For posts looking for feedback or verification of a proposed solution.

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1answer
62 views

Prove that $f'(c ) = 0$

Let $f: (a,b) \to \mathbb{R}$ be a function defined on $(a,b)$. Let $c \in (a,b)$ be a local maximum and $f'( c)$ exists. Prove that $f'(c ) = 0$ Sometihng I have thought so far: For some $\delta ...
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1answer
31 views

$f,g \in R(T)$ such that $\hat{f} \cdot n^{2/3} = \hat{g}$ prove that $f$'s Fourier series converges absolutely.

Can someone help me by checking my solution. Is there a shorter More elegant solution ?(i'm almost sure you can some how express $f$'s Fourier series using $|\hat{g}|^2$ + constant, i saw someone do ...
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1answer
34 views

Roulette with p=$\frac{2}{3}$. What is the probability of not going home?

I'm learning about the gamblers ruin. The problem is that I don't know how to calclate the formula. I got two exercise questions in my book. Both of the questions will be about a strange roulette ...
2
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1answer
35 views

Intermediate value theorem

Suppose $f$ is a continuous function on $[a,b]$ and $\lambda$ is a value between $f(a)$ and $f(b)$. Prove that $\exists c \in [a,b]$ s.t. $f( c) = \lambda$ Let, $$g(x) = f(x) - \lambda$$ $g$ is ...
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1answer
16 views

Evaluating the volume of a torus formed by rotating a region about a horizontal axis using shells.

Using the method of cylindrical shells, find the volume of the shape created by revolving the region $x^2+(y-5)^2=4$ about $y=-1$. A cylindrical shell is given by: $2\pi v f(v) \ dv$ I solve ...
2
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1answer
31 views

Please check my proof of this elementary covector result

I would appreciate it if someone could look over my proof and verify that it's correct. The question: Let $f$ be a $k$-covector on vector space V. Let $u_1,\dots u_k\in V$ and $v_1,\dots,v_k\in V$ ...
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1answer
14 views

Evaluate the volume of a solid of revolution using shells.

A cylindrical shell $S$ formed by some revolution about the $y$-axis is given by the equation: $S=2\pi x f(x)dx$, where the circumference $C$ of the shell is $C=2\pi x$, the height of the shell ($H$) ...
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1answer
31 views

confirming solution - elementary probability

Box 1 contains 1000 transistors, of which 100 are defective, and box 2 contains 2000 transistors, of which 100 are also defective. A box is taken at random and two transistors are drawn from it, at ...
3
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2answers
139 views

Show that $f(x) = x^2$ is not uniformly continuous on $[0,\infty)$

Ok, I know the same question has already been asked here, and I am not looking for an answer even though my proof looks kind of the same. But, I need to know whether or not I am on the right track. ...
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1answer
36 views

Is it a solution of Heat Equation?

I find in a Centrale's school document this solution for "short" time of the heat equation, I have not MAPLE or other calculus softwares, and I just want to be sure if my hand verification is correct ...
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1answer
26 views

Chromatic number of a hypercube

What is the chromatic number $\chi(Q_4)$ of a four-dimensional cube. I know that all Hypercubes $Q_d$ are bipartite, so then this would yield $\chi(Q_4) = 2$, because every bipartite graph has ...
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1answer
23 views

Calculating the volume of a solid of revolution about a line.

A figure is formed by revolving the region bounded by $f(x) = \cos{(x)}$ and $g(x) = \sin{(x)}$ from $0$ to $\dfrac{\pi}{4}$ about the line $y=-1$. This figure is formed by integration of two ...
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1answer
23 views

Multivariable calculus: what principle is this step based on?

The background is that I was asked to solve the following problem using Green's formula $L$ is a Jordan curve (smooth and closed) which encloses the origin point in $xOy$ plane. Caculate this ...
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1answer
17 views

Finding the volume of a cone with and oblique base.

The base of $S$ is an elliptical region with boundary curve $9x^2+4y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base. The base of $S$ is ...
1
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1answer
38 views

Optimizing a function of a matrix

Let \begin{equation} \begin{aligned} W= & \underset{X}{\mathrm{maximize}} & & \log \left|X + K_1\right|- \alpha \log \left|X + K_2\right|\\ & \mathrm{subject \; to} & & 0 ...
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0answers
38 views

Integral of a function with an exponentiated inner product

Let $f:\Bbb R^n\to \Bbb R^n$ be a continuous function such that $\int_{\Bbb R^n}|f(x)|dx\lt\infty$. Let $A$ be a real $n\times n$ invertible matrix and for $x,y\in\Bbb R^n$, let $\langle x,y\rangle$ ...
1
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1answer
32 views

For which values of $x \in I$ is $(f_n)$ differentiable term by term?

