0
votes
0answers
10 views

How can we represent $10$ in a decimal system?

This may sound like a silly question to begin with but I'm having problems finding a proper answer. The question is generally targeting numeral systems of any base, but for simplicity, I will ...
0
votes
0answers
5 views

Properties of the inverse of unit (lower) triangular matrix

Is there any special properties about the inverse of a unit lower triangular matrix? I'm trying to prove this: $$L^{-1}=I_n + N + N^2 + ... + N^{n-1}$$ where $L$ is a unit lower triangular matrix ...
0
votes
0answers
10 views

how to prove the following recursive sequence produce relatively prime numbers

Sequence an is defined recursively: $a_1=2$ $a_{n+1}=a_n^{2}-a_n+1$ Prove that $a_i$ and $a_j$, $i\ne j$ are relatively prime. Hint: Prove that $a_{n+1} = a_1a_2…a_n + 1$ and use Euclid’s theorem. ...
0
votes
0answers
6 views

Finding an angle on a triangle inscribed in a circle

How is angle $\angle OBW_2$ equal to $\theta$? $\angle AOB$ is given as $2\theta$. Is there a theorem? or simple geometry I'm not using. Would appreciate if someone drew out how $\angle OBW_1$ is ...
-1
votes
0answers
10 views

Probability related question, Permutations, combinations

Im doing a practice problem for an upcoming test, I had a hard time figuring out this question, could anyone walk me through it?
-2
votes
0answers
16 views

Complex analysis problem (entire functions)

How to solve following problems: Find all entire functions $f$ for which $|e^z-f(z)|\ge 2$, for every $z\in\mathbb{C}$ and $f(0)=-1$. If $f$ is entire function and $|f(z)|\le |z|^{2013}-|z|+2013$ ...
-2
votes
1answer
23 views

Can anybody provide some steps on how to do this?

How do I find a basis for the given plane $x+y+z=1$ in $\mathbb{R}^3$?
0
votes
1answer
34 views

What is an intuitive way to think of Cauchy's theorem?

I am looking at a problem which involves an understanding of why a finite group $G$ has an element with order $p$ if $p$ is a prime factor of $|G|$. I have looked at several resources and proofs ...
2
votes
0answers
9 views

Understanding averaging of symplectic matrices via Haar measure

In McDuff and Salamon's Intro. to Symplectic Topology (2nd edition), there's a proof that $U(n)$ is a maximal compact subgroup of $Sp(2n)$ which I'm trying to understand. The proof uses the Haar ...
-3
votes
0answers
15 views

Product Spaces and axioms of countability

Prove that a countable product of second countable spaces is second countable
0
votes
1answer
9 views

Birational map between singular variety and smooth variety

$A$ is singular and $B$ are smooth algebraic varieties. Is it possible that $A$ is birationally equivalent to $B$? (over $\mathbb{C}$)
1
vote
0answers
9 views

Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
1
vote
0answers
6 views

Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.

Let $GL_2(Z_m)$ denote the multiplicative group of invertible $2 * 2$ matrices over the ring of integers modulo m. Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.
0
votes
0answers
14 views

reference needed for a property of the Riemann sphere

I need a reference citation for this fact, which I think is a common sense The bijective conformal mappings from the Riemann sphere to itself are Möbius transformations.
2
votes
2answers
19 views

A property of the fourier series

Show that any periodic function $f(x)$ with period $2\pi$ which is both odd and satisfies $f(\pi-x)=f(x)$ has $b_{n}=0$ for $n$ even and so has a fourier series of the form $$f(x) = ...
0
votes
0answers
7 views

Evaluating minimal polynomial over a field $F$ as a characteristic polynomial for a $F$-linear map.

I'm considering $K/F$ to be an extension of degree $n$. I've shown that for any $a\in K$, the map $\mu_a : K → K$ defined by $\mu_a(x) = ax$ for all $x\in K$, is a linear transformation of the ...
2
votes
0answers
30 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all the positive integers $m$ such that both the ratios $$ \frac{2(5^m+5)}{3^m+1}, \frac{9^m+1}{5^m+5}$$ are integers. Attempt to a solution: If the ratios are both integers, than their ...
0
votes
1answer
23 views

Inconsistent equations when using the scalar product?

