0
votes
1answer
22 views

A question about manifolds with boundaries.

My topology textbook says the following: Let $S\subset \Bbb{R^2}$ be a closed disc. Then every point in $S$ is contained in a neighbourhood which is homeomorphic to that portion of a ball in ...
8
votes
1answer
278 views

Can all integration be thought of as projections?

For example, the integral of the function f(x) could be thought of the projection of f on the function g, where g is identically 1. Following this logic, can we think of the multiplication of f and ...
1
vote
0answers
22 views

Geometric definition of the stable commutator length

In his book, D.Calegari proves the equivalence of the algebraic and geometric definitions of stable commutator length (Proposition 2.10, p. 15). I actually have some difficulties in understanding the ...
1
vote
2answers
19 views

Rewriting a trigonometric inequality (including a parameter)

How is it possible to rewrite these equations? $\sin{x}- \cos{x} ≤ \mu(\cos{x} + \sin{x}) \implies \tan{}x ≤ \frac{1 + \mu}{1 - \mu}$ and $\cos{x}- \sin{x} ≤ \mu(\cos{x} + \sin{x}) \implies \tan{}x ...
0
votes
1answer
17 views

Find total yards with LXWXH?

I have a box I need to cover all 6 sides with a single layer fiberglass fabric. The fabric comes in 50" or 28" widths. The dimensions of the box are 34" long; 15" wide; 12" High. How do I determine ...
0
votes
0answers
15 views

Solving wave equation using Green's function

I am currently self studying/ self teaching myself this great book Mathematical methods of physics by Mathews and Walker and must tell you that I am really struggling with the explanation of the ...
2
votes
1answer
39 views

Unique Products on a Times Table

I was looking at a 10x10 multiplication table, and I decided to count the unique products. There are 42 out of a possible 100 numbers represented. I had to wonder, why 42? I counted the 58 non-listed ...
2
votes
0answers
33 views

The 3rd and 4th Critical Point?

I must find and classify all the critical points in the following function: $$ f(x,y)= x^2 + y^2 +x^2y +4$$ I have said that $$f_x=2x+2xy=0$$ $$ 2x = -2xy$$ $$ \frac{ 2x}{\ -2x}=y $$ $$y=-1$$ $$f_x = ...
3
votes
1answer
17 views

Regularity for a parabolic problem with nonsmooth coefficients

I'm looking for references on the regularity of the (weak) solution to the parabolic problem with nonsmooth coefficients. In most literature, like Evans, the coefficients are often assumed to be ...
1
vote
0answers
9 views

Unable to get a particular solution for a system of ODE equations with the method of undetermined coefficients

so I have solved this problem using another method (the method of diagonalisation), but I now want to try with the method of undetermined coefficients and cannot get the right result for $\vec{b}$. ...
0
votes
0answers
11 views

$P_{x}(T_{B_{0,r}}<\infty)$ in integral form

For example for $$P_{x}(T_{B_{0,r}}<\infty)\tag1$$ and $$\frac{1}{c}\int_{0}^{\infty}\int_{\partial B_{0,r}}p(t,x,y)\,dy\,dt\tag2$$ for $x\in (B_{0,r})^{c}$ in three dimensions, c is a ...
4
votes
2answers
44 views

What is a finitary proof?

I started reading "mathematical logic", by J.R.Shoenfield, but I cannot quite understand a sentence in the very first chapter: Proofs which deal with concrete objects in a constructive manner are ...
2
votes
0answers
18 views

Correct math notation to show min and max of a list of numbers and sign?

I am working on a paper that uses a calculation from DIN 15018. The calculation is only described in text and not as a math equation. I would like to display as an actual equation if possible. The ...
2
votes
1answer
22 views

How many terms of the Taylor expansion should I develop?

How do I know how many terms of the power series should I develop to evaluate a limit? Example: Given this limit: $L:=\displaystyle\lim_{x\to0}\left(\frac{\ln(1+x)}x-e^{-x/2}\right)\frac1{\cosh x - ...
0
votes
1answer
21 views

Fundamental group of disjoint union of two 2-tori identifying them along pairs of points

I'm trying to solve the following problem: let $X$ be the space obtained from the disjoint union of two 2-tori $A,B$ be identifying them along 2 pairs points (resp. three pairs of points). If we ...
2
votes
1answer
33 views

Linear recurrence relation in Cantor-like sets

I have a linear recurrence relation $$a_i = \alpha_0 a_{i-1} + \beta(i)$$ Where $\beta(i) = \beta_0b_i$ with $b_i \in \{0,1\}^\mathbb N$ I know that $0 < \alpha_0 < \frac{1}{2}$, $\beta_0 = 1 ...
2
votes
0answers
55 views

$\phi(f) = 0$ on all positive functionals implies $f=0$?

