1
vote
1answer
66 views

Find the matrix $[T]$ with respect to the given basis, where $T$ is defined as $T:\mathcal{V}_2 \rightarrow \mathcal{V}_3$ by $Tv(x)=\int_0^x v(t)dt$

The information given to me is that $\mathcal{V_n}$ is the vector space of real polynomials of degree $\leq n$ and that the basis if given by $v_i(t)=t^{i-1}.$ $T$ is defined as $T:\mathcal{V}_2 ...
0
votes
3answers
117 views

Domain/range of composition function

$$f: \mathbb{R} \to \mathbb{R}, f(x) = sin x. \\g: \mathbb{R} \to \mathbb{R}, g(x) = e^x.$$ Need domain/range of compositions. That's what I got, but I'm not 100%: $$f \circ g = sin(e^x)$$ Domain ...
1
vote
1answer
82 views

K-algebra isomorphic to a polynomial ring

I am trying to understand why this is true: Let K be a field, and let $K[a_1,\ldots,a_r]$ be a finitely generate $K$-algebra. If $a_1,\ldots, a_r$ are algebraically independent, then ...
1
vote
3answers
295 views

Prove or disprove that if one root of a quadratic equation is rational, then the other root must be rational as well.

I'm taking an introduction to discrete math course and I'm having some trouble with this homework problem. I think we're supposed to assume that the coefficients are integers based on other examples ...
1
vote
3answers
546 views

How to properly construct an $\epsilon-N$ proof

I've asked a couple of questions now on this type of proof. My question is: could someone give me a step-by-step set of steps to follow in the general case (e.g. for proof by induction, I'd say the 3 ...
4
votes
2answers
287 views

Do two distinct level sets determine a non-constant complex polynomial?

Let $f$ and $g$ be non-constant complex polynomials in one variable. Let $a\neq b$ be complex numbers and suppose $f^{-1}(a)=g^{-1}(a)$ and $f^{-1}(b)=g^{-1}(b)$. Does this imply $f=g$? If we think ...
28
votes
8answers
3k views

Active learning vs Passive learning in Math

I am trying to improve how I learn in general but specifically in math and a common suggestion I keep coming across is the difference between active learning and passive learning. The problem is, most ...
1
vote
0answers
50 views

Problems with the integration theorem of stokes.

i got problems with the next question: Let $w(x,y,z) = \left(zx+z^2y+x.z^3xy+y,z^4x^2\right)$ and let $P = P_1 \cup P_2$. $P_1 = \{(x,y,z) \in \mathbb{R}^3|x^2+y^2+(z-1)^2=1, z \geq 1 \}$ $P_2 = ...
3
votes
1answer
148 views

Prove that for every integer greater than 1. $\dbinom{n}{1}-2\dbinom{n}{2}+3\dbinom{n}{3}+…+(-1)^{n-1}n\dbinom{n}{n}=0$

How do you show the following. Prove that for every integer greater than 1. $\dbinom{n}{1}-2\dbinom{n}{2}+3\dbinom{n}{3}+.....+(-1)^{n-1}n\dbinom{n}{n}=0$ My idea is that is that the whole sequence ...
0
votes
1answer
168 views

Determine whether each of the following sets is well ordered?

A set is well ordered if every nonempty subset of this set has a least element. Determine whether each of the following sets is well ordered. a) the set of integers b) the set of integers greater ...
2
votes
2answers
111 views

How to prove from the definition that $X_n \xrightarrow{\mathbb{P}} X$ implies $\frac{1}{X_n} \xrightarrow{\mathbb{P}} \frac{1}{X}$?

Let $X, X_1, X_2, \ldots : \Omega \to (0,\infty)$ be random variables such that $X_n \xrightarrow{\mathbb{P}} X$. I'd like to show from the definition that $\frac{1}{X_n} \xrightarrow{\mathbb{P}} ...
5
votes
2answers
91 views

Is there a continuous injective map from $\mathbb{R}$ that has compact image?

Suppose I have a function $f: \mathbb{R} \to X$, where $X$ is some non-compact metric space. Is it possible that $f$ is injective yet has compact image? If the answer is yes, what characterizes ...
0
votes
1answer
97 views

Erdös-Rényi random graphs: Binomial approximation

I am working on extending the Erdös-Rényi paper "On the evolution of random graphs" (http://www.renyi.hu/~p_erdos/1960-10.pdf). In theorem 2a they calculate the number of isolated trees of order k. I ...
4
votes
3answers
99 views

limit of rational sequence

Let $a$ and $b$ be two real number. Assume that there exits two real sequences $a_n, b_n$ such that $$ \lim a_n=1, \lim b_n=1$$ and all $$\frac{a- a_n}{b-b_n} $$are rational numbers. Is it true ...
0
votes
1answer
31 views

Computing the probability of a collection summing above a given number.

