1
vote
2answers
86 views

Question on rank function.

In a previous question I asked about the fiber $M(P)=M_P / PM_P$ where $M$ is an $A$-module and $P$ a prime ideal of $A$. Later I introduced the rank function $$rk_M : \text{Spec} A \to \mathbb{N} ...
5
votes
1answer
180 views

Moscow puzzle. Number lattice and number rearrangement. Quicker solution?

I have already considered chains of numbers like $4-19, 19-9, 9-22$, to solve the problem and got the answer. However just out of curiosity, can anyone think of a better/quicker solution? (answer ...
3
votes
1answer
170 views

A function $f$ such that $f \in L_1$ but $f \notin L_p$ for $p>1$ [duplicate]

I want find a function $f: [0,1] \mapsto \mathbb{R}$ such that $f \in L_1[0,1]$ but $f \notin L_p[0,1]$ for all $p>1$. My attempts: First I thought in the family of functions $\frac{1}{x^\alpha}$ ...
1
vote
2answers
45 views

Precomposing a rational function $f$ with a birational isomorphism $g$ to make $f\circ g$ regular?

I think I have a gap in my understanding, Suppose $Y$ and $Z$ are quasi-projective varieties, and that $Y$ is irreducible. Suppose we have a rational map $f\colon Y\to Z$. Then we know there ...
1
vote
3answers
116 views

Integral $\int \frac{dx}{\sqrt{x^2+x+1}}$

$$\int \frac{dx}{\sqrt{x^2+x+1}}$$ I tried to use trigonometric substitution, let $x=\tan^2 U$,but it is difficult to proceed, can anyone help me how to integrate this indefinite integral?
1
vote
1answer
58 views

Why $X$ contains a countable $\pi$-basis?

I don't understand the following statement. First, I write what a $\pi$-bases means. Let $X$ be a topological space and $\mathcal{B}$ a family of non-empty open sets. We call $\mathcal{B}$ a ...
7
votes
1answer
190 views

What can we say about a vector bundle with trivial unit sphere bundle?

If you want to avoid the back story to this question, feel free to skip the paragraphs between the horizontal lines. Yesterday Georges Elencwajg asked me the following question (I'm paraphrasing): ...
2
votes
1answer
78 views

Convergence of series in a Hilbert Space

I'm hoping for some help on the following question. I haven't gotten very far: Let $\{h_n\}_{n\geq 1}$ be a sequence of vectors in a Hilbert space $H$ with the property that $(h_n-h_m)\perp h_m$ ...
4
votes
2answers
83 views

$\{p_i\}$ generate the $k$-algebra of symmetric polynomials in $k[t_1, \dots, t_n]$ and are algebraically independent over $k$

Let $k$ be a field of characteristic $0$. For $j \ge 0$, let $p_j = t_1^j + \dots + t_n^j \in k[t_1, \dots, t_n]$. Prove that $p_1, \dots, p_n$ generate the $k$-algebra of symmetric polynomials in ...
1
vote
2answers
81 views

$A_n$ is an increasing set. Then what is $\ A_1\setminus A_0$ if $A_0=\emptyset$ .

Highlighted in green. I am not sure what $\ A_1\setminus A_0$ equals to... if $A_0=\emptyset$. I suspect that $\ A_1\setminus A_0$ equals $A_1$. The fact that $A_0=\emptyset$ makes me reason that ...
5
votes
1answer
97 views

Find the Poincare Dual of a ray in $\mathbb{R}^2-\{0\}$

This is example-exercise 5.16 in Bott and Tu (which I'm independently reading through.) The problem states: Let $M=\mathbb{R}^2-\{0\}$, and $X\subseteq M$ be the closed submanifold ...
0
votes
2answers
266 views

Problem in game theory related to traffic networks

I have learnt game theory for a short period of time and I am not familiar with multi-player non-zero sum games. Here is a problem from my book which I am stuck: In this road network below each of ...
1
vote
2answers
92 views

Write ODE in Polar Coordinates [closed]

I want to write this ODE system in polar coordinates (r,$\theta$). $$\dot x =x-y-x^3 $$ $$\dot y = x+y-y^3$$
2
votes
1answer
70 views

Field Extensions and Number of Isomorphisms

The picture above is from Dummit and Foote, Third Edition. Later in the book, we find Clearly, the condition of equality is not necessary, as seen by taking the polynomial $ f(x) = ( x ^{2} ...
2
votes
1answer
124 views

Prove an infinite sum to be holomorphic.