I have a question, but i don't know for sure if it's true. If $E$ is a partially ordered normed vector space, and $E^*$ the norm-dual. ($E$ is a Riesz-space). Let $f\in E$. What if $\phi(f)=0$ for ...
0
votes
1answer
26 views

Canonical inclusion $L^q(0,1) \to L^p(0,1)$ is compact?

Does there exist $q>p$ such that the canonical inclusion $L^q(0,1) \to L^p(0,1)$ is compact? My answer is no. Since we know that $L^\infty (0,1) \to L^p(0,1)$ is not compact, take $\{\sin(nx)\}$ ...
1
vote
0answers
17 views

efficient least squares A = BX+CXD (solving for matrix X)

I am interested in solving a least-squares solution of the form $$ \operatorname{argmin}_X \| A - BX - CXD \|_F^2 $$ for large (rank in hundreds to thousands) matrices $A,B,C,D,X$ I know this is ...
0
votes
0answers
24 views

How to evaluate the following integral? $∫_{-β}^{2π-β}\exp⁡(ix\cos(φ-β))dφ.$

I'm trying to calculate the following integral: $$∫_{-β}^{2π-β}\exp⁡(ix\cos(φ-β))dφ.$$ I tried by parts with no success and also by writing $\exp (ix)$ in terms of $\sin$ and $\cos$, with no ...
0
votes
0answers
24 views

how would you define the term “elementary” in the context of categories and sets?

I was just reading P.T. Johnstone's introduction to his book "Topos Theory", where he uses the term "elementary" many times to classify the nature of theorems and definitions, examples below. I ...
0
votes
1answer
21 views

Symmetric Sum Notation

From here is the following excerpt Suppose one is given a homogeneous symmetric polynomial $P$ and asked to prove that $P(x_1, \ldots , x_n) ≥ 0$ How should one proceed? Our first step is ...
1
vote
4answers
84 views

The probability that after repeated random drawing from an urn, all balls left in the urn will be red

Problem An urn contains $p$ red and $q$ green balls. Balls are drawn one by one till balls left in the urn are all red. Prove that the probability of this event is $\dfrac {p}{p+q}$. Please note that ...
0
votes
0answers
11 views

abstract conceptual usage of power series, advice on how to approach similar problems

This problem is a bit strange, I have the solution for this particular one, I just think that it a very ambiguous question. How would you go about solving it? My answer is that b is larger because ...
3
votes
1answer
62 views

Evaluating $\int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$

I'm trying to find the general formula for the following: $$I_n = \int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$$ I remember doing it a while back but for the life of me, I can't remember right now. ...
1
vote
0answers
24 views

How to see that $\text{dim}(L)=k-1$?

Consider $L:=\left\{x\in\mathbb{R}^k: cx=\delta\right\}$ with $c=(c_1,\ldots,c_k)$ and $\delta\in\mathbb{R}$. Show that $\text{dim}(L)=k-1$. Do not know how to show that. Anyhow my first ...
3
votes
0answers
68 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
1
vote
1answer
30 views

Computational complexity and the big $\mathcal{O}$

I have a question about this Big $\mathcal{O}$ problem. I have the question down $90\%$, but the other $10\%$ isn't getting to me. I will write out the entire question and I'll point out the step, ...
0
votes
0answers
16 views

Bezout's bound and resultants - reference request

In Terry Tao's blog post about Bezout's inequality, he writes: In our notation*, this theorem states the following: Theorem 1 (Bezout’s theorem) Let $d=m=2$. If $V$ is finite, then it has ...
0
votes
1answer
28 views
1
vote
2answers
36 views

Cardinality of the set of multiples of “n”

I've yet another question about the cardinality of sets. Apologies, but I just can't seem to fully grasp it. For what it's worth, I have tried searching the site for a solution to this problem. Let ...
1
vote
1answer
35 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
0
votes
2answers
20 views

