0
votes
1answer
12 views

Steady-state solution

Obtain the steady-state solution of the problem $$\frac{\partial^2u}{\partial x^2}+\gamma^2(u-T)=\frac{1}{k}\frac{\partial u}{\partial t}, \ 0<x<a, \ t>0,$$ $$u(0,t)=T, \ u(a,t)=T, \ ...
0
votes
0answers
8 views

Is the conjugacy map between two distinct circle homeomorphisms unique?

Suppose $f,g,h_1,h_2$ are circle homeomorphisms with $f≠g$ and $fh_i = h_ig$ for $i=1,2$. Does it follow that $h_1 = h_2$? I restrict $f≠ g$ because I noticed that if $$f(x) := g(x) := R_\alpha(x) ...
0
votes
1answer
11 views

Orthonormal Basis and Matrix of a Linear Operator Proof.

Let B = $\{v_1, \dots, v_n\}$ be an orthonormal basis for Rn Let p = $[v_1, \dots, v_n$]. Prove that for any x, we have that the B-matrix of x is equal to the tranpose of P times x. I am unsure as to ...
0
votes
1answer
12 views

Maximal value of $\vert r^2-n\vert$ with a special condition

let $M,n\in \mathbb{N}$ and $R=\lbrace r\in \mathbb{N} \mid \vert r- \sqrt{n}\vert <M<2\sqrt{n}\rbrace $. I have to show that the maximal value of $\vert r^2-n\vert $ for $r\in R$ is at most ...
0
votes
1answer
9 views

Prove that this extension is Galois with galois group the quaternions

Let $M=\mathbb Q(\sqrt{2},\sqrt{3})$ and let $E=M(\sqrt{(\sqrt{2}+2)(\sqrt{3}+3)})$. Prove that: M is Galois over $\mathbb{Q}$ Show that $E$ is Galois over $\mathbb Q$ with Galois group the ...
0
votes
0answers
32 views

Determinant Question

This is not a homework question. Let $A$ be an $n\times{n}$ matrix with entries in the field $F$ such that each entry is relatively prime to any other entry. What is the number of elementary matrices ...
0
votes
0answers
15 views

Implicit Differentiation and Tangent Lines (sin and cos)

a. verify that the given point lies on the curve. b. determine an equation of the line tangent to the curve at the given point function : $\cos{(x-y)}+\sin y = \sqrt{2}$ a. I derived it using ...
1
vote
0answers
14 views

Chapter II Example 3.3 -p.28 Silverman

I have a question about Chapter II Example 3.3 -p.28 in Silverman "Arithmetic of Elliptic Curves". I feel like I'm misreading it and would like clarification. Let $K$ be a field such that ...
0
votes
0answers
17 views

Is the projection of two isomorphic direct sum of modules isomorphic?

If A and B are R-modules, and if A x A is isomorphic to B x B, does this imply that A is isomorphic to B?
0
votes
2answers
15 views

question about vector spaces and linear algebra

LEt $V = \{ (x,y) : x,y \in \mathbb{C} \} $. The addition and scalar multiplication are the usual ones. We know $V$ is a vector space over $\mathbb{C}$ with dimension $2$. My question: Why does $V$ is ...
2
votes
2answers
35 views

Area between $y = \frac{2}{1+x^2}$ and $y = |x|$

The exercise is from Anton 8th, page 443, question 15. The problem asks for the area between: $f(x) = \frac{2}{1+x^2}$ and $g(x) = |x|$ It is not said what interval the area should be calculated, so ...
0
votes
1answer
13 views

Intersection of Level Curves and a Ellipse at a given angle

I am preparing for an exam and I'm going over previous administered tests. I have come across the following problem and have little idea how to tackle it. It goes as follows: Let ...
-4
votes
0answers
25 views

SELF COMPLEMENTARY GRPAHS [on hold]

A simple graph is called self-complementary if it is isomorphic to its own complement. Let G be a simple graph on n vertices. Prove that if G is self-complementary, then either n = 4t or n = 4t + 1 ...
1
vote
1answer
20 views

Using recursion tree to solve recurence T(n) = 3(n/2)+n

I am trying to solve the recurrance of the function, T(n) = 3(n/2)+n where T(1) = 1 and show it's time complexity. n can be ...
1
vote
2answers
35 views

So why isn't $\Bbb R^n = \oplus _{n = 1}^{m}\Bbb R^n$

I thought that direct sum means each component of $V = \oplus U_i$ can be decomposed into elements of $U_i$. But if $U_i$ is replaced by the whole space, doesn't it mean the everything else in the ...
-1
votes
0answers
20 views

Let G be a group and let x be a fixed element of G. Define Γ(x) = {g ∈ G : gx = xg} [on hold]

(a) Prove that Γ(x) is a subgroup of G. (b) Let G = A4, let x = (1 3)(2 4) and let y = (2 4 3). Find (i) Γ(x) and (ii) Γ(y).
0
votes
0answers
18 views