I'm given that two lines $m_1$ and $m_2$ pass through the origin and have directions $i+j$ and $i+k$ respectively. I am required to find the directions of the two lines $m_3$ and $m_4$ that pass ...
0
votes
0answers
2 views

showing something supposedly obvious in the proof of the fletcher-powell algorithm

I read the paper below and understand most of it. But there is one statement that I don't follow that I guess is straightforward to show. It's not necessary to look at the paper but it has to do with ...
0
votes
0answers
25 views

Proving that the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ is irreducible modulo $p$ for every prime $p$. [duplicate]

I've computed the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ to be $x^4+10x^2+1$. I want to show that this polynomial is irreducible module $p$ for every prime $p$. How do I do ...
1
vote
1answer
6 views

Separability of the Space of all Real-Valued functions over $[a,b]$ with a Continuous First Derivative

I'm reading Neal Carothers' Real Analysis and I'm stuck on the following question: Let $f$ be real-valued, continuously differentiable function over $[a,b]$ and let $\epsilon>0$. Show that there is ...
0
votes
0answers
7 views

Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$

Let $z_1$ and $z_2$ be 2 elements of $C _∞$ . Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$ (the ...
5
votes
2answers
66 views

$ \sum_{n=1}^{\infty}a_{n} $ diverges but $ \sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}^{2}} $ sometimes converges and sometime diverges.

Let $ \lbrace a_{n}\rbrace $ be a sequence of positive terms such that $ \sum \limits_{n=1}^{\infty}a_{n} $ diverges. I am going to show that the series $$ \sum ...
0
votes
0answers
5 views

A question of quasiconvexity

Let $X$ be a $\delta$-hyperbolic space. If $g \in \mathrm{Isom}(X)$ is a hyperbolic isometry, let $\| g \|= \inf\limits_{x \in X} d(x,gx)$ denote its translation length and $\mathrm{Axis}(g)= \{ x \in ...
1
vote
0answers
8 views

Difference Operators

Let $K$ be a field. Given a map $f\colon K\longrightarrow K$, and $h\not=0$ define $\Delta_h f$ to be the map $x\longmapsto\dfrac{f(x+h)-f(x)}{h}$. Then $\Delta_h^j f$ is defined for $j=0,1,2,\dots$. ...
1
vote
0answers
9 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...
0
votes
1answer
10 views

$V_1=V(x-y)$ and $V_2=V(x+y)$ are algebraic sets

I am looking at irreducible algebraic sets. $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. At my lecture ...
-2
votes
0answers
11 views

a question in GTM 42:linear representations of finite group

I am now reading Serre's book,GTM42;and I have a problem in Page19,Theorem 6. I cannot understand the proof of this theorem, especially the second sentence in the proof has confused me:Why not X but ...
-3
votes
1answer
19 views

What is the effect in the time required to solve a problem when you double the size of the input from n to 2n?

What is the effect in the time required to solve a problem when you double the size of the input from $n$ to $2n$, assuming that the number of milliseconds the algorithm uses to solve the problem with ...
0
votes
1answer
11 views

Algebraic set - Radical Ideal - $Rad(Rad(I))=Rad(I)$

In my lecture notes we have the following: $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. It stands that ...
1
vote
1answer
25 views

Puzzle - Finding which balls are heavy

Puzzle my sister told me about, I've yet to solve it and im open to ideas. You have 6 balls, 2 red ones, 2 blue ones, and 2 green ones. Out of each pair, 1 is heavy and 1 is light (so overall you ...
1
vote
1answer
18 views

Understanding proof of The Ratio Root test

Now this is how I reason. I first try to identify which method that is used to give the proof. I am however so bad at identifying if there are any "hidden" quantifiers in the text. (if there are ...
0
votes
0answers
4 views

Decision Boundary of A Single Perception with Logistic Function

I am currently studying neural networks and have been trying to reason about this for a while to no avail. I understand that given a perceptron(such as above) with f as a step function, any ...
0
votes
1answer
9 views

Nonnegative solution to underdetermined linear system

I would like to show that the underdetermined system $Ax=b,\; x\ge 0$, with $b$ being a positive vector and $A$ being a binary matrix, has at least one solution. I've seen several other related ...
1
vote
0answers
22 views

All triangles are equilateral

So I was watching a video in which a man has "proven" that all triangles are equilateral. He also said that there is somewhere an error but I really cant find. So can someone show me it or give me a ...
1
vote
1answer
9 views

Algorithm to minimize number of trips?

