2
votes
2answers
168 views

When is this set a field?

For which $c$ will the set $K(c) = \{a + b\sqrt{c} : a, b \in \mathbb{Q}\}$ be a field? I know for example, that $K(\frac{2}{3})$ will be one, I am just wondering what the most general result of this ...
1
vote
1answer
330 views

Proving a function is Lipschitz

I have a homework problem which consists of two parts, the first of which I have been staring at for several days with very little (constructive) progress. I need to show that the function $$f(t) = ...
5
votes
2answers
349 views

On the automated solution of Olympiad problems

This is a soft question on developing software to solve exam/contest problems. Imagine the following scenario: You, as user, have (say) an IMO exam paper. You type one question at a time, perhaps ...
3
votes
1answer
152 views

Starting out with functional equations

I am thinking of starting learning about various functional equations and ways to solve them, any help as to which books could be of help to me? I have some knowledge about some basic functional ...
1
vote
2answers
96 views

ODE question: $y'+A(t) y =B(t)$, with $y(0)=0, B>0$ implies $y\ge 0$; another proof?

I am trying to prove that the solution for the different equation $$y'+A(t) y =B(t)$$ with initial condition $y(0)=0$ and the assumption that $B\ge 0$, has non-negative solution for all $t\ge ...
1
vote
4answers
2k views

Differential to estimate a number

I am trying to estimate $(1.999)^4$, so I set up the problem like this. $$y=x^4 ,$$ $$x=2 ,$$ $$dx=.001.$$ Then I find the derivative of $f(x)$ which is $4x^3$ and multiply that by $dx$ which is ...
2
votes
2answers
2k views

Geometric Interpretation of Solutions to Linear Systems

I've been reading Linear Algebra by Jacob for self study and I'm wondering about the geometric interpretation of a unique solution to systems of equations in $\mathbb R^3$. For example, the ...
2
votes
1answer
1k views

Taking powers of a triangular matrix?

So there is a formula for the $n$th power of a matrix in Jordan normal form. Is there a formula for the $n$th power of a general triangular matrix? If not, are there known formulas for "nice" upper ...
2
votes
0answers
100 views

An estimate on the $p$ norm of a convolution integral

The theorem is that for any summability kernel $\{\phi_{n}\}$, if $f\in L_{p}(\mathbb{T}^{d})$, then $||f*\phi_{n} - f||_{p}\rightarrow 0$. The step that I cannot follow is this: ...
2
votes
1answer
321 views

Convolution of an $L_{p}(\mathbb{T})$ function $f$ with a term of a summability kernel $\{\phi_n\}$

... is the result in $L_{p}$? A remark in my notes says yes but I can't see how to verify it. As was pointed out to me in a previous question I asked last night, I need to show that the following ...
1
vote
1answer
117 views

Projective plane and some curves

We define a line in the projective plane as a set of the form $$ L_{a,b,c} = \left\{ {\left[ {x,y,z} \right] \in P_R^2 :ax + by + cz = 0} \right\}\text{ or just }L $$ Let a finite collection of ...
14
votes
5answers
783 views

Why isn't $\mathbb{R}^2$ a countable union of ranges of curves?

I came across this question on the topology board at AoPS, where it hadn't really received an answer. It seems interesting, but I'm not sure how to solve it. Hopefully an answer will be found here. ...
1
vote
3answers
1k views

Linear Algebra Question (Polynomial Interpolation)

Given the data for an experiment: Velocity: 0, 2, 4, 6, 8, 10 Force: 0 , 2.9, 14.8, 39.6, 74.3, 119 (One force value listed below one velocity value in a table) Find an interpolating ...
1
vote
2answers
329 views

What is the generalization of 'theorem' and 'conjecture'?

What is the most specific word that describes both? As in "all theorems and conjectures are ..." ?
6
votes
3answers
2k views

The Cantor set is homeomorphic to infinite product of $\{0,1\}$ with itself - cylinder basis - and it topology

I know the Cantor set probably comes up in homework questions all the time but I didn't find many clues - that I understood at least. I am, for a homework problem, supposed to show that the Cantor ...
0
votes
1answer
124 views

Morphism from Spec DVR to variety

Take an integral, proper variety $X$ over $k$ with function field $k(X)$. Let $A$ be a DVR containing $k$ having field of fractions $k(X)$. Take $P \in X$. Does there always exist an injection ...
1
vote
2answers
538 views

A coin is flipped when dice hits 6 (conditional probability)

Prof gave us homework on conditional probability that is due on the day of the lecture on conditional probability. Yeah, this has been a bad week and I've no idea what I'm doing. Q: 3 dice are ...
1
vote
2answers
75 views

Bounds on expected value and distribution of a product of beta random variables

Let $V_1,...,V_n$ be random variables distribution according to the Beta distribution with parameters $\mathrm{Beta}(1,\alpha)$. Define $X_i = V_i \prod_{j=1}^{i-1} (1-V_j)$ for $i=1,...,n$. Is ...
3
votes
1answer
62 views

I don't get the same graph, before and after solving for Y?

