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H: Partial fractions for geometric probability-generating function wrong
Let $X\sim \text{Geo}(1/4), Y\sim \text{Geo}(1/2)$ be given. First I have to compute $\mathbb{E}[z^{X+Y}]$:
$$\mathbb{E}[z^{X+Y}]=\mathbb{E}[z^{X}]\cdot\mathbb{E}[z^{Y}]=\frac{\frac{1}{4}z}{1-\left(1-\frac{1}{4}\right)z}\cdot\frac{\frac{1}{2}z}{1... |
H: Area of a revolution of $x=\frac{1}{3}\left(y^2+2\right)^\frac{3}{2}$
I think my biggest problem here is I can not find a good way to find the square root in this problem
$$x=\frac{1}{3}\left(y^2+2\right)^\frac{3}{2} \ \ \ \ 1 \le x \le 2$$
$$\int_1^2 2 \pi \cdot {\frac{1}{3}\left(y^2+2\right)^\frac{3}{2}} \sqrt{1 ... |
H: Double integrals: finding a volume of a solid
I am trying to determine the bounds given a solid bounded by the plane $y+z=4$ and the cylinder $y=x^2$, and $xz$ and $yz$ planes in the first octant. I am not sure if I am on the right track, but to find my bounds for $y$, I set $z$ equal to zero and calculated $y=4$. ... |
H: How can I alternately solve or otherwise optimize the solution to finding paths up a ladder 1 or 2 steps at a time?
I got thrown off by an interview question recently,
What structure do you use to find all the ways up a ladder with N
steps in 1 or 2 steps moves?
It occured to me today that a graph expansion co... |
H: If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected
I have an exercise found on a list but I didn't know how to proceed. Please, any tips?
Let $X$ be a connected subset of a connected metric space $M$. Show that for each connected component $C$ of... |
H: A simple Integral Question
What are the steps to calculate the value of $c$ in the following integral equation?
$$
\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}c.e^{-(x_1+2x_2+3x_3)}\,dx_1 \, dx_2
\, dx_3 = 1
$$
AI: Since $e^{-(x_1+2x_2+3x_3)}=e^{-x_1}e^{-2x_2}e^{-3x_3}$, we have
$$
\int_0^\infty\int_0^\infty... |
H: Closure of the interior of another closure
Let $X$ be a topological space and let $A \subset X$.
Is it true that $\overline{\rm{Int}(\overline{A})}=\overline {A}$?
This question arose when I try to show$\overline{X-\overline{\rm{Int}(\overline{A})}}=\overline{X-\overline{A}}$
AI: The statement is false in general. ... |
H: Algebraic proof of a binomial sum identity.
I came across this identity when working with energy partitions of Einstein solids. I have a combinatorial proof, but I'm wondering if there exists an algebraic proof.
$$\sum_{q=0}^N\binom{m + q - 1}{q}\binom{n + N - q - 1}{N - q} = \binom{m + n + N - 1}{N}$$
I've tried i... |
H: Square three digit numbers, the efficient way
I would like to square a three digit number in my head.
Now I know that the formula is
$$ ( X + r ) ( X - r ) + r^2 = X^2 - rX + rX - r^2 + r^2 = X^2
$$
Where $\,r\,$ is a number such that $\,X + r\,$ is divisible by $10$ and/or $100$
Now the problem is that, I would l... |
H: Extreme boundary of a compact, convex, metrizable set is $G_\delta$
Let $X$ be a topological vector space (no assumptions about local convexity are made in the question, though I am worried they might be required). Suppose $K\subset X$ is a compact, convex, metrizable subset of $X$, and denote by $\partial_e K$ the... |
H: If $p:E\to B$ is a covering space and $p^{-1}(x)$ is finite for all $x \in B$, show that $E$ is compact and Hausdorff iff $B$ is compact and Hausdorff
I can show that if $E$ is compact and Hausdorff $B$ has the same properties, also I can show that if $B$ is compact and Hausdorff $E$ is Hausdorff, but I have troubl... |
H: What did Cantor take to be the relationship between the countable ordinals and the power set of the naturals?
I've been told that Cantor sees a relationship between the countable ordinals (Cantor's second number class) and the powerset of the natural numbers.
I've read the "Grundlagen" a few times, but can't seem t... |
H: Center of mass of a semi-annular plane
I am trying to find the $y$-coordinate of the center of mass of a semi-annular plane region bounded by $1\leq x^2+y^2 \leq 4$ and $ y > 0$. I know that the sketch should look similar to half a donut above the y axis, but I am not sure how to set up the integral with the approp... |
H: Liouville's theorem for Banach spaces without the Hahn-Banach theorem?