A Contradiction of Riemann Zeta Residues

We can show (1+2+3...+n)^2 = 1^3 + 2^3 + ... +n^3, which holds for any finite n, shouldn't this imply Z(-1)^2 = Z(-3)? However, this does not hold if we look at the residues of the zeta function ...
1
vote
1answer
22 views

Translate the following sentence into conjunctive normal form

"Anyone who has cats as pets will not have mice": $$\forall x[\exists zHave(x,cat(z))]\rightarrow \forall y[\neg Have(x,mouse(y))]$$ I need to translate this into conjunctive normal form. So the ...
0
votes
0answers
13 views

Conditional Probabilities with Given Data

I posted this question the other day but I believe it got deleted? In any case, I also wasn't very specific with my notation and I apologize for that! Here is the question again. Given event A, B, ...
0
votes
1answer
13 views

Irreducibility of markov chains

Let $A=(a(x,y))_{x,y\in X}$ be a finite irreduzibel nonnegative matrix. Let $b,c >0$, and $\alpha=a^{(b+c)}(x,x)>0$. So $a^{(n(b+c))}(x,x)\ge \alpha^n$. And therefore $\lim ...
1
vote
2answers
13 views

Convergence of functions in a metric space

Let $C([0,1])$ be the space of all continuous functions from $[0,1]$ to $\mathbb{R}$ under the metric $$ \lVert f \rVert_1 = \int_0^1 \lvert f(x) \rvert \, dx. $$ Now consider $f_n(x) = e^{-nx}$. I ...
1
vote
2answers
17 views

How is it sometimes helpful to use cross multiplication in order to complete proportions with a variable?

How can it be helpful to do cross multiplication with proportions with variables such as ${2\over 4}={3\over x}$? In this one, the value of x has to be found. It can be found this way: 1. Do the ...
1
vote
2answers
40 views

Why exactly is this injective? Algebraic Topology.

Let $(A^*, d^i)$ be a chain complex of finite dimensional vector spaces, i.e, $$0 \to A^0 \to A^1 \to \dots \to A^n \to 0.$$ Show the sequence $$0 \to H^i(A^*) \to A^i/Im(d^{i-1}) \to Im(d^i) ...
1
vote
1answer
22 views

The number of ways to paint 3 cubes using 3 cans of paint, so that two cubes are blue

I have a question about the usage of the probability formula (I believe that is what it is called). So, I have 3 cubes and 3 differently colored cans of paint (Let's say, Red, Yellow and Blue). I am ...
1
vote
2answers
13 views

Probability Distribution Function?

An urn contains 8 green balls and 17 yellow balls. A ball is drawn from the urn and its color is noted and then the ball is placed back in the urn. 5 balls are drawn this way. Let $X$ denote the ...
0
votes
1answer
14 views

quadratic algebra word problem

Can someone please help me with this problem I have had an attempt at it but don't think i'm anywhere near right. A ship covers 480 miles at a uniform speed. If her speed had been 4 knots less, she ...
1
vote
0answers
10 views

$\|\nabla f\|_p\leq C (\|\nabla \times f \|_p +\|\nabla \cdot f\|_p)$

Let $f\colon\mathbb{R^3}\to \mathbb{R^3}$ have compact support. The identity $$ -\Delta = \nabla\times\nabla \times - \nabla \nabla \cdot, $$ and two integration by parts shows that $$ \|\nabla f\|_2 ...
0
votes
3answers
20 views

Proof by Induction: Series of binomial coefficients with same k-length subsets

I have no idea how to prove this binomial equation identity. For reference this is included in Discrete Mathematics for Computer Scientists by Clifford Stein, Robert Drysdale and Kenneth Boggart, ...
2
votes
1answer
26 views

“Evaluated at” or “at” notation

Normally a variable that is a function another variable would be represented as in the following fashion: $ V(t) $ (voltage as a function of time). However, my engineering professor (who also wrote ...
0
votes
3answers
41 views

catch fish problem

There are 20 fish in a lake, among them 5 are trouts. What is the probability that a fisherman have to catch more than 6 fish in order to obtain 3 trouts? Assume he keeps every fish he caught. How ...
1
vote
4answers
25 views

systems of linear equations intuition

I want to know why in a system of linear equations I'm allowed to sum or subtract the equations. I can't get the intuition of why I can do that to solve for the equations.
0
votes
1answer
25 views

Finding maximum number of solutions in a matrix

Given x+y+5z=2 x+2y+7z=1 2x−y+4z=a a) Determine the value of a which will make the given system have many solutions. Explain your answer. b) Choose a value of a which will make the given system ...
1
vote
3answers
29 views

How to find inverse laplace transform

$$ F(s) = \dfrac{6s+9}{s^2-10s+29} $$ How do you solve the inverse Laplace transform of this above equation?
2
votes
0answers
27 views

How would I solve this congruence?

