Recent Questions - Mathematics Stack Exchange |
- Solve generating function for $a^n$
- Calculate coordinate of center of gravity
- Show that the set of sentences of Propositional Logic over {A,B} is not a regular language.
- Integral of rational function defined over $\mathbb{C}$
- Frustrum. How to solve for either bottom radius or height
- Finding specific solution to a system of ODE
- Let $A\in\mathbb C^{n\times n}$ be normal with $n$ eigenvalues and singular values. Show that $\sigma_i(A) = |\lambda_i(A)|$ for $i\in n$.
- Combi Problem - Proving Existence of a row
- Parseval's identity on Fourier series of $f(x)=e^x$
- Does equal Bell series imply Dirichlet convolution?
- Find critical paths of a project having multiple ending activities
- Show that $\sum_{k=1}^{n}\frac{1}{k} $ is between $\log(n)$ and $\log(n)+1$
- Find n such that the equations has at least x solutions in Z/Zn
- Branch line and integrals. Not finding the correct answer.
- Find the smallest prime that is of the form $709x^2 + 1061xy + 397y^2$ for some $x,y\in\mathbb{Z}$
- How to determine the probability that at least two dice will match when rolled together
- Hodge decomposition for complex manifold
- Quantile function and distribution composite
- Find the coefficient of $x^n$ in $(x^2 +x^3 +x^4 +\cdots)^5$
- Voigt Limiting Distributions (How to calculate)
- Finding $\lim_{n\to\infty} \frac{z_{n}}{z_{n-1}}$ where $3z_{n}=z_{n-1}+z_{n-2}+z_{n-1}z_{n-2}$
- Finding p-value in two-tailed F-test [migrated]
- If the image of an operator is closed, is the image of the powers of the operator also closed?
- If $x_0$ is limit point for $A⊂X$ then y0=f(x0) is limit point for f(A).We have this.Want to prove from this follows that f is continuous at point x0.
- Is it already known that $\sum_{i=1}^x\cos(S(i))\sim ax\cos(b\ln x)$, as $x\to\infty$, where $S(i)$ is the number of Collatz steps from $i$ to $1$?
- Area of Projection of Tetrahedron
- How to solve this integral (delta function)
- if $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of $I= \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) $
- Rank of two linear functions
- Mean & SD of Sampling Distribution
| Solve generating function for $a^n$ Posted: 05 Dec 2021 03:26 PM PST I am not sure how to solve questions like this. I am aware of both recurrence relations and generating functions and how two of these concepts work, but I find it hard to combine them. I would really appreciate if you could explain how to solve it. Thank you! Three standard six-sided dice are thrown. Write down a closed-form expression for the generating function for an, the number of ways in which a total of n can be thrown using the three dice, assuming that the dice are distinguishable. You should express the function as compactly as possible, but are not required to identify the coefficients in power series expansion. |
| Calculate coordinate of center of gravity Posted: 05 Dec 2021 03:26 PM PST Question: Let it be $\Sigma$ homogeneous surface, which ia given as a graph of the function $f(x,y)=\frac{a}{\sqrt2}\cosh(\frac{x+y}{a})$ above square $[-a,a]\times[-a,a], a>0.$ Calculate coordinate of center of gravity. So I define $\vec r(x,y)=(x,y,\frac{a}{\sqrt2}\cosh(\frac{x+y}{a}))$ and I define $x = au, y=av$, where is $u \in [-1,1], v \in [-1,1]$. Then I got $\vec r(u,v)=(au,av,\frac{a}{\sqrt2}\cosh({u+v}))$. Next I calculate the mass of surface like this $$mass = \int_{-1}^{1} du\int_{-1}^{1}a^2\sqrt{1+\sinh^2(u+v)}$$, where I got for result $$mass = 2a^2(\cosh(2)-\cosh(0))$$ In the end I calculate each coordinate of center of gravity and I got these results $$x_{T}=\frac{1}{mass} \int_{-1}^{1}du \int_{-1}^{1}a^3u\cosh(u+v)dv = 0$$ $$y_{T}=\frac{1}{mass} \int_{-1}^{1}dv \int_{-1}^{1}a^3v\cosh(u+v)du = 0$$ $$z_{T}=\frac{1}{mass} \int_{-1}^{1}dv \int_{-1}^{1}\frac{a^3}{\sqrt2}\cosh^2(u+v)du = \frac{7a+a\cosh(4)}{8\sqrt2(\cosh(2)-1)}$$ I just want to know if this is correct. |