Let $B$ be a (complex) Banach space. A function $f : \mathbb{C} \to B$ is holomorphic if $\lim_{w \to z} \frac{f(w) - f(z)}{w - z}$ exists for all $z$, just as in the ordinary case where $B = \mathbb{C}$. Liouville's theorem for Banach spaces sa... |
H: The boundary is a closed set
A point $p$ in a metric space $X$ is a boundary point of the set $A$, if any neighbourhood of $p$ has points of both $A$ and $X-A$.Prove that the set of all boundary points of $A$ is closed.
My attempt:
By definition of an open set this means that for every $x$ in the boundary there is ... |
H: Exhibiting a ring isomorphism between a ring and itself.
I recently proved to myself that if $R$ is a ring, and $R'$ a set in bijection with $R$, say by $f\colon R'\to R$, then one can turn $R'$ into a ring by defining $0'=f^{-1}(0)$, $1'=f^{-1}(1)$,
$$
r'+s'=f^{-1}(f(r')+f(s')),\qquad r's'=f^{-1}(f(r')f(s')),
$$
a... |
H: Matrix commutator question
Here's a nice question I heard on IRC, courtesy of "tmyklebu."
Let $A$, $B$, and $C$ be $2\times 2$ complex matrices. Define the commutator $[X,Y]=XY-YX$ for any matrices $X$ and $Y$. Prove
$$[[A,B]^2,C]=0.$$
AI: Since the trace is additive, and $\mathrm{trace}(XY)=\mathrm{trace}(YX)$, it... |
H: Prove equality of integrals
Let $f(x,y) = \text{sgn}(x-y)e^{-|x-y|}$ (Where $\text{sgn}(t)$ is the sign of $t$)
I want to prove the equation below.
$$\int^\infty_0dx \int^\infty_0 f(x,y)dy = -\int^\infty_0 dy \int^\infty_0 f(x,y)dx =-1$$
I don't know how can I start to prove this. Please give some outline for that.... |
H: How do I derive these roots
Let $z = \cos(\frac{\pi k}{5}) + i\sin(\frac{\pi k}{5})$
Consider the imaginary part of $z^5$, and deduce that $x^4 - 3x^2 + 1
= 0$ has solutions:
$$2\cos(\frac{\pi}{5}), ~2\cos(\frac{2\pi}{5}),
~2\cos(\frac{3\pi}{5}), ~2\cos(\frac{4\pi}{5})$$
So, 'considering' the imaginary part of ... |
H: Birational morphism
I have a question on rational and birational maps:
Is the map $$\mathbb{P}^1\rightarrow \mathbb{P}^2, (x:y) \mapsto (x:y:1)$$ rational? Birational? If birational what is its inverse?
Same questions for map $$\mathbb{P}^1 \rightarrow \mathbb{P}^2, (x:y) \mapsto (x:y:0).$$
My guess is that both ... |
H: Tensor product of $R$-algebras
Let $f: R \to S$ and $g: R \to T$ be two $R$-algebras. To show that $S \otimes_R T$ is an $R$-algebra I need to define a ring structure (multiplication) on it and a ring homomorphism $h : R \to S \otimes_R T$.
Using the universal property of the (multi-)tensor product, defining multip... |
H: Limit finding of an indeterminate form: $\lim\limits_{x\to0} \frac{x^3}{\tan^3(2x)}$
Here is the limit I'm trying to find out:
$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$
Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with:
$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \... |
H: Does $x\cos(x)$ have oblique asymptotes?
Looking at the graph of $x\cos(x)$ or $x\sin(x)$ etc., it looks like the magnitude of the waves are following a line. Are they oblique asymptotes or something else?
I am familiar with finding the oblique asymptotes of a rational function like $\frac{P(x)}{Q(x)}$ by dividing ... |
H: Analytic proof for Circles of Apollonius
I'm looking for an analytic proof the statement for a Circle of Apollonius (I found a geometrical one already): If $\overline{AC}:\overline{BC}=s$, then $P \in k_s$. $s \in (0,1)$.
