0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
3
votes
2answers
39 views

Centre of a group and normalizers

Let $G$ be a group and let $A \subset G$ be a non empty subset of $G$.Define the following subsets of $G$ $$Z(G) = \{z \in G \space | \space zx =xz \space \space \forall x \in G \}$$ $$N_G(A) = \{h ...
1
vote
0answers
15 views

Is this theorem equivalent to “existential instantiation” rule?

In Enderton's, There is a theorem called "existential instantiation", it says: Assume that the constant symbol $c$ does not occur in $\alpha ,\beta , \Gamma$ and that: $$ ...
0
votes
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 ...
-1
votes
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?
0
votes
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 ...
0
votes
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 : ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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: $a_1 = 2^1 + 1 = 2 + 1 = 3$ $\checkmark$ $a_2 = 2^2 + 1 = 4 + 1 = 5$ ...
3
votes
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 ...
1
vote
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 $K^m \times K^n$ ($\overline{x} \in K^m, A\subset K^n$). ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
3
votes
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
4
votes
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 ...
0
votes
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 ...
0
votes
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) ...
0
votes
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: ...
2
votes
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 ...
0
votes
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.
1
vote
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 ...
1
vote
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 ...
5
votes
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.
1
vote
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 ...
2
votes
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 ...
4
votes
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 ...
0
votes
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 ...
0
votes
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.
0
votes
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 ...
0
votes
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 ...
3
votes
1answer
35 views

How to indentify the power series I need to use?

Let $$ f(z) = \frac{1}{(z - 4)(z + 8i)} $$ a) Find the domains where f(z) is valid b) Find its power series at such domains Considering three singularities, I believe the domains are: $$ D_{1} = ...
1
vote
1answer
16 views

Martingale convergence without $L^p$ boundedness

I read the following example in the book titled "COUNTEREXAMPLES IN PROBABILITY AND REAL ANALYSIS" (by GARY L. WISE, and ERIC B. HALL): Does anyone know simpler examples? I do have one! I would be ...
9
votes
1answer
47 views

Existence of a sequence of positive continuous functions on $\mathbb R$ which is unbounded precisely on $\mathbb Q$

Is it possible to find a sequence of positive continuous functions $g_n$ on the real numbers such that $( g_n(x) )$ is unbounded if and only if $x \in \mathbb{Q}$ ?
0
votes
0answers
7 views

The distance of point from hyperboloid

I need to find the distance from the point $(0,0,0)$ to the hyperboloid $c=ax^2+by^2-z$. I have tried it with Lagrange multipliers and I have wierd outcomes.. I need help in this one.
2
votes
2answers
30 views

Polar Plots and square roots

When I plot a polar plot of $r=\sin (3 \theta)$, and $r=\sqrt{\sin (3 \theta)}$ I get nearly identical graphs, both $3$ pedal rose type plots. In the case without the square root, it is easy to ...
10
votes
3answers
302 views

Can Number Theory be visualized?

So I was thinking about a hard euclidean geometry problem, when it hit me just how much more difficult it would become without the aid of a diagram. This got me thinking: Wouldn't it be great if we ...
0
votes
1answer
14 views

Which Points are not Contained in the Line

The Circle $$x^2+y^2-4x=0$$ is cut by a line $AB$ at two points. If $A$,$B$ and two other points $C(1,0)$ and $D(0,1)$ are Concyclic, Then which of the Following points are not contained by the line. ...
0
votes
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?

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