Composition of a convex function and a convex decreasing function is quasi-concave

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h(g(x))$ is a quasi-concave function?
1
vote
2answers
36 views

Bijection between measurable sets

I was doing some self-studying and came upon the following question. Suppose $a < b$ and let $M([a,b])$ denote the Lebesgue measurable subsets of $[a,b]$. Define the function $f: [0,1] ...
-3
votes
0answers
20 views

Finite Blaschke product [on hold]

I want to do my research in Finite Blaschke products. Also, I want to enter graduate school in the US. I'm a final year mathematics student in Sri Lanka.
0
votes
0answers
34 views

Positively graded k-algebra

Suppose we have a positively graded $k$-algebra $A=\bigoplus_{i\ge 0}A_i$, such that $A_0$ has finite global dimension. Furthermore, all $A_i$ are finite dimensional and $A$ is generated in degree $0$ ...
2
votes
1answer
40 views

A double sum with combinatorial factors

Let $n$, $p$ and $j$ be integers. As a byproduct of some other calculations I have discovered the following identity: \begin{equation} \sum\limits_{p=0}^{j} \sum\limits_{p_1=0}^j \binom{p+p_1}{p_1} ...
1
vote
1answer
47 views

Projection of a hypersurface from a point

Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X_n=0\}$ and a hypersurface $X$. We want ...
1
vote
2answers
25 views

Convergence, interval, radius of power series, conceptual explanation please [on hold]

Could someone explain how to solve the problem. A very basic and broad understanding is what I am looking for so that if I were to have to approach this problem with different numbers I would know ...
2
votes
0answers
30 views

Graded rings and modules - which book to refer to?

Could you recommend a book with a good treatment of the theory of graded rings and modules?
5
votes
4answers
328 views

Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
1
vote
1answer
34 views

Solving the second order differential equation $d^2u /dt^2 =a$

Let $\frac{\partial^2 u}{\partial t^2}=a$ which $ a$ is constant, then $u=\frac{a}{2}t^2+bt+c$ on interval $ [0,T)$. Let's say we have $a, c$. Then how can we find $b$?
1
vote
2answers
42 views

To find the center of gravity of a homogeneous tetrahedron

The center of gravity coordinates of a triangle can be calculated $O(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3})$ where $P_1,P_2, P_3$ are the corner points of a homogeneous ...
0
votes
0answers
10 views

Heuristic Algorithm for integrating algebraic functions

Is there any heuristic "algorithm" or a good technique for integrating algebraic functions? The general algebraic case was solved by Trager and Davenport. But their algorithms are rather complicated ...
0
votes
0answers
21 views

Relation between girth and lower bound of the number of vertices

Let $G$ be a graph with girth $g > 3$. Suppose that every vertex in $G$ has degree at least $k > 1$. Can we found a nice lower bound for $|V(G)|$? Let me be more specific on what I want: ...
1
vote
2answers
42 views

Help on an integration by substitution

In a proof to show that $\int_{0}^{1} f \left(\left\{1/x\right\}\right) \frac{ \mathrm{d}x}{1-x}=\int_{0}^{1} f(v) \frac{ \mathrm{d}v}{v}$, i found this line : ...
5
votes
3answers
210 views

Is symmetry a valid option in inequalities?

Consider two questions: Q1. $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ $$a,b,c,d,e\in\mathbb{I^+_0}$$ Find maximum value of 'e'? My answer: Since when e is maximum when all other variables are equal ...
0
votes
1answer
31 views

I am working on basic functions, I am asked is x-5=y^2 a function,

i use the square root property and get plus or minus the sqaure root of x-5=y, then I come to my question, for any value of x greater than 5, how many values of y result? I need some insight to fully ...
0
votes
0answers
12 views

Negative Weight meromorphic modular forms/ Sections of Line bundles

it is known, that we can see modular forms as section of line bundles on a Riemann surface. Especially, we know that a meromorphic modular form of weight 2 on SL(2,Z) corresponds to a meromorphic ...
2
votes
0answers
40 views

Example for divisors, line bundles and meromorphic functions on $\mathbb{CP}^2$

I have been studying divisors using Griffiths/Harris (chapter 1.1) as well as Huybrechts (chapter 2.3). However, I cannot seem to find any very easy worked examples - i.e. $\mathbb{CP}^1$ or ...

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