1
vote
1answer
57 views

The Real solution of the equation $x^\frac13= -1$

Surely, $x=-1$ is the only solution. But, wolframalpha says there is no solution in the real field. I think this is because they transformed $x^\frac13$ to $e^{\frac13(lnx)}$. But why did they ...
5
votes
2answers
55 views

An integration question to be solved without using differentiation under the integral sign.

$$I(\alpha)=\int_0^1 \frac{x^\alpha-1}{\ln x}dx.$$ As the title says, if someone could solve this without using the differentiation under the integral sign technique, I would be very grateful.
1
vote
0answers
26 views

Highschool inequality 2

Given a real number $a>0$ find a $b>0$ such that $\sqrt{(x-2)^2+(y-1)^2+(z-1)^2+(ω-3)^2}<b\Longrightarrow |xyzω-6|<a$ I tried the procedure followed in highschool inequality 1 but it ...
0
votes
4answers
42 views

Are primary ideals always contained in unique maximal ideal?

Just wondering, is this a standard fact? I notice a couple Banach algebra texts define primary ideals in this way. Another question: does this property, i.e. being contained in a unique maximal ideal, ...
-4
votes
1answer
28 views

Lift-club rates (This should be really easy)

Right, this is actually a real-life problem. I want to join Bob and Joe's lift club. Joe usually pays about \$40 a week (in total) to drive between A and B (for fuel). (Driving from A to B and back is ...
2
votes
2answers
29 views

How to prove linearity?

Let suppose third-order differential equation, that solved for highest derrivative admits solution: $Y(t) = y(t) + C_1 f_1(t) + C_2 f_2(t) + C_3 f_3(t),$ where $y(t)$ is some solution, $f_1(t), ...
0
votes
0answers
15 views

Computing efficiently a small base to the power a large number

Is there a fast algorithm to compute an exponential with a small base, (namely , close to 1) For example, computing 1.01 to the power 100?
-1
votes
0answers
21 views

Is the flux through $A$ the same as the flux through $B$?

In the figure below, the amount of field lines through $A$ is the same as the amount of field lines through $B$, but can you say the flux through $A$ is the same as the flux through $B$ as well?
8
votes
1answer
55 views

How to prove a function is a matrix exponential?

If $F(x) = \exp(x A) = \sum_{i = 0}^\infty \frac{1}{i!} x^i A^i$ where $F(x), A \in \mathbb{R}^{n \times n}$, then \begin{equation} F(x + y) = F(x) F(y) = F(y) F(x) \end{equation} holds. When is the ...
0
votes
1answer
24 views

Prove if $E_1$ and $E_2$ are measurable then$ m(E_1\cup E_2)+(E_2\cap E_2)=m(E_1)+m(E_2)$

by additivity $m(E_1\cup E_2)=m(E_1)+m(E_2)$ (because $E_1,E_2$ are measurable) but i don't know what to do with $E_1\cap E_2$. I tried to use demorgan's identity to solve this part but this is not ...
3
votes
1answer
15 views

Self complementary graph with a pendant vertex

Show that if a self-complementary graph contains a pendant vertex, then it must have at least another pendant vertex. Let $G$ be a graph of order $n$, so it has $n(n-1)/4$ edges, just like its ...
-3
votes
2answers
45 views

What is the pattern of this sequence of numbers? [on hold]

$5,\,7,\,14,\,16,\,33,\,55,\,82,\,114$ I was given this sequence of numbers, but I can't find the pattern to it. I have put them into different online sequence calculators, but they did not work. ...
0
votes
1answer
9 views

Variance of not quite the product of two independent, normally distributed random variables

Let's say I have two independent variables, $X\sim N(10,9)$ and $Y\sim N(5,4)$. $X$ represents the number of orders received in a month, and $Y$ represents the size of each order. For this example, a ...
1
vote
1answer
9 views

Poisson process Probabilities

If I assume that {$N(t)=: t >= 0$} is a Poisson process with intensity $\lambda$. For $0<s<t$, how would I find the Pr{$N(t)>N(s)$}?
1
vote
2answers
17 views

point of intersection of a line l and the plane p, i get 0 somehow

$L: x=\frac {y-1}{2}=\frac {z+1}{3}$, $P= x − 2y + z = 1$. Find the point of intersection of the line L and the plane P.
1
vote
1answer
21 views

Arrangement of the word MATHEMATICS if last spot must have the letter 'T'

How many ways can the word MATHEMATICS be arranged if the last letter must be a T? My solution: There are $2$ possible choices for the last letter (There are $2$ different T's), which leaves $10$ ...
-1
votes
1answer
28 views

why does $\frac{d}{dx} log_b(x)$ not = $\frac{lnb}{x}$?

