0
votes
0answers
35 views

Formula or function to generate a score between 1 to 10 based on 2 factors.

I'm conducting a model auction as part of an event at my college. 25 teams would be participating and I need to give scores to each of them to decide the five highest scoring teams to promote them to ...
12
votes
2answers
311 views

Different ways of representing a second cohomology class

There are probably many ways of talking about a second (integral) cohomology class of a smooth, closed, orientable manifold $M$ of dimension $n$. Here are a few, with $\alpha\in H^2(M,\mathbb{Z})$: ...
2
votes
4answers
499 views

Do these axioms fully describe the integers?

Here, I use Peano-like axioms to describe the set of integers $Z$. They are based on two successor functions, each starting with a common point of $0$, and a principle of induction for the integers. ...
-1
votes
1answer
85 views

“Question about measure theory” [closed]

If $f\in M^{+}(X,$ X $)$ and $\int f\, d\mu<{\infty}$, then the set $N = \{x\in X: f(x)>0\}$ is $\sigma$-finite (that is, there exists a sequence $(F_n)$ in X such that $N \subset \bigcup F_{n}$ ...
4
votes
2answers
147 views

Integration and fundamental Theorem of Calculus

I need some help with the following integration/use of fundamental theorem of calculus: $\displaystyle x(t) = \int_{0}^{t} \exp (-2s)a(s) \ ds$, where $a(x) = \left\{ \begin{array}{lr} ...
0
votes
1answer
89 views

Problem about a matrix on the ring $\mathbb Z$.

Suppose $A$ is a $m\times n$ ($n\geq m$) matrix on the ring $\mathbb Z$ of integers and the greatest common divisor of its $m\times m$ minor determinants is $1$. Prove that there is a $n\times m$ ...
5
votes
1answer
254 views

Is the set $\{1\}$ a member, and not a subset of the power set of $\{1, 2\}$?

Is the set $\{1\}$ a member, and not a subset of the power set of $\{1, 2\}$? I just want to be sure I am not making a mistake here, my reasoning: A is a subset of B if for every $x \in A$, $x \in ...
6
votes
2answers
701 views

Can you prove why Popsicle Stick Multiplication works?

This is a unique way of multiplying numbers by using sticks. Let's call it "Popsicle Stick Multiplication". Or maybe "Linear Algebra" quite literally. Take a look at both images that I've drawn ...
0
votes
1answer
264 views

Exterior, Interior, Boundary

If we denote the general point of $\mathbb{R}^2$ by $(x,y)$, determine $\operatorname{Int}A$, $\operatorname{Ext}A$, and $\operatorname{Bd}A$ for the subset $A$ of $\mathbb{R}^2$ specified by each ...
2
votes
3answers
2k views

Can someone explain the Borel-Cantelli Lemma?

I’m looking for an informal and intuitive explanation of the Borel-Cantelli Lemma. The symbolic version can be found here. What is confusing me is what ‘probability of the limit superior equals $ 0 ...
6
votes
2answers
75 views

What are the sets of integers obtained from multiplication/division from a given set of primes called?

We are given some (finite) set of primes $P=\{p_1, p_2, \ldots, p_n\}$. Define the following two sets: $S$ is the set of all integers that can be generated from $P$ by multiplying members of any ...
2
votes
2answers
1k views

Dirac Delta Function as Initial condition for 1D Diffusion PDE: ONE or TWO equations(conditions)?

I have 1D diffusion (u(t,x)) PDE with Dirac Delta initial condition. Question is regarding it's implementation: Dirac delta func is formally defined as an encapsulation of 2 conditions: 1st ...
76
votes
7answers
5k views

How to read a book in mathematics?

How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs? I have been reading Munkres, Artin, ...
0
votes
3answers
147 views

a problem in geometric probability

Inside a square of side $2$ units , five points are marked at random. What is the probability that there are at least two points such that the distance between them is at most $\sqrt2$ units? ...
2
votes
1answer
64 views

Four times the distance from the $x$-axis plus 9 times the distance from $y$-axis equals $10$.

What geometric figure is formed by the locus of points such that the sum of four times the distance from the $x$-axis and nine times its distance from $y$-axis is equal to $10$? I get ...
6
votes
2answers
978 views

Classify groups of order 27

Let $|G|=27$. Prove that all subgroups of index 3 are normal. Classify all groups of order 27. I can do the first one, but the classification is overwhelming. I don't even know where to start. ...
1
vote
2answers
47 views

Is my answer correct to this homework involving sets?

