0
votes
0answers
30 views

Question about proving a real number

If we know that for any, $\alpha \in \{0, 2\}^\mathbb{N}$ that $0 \le \sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \le 3$, then what property of real numbers do we have to use to prove that ...
7
votes
1answer
64 views

Applications of Geometry to Computer Graphics

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of a friend of mine has only a bachelors degree in pure math and got ...
1
vote
1answer
20 views

Is a ruled surface of degree>2 always singular?

Let $X=\mathbb{C}\mathbb{P}^3$ and let $V\subset X$ be a closed algebraic sub variety. By V-is ruled, I mean that for every point in $V$ there is a line passing through it which also lies in V. ...
1
vote
1answer
26 views

Strong epimorphic counit iff conservative right adjoint?

On page 13 of Lack and Street's Combinatorial Categorical Equivalences, it is written (but not proven) that: A right adjoint is conservative if and only if the components of the counit are strong ...
5
votes
1answer
57 views

Almost pointwise inner automorphism of free products of groups.

Let $A,B$ be groups, let $G = A\ast B$ be their free product and let $\phi \in \text{Aut}(G)$ be a automorphism of $G$. We say that $\phi$ is pointwise inner if $\phi(g) \sim_G g$ (there is $w \in G$ ...
3
votes
3answers
455 views

How to find the minimum value of the expression?

Let $x$, $y$, $z$ be three nonegative real numbers and $x^2 + y^2 + z^2 = 5.$ Find the minimum of the expression $$E=\dfrac{1}{2}(x^2 y^2 + y^2 z^2 + z^2 x^2) + \dfrac{96}{x + y + z + 1}.$$ What ...
2
votes
0answers
18 views

Composing a smooth even function and square root

Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $f(-x)=f(x)$ for all x. Define $g:[0,\infty)\to\mathbb{R}$ by $g(x)=f(\sqrt{x})$. Is $g$ necessarily smooth at $0$? I guess the answer is ...
4
votes
1answer
51 views

Decision and the Uncountable Spectrum

In 2000, Hart, Hrushovski, and Laskowski classified all complete first order theories in a countable language up to their uncountable spectra. However, does this also imply that given a $any$ ...
0
votes
1answer
20 views

Calculate the supremum of $\frac{e^{|\gamma_n+1|}-1}{|\gamma_n+1|}$

If $\{\gamma_n\}$ is a sequence of real number and $\exists M>0$, finite, such that $|\gamma_n|\leq M$, find the supremum of the following sequence: $$\frac{e^{|\gamma_n+1|}-1}{|\gamma_n+1|}$$
2
votes
1answer
68 views

Probability Question: Who's right, me or the book?

I'll be giving some classes on probability theory later this year, and so I've been going through the textbook to check that I'm up to speed. I came across the following question: The discrete random ...
0
votes
2answers
18 views

Conditional joint probabilities with CDFs

Let X and Y be independent continuous random variables with marginal CDFs given by $F_X(x) =\begin{cases} 0 & \text{for }x < 0\\ x/3 & \text{for }0 \leq x \leq 3\\ 1 & \text{for }x ...
1
vote
1answer
38 views

Find the equation of the linear transformation of an orthogonal projection on the line y=mx.

Let $T : \mathbb R^2 → \mathbb R^2$ the orthogonal projection on the line $y = mx$. Prove that for all $a, b \in \mathbb R$, $$\begin{align}T((a,b)) = {\frac{1}{m^2 + 1}}(a+mb, ma + ...
3
votes
1answer
38 views

Orientability of a product of smooth manifolds implies orientability of each factor

I've been learning a bit about orientability on smooth manifolds. I'm having torubles with this exercise: Given two smooth manifolds $M$ and $N$, show that the product manifold $M \times N$ is ...
2
votes
1answer
29 views

Does differentiability imply absolute continuity? [duplicate]

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a function which is (i) differentiable at all $x \in (a,b)$ (ii) the right-derivative at $x=a$ exists and the left-derivative at $x=b$ exists. Does it ...
1
vote
0answers
8 views

Basis representation for non-negative, compact support, reasonably smooth spectral function