I know that $log_b(x) = \frac{lnx}{lnb}$, and that differentiating $$\frac{d}{dx}(\frac{lnx}{lnb}) = \frac{1}{lnb}\frac{d}{dx}(lnx)=\frac{1}{xlnb}$$, so where is my mistake when I do it this way: ...
0
votes
4answers
30 views

Show that A*B and B*A have the same order [duplicate]

How can I show that the elements A * B and B * A have the same order? where A, B belong to a finite group G How can I prove that 2 elements have the same order? I was thinking of showing that ...
3
votes
2answers
59 views

Problem in deducing the number of onto functions

Let $A, B$ have $m, n$ elements ($m > n$). Therefore, the number of onto functions from $A$ to $B$ is: $$\sum_{k = 0}^n (-1)^k \binom{n}{k} (n - k)^m.$$ How can one use the IE (Inclusion/Exclusion) ...
0
votes
3answers
26 views

Find some n such that $|s-s_n|< 10^{-3}$

Consider the series $\sum_{n=1}^\infty \frac{1}{n^2}$. Let $s_n$ be the $n$th of the series and $s$ be the sum of the series. Find some $n$ such that$$|s-s_n|< 10^{-3}$$ Can someone please ...
0
votes
1answer
26 views

How do I make my TI-89 evaluate a recursive function?

On my TI-89 I can assign variables recursively such as: $1\to x$ returns 1 $x+1 \to x$ returns 2 $x+1 \to x$ returns 3 etc. How could I do functions the same way: $x \to f(x)$ returns Done $2\cdot ...
0
votes
0answers
13 views

Tensor algebra becomes a graded $R$-algebra short proof

I had a post proving that the tensor algebra becomes a graded ring, i have come up with a simple approach that goes as follows: Proposition: The tensor algebra $T(M)$ with multiplication defined ...
1
vote
1answer
44 views

the blood test riddle (number theory)

A microbiologist has been given a set of $100$ blood vials. Exact one of those $100$ vials is positive to a concrete disease X. The microbiologist desires to send only $7$ vials for analysis. He can ...
1
vote
1answer
39 views

What are the components of binary strings?

$C_{9}$ is the graph with vertices representing all binary strings of length nine. Two strings are adjacent if and only if they differ in exactly three positions. How can I compute how many components ...
0
votes
1answer
11 views

Brownian Motion Finding M(t)

If I have that {$B(t); t >=0$} is a standard Brownian motion, with $B(0)=0$, and I let $M(t)$ = max{$B(u) ; 0 \leq u \leq t$} and I am supposed to: a) Evaluate Pr{$M(4) \leq 2$} and b) Find the ...
0
votes
1answer
38 views

How to solve equation $ x=W(a+bx^{n})+1 $?

How i can resolve the equation $x=W(a+b x^n)+1$, where $W$ is the Lambert $W$ function? thanks
0
votes
1answer
14 views

Does $\langle \Psi, \mathbb{I} \rangle_G=\langle Res_H\Psi, \mathbb{I} \rangle_H$ always hold?

Let $G$ be a group and $H < G$. Let $\Psi$ be a character. Let $\mathbb{I}$ be the trivial representation Does $\langle \Psi, \mathbb{I} \rangle_G=\langle Res_H\Psi, \mathbb{I} \rangle_H$ always ...
1
vote
3answers
31 views

finding a third 3d point in a series

Given two three dimensional points. find the z coordinate of a third point that has two known coordinates. I'm not entirely sure how to solve this system. I'll be implementing this into an algorithm ...
0
votes
1answer
26 views

Question about counting cards

A standard deck of cards contains $52$ cards divided into four suits: the red suits, hearts and diamonds, and the black suits, clubs and spades. Each suit, in turn, is divided in 13 ordered ranks: ace ...
7
votes
1answer
44 views

$p$-series divided by alternating $p$-series = geometric series? Why?

I thought the following equation was interesting: $\dfrac{1 + \frac{1}{2^p} + \frac{1}{3^p} + ... }{1 - \frac{1}{2^p} + \frac{1}{3^p} - ...} = \dfrac{1}{1-2^{1-p}}$ for $p>1$, where $p$ is a real ...
1
vote
0answers
10 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
0
votes
2answers
39 views

For a normed vector space $ E $ and an element $ x \in E $, prove that if $ L(x) = 0 $ for every continuous linear functional $ L $, then $ x = 0 $.