Of a group of people, each person is wearing green, blue, or both. One-fifth of those wearing green are also wearing blue. One-eighth of those wearing blue are also wearing green. Are more than ...
1
vote
4answers
398 views

Is there an abbreviation for “almost all $x\in X$”?

Is there an abbreviation for "almost all $x\in X$? I have "$\forall a.e. x\in X$" in my mind, but i see nobody uses this..
8
votes
2answers
428 views

Why should Gaussian noise have fractal dimension of 1.5?

In a paper I'm trying to understand, the following time series is generated as "simulated data": $$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,...,N)$$ where $Z(j)$ is a Gaussian noise with ...
3
votes
1answer
411 views

Finding the equation of a line entirely defined by a three variable equation.

How can you find the equation of a line that lies completely in a set defined by a three variable equation. For instance, the equation of a line entirely in the set defined by $x^2 + y^2 - z^2 = 1$ ...
0
votes
0answers
52 views

The $I$-torsion submodules of an injective module [duplicate]

Possible Duplicate: Prove that the following module is injective Prove that $I$-torsion submodules of injective modules are injective (the ring is Noetherian). Please help me.
0
votes
1answer
3k views

When does Schwarz inequality becomes an equality?

In Spivak Calculus you are asked to prove that in Schwarz inequality, equality holds only when $y_1 = y_2 = 0$ or when there is a number $\lambda$ such that $x_1 = \lambda y_1$ and $x_2 = \lambda ...
-4
votes
2answers
93 views

What is the solution to the equation below? [duplicate]

Solve the equation below. $$x^2+\frac{81x^2}{(9+x)^2}=40$$ I couldn't solve it after trying many time.
3
votes
2answers
294 views

integration inequality [duplicate]

Possible Duplicate: Proving Integral Inequality Suppose $f(x)$ is differentiable on $[0,1]$ , $f(0)=0$ and $1\geq f'(x) >0 $ Prove that $\displaystyle\left(\int_{0}^{1} ...
2
votes
4answers
885 views

Can you show why zero divided by zero does not equal zero? [duplicate]

I was talking about division by zero with my discrete math instructor, and it was explained to me that dividing can be broken down into simpler terms, i.e: Consider 6 divided by 3. To reach the answer ...
0
votes
1answer
144 views

An exercise of Boolean algebras

On page 87, Introduction to Boolean Algebras,Steven Givant,Paul Halmos(2000) Give an example of a subalgebra $B$ of a Boolean algebra $A$ and of a subset $E$ of $B$ such that $E$ has a supremum ...
4
votes
5answers
695 views

Determining the next Twin Prime?

A really simple I question I guess. Is there an algorithm or method such that given an integer $N$ there is a way to determine the next twin prime pair greater than $N$? If yes, then could you please ...
1
vote
2answers
466 views

Set with empty interior

What is the name of a set with empty interior? Wikipedia in a older version, say that is a hollow set, but i think that it is false. It is true? thanks in advance!
5
votes
2answers
171 views

General counting problem. Not sure how to proceed.

Mr. Popular has seven friends. He wants to count the number ways that he can invite a different subset of 3 of his friends for a dinner on 7 straight evenings s.t. each pair of friends are together ...
0
votes
1answer
510 views

Are there sets in the K-Topology that aren't open in the standard topology?

It seems to me that the basis for the K- Topology and the basis for the standard topology generate the same open sets. For instance, the open sets in the K-topology's basis that are different from ...
7
votes
2answers
577 views

Visualizing Concepts in Mathematical Logic

If you were forced to speculate or offer anecdotal evidence, how would you say excellent practicioners of mathematical logic coneptually grasp statements like: $$ \vdash ((P \rightarrow Q) ...
66
votes
11answers
10k views

Results that came out of nowhere.

Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. ...
0
votes
2answers
355 views

Nilpotent blocks of matrices

I'm trying to read through Hirsch and Smale's "Differential Equations, Dynamical Systems, and Linear Algebra", and I don't understand how this theorem follows from this other theorem. The first ...
0
votes
0answers
75 views

How do i calculate the probability

I have this question but i have difficult to understand and which probability rules i use Assume that an old used car fails to start 10% of the time. The other 90% of the time it drives for a time ...
1
vote
4answers
882 views

Finding coefficient of a complicated binomial expression?

