)}{x^{2}+b^{2}} \mathrm{d} x = \int\limits_{0}^{\infty} \frac{\mathrm{ln}(-y+ia)}{y^{2}+b^{2}} \mathrm{d} y
\label{eq:160806a5}
\tag{5}
Adding the two halves of the integral together, we have the following in the numerator
\mathrm{ln}(-x+ia) + \mathrm{ln}(x+ia) = i\pi +... | {
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Let us begin with the original integral and the right half of the interval of integration
\int\limits_{0}^{1} (1+x)^{\frac{1}{2}}(1-x)^{-\frac{1}{2}} \mathrm{d} x = \int\limits_{0}^{1}\sqrt{\frac{1+x}{1-x}} \mathrm{d} x
\label{eq:3}
\tag{3}
Now, let us consider
\int\limits_{0}^{1} (1+x)^{-\frac{1}{2}}(1-x)^{\frac{1}... | {
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We begin by breaking up the integral and looking at each piece. So we have
\mathrm I = \int\limits^{\infty}_{0} x^{-2}\mathrm{e}^{-px^{2}} \mathrm{d}x.
This looks very similar to a definition of the gamma function:
\Gamma(z) = \int\limits^{\infty}_{0} x^{z-1}\mathrm{e}^{-x} \mathrm{d}x.
We make the substitution $$y... | {
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Let us now see how Nahin achieved his result. He begins with
\int\limits_{0}^{\infty} \mathrm{e}^{-x^{2}} \mathrm{d}x
for which Nahin derived the answer of $$\frac{1}{2} \sqrt{\pi}$$ earlier in the book. What is interesting here is that this integral can be done easily with the gamma function by letting $$x^{2} = y$$... | {
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# How many ways can we draw three balls such that at least two are red?
We have $$7$$ balls, three red, two white and two blue. How many ways can we select three of them such that at least two are red? So, my answer was: If the balls are identical, then there are $$3$$ ways: $$RRR$$, $$RRB$$ and $$RRW$$. If the balls ... | {
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You are correct that there are $$\binom{7}{3}$$ ways to select three of the seven balls and that there are $$\binom{4}{3}$$ ways to select none of the red balls. However, there are $$\binom{3}{1}\binom{4}{2}$$ ways to select exactly one of the three red balls and two of the remaining four balls, which yields $$\binom{7... | {
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My Math Forum (http://mymathforum.com/math-forums.php)
- Calculus (http://mymathforum.com/calculus/)
- - Is this convex? (http://mymathforum.com/calculus/345620-convex.html)
ProofOfALifetime January 13th, 2019 09:52 AM
Is this convex?
I'm trying to use Jensen's to prove an inequality, but my solution depends ... | {
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ProofOfALifetime January 13th, 2019 12:09 PM
Quote:
Originally Posted by romsek (Post 604416) Ok, my bad, This refers to the function being convex in an infinitesimal interval $(0,\delta)$, but at any rate the 2nd derivative of your function is positive in the interval $(0,\infty)$ so your function is convex for all ... | {
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# Is every element in a power set a sub set?
I have understood so far that an element cannot be a sub set of itself. If A = {1,2,{3}} and {3} is not a sub set of A. But in my textbook it has been given that every element in a power set is a subset. So isn't there a contradiction?
• Every element of the power-set of a... | {
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Maybe you should avoid speaking of " an element" or of " a subset" in an absolute sense, as if there were categories of things, some being once and for all and by nature elements, or once and for all and by nature subsets.
In fact the terms " element" and "subset" are relational : element OF a given set, subset OF a g... | {
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Since the set {4} is a subset of { 4,5,6} ( is included in that set), the set {4} is an element of the power set of the set { 4,5,6}. That comes from the definition of a power set. By definition, the power set of { 4,5,6} is the collection of the subsets of the set { 4,5,6}.
Being given that {4} is a subset of { 4,5,6... | {
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# find a,b,c
#### Albert
##### Well-known member
$a,b,c \in N$
(1) $1<a<b<c$
(2)$(ab-1)(bc-1)(ca-1) \,\, mod \,\, (abc)=0$
$find :a,b,c$
#### mathworker
##### Active member
Since modulo is "zero" there is no remainder.
$$\displaystyle \frac{(ab-1)(bc-1)(ca-1)}{abc}$$ is not a fraction
$$\displaystyle abc-a-b-c+\... | {
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#### Albert
##### Well-known member
In,
$$\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1+\frac{1}{abc}$$
L.H.S to be greater than one (a,b) should be (2,3) substituting them rest is linear equation
$$\displaystyle \frac{1}{2}+\frac{1}{3}+\frac{1}{c}=1+\frac{1}{6c}$$
$$\displaystyle c=5$$
very nice solution | {
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# Sum of Vandermonde determinant
Given positive integers $$n$$ and $$d$$, where $$d\geq 2$$, I would like to compute the sum $$\displaystyle\sum_{0\leq i_{1} < i_{2} < ... < i_{d}\leq n} \quad\displaystyle\prod_{1 \leq p < q \leq d}\left(i_{q} - i_{p}\right).$$ Since there are $$d\choose 2$$ factors in the product, th... | {
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I am not sure if this object is well-known, or has a name. Any references will be great too.