Let $\{a_n \}$ be a sequence of real numbers such that for any $z \in \mathbb{C}$ with $\operatorname{Im}z>0$, the series $$ h(z)=\sum_{n=1}^\infty a_n \sin(nz) $$ is convergent. Show that $h$ is ...
0
votes
0answers
54 views

Conjectured optimal running time for integer factorization

While detecting prime numbers is computationally fast ($O(\log^3 n)$), the fastest known algorithms to split a composite number into its prime factor are very slow (RSA cryptography relies on this ...
1
vote
2answers
78 views

An improper integrals related to probability, $\int_0^\infty\frac1y \exp(\frac{-x_0}y-y)\,dy$

How can I calculate the integral $$\int_0^\infty{\frac1y e^{\frac{-x_0}y-y}}dy$$ in terms of well-known constants and functions? I used some fundamental techniques of integration but got nothing.
0
votes
0answers
66 views

What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
2
votes
1answer
47 views

Why a convergent succesion does not have the same homotopy type of a CW-Complex?

The question is pretty much in the title; If my space is $\{1/n\}_{n\in \mathbb{N}} \cup \{0\} $ why it isn't homotopically equivalent to a CW-Complex?
3
votes
2answers
98 views

Infinitely many systems of $23$ consecutive integers

Prove that there are infinitely many systems of $23$ consecutive integers whose sum of squares is a perfect square. My try: $$(n-11)^2+\cdots+(n+11)^2=23n^2+1012=23(n^2+44)=m^2$$ so $m=23k$ , ...
1
vote
1answer
21 views

Question regarding cyclic subspaces

Let $T:V\to V$ be a linear operator on a finite dimensional field and let $\vec v \in V$ ($\vec v \neq \vec 0$) such that $\operatorname{Span}${$\vec v, T(\vec v), T^2(\vec v),...$}$\neq V$ If $\vec ...
1
vote
1answer
55 views

Generators of $H^1 (T)$: take two

Previously, I asked about how to prove that $dx + dy$ is a generator of the de Rham cohomology group of the torus. Now it occurred to me that $dx$ and $dy$ are both also generators of $H^1(T)$. ...
3
votes
1answer
199 views

Using Extended Rice's Theorem to Prove Decidability

I have a Turing Machine M. Let L be the set of all strings representing the encoding of M that has input alphabet {1,2}, where M accepts infinitely many strings that start with 1 and finitely many ...
0
votes
3answers
61 views

Show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$ [closed]

How could we show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$ ?? Could you give me some hints??
0
votes
1answer
590 views

Optimization problem: rectangle inscribed between two parabolas

Consider the region R in the 1st quadrant bounded on the left by $y=x^2$, on the right by $y=(x-5)^2$ and below by the x-axis. Find the area of the largest rectangle inscribed inside region $R$. I ...
3
votes
1answer
69 views

Density of a set around $0$ and on $\mathbb{R}$

In this question, we prove that $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ is dense in $\mathbb{R}$ by proving that is it dense around $0$. Why is that enough to prove that it is dense on $\mathbb{R}$ ?
1
vote
1answer
32 views

Obtaining a Transformed Matrix

I have a matrix $$m = \begin{bmatrix} 0 & 2 & 1 & 4 & 3 \\ 1 & 0 & 3 & 2 & 4 \\ 3 & 1 & 0 & 2 & 4 \\ 4 & 3 & 1 & 0 & 2 \\ 4 & 3 ...
7
votes
3answers
885 views

Is there a generalization of the Lagrange polynomial to 3D?

What is a way to construct a smooth polynomial surface ($\mathbb{R}^2 \rightarrow \mathbb{R}$) with Lagrange-polynomial properties in every partial derivative? I want to try this for image ...
3
votes
1answer
78 views

Compact Lie group with non discrete center?

Could someone please give me an example of a compact Lie group with non discrete center which is not just a product of a group with a torus?
1
vote
1answer
101 views

SOCP with a norm constraint

Is it possible to convert this optimization problem into a SOCP: \begin{eqnarray} \min && c^T x \\ s.t. && \|A_ix + b_i \|_2 \le c_i^T x + d_i \\ && \| Dx \|_2 = g ...
3
votes
1answer
207 views

How to solve $ \left \lfloor x^2 - x - 2 \right \rfloor = \left \lfloor x \right \rfloor $

I need some help to solve the next equation: $$ \left \lfloor x^2 - x - 2 \right \rfloor = \left \lfloor x \right \rfloor $$ Where $ \left \lfloor \cdot \right \rfloor $ is the floor function. What ...
2
votes
1answer
99 views

If 10 is not a solitary number, what properties would a friend of 10 have

It is of course an unsolved problem if 10 is solitary or not, but it is conjectured that it is. (See definition of friendly and solitary number on wiki: http://en.wikipedia.org/wiki/Friendly_number) ...
4
votes
0answers
48 views

Newton series and Fourier transform, is there an analogy?