Say we have something like $$ (x+2+y)^{23} $$ How does one go about finding the co efficient of say $$ x^6y^7 $$
3
votes
0answers
83 views

Seeing $1/x$ as a distribution

I have to show that by defining $$\langle u, f\rangle=\lim_{\varepsilon\rightarrow 0}\int_{-\infty}^{-\varepsilon}+\int_{\varepsilon}^{\infty}\frac{f(x)}{x}dx$$ with $f\in\mathcal{D}(\mathbb{R})$, ...
1
vote
1answer
262 views

full hessian, spherical coordinates

The question itself is pretty simple. I am running into confusion. Seems like there is a typo in the book. I wanna check myself. Maybe I am doing something wrong. Suppose we have the function (which ...
1
vote
0answers
268 views

Ramanujan's sums

Are the series expansions of arithmetic functions in terms of Ramanujan sums computationally useful? I didn't think they would be, but they seem to be good approximations even when summed with few ...
1
vote
1answer
167 views

What is the characteristic property of surjective submersions?

In Lee's 'Introduction to smooth manifolds' he states that given smooth manifolds $X,Y$ and a surjective submersion $f:X\to Y$, then $f$ is a smoothly final map, that is for any further smooth ...
1
vote
1answer
138 views

Showing that the following statements are equivalent with regard to algebraic topology

Suppose we have a space $X$ that is path connected and locally path connected, and let $f: X \rightarrow Y$ be continuous. How do we show that the following statements are equivalent? (A) $f$ lifts ...
5
votes
2answers
303 views

How to think about ordinal exponentiation?

I'm just trying to understand better how to see $\alpha^{\beta}$ for an arbitrary ordinal. I've already know that one can think about $\alpha . \beta$ as $\langle \alpha \times \beta, AntiLex\rangle$ ...
0
votes
1answer
151 views

inverse function to parametrisation of unit circle

I was asked to prove that the inverse function of the function $f$ isn't continuous. $f: [0,2\pi)\rightarrow S^1$; $f(t)=(\cos(t),\sin(t))$ $S^1$ denotes the circle of radius one centered in the ...
3
votes
1answer
150 views

An analogue of degree-genus formula for surfaces.

I have recently learnt Riemann-Roch formula for surfaces. Roughly speaking, the theorem says that on a reasonably nice surface we have the relation: $$ \chi(D) = \frac{1}{2}(D.D - D.K) + p_a + 1 $$ ...
1
vote
3answers
188 views

What academic level would one need to be at to fully understand papers published on the twin prim conjecture?

Specifically, what academic level would one need to be at to fully understand Goldston-Pintz-Yildirim's work on twin primes? Undergraduate, Graduate, or PhD?
2
votes
1answer
493 views

Vector Geometry Proof

Hello, this problem states to prove that the line segments drawn from one vertex of the parallelogram to the midpoints of the opposite sides trisects the other diagonal. Only vector addition, ...
0
votes
4answers
9k views

Prove that 1+1=2 [duplicate]

Possible Duplicate: How do I convince someone that $1+1=2$ may not necessarily be true? I once read that some mathematicians provided a very length proof of $1+1=2$. Can you think of some ...
11
votes
2answers
713 views

How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis if at all?

How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?
5
votes
3answers
252 views

Finding a topology on $ \mathbb{R}^2 $ such that the $x$-axis is dense

The problem is the following Put a topology on $ \mathbb{R}^2$ with the property that the line $\{(x,0):x\in \mathbb{R}\}$ is dense in $\mathbb{R}^2$ My attempt If (a,b) is in $R^2$, then define an ...
2
votes
1answer
69 views

Is it possible to choose a function to zero an integral?

Let $I(\omega,L)=\int_{0}^{L}xf(x,L)\cos(\omega x/L)dx$. I was wondering whether or not it was possible to find a nontrivial $f(x,L)$ (which can't depend on $\omega$) such that $I(\omega,L) = 0$. I ...
2
votes
2answers
71 views

An equation in natural numbers

Given $a,b,n\in \mathbb{N}$. What is the easiest route to find a pair of integers $x,y$ such that $(a^2+b^2)^n = x^2 + y^2$?

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