2
votes
1answer
22 views

Can a discrete random variable be both larger and smaller than another?

Suppose we have two discrete random variables $A$ and $B$ with a finite number of possible outcomes. Then the expression '$A < B$' can be interpreted as a random variable itself, taking the values ...
1
vote
0answers
24 views

Optimal control for hitting a random point with gaussian distribution

A particle $X$ starts from the origin $X_0=0$ of the real line and can move to the right or the left with speed $\pm 1$ and should hit a point, $\xi$, normally distributed (mean zero, and variance 1) ...
1
vote
2answers
274 views

How to Represent a 3D Line under Polar Coordinates

In one of my applications, I need to represent a line under 3D polar coordinates system. In 2D, we can define a line by a distance to the origin and then a angle indicating the direction of the line ...
0
votes
1answer
43 views

Explanation for the orders of subgroups and number of groups with these orders.

This was a question on my exam that was just given back and I need help understanding why both (a) and (b) and (c) are the answers they are. In each group listed below, give the orders of the ...
0
votes
2answers
62 views

Integrals on unlimited sets

How do you evaluate this expression $$ \left| \int_{1}^{\infty} 1 \; dx - \int_{1}^{\infty} 1 \; dx \right| \quad ? $$ Using improper integral definition, this should be an indeterminate $\infty - ...
0
votes
1answer
40 views

What does this mean in derivatives?

I have seen this when I am integrating things, but I haven't seen this in this context. What does it mean? To be specific, what does $\alpha_0$ do in this calculation.
0
votes
1answer
56 views

An example of a morphism which does not preserve normality.

While doing an exercise, I was prompted to tell what I could say about the image of a normal subgroup under a morphism. While trying to reach a conclusion, I was able to demonstrate that if it is ...
3
votes
1answer
72 views

An immersive map is locally left invertible

Question: Suppose that $m < n$, that $U$ is an open set in $\mathbb R^m$ and that $f : U \rightarrow \mathbb R^n$ is a $C^1$ function that has rank $m$ everywhere in $U$. Show that for every ...
2
votes
1answer
70 views

Solve P(x) ≡ x^3 - x^2 + x -1 ≡ 0 mod(2^i), for i=1,2,3,4.

Here is what I got so far: For i=1: x ≡ 1 mod (2) is a solution to P(x)≡0 mod (2). P'(x) ≡ 3x^2 -2x + 1 So, P'(x) ≡ P'(1) = 3-2+1 ≡ 1+1 ≡ 2 mod (2) and P(1) ≡ mod (2). Then, by Hensel's Lifting, ...
4
votes
2answers
120 views

When are stable continuous time Markov chains Feller? Always?

This is a question is similar to this 2 year-old one that never got answered (truthfully it's pretty much the same question except that I'm adding a bit more detail and the assumption that the $Q$ ...
3
votes
0answers
37 views

Subgroups and Cosets Question

I'm having trouble establishing this proof for my abstract algebra class. The question is Suppose that $H$ is a subgroup of $G$ and $g$ is an element of $G$. Prove that the following subset of ...
4
votes
2answers
163 views

Fermat's Last Theorem with negative exponent

FLT says that the Diophantine equation $a^n+b^n=c^n$ isn't satisfied by any triplet $(a,b,c)$ where $n\in\mathbb{N}$ and $n>2$. But what happens if $n\in\mathbb{Z}$ and thus can be negative? ...
4
votes
2answers
3k views

For all square matrices A and B of the same size, it is true that (A+B)^2 = A^2 + 2AB + B^2

The below statement is a true/false exercise. Statement: For all square matrices A and B of the same size, it is true that (A + B)2 = A^2 + 2AB + B^2. My thought process: Since it is not a proof, I ...
0
votes
2answers
64 views

What is series coefficient for $f(x)=\csc^2 x - \frac1{x^2}$?

What is general formula for Maclauren series expansion for $f(x)=\csc^2 x - \frac1{x^2}$ ?
0
votes
2answers
61 views

HK are subgroups of G

I'm proving that if $H$ is a subgroup of a group $G$ and $K$ is a normal subgroup of $G$, then $HK$ is a subgroup of $G$. I've tried pondering the fact that $HK$ is a subgroup of $G$ iff $HK = ...
2
votes
1answer
79 views

Show an iterative fixed point method does not converge

I was asked the following question: let $g(x)=\frac{30}{1+x}$. Notice that $g(5)=5$. Is there an $\epsilon >0$ such that the series $\{x_k\}_{k=0}^{\infty}$ defined by $x_{k+1}=g(x_k)$ and ...
1
vote
0answers
138 views

Markov Chains: Limiting probabilities of positive recurrent states sum to one?

