1
vote
2answers
805 views

Analytic Geometry: Point coordinates, same distance from two points

Given are two points, $P1(x_1, y_2)$ and $P2(x_2, y_2)$, and distance $a$. Now I want to find the two points $T1$ and $T2$. $$d(P1,T1) = d(P2, T1) = a = d(P1,T2) = d(P2, T2)$$ Eg: (T1 and T2 are my ...
19
votes
1answer
783 views

Intuition for Little Picard's theorem

Little Picard's theorem is the following: Suppose $f:\mathbb{C}\rightarrow \mathbb{C}$ is entire. Then either 1) $f$ is constant 2) $f$ is surjective or 3) $f$ is onto $\mathbb{C}-\{p\}$ for ...
11
votes
3answers
784 views

Proving the countability of algebraic numbers

I am trying to prove that algebraic numbers are countably infinite, and I have a hint to use: after fixing the degree of the polynomial, consider summing the absolute values of its integer ...
1
vote
1answer
77 views

Formula for $\sum_{i = 1}^{n} \frac{i}{K - i}$

How can I find a formula for $\sum_{i = 1}^{n} \frac{i}{K - i}$, $K \in \mathbb{Z}$, $K > n$?
3
votes
1answer
251 views

Entropy of $X =\{1,2,\ldots,\infty\}$ with the probability of $\{1/2^1,1/2^2,\ldots,1/2^\infty\}$?

I'm studing for an information theory exam, maybe some of you can help me here with an exercise. What's the entropy of $X$ as $\{1,2,\ldots,n\}$ ($n$=infinity) where the probabilities are $P \{1/2^1, ...
28
votes
4answers
735 views

How should I approach taking math tests?

I always do bad on all my math tests yet I do great on projects and homework. Also I like doing research. For exams that have proof based questions I just freeze up under pressure. I just can't do ...
5
votes
1answer
275 views

Dominant finite morphism and finite algebraic extension

I don't know how to prove the following proposition. If two varieties $X$ and $Y$ are irreducible, a morphism $\phi: X \rightarrow Y$ is dominant and finite, then $K(X)$ is a finite algebraic ...
2
votes
2answers
228 views

Scoring system for two-party game

Suppose we have a game that operates with '$a~$' players for one party, and '$b~$' players on the opposing party. Each party has a long-run average win percentage for this game, with $A\%$ and ...
22
votes
3answers
1k views

Math without infinity

Does math require a concept of infinity? For instance if I wanted to take the limit of $f(x)$ as $x \rightarrow \infty$, I could use the substitution $x=1/y$ and take the limit as $y\rightarrow 0^+$. ...
3
votes
2answers
423 views

Algorithm to determine if a Diophantine Equation has an infinite number of solutions

In their paper , Marker and Slaman, proved the decidability of the the theory of the natural numbers with the quantifier "for all but finitely many", One can obviously encode the question of whether ...
1
vote
1answer
366 views

Maximal subspace that a quadratic form is non-negative on

Let $q: \mathbb{R^3}\to \mathbb{R},\ q(x_1, x_2, x_3)=-5x_1^2-x_2^2-x_3^2+2x_1x_3+2x_2x_3-4x_1x_2$ be a quadratic form. Find a maximal subspace $W \subseteq \mathbb{R^3}$ such that $\forall w ...
4
votes
2answers
585 views

Continuous representatives in Sobolev Spaces

My question arise from the study of the possible extensions of Rademacher's Theorem to the Sobolev Space $W^{1,p}(\Omega)$, with $\Omega\subset \mathbb{R}^n$. In specific I'm studying the proof of the ...
3
votes
4answers
1k views

particular solution of $4y''-y= \sin(x)\cdot \cos(x/2)$

So I'm working with a nonhomogeneous second order differential equation: $$4y''-y=\sin(x)\cos(x/2).$$ I know that the general solution, $y$, equals $y_c + y_p$ where $y_c$ is the general ...
15
votes
5answers
668 views

Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
-4
votes
2answers
1k views

The last number of infinity [closed]

First of I am not a math guru, so it might be a dumb question, but today in the metro I was bored, and started thinking what is the last number of the infinity, and came up with this theory, that the ...
2
votes
4answers
516 views

Geometry/ Similar Triangles Problem

Consider the trangle shown below with vertices A, B, C where point D lies on the side AB, point E lies on the side BC and point F lies on the side AC and the three lines AE, BF, and CD intersect at a ...
4
votes
1answer
361 views

