# All Questions

1answer
40 views

### Precalculus - connect 2 towns

A state highway department plans to construct a new road between towns $A$ and $B$. Town $A$ lies on an abandoned road that runs east-west. Town $B$ is $20$ miles north of the point on that road that ...
0answers
18 views

### Quadratic form - non-degenerate

(The order of a quadratic form is defined to be the order of the matrix $A$) Definition: $Q(x_1, x_2, \dots , x_n)$ is called non-degenerate $\Leftrightarrow (a)$A=$invertible (b) At each$v \in ...
1answer
29 views

### How to simplify logs and powers?

Is there any way to simplify $(\log a)^{\log b} = c$? And even this $(\log x)^y = z$? And also this $(\log m)(\log n) = p$ (which is essentially $\log m^{\log n} = p$) I was trying to simplify some ...
1answer
19 views

### Connectedness, continuous functions, and the intermediate value theorem

I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never equals s, it follows that f(p) < s for all p or f(p) > s for all s. So here's what I know so ...
2answers
39 views

2answers
23 views

### What does it really mean by a derivative in a sense of something per unit.

Suppose we are given the differential equation $\frac{dP(t)}{dt}=kP$ where $P(t)$ is a function of population with variable time measured in years. And say $k>0$ is the relative growth rate of the ...
0answers
37 views

### Given automatic equation solvers exist, should one know how to solve equations by hand? [on hold]

Automatic equation solvers seem much faster and less error-prone. If one should learn how to solve them manually, why? If one shouldn't, why is doing so still taught?
1answer
18 views

### Find the Domain and Sketch the Graph of the Function $h(x)= \frac{3x+|x|}{x}$

$(3x+|x|) /( x)$ $|x| = (3x+|x|) /( x) \,\,\,if\;\; x >0$ $(3x+|-x|) /( x) \,\,\, if\;\; x < 0$ I am confused as to whether the $-x$ replaces $x$ for all $x$'s for the lower part of the ...
0answers
5 views

### Natural Cubic Spline Unequal Spacing

I'm currently trying to create a natural cubic spline by setting up linear equations to solve for the coefficients of the spline. Where the spline follows as : ...
0answers
25 views

### Need help determining coefficients of a cubic equation. [on hold]

I posted this in a LinkedIn group and now I put it here as per the suggestion of a group member. I have a colleague trying to finish his thesis so he can graduate. He's stuck at solving the ...
1answer
11 views

### equation system: solve $L_{1,1}$

a and b is two given constants, let L_(a,b) denote the system x1 2x2 3x3 = 0 2x1 4x2 ax3 = 0 3x1 bx2 9x3 = ab i) solve $L_{1,1}$ ii) find the pairs ...
1answer
22 views

### Why is Poisson's equation useful?

Ever since joining SE, I have heard many people mention Poisson's equation and the Laplacian. I have also started to encounter these terms more in resources I have been directed to. I am consumed ...
0answers
11 views

### why are ill conditioned system of equation hard to solve iteratively (intuition)

Is there some intuition as to why ill conditioned system of equations hard to solve iteratively ( i.e. the convergence is slow) ? I've read convergence proofs of several methods, but still don't have ...
0answers
25 views

### Determine all t ∈ ℝ for which At is diagonal

so I have this matrix: At= 2+t 4 2+t 2+t t-2 0 -6+t -2-t -t+2 -4 -t+2 -2-t 0 0 0 2t I must determine all t ∈ ℝ for which At is ...
1answer
15 views

### Question about asymmetry of chi-square distribution

Let $X_1,\dots,X_n$ be a set of i.i.d. chi-square random variables with $k$ degrees of freedom. Consider the statistic $\arg\max_i\{|X_i/k - 1|\}=X_{\alpha}$. I wonder about the probability that ...
2answers
20 views