$k_s$ is the circle. I made the following drawing:
WLOG I can set $A=(0/0)$ and $B=(0/1)$. I ... |
H: Applications of graph colorings to discrete math
I'm looking for interesting examples of questions in discrete math that can be proved using graph colorings (as, for example existence of a particular division of people into $k$ distinct groups with given conditions, using, in the proof, that an associated graph is ... |
H: Examples of rings of fractions
I wanted to come up with a few examples of rings of fractions $S^{-1}R$. Can you tell me if these are correct:
1.Let $R = \mathbb Z$, $S = (2 \mathbb Z \setminus \{0\}) \cup \{1\}$. Then every $[x] = \frac{r}{s} \in S^{-1}R$ consists of the elements: $[x] = \{ 2x, \frac12 x\}$.
The r... |
H: some question of combination
we know that hilbert seris of n- variables polynomial ring is $\Sigma_{i} \binom{n-1+i}{i}t^{i}$
But, I don't know $\Sigma_{i} \binom{n-1+i}{i}t^{i}=(1-t)^{-n}$.
I wonder to prove in detail.
AI: After the modifications in my comment, we can write the right hand side as
$\frac{1}{(1 - t... |
H: Limits of Subsequences
If $s=\{s_n\}$ and $t=\{t_n\}$ are two nonzero decreasing sequences converging to 0, such that $s_n ≤t_n$ for all $n$. Can we find subsequences $s ′$ of $s$ and $t ′$ of $t$ such that $\lim \frac{s'}{t'}=0$ , i.e., $s ′$ decreases more rapidly than $t ′$ ?
AI: Yes, we can. So we hav... |
H: Elementary results from Algebraic Number Theory
The purpose of this question is to motivate me to study algebraic number theory.
Let me explain. My motivation for studying number theory is to learn about beautiful results with simple, accessible statements. For example, the theorem that a prime can be written as th... |
H: Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$
How to prove the following product?
$$\frac{\sin(x)}{x}=
\left(1+\frac{x}{\pi}\right)
\left(1-\frac{x}{\pi}\right)
\left(1+\frac{x}{2\pi}\right)
\left(1-\frac{x}{2\pi}\right)
\le... |
H: Inner product space over $\mathbb{R}$
Definition of the problem
I have to prove the following statement:
Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an
inner product space over $\mathbb{R}$. prove that for all $x,y\in E$
we have
$$
\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)\l... |
H: is the approximation of the sum true?
Someone commented under my question Calculation of the moments using Hypergeometric distribution that
$$
\sum_{k=0}^l\frac{{l \choose k}{2n-l \choose n-k}(2k-l)^q}{{2n\choose n}}\sim \sum_{k=0}^l (2k-l)^q {l \choose k}.
$$
I've tried to use the Stirling's approximation formula... |
H: probability of passing an exam
I have an exam that I can pass with a probability p.
If I fail the exam, I can retry as many times as I want until I do (the chances to succeed at the 3rd try is still p).
What's the probability that I succeed at the n th retry (which is to say that I took n-1 exams, failed them, and ... |
H: Trigonometry in Simple Harmonic Motion
In one of my high school maths questions the example given to find the maximum displacement of a Simple Harmonic Motion where
$ x=2+4\cos \left (2t + \frac{\pi}{3} \right ) $
and the motion lies in the interval:
$-2 \leq x\leq 6$
is:
let $x=2+4\cos \left (2t + \frac{\pi}{3... |
H: Regular polygons that touching to a sphere surface
What is the possible number of n sided polygons(every face is the same regular polygon) that touching their corners to sphere surface and also touching each other ?
I would like to know the relation between n sides polygon and possible placing number.
And also I ... |
H: proof that translation of a function converges to function in $L^1$
Let $f \in L^1(\mathbb{R})$, for $a\in \mathbb{R}$ let $f_a(x)=f(x-a)$, prove that:
$$\lim_{a\rightarrow 0}\|f_a -f \|_1=0$$
I know that there exists $g\in C(\mathbb{R})$ s.t $\|f-g\|_1 \leq \epsilon$, this is also true for $f_a$ and $g_a$.
Now I h... |
H: Recurrence relation $T_{k+1} = 2T_k + 2$
I have a series of number in binary system as following:
0, 10, 110, 1110, 11110, 111110, 1111110, 11111110, ...
I want to understand : Is there a general seri for my series?
I found this series has a formula as following:
(Number * 2) + 2
but i don't know this formula is c... |
H: Do improper integrals like $\int_{-\infty}^{+\infty} f$ converge if $xf(x)\rightarrow 0$?