Let $f_n(x) = \frac{1}{n}e^{-nx}$ on $x\in[0,1] =I$. Discuss the pointwise and uniform convergence of $(f_n)$ on $I$. For which values of $x \in I$ is $(f_n)$ differentiable term by term? ...
2
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1answer
33 views

How to calculate $\nabla \ln(|\mathbf{u}-\mathbf{v}|)$

I need to calculate:$$\nabla \ln(|\mathbf{u}(\alpha)-\mathbf{v}(\gamma)|)$$where $\mathbf{u}$ and $\mathbf{v}$ are vector valued functions with $\alpha$ and $\gamma$ as independent values and ...
2
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1answer
44 views

Every continuous map of a closed interval into itself has a fixed point

The Question: Please show this theorem: Let $f: I=[a,b] \rightarrow \mathbb{R}$ be a continuous map such that $f(I) \supset I $. Then $f$ has a fixed point on I. My Attempt: Suppose there is a ...
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1answer
43 views

Prove that if $f$ is entire and $|f(z)| \leq |z|^2+12$ then $f$ is a polynomial of deg $\leq 2$

[Solution Verification] Prove that if $f$ is entire and $|f(z)| \leq |z|^2+12$ for all $z \in \mathbb C$ then $f$ is a polynomial of degree $\leq 2$ So here's my proof, and there is a ...
3
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1answer
17 views

Finding the determinant of the adjoint of matrix of A whose determinant is 2.

Question Let $A$ be a 4 x 4 matrix with determinant 2. Recall that we can write $A^{-1}=\frac{1}{\det A} \text{adj} A$. Find the determinant of the adjoint matrix of A. ...
3
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1answer
52 views

Are these subrings of $\Bbb Q$?

Are the following subrings of $\Bbb Q$? 1) The set of non-negative rational numbers. No since we don't have any additive inverses, and the subring should be armed with an Abelian group for ...
2
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1answer
18 views

Have I approached this question correctly on conditioning the components of a vector that spans a subspace?

Can someone please check if I have answered correctly for this question (consisting of two parts). I don't have a solution set to check my working with, that is why I need your help. ...
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1answer
29 views

Subnormal series and indices

I'm trying to solve this problem: Let be $H, K$ subgroups of a finite group $G$. Suppose that exists one serie of subgroups such that $G=G_{0}\triangleright G_{1} \triangleright \ldots ...
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2answers
33 views

Bounding a complex integral over a square

I'm solving the following exercise: Use the estimate lemma to prove that $$\left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right| \leq 4\sqrt{10},$$where $\gamma$ is the square with vertices $\pm 1 ...
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1answer
19 views

Special Form of Combination - Formula Verification

Giving abc how many combination that involves a ? answer is a ab ac abc equal to 4 I came up with the following formula but I would like to know of its correctness plus if there is simpler form ...
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0answers
22 views

piecewise defined function finding at which points it is continuous

We have: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ $$ f(x) = \begin{cases} x^3 - 3x + 2 &\text{if }x \in \mathbb{Q} \\ x^3 + x^2 + 4 & \text{if }x \in\mathbb R\setminus\mathbb Q \\ ...
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2answers
46 views

Use a numerical method to approximate the roots of $x^2-1000.01 x+10=0$.

Problem: Find the roots of the following equation with calculations of four significant digits. Then use a method to find the roots of the equation with the maximum accuracy. $$x^2-1000.01 ...
2
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0answers
33 views

Probability that at least one ticket is available

I have an exam coming up and I'm working on exercises given in the textbook. Some of the problems had solutions but this one doesn't and I'm wondering if anyone can check my answer and help me out if ...
3
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1answer
26 views

Solving cauchy riemann equations, finding all analytic functions

I need someone to check my work! I tried doing this as properly as possible, but I have no way to check whether this is correct. Find $\textit{all}$ analytic functions $f = p(x,y) +iq(x,y) $ such ...
5
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2answers
571 views

Multivariable caculus: Am I or is Maple wrong?

Well, the problem goes as follows: $$\int_{\Omega}(x^2+y^2)dxdy\quad\Omega:=\{(x,y)\mid x^4+y^4\le 1\}$$ My approach: By symmetry I needed only to find ...
1
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1answer
48 views

If $f:\mathbb{R}\rightarrow\mathbb{R}$, $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0$ then $\frac{f(x)}{x}\in C^{\infty}(\mathbb{R})$

If $f:\mathbb{R}\rightarrow\mathbb{R}$, $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0$ then $\frac{f(x)}{x}\in C^{\infty}(\mathbb{R})$. Following ther is what i did: the Maclaurin series for $f(x)$ is ...
5
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0answers
64 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
1
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1answer
39 views

Find the murderer by using truth table for formal logic (formal disjunction or formal implication)

I'm studying formal logic and i was wondering if you can check whether I've solved this task correctly. TASK. Two people are arrested as suspects for a murder case, Stan and Peter. Four witnesses ...
1
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1answer
22 views

partial differential equations , first order help

Consider the linear first order non-homogeneous partial differential equation $U_x+yU_y-y=ye^{-x}$ By using the method of characteristics show that its general solution is given by ...
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1answer
22 views

Sol. verfication: Prove that if $f$ is entire and bounded on the open disk $|z|<R$ by $K$, and $f'(0)=f(0)=0$ then $|f(z)| \leq \frac{K |z|^2}{R^2}$

$f$ is entire. Let $D= \{z \in \mathbb C| |z|<R \}$, $f(0)=f'(0)=0$ and there is a real $K$ so that $|f(z)|\leq K$ for every $z \in D$. Show that for every $z \in D$, $|f(z)| \leq ...
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0answers
25 views

Circles under inversions; mobius trans.