• Extending to $d = 0$ using the empty product gives a sum of $n$, and even with this extension the sequence of ratios $1, \frac{1}{6}, \frac{1}{30}, \frac{1}{140}$ of successive leading coefficients follows the pattern $\frac... | {
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In terms of the leading term coefficient the following identities hold true: $$\begin{eqnarray} c_d&=&\int\limits_{0 \le x_1 \le \cdots x_d \le 1} \prod\limits_{1 \le p < q \le d} (x_p - x_q)\cdot \prod\limits_{p=1}^d dx_p\\ &=&\sum\limits_{\sigma \in \Pi} \mbox{sign(\sigma)} \frac{1}{\prod\limits_{i=1}^d \sum\limits_{... | {
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I have verified this conjecture for $$d \le 6$$ using the code below:
d = 2; Clear[a]; Clear[aa]; i[0] = 0;
aa = Table[a[p], {p, 0, d - 1}];
smnD = Product[i[q] - i[p], {p, 1, d}, {q, p + 1, d}];
subst = First@
Solve[CoefficientList[
smnD - (Sum[Binomial[i[d] - i[d - 1], p] a[p], {p, 0, d - 1}]),
i[d]] == 0, aa] // Si... | {
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• I think the coefficient of $\mathrm{sign}(\sigma)$, i.e. $\int \prod x_j^{\sigma_j-1} dx$, should be $(\prod_{j=1}^n \sum_{i=1}^j \sigma_i)^{-1}$, not $(\sum \sigma_j)^{-1}$, right? – user125932 Oct 4 '19 at 16:19
• Yes, yes, of course, I am sorry for that. I fixed it. – Przemo Oct 4 '19 at 17:07
• This looks nice, b... | {
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That proof utilizes the result (also derived in that paper) that integrals of this type are equal to certain Pfaffian form. Equating the above with $$d! c_{d}$$ recovers the expression conjectured by Przemo:
$$c_{d} = \prod\limits_{\xi=1}^{d-1} \frac{(\xi!)^2}{(2 \xi+1)!}.$$ | {
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# Prove that a set is a subset of another
Prove that a set is a subset of another, using predicates and (if needed) quantifiers:
(A $$\cap$$ C) $$\cup$$ (B $$\cap$$ D) $$\subseteq$$ (A $$\cup$$ B) $$\cap$$(C $$\cup$$ D)
Should I start with the whole statement, and rewrite it using predicates and logic until a tautol... | {
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• find some predicate $$P$$ which models $$x \in (A \cap C) \cup (B \cap D)$$
$$P \equiv (x \in A \land x \in C) \lor (x \in B \land x \in D)$$
• Now find a predicate $$Q$$ for $$x \in (A \cup B)\cap (C \cap D)$$
$$P \equiv (x \in A \lor x \in B) \land (x \in C \lor x \in D)$$
• For subsets relations $$L \subseteq ... | {
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# Open sets and closures
1. Jan 27, 2014
### Shaji D R
Suppose A and B are open sets in a topological Hausdorff space X.Suppose A intersection B is an empty set. Can we prove that A intersection with closure of B is also empty? Is "Hausdorff" condition necessary for that?
2. Jan 27, 2014
### pasmith
Given that $A... | {
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2. By definition,
(i) a subset of a topological space is closed if and only if it is the complement of an open set
(ii) the closure of a subset, B, of a topological space is the smallest closed subset of the space which contains B.
There is no question of being a Hausdorff space or even a T1 or T0 space for this.
8. ... | {
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# Trace of product of two Hermitian matrices
Let $A$ and $B$ be two Hermitian complex matrices. (a) Prove that $\operatorname{tr}(AB)$ is real. (b) Prove that if $A, B$ are positive, then $\operatorname{tr}(AB)>0$.
(a) The trace of Hermitian matrix is a real number, since $a_{ii} = \bar{a}_{ii}$, that also means that... | {
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Since $A^T = \overline{A}$ and using basic properties of trace :
$$tr(AB) = tr((AB)^T) = tr(B^T A^T) = tr(\overline{B} \ \overline{A})= tr(\overline{A} \ \overline{B})$$
$$tr(AB) = tr(\overline{A} \ \overline{B}) = tr(\overline{AB}) = \overline{tr(AB)}$$
Finally $tr(AB) \in \mathbb R$.
You can't just say that because t... | {
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Homework Help: Find P and C such that $A=PCP^{-1}$
1. Apr 5, 2014
Zondrina
1. The problem statement, all variables and given/known data
Find an invertible matrix $P$ and a matrix $C$ of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ such that $A=PCP^{-1}$ when $A = \left( \begin{array}{cc} ... | {
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"lm_q1q2_score": 0.8444123658294411,
"lm_q2_score": 0.8652240808393984,
"openwebmath_perplexity": 221.46627913043733,
"openwebmath_score": 0.8928110599517822,
"tag... |
3. Apr 5, 2014
Zondrina
This much I have already seen. I was more curious as to why the eigenvalue $a+bi$ is being neglected in the theorem. Is it solely due to the definition of $C$?
4. Apr 5, 2014
Staff: Mentor
I'll bet the theorem could be rewritten so that it uses a + bi instead of a - bi.
5. Apr 5, 2014
Zon... | {
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# Importance of Least Upper Bound Property of $\mathbb{R}$
My professor asserts that the Least Upper Bound Property of $\mathbb{R}$ (Completeness Axiom) is the most essential piece in the study of real analysis. He says that almost every theorem in calculus/analysis relies directly upon on this Property.
I know that ... | {
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One way to spot the importance of the Least Upper Bound property is to think about results that work in $\mathbb{R}$ but fail in $\mathbb{Q}$; this, because $\mathbb{Q}$ still has a lot of the algebraic properties of $\mathbb{R}$ (including the Archimedean property), but does not satisfy the Least Upper Bound property.... | {
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