My teacher assumes without proof in his notes that, given a rational function $R(x)$, the improper integral $\int_{-\infty}^{+\infty} R(x)dx$ converges if $\lim_{|x|\rightarrow\infty} xR(x) = 0$. He then proceeds to explain th... |
H: On the $PSL(2, p)$, $p$ a Mersenne prime
We know $|PSL(2,p)|=p(p+1)(p-1)/2$. Let $p$ be Mersenne prime (that is $p+1=2^{n}$) and $r$ be prime divisor of $(p-1)/2$.
My question: What is the number of Sylow $r$-subgroups of $PSL(2,p)$?
AI: Let $G=\operatorname{PSL}(2,q)$ for q an odd prime power. If r is an odd prime... |
H: A "prime-mapping" polynomial
Suppose that $f$ is a polynomial with integer coefficients with the property that for any prime $p$, $f(p)$ is a prime. Is there any such polynomial $f$ other than $f(x)=x$ of course?
My approach was that if the leading coefficient $a_{0}$ of $f$ is $0$, then $f(p)=p$ for any prime $p$,... |
H: Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.
Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.
where $\sigma (n)$ is the sum of all the divisors of $n$
and $\sum\nolimits_{d|n} f(d)$ is the summation of $f$ at e... |
H: How many numbers $\in [1 .. 10^9]$ that are not of the form $ x^2, x^3$ or $x^5$?
We've started with $x^2$, saying that there are $\sqrt{10^9}$ numbers that are not $\in 10^9$
i'm thinking that if we add $\sqrt[3]{10^9}$ and $\sqrt[5]{10^9}$ to $\sqrt{10^9}$ and subtract them from $10^9$, we would have gone too far... |
H: The limit of integral
Let $1 \le p < \infty$ and assume $f \in L^p(\mathbb{R})$.
I'm trying to prove the limit of integral
$$\lim_{x \to \infty} \int^{x+1}_x f(t)dt =0.$$
Can I use Riesz Theorem for Banach spaces?
AI: Note that we need only prove this for real-valued $f\geq 0$, since
$$\left|\int_x^{x+1}f(x)dx\rig... |
H: Measurable function on the interval $[0,1]$
Assume that $f$ is a measurable function on the interval $[0,1]$ such that $0<f(x)<\infty$ for $x \in [0,1]$. Then, how can I prove the inequality below?
$$\int^1_0 f(x)dx \int^1_0 {1 \over {f(x)}} dx \ge 1$$
AI: Here is another way. What you have written is nothing but t... |
H: Minimal polynomial of $\sqrt2+1$ in $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$
I'm trying to find the minimal polynomial of $\sqrt2+1$ over $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$. The minimal polynomial of $\sqrt2+1$ over $\mathbb{Q}$ is
$$ (X-1)^2-2.$$
So I look at $\alpha = \sqrt2 + \sqrt3$
$$ \alpha^0 = 1$$
$$ \alpha^1 = \sqrt2 ... |
H: probability of passing an exam (continued)
Following my previous question, which can be found here:
probability of passing an exam,
I found out that the probability of passing an exam at the nth try is $p(1-p)^{n-1}$.
If I now assume that taking an exam takes me one hour of work, how many hours on average will I ha... |
H: Show that $\frac{n}{\sigma(n)} > (1-\frac{1}{p_1})(1-\frac{1}{p_2})\cdots(1-\frac{1}{p_r})$
If $n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}$ is the prime factorization of $n>1$ then show that :
$$1>\frac{n}{ \sigma (n)} > \left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right)\cdots\cdots\left(1-\frac{1}{p_r}\right)$$
... |
H: Is there any permutation $x≠1$ leaving at least $n-2k$ letters fixed at this group?
This question has an answer which I am noting both here.
Q: Suppose that $G$ is permutation group of degree $n$. If for an integer $k$ where $4≤2k≤n$ we have $|G|≥(n-k)!k$ then $G$ contains a permutation $x≠1$ that leaves at least ... |
H: Is $\mathbb{Q}[\sqrt2]$ = $\mathbb{Q}[\sqrt2+1]$?
Is $\mathbb{Q}[\sqrt2]$ = $\mathbb{Q}[\sqrt2+1]$?
I think so because
$$\mathbb{Q}[\sqrt{2}+1] = \{\sum_{i=0}^{n}c_i(\sqrt{2}+1)^i\mid n\in\mathbb{N}, c_i\in\mathbb{Q}\}$$
$$= \{\sum_{i=0}^{n}c_i(\sqrt{2})^i\mid n\in\mathbb{N}, c_i\in\mathbb{Q}\} = \mathbb{Q}[\sqrt{2... |
H: Bounded operator that does not attain its norm
What is a bounded operator on a Hilbert space that does not attain its norm? An example in $L^2$ or $l^2$ would be preferred.