Could you please confirm or refute this argument: I'm trying to apply inversion (the map $z \mapsto \frac{1}{z}$) to the disk centered at $1+i$ and radius $\sqrt2$, so the disk whose boundary is ...
0
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1answer
33 views

Find the residue of $e^{1/z}\sin(z)$ at $z=0$

I am looking for the residues of $f(z)=e^{1/z}\sin(z)$ at its singular points. Found that only $z=0$ is an essential singularity, where $$a_{-1}=\sum\limits_{n=1}^{\infty} ...
0
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1answer
15 views

Residue of $f(z) = \frac{\cos(z+i)-1}{(z+i)^4} $

The function $$f(z) = \frac{\cos(z+i)-1}{(z+i)^4} $$ has one singular point in $\Bbb C$. i) Identify the point: The point is $z+i=0$ hence $z=-i$ is the point. ii) Show the singularity is a pole ...
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0answers
22 views

More on the implicit function theorem: is this example correct?

I am trying to understand the implicit function theorem so I thought it would be a good idea to work out an example. Please could someone look at this and tell me if it is correct? Consider the ...
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0answers
50 views

Prove: $\int_C \frac{dz}{z^2+1} =0$ on the annulus $6\lt |z| \lt 8$

Let $D$ be the annulus $6\lt |z| \lt 8$ and let $C$ be any simple closed contour inside $D$. Show that there holds: $$\int_C \frac{dz}{z^2+1} =0$$ This has two singular points, $z=\pm i$, these are ...
0
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0answers
18 views

Determine the Mobius transform from $\infty$ to $0$, $0$ to $-i$ and $i$ to $\infty$

Determine the Mobius transform from $\infty$ to $0$, $0$ to $-i$ and $i$ to $\infty$ I have done the following: 1) $$\lim \limits_{z\to \infty} \frac{az+b}{cz+d}=0$$ $$\iff \lim_{z\to 0} ...
0
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1answer
16 views

Answer verification: Power series expansion of $\frac{1}{3-z}$ and radius of convergence about $3i$

Find a power-series expansion of the function $f(z)=\frac{1}{3-z}$ about the point $3i$ and calculate the radius of convergence, my attempt: $$f(z)=\frac{1}{3}\left(\frac{1}{1-(\frac ...
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0answers
22 views

stability of $ \dot x = (x-1)(y-2), ~~~\dot y=(x-3)(y-2)$

Question: I want to determine the point of equilibrium and the stability (asymptotically stable, stable, or intable) $$ \dot x = (x-1)(y-2), ~~~\dot y=(x-3)(y-2)$$ Attempted solution: So it has to ...
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3answers
68 views

The Hawaiian language has only 12 letters: what is the probability a randomly selected 3 letter “word”

The Hawaiian language has only 12 letters: the vowels a, e, i, o and u and the consonants h, k, l, m,n,p and w what is the probability a randomly selected 3 letter "word" begins with a consonant and ...
1
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1answer
31 views

Reviewing Probabilities and Bayes Rule

I am reviewing probabilites and I have a couple of questions that have come up... I have attempted answers and will share my thinking but I am stuck in a couple of places and would like some ...
0
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1answer
28 views

Verifying An Integral Problem

Okay, so basically I thought I got my answer fully correct, but seeing the correction, it seems I'm not. Either I'm wrong or the one who corrected the exam and sent the correction is. (It's a board ...
4
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0answers
74 views

$\operatorname{Ext}^n$: computation verification

I would like someone to verify my computation of $\operatorname{Ext}^n$. Problem: Let $p$ be a prime, $k$ a field of characteristic $p$, $G = \langle x \mid x^p = 1 \rangle$, $B = kG$, $S = k(1 ...
0
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0answers
28 views

Computer Networks - Queues (Check my math)

A buffer is filled over a single input channel and emptied by a single channel with a capacity of 64 kbps. Measurements are taken in the steady state for this system with the following results: ...
0
votes
2answers
29 views

Find parameters so that random variables (connected to Brownian movement) are independent.

$W_t\sim\mathcal{N}(0,t)$ is Brownian movement, find values of parameters $a, b$ for which $aW_1-W_2$ and $W_3+bW_5$ are independent. I don't even know where to start, so any hint is highly ...