| Show that the set of sentences of Propositional Logic over {A,B} is not a regular language. Posted: 05 Dec 2021 03:26 PM PST
I currently am going through a computer science and logic course and this was a suggested problem in the book. My guess is we start by supposing L=L(M) where M has k states. Then, we let s0, s1,...,sk be the states of M on reading Ak. But, I am not entirely sure how to continue the proof. We do not have access to advanced proofs for this class. |
| Integral of rational function defined over $\mathbb{C}$ Posted: 05 Dec 2021 03:21 PM PST I was just thinking of the following problem while evaluating some integrals. I can't find an easy solution to this at the moment. Assume we are given two rational functions $f,g$ such that $$\int_{\mathbb{C}} log|f| dz = \int_{\mathbb{C}} log|g| dz = 0.$$ Is it possible to construct a rational function h from f and g such that $\int_{\mathbb{C}} log|h| dz \neq 0$. |
| Frustrum. How to solve for either bottom radius or height Posted: 05 Dec 2021 03:17 PM PST Given Volume=72, slant height 5.88, and radius (top) = 2.12. Find the bottom radius |
| Finding specific solution to a system of ODE Posted: 05 Dec 2021 03:12 PM PST Let $\alpha\in R$ and homogenous system $\textbf{x'}(t)=A\textbf{x}(t)$ where $$\left[\begin{matrix}0 & -1 & -3\\ \alpha & -2 & -4 \\ 0 & 0 &\alpha \end{matrix} \right] $$ I need to show that the system has a non trivial solution where all the elements of the solution vector $\textbf{v}$, i.e $v_1, v_2, v_3$ uphold $\lim_{t\to\infty} v_i = 0\space (i=1,2,3)$ and another non trivial solution where at least one of $v_1, v_2, v_3$ uphold $\lim_{t\to\infty} v_i = \infty$ The characteristic equation of $A$ is $$|\lambda I-A|=(\lambda -\alpha)(\lambda^2+2\lambda+\alpha)$$ so the eigenvalues are $$\lambda_1=\alpha, \lambda_{2,3}=-1\pm\sqrt{1-\alpha}$$ now I know that:
But I'm having a problem with the second part, i.e finding a solution where at least one of the solution vector's elements' limit at infinity is infinity, I think I need to keep the same cases for $alpha$ but I just wasn't able to come up with something |
| Posted: 05 Dec 2021 03:11 PM PST For some unitary $U$, we have $$U^HAU = D = \text{diag}(\lambda_1,\dots, \lambda_n)$$ Then $$ DD^H = U^HAUU^HA^HU = U^HAA^HU $$ how do we then conclude that $\sigma_i^2(A) = \lambda_i(A)\overline{\lambda_i}(A)$ and thus proves the assertion. |
| Combi Problem - Proving Existence of a row Posted: 05 Dec 2021 03:09 PM PST The following problem comes from a Problem Set that concluded recently - Problem: $50$ girls and $50$ boys stand in line in some order. There is exactly one stretch of $30$ children next to each other with an equal number of boys and girls. Show that there is also a stretch of $70$ children in a row with an equal number of boys and girls. I've tried a couple of approaches and seem to be going nowhere. One approach I tried was applying the Pigeonhole Principle by defining the $71$ possible rows of $30$ children as pigeons, and another thing I attempted was considering the contrapositive, but nothing seems to work. I would appreciate any hints (not a full solution) towards solving this problem. |
