3
votes
2answers
104 views

Find the formula of a sequence using generating functions

The given sequence is $\sum_{k=1}^{n}kx^{k}$. How can I do this? I think I should start with the already known generating function: $A(x) = \sum a^{n}x^{n} = \frac{1}{1 - ax}$ And then I should do ...
51
votes
2answers
2k views

Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, ...
1
vote
1answer
53 views

How to verify whether $(C_{00},\|\cdot\|_p)$ is complete

How to verify whether $C_{00}=\{(x_n):\text{All but finitely many terms are }0\}$ is complete with respect to $p$-norm given by $$\|(x_n)\|_p=\left(\sum_{n=1}^\infty|x_n|^p\right)^{1/p}$$ where $1\le ...
0
votes
2answers
821 views

Calculate distance from plane to parallel plane in O using vector and normal

I'm trying to figure out what's the best method to get the distance between two planes where i have the normalized vector of the plane and a point in the plane. What I want to do is to create a ...
0
votes
1answer
189 views

Finding the value of n if $2^{200}-2^{192}31 +2^n$ is perfect square

Problem : Finding the value of n if $2^{200}-2^{192}31 +2^n$ is perfect square Solution : $2^{200}-2^{192}31 +2^n = 2^{192}(2^8-31)+2^n$ = $2^{192}(256-31)+2^n = 2^{192}(225)+2^n$ $\therefore$ ...
4
votes
0answers
140 views

Differential calculus on Banach space

I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ...
4
votes
1answer
108 views

Does there exist such a sequence?

Does there exist a infinite sequence of positive integers $a_n$. Such that $ ((n|a_n) | \forall n)$ and $\left(\sum_{n=1}^\infty \frac {1}{a_n} =1\right)$ , what if we replace 1 with a positive real ...
1
vote
2answers
64 views

Finding unknown when $0/0$

Here Inderteminate(variable) written that $$"x={a-2\over 3-b}$$ while for b = 3 the problem either has no solution at all, or, if also a = 2, it admits any value for x as solution." $$$$My question ...
2
votes
2answers
78 views

Let $G$ be a planar (simple) graph with $m (≥2)$ edges and $f$ faces. Prove that $f ≤ \frac{2}{3} m$.

I am a first year math student, and I'm studying some exercises for my graph theory exam. This exercise I didn't understand very well: Let $G$ be a planar (simple) graph with $m (≥2)$ edges and ...
1
vote
1answer
627 views

Expansion of power series for $\frac{\ln(1-x)}{1+x}$

My Problem is to expand $f(x)=\dfrac{\ln(1-x)}{1+x}$ into a power series. My Approach: from looking onto the Graphs of this function, i know, for rising x the y is falling towards Zero, without ...
2
votes
1answer
173 views

Antichain and predense set

I am trying to prove $A$ is a maximal antichain i ff $A$ is predense. It is problem 16 from link here. If $A$ is maximal antichain then is easy to prove that $A$ is predense. I stuck on proof ...
3
votes
2answers
313 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
4
votes
1answer
138 views

Why does $ \frac{1}{2\pi i} \int_{\gamma} \frac{f'(z)}{f(z)} dz = \eta( f \circ \gamma, 0) $ ($ \gamma $ is a regular closed curve)

My teacher wrote the above equation on the board the other day and acted like it was obvious. Here, $ \eta $ is the winding number of $ f \circ \gamma $. Here is the justification he gave: $$ ...
4
votes
1answer
202 views

Volume of a hypersphere

We know that the area of a circle (2-D) =$\pi r^{2}$ and the volume of a sphere (3-D)= $\dfrac{4}{3}\pi r^{3}$. Question:What is the "volume"(or whatever that is called) of a n-dimensional sphere? ...
3
votes
1answer
49 views

Elimination of tickets

There are 8 tickets numbered from 1 to 8. They are randomly paired, and larger numbered ticket of each pair is eliminated, leaving only four tickets (the smaller ones in each pair). This process is ...
3
votes
2answers
70 views

How to show $\sum_{d\mid n} \frac{\mu^2(d)}{d} =\prod_{p|n} \left(1+\frac{1}{p}\right)$?

how to prove: $$\sum_{d\mid n} \frac{\mu^2(d)}{d} =\prod_{p|n} \left(1+\frac{1}{p}\right)$$ $\mu : \Bbb N\rightarrow \Bbb R$ $\mu(1)=1$ $ \mu(n)= \begin{cases} 0 &,\;\;\; \text{if $\,n\,$ is ...
4
votes
2answers
3k views

Are regular languages necessarily deterministic context-free languages?

