2
votes
2answers
98 views

Last step of the proof of the Chinese remainder theorem.

For the Chinese Remainder Theorem for rings we have: $$ A/(I\cap J) \cong A/I\times A/J $$ So far I have proven that there is a ring homomorphism from $\phi :A \rightarrow A/I\times A/J $ and the ...
2
votes
1answer
188 views

Gamma Function Contour Integration

So, I've been trying to prove the following integral related to the gamma function, and I'm really banging my head against the wall over this: ...
2
votes
1answer
209 views

Is complex symplectic structure equivalent to a quaternionic structure?

Let $S$ be a vector space over $\mathbb{C}$ of dimension $\operatorname{dim} S =2n$, let me denote by $S_\mathbb{R}$ real form of $S$ i.e. vector space over $\mathbb{R}$ of dimension $4n$. Is it true ...
6
votes
8answers
97 views

Functions of the form $\int_a^x f(t) dt$ that are commonly used.

I am a graduate student and teaching assistant, and I am teaching Calc 1 for the first time. In a few weeks I will be covering the Fundamental Theorem of Calculus. I'm using James Stewart's Calculus ...
2
votes
1answer
435 views

Convert a boolean function into K-map

I would like to know how can I convert the following boolean function into a truth table and accordingly construct the k-map $$F = A'B'C'+B'CD'+A'BCD'+AB'C'$$ thanks in advance :)
2
votes
1answer
293 views

Bisection Method Question, Multiple Roots

I understand how to do the bisection method and how to do it with a point of intersection. My question is should this not actually have multiple points of intersection? and if you're not given any ...
2
votes
1answer
811 views

Is $\sum_{n=1}^{\infty} \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$convergent or divergent

I have worked on this my answer is L= Div and B Consider the series $\displaystyle \sum_{n=1}^{\infty} a_n$ where $$a_n = \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$$ In this problem you must ...
0
votes
2answers
115 views

How many trials until I each my desired outcome

I'll just get straight to the point; I want to know how many independent trials are required to get an outcome(?) of 10. The situation is that I have an 80% success rate and I increment x by 1, ...
1
vote
3answers
81 views

Showing a subset is uncountable [closed]

How do I show if $A \subseteq B$, and $A$ is uncountable then $B$ is uncountable?
0
votes
1answer
135 views

Cardinality of the union of all repeated Cartesian products of N with itself [duplicate]

Here is a silly question, but I am a silly person. Consider the: Natural Numbers. Natural Numbers X Natural Numbers. Natural Numbers X Natural Numbers X Natural Numbers ... Now take the union of all ...
1
vote
0answers
48 views

Finding the $\log$ of a matrix by contour integration

My teacher presented this way of determining the logarithm of a matrix $\Omega$ in class today: $$\log \Omega = \frac1{2\pi i}\oint_{\Gamma} (\zeta I - \Omega)^{-1} \log \zeta \,d\zeta.$$ Does ...
0
votes
0answers
57 views

Why does the limit of a subsequence lie in a closed set?

I am reading through a proof where I encountered the following: Let $K_0\subset K$, where $K$ is a compact metric space, and $K_0$ is closed. If $x_{n}\subset K$ has a convergent subsequence, ...
1
vote
1answer
306 views

Paths on a Rubik's cube

Here is the Question i'm trying to solve: An ant is initially positioned at one corner of the Rubik's cube and wishes to go to the farthest corner of the block from its initial position. Assuming ...
1
vote
1answer
28 views

Proof, wheather a subset of a Group is a Subgroup

I have to check, weather the following subset of a group is also a subgroup: $$U = \left\{ \begin{pmatrix} a & -b \\ \overline{b} & \overline{a} \end{pmatrix} \in GL(2, \mathbb{C}) \bigg\vert ...
1
vote
1answer
49 views

Finding a function $g$ which is analytic on the region $E=\{ z \in \mathbb{C}: |z| >1 \}$

Find a function $g$ which is analytic on the region $E=\{ z \in \mathbb{C}: |z| >1 \}$ and maps $E$ one-to-one onto $H= \{ w \in \mathbb{C}: Re w<0 \}$. My approach: The region $E=\{ z \in ...
3
votes
3answers
2k views

Number of ways to form a 3-letter word with repetition allowed?

The additional rule is: no letter can be used more often than it appears in MILLENNIUM? (Which is pretty logical I guess) MILLENNIUM = MM, II, LL, NN, E, U My logic: Case 1: Double letters + 1 ...
1
vote
1answer
2k views

Can someone help explain how to do these two work problems?