| Parseval's identity on Fourier series of $f(x)=e^x$ Posted: 05 Dec 2021 03:05 PM PST Let $\{\varphi_k\}_{k=1}^\infty$ be an orthogonal system, and $\{\alpha_k(f)\}$ the Fourier coeffitients for a function $f\in L^2([a,b])$. Then the Parseval's identity is given by the formula \begin{equation} |f|_2^2 = \sum_{k=1}^\infty \alpha_k(f)^2|\varphi_k|_2^2, \end{equation} where $|f|_2 = \left(\int_a^bf^2\right)^{1/2}$. I'm computing the complex Fourier series for $f(x) = e^x$, and if nothing is wrong the series goes as follows: \begin{equation} f(x)\sim\sum_{k=-\infty}^\infty\frac{(-1)^{k}\sinh(\pi) i}{(i-k)\pi}e^{ikx}. \end{equation} I also computed it with the real trigonometric coeffitients, and if I haven't messed up anywhere, this is what I got: \begin{equation} f(x)\sim\frac{\sinh(\pi)}{\pi}+\sum_{k=1}^\infty\left(\frac{2(-1)^k \sinh(\pi)}{(1+k^2)\pi}\cos(kx)-\frac{2(-1)^k k \sinh(\pi)}{(1+k^2)\pi}\sin(kx)\right). \end{equation} I'm trying to see what I can get from the Parseval's identity, but none of the series seem to give me any equality: \begin{equation} \int_{-\pi}^\pi e^{2x}\;dx = \sinh(2\pi),\;\;\int_{-\pi}^\pi e^{2kix}\;dx = 0,\;\;\int_{-\pi}^\pi\;dx = 2\pi,\;\;c_0(f)^2 = \frac{\sinh^2(\pi)}{\pi^2} \end{equation} \begin{equation} \sinh(2\pi) = 2\frac{\sinh^2(\pi)}{\pi}, \end{equation} but these two are not equal. Something similar happens with the other series (I took this one directly from Mathematica, it was too long to write): \begin{equation} \sinh(2\pi) = \sin\frac{\sinh^2 (\pi )}{\pi^2 }+\frac{2 \sinh ^2(\pi ) (\pi \coth (\pi )-1)}{\pi }, \end{equation} which again, the equality doesn't hold. Did I commit a mistake while writing Parseval's identity with the correspondant variables? Or perhaps $e^x$ doesn't hold the Parseval's identity (I think it should, since $f\in L^2([-\pi,\pi])$). Could anyone please help? |
| Does equal Bell series imply Dirichlet convolution? Posted: 05 Dec 2021 03:12 PM PST Apostol states in Ch.$2$ , section $2.17$ of his "Intro to Analytic NT" book;
Let me explain some of the notation in the more common/known (?) setting, here, arithmetical functions just means $f: \mathbb{N} \to \mathbb{R}$ , here, the symbol $*$ is referring to the Dirichlet convolution, and $f_p(x) $ is referring to the Bell series of $f$ modulo $p$. I know how to prove this, it's quite routine in fact, I want to know if the converse of this holds as well, ie. If $h_p(x)=f_p(x) \cdot g_p(x)$ for every prime $p$, does this imply $h=f*g$ ? It is easy to show this is equivalent to asking: If $h(p^n)= f*g \ (p^n)$ for all prime $p$ and $n \in \mathbb{N_{0}}$, does this imply $h=f*g$ ? I also know that if $f$ and $g$ are multiplicative functions, $f=g \iff f_p(x)=g_p(x)$ for all primes $p$. So does this mean the answer to my question (in bold) is that the converse holds if (and only if?) $h$ is multiplicative? I have learnt all this in the past 12 hours only, so I'm not sure if I understand it well at all, but after reading the theorem in Apostol's book, the converse seemed like a natural question to ask, so I did ask myself that question, and I tried to come up with an answer, but I don't think it is satisfactory, mainly because it seems like the thing that the book should have mentioned, as it is very strong(?) and cool and also because I don't think I understand this stuff well right now. Any answer will be appreciated, Thanks! |
