0
votes
1answer
17 views

Baby version of Sturm Comparison Theorem

In Problem 15-32 of Spivak's Calculus, 4th edition, he proves the following: Suppose $\phi_1$ and $\phi_2$ satisfy $$\phi_1''+g_1\phi_1=0, \\ \phi_2''+g_2\phi_2 = 0,\\[10pt] g_2>g_1, \\[10pt] ...
1
vote
0answers
37 views

Counting the roots of a polynomial over a finite field

Let $\mathbb{F}_{11}$ be the field of 11 elements and let $\mathcal{K}$ be the splitting field of $x^{3} - 1$ over $\mathbb{F}_{11}$. How many roots does $(x^{2} - 3)(x^{3} - 3)$ have in ...
-2
votes
0answers
11 views

correlation coefficient between ordinal and binary variables spss or stata [on hold]

First of all, I know only the basics regarding correlation. I have two variables, the one is binary (Yes/No) and the other is likert scale (1=not at all, 2, 3, 4, 5=very much). I want to find the ...
0
votes
1answer
37 views

Finding the cardinality of $\{ X \subseteq \mathcal{P}(A) : |X| \leqslant 1 \}$.

Given that $|A| = m$, my task is to find the cardinality of the set $Q = \{ X \subseteq \mathcal{P}(A) : |X| \leqslant 1 \}$. Since this is an even-numbered exercise in the text I'm working through, ...
1
vote
1answer
27 views

Showing the image of $H^j(X;\mathbb C^\times)$ lies in the torsion subgroup of $H^{j+1}(X;\mathbb Z)$

Let $X$ be a (compact, if necessary) topological space. Then from the short exact sequence of constant sheaves $$ 0 \to \mathbb Z \to \mathbb C \to \mathbb C^\times \to 0 $$ we have a connecting ...
1
vote
0answers
10 views

Deriving an estimate in regularity theory of the heat equation

I have another question from PDE Evans 2nd edition, this time from pages 380-381. It's about a step in the formal derivation of estimates. Given the initial-value problem for the heat equation ...
2
votes
1answer
33 views

Calculate the area of the ellipsoid that rotates around the $x$-axis

So we are about to calculate the area of the ellipsoid around the $x$-axis. $$ \frac {x^{2}}{2}+y^{2} = 1 \implies x=\sqrt{2-y^{2}}$$ We are squaring it so the sign shouldn't matter. I was ...
0
votes
2answers
38 views

What is the difference quotient for the function? [on hold]

The difference quotient of a function $f(x)$ is defined as $$\frac{f(x+h)-f(x)}{h}.$$ Determine the difference quotient for the function $f(x)= \frac{4}{x}$, in its most simplified form.
2
votes
1answer
64 views

Why is the powerset axiom more acceptable than the axiom of choice?

The key step in Zermelo's proof of the well ordering theorem is to use $\text{AC}$ to simultaneously choose the next elelment for all possible partial chains in prospective well orderings, but that ...
0
votes
1answer
64 views

Why is the expected average displacement of a random walk of N steps not $\sqrt N$?

Let $D_N$ be the expected average of the displacement of a random walk on $\mathbb Z$ from the origin, where $N$ is the number of steps, each of which is either $-1$ or $1$. We take the definition of ...
1
vote
2answers
20 views

gcd and lcm from prime factorization proof [on hold]

How should I approach obvious proofs like these I have been trying but couldn't find an elegant way to work these. Any help is highly appreciated ! Especially looking for a nice proof/hint for ...
0
votes
1answer
33 views

Why does the $\tan$ reduction formula have a restriction?

My book says the reduction formula is only valid for an integer $n > 1$. Why? This derivation doesn't require $n$ to be an integer or greater than $1$.
0
votes
2answers
30 views

Finding the partial derivatives of $V (x, y) = U (x, y)e^{−ax−by}$

I think I did something wrong, so I was hoping someone might be able to show me the solution Two functions $V (x, y)$ and $U (x, y)$ are connected by the equation $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
1
vote
1answer
54 views

Are there at least denumerably many distinct group operations on any denumerable set?