I was wondering if anyone has ideas on representing a non-negative, compact support (from x=-1 to 1 on the real axis) spectral function as a superposition of basis elements. Ideally, the basis ...
1
vote
0answers
19 views

How do I show the probability of failure in my program? (using a lock-free stack)

I've got a question about code I'm writing here, and am looking for a way to see into the math of what's happening. Thought you guys could help :) I'll see what I can do about describing the ...
1
vote
3answers
33 views

Transformation of two independent uniform random variables

Suppose $X,Y \sim \text{Uniform} \left(0,1 \right)$ are independent. Then I need to find the PDF for $W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ...
5
votes
1answer
466 views

Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11

Let $K$ be the rational function field $k (x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u$ in $K$, and write $u = \frac {f(x)}{g(x)}$ with $f$ and $g$ ...
0
votes
2answers
26 views

Question about partial sum

I'm confused on the partial sums formula Why is $$\sum_{i=m+1}^\infty \frac{2}{3^i}=\frac{1}{3^m},$$ if $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$
2
votes
0answers
27 views

What do rank-2 tensor entries represent? What's their geometrical meaning?

I just followed a course on tensor analysis and I think I understood almost everything. Except the following: given a vector (with, say, 3 dimensions), if I multiply that with a second-rank tensor, I ...
-1
votes
1answer
36 views

Find the volume generated! [on hold]

Find the volume generated by revolving the curve bound by $y=4-x^2$, the $y$-axis and $x$-axis about the $x$-axis using the disk method.
0
votes
0answers
39 views

Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
4
votes
3answers
229 views

Example of space that is separable but not a Hausdorff space.

Give an example of a topological space that is separable but not a Hausdorff space. I have not been able to discover an example, I thought of the Arens-Fort space because this is separable, ...
0
votes
1answer
23 views

How to get polarised electromagnetic TE wave differential equation from Maxwell's Equations?

I wish to understand how the following equation: $\frac{\partial^2 E_x}{\partial y^2} + \frac{\partial^2 E_x}{\partial z^2} + n^2 k_0 E_x = \frac{\text{d} (\ln \mu)}{\text{d}z}\frac{\partial ...
0
votes
1answer
28 views

Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
0
votes
1answer
42 views

Chain rule application in fundamental Theorem of Calculus

I have attached a question that I came across in trying to understand the fundamental theorem of calculus. The solution (highlighted with an arrow). I have difficulty understanding why the chain rule ...
5
votes
1answer
50 views

Example of first order logic without equality.

Most logic texts say that = is a special symbol which is always part of our language. It is my understanding, though, that it is perfectly acceptable to consider = to be an ordinary binary relation ...
0
votes
0answers
49 views

$H_I^n(R)=0$ and $H_I^n(M)\neq 0$ [on hold]

Find R and M as an R-module such that $H_I^n(M)\neq 0$ and $H_I^n(R)=0$, where I an ideal of R and $n\in N$. I found it in Cohen Macaulay rings. there`s nothing to find.
3
votes
0answers
34 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
2
votes
5answers
46 views

How to show if A is denumerable and $x\in A$ then $A-\{x\}$ is denumerable

My thoughts: If $A$ is denumerable then it has a bijection with $\mathbb{N}$ So therefore $A\rightarrow \mathbb{N}$. Then x is a single object in A and A is infinite. So if a single object is ...
0
votes
1answer
62 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
9
votes
1answer
106 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
6
votes
2answers
152 views

Value of the sum $\sum_{n=1}^\infty {\frac{1}{10^n-1}} $ and location of its second digit $1$

what is a value of this series? $$\sum_{n=1}^\infty {\frac{1}{10^n-1}} $$ Can you find a closed form of following series? $$\sum_{n=1}^N {\frac{1}{10^n-1}} $$ I found that $\sum_{n=1}^N ...
4
votes
2answers
74 views

Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.