### Prove that a recurrence relation (containing two recurrences) equals a given closed-form formula.

Prove that $a_n = 3a_{n-1} - 2a_{n-2} = 2^n + 1$ , for all $n \in \mathbb{N}$ , and $a_1 = 3$ , $a_2 = 5$ , and $n \geq 3$ Basis: $a_1 = 2^1 + 1 = 2 + 1 = 3$ $\checkmark$ $a_2 = 2^2 + 1 = 4 + 1 = 5$ ...
0answers
9 views

### Sequence of non-collinear integer points.

This is a question from a British Olympiad, I've completed the first 3 but this one had me rather stumped. Given two points $P$ and $Q$ with integer coordinates, we say that $P$ sees $Q$ if the ...
0answers
12 views

### Closed slice of affine space has closed projection

Let K be an algebraically closed field and suppose a set $\{ \overline{x} \} \times A$ is closed in the Zariski topology of the affine space $K^m \times K^n$ ($\overline{x} \in K^m, A\subset K^n$). ...
0answers
6 views

### Shift operators and c0 semigroups

I am asked where the bounded continuous functions on $\mathbb{R}$ (with sup norm) with the right shift operators ( $T_t(f)(x)=f(x-t)$ ) form a c0 semigroup. I believe the answer is no, but I am ...
1answer
22 views

### If $\phi$ is a $\Sigma_2$ sentence and $H_{\kappa} \models \phi$, then $V \models \phi$?

In the title question, $\kappa$ is any infinite cardinal. It's easy to see that the result is true if $\phi$ is $\Delta_0$ or $\Sigma_1$. I first tried proving the result for $\Pi_1$, but I don't see ...
0answers
13 views

### When is a total $F:\omega^\omega\rightarrow\omega^\omega$ said to be recursive?

Let $F:\omega^\omega\rightarrow\omega^\omega$ be a total function. According to definitions given by Sacks (Higher Recursion Theory) and Rogers (Theory of Recursive Functions) regarding recursive ...
0answers
35 views

### Bachmann's construction of the real numbers

On page 44 of this book an approach to constructing the real numbers as equivalence classes of nested rational intervals is outlined and attributed to Bachmann. The outline in the book is very ...
2answers
23 views

### Finding a good comparison/bound for determining the convergence of a series

The series is defined as follows: $b_0=1,b_1=-7,b_k=2b_{k-1}+b_{k-2}$. I need to find a good comparison sequence to determine whether $\sum_{k\geq0}1/b_k$ converges. I considered using $1/k^2$, which ...
1answer
9 views

### Closed linear operator

I am having some trouble in showing the following map is closed: For $f\in L^2(\mathbb{R^2})$ with $(x+iy)f(x,y)\in L^2(\mathbb{R^2})$, $M(f)(x,y)=(x+iy)f(x,y)$. I am also asked to find the ...
1answer
14 views

### Cubic: Finding turning point when given x and y intercepts

I have tried substituting in the two points (-4,0) and (0,28) and solving simultaneously for b and c with no success, and the book gives two separate but equally correct solutions for b and c that ...
2answers
41 views

### Is simple module over commutative ring always a field?

M is a simple module if and only if $M\cong R/I$ for some I maximal ideal in R. If $R$ is commutative, can I say M is a field? I'm confused about this fact because when proving it I use the fact that ...
1answer
75 views

### Do every math operation derive from sum?

I've been told sometimes, that every math operation (sum, subtraction, exponentiation, square rooting, so on) can be transformed to a sum of operands. For example, subtraction can be made as ...
0answers
7 views

### Difficulty in understanding a solution: Constraint minimization of sum of Non-symmetric matrices

I am trying to understand why there is significance difference in the performance of two proposed solutions. Original question (Constraint minimization of sum of Non-symmetric matrices) ...
0answers
19 views

### Lie Groups: Infinitesimal Operations

Given a Lie group. At the identity multiplication acts infinitesimally by: $$\mathrm{d}_{(e,e)}\mu:\mathrm{T}_{(e,e)}(G\times G)\to T_eG:(u,v)\mapsto u+v$$ This exploits the identification: ...
0answers
22 views