What is the best way to solve this congruence: $r \cdot a^2 = b^3\bmod p$, where $p$ is prime in general? $r$ and $p$ are known, and I want to solve for $a$ and $b$.
0
votes
1answer
16 views

Negative binomial with conditional probability

Let X be a random variable that follows a negative binomial distribution: NB(r=4, p=0.4) Calculate P(X = 8 | x > 6) I know how to calculate P(X = 8): $$ \binom{7}{3} \cdot (1 - 0.4)^{7-3} \cdot ...
-1
votes
0answers
18 views

Number of chains having k subsets [on hold]

In a partition of the subsets of {1,2, ... ,n} into symmetric chains, how many chains have k subsets in them?
1
vote
0answers
13 views

Closure in a Hilbertspace

Define for a pure contraction $S$ (remember: $\|S\|\leq1$ and $\pm1\notin\sigma_p(S))$ the following set: $C_c^*(S):=\{g(S):g\in C_c(\hat{\sigma}(S))\}$ with $\hat{\sigma}(S)=\sigma(S)\cap(-1,1)$. Now ...
2
votes
0answers
46 views

The Set of All Integers is NOT a Variety; How Come?

My understanding is that a variety is, essentially, a set of common "zeros" of some given functions in the given ring. My professor told us that a finite set of integers form a variety; however, the ...
1
vote
0answers
4 views

Number of samples with replacement to reach expected coverage of population under non-uniform sampling

I am interested in finding the number of times $n$ I need to draw with replacement from a population of size $N$ such that the expected proportion of the population seen is at least $P$. From this ...
0
votes
0answers
17 views

T is not compact and orthonormal sequence

I want to show that if $\,T$ is not compact then there exists an orthonormal sequence $x_{n}$ and $R>0$ such that $ \forall n\in \mathbb{N}\,\,\,\,\|T(x_{n})\|\geq R$. It is obvious by the ...
0
votes
0answers
15 views

Problems about Linear Extensions

The following are exercises 57 and 58 from R. Stanley's Enumerative Combinatorics. I can't see to figure out how to explain an answer to 57, and I don't know where to begin with 58. $e(P)$ denotes the ...
1
vote
3answers
48 views

n! v.s. $a^{n}$ How can we know which one is faster without graphing?

n! v.s. $a^{n}$ If we are given an arbitrary number a (a>1). How can we know which one is faster as n->INFINITY without graphing?
-2
votes
0answers
17 views

Analyse existence and uniqueness for solutions of BVP

$$u''-u+λ \arctan (ux^2) = 0,\: 0≤x≤1$$ $u(0) = 0,\: u(1) = 1, \:u ∈C^2([0,1])$ Here $λ$ is a real number and all functions are real-valued. What can be said for different values of $λ$?
14
votes
0answers
80 views

What did mathematicians study as an undergraduate/graduate before modern mathematics such as modern algebra and analysis?

I am curious as to what mathematicians such as Leibnitz and Gauss and the Bernoulli's studied when they were students in university. I find it fascinating how we are taught calculus and abstract ...
3
votes
3answers
44 views

What is $\bigcup_{n=1}^{\infty}[0,1-\frac{1}{n}]$?

I often read that: $\bigcup_{n=1}^{\infty}[0,1-\frac{1}{n}]=[0,1)$. But why? My intuition would say that the result would be $[0,1]$ because $\lim_{n\rightarrow \infty}[0,1-\frac{1}{n}]=[0,1]$
1
vote
2answers
24 views

Rudin's Chapter 3: Numerical sequences and series

In the Rudin's Principles of Mathematical Analysis 3rd edition, Chapter 3, page 56, there is a definition of a set $E$ that, in my point of view, is very doubtful. What is the set $E$? I couldn't also ...
6
votes
0answers
22 views

Question on the paper Donal F. Connon, “Some integrals involving the Stieltjes constants”

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
-1
votes
0answers
31 views

Sup and inf of bounded sequence [on hold]

Let $(x_n)$ be a bounded sequence, let $L$ be the set of all accumulation points of $(x_n)$, and $S=\inf{\{\sup{\{x_k:k\geq n\}}\}}$, prove that $S=\sup{L}$
0
votes
1answer
18 views

Finding permutation $a$ given $b$ and conjugate $a^b$

Normally we define a conjugate relationship as $$a^b = b~a~b^{-1}$$ But I don't know how to find $a$ given that we know $b$ and $a^b$.
1
vote
0answers
30 views

Big O-notation proof: show that $x^{2}+5x+11$ is $O(x^{3})$

Show that $x^{2}+5x+11$ is $O(x^{3})$ by providing the smallest value of the witness $C$ such that $|f(x)|≤C|g(x)|$ whenever $x>11$. What's the value of $C$?

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