Suppose I have a sequence of numbers $x_1, x_2,x_3...,x_{49}$ where $x_i \in [12..17]$. To compute something like the probability of getting the maximum value I can perform a calculation like ...
0
votes
1answer
43 views

Continuity of difficult vector function

I have a question about the continuity of a vector function. I have proved sufficiency for a function to be continuous but I am wondering if this is also a necessity. Assume $f(t,z(t))$, is a funtion ...
0
votes
1answer
66 views

On the isomorphism between bounded sesquilinear forms and bounded operators between two Hilbert spaces

Let $H$ and $K$ be two Hilbert spaces. Let $S(H,K)$ be the vector space of bounded sesquilinear forms $u:H\otimes \overline{K}\to\mathbb{C}$, and let $B(H,K)$ be bounded linear operators from $H$ to ...
5
votes
2answers
111 views

Integral $\int_0^4 \int_\sqrt{y}^2 y^2 {e}^{x^7} \operatorname d\!x \operatorname d\!y\,$

I have to evaluate this integral: $$ \int_0^4 \int_\sqrt{y}^2 y^2 {e}^{x^7} \operatorname d\!x \operatorname d\!y\, $$ I have no idea what to do with $\;{e}^{x^7}$. I have even tried $\int{e}^{x^7} ...
0
votes
0answers
24 views

How to shorten the list of enough conditions of a simple theorem?

Let $f$ be a symmetric binary relation. Let it's known that $f \subseteq \nu$ for a binary relation $\nu$. We need to prove $f \circ f \subseteq \nu$. For this it is obviously enough any one of the ...
1
vote
1answer
49 views

The most optimal way to solve this set of non-linear equations in high dimensions

So I have a series of non-linear equations which I wish to solve as fast as possible, to illustrate for the case of $n = 4$, I have the following equations: \begin{gather*} ...
2
votes
2answers
60 views

solution set of linear equation

Is there a system of linear equations with 3 variables and solution set $\{(a, b, c) | a^2 = b\}$? My answer is no but I'm not sure how to explain it. Edit: can't I just say that if ...
2
votes
2answers
94 views

Question about the logic behind hypothesis testing

Let us say that we have this following problem: "A government agency claims that more than 50% of US tax returns were filed electronically last year. A random sample of 150 tax returns for last year ...
1
vote
2answers
699 views

Discrete or Continuous Variables

I am having a hard time determining if these are either discrete or continuous. The number of visits to a website in a particular week. I said this was discrete because it is finite. The height ...
1
vote
0answers
32 views

Find all homogeneous ODE with continuous coefficients with a peculiar property.

My problem is to find all homogeneous ODE with continuous coefficients the such that for any solution $f(t)$ and any $c \in \Bbb R$, $f (t + c)$ is also a solution. Let ...
4
votes
1answer
66 views

Deciding to place a bet on outcome of a dice roll based on the probability

I have encountered several question of the following format. I have no trouble answering the first half but second half I have no clue on how to proceed. a: If you roll 5 standard six-sided dice, ...
1
vote
1answer
57 views

Buying the best hard drive, or: Confidence interval for difference between means, variance of ordinal data, etc.

I think my hard drive is on the fritz, and so I've been sorting through my options on Amazon. However, the ratings on Amazon don't really make distinctions as nicely as I'd like; the only readily ...
1
vote
1answer
96 views

Proof that a $\sigma$-field contains the sets $A \cap B$, $A \setminus B$, and $A \triangle B$.

I would like to get feedback on my proof. If it is not correct, I would appreciate it if you could only provide hints and not the full answer. Proof that $A \cap B \in \mathcal{F}$. $A,B \in ...
1
vote
3answers
120 views

How can this trig equation be simplified?

We have $9+40\sin^2x=-42\sin x\cos x$. I know this simplifies to $7\sin x+3\cos x=0$, but how?
3
votes
1answer
114 views

Algorithm to find solutions $(p,x,y)$ for the equation $p=x^2 + ny^2$.

As the classical book of David Cox argues, Assume the conditions are satisfied and $p$ can be represented as $x^2 + ny^2$. What would be a way to find solutions $(p,x,y)$ efficiently? Ideally, one ...
7
votes
2answers
19k views

U-substitution for integral of 1/(1+e^x)dx. What am I doing wrong?

Here is my work, witth the right answer. I feel like every step is right, but somehow I am getting the wrong answer. How? $$ \int \frac{1}{1+e^z}dz = \int\frac{1}{e^z(\frac{1}{e^z} + 1)}dz ...
0
votes
1answer
92 views

Differential equation, why use x instead of f(x)

I'm struggling with understanding a simple ODE. Let's say $x(t) = \exp(-t)$ so $x'(t) = -\exp(-t)$. When using Euler integration to solve the equation numerically, starting from $x_0 = x(t)$, we take ...
4
votes
1answer
2k views

Finding the max flow of an undirected graph with Ford-Fulkerson

Given the following undirected graph, how would I find the max-flow/min-cut? Now, I know that in order to solve this, I need to redraw the graph so that it is directed as shown below. However, ...
2
votes
1answer
293 views

Conclusion from trigonometric identity

Let $\alpha$ and $\beta$ be angles in triangle, i.e $\alpha, \beta \in \left(0,\pi\right)$ can we conclude that $\alpha = \beta$ if the following statement is true: $$\left(\frac{\sin \alpha}{\sin ...
1
vote
2answers
515 views

Use the Shell Method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis?