Finding eigenfunctions and eigenvalues

Let $K$ be the integral operator defined by $$ (Kf)(x)=\int_0^1 u(x)v(y)f(y) dy $$ for some continuous functions $u,v$ in the Hilbert space with inner product $\langle f,g \rangle = \int_0^1 f(x)^* ...
2
votes
2answers
109 views

composition of limits it´s not always valid in the final value

Suppose we have a function $ f: X \to Y $ such that $ d_1 (f (x), y_0) <A_1 $ whenever $ d_0(x, x_0) <A_0 $ and so such a function $ g:Y \to W $ such that $ d_2 (g (y), w_0) <A_3 $ if $ d_1 ...
3
votes
0answers
518 views

What it takes to a mathematician [closed]

I am a junior in Caltech's mathematics department. I keep being harassed by some non-mathematicians and fellow students that I do not have what it takes to do mathematics. They say that one must ...
0
votes
2answers
181 views

When are two modular functions equal?

Suppose $f, g \in M_{k}(\Gamma)$ for some $\Gamma$ a congruence subgroup of $PSL_{2}(\mathbb{Z})$. If $k \geq 1$ then it is a finite computation to determine whether or not $f = g$. What if $k = 0$, ...
3
votes
4answers
416 views

Help with integrating 101: $\int y \ln{y}\,dy$

Would appreciate it if someone would please help me solve this $$\int y\;\ln y\, dy$$ taking time to explain reason for each step taken. Thanks in advance!
5
votes
3answers
176 views

Does solution of $ x=\sum_{n=0}^\infty e^{-A_n/x}$ exist?

Usually when working with indefinite sums, I want to work out the sum or whether it convergence. But now I encountered a problem they other way around and I'm clueless... Is there even a general ...
1
vote
1answer
128 views

Help me formalize this calculation

I needed to find the number of five digits numbers that are made of numbers from $0,1,2,3,4,5$ and are divisble by 3. One of the proper methods can be, that $0+1+2+3+4+5 = 15$ So we can pick out ...
3
votes
2answers
316 views

Calculate measurements for a diagonal fence beam

Given the width W and the height H of a rectangle, and the thickness T of a beam extending exactly from the upper left corner to the lower right corner as shown, how do I solve for length X and angle ...
3
votes
1answer
209 views

fundamental representation of $\ sl(l+1,F)$

This problem concerns the topic representation theory of Lie Algebras. The main purpose of the exercise is to study the form of the fundamental dominant weights of a Lie Algebra. I would be very ...
1
vote
1answer
294 views

Unbiased estimator and Variance

I am having a hard time trying to solve this problem. I don't know how to start it. Any help would be greatly appreciated. Let T be any unbiased estimator of $\tau(\theta),$ and let $W$ be a ...
3
votes
0answers
192 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
6
votes
3answers
521 views

Finding a Möbius transformation

Let $f$ be a holomorphic mapping from {$z:\Re(z)>0$} into itself. Let $1$ be a fixed point of $f$. In addition suppose that $\left|\frac{f(2)-1}{f(2)+1}\right|=\frac13$. I want to show that ...
8
votes
3answers
874 views

Is the empty graph connected?

Is the empty graph always connected ? I've looked through some sources (for example Diestels "Graph theory") and this special case seems to be ommited. What is the general opinion for this case ? As ...
6
votes
3answers
363 views

Given a real function $g$ satisfying certain conditions, can we construct a convex $h$ with $h \le g$?

The following is Exercise 8 from Chapter 3 of Rudin's Real and Complex Analysis (not a homework problem, just for fun). Let $g$ be a positive function on $(0, 1)$ such that $g(x) \to \infty$ as $x ...
0
votes
2answers
143 views

show some subset of a manifold is a neighborhood

let $X$ be a $n$-manifold. let $A=\{(x,y,z) \, |\,x=y\}$. I want to see if $A$ is a submanifold of $X^3$. Consider the map $\Delta\times 1:X\times X\rightarrow X \times X\times X;\, (x,y)\mapsto ...
3
votes
1answer
372 views

Indefinite integral $\int^{\infty}_{0}\frac{x}{x^4+1}dx$ via residues

I want to compute $\displaystyle \int^{\infty}_{0}\frac{x}{x^4+1}dx$ using the residue theorem. The poles in the upper half plane are: location: $\large e^{\frac{\pi i}4}$, order: 1, residue: ...
12
votes
1answer
2k views

Proof of Euler's Theorem without abstract algebra?

Every proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. ...
5
votes
3answers
733 views

Questions on perfect squares

I recently attended a test which ask me two question based on perfect squares; here they are: $1.$ How many even perfect squares between $1000$ and $5000$ are divisible by both $5$ and $9$? $2.$ Can ...
2
votes
0answers
222 views

Is the product of Borel spaces a Borel space?