### An algorithm to find isometry between surfaces in $\mathbb{R}^3$?

Given two surfaces in $R^3$, i would like to find isometry between these two. Usually, in class, we did some examples, like bending the plane into a cylinder, or cone, and they were not hard, quite ...
1answer
8 views

### Number of Binary Operations(Compositions) with a specific neutral element

Let $E$ be a finite set of $n$ elements. If $a\in \mathbb{E}$, for how many compositions on $\mathbb{E}$ is is $a$ the neutral element.
1answer
39 views

### Is the way I simplify my notation?

Do you agree with how I simplify this equation notation $$\sum_{i=1}^n \phi_{v_i} \left(\sum_{i=1}^m \phi_{l_i}\right)=\sum_{(i,j)\in \mathcal{S}}\phi_{v_i}\phi_{l_j},$$ where I define ...
0answers
54 views

### Exercise 3.4 in Rotman's An Introduction to Algebraic Topology

I am self-learning algebraic topology by reading Rotman's An Introduction to Algebraic Topology. I am stuck on Exercise 3.4 on page 41. I'd be grateful for any hints or solution. Exercise 3.4: Let ...
4answers
113 views

### Can someone explain why $x^{\log(a)} = a^{\log(x)}$?

I'm trying to see why the below is true. $$x^{\log(a)} = a^{\log(x)}$$ Anyone here know why this is? Thank you.
0answers
18 views

### Open set in $\mathbb{R}$ as countable union of open intervals and which version of Choice [duplicate]

In proving, every non-empty open set in $\mathbb{R}$ is union of a countable collection of disjoint open intervals in $\mathbb{R}$. It seems to me this result is using some version of Choice(probably ...
3answers
39 views

### Determine the integers $a$ such that the congruence $x^4 \equiv a \pmod p$ has a solution for $p = 7, 11, 13$

Determine the integers $a$ such that the congruence $x^4 \equiv a \pmod p$ has a solution for $p = 7, 11, 13$ This looks similar to previous problem but kinda tricky. I'm not sure where to ...
1answer
161 views

### Pythagorean triples.

We are given $$a^2 + p^2 = b^2$$ where $a,b\in\mathbb{Z}$ and $p$ is prime. We are to show that $$2(a+p+1)$$ is a perfect square. Is there any elegant ways to go about this problem? Struggling to ...
1answer
21 views

### proof detail concerning bijection between a set and its power set

Theorem: If $X$ is a set, then $X$ is not equivalent to its power set. Proof: suppose for a contradiction that $f:X\to P(X)$ is a bijection. Define $B:=\{x \in X, x\not\in f(x)\}$. Because $f$ is ...
0answers
31 views

### Help solving the following equation system:

Let $p, q$ be any two integers, and $a, s, t, n \in\mathbb{Z}$. How do I solve the following system for $s, t$: $$p - q^{n} = ta(n + 1)$$ $$p - (n + 1) a^{n} = sq$$ Please help.
1answer
32 views

### Order of a subgroup formula

So I need to proof For cyclic group G of order n with a generator g, then for $x = g^m \in G$ we have |x| = $n/gcd(n,m)$. However I seem to have something wrong in my proof. Suppose gcd(n,m) = d I ...
0answers
17 views

### Do these claims imply each other?

$T$: A set of natural numbers. $C_1: 2$ is the only prime number that divides elements of $T$. $C_2 :$ If $i, j \in T$, and $i < j$, then $i$ divides $j$. For $C_1 \rightarrow C_2$, I think ...
1answer
35 views

0answers
6 views

### Transform Confocal Ellipsodal to Spherical Coordinates

I am having trouble transforming confocal ellipsoidal coordinates to spherical coordinates. How does one perform such a transform?

15 30 50 per page