All Questions

6
votes
1answer
145 views

When do boundedness wrt metric and wrt order agree?

In Wikipedia: A subset $S$ of $\mathbb{R}^n$ is bounded with respect to the Euclidean distance if and only if it bounded as subset of $\mathbb{R}^n$ with the product order. More generally, I ...
2
votes
1answer
4k views

How to calculate the gradient of matrix equation

Short question: How do I calculate the gradient of the $MSE(a, b)$ equation below? Longer explanation: This problem arises, while I'm following a derivation of a term for an optimal beamvector $a$ in ...
2
votes
2answers
273 views

Density Question - Statistics

A point is picked randomly in space. Its three coordinates $X$, $Y$, and $Z$ are independent standard normal variables. Let $R = \sqrt{X^2+Y^2+Z^2}$ be the distance from the point from the origin. ...
1
vote
1answer
3k views

splitting arrays in matlab

Greetings All I'm trying to 1)split an array into multiple parts 2)export each part to separate wave files 3)re-import wav files and join them together to make sure the array data that was split ...
4
votes
3answers
367 views

Finitely generated free group is a cogroup object in the category of groups

I am trying to show that every finitely generated free group is a cogroup object in the category of groups. (Note I believe that this is also true for non-finite free groups, but that is probably much ...
18
votes
1answer
2k views

Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$

Here is the problem: Find all continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x+y)+f(x-y)=2[f(x)+f(y)]\;\;\;(1).$$ Here is my attempt: Fix $\delta>0$ and let $C=\int_{0}^{\delta}...
4
votes
1answer
379 views

Meaning of a partial derivative here?

I am given a 'tariff' function for two countries, $i=1, 2$. Both players can select a tariff between 0 and 100. If player $i$ selects $x_i$ and player $j$ selects $x_j$, country $i$ gets a payoff of $$...
1
vote
2answers
11k views

With a few data points can a generate a close equation to meet them?

I have 1x = -40, 2x = -41 , 3x = -54 and getting a few more. Can I generate a equation for a graph that gets close to this? I was trying to get Wolfram Alpha to ...
4
votes
2answers
5k views

eigenvalues and eigenvectors for rectangular matrices

We can generalize matrix inverses from non-singular square matrices to rectangular matrices in general, for example, the well-known Moore–Penrose pseudoinverse. I am wondering how this can be done for ...
8
votes
3answers
8k views

CDF of a ratio of exponential variables

Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$. I tried out the problem, and wanted to check to see if my answer of: $\frac{\...
5
votes
2answers
1k views

Finding Nash equilibrium aka finding where lines intersect

I am tagging this as multivariable calculus because it potentially involves taking partial derivatives. I am working on some mathematical treatment for Cournot duopoly models (not homework, just ...
2
votes
3answers
6k views

Density and expectation of the range of a sample of uniform random variables

If the variables $\alpha_1$...$\alpha_n$ are distributed uniformly in $(0,1)$, How do I show that the spread $\alpha_{(n)}$ - $\alpha_{(1)}$ has density $n (n-1) x^{n-2} (1-x)$ and expectation $(n-1)...
3
votes
1answer
487 views

A question about why a space under a certain norm is complete

A theorem I am reading (about the existence and uniqueness of solutions to Sturm-Liouville intial-value problems) defines a space $B$ consisting of the continuous functions defined on a closed real ...
4
votes
1answer
272 views

Does the Robertson-Seymour theorem apply to vertex-labeled graphs?

Does the Robertson-Seymour theorem apply to vertex-labeled graphs? A minor as I understand it is a graph which can be reached by a sequence of edge contractions and non-disconnecting edge deletions. ...
7
votes
2answers
2k views

Field Norm Surjective for Finite Extensions of $\mathbb{F}_{p^k}$

I'd rather not have the answer, because I feel like this should be a relatively easy question, and I'm just missing some key step, but could anyone give me a hint on showing that the norm (defined as $...
1
vote
3answers
53k views

Drawing a card from a deck

A single card is drawn from a standard 52-deck of cards with four suits: hearts, clubs, diamonds, and spades; there are 13 cards per suit. If each suit has three face cards, how many ways could the ...
1
vote
5answers
286 views

Fix $k \geq 2$: convergence of $\sum \frac{1}{n^k}$?