I have a question about Markov chains. I am trying to understand the proof of Proposition 2.6 of http://www.columbia.edu/~ks20/stochastic-I/stochastic-I-MCII.pdf. The setting is: we have a positive ...
3
votes
1answer
46 views

Integral with absolut-value function

How do I seperate the following integral? The integral of $|x^2-y|$ with $|y| \leq 1$ and $|x| \leq 1$. I know that the absolute value is positive for $x^2 \geq 1$ and negative for $x^2$ but I am ...
4
votes
2answers
64 views

Let $a$ be a group element of order $n$. Prove the order of $a^{-1}$ is $n$.

Prove the following: The order of $a^{-1}$ is the same as the order of $a$. Would this be the correct proof? $a^n=e$ then $e=(a^{-1})^n$ and let $m=$ord$(a^{-1})$. Then $m \le n$. and $(a^{-1})^m ...
0
votes
1answer
41 views

What is an example of an infinite group (say $G$) and a subgroup $H$ of $G$ which has index $2$?

So I know an infinite group has an order which, in a sense cannot be found. But would would be an example of a subgroup which has an index 2?
6
votes
3answers
743 views

Relation between Right Riemann sum and definite integral

Let a partition $\{t_0,\ldots,t_n\}$ of the interval $[a,b]$ and let $f$ an integrable function. (we may also assume that $f$ is differentiable on $[a,b]$) We know that the Right Riemann sum is ...
0
votes
1answer
40 views

Non-archimedean valuation

In the definition of non-archimedean valuation the value group is the group of real numbers $\mathbb R$. Can we replace $\mathbb R$ by any totally ordered group different from $\mathbb R$ ? Thanks
5
votes
1answer
167 views

Distinguished open sets in $\mathbb{P}^n$

I have given a homogenous polynomial $F\in\mathbb{C}[x_0,\ldots,x_n]$ of degree $d>0$ and consider the open set $U_F=\{p\in\mathbb{P}^n; F(p)\neq 0\}$. I'd like to prove that $U_F$ is isomorphic to ...
1
vote
3answers
196 views

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$.

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$. First of all, I solved $1-x<(x-1)(5-x)<3$ which gives me $(x-1)(x-6)<0$ and $(x-4)(x-2)<0$. How to find the range, ...
0
votes
2answers
48 views

What is the expected value of the score?

A card is drawn from a deck of 52. The score equal to its rank unless it is a court card (Jack, Queen or King) with a score of 10, otherwise equal to its rank and Ace counts as one. What is the ...
2
votes
1answer
63 views

Linear Algebra - Determine if a linear transformation is one-to-one

I have been faced with this question: $T:\mathbb R^3 \to\mathbb R^3$ defined by $T(X) = AX$ where $A = \begin{bmatrix}1&-1&0\\0&1&2\\2&-1&1\end{bmatrix}$ How do I tell if ...
0
votes
2answers
182 views

Conditional multivariate normal pdf with inequality $f(x_1 | x_2 > a)$

Let $$\begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim\mathcal{N}\left[\begin{pmatrix} 0\\ 0 \end{pmatrix} ,\begin{pmatrix} \sigma_{1}^2 & ...
0
votes
1answer
71 views

Linear Algebra - Understanding how to determine if a transformation is linear

I'm new to linear transformations in linear algebra and I can't quit understand how to find out if a transformation is linear. Any help would be much appreciated! a) $T:\mathbb R^3 \to\mathbb R^2$ ...
0
votes
3answers
83 views

If I know that $S$ is a square but not its square root, how do I find the next square number?

Given a square number $s = n^2$, is there a way to find the next square $s'$ without knowing $n$? (That is, if you can't take the square root of $s$ to determine $n$, can you compute $s'$?)
0
votes
2answers
122 views

Polynomial Solutions for Differential Equations

Suppose we have a set of polynomials where $\deg(Q_k(x))\le k$, and consider the following differential equation, $$W:=\sum_{k=0}^n Q_k(x)\frac{d^k}{dx^k} .$$ It is known that if there is a ...
0
votes
1answer
33 views

Create a general NFA for $M_n$

I need to create a general NFA $M_n$ where $n \in \mathbb{N_0}$ with the following language defined: $$L(M_n) = \left\{ w \in \{0,1\}^* \big | x1y \textit{ for } x \in \{0,1\}^* \textit{ and } y \in ...
4
votes
3answers
241 views

Exercise 3.7 Hartshorne

Problem. Show that any two curves in $\mathbb{P}^2$ have a nonempty intersection. This seems to follow immediately from the Projective Dimension Theorem, but I was wondering if anyone could provide a ...
0
votes
1answer
119 views

Dice Probability of rolling at least one four?

One dice is rolled three times. What is the probability of getting at least one four? I've been getting stuck on what to do next I know that it could be one four, two fours, or all three rolls could ...
0
votes
1answer
123 views

Finding the remainder of a polynomial P when divided by $x^2 -1$

I need help please to answer this problem: The remainder of a polynomial P (in one variable $x$) when divided by $x^2 -1$ is a polynomial of degree at most 1, that is, it has the form $ax + b$ ...
9
votes
2answers
155 views

Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?