Now, if I draw the following: $3x+y=3$ $2x^2-y^2=-1$ With Wolfram I get the following graph. But if I draw the following functions that I have solved for Y $y=\sqrt{1+2x^2}$ $y=3-3x$ I seem to be ...
11
votes
3answers
772 views

Baby Shower Problem. Too hard for 1st grader but got parents thinking

So our six year old son comes home from 1st grade with the following math puzzle. Your Aunt is having a baby. You have created a party game for a baby shower. It is called pick the gender. You ...
1
vote
1answer
156 views

Stereographic projections

Simple question: Are there stereographic projections from the circle on the to x-y plane? If yes, how are they formulated? I've read the nice article on wikipedia and it mentions stereographic ...
1
vote
3answers
598 views

Counting Number of k-tuples

Let $A = \{a_1, \dots, a_n\}$ be a collection of distinct elements and let $S$ denote the collection all $k$-tuples $(a_{i_1}, \dots a_{i_k})$ where $i_1, \dots i_k$ is an increasing sequence of ...
3
votes
1answer
112 views

Exact preimage of an interesting open ball

Consider the function $$g: \mathbb{R}\to \mathbb{R}^\omega$$ given by $$g(t)=(t,t,t,...)$$ where $ \mathbb{R}^\omega$ is in the uniform topology. Can we find the exact answer to ...
2
votes
0answers
74 views

Dimension of space of k-forms proof [duplicate]

Possible Duplicate: Counting Number of k-tuples Let $V$ be a vector space of dimension $n$ and let $\Lambda^{k} (V)$ denote the set of all $k$-forms on $V$. If $\{v_1, ..., v_n\}$ is a ...
5
votes
2answers
612 views

Is the determinant of a zero divisor zero?

Suppose that $A$ is a zero divisor in the ring of $(n\times n)$-matrices over the ring $R$. Is $\det(A) =0$ if $R$ is a field? Is $\det(A) =0$ if $R$ is an integral domain? It's not necessarily ...
2
votes
2answers
388 views

Maximization over vectors (seen as column matrices)

I am trying to solve the following question: $$\text{Maximize } f(x_1,x_2,\ldots, x_n)=2\sum\limits_{i=1}^n x_i^t A x_i+\sum\limits_{i=1}^{n-1}\sum\limits_{j>i}^n (x_i^tAx_j+x_j^tAx_i)$$ subject ...
2
votes
1answer
85 views

Puzzle: Generate the Highest Bounded Number Using a Limited Number of Characters

A friend and I were sitting in our cubes at work and trying to create the greatest bounded number we could using only a few characters. We came up with $A(G,G)$, which is the Ackermann function with ...
0
votes
1answer
88 views

Smooth structure on solution space

Broadly speaking the question in this post is : How to check that the solution space of a set of equations is a smooth manifold? This happens in physics all the time, say solution set of Hamiltonian ...
5
votes
1answer
423 views

Relative entropy is non-negative

Let $p=(p_1,\dotsc,p_r), q=(q_1,\dotsc,q_r)$ be two different probability distributions. Define the relative entropy $$h(p||q) = \sum_{i=1}^r p_i (\ln p_i - \ln q_i)$$ Show $h(p||q)\geq 0$. I'm given ...
1
vote
2answers
242 views

Graphing Transformations. Why does the +2 in $f(x) = \sqrt{-x + 2}$ not work as expected when done out of order?

Graphing $f(x) = \sqrt{-x + 2}$ from the graph of $y=\sqrt{x}$. Correct Method First graph $f(x) = \sqrt{x}$. then $f(x) = \sqrt{x+2}$ (shift left 2) then $f(x) = \sqrt{-x+2}$ (Reflect over ...
3
votes
5answers
1k views

Complete implies locally compact in length metric space?

I am confused. The way I see it, in a complete metric space, closed balls of finite diameter are compact since they are complete and totally bounded. Consequently a complete metric space is locally ...
7
votes
2answers
599 views

What is required to learn about algebraic geometry?

I want to learn about classical algebraic geometry. So what are subjects that are required to start learning about it? (Some preknowledge of algebra, commutative algebra?)
0
votes
1answer
445 views

How many solutions does an equation system with binary values have?