| Find critical paths of a project having multiple ending activities Posted: 05 Dec 2021 03:04 PM PST I am currently working on a project scheduling problem that gives me an interesting situation of two ending activities. The objective is to determine the critical path of this project. Although there are two ending activities separate from each other, I got the result that there is exactly one critical path in this case (because there is only one longest path). Below is the data on the project composed of $8$ activities: Activities |Time |Precedence Constraints A | 3 | B | 5 | A C | 1 | A D | 4 | B, C E | 3 | B F | 3 | E, D G | 2 | F H | 4 | E, D My attempt. I first construct the activity-on-node network as follows
Now, we compute the earliest finished time (EF), earliest starting time (ES), latest finished time (LF) and latest starting time (LS) for each activity $i = A,B,C,D,E,F,G,H$, we have: $ES_A = 0, EF_A = 3, LF_A = 3, LS_A = 0$ (this last result confuses me) $ES_B = 3, EF_B = 8, LF_B = 8, LS_B = 3$ $ES_C = 3, EF_C = 4, LF_C = 8, LS_C = 7$ $ES_D = 8, EF_D = 12, LF_D = 12, LS_D = 8$ $ES_E = 8, EF_E = 11, LF_E = 12, LS_E = 9$ $ES_F = 12, EF_F = 15, LF_F = 15, LS_F = 12$ $ES_H = 12, EF_H = 16, LF_H = 16, LS_H = 12$ $ES_G = 15, EF_G = 17, LF_G = 17, LS_G = 15$ Finally, computing the slack $i$ for each activity $i$ above, we obtain $Slack_A = Slack_B = Slack_D = Slack_H = Slack_F = Slack_G = 0$. Thus there is only one critical path: \fbox{$A-B-D-H-F-G$}. My question. Could anyone please help confirm if my above solution is correct? It is weird to me that we have two ending activities but only one critical path. Also, if we want to draw a network where each arc represents an activity above using only one pseudo activity, is the construction of the following network valid?
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| Show that $\sum_{k=1}^{n}\frac{1}{k} $ is between $\log(n)$ and $\log(n)+1$ Posted: 05 Dec 2021 03:06 PM PST The lower bound is pretty simple, $\sum_{k=1}^{n}\frac{1}{k} \ge \int_{1}^{n}\frac{1}{x}dx = \log(n)$. Could I please ask for pointers for the upper bound? |
| Find n such that the equations has at least x solutions in Z/Zn Posted: 05 Dec 2021 03:22 PM PST Consider over Z/nZ x^2 + x = 0 (a) Find an n such that the equation has at least 4 solutions. (b) Find an n such that the equation has at least 8 solutions. I figured out (a) by doing Trial and Error, and got 4 unique solutions but for (b) this seemed to be pretty hard so I'm just wondering if you have an easier way on how to approach this problem. |
| Branch line and integrals. Not finding the correct answer. Posted: 05 Dec 2021 03:09 PM PST $$ \int_{0}^{\infty } \frac{ln x}{x^{3/4}(1+x)}dx$$ Integrating the real axis from 0 to infinity, so go around a big circle and close from - infinity to 0, around 0 close the path with a little circle. $x = r$ along the real axis and Re>0. $I = \int_{0}^{\infty } \frac{ln x}{x^{3/4}(1+x)}dx$ Along the big circle, the integral vanishes. along the real axis and Re>0, $ \int_{\infty}^{0} \frac{e^{i \pi /2 }ln r}{r^{3/4}(1+r)}dr = - \int_{0}^{\infty} \frac{e^{i \pi /2 }ln r}{r^{3/4}(1+r)}dr = -iI$ Along the little circle it will diverge ... Along a circle around minus one, it diverges too. So what should i do? I can't see any other point . |
| Find the smallest prime that is of the form $709x^2 + 1061xy + 397y^2$ for some $x,y\in\mathbb{Z}$ Posted: 05 Dec 2021 03:16 PM PST Find the smallest prime that is of the form $709x^2 + 1061xy + 397y^2$ for some $x,y\in\mathbb{Z}$. Which of the following equations have integer solutions?