I have a lot of sum questions right now ... could someone give me the convergence of, and/or formula for, $\sum_{n=2}^{\infty} \frac{1}{n^k}$ when $k$ is a fixed integer greater than or equal to 2? ...
12
votes
1answer
1k views

Measurability of the composition of a measurable map with a surjective map satisfying an expansion condition

I am trying to figure out the following problem in measure theory and am stuck. It seems like it should be very easy, so I must be missing something. Let $g: \mathbb{R} \to \mathbb{R}$ be a mapping ...
4
votes
2answers
492 views

Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$ [duplicate]

Let $n$ be a positive natural number, $n\ge 2$. Then $\displaystyle\sum_{k=1}^n \frac{1}{k^2} \lt 2 - \frac{1}{n}.$ The basis step was easy but could someone give me a hint in the right direction as ...
0
votes
2answers
156 views

Convergence of $\sum \frac{n}{n-1}$?

This is a piece of a much tougher infinite sum I'm trying to get. I think it should be a simple answer but having trouble knowing how to approach it. Thanks for the help! Does this sum converge? $$\...
14
votes
6answers
4k views

Prove $\sin(\pi/2)=1$ using Taylor series

Prove $\sin(\pi/2)=1$ using the Taylor series definition of $\sin x$, $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ It seems rather messy to substitute in $\pi/2$ for $x$. So we have $$\sin(\...
13
votes
2answers
2k views

Do infinitely many points in a plane with integer distances lie on a line?

Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid ...
11
votes
5answers
903 views

What is a good language to develop in for simple, yet customizable math programs?

I'm writing to ask for some guidance on choosing a language and course of action in learning programming. I've seen thread after thread with questions from newbies, asking, "What is the best language ...
4
votes
1answer
989 views

Distributional derivative coincides with classical derivative?

I am having some trouble understanding the precise meaning of the following statement: "if $f \in C^1 (\Omega)$ for some $\Omega \subset \mathbb R^n$, then the distributional derivative of $f$ ...
1
vote
0answers
750 views

Find the first odd multiplicity root of a function

I'm trying to find the "first" (greater than some initial $t_0$) odd root (that is, a root after which the sign of the function changes) of a function $f(t)$, if there is one, that is also less than ...
16
votes
4answers
1k views

Finding an Explicit Formula from the Recurrence: $na_{n}= 2 ( a_{n-1}+a_{n-2})$

Here is the recurrence: $$na_{n}=2(a_{n-1}+a_{n-2}) \qquad\text{ where } a_{0}=1\text{ and }a_{1}=2$$ At first I thought that this could be easily solved by simply multiplying the Fibonacci ...
2
votes
1answer
48 views

Is there a particular name for a'long-small-small' tensor/array?

I'm thinking of a 3D array, with dimensions small,small,large. I've taken to saying 'sausage' as shorthand (and I'm sure there are worse NSFW descriptions) but is there a 'legitimate' description for ...
3
votes
3answers
620 views

The probability that a polynomial has complex roots

Find the probability that $x^2 - 2ax + b$ has complex roots if the coefficients $a$ and $b$ are independent random variables with the common density uniform, that is $1/h$, and exponential, that is $\...
4
votes
2answers
568 views

Two ideals with equal radical in a noetherian ring

Let $A$ be a commutative noetherian ring with two ideals $I,J$ such that $\sqrt{I}=\sqrt{J}$. Does there always exist integers $p,q,r$ such that $$ I^p \subset J^q \subset I^r? $$
3
votes
2answers
2k views

Probability question about chords on a circle

Four points are chosen independently and at random on a circle. Find the probability that chords X1X2 and X3X4 intersect a) without calculation using a symmetry argument and b) from the definition by ...
0
votes
1answer
186 views

Finding derivative in an implicit function

I searched and couldn't find anything specific to my question, so I'll ask it here. I'm asked to find the indicated derivative: $${\operatorname{d}y\over\operatorname{d}x} \sin(xy^2)-x^2 = x+5$$ ...
7
votes
1answer
2k views

Stochastic integral with a Poisson process

I have a Poisson process $X_t$ for $t\ge0$. How I can find a process $b_t$ such that $$\exp ({\alpha X_t})=1+\int_0^t b_{s^{-}}dX_s$$ where $\alpha\in\mathbb{R}$ and what would be the expectation of ...
6
votes
1answer
280 views

Is there a standard category-theoretic way to express a loop or quasigroup?

The standard way to encode a group as a category is as a "category with one object and all arrows invertible". All of the arrows are group elements, and composition of arrows is the group operation. ...
5
votes
0answers
4k views

Is there an equivalent to the distributive law for division over subtraction and/or addition?

