# All Questions

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### Definite Integral problem.

We're given : $\int_0^{\infty}e^{-sk}sinkx\:dk$ = $\dfrac{x}{x^{2}+s^{2}}$ We need to evaluate : $\int_0^{\infty}\dfrac{e^{-sk}sinkx}{k} \: dk$ I tried as follows : ...
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### Finding vector equation of a plane from its Cartesian equation

The Cartesian equation is $x-3y-4z=1$. Here is what I have tried: Finding three points on the plane by setting two variables equal to 0: $x=0$, $y=0$; $z=\frac{-1}{4}$ $y=0$, $z=0$; $x=1$ $x=0$, ...
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### How does the Fourier transform of a “zero avoiding” function look?

Let $n$ be a very large positive integer. Let $f \in\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function, satisfying $0\leq f\leq1$, and supported on $[-n,-\frac{1}{n}]\cup[\frac{1}{n},n]$ such ...
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### Solving without induction show that $a_{n}=2n-1$

Let $a_{1}=1$,and such $$4S_{n}=n(a_{n}+a_{n+1})$$ where $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ find $a_{n}$ since $a_{2}=3$,and we can easy to prove $a_{n}=2n-1$ Induction Methods Assume $a_{k}=2k-1$, ...
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### Evaluate $\int_{ - \infty}^{\infty} \frac{dx}{1+x^2}$ using complex integration

I'm trying to evaluate the real integral $$\int_{ - \infty}^{\infty} \frac{dx}{1+x^2}$$ Denote $\Gamma_{1}=\left[-R,R\right]\ \Gamma_{2}=Re^{it}$, for $t\in\left[0,\pi\right]$, and let $\gamma$ be a ...
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### How does one find the automorphism group of the following groups?

An automorphism of a group G is an isomorphism of G in itself. I am trying to find the automorphism groups of: $\mathbb{Z}; \mathbb{Z}/p\mathbb{Z}$ p prime; $\mathbb{Z}/6\mathbb{Z}$ I know that any ...
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### $H$ is a subgroup of $G$ with finite index. Prove tat G has finitely many subgroups of form $xHx^{-1}$

$H$ is a subgroup of $G$ with finite index. Prove tat $G$ has finitely many subgroups of form $xHx^{-1}$. Let $h\in H$, $x\in G$ Since H is a subgroup of G $h \in G$ $\rightarrow he \in G$ ...
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### Estimating distribution from two distributions

I have been doing a survey on Family Incomes in India. The income of male and females are denoted by x and y. x and y are strictly positive. Per chance, individual values of y were deleted. I only ...
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### If ideal quotients of a ring are isomorphic, are these ideals isomorphic?

Suppose that $R$ is a ring, $I$ and $J$ are ideals in $R,$ and $R/I\cong R/J.$ as rings. When does $I\cong J$ as $R$-modules hold?
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### Formula for the gradient of $f(A) = u^TA^kv$

Given a function of the form $$f(A) = u^TA^kv,$$ where $A$ is an $n\times n$ real-valued matrix, $u$ and $v$ are real vectors, and $k$ is some positive integer power. Does there exist a general ...
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### mean time question [on hold]

A sample of 600 units yields a mean of 7.2 days.since the population standard deviation of s=1.9 days must be used. Calculate & interpret the 90 percent interval for the mean completion time for ...
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### For a given non-constant polynomial $f(x)$ with integer coefficients, how many solutions are there to $f(x)\equiv 0 \mod(n)$ where $n$ is composite?

For a given non-constant polynomial $f(x)$ with integer coefficients, how many solutions are there to $f(x)\equiv 0 \mod(n)$ where $n$ is composite? Is there a general way to determine the number of ...
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### Singularities, essential singularities, poles, simple poles

Could someone possible explain the differences between each of these; Singularities, essential singularities, poles, simple poles. I understand the concept and how to use them in order to work out ...
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### Find the conditional pmf of $Y$ given $X = 0$

Let $X$ and $Y$ have the joint pmf defined by $f(0, 0) = f(1, 2) = 0.3$, $f(0, 1) = f(1, 1) =0.2$ $(a)$ Tabulate the conditional pmf of $Y$ given $X=0$ $(b)$ Tabulate the conditional pmf of $X$ ...
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### Number of solutions to $f(x)\equiv 0 \mod(11\cdot 19^{2})$

I have been asked to explain why the number of solutions of the polynomial congruence $f(x)\equiv 0 \mod (11\cdot 19^{2})$ cannot be 121, where $f(x)=x^{10}+10x^{8}-17x+12$. Any ideas?
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### Decrypting RSA message

I need help with a practice problem for an upcoming test. I've learned the answer to the problem is "well done", but don't know how to get there. Any help is greatly appreciated. Suppose that the RSA ...
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+50

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### Number of outcomes with 3 distinct numbers rolling 4 dice.