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| How to determine the probability that at least two dice will match when rolled together Posted: 05 Dec 2021 03:11 PM PST I'm very unlearned in probability and would appreciate learning how to do this. I play a silly game where I roll several different dice. There is a d30, d28, d26 and two d24s. I am frequently astounded by how often these dice, when rolled, result in two or more of the five matching in value. How can I determine the probability that two (or more) dice of different values will match, when rolled together? I tried modifying a solution to the birthday paradox, but I'm almost certain I did something wrong as it gave me a chance of 54%. |
| Hodge decomposition for complex manifold Posted: 05 Dec 2021 03:03 PM PST Let $X$ be a compact oriented Riemannian manifold. By Hodge decomposition, we can decompose $$\Omega^k(X)=\mathrm{im}(d)\oplus\mathrm{im}(d^*)\oplus\ker(\Delta).$$ Now, if further $X$ has a complex structure, I read on a book that $$\Omega^{p,q}(X)=\partial(\Omega^{p-1,q}(X))\oplus\partial^*(\Omega^{p+1,q}(X))\oplus\ker(\Delta_{\partial}),$$ or $$\Omega^{p,q}(X)=\bar\partial(\Omega^{p,q-1}(X))\oplus\bar\partial^*(\Omega^{p,q+1}(X))\oplus\ker(\Delta_{\bar\partial}).$$ My question is: given that I know the theorem for the real case and $d=\partial+\bar\partial$, shouldn't the result be $$\Omega^{p,q}(X)=(\partial(\Omega^{p-1,q}(X))+\bar\partial(\Omega^{p,q-1}(X)))\oplus(\partial^*(\Omega^{p+1,q}(X))+\bar\partial^*(\Omega^{p,q+1}(X)))\oplus\ker(\Delta)?$$ |
| Quantile function and distribution composite Posted: 05 Dec 2021 03:19 PM PST Let $F$ be a distribution for a probability measure and let $F^{-1}$ be the quantile-function for $F$; Show that
My thoughts and effort so far is as follows. $F$ is non-decreasing and right continous which makes the quantile function be non decreasing and left continous. $F$ also has limits $0$ for $x\to-\infty$ and $1$ for $x\to\infty$. As this is the case I have to look at what happens for the quantile function as $p\to 0,1$ respectively. Does the limits of the quantile function act such that $p\to 1$ then $F^{-1}(p)\to \infty$ and thus the result? |
| Find the coefficient of $x^n$ in $(x^2 +x^3 +x^4 +\cdots)^5$ Posted: 05 Dec 2021 03:16 PM PST I have got stuck on this question, though I realise that I have probably got really close to an answer. This is how I approached it: \begin{align*}f(x) &= (x^2+x^3+x^4+\cdots)^5\\ &= x^{10}(1 + x + x^2 + \cdots)^5\\ &= x^{10}\left(\frac{1-x^{m}}{1-x}\right)^5\\ &= x^{10}(1-x^{m})^5(1-x)^{-5}.\end{align*} Then I have used the binomial theorem: \begin{align*}(1-x^{m})^5 &= \sum^5_{i=0}\binom{5}{i}(-1)^i(x^m)^i,\\ (1-x)^{-5} &= \sum^\infty_{j=0}\binom{4+j}{j}(x)^j.\end{align*} Therefore, $$f(x) = x^{10}\cdot\left(\sum^5_{i=0}\binom{5}{i}(-1)^i(x^m)^i\right)\cdot\left(\sum^\infty_{j=0}\binom{4+j}{j}x^j\right),$$ and as I work out coefficient of $x^n$ I arrive to: $$[x^n] = \binom{5}{0} \binom{n-6}{n-10} $$ I think my answer is incomplete and unfortunately, I don't have a solution to check it. I would really appreciate your help. Thank you |
| Voigt Limiting Distributions (How to calculate) Posted: 05 Dec 2021 03:11 PM PST I wanted to verify some specific parameter values on the Wikipedia page for the Voigt distribution. It should have been a simple exercise but it seems that either I am lacking in a correct understanding of how to obtain those specific parameter values or those listed on the website are incorrect. The description of the Voigt distribution and the information pertinent to my particular question can be found here. The specific values that I attempted to verify are those listed in the first and second graphs of the distribution. Under the first graph, we have the following text: "Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively." The picture of the plot is below. I am only interested in the limiting cases (black and red). I tried to verify the numbers 1.53 and 1.8 numerically by writing a small Matlab program that should build a Voigt distribution. Since the Voigt distribution is the convolution of Normal and Cauchy distributions (i.e. we add those random variables to each other), we should then be able to take the resulting data points and estimate its parameters for the Normal and Cauchy distributions, respectively. The snippet of code is below. Notice that the snippet is attempting to obtain the 1.8 value: I'd appreciate any insights the community could provide me with. I feel as if I am missing some piece of understanding. Thanks again!