All of the simple examples I have looked at (the identity operator, the shift operator) attain their respective norms.
AI: For an example in $L... |
H: Evaluating $ \int_0^{\infty } \exp\left(-g x-\frac{x^2}{2}-\frac{x^2 z}{1-z}\right) x^k \sin(hx) \, dx $
I'm attempting to evaluate the following integral, so far, with little success. Any help would be appreciated:
$$
\ \int_0^{\infty } \exp\left(-g x-\frac{x^2}{2}-\frac{x^2 z}{1-z}\right) x^k \sin(hx) \, dx
$$
Al... |
H: L-measurable function and integral
Assume that $f:E \to [0,\infty]$ where $E \subseteq \mathbb{R}^n$ is a measurable set, and $f$ is $\mathbb{L}$-measurable. And use $x \in \mathbb{R}^n$ and $y \in \mathbb{R}$.
First I'm wondering why the subsets A and B stated below are measurable.
$$A=\{(x,y) \in \mathbb{R}^{n+1... |
H: Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$
Evaluate the sum:
$$\sum_{n=0}^{\infty} \frac{1}{F_{(2^n)}}$$
where $F_{m}$ is the $m$-th term of the Fibonacci sequence. I need some support here. Thanks.
AI: As wikipedia claims the result follows from the identity
$$
\sum\limits_{n=0}^N\frac{1}{F_{... |
H: Multi variable integral : $\int_0^1 \int_\sqrt{y}^1 \sqrt{x^3+1} \, dx \, dy$
$$\int_0^1 \int_\sqrt{y}^1 \sqrt{x^3+1} \, dx \, dy$$
Here is my problem in my workbook. If I solve this problem by definition, that find integral for $x$, after that solve for $y$. so $\int_\sqrt{y}^1 \sqrt{x^3+1} \, dx$ is so complicate... |
H: Proof that $A \otimes B \cong B \otimes A$
Possible Duplicate:
There exists a unique isomorphism $M \otimes N \to N \otimes M$
I want to show that for Abelian groups $A$ and $B$ that the tensor product $A \otimes B$ is isomorphic to $B \otimes A$. I believe that I have accomplished this and have posted my attemp... |
H: A continuity condition for a bilinear form on a Hilbert space
Let $H$ be a real Hilbert space, and let $B : H \times H \to \mathbb{R}$ be bilinear and symmetric. Suppose there is a constant $C$ such that for all $x \in H$, $|B(x,x)| \le C \|x\|^2$. Must $B$ be continuous?
This seems like it should just be a simpl... |
H: Cyclic refinements of abelian towers
I was looking through Lang's Algebra and found the following statement,
Let $G$ be a finite group. An abelian tower of $G$ admits a cyclic refinement.
After some work, I understand the proof, and now I want to show that we cannot drop the hypothesis that $G$ is finite. Its en... |
H: Very Important question. Limiting distribution
I have an exam in the morning and there is still one question I cannot do.
$X_1, \ldots, X_n$ are iid random variables each having distribution with density $f_{X_i}(x;\theta)= 1/\theta$, for $x \in [0,\theta]$ where $\theta>0$ compute the CDF of the random variable $... |
H: $\wedge^k(V)^* \cong \mathrm{Alt}^k(V)$
Let $V$ be a finite dimensional real vector space, let $\mathrm{Alt}^k(V)$ denote the space of
alternating $k$-linear forms on $V$ and let $\wedge^k(V)$ denote the $k^{th}$ exterior power of $V$.
I am trying to see why the algebraic dual $\wedge^k(V)^* := (\wedge^k(V))^*$ is... |
H: Collection of converging sequences determines the topology?
Is it the case that the set of converging sequences uniquely determines the open sets in a topological space?
In other words: Given a space $X$ and two topologies $T_{1}$, $T_{2}$ on $X$. such that the set of converging sequences under $T_{1}$ equals the s... |
H: Probability of an observation message
I want to do inference in a Hidden Markov Modell (Gaussian Mixture), given observed continoues variables $Y$ and latent discrete variables $X$. For this I need to compute the probability of an observation message $\mu_{Y \rightarrow X}(x_t) =: \varrho_t)x_t = P(y_t|x_t)$.
But h... |
H: Probability of observed data in a HMM
Possible Duplicate:
Probability of an observation message
In a given Gaussian mixture model with observed continues variables $Y$ and latent discrete variables $X$ I want to apply the forward-backward algorithm in order to compute the marginal posteriors $P(x_t|y_{1:T})$.