I understand that the the distributive law cannot be applied to division over addition/subtraction, but is there an equivalent law to expand it out. For example, I know: $$100 \times (5 + 3) = (100 \...
5
votes
1answer
636 views

Algebraically finding a Nash equilibrium

Here's the problem that relates to a whole class of problems to which I am trying to figure out a general solution. Given two players 1 and 2 who can select a number from the interval $[0, 1]$, ...
1
vote
1answer
530 views

$L^{\infty}$($E$) could be separable for a measurable set $E$

I know that in general, $L^{\infty}$($E$) is not separable, like for example, if $E$ = [$a$,$b$]. But wouldn't $L^{\infty}$($E$) be separable if $E$ = $\mathbb{Q}$, i.e. the set of rational numbers? ...
1
vote
2answers
541 views

Some complex analysis problems

Here are my problems: Find an entire function $f(z)$ such that $|f(z)| < e^{\operatorname{Im} f(z)}$ for all $z \in \mathbb{C}$ and $f(0)=2$. I am trying to guess $2\cos(z), 2e^z$, etc, but they ...
4
votes
2answers
1k views

elementary question about tensor product of modules

I'm a bit embarrassed to ask this, but I've gotten myself confused over what I think is a simple issue. Let $A$ be a local ring, $k$ its residue field, and $M,N$ finitely generated $A$-modules. An ...
7
votes
1answer
637 views

ZFC + $\exists$ Standard model $\rightarrow$ Con(ZFC + $\exists \omega$-model)

$ZFC + \exists V_\alpha$ model of $ZFC \vdash Con(ZFC + \exists$ transitive standard model of $ZFC)$ and then $ZFC + \exists$ transitive standard model of $ZFC \vdash Con(ZFC + \exists \omega-model$ ...
13
votes
2answers
2k views

What does “splitting naturally” mean in the Universal Coefficients Theorem

The Universal Coefficients Theorem states that $0\rightarrow H_n(X)\otimes G\rightarrow H_n(X;G)\rightarrow\operatorname{Tor}(H_{n-1}(X),G)\rightarrow 0$ splits, but not naturally. In all the ...
1
vote
4answers
2k views

Finding distribution functions of exponential random variables

Find the distribution functions of X+Y/X and X+Y/Z, given that X, Y, and Z have a common exponential distribution. I think the main thing is that I wanted to confirm the distribution I got for X+Y. I'...
1
vote
1answer
119 views

On maximizing a function

I have a function as below, $$ f(n) = \sum_{i}{-\Big( (1-q_i)^{n-1} - b \Big)^2} $$ how to find an integer $n\in[0,N]$ that maximizes function $f(\cdot)$. Here, $q_i\in[0,1]$, $b\in[0,1]$ and $N \gg ...
2
votes
1answer
2k views

Prove that $a$ is quadratic residue modulo every prime if and only if $a$ is perfect square [duplicate]

Possible Duplicate: Proving that an integer is the $n$ th power Prove that $a$ is quadratic residue modulo every prime if and only if $a$ is perfect square My attempt was, Since $a$ is ...
2
votes
3answers
3k views

How to determine the degree of a polynomial?

If $$g(x) = x^4 + x^3$$ From my understanding, the degree of the above polynomial i.e. $g(x)$ is 4. However, for this polynomial, $$f(x) = (x-1)(x-2) \cdots (x-p+1)$$ What degree does $f(x)$ have? ...
1
vote
1answer
96 views

Lipschitz contradiction

Assume that $\phi:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is a smooth vector field, and assume that we can find vectors $y_k,x_k$ ($k$ positive integer) such that $(\phi(x_k)-\phi(y_k),x_k-y_k)\geq k \...
2
votes
1answer
3k views

Trajectory of a projectile meets a moving object (2D)

First of all, I asked this question on Stackoverflow, but I realize this is a better place to ask the question. So i moved it here. I've looked for quite some time now to find a nice math solution ...
3
votes
2answers
347 views

Concerning the definition of effective quotient orbifold

I've been trying to figure out orbifolds, and in all of the sources I seem to be confused with the orbifold structure on quotient orbifolds. A quotient orbifold is defined as follows. Let $M$ be a ...
1
vote
1answer
88 views

an estimate for derivative

let $F$ a closed convex subset of $\mathbb{R}^n$, let $x,y\in F$ and assume that for any $s\in[0,1]$ we have $f(s):=\mid sx+(1-s)y-z\mid\geq \mid y-z\mid$ why is it true that $\frac{\partial}{\...
1
vote
2answers
891 views

Relations: is the composition of an empty set with a nonempty set empty? And the other way around

Given $T\circ S=\emptyset$ and $R$ nonempty, would $$(T \circ S) \circ R$$ be anything other than the empty set? I'm also curious the other way around. I think that it would be just empty.
3
votes
4answers
470 views

Galois Theory and Number of Solutions

Is the Galois group of an irreducible separable polynomial of degree $n$ isomorphic to a group on exactly $n$ letters? Is that enough to prove that a degree $n$ polynomials has $n$ roots. What I mean ...

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