A trough is 8 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of y=x^2 from x=−1 to x=1. The trough is full of water. Find the amount of ...
0
votes
4answers
97 views

Is a trig function with a constant in its parenthesis a constant?

For example: Is $\sin(8)$ a constant? I want to know because my professor differentiated it to $0$ and that was the explanation he gave. Thanks
2
votes
2answers
58 views

Kernel of $\phi:G \rightarrow \operatorname{Sym}(S)$ Group actions

$\operatorname{Sym}(S) == \text{All permutations of the set }S$. Prove $\ker(\phi)=\bigcap_{x\in S}G_x$ where $G_x$ is the stabilizer of $x$. Let $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
0
votes
2answers
254 views

No. of 5-digit monotonic numbers

The monotonic number is made of digits 1, 2, …, 9, such that each subsequent number equal to or greater than the previous number. Examples: 11119, 12369, 18999 etc. I understand that I can isolate ...
4
votes
1answer
272 views

Violation of the irrelevant alternative criterion of fairness in a pairwise comparison

I am teaching my students about the fairness criteria for voting system, working up towards arrow's impossibility theorem. One of the voting methods is called the pairwise comparison method: voters ...
1
vote
1answer
62 views

Minimal prime ideals consist of zerodivisors [duplicate]

I don't find the proof for this little demonstration ... Let $P$ be a minimal prime ideal of $A$. Show that $P$ is contained in the set of zero divisors of $A$.
1
vote
1answer
78 views

Any odd > 1 is the average of three primes

I think that any odd integer is the average of three primes. My first question is if this is equivalent to some other conjecture/theorem in number theory (I suspect it is). But more importantly, I ...
1
vote
1answer
36 views

Why are isotropy groups named as such?

Why are isotropy groups, also known as stabilizers, named as such? In physics, the word isotropy means having the same property in all directions. Can one draw an analogy from this to interpret the ...
2
votes
2answers
247 views

Recommendation for Number Theory Textbook

. Greetings, every mathematicians! I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied ...
0
votes
1answer
114 views

Calculating incremental coordinate change along a 3D vector

This is pretty trivial, but I've not managed to resolve it to my satisfaction and it has reached the stage where fresh eyes are definitely useful! I have an xyz point, and a 3D vector originating at ...
3
votes
3answers
409 views

What is the probability that you will see an odd number of heads?

You have $100$ biased coins. The probabilities of seeing heads when you toss these coins are equal to $1/3$, $1/5$, $1/7$, $1/9$, and so on, up to $1/201$ for the last coin (in general, for the $k$-th ...
0
votes
0answers
39 views

Question concerning the gamma function in relation to other holomorphic functions when $Re(\xi) > 0$

Let $f$ be indefinitely differentiable on $\mathbb R$ that has compact support. $\implies f$ belongs to the Schwartz space. Consider: $$I(\xi) = \frac1{\Gamma(\xi)} \int_0^\infty f(x)x^{-1+\xi}dx$$ ...
0
votes
0answers
35 views

Mapping a set of sets to a partitioning.

I've been experimenting with the following idea, and I wondered if there's a name for it: Suppose $S_0, S_1, ... S_{n-1}$ is an array of $n$ sets of elements in $U$. Now for any element $e \in U$ we ...
0
votes
1answer
45 views

Basis of row space and subspace

If given some vectors, and asked to find a basis for the subspace of R(n), will the basis for the subspace be the same as the basis for the row space? example: given the following vectors: ...
2
votes
0answers
154 views

Ideal of a Vanishing set $I(V(F[X,Y]))$ and how to repeat the computation.

The video I am getting this from is found here: https://www.youtube.com/watch?v=spHxUPvrkXw, it is around 5 minutes in. The first part of the question is: for $F[X,Y] = Y^2 - X^3 = 0$ find ...
0
votes
1answer
50 views

Explanation on how this solves for the real part of a complex fraction

I'm trying to solve the following fraction to find out what omega $\omega$ will leave me with only the real parts, assuming I know the values L, C, and R. $z = \dfrac{ \dfrac{L}{C} + \jmath\omega ...
0
votes
1answer
16 views

Get overall sum having partial data

I have partial data of online game ingame sales. I know the data for N users. The overall game users count is M which is about 6 times greater than N. I can calculate overall revenue for N users. But ...
2
votes
1answer
120 views

Well-ordering the reals: finding a certain model of $\mathsf{ZFC}$

How would one go about constructing a model $\mathfrak{M}$ of $\mathsf{ZFC}$ such that under $\mathfrak{M}$, no formula defining a well-ordering of $\mathbb{R}$ exists? I am certain such models are ...
1
vote
1answer
67 views

