2
votes
2answers
21 views

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 : ...
0
votes
3answers
25 views

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: $x=0$, $y=0$; $z=\frac{-1}{4}$ $y=0$, $z=0$; $x=1$ $x=0$, ...
2
votes
0answers
15 views

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 ...
0
votes
0answers
7 views

Partial converses to extreme value theorem

Under what conditions can we establish a converse to the extreme value theorem? That is, for what topological spaces $(X, \tau)$ can we say that if $(\forall f \in C(X))(\exists c \in E) \left( f(c) = ...
0
votes
0answers
22 views

Solving the integration problem by use of fundamental theorem of calculus and chain rule

In one test the question said $$f(x,t)=\int_{0}^{g(x,t)} e^{-u^2} du$$ Now how I can calculate $\partial^2f/\partial t^2$ ? I have this idea: $$\int_{0}^{g(x,t)} e^{-u^2} du=F(g(x,t))-F(0)$$ ...
0
votes
1answer
29 views

How do I find the limit of this specific function

$$\lim_{x\to-1} \frac{x^2+2x+1}{x+1}$$ It works for some questions, I've tried substituting and factorising, but the correct answer is 0 and I don't know how to get that.
3
votes
2answers
23 views

An average of three calls arrive every $5$ min. Find the probability that exactly four calls will arrive during a $5$ minute interval.

An average of three calls arrive every $5$ min. Assuming a Poisson arrival rate, compute the probabilities of the following events: (a) exactly four calls will arrive during a $5$ minute interval. ...
0
votes
0answers
8 views

Conditions for the trace of a curve to belong to a line

I want to find the conditions so that the trace of the curve $\vec{x}\in\mathbb{R}^3$ belongs to a line. I know that the equation of a plane $\alpha$ in $\mathbb{R}^3$ is given by $ax+by+cz+d=0 ...
3
votes
1answer
69 views

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 $a_{k}=2k-1$, ...
3
votes
1answer
32 views

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 ...
1
vote
1answer
18 views

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 ...
2
votes
0answers
23 views

$H$ 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 $h \in G$ $\rightarrow he \in G$ ...
0
votes
0answers
27 views

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 ...
4
votes
2answers
404 views

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?
1
vote
1answer
15 views

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 ...
-5
votes
0answers
22 views

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 ...
0
votes
0answers
17 views

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 ...
1
vote
2answers
404 views

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 ...
0
votes
1answer
12 views

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 $X$ ...
0
votes
0answers
9 views

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?
0
votes
0answers
12 views

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 ...
0
votes
0answers
31 views
+50

Change in eigenvalues by changing only one entry of a square matrix

Consider following square matrix $A$ of order $n$ $A=\begin{bmatrix} 0 & a_{12} & a_{13} & a_{14} & \cdots & a_{1n} \\ a_{21} & 0 & a_{23} & a_{24} & \cdots ...
1
vote
0answers
121 views

Can Hamiltonian Circuit be solved in Quasi-polynomial time?

Consider a graph $G$. Let’s assume it has a Hamiltonian Circuit . Now, we divide the graph $G$ in to $x$ subgraphs $G_1, G_2, ….. G_x$ in a way such that $|G_i|$ is the power of a prime number where ...
1
vote
3answers
49 views

What is the shortest way to compute $\int \sqrt{x-x^2}$?

I can solve this problem after $\int \sqrt{x}\sqrt{1-x} \text{dx}$ by substitution $x=u^2$ and then $u=\sin v$, and at last I get $\int \cos^4 x$. But it looks a very long way, is there any shortest ...
0
votes
0answers
8 views

augmentation ideal for universal enveloping algebras

Let $L$ be a restricted Lie algebra with the restricted enveloping algebra $u(L)$ over a field $F$. Let $ω(L)$ denote the augmentation ideal of $u(L)$ which is the kernel of the augmentation map ...
4
votes
3answers
45 views

If a function maps an input to its inverse, is it bijective?

I read in my textbook that a function is a bijection if and only if it has an inverse. Is it the same thing to say a function $f: X → X$ is a bijection if $f(x) = x^{-1}$? If $a = x$ and $b = x^{-1}$, ...
2
votes
2answers
46 views

Find triples $(a,b,c)$ of positive integers such that…

Find the triples $(a,b,c)$ of positive integers that satisfy $$\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=3. $$ I found this on a local question paper, and I am ...
0
votes
0answers
12 views

Complex shifting of Fourier transform

Is there a rule for calculating the complex shifting in Fourier's series? in the other words, How do we obtain the following values? $\mathcal{F}[u(x \pm i x_0)]$ where $x,x_0 \in \mathbb{R}^n$.
0
votes
1answer
20 views

Proof for $\forall x \in R^+, x^2 + y^2 + z^2 \geq xy + xz + yz$

I am doing a question to practice doing proofs with real numbers as I am still not so good at it. I ran into some problems for the following question where it is as given: $\forall x \in R^+, x^2 + ...
0
votes
2answers
45 views

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 ...
1
vote
0answers
11 views

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 ...
1
vote
0answers
20 views

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 $p_2=7$, ...
3
votes
0answers
129 views

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 ...
0
votes
2answers
27 views

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?
1
vote
1answer
17 views

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 ...
2
votes
2answers
27 views

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 ...
1
vote
0answers
15 views

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 ...
0
votes
0answers
9 views

Let V be an inner product space over F.Prove the Polar Identities:For x,y$\in$V

$\langle x,y\rangle=1/4\sum_{k=1}^4 i^k\lVert x+i^ky\rVert^2 $;if F=$\mathbb C$,where $i^2=-1$. PROOF:=>We have,$\lVert x+i^ky\rVert^2 $=$\langle x+i^ky,x+i^ky\rangle$=$\langle x,x\rangle$+$\langle ...
3
votes
1answer
37 views

Show that $\frac{z^2}{z-3}$ is analytic.

Explain why $\displaystyle \int_{C_1(0)} f(z) dz =0$ for the function $\dfrac{z^2}{z-3}$. In case there's some confusion with the notation, $C_1(0)=$ circle of radius $1$ centered at $0$ in ...
1
vote
0answers
16 views

Show that if $f^{-1}(A)$ is Borel measurable, then $f$ is also Borel measurable

Let $A$ be a Borel measurable set. I want to show that if $f^{-1}(A)$ is Borel measurable, then $f$ is also Borel measurable. I know the other direction is true, but I'm not sure if my claim is ...
0
votes
0answers
12 views

Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd S_\mathbb{N}$.

Let $S_\infty \subset S_\mathbb{N}$ be the set of permutations of $\mathbb{N}$ which are the identity on all but a finite number of elements. Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd ...
0
votes
1answer
42 views

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 $-\frac{3}4 \sin 2x - \frac{1}2 x + C$
0
votes
0answers
11 views

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) ...
0
votes
1answer
21 views

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. ...
2
votes
2answers
45 views

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 ...
0
votes
0answers
7 views

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: ...
0
votes
0answers
12 views

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 ...
1
vote
3answers
34 views

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 ). ...
1
vote
1answer
30 views

Consecutive balls of the same color on a line

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 ...
0
votes
1answer
30 views

Mean value for a simple random variable

From a box with numbers from 1 to 90, 6 numbers are extracted without reintroduction. To play this "game", you have to pay 1 and you win 15 millions if you predict the 6 numbers (nothing in all the ...

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