Suppose you roll 4 distinct dice. I am trying to find: a) The number of outcomes with 3 distinct numbers b) The number of outcomes with 2 distinct numbers I just want to check that my reasoning is ...
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### If $p$ is an odd prime show that $2^{p-1}(2^p-1) \equiv 1 + 9p(p - 1)/2\pmod {81}$

If $p$ is an odd prime show that $$2^{p-1}(2^p-1) \equiv 1 + 9p(p - 1)/2\pmod {81}$$ This is an exercise from Elementary Number Theory, 2nd Edition by Underwood Dudley. I know that the expression ...
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### Modular Algebra

I am devising an algorithm to solve equations like the following: $$10^{\lfloor\log(p1)\rfloor}x+p_1\equiv0\pmod{p_2}$$ In the scenario: $10^{1}x+5\equiv0\pmod{7}$, where $p_1=5$ and $p_2=7$, ...
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### What are the Big Theorems in Information Geometry?

I am working on preparing a talk on information geometry to a young finance/applied math audience. Motivating this area is turning out to be a little difficult. What are some big theorems or results ...
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### Is there a graph that has 7 vertices and each vertex has a degree of $2,2,3,5,5,5,6$?

Is there a graph that has 7 vertices and each vertex has a degree of $2,2,3,5,5,5,6$? Any ideas on how to solve this one?
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### Finding the Probability of a Normal Distribution

The mean IQ scores of 30 primary school students is 108.56 and the Standard deviation is 12.33. Assume that IQ scores for primary school students that have been kept for 50 years illustrate a normal ...
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### Show $3\cos 2x + 1 = 4\cos^2 x - 2\sin^2 x$

Show $3\cos 2x + 1 = 4\cos^2 x - 2\sin^2 x$ Using the formula $\cos 2x = \cos x - \sin^2 x$ I can say $3\cos 2x + 1 = 3(\cos^2 x - \sin^2 x) + 1$ $\Rightarrow 3\cos x^2 - 3\sin^2 x + 1$ But from ...
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### Proof That For a Group $G$, All Automorphism of G Is Defined by The Image of The Generating Set Of G.

Let (G,*) be a group. Let $\Phi: G \rightarrow G$ be the automorphism of $G$. I want to show that all automorphism of $G$ is defined by the image of de generating set of $G$. My proof Let ...
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### Find $\int (\sin^2 x - 2\cos^2 x)\,dx$

Find $\int (\sin^2 x - 2\cos^2 x)\,dx$ => $\frac{1}3 -\cos^3x - \frac{1}4 \sin^3 x$ This is of course not the right answer which is $-\frac{3}4 \sin 2x - \frac{1}2 x + C$
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### Solid Volume Enclosed by An Cone and An Incline Plane

I am wondering whether the exercise is correct.... Find the solid volume enclosed by $x^2+y^2=\frac{1}{3}z^2$ and $x+y+z=2a$. (A) $\frac{4}{\sqrt{3}}\pi a^3$ (B) $\frac{8}{\sqrt{3}}\pi a^3$ (C) ...
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### Show that uniformly continuous functions on subsets of $\mathbb R$, are transitive

Let $X, Y, Z$ be subsets of $\mathbb R$. Let $f : X \rightarrow Y$ be a function which is uniformly continuous on $X$, and let $g : Y \rightarrow Z$ be a function which is uniformly continuous on Y. ...
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### Suppose $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$?

Suppose $A,B\in\mathbb{R}^{n\times n}$ are matrices such that $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$. I couldn't come up with a ...
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### Characteristic of a pseudo-upper-triangular matrix.

From number 8b of this released exam: Let $W$ be a $T$-invariant subspace of $V$. Prove that the characteristic polynomial of $T_W$ divides $T_V$. In part a I showed that $T_V$ was of the form: ...
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### Fourier transform of dual function and real function

I need to proof that $$f*(t) \leftrightarrow \hat{f}*(-\omega)$$ and $$f(t) \in \mathbb{R}\leftrightarrow \hat{f}(-\omega) =\hat{f}*(w)$$ I tried to write a formula of Fourier transform and then ...
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### Demonstrating that 1! is = 1

The problem with this explanation is that it's using n = 2 instead of n = 1. Please read the explanation I found on "Math Forum - Ask Dr. Math" ( http://mathforum.org/library/drmath/view/57128.html ). ...
Fix a positive integer $k\geq 1$. $2k+1$ red balls and $2k+1$ blue balls are on a line in some order. What is the least $n$ (in terms of $k$) such that we are always able to remove $n$ red balls and ...