Fourier expansion for a function: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$ Newton series expansion of a function: ...
1
vote
1answer
34 views

The set where $|g|>\|g\|_{L^\infty}$ has measure zero

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions in $L^\infty$ where $f_{n}\longrightarrow f\in L^{\infty}$. Set $E_{n_0} = \{x \in [0,1], |f_{n_{0}}(x) - f(x)| > \|f_{n_0} - ...
0
votes
0answers
35 views

How to prove that this stochastic matrix has a limiting distribution

I have the following stochastic matrix with $p_{ij} > 0$ and $\sum_j p_{ij} = 1$ $$ P = \begin{bmatrix} p_{11} & p_{12} & 0 & 0 & 0 & 0 \\ p_{21} & ...
1
vote
0answers
43 views

Determining functions involved in a definite integral using substitution;

The question is the following: Consider the continuous function $\phi \colon\Bbb R\to \Bbb R$ and another function $f \colon\Bbb R\setminus\{0\}\to \Bbb R$ defined as follow: $$ f(x) = \int_0^{1/x} ...
6
votes
2answers
129 views

Why isn't it necessary to postulate the existence of $1$?

These are the Peano axioms, I'll focus on the second one now: If $a$ is a number, the successor of $a$ is a number. Basically, here is defined the successor function $S(n)=n+1$. My question is, ...
1
vote
0answers
93 views

Proof using Gronwall inequality

Using the Gronwall inequality, how do I show that if: $$\partial_t \log (1 + u(t) ^2) <1$$ Such that $u(t)$ is a solution of the differential equation : $$\dot x(t) = - ax(t)+ \log ...
2
votes
1answer
153 views

Wikipedia article about T-joins

The Wikipedia article about T-joins explains: Let T be a subset of the vertex set of a graph. An edge set is called a T-join if in the induced subgraph of this edge set, the collection of all the ...
0
votes
1answer
82 views

Question on exercise of ideal of a point

The question was to find the ideal of a point $(\sqrt{2},\sqrt{3})$ in $\mathbb{Q}[X,Y]$ and its conjugates in $\mathbb{C}^2$. Is is correct to say that the ideal of a point is ...
2
votes
1answer
114 views

if $f(x)$ is differentiable at a x, prove that: $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$

If $f(x)$ is differentiable at x, I need to prove that $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$ exist and is finite. so if $f(x)$ is differentiable at a $x$, the difference quotient exist for this ...
2
votes
1answer
141 views

Countable state Markov chain with multiple transitions

I'm searching for hints on how to analyze the following Markov chain. I can solve for the steady state probabilities numerically by using a finite transition matrix. However, I would like to have an ...
0
votes
1answer
86 views

Complexity of polynomial simplification into standard form

I am curious to know if any given $n$-variable polynomial in $\mathbb{R}[\mathbf{x}]$, not in standard form, can be simplified by an algorithm in polynomial time. The polynomial is $$ p(\mathbf{x}) = ...
1
vote
0answers
43 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
1
vote
1answer
65 views

Check for valid statistics

Alice conducted a voting about N of his opinions. A[i] percent of people voted for opinion number i. This statistics is called valid if sum of all A[i] is equal to 100. Now let us define rounding up ...
0
votes
0answers
46 views

Dimension of generalized eigenspace.

Let $T \in \text{Hom}_F(V,V)$, suppose the characteristic polynomial of $T$, $c_T(x) = (x- \lambda)^kp(x)$, where $p(\lambda) \neq 0$, show that $\text{dim}_F (E_{\lambda}^\infty) = k$, where ...
1
vote
1answer
169 views

Super conic sections?

I know graphs of the form $A x^2 + B xy + C y^2 + D x + E y + F = 0$ are conic sections. But what would happen if I changed the highest power to 3? Would this be a new 3D shape, a 4D version of it, or ...
6
votes
2answers
113 views

Show that $\bigl| e^x + e^{-x}-2-x^2\bigr| \le {x^4 \over 6} $ for $|x| \leq 1$

My try at it $$ \left| e^x + e^{-x}-2-x^2\right| \iff | f(x) - p_2(x)| = |R_3(x)| $$ where $ f(x) = e^x + e^{-x} $ and $ |x| \le 1 $ This gets me $$ |R_3(x)| \le (e-e^{-1}) {x^3 \over 6} $$ This ...
3
votes
3answers
108 views

Orthogonal complement propertiss

I'm required to proove that $W^{\perp} + U^{\perp} \subseteq (W \cap U)^{\perp}$. I've already proven $U \subseteq W \to W^{\perp} \subseteq U^{\perp}$ and $(U+W)^{\perp} = W^{\perp} \cap U^{\perp}$. ...
2
votes
1answer
65 views

Continuous functions and existence of a root

Let $\, f:[1,2] \rightarrow \mathbb R$ be a continuous function such that for every $n$ $\in$ $\mathbb N, \exists$ $x \in [1,2]$ with $\ |f(x)| < \frac 1n$ Show that $ \exists \;c \in [1,2]$ such ...

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