I'm confused. Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module? We know that $\Bbb Z_{p^{\infty}} \subset \Bbb Q/\Bbb Z$ is artinian. The following argument is true or not ? $\mathbb Q / ...
2
votes
0answers
70 views

Isomorphism,on ${R}^4$

I dont understand what the function is for part (a) such that a mapping from $X\in T_p{R}^4$ to $w(X,-)\in T^{\star}_p{R}^4$ be an isomorfism!. So Consider on ${R}^4=(x_1,y_1,x_2,y_2)$ the ...
2
votes
1answer
60 views

Submultiplicativity stronger than triangle inequality?

I would like to ask a question about matrix norm. Is the submiltiplicativity property always stronger than the triangle inequality? So, if i prove for a matrix norm that it's submultiplicative, i ...
1
vote
1answer
30 views

Concrete representation of the annihilating algebra

Suppose $\mathfrak{M} = A^{\prime\prime}$, where $A$ is a concretely described subalgebra of $\mathcal{B}(\ell^{2}(\mathbb{N}))$. In some instances, it is possible to provide a concrete description of ...
0
votes
1answer
16 views

Expressing array response $A(Z) = \sum_{-N}^{N} w_n Z^n$ as sine-function

The array-response of an antenna can be defined as: $$A(Z) = \sum_{-N}^{N} w_n Z^n$$ where $Z = \exp(-i \omega \Delta t) = \exp(-ik\Delta x \sin \alpha)$ According to my textbook, if we let $w_n = ...
0
votes
1answer
36 views

Showing that $\underset{n \rightarrow \infty}{\lim} |a_n| / |a_{n+1}| = R$ implies that the radius of convergence of $\sum a_n z^n$ is also $R$

Hypothesis: Suppose that $\underset{n \rightarrow \infty}{\lim} |a_n| / |a_{n+1}| = R$. Goal: Show that $\sum a_n z^n$ has radius of convergence $R$. Attempt: The radius of convergence of $\sum ...
0
votes
1answer
29 views

Extending a polynomial function on an interval to be infinitely differentiable on all of R

If $ f:(a,b) \to \mathbb{R}$ is a polynomial function, can it be extended to $ g:\mathbb{R} \to \mathbb{R}$ such that g is infinitely many times differentiable and it is NOT the same polynomial? What ...
0
votes
1answer
65 views

Coprimes and Congruence

I need help getting a proof, I don't want solution to the problem just help guiding me to complete the proof. Suppose $m,n$ are coprime, Prove that $a \equiv b \mod{mn}$ if and only if: $a ...
3
votes
2answers
103 views

Finding the second derivative; What am I doing wrong?

Original Question: $xy+y-x=1$ Find the second derivative; $d^2y\over{dx^2}$$(xy+y-x=1)$ We are allowed to use either notation as far as I know: ${dy\over{dx}}$ or ${y'}$. Because ...
0
votes
3answers
34 views

Coin and steps -Probabilty and Statistics

Turn a coin and if it falls heads move three places to the right otherwise move 2 places left. After 20 times you turn the coin, in what positions might you be and what is the probability to be in ...
3
votes
2answers
268 views

Closedness and convexity of half spaces $\mathbb{R}^n$ determined by hyperplanes

Every hyperplane divides $\mathbb{R}^n$ into two "half space": the set of points "on and above" the hyperplace, $H^+ = \{ \mathbf{x} \mid \mathbf{a} \cdot \mathbf{x} \geq \alpha \}$, and the set of ...
0
votes
3answers
72 views

Why does $aH = Ha \neq ah = ha$

For normal subgroups...How come $aH = Ha$ does not imply that $ah = ha$ for all $h \in H$? I'm showing that every subgroup of a commutative group is normal and I thought I had it on that one but ...
3
votes
1answer
58 views

Nonconvex set converging to a convex set despite holes

I'm looking at the example in Figure 4-7 of "Variational Analysis" (Rockafellar and Wets). Basically, there's a sequence of sets $C_{\nu}$ riddled with holes, and it states that the sequence ...
3
votes
4answers
120 views

How to show that $x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big)$ is decreasing?

I am a non-mathematician who knows some elemententary calculus ans I want to prove that the sequence $(x_n)$ given by $$ x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big) $$ is decreasing. Is there ...
1
vote
1answer
62 views

Where can I find good examples about Algebra (but not only): Usual counter-examples, but also limit cases, rare ones, etc [duplicate]

I recently discovered the importance of examples and couter-examples in mathematics. Where could I find good examples books or anything related to it ? I am particularly looking for rare limit-cases, ...
2
votes
1answer
59 views

SOLVED: Green's theorem result and line integral result are not equal! What am I doing wrong?

I have this line integral: $\oint 3ydx+x^2dy$ and the path is a line from $(0, 0)$ to $(1, 0)$ (so this is $y=0$), another line from $(1, 0)$ to $(1, 1)$ (so this is $x=1$) and a curve $y=x^2$ from ...

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