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| Finding $\lim_{n\to\infty} \frac{z_{n}}{z_{n-1}}$ where $3z_{n}=z_{n-1}+z_{n-2}+z_{n-1}z_{n-2}$ Posted: 05 Dec 2021 03:16 PM PST Question.
Observations.
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| Finding p-value in two-tailed F-test [migrated] Posted: 05 Dec 2021 03:09 PM PST
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| If the image of an operator is closed, is the image of the powers of the operator also closed? Posted: 05 Dec 2021 03:11 PM PST Say $T$ is a bounded linear operator in a normed space that maps to itself (Banach or Hilbert space is fine). If the image $\text{Im}(T)$ is closed, then is it true that $\text{Im}(T^n)$ is closed? If not, what is a counterexample? |
| Posted: 05 Dec 2021 02:58 PM PST $f$:$X \to Y$ $A$ is any set such that $x_0$ is a limit point of A, then $y_0$ is a limit point of $f(A)$.We have this.Want to prove from this follows that $f$ is continuous at point $x_0$. For $y_0=f(x_0)$ let's choose any $V$ open neighbourhood and show that $U=f^{-1}(V)$ subset is open neighbourhood for $x_0$.$W=X$ \ $U$.If $W=\emptyset$ then $U=X$ is open in $X$.So $W\neq\emptyset$.Choose $x_1 \in \overline{W}$(closure of W).$f(x_1)\in f(\overline{W})\subset \overline{f(W)}\subset\overline{Y/V}=Y/V$ Can you explain how author got $\overline{f(W)}\subset\overline{Y/V}=Y/V$. I am thinking about $f(X$ / $U$)$\subset$ $f(X)$ \ $f(U)$ but this is true if $f$ is injective and even if was true I am not sure this will help. |
| Posted: 05 Dec 2021 03:10 PM PST I was playing with the Collatz Conjecture today, and empirically found a curious behaviour: Let $S(i)$ be the function that calculates the number of steps needed for $i$ to reach $1$: It seems that $\sum\limits_{i=1}^{x} \cos(S(i)) \sim ax\cos(b\ln(x)), \,\, x \rightarrow \infty, \,\ x \in \mathbb{N}$ Where $a \approx \frac{1}{4\pi + \zeta(3)}, \,\, b \approx \frac{52191}{5000}$ The constants have been found empirically while trying to minimize the error between the functions, and do not necessarily correspond to the correct ones. Graphical evidence up to $10^8$: $$\sum\limits_{i=1}^{x} \cos(S(i))$$
$$ax\cos(b\ln(x))$$
Was this fact already known? If not, I conjecture this is true, it's just a matter of finding the correct constants $a$ and $b$. Why do you think it behaves likes this? Interesting. |
| Area of Projection of Tetrahedron Posted: 05 Dec 2021 03:07 PM PST this is my first post. Here is a question I found in a handout I am reading on 3D geometry. A plane passes through the midpoints of two skew lines of a regular tetrahedron. The projection of the tetrahedron to the plane produces a quadrilateral with an angle of $60$. What is the ratio of the area of this quadrilateral to the surface area of the original tetrahedron? I was able to solve the problem using geogebra to model the diagram, but I am unable to prove it. The answer that I found was about $0.3145$. I am really stuck on how to prove an exact answer. Please help. *Here is the link to my geogebra model: https://www.geogebra.org/3d/tjpgzaqb |