Let $(S_i,\mathbf{S}_i)$ be a sequence of Borel spaces, i.e. such that for all $i$ there is a 1-1 bimeasurable map $\varphi_i:S_i\to T_i$, where $T_i$ is a Borel subset of [0,1]. Is $\prod_{i=1}^n ...
3
votes
0answers
233 views

Understanding orientability of vector bundles

I'm having trouble understanding how orientability of vector bundles work. The book I'm reading, Spivak's A comprehensive introduction to differential geometry, is not very clear on this. Edit: ...
2
votes
2answers
324 views

is any subset of a manifold a submanifold?

by definition a submanifold is a subset of a manifold which is itself a manifold. consider $A$ a subset of an $n$-manifold $M$. a neighborhood of $x\in A$ is $\mathbb R^n$ since $x$ is an element of ...
5
votes
1answer
773 views

Career advice for MSc [closed]

I got into an interesting situation and I would love if you could help me with an advice, opinion or something. I have an offer for an MSc study (one year) in applied mathematics (with a scholarship) ...
4
votes
1answer
554 views

What is the connection between the definition of complete intersection variety and complete intersection ring?

An algebraic variety is called a complete intersection if its defining ideal is generated by codimension many polynomials. A Noetherian local ring $R$ is called a complete intersection if its ...
0
votes
1answer
325 views

Complexity of $T(n)=\sqrt{n}T(\sqrt{n})+n$

I tried to find the complexity of this recursion equation: $T(n)=\sqrt{n}T(\sqrt{n})+n$, by doing couple of iterations and getting a general idea, but I completely got lost. I'd really love your ...
56
votes
2answers
4k views

Open problems in General Relativity

I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. I mean is there something that still needs to be ...
2
votes
4answers
221 views

Need help in Taylor series expansion

In this question, I have to write Taylor's series expansion of the function $f(x) = ln(x+n)$ about x = 0, where n ≠ 0 is a known constant. I have done the following: But my professor handed me back ...
0
votes
1answer
277 views

non transitive subgroups of the symmetric group

How to show that any non transitive subgroup of the symmetric group $S_n$ is up to conjugation contained in a Young subgroup $S_k\times S_{n-k}$? Take $\mathbb Z_3$ the subgroup of $S_3$ generated by ...
0
votes
1answer
64 views

Conditions for some inequality

Suppose I have $f(x)A+g(x)B+h(x)C \ge 0$. Here $A,B,C$ can be positive or negative and $f,g,h$ are nonnegative. I would like to obtain a condition for $f,g,$ and $h$ such that ...
1
vote
1answer
121 views

$T_{3}=\Theta(n^{0.99}) ,T_{2}=\Theta(n^{\log\log n}),T_{1}=\Theta\left(\frac{n}{\log n}\right)$

$T_{3}=\Theta(n^{0.99}),\quad T_{2}=\Theta(n^{\log\log n}),\quad T_{1}=\Theta \left(\frac{n}{\log n}\right)$ I need to decide what is the relation (ratio?) between $ T_{1},\, T_{2},\, T_{3}$? So by ...
10
votes
3answers
740 views

Proof that $\dim(U \times V) = \dim U + \dim V$.

The following theorem in Serge Lang's Linear Algebra is left as an exercise, namely, Let $U$ and $V$ be finite dimensional vector spaces over a field $K$, where $\dim U = n$ and $\dim V = m$. Then ...
10
votes
1answer
560 views

Prove that minimum of $\lambda \sin \theta + (1 - \lambda) \cos \theta \le -\dfrac{1}{\sqrt 2}$

I need a little nudge to the finish for the last bit of this problem. Express $\lambda \sin \theta + (1 - \lambda) \cos \theta$ in the form $R \sin (\theta + \phi)$, where $R(R>0)$ and $\tan ...
0
votes
2answers
158 views

Combinations Help

I have an application where I iterate through all k-combinations of a set of size n. For example here I have listed all k-combinations for when n is 4. Also I have separated each list of combinations ...
8
votes
1answer
396 views

Numb3rs Challenge

I am by no means a mathmatician. I guess you could say that I am a mathematically inclined individual, but never made anything of it until I was in my 30's and became a software engineer. Although ...
1
vote
1answer
84 views

Simple estimation $e^{a\sqrt{r}} > r$

I want to prove a simple theorem about contour integration via residues and I need the following estimation: $e^{a\sqrt{r}} > r$ for any real a > 0 and r >> 0. Is this true? If so, what is an ...

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