I have the following two questions: The Fourier transform defines a unitary (provided that it is normalized properly) map $\hat{\cdot}:L^2(\mathbf{R})\rightarrow L^2(\mathbf{R})$. I figured out its ...
1
vote
2answers
68 views

Polynomial Division - “Define the largest natural number…” [on hold]

Would someone mind helping me with this question? The more detailed possible so I can have 100% of understanding. Thanks. Question: Define the largest natural number m such that the polynomial ...
4
votes
1answer
70 views

How to find the height of a tower given the distance, angle of elevation, and angle of depression? [on hold]

A building is 16m from a television tower from the top of the building, the angle of depression from the base 43 degrees, and the angle of elevation to the top of the tower is 24 degrees. Find the ...
2
votes
1answer
16 views

Can I convert between a rotation about an axis and a rotation according to two angles (all in 3D) without solving a system of nonlinear equations?

I am writing a program that needs to be able to switch between a rotation described by 2 angles to a rotation described an axis and one angle. I found one way to do this from this question, which ...
2
votes
2answers
91 views

Differentiating an integral with respect to a function

If we have $Q(z)$ and $P(z)$ functions of $z$ and $a, b, \lambda, I$ constants. How would you differentiate $$\int_0^1 aQ(z) - \frac{1}{2} bQ(z)^2 \, dz - \lambda \left(I - \int_0^1 P(z)Q(z) \, dz ...
1
vote
4answers
53 views

Trigonometric functions of the acute angle

Find the other five trigonometric functions of the acute angle A, given that: \begin{align} &\text{a)}\ \ \sec A = 2 \\[15pt] &\text{b)}\ \ \cos A = \frac{m^2 - n^2}{m^2 + n^2} \end{align} ...
2
votes
1answer
16 views

Rounding Percentages

Okay so I have been having a problem and would really appreciate any help. I have posted a table below with the aim of making this as simple/precise as possible. ...
3
votes
1answer
143 views
+50

An arithmetic sequence whose members do not contain the digit ‘9’

There is a non-constant arithmetic progression made of natural numbers only; none of them contains the digit $9$. Prove that such an arithmetic progression has no more than $72$ terms.
3
votes
2answers
36 views

The union of all the open sets in a family of topologies

I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let $\{\tau_\alpha\}$ be a family of topologies on X. [...] To say that ...
4
votes
1answer
294 views

Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$

Question : Is the following $(\star)$ true for $a,b,c\in\mathbb Z$ ? $$\begin{align}\int_{0}^{\frac{\pi}{2}}(ax^4+b\pi x^3+c{\pi}^{2}x^2)\log(\sin x)dx=0\Rightarrow a=b=c=0\qquad(\star)\end{align}$$ ...
3
votes
2answers
30 views

How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
1
vote
2answers
32 views

P/1 Actuary Exam Question

I was doing problems and came across this one and was wondering why the P[1<=x<=2] is F(2) - lim (x->1-) F(x) rather than F(2)-F(1)? Could someone please explain this for me?
1
vote
0answers
29 views

Discontinuous linear operator on $\ell^{2}$

Let $e_{n} = (0, 0, \ldots, 0, 1, 0, \ldots)$ where $1$ is in the $n$th position. Then $\{e_{n}\}$ is an orthonormal basis for the Hilbert space $\ell^{2}(\mathbb{N})$. Does there exists a linear ...
1
vote
2answers
41 views

Division with Remainder

When I divide 48/12 by hand, it's 0 Remainder 12. But when I divide it in a calculator, it's .25. Why is this? How does 0R12 turn into .25? Thanks. ANSWER (system won't let me answer my own ...
-4
votes
0answers
27 views

Geometry Altitudes Triangles [on hold]

Points $D$, $E$, and $F$ are the midpoints of sides $BC$, $CA$, and $AB$ of $ABC$, respectively, and $CZ$ is an altitude of the triangle. If $\angle BAC=71$, $\angle ABC=39$, and $\angle BCA=70$, ...
1
vote
1answer
38 views

How many ways it can be done?

There are 5 chairs. Bob and Rachel want to sit such that Bob is always left to Rachel. How many ways it can be done ? I have solved this in the following way: seats: * * * * * ...
2
votes
1answer
18 views

finding parallel sides from a irregular decagon?

Is it possible to find out that which of two sides are parallel in this irregular decagon.If,it is yes;then how can I proceed? I have tried with "Consecutive Interior Angles".but can't come to a ...

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