Combinatorics Parity Problem

There are 229 girls and 271 boys at a school. They are divided into 10 groups of 50 students each, with numbering 1 to 50 in each group. A quartet consists of4 students from 2 different groups so that ...
2
votes
3answers
47 views

How do I find the $x$ intercepts for $-x^2-3x+3$

How do I find the $x$ intercepts? $-x^2-3x+3$? I converted the function into vertex form but I am stuck at $3/4= -(x^2+1.5)$. Can someone give me and idea what I can do?
1
vote
5answers
93 views

Limits: What if it evaluates to $-\infty + \infty$

I would like to evaluate the following limit: $$ \lim_{x\to 1^{-}}\left[% \ln\left(1-x \over 1 + x\right) - \ln\left(1 - x^{2}\right)\right] $$ I put this into a limit calculator and the calculator ...
2
votes
2answers
69 views

Proving a sequence to be divergent

I'm trying to prove this sequence: $a_n = \sqrt{n}-\sqrt{n^2-1}$ to be divergent. How would I do this? I'm thinking of proving that it's not bounded below, but I'm not sure how to do that with ...
3
votes
0answers
85 views

Spectral sequence differentials

Let $\mathcal{F}^{\bullet}$ and $\mathcal{G}^{\bullet}$ be complexes of coherent sheaves on a variety $X$. There is a spectral sequence ...
4
votes
2answers
570 views

Multivariable calculus - Implicit function theorem

we are given the function $F: \mathbb R^3 \to \mathbb R^2$, $F(x,y,z)=\begin{pmatrix} x+yz-z^3-1 \\ x^3-xz+y^3\end{pmatrix}$ Show that around $(1,-1,0)$ we can represent $x$ and $y$ as functions of ...
1
vote
1answer
20 views

Need help with this conditional probability question

There are 4 alternatives on a multiple choice test. Let suppose that a student learned 70% of the material. If she doesn't know the answer, she picks one randomly. If she picked the right answer, ...
1
vote
1answer
255 views

Use Ito's Lemma to show:

I am somewhat unsure how to go about showing this: Use Ito's Lemma to show for any deterministic differentiable function, $f$: $$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$ Where $B(t)$ ...
0
votes
2answers
121 views

How to rigorously find range of a function?

What is a rigorous method/mechanism to find the range of a function $f(x)$? Is it acceptable to find $f^{-1}(x)$ and then its domain? I understand that $f^{-1}(x)$ does not always exist (at least not ...
7
votes
2answers
395 views

Integration in Banach spaces - interesting, nice and non-trivial examples needed

I am interested in $\textbf{Integration in Banach spaces}$. Here is a little motivation for my question: Let $\left(X,\|\cdot\| \right)$ be a Banach space, $a,b \in \mathbb{R}$ with $a<b$ and $f ...
0
votes
1answer
53 views

Chain rule application to find second partials

$x = 2r-s$, $y= r + 2s$, $V = f(x,y)$. Find $\frac {\partial^2 V}{\partial y \partial x}$ in terms of derivatives of $V$ with respect to $r$ and $s$. My work so far: $$\frac {\partial x}{\partial r} ...
1
vote
1answer
59 views

Convergence of an alternating series (from exam Q)

State whether $\sum_{n=1}^{\infty} a_n$ converges or diverges and prove your result: iv) $$a_{2^n + r} = (-1)^n(2^n + r)^{-1}\text{ for }0 \leq r \leq 2^n -1\text{ and }n \geq 0$$ I tried ...
0
votes
1answer
97 views

GAP-Character table

I the following link I have found the character table of $S_8$ which is computed with the program GAP. http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_symmetric_groups But I ...
0
votes
1answer
114 views

On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
1
vote
0answers
89 views

Find the abs max and min values of $f(x,y,z)=x-2y+3z$ on the ellipsoid $\frac{x^2}{6}+\frac{y^2}{3}+ \frac{z^2}{2}-1=0$

How do I do the following Find the abs max and min values of $f(x,y,z)=x-2y+3z$ on the ellipsoid $\frac{x^2}{6}+\frac{y^2}{3}+ \frac{z^2}{2}-1=0$ I found my two gradient ofcourse. $<1,-2,3>$ ...
1
vote
2answers
23 views

Binomial probability on ports

This problem appears very simple, but I am almost positive that it should not be so simple. 10 ports. P1,P2,P3...P10 are connected to a computing device which polls them in order to check which ...

15 30 50 per page