Si... |
H: What other substitutions could I use to evaluate this integral?
Consider the integral
$$ \int x^2\sqrt{2 + x} \, dx$$
I need to find the value of this integral, yet all its (seemingly) possible substitutions don't allow me to cancel appropriate terms. Here are three substitutions and their outcomes, all of which co... |
H: Centroid of a region
$$y = x^3, x + y = 2, y = 0$$
I am suppose to find the centroid bounded by those curves. I have no idea how to do this, it isn't really explained well in my book and the places I have looked online do not help either.
AI: Say $f(x)$ and $g(x)$ are the two bounding functions over $[a, b]$
The ma... |
H: hausdorff, intersection of all closed sets
Can you please help me with this question?
Let's $X$ be a topological space.
Show that these two following conditions are equivalent :
$X$ is hausdorff
for all $x\in X$ intersection of all closed sets containing the neighborhoods of $x$ it's $\{x\}$.
Thanks a lot!
AI... |
H: Minimum sphere containing a tetrahedron
Is there an equation which would give me the radius of the smallest sphere containing a certain tetrahedron (no need to touch all vertices); given that I know the insphere, circumsphere radii and the longest edge of the tetrahedron.
For 2D example of a triangle:
http://demons... |
H: Hydrostatic pressure on a triangle
I am attempting to follow Paul's calculus notes, but am having trouble, in particular at this page: http://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx
I get to the part with the "The height of this strip is $\Delta x$ and the width is $2a$. We can use similar ... |
H: The comparison theorem for matrices
Let matrix $X$ satisfy a differential equation
$$
\dot X = f(t,X)
$$
where right side is real and symmetric. Let $X(0) = M = M^{T} \succeq 0$ and matrix $Y$ satisfy differential inequality
$$
\dot Y \succeq f(t,Y), \;\;\; Y(0) = M
$$
where $A \succeq B$ means that for any ve... |
H: How to prove that $L=\{w \mid \#a(w)=\#b(w)=\#c(w)\}$ is not context free using closure
How can I prove that the language $L = \{w \mid \#a(w)=\#b(w)=\#c(w)\}$ is not context free using closure?
EDIT :
I know that the language $L_1 = \{a^i b^i c^i \mid i\geq 0\}$ is not a context free language.
Now I'm trying to... |
H: Terminology for a function computed by a finite-state transducer?
A finite-state transducer is a generalization of a finite state machine that accepts an input string and produces an output string (instead of just accepting or rejecting). Is there a name for a function $f : \Sigma_1^* \rightarrow \Sigma_2^*$ that ... |
H: Why is it undecidable whether two finite-state transducers are equivalent?
According to the Wikipedia page on finite-state transducers, it is undecidable whether two finite-state transducers are equivalent. I find this result striking, since it is decidable whether two finite-state automata are equivalent to one a... |
H: Cartan Theorem.
Cartan Theorem: Let $M$ be a compact riemannian manifold. Let $\pi_1(M)$ be the set of all the classes of free homotopy of $M.$ Then in each non trival class there is a closed geodesic. (i.e a closed curve which is geodesic in all of its points.)
My question: Why free classes? Why the theorem does ... |
H: Union of compact sets in a convergence space
Let $X$ be a convergence space and let $K_1, K_2, \ldots, K_n$ be compact subsets of $X$. I'm trying to prove for myself that the union $K$ of the $K_i$ is compact. By definition, $K$ is compact if every ultrafilter on it converges, so given an ultrafilter $\mathcal U$ o... |
H: Sequence in $L^p(X,M,\mu)$
I have two question.
Suppose that {$f_k$} is a sequence in $L^p(X,M,\mu)$ such that
$f(x) = \lim_{k \to \infty} f_k(x)$
exists for $\mu$ -a.e. $x \in X$.
Assume $1\le p<\infty$,
$\liminf_{k\to \infty} ||f_k||_p = a$
is finite.
First one is proving that $f \in L^p$ and $||f||_p \le a$.
A... |
H: $L^p$ measurable functions equality
Suppose $1<p<\infty$ and $f,g \in L^p (X,M,\mu)$. Where $||f||_P$ and $||g||_p$ are non zero, and
$||f+g||_p = ||f||_p +||g||_p$ .