| How to solve this integral (delta function) Posted: 05 Dec 2021 03:02 PM PST $$ I=\int_{-\infty}^{+\infty} dx \bigg(a+\frac{x}{2}\bigg)^2 \delta\big(a^2-(a+x)^2\big) $$ here $\delta(x)$ is a Dirac delta function. According to Mathematica, the answer is $\frac{a^2}{2|a|}$. I tried to solve it using the method of substitution. Put $u=(a+x)^2\to x=\sqrt{u}-a$. And $$ \frac{du}{dx}=2(a+x) \to \frac{du}{2(a+x)}=dx $$ so, the integral becomes $$ I=\int du\frac{1}{2(a+\sqrt{u}-a)} \bigg(a+\frac{\sqrt{u}-a}{2}\bigg)^2 \delta\big(a^2-u\big) \\ I=\int du\frac{1}{2\sqrt{u}} \frac{1}{4}\bigg(a+\sqrt{u}\bigg)^2 \delta\big(a^2-u\big) \\ I=\frac{1}{8\sqrt{a^2}} \bigg(a+\sqrt{a^2}\bigg)^2 \\ I=\frac{4a^2}{8a} =\frac{a}{2} $$ But, the correct answer is $\frac{a^2}{2|a|}$. Can someone please help me, where am I going wrong? |
| Posted: 05 Dec 2021 03:25 PM PST Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of $$I= \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) $$ I try: since $(a-b)^3=a^3-3a^2b+3ab^2-b^3$,and $\sum_{n=1}^{10}n^3=\sum_{n=1}^{10}a^3_{n}$so we have $$3I=\sum_{n=1}^{10}(3na_{n}^2-3n^2a_{n})=\sum_{n=1}^{10}(n-a_{n})^3$$ take $b_{n}=n-a_{n}$,and we need to maxumize $\sum_{n=1}^{10}b^3_{n}$ with the constraint $\sum_{i=1}^{10}b_{i}=0$ and $-9\le b_{i}\le 9$,and I can't,somedays ago,it is said can use the Karamata inequality to found it,and to day said the reslut is $336$,But I consider sometimes,can find it,Thank you for your help |
| Posted: 05 Dec 2021 03:16 PM PST I have two linear functions $f$ and $g$ in an finite Vectorspace with dimension of n. Also $ f $ and $g$ both $V \to V $ I should proof that if $ f \circ g = 0 $ then $ rank(f) + rank(g) \le n $ Does this mean $f = 0$ and $ g = 0 $ well then the rank always have to be 0, but this what about $f=-1$ and $g = 1 $ what could I say about the rank of the two functions? It has be $ \le n $ for each but how does the operator $ \circ $ affect it? Normaly I would say $ Rank(f+g) = rank(f) + rank(g) $ |
| Mean & SD of Sampling Distribution Posted: 05 Dec 2021 03:04 PM PST A population consists of $4$ numbers $\{0, 2, 4, 6\}$. Consider drawing a random sample of size $n = 2$ with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? Since $\bar x $~ $N\left(\mu, \dfrac{\sigma^2}{n}\right)$? (b) Calculate the mean & standard deviation of the sampling distribution of $\bar x$. I got the answer of mean $\mu$ by $\frac{0+2+4+6}{4} = 3$ Thereafter, I proceed to calculate $\sigma$ $\sigma = \frac{(0 - 3)^2 + (2 - 3)^2 + (4 - 3)^2 + (6 - 3)^2}{4} = 5$ Substituting it back into the sample distribution gives: $\bar x $~ $N\left(3, \dfrac{5^2}{4}\right)$ Thus, I derive the standard deviation to be: $\sqrt{\frac{\sigma^2}{n}} = \dfrac{5}{2}$. However, the answer given was $\dfrac{\sqrt{5}}{\sqrt{2}}$. Can someone explain why is this so? I'm really quite confused with the whole concept of sampling distribution.. Thanks a lot! |
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