Question. Let $ E $ be a normed vector space. Is it true that for a given $ x \in E $, if $ L(x) = 0 $ for every $ L \in E' $, then $ x = 0_{E} $? One way to prove this is to find an $ L \in E' ...
-1
votes
1answer
13 views

How to measure reaction time when given a right or wrong answer test? [on hold]

Here is an example of some data. test 1 test 2 test 3 correct 20 10 13 wrong 4 8 12 Lets ...
0
votes
1answer
10 views

If a quadratic form $f$ takes the minimum on a triangle in a vertex, what can I say about min of $f$ on edges of a subdivision?

Let $f(x)=x^2+y^2$ be the Euclidean square-norm and $A,B,C\in\mathbb{R}^2$ be vertices of a triangle $\Delta$ such that $f$ takes the maximum on $\Delta$ in $C$, the minimum in $A$ and takes the ...
-3
votes
0answers
14 views

Find preimage of a function given that the image of the function [on hold]

I have a following question that I need help on. Let $F(s,t)=\left( s^2 \cos(t), s^2 \sin(t) \right).$ Find the $F$-preimage of $[0,1] \times [0,1].$
2
votes
1answer
32 views

Does a simplex with equal altitudes have to be equilateral?

Consider a simplex in $\mathbb{R}^d$. Assume that all its altitudes have the same length. Does it necessarily mean that the simplex is equilateral, i. e. all distances between its vertices are equal ...
0
votes
0answers
29 views

Group Theory: Finding Homomorphisms From a Cyclic Group to an Automorphism Group.

I have to find all the homomorphisms, $$h:C_{5}\to Aut(C_{31})$$ Given that there are thirty elements in $Aut(C_{31})$, do I have to find the order of each of the elements and then see which of them ...
2
votes
3answers
67 views

finding the probability to get a diploma

For getting a diploma a person needs to go to $3$ interviews at $3$ teachers: $A,B,C$. In each interview a teacher can give a positive opinion or negative opinion. The person will go to interview at ...
1
vote
0answers
23 views

Is the discriminant of a polynomial surjective onto $\mathbb Z$?

Consider polynomials of degree two over $\mathbb Z$: $f = ax^2+bx+c$ The discriminant is $D = b^2-4ac$ And we can show that $D=2$ is not a possible value for $D$. I wonder if the value $D=2$ ...
0
votes
0answers
14 views

Inducing $A_4$ from $\langle (123) \rangle$

Let $G=A_4$ and $H=\langle (123) \rangle < G$. Compute $Ind_{H}^G \chi$ for every irreducible $\chi$ of $H$. Choose the right transversal of $H$ in $G$ as $V_4=\{ 1, (12)(34),(13)(24),(14)(23) ...
1
vote
1answer
17 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
1
vote
1answer
26 views

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)\hspace{1mm}|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$ As you can see two forms are easy. $$\iiint_E ...
1
vote
1answer
21 views

Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
2
votes
1answer
25 views

Tower of Hanoi variation from Concrete Mathematics - possible arrangements

From Concrete Mathematics, there is a problem that describes a variation of the Towers of Hanoi, where the disks can not move directly from peg $A$ to peg $B$, but must go through a middle peg. ...
0
votes
0answers
15 views

Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
1
vote
1answer
22 views

Dense sets and Empty Interior

if $A$ is dense in $X$, is there a relation which shows in which cases $A$ has empty interior ? $\mathbb{Q}$ has an empty interior as a dense set in $\mathbb{R}$, so does its complementary in ...
1
vote
1answer
13 views

Probability function and distribution - taking out fish from a pool

In a pool of fish there are 4 fish of type A, 3 fish of type B, 2 fish of type C, 1 fish of type D. We take out fish without returning them until we get fish of type C for the first time. ...
0
votes
1answer
11 views

Does a bounded countably infinite union of sets with volume have volume?

If $ A_1, A_2,...$ are sets with volume and $A= \cup_{i=1}^\infty A_i$ is a bounded set, must $A$ have volume? This was a homework problem that we went over in class, and if I remember correctly the ...
0
votes
2answers
15 views

vectors and cartesian equation on the line in 3d

Find in scalar parametric form an equation for the line of intersection of the plane $P$ and the plane with Cartesian equation $2x + y − z = 0$. $P= x − 2y + z = 1$.
0
votes
0answers
2 views

Sampling specific unit vectors

Given a unit vector $A\in \Bbb{R}^N$ and an angle $\theta$, the unit vector $P$ needs to satisfy $\left<A,P\right>=\cos\theta$. How to sample $P$ uniformly? For example: If $A = ...

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