Proving that equality:
$${f \over ||f||_p} = {g \over ||g||_p} \text{ }\mu -a.e.$$
What theorem is available for that prove? I can't find start point... |
H: Measure space inequality
$X,M,\mu$ is a measure space for which
$$\mu(A)>0 \to \mu(A) \ge 1.$$
If $1 \le p<q\le\infty $, then $L^p \subset L^q$ , and then
$$||f||_\infty \le ||f||_q \le ||f||_p \le ||f||_1$$
This inequality is introduced on my book, and very useful. So I'm trying to prove that inequality. How can I... |
H: Is there a differential limit?
I'm wondering if there's such a concept as a "differential limit". Let me give an example because my nomenclature is my own and unofficial, but hopefully indicative of the concept.
For some function f(x), there may exist a derivative of that function f'(x) which we call a first order ... |
H: Generalization of Hölder's inequality
Assume $1<p_k< \infty$ for $k=1,\ldots,N$ , and $\displaystyle\sum^N_{k=1}\frac{1}{p_k} =1$.
I want to prove that
$$\left|\int_X f_1 f_2\cdots f_N\; d\mu \right| \le \lVert f_1\rVert_{p_1} \lVert f_2\rVert_{p_2} \cdots \lVert f_N\rVert_{p_N}.$$
How can I directly adjust Hölder'... |
H: Simple Harmonic Motion with trigonometry
I need some help with my high school maths question:
A particle is moving in simple harmonic motion has speed 12m/s at the origin. Find the displacement-time equation if it is known that for positive constants a and n:
$x=a\cos 8t$
$x=16\cos nt$
So far this is what I know:... |
H: How can the jth level of a binary tree with n nodes has problems of size $({\frac{n}{2}})^j$?
I read from a book that the jth level (starting from j=0 or the root) of a binary tree with n nodes divides a problem into $2^j$ subproblems, each of size $\frac{n}{2^j}$. I understand where $2^j$ comes from, but where doe... |
H: Cardinality, $|x|=|y|$ implies $|A^x|=|A^y|$
I am trying to prove, without using the Schroder-Bernstein theorem, (where a modulus defines cardinality) that
i.) $|x|=|y|$ implies $|A^x|=|A^y|$
and
ii.) $|x|=|y|$ implies $|x^A|=|y^A|$.
Thank you!
AI: To say that $|x|=|y|$ is to say that the difference between $x$ ... |
H: Understanding the $L^\infty$ norm
I'm very confused about $L^\infty$. So I'm trying to prove this:
Is $\|f\|_{\infty}$ the smallest of all numbers of the form $\sup\{|g(x)| \,:\,x \in X\}$, where $f=g $ $\mu$-a.e.?
AI: The answer to your question is yes. Recall that the definition of $\|f\|_\infty$ is
$$\|f\|_\inft... |
H: An inequality problem.
Possible Duplicate:
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
In the condition $a,b \in[0,\infty)$, $1\le p<\infty$,
How can I conclude this inequality?
$$(a+b)^p \le 2^{p-1} (a^p + b^p)$$
AI: you could solve it using Hölder inequality.
$ (a+b)^p \leq... |
H: On the gaps between consecutive primes
I have observed something, that either:
Given any natural number $n$, there exists some natural number $k$, such that above $k$, the difference between any two consecutive primes is $> n$ ($\implies$ the prime gap increases steadily, having limit infinity), or
For some $n$, ... |
H: Why do $\mathbb{C}$ and $\mathbb{H}$ generate all of $M_2(\mathbb{C})$?
For this question, I'm identifying the quaternions $\mathbb{H}$ as a subring of $M_2(\mathbb{C})$, so I view them as the set of matrices of form
$$
\begin{pmatrix}
a & b \\ -\bar{b} & \bar{a}
\end{pmatrix}.
$$
I'm also viewing $\mathbb{C}$ as ... |
H: Quick question about lim and sup
I've got a question regarding a step in a proof, the situation is following:
Let $X_{1},\dots,X_{n}$ be independent, symmetric stochastic variables so that $\sum\limits_{n=1}^{\infty}X_{n}$ exists in probability.
If I use the fact that convergent in probability implies the same conv... |
H: Basic fact in $L^p$ space
I'm studying $L^p$ space.
$1 \le p < r <q < \infty$ then $L^p \cap L^q \subset L^r$.
More over $L^p \cap L^\infty \subset L^r$
I'm trying to prove that fact. Which theorem is useful for proving that?
AI: For $f(x)\ge0$, Jensen's Inequality yields
$$
\left(\frac{1}{\int_X f^p(x)\,\mathrm{... |
H: How are the limits of this integral transformed?