The original problem Suppose M is DCFL (Deterministic Context Free Language) and N is a regular language. Answer the following questions and justify your answers. a) Is M-N necessarily context-free? ...
3
votes
1answer
239 views

About Henstock integrable vector-valued function

In what follows, $X$ is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family $P$ of all continuous seminorms on $X$. We consider the following ...
0
votes
1answer
37 views

Number of Jordan cells size $j+1$

Could you tell me how to prove that the number of Jordan cells of size $j+1$ is equal to $\dim V^j/V^{j-1}$(quotient space)? $V^j = \ker (f-\lambda)^j$ I know that for $U_1, U_2, ..., U_s$ - Jordan ...
3
votes
3answers
310 views

Book recommendation for associative algebras

Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment. ...
35
votes
1answer
540 views

A ring isomorphic to its finite polynomial rings but not to its infinite one.

I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious: ...
0
votes
2answers
58 views

Proof: let $f:A \to B$ with $f$ bijective, then $f{\restriction_{C} }: C \to B$ is bijective

I need the proof of following: "let $f:A \to B$ with $f$ bijective, then $f{\restriction_{C} }: C \to B$ is bijective" Thanks in advance
1
vote
0answers
111 views

Generators in $p$-groups

Let $G$ be a non-abelian finite $p$-group such that $G/[G,G]$ is elementary abelian group of order at least $p^3$. Let $S=\{\bar{x_1},\bar{x_2},\bar{x}_3,\cdots,\bar{x}_n\}$ be independent generators ...
3
votes
1answer
69 views

how to find $n| 3^n+1$

this a problem in: Waclaw Sierpinski 250 problems in elementary number theory 1971 if $n\in \Bbb N , n \gt1$ ,how to find : $$n\mid3^n+1$$ the solution of book is : There is only one such odd ...
1
vote
2answers
381 views

is $\sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right)<\infty$?

I don't know why but I'm having a hard time determining whether this series $$ \sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right) $$ converges to a real limit. ...
4
votes
2answers
980 views

Having trouble understand the proof of Rouché's Theorem

I am trying to understand this proof of Rouché's theorem, but I am missing the logic of the last and most crucial step. Here are the assumptions: Suppose that $ f $ and $ g $ are analytic inside ...
1
vote
1answer
35 views

Partitions of an interval and convergence of nets

Let $\mathscr{T}$ be the set of partitions $\tau = (\tau_0 = 0 < \tau_1 < \dots < \tau_N = 1)$ of the interval $[0,1]$ (where $N$ is not fixed). This becomes a directed set by setting $\tau ...
0
votes
1answer
213 views

Canonical Markov Process

Let $X$ be a canonical, right-continuous Markov process with values in a Polish state space $E$, equipped with Borel-$\sigma$-algebra $\mathcal{E}$ and we assume that $t\rightarrow E_{X_{t}}f(X_{s})$ ...
1
vote
0answers
109 views

(Basic) question regarding Einstein-Hilbert-functional / total scalar curvature

I got a question regarding the total scalar curvature / Einstein-Hilbert-functional. I found it in Peter Topping's book "Lectures on the Ricci flow" (you can download it for free here: ...
3
votes
3answers
163 views

How does the differential $df$ act on an element of $T_pM$?

Let $f$ be a smooth real valued function on a smooth manifold $M$. The differential of $f$ is the covector field $df$ defined by $$df_p(v) = v(f)$$ where $v \in T_pM$ and where we are now thinking ...
1
vote
3answers
234 views

Find the inverse Laplace transform of $f(t) = \int_t^\infty \frac{e^{ - u}}{u}du$

Find the inverse Laplace transform of the integral:$$f(t) = \int_t^\infty {\frac{{{e^{ - u}}}}{u}du} $$ If the integral: $$f(t) = \int_0^\infty {\frac{{{e^{ - u}}}}{u}du} $$ I had done. However the ...
1
vote
1answer
111 views

Understanding topological and manifold boundaries on the real line

Let $M$ be the subset $[0,1)$ $∪ $ {$2$} of the real line. Find its topological boundary $\mathrm{bd}(M)$ and its manifold boundary $\partial M$. I know that to find the topological boundary, I ...
1
vote
0answers
99 views

Girsanov kernel moments

Let $Z_t=e^{\int_0^tq_tdB_t-\frac{1}{2}\int_0^tq^2_tdt}$, where $(q_t)_{t\geq0}$ is a predictable process, and $(B_t)_{t\geq0}$ a $\mathbb{P}$-Brownian motion. In particular, Novikov's condition ...
5
votes
1answer
138 views