$y= 8x^2$, $y=8 \sqrt {x}$ I know the limits are $0$ and $8$. But do I change the function so it's "$x=$" or leave it as it is?
3
votes
1answer
650 views

Laplace operator and Fourier transform

Let $f$ be a function with compact support in $D\subset \mathbf{R}^2$. Is is known that $$-\Delta f=F^{-1}|x|^2 F f,$$ where $F$ is the Fourier transform. Which is the $n-$dimensional analogous ...
6
votes
1answer
114 views

A number of men enter a disreputable establishment

A number of men enter a disreputable establishment and each one leaves a coat and an umbrella at the door. When a message is received saying that the establishment is about to be raided by the police, ...
1
vote
0answers
36 views

Define a plane with double integral

For my transport physics I have to describe how much volume through a rectangular pipe goes. The speed is given by $\displaystyle V_x=V_0 \frac{y}{h}$. But I have to define the plane with a double ...
1
vote
2answers
70 views

Venn diagram, is C = (A ∩ B ∩ C)?

I just saw a Venn diagram that has: A = All integer numbers between 1 and 100 that is dividable by 2. B = All integer numbers between 1 and 100 that is dividable by 3. C = All integer numbers between ...
0
votes
1answer
91 views

Complexity of Recursively Inseparable Sets

I am interested in examples of recursively inseparable sets. A standard example is the set of positive integers encoding a Turing machine that halts in an odd number of steps on blank input versus ...
0
votes
2answers
140 views

Probability questions

A policy requiring all hospital employees to take lie detector tests may reduce losses due to theft, but some employees regard such tests as a violation of their rights. To gain some insight into ...
7
votes
1answer
174 views

The limit of sin(n!)

It is known that $\lim\limits_{n\to\infty}\sin n$ does not exist. $\lim\limits_{n\to\infty}\sin(n!)$ exists or not?
0
votes
2answers
411 views

Manifold that is NOT smooth

Could someone provide an example of a manifold that is not smooth? All manifolds that come to mind are smooth! By a manifold, I mean a hausdorff, second countable locally euclidean space.
3
votes
0answers
67 views

Is there a topos-like category that classifies regular subobjects?

A quasi-topos is a category that characterised by being finitely complete and finitely cocomplete that is also locally cartesian closed and has a strong sub-object classifier. A topos is finitely ...
0
votes
1answer
46 views

asymptotics of $F_n = F_{k-1} F_{n-k-1} + F_{k-1} F_{n-k} + F_{k-2} F_{n-k} $

Starting from $F_n = F_{k-1} F_{n-k-1} + F_{k-1} F_{n-k} + F_{k-2} F_{n-k} $ and letting $F_k \approx \phi^k$, I am hoping to find the corresponding statement for the Golden ratio: $\phi^n = 2 ...
0
votes
1answer
36 views

Can one find in this specific setting an extension of a given ring map?

All rings in this question are unitary and commutative and all maps are homomorphisms of commutative rings sending $1$ to $1$. Let $R$ and $S$ be regular local rings and let $$ ...
2
votes
1answer
100 views

Does the natural bijection between the set of prime ideals in A disjoint from S and Spec$(S^{-1}A)$ restrict to maximals?

I was studying rings of fractions, and I was wondering about the problem of restricting the canonical bijection (induced by retraction and extension of ideals) $\{p\in \text{Spec}(A) \mid p\cap ...
0
votes
3answers
380 views

Proving a set is a subset

Haven't done this for a long time, just want to know if this is the right method for a really simple example. Say we have two (obviously equal) sets $$A= \Big\{\begin{bmatrix}a & b\\c & ...
0
votes
0answers
162 views

Why is a graph $2$-connected if and only if it has an ear decomposition?

Why is a graph $2$-connected if and only if it has an ear decomposition? Reading the book "Introduction to Graph Theory" I have come across the following definition and statement: why is this ...
2
votes
1answer
303 views

How can I find a non-negative interpolation function?

In numerical mathematics I have learnt about some interpolation methods, however today I've come across some sort of interpolation problem which I don't know how to solve or even work with: Let ...
1
vote
0answers
67 views

Error Term of the Taylor series of cosh

I have the Taylor series of cosh $$\sum_{n=0}^\infty \frac {x^{2n}} {(2n)!}$$ and I know that this series converges for all x, but now I want to know if the series represents the function, in other ...

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