I'm working on a proof of the following statement: For any denumerable set $D$, there exist at least denumerably many distinct group operations on $D$. My argument is looking fairly messy, so I'm ...
3
votes
3answers
55 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
1
vote
0answers
35 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [on hold]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
4
votes
4answers
40 views

Basic understanding of Log and $2 \log _3(x)+\log _9(x)=10$

So this is what I have done so fare $$2 \log _3(x)+\log _9(x)=10$$ I know that $$\log _9(x)=\log _3\left(\sqrt{x}\right)$$ I therefore have $$\log _3\left(x^{5/2}\right)=10$$ However here is ...
1
vote
2answers
50 views

Questions regarding Cantors' Theorem

The proof of Cantor's Theorem in the Wikipedia Article goes like this: Two sets are equinumerous (have the same cardinality) if and only if there exists a one-to-one correspondence between them. ...
1
vote
3answers
20 views

refactoring binomial with negative power

I am reading Calculus Made Easy where in Chapter IV: $$(x+dx)^{-2}$$ Is refactored as: $$x^{-2}\left(1+\frac{dx}x\right)^{-2}$$ Could someone give me an insight into this refactoring? I can see from ...
5
votes
1answer
39 views

Geometric interpretation of complex path integral

Let's say that we want to make sense of integrating a function $f: \mathbb{C}\rightarrow\mathbb{C}$ over some path $\gamma$. I can imagine two reasonable ways of doing it. First, there's the way ...
0
votes
1answer
44 views

Why this order is well-defined?

I'm trying to understand why this order defined in Fulton's algebraic curves book is indeed well-defined: I have two questions: Why $u\in \mathcal{O_P}(X)$ (he is implicitly assuming this fact so ...
1
vote
0answers
21 views

A technical step in the proof of Atiyah-Bott fixed point formula

From John Roe, Elliptic operators, topology and asymptotic methods, page 135. Here Roe tried to prove that $$ \DeclareMathOperator{\Tr}{Tr} \sum_{q}(-1)^{q} ...
0
votes
1answer
26 views

Solid Angle Integration

Can somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a ...
3
votes
0answers
29 views

Rank of the product of 3 matrices

Suppose I have 3 n by n matrices $A,B,C$ with $ABC=0$, what could be the maximal rank of $CBA$? I guess the answer would be n but I failed to prove it( tried to use Rank-Nuillity Theorem but I don't ...
3
votes
0answers
25 views

Simplify exponential sum over $\mathbb{F}_p$ to prove identity

I have a sum of powers of prime ($p>3$) roots of unity (where $\frac{1}{t}$ is to be understood as the field inverse $t^{-1} \bmod p$ etc.) $\begin{align*} &d_{j,k}=\sum_{a,b,c \in ...
1
vote
0answers
26 views

Formally evaluating integral to calculate electric or gravitational field.

I never understood how such integrals are calculated, formally. In a line is easy, just a line integral. In a surface, sometimes is easy, like in a disc. But, some surfaces, like sphere, it gets ...
0
votes
1answer
19 views

Does anybody know about an ebook version of Saaty The Analytic Hierarchy Process?

I'm looking for Thomas L. Saaty The Analytic Hierarchy Process in pdf, but I only found hardcover versions to order.
3
votes
1answer
30 views

How do I find the relative extrema of a function in spherical coordinates?

I want to find the relative extrema for the following function. $f(\theta,\phi)=AR\cos\theta\sin\phi + BR\sin\theta\sin\phi + CR\cos\phi $ $A,B,C,R$ are constants In a function $g(x,y)$ using ...
1
vote
1answer
33 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
0
votes
1answer
29 views

Rearranging $sk^{0.5} - \delta k = 0$ to isolate $k$

I'm going through an examination and the answer booklet and it gives the answer simply as $$sk^{0.5} - \delta k = 0$$ rearrange to $$k = (s/\delta)^2$$ Could anyone explain how to get to this ...
-2
votes
0answers
9 views

Does $\sum_B p(L|B)p(B|G) = \sum_B p(L,B |G) = p(L|G)?$ [on hold]

Does $\sum_B p(L|B)p(B|G) = \sum_B p(L,B |G) = p(L|G)?$ By chain rule, does $p(L,B |G) = p(L|B,G)p(B|G)$? Does $\sum_B p(L|B)p(B|G) = \sum_B p(L|B,G)p(B|G)$
1
vote
1answer
37 views

A continuous bijection from a Hausdorff space to a non-compact space which is not a homeomorphism

Recall the following theorem: Let $X$ be a compact space and $Y$ a Hausdorff space. Suppose that $f:X \rightarrow Y$ is a continuous bijection. Then f is homeomorphism. Prove that the ...
0
votes
2answers
25 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
2
votes
3answers
28 views

Finding the change of variables to transform $u_{tt} - u_{xx} = 0$ into $u_{rs} = 0$

I'm just beginning to introduce myself to partial differential equations and one of the first problems presented in the textbook I have literally no idea how to do. I think the author intended the ...
3
votes
0answers
41 views

Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
0
votes
1answer
16 views

Is this symmetric, block-diagonal matrix positive semi-definite?