Using $$\ln(x) = \int_1^x \frac{1}{t} dt$$
Show that for $x > 0$, $\ln\left(\frac{1}{x}\right) = -\ln(x)$
I am following a provided answer and didn't quite understand the following transformation and why/how it is done:
$$\ln\left(\frac{1}{x}\right) = \int_1^{\frac... |
H: Show that $f'$ is not continuous at 0 for the following function:
$$ f(x) = \begin{cases} x + 2x^2\sin(1/x) & \text{ for }x \neq 0 \\ 0 & \text{ for } x = 0\end{cases} $$
This is another exam practice question I am working on.
I simply took the derivative:
$$f'(x) = 1 + 4x\sin(1/x) - 2x^4\cos(1/x) $$
Now we see ... |
H: How large need $n$ be taken to ensure that $T_n(x)$ gives a value of $\ln(1.3)$ which has an error of less than $0.0002$?
$$ f(x) = \ln(1+x)$$
The previous part of this question required me to write down the remainder term for the taylor polynomial of order n.
My remainder term worked out to be:
$$R_n(x) = (-1)^n ... |
H: Parametric equation involving exponents and e
How do you go about solving this problem?
For each plane curve given below, find a rectangular equation. State the appropritate interval for $x$ and $y$.
$x(t) = e^{5t}$, $y(t) = e^t$, $t \in (-\infty, \infty)$.
Which is the correct rectangular equation?
(a) $x = \frac... |
H: Characterization of ideals in rings of fractions
Let $R$ be a commutative unital ring. Let $S$ be a multiplicative subset.
Is there a characterisation of the ideals in the ring of fractions $S^{-1}R$ in terms of ideals $I$ in $R$ and $R$?
AI: I don't believe in general you can say much about how ideals in $R$ are... |
H: Prove that $\frac{{a}^{2}}{b-1}+\frac{{b}^{2}}{a-1}\geq8$
I need to prove that for any real number $a>1$ and $b>1$ the following inequality is true:
$$\frac{{a}^{2}}{b-1}+\frac{{b}^{2}}{a-1}\geq8$$
AI: Let $a = 1+x$ and $b = 1+y$. Then we need to prove that $$\dfrac{(x+1)^2}{y} + \dfrac{(y+1)^2}{x} \geq 8$$ i.e. $$... |
H: Bijective holomorphic map
A bijective holomorphic map from unit disk to itself will be rotation? That mean $f(z)=e^{i\alpha}z$? How do I approach to solve this problem?In addition I want to know how one can remember the conformal maps which sends unit disk to upper half plane or conversely,and all possible known ... |
H: direct product of center of group
Let $Z(G)$ denote the center of a group $G$, let $J_n=Z(G)\times\dots \times Z(G)$, is it true that as a subset of external direct product $G\times\dots\times G$, $J_n$ is a subgroup?normal sybgroup?is it isomorphic to $Z(G)\times\dots \times Z(G)$ ($(n-1))$ times ?
I know $Z(G)$... |
H: Loopspace adjunction: when are unit or counit equivalences?
For (nice?) pointed spaces, the reduced suspension $\Sigma$ is left adjoint to the loop space $\Omega$. This adjunction is given by the unit maps
$\eta_X : X \to \Omega \Sigma X$, $x \mapsto (t \mapsto [x,t])$
and the counit maps
$\varepsilon_X : \Sigma \O... |
H: Does a solved sudoku game always have same sum? Is this sum unique to solved game?
Fundamentally, I'm looking for help on two things:
Verification that my math is correct for the assumption that all Xs are Y.
Proof that are the inverse is true, that all Y's are X, or, if it's not true, example of X that is outside... |
H: Inequality between volume and its projections
Let $A \subset \mathbb{R}^3$ be connected and let's define $A_1, A_2, A_3 \subset \mathbb{R}^2$ as projections of $A$ onto three perpendicular (to each other) planes. Show that:
$$|A| \le \sqrt{|A_1| |A_2| |A_3|}\;,$$
where $|\cdot|$ is volume when applied to $A$ and ar... |
H: Definition of neighborhood and open set in topology
I am a Physics undergrad, and just started studying Topology. How do you define neighborhood and open set in Topology.Wikipedia gives a circular definition.
An open set is defined as follows.
In topology, a set is called an open set if it is a neighborhood of ... |
H: How to check that whather a Polygon is completly inside of another Polygon?
Let's say I have two polygons. I know the co-ordinates of both polygons. Now, I need to check whether the first Polygon is completely inside of second polygon? IN this figure only 1 polygon is completely inside of red polygon.
AI: One way w... |
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