Tom the shepherd and his herd of $n$ sheep have decided that it’s time they leave their home and head to a new home across the state as food is low. However, there are two issues. First off, Tom only ...
1
vote
0answers
8 views

Rank of a coherent sheaf in terms of coefficients of the Hilbert polynomial

Let $X$ be a projective scheme over a field $k$. Let $\mathcal{O}(1)$ be an ample line bundle on $X$, then the Hilbert polynomial $P(E)$ is given by $m\mapsto\chi(E ⊗ O(m))$. The explicit polynomial ...
-2
votes
0answers
14 views

compact manifold, curve,example,

Is the following true? On any compact smooth manifold $\cal{M}$, embedded in $\mathbf{R}^n$,for some $n$ by Nash theorem, for any sufficiently small $\epsilon>0$, there does not exist an infinite ...
5
votes
1answer
94 views

“Solving challenging problems does not make a good mathematician” - A Small Discussion I wanted to have with Professional Mathematicians

My professor is a really learned man, having written and published more than 18 research papers within 20 years. Hence I greatly admire him and value his opinion. He is, of course, a PhD in ...
3
votes
0answers
70 views

Is there a real problem to which $1$ radian is the answer?

I can't recall if I've ever seen any problem related to angles, in math or engineering books, that would result in an answer like $$\alpha=1 \ \ \text{radian}.$$ The answers to such questions, I ...
0
votes
4answers
33 views

Does Induction theorem fails here at Euler's conjecture?

I read in a Book written by Raymond A. Barnett and Micheal R. Ziegler the way to prove conjectures for infinite members of a given set and that is, Mathematical Induction. When I read Induction, I ...
0
votes
2answers
44 views

For which prime numbers $p$ does the congruence $x^2 + x + 1 \equiv 0 \pmod{p}$ have solutions?

For which prime numbers p does the congruence $x^2 + x + 1 \equiv 0 \pmod{p}$ have solutions? We've recently learnt about quadratic reciprocity in class, however I am not sure how to tackle this ...
1
vote
1answer
15 views

How to prove elementary identities for binomial coefficients using combinatorial arguments?

I'm in a second year discrete mathematics course, and we have identities like this $$\binom{n}{k}(n-k) = \binom{n-1}{k}n$$ and Pascal's Triangle law. Our professor said that algebraic proofs are ...
0
votes
1answer
12 views

Integral evaluation involving trignometric functions

How to explain the following equality? (Part of an integral calculation): $$\frac{2}{2\pi}\int_{-\pi}^\pi \left| \sin x \right| (\cos nx + i\sin nx) dx = \frac{4}{2\pi}\int_0^{2\pi} \sin x \cos nx ...
-1
votes
0answers
31 views

Prove that $S^{n}*S^{m}=S^{n+m+1}$ [on hold]

Prove that $S^{n}*S^{m}=S^{n+m+1}$ where $*$ is the join operation on the spheres. I think that it's intuitively clear why it's true but I don't know a formal proof of this .
1
vote
2answers
11 views

Integral extension is a finitely generated $R$-module?

Let $R$ be a commutative ring. If $b_1,\ldots,b_n$ are elements of a ring $R'$ (commutative) which are integral over $R$ then $R[b_1,\ldots,b_n]$ is a f.g. $R$-module. My question is: If ...
2
votes
0answers
12 views

Boundary on a manifold

I was wondering how I can discover if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I ...
-5
votes
0answers
15 views

Possible $C^1$ solutions of a certain Cauchy problem [on hold]

Consider the Cauchy problem of finding $u=u(x,t)$ such that \begin{align} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} &= 0 \quad (x\in \mathbb{R}, \, t>0) \\ u(x,0) &= ...
1
vote
1answer
11 views

Freely generated modules and bases

I'm trying to show that a subset $S=\{m_1,\dots,m_k\}$ of an $R$-module $M$ generates $M$ freely if and only if $S$ is a basis for $M$. I think I can see the 'if' - using the unique expression of a ...
1
vote
1answer
28 views

What does it mean that an expected value does not exist?

$X$ is a random variable with pdf $f$ and $g: \mathbb R \to \mathbb R$ is a measurable function. Before I start operating with $E[g(X)]$ I need to show that it exists. What does it take to show it? ...

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