Showing that $\int_{0}^{\pi/2}\frac{1+\sin(x)-\sin^2x}{e^{\sin(x)}+\cos(x)}=\log 2$

I don't see how to prove that $$\int\limits_{0}^{\pi/2}\frac{1+\sin(x)-\sin^2(x)}{e^{\sin(x)}+\cos(x)}\,\mathrm dx=\log 2$$ Integrating by parts is of no help. The same with variable change. Should I ...
1
vote
0answers
93 views

Proof of Lie theorem on solvable Lie algebra

I am reading a book of Helgason. As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that $v$ is an eigenvector of any element of $g$. I can follow the proof in ...
2
votes
2answers
286 views

Conditions that Roots of a Polynomial be Less than Unity

Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
1
vote
1answer
125 views

The infinity version of Blumenthal's 0-1 law

Blumenthal's 0-1 law states that on the space of continuous maps with domain $[0, \infty)$ with the appropriate (Wiener) measure making the coordinate maps Brownian motions starting at $x$, any event ...
4
votes
3answers
224 views

Is $\sqrt{-1}$ positive or negative?

Do the concept of positive or negative make sense in this case? I remember that $\mathbb{R}^2$ has four quadrants thus ordered pairs of numbers could be $(+,+),(+,-),(-,-)(-,+)$, I presume that ...
2
votes
1answer
107 views

$xy$ itself square in this particular logic

I would like to know the solution or procedure to find the exact analysis/solution of one of my observation. let $x = a^2$ and $y = b^2$, then can we express $xy$ (concatenation of $x$ and $y$) as ...
3
votes
3answers
65 views

A problem on recurrence relation

Consider the sequence $$a_n = a_{n-1} a_{n-2} +n$$ for $n \geq 2$, with $a_0 = 1$ and $a_1 = 1$. Is $a_{2011}$ odd? By writing all the terms of the sequence I see that $a_n$ is odd when $n$ is odd ...
8
votes
2answers
489 views

Finding $\lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$ for continuous $f:[0,1]\to[0,\infty)$ [duplicate]

Find $$\lim_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$$if $f:[0,1]\rightarrow(0,\infty)$ is a continuous function. My attempt: Say $f(x)$ has a max. value $M$. ...
0
votes
2answers
553 views

expected number of edges in a random graph

If we have a random graph $G \in g(n,\frac{1}{2})$ how do we show that the expected number of edges is $\frac{1}{2} {{n}\choose{2}}$ Thanks in advance
0
votes
1answer
141 views

Rice’s theorem and recursion theorem

Prove Rice’s theorem using recursion theorem. I need some hints as to what must be done about it. Please use Davis' book notation: Computability, Complexity, and Languages, Second Edition: ...
2
votes
1answer
604 views

Prove a non-empty subset is closed in an inner product space

I hope someone would be able to help me with the finer details of this proof. Problem: $M$ is a non-empty set in an Inner Product Space (IPS) $X$. I need to show that the annihilator of $M$ which is ...
4
votes
1answer
363 views

Strength of attraction of fixed points

Consider a smooth map $f: \mathbb{R} \rightarrow \mathbb{R}$ with an attracting fixed point $F$. Then, we have if $f'(F) \ne 0$, $F$ is a "simple" attracting fixed point, if $f'(F) = 0$, $F$ is a ...
6
votes
2answers
211 views

Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related ...
1
vote
0answers
97 views

How to predict order of a set of fractional differential equations?

I have a set of differential equations of the form: $$\frac{dv}{dt} = a[b-c*m-d*n-e*h]$$ $$\frac{dm}{dt} = p(v)$$ $$\frac{dn}{dt} = q(v)$$ $$\frac{dh}{dt} = r(v)$$ Using fde12 in MATLAB I can ...
0
votes
1answer
92 views

uniform convergence of analytic function becomes analytic in a region

Suppose that, $G$ is a region in the complex plane and $f_n : G \to \mathbb{C}$ is analytic for each $n\ge1$. Suppose further that, the sequence {$f_n$} converges uniformly on $G$ to a function $f:G ...
3
votes
3answers
149 views

About Linear algebra question.

Let $F$ be a field, $n$ a positive integer, and $V$ be the space of $n \times n$ matrices over $F$. If $A$ is a fixed $n \times n$ matrix over F, $ T_A(B) = AB - BA$. Consider the family of linear ...
2
votes
1answer
170 views

Normal form of a vector field in $\mathbb {R}^4$.

EDIT (In responce to xpaul's answer): I'm looking for the exact normal form, not the one up to $O(|x^5|)$ Those are two analogous problems, the first one of which I have already accounted for. Find ...

15 30 50 per page