I have a matrix of the following form, where $a,b,c>0$ \begin{align*} A = \left[ \begin{array}{cccccc} aM_{12}^2 & aM_{12}M_{13} & 0 & 0 & 0 & 0 & 0 \\ aM_{13}M_{12} ...
1
vote
2answers
46 views

If $\operatorname{rank}(A)=n$ then $\operatorname{rank}(AB)=\operatorname{rank}(B)$

I have looked here, but still I cannot understand how to get to equality. Let assume that the matrices are squared $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ is easy to show, but how can I ...
0
votes
0answers
32 views

Uniform convergence and Weakly convergence

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , ...
-2
votes
0answers
38 views

Kahler condition [on hold]

Let $(M,\omega) $be a Kahler manifold. Why is the Kahler condition $$d \omega = 0$$equivalent to: $$\partial_kg_{i\bar j} = \partial_ ig_{k\bar j}$$ for all $i; j; k$?
1
vote
2answers
38 views

How can I show that this space is not locally compact but normal?

Let $E$ be the set of all ordered pairs $(m,n)$ of non-negative integers. Topologize $E$ as follows: For a point $(m,n)\neq (0,0)$ any set containing $(m,n)$ is a neighbourhood of $(m,n)$. A set $U$ ...
3
votes
0answers
17 views

Integral form of Bessel's (n = 0)

While studying for a comprehensive exam, I have come across this old problem: Consider the Helmholtz equation in the $\mathbb{R}^2$ plane $$u_{xx} + u_{yy} + \omega^2u = 0$$ Derive an ...
0
votes
0answers
32 views

How many Geese were there before any flew away? [on hold]

This equation represents geese flying away in one hour intervals. How many geese were there before any flew away? The first part x - [1/5 = x] represents the quantity of geese flying away at 1:00pm ...
4
votes
1answer
55 views

Characterization of measurability by closed sets.

If $E \subseteq \Bbb R$ is measurable, then for all $\epsilon > 0$, exists $F \subseteq \Bbb R$ closed such that $F \subseteq E$ and ${\frak m}^*(E \setminus F) < \epsilon$. I have already ...
1
vote
2answers
45 views

Finding a condition on the real $a$ such that $P$ is divisible by $(x-a)^2$

Let $P(x)=\frac{x^3}{6}+\frac{x^2}{2}+x+1$. I have to find a condition on the real $a$ such that $P$ is divisible by $(x-a)^2$. I tried to use Polynomial long division and solve a system (we need ...
2
votes
3answers
50 views

Why is $ A_1 x + … + A_n x^n $ a solution of $ \sum_0^{n} (-1)^n \frac{x^n}{n!} \frac{d^n y}{d x^n} = 0 $?

I was playing(/fiddling) around with some maths and I saw this pattern( where $ A_n $ is a constant.): $ A_1 x $ is a soultion of: $$ \frac{y}{x} - \frac{dy}{dx} = 0 $$ $ A_1 x + A_2 x^2 $ is a ...
2
votes
2answers
44 views

What is the simplest way to understand $\nabla \cdot (\nabla \times \textbf{A} ) =0$?

Of course one can prove it by brute force. But how to Understand it, in the simplest way? Another issue is $\nabla \times (\nabla f) =0 $.
2
votes
3answers
52 views

Irreduciblility of $x^3 + 9x + 6 $ in $\mathbb{Q}[x]$

I am trying to prove the irreducibility of $x^3 + 9x + 6 $ in $\mathbb{Q}[x]$ without using Eisenstein's criterion. What I have done is -- Let assume it is reducible in $\mathbb{Q}[x]$, then it can ...
1
vote
4answers
59 views

Does $(m+1) + m2 + (m - 1)2^{2} \ldots + 2^{m}$ equal something simpler?

Does $(m + 1) + m2 + (m - 1)2^{2} \ldots + 2^{m}$ equal something simpler, where $m\in \mathbb{N}$? Excuse me if it is too simple, I am bit tired. Thanks.
1
vote
5answers
29 views

If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c

I'm trying to prove that if $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c. I tried using Bezout identity but couldn't get anywhere.
1
vote
2answers
35 views

$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group

Let $G$ be a finite abelian group. Prove that $(a_1 a_2 \cdots a_n)^2=e$. My proof: $$\forall a \in \{a_1,a_2,\cdots,a_n\} \exists ! a^{-1} \in \{a_1,a_2,\cdots,a_n\}:a^{-1}a=aa^{-1}=e$$ hence, ...

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