I have an equation system with binary values ($0$ and $1$). After doing a gauss-elimination, I can calculate the determinant by anding the entries of the main diagonal. If it is $1$, it's trivial to ...
5
votes
1answer
120 views

Convergence of $\frac{\sin(g(n))}{f(n)}$

What do we require of $g(n)$, if for every positive strictly increasing unbounded $f(n)$, this sum converges? $$\sum_{n=1}^{\infty} \frac{\sin(g(n))}{f(n)} .$$ Does it converge for $g(n)=n$?
12
votes
5answers
7k views

Preparing for Spivak

For financial reasons, I dropped out my senior year of college as a piano performance major. I will be returning to college to dual major in mathematics and computer science. I've taught myself to ...
1
vote
2answers
226 views

Complex manifolds without compact submanifolds

It is well known that $\mathbb{C}^n$ does not admit any compact complex submanifold, I was wondering if this can happen for compact manifolds, i.e., does there exist an example of compact complex ...
3
votes
1answer
230 views

Conceptually, what is the difference between the Beta function and the Beta distribution?

I have read the Wikipedia pages on the Beta function and the Beta distribution, but I'm still not sure I have a good intuition for what's going on. I'm am hoping someone will be kind enough to ...
0
votes
1answer
88 views

Name that function property

I am encountering functions of real variable with the following property: $$ f(x) = f(1/x) $$ For example, $$ f(x) = \left(x - \frac{1}{x}\right)\log^{3}{x} \qquad x > 0 $$ Is there a name for this ...
0
votes
0answers
61 views

For which $f$ does $\sum_{n=0}^{\infty} f(a_{1},…,a_{k};n) = \int_{0}^{\infty} f(a_{1},…,a_{k};x) dx$?

Do there exist real valued functions $f(a_{1},...,a_{k};x)$ with real parameters $\{a_{1},...,a_{k}\}$ such that" $$\sum_{n=0}^{\infty} f(a_{1},...,a_{k};n) = \int_{0}^{\infty} f(a_{1},...,a_{k};x) ...
5
votes
3answers
1k views

Understanding direct sum of matrices

I read the definition of direct sum on wikipedia, and got the idea that a direct sum of two matrices is a block diagonal matrix. However this does not help me understand this statement in a book. In ...
1
vote
3answers
360 views

How to prove that proj(proj(b onto a) onto a) = proj(b onto a)?

How to prove that proj(proj(b onto a) onto a) = proj(b onto a)? It makes perfect sense conceptually, but I keep going in circles when I try to prove it mathematically. Any help would be appreciated.
2
votes
1answer
88 views

One-reducibility extending to onto function

I'm working on the following problem from Soare: If $A$ is one-reducible to $B$ ($A \leq_1 B$) and $A, B$ c.e., $A$ infinite then $A$ is one-reducible to $B$ via some $f$ such that $f(A)=B$. I know ...
11
votes
2answers
419 views

Asymptotics of LCM

Let $\operatorname{LCM}(x_1,x_2,\ldots,x_n)$ be the least common multiple of the integers $x_i$. How can one find the asymptotics of $\operatorname{LCM}(f(1),f(2),\dots,f(n))$ as $n$ approaches ...
2
votes
1answer
99 views

Branching process: show $P(Z_m>k|Z_n=0) \le (G_n(0))^k, m<n,k\ge 0$

Let $(Z_n)_{n \ge 0}$ be a branching process with generating function $G_n(s)=E(s^{Z_n})$. Let $m<n$, show that $P(Z_m>k|Z_n=0) \le (G_n(0))^k, k \ge 0$. I know that the event $\{Z_n=0\}_{n \ge ...
1
vote
2answers
109 views

A problem in Poisson Processes

Let $X_n$ be the interarrival times for a Poisson process $\{N_t; t \geq 0\}$ with rate $\lambda$. Is it possible to calculate the probability $P\{ X_k \leq T \text{ for } k \le n, \sum_{k=1}^{n}{X_k} ...
1
vote
0answers
99 views

How can I calculate $\sum\limits_{i=1}^n\cfrac{n}{2^n}$ [duplicate]

Possible Duplicate: How to find the sum of this infinite series How can I calculate result of $\sum\limits_{i=1}^n\cfrac{n}{2^n}$ ?
4
votes
1answer
127 views

A sequence pointwise convergent and equibounded in $L^2(\mathbb R)$ norm is weakly $L^2(\mathbb R)$ convergent

AS is said in the title, I'm given a sequence $\{f_n\}\in L^2(\mathbb R)$ and the following hypothesis: $\{f_n\}\to 0$ pointwise and there exists a constant $C$ such that $\|f_n\|_{L^2(\mathbb ...
6
votes
4answers
2k views

Is the ideal generated by irreducible element in principal ideal domain maximal? [duplicate]

Possible Duplicate: Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible I am trying to see whether the ideal generated by irreducible element in a ...
1
vote
2answers
285 views

What is the origin of the prefix logic notation used in WFF 'N PROOF?

The classic "modern logic" game of WFF 'N PROOF uses a set of symbols to represent logical relations that I've seen used nowhere else: $C$ for then; $A$ for or; $K$ for and; $E$ for if and only if; ...
1
vote
2answers
173 views

Extended James space

The discussion in Convergence in topologies, especially the comments of GEdgar, led me to another (converse) question concerning convergence. In the paper G. A. Edgar, A long James space, in: ...

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