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H: How does a Class group measure the failure of Unique factorization?
I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a hell.
The Class group is given by $\rm{Cl}(F)=$ ... |
H: Evaluate $\lim_{n \to \infty }\frac{(n!)^{1/n}}{n}$.
Possible Duplicate:
Finding the limit of $\frac {n}{\sqrt[n]{n!}}$
Evaluate
$$\lim_{n \to \infty }\frac{(n!)^{1/n}}{n}.$$
Can anyone help me with this? I have no idea how to start with. Thank you.
AI: Let's work it out elementarily by wisely applying Cauchy-d... |
H: Calculate $I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}.$
I want to calculate this integral with singularity:
$$I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}. $$
I hope to obtain a ... |
H: Complex inequality
How can I show this inequality $\sqrt{2}|z|\geq |\mathrm{Re} (z)|+|\mathrm{Im}(z)| $
please give me some hint. Which result is useful to show this. please help me out.thanks in advance.
AI: You need to think about the geometric representation of complex numbers. Given a rectangle with sides $a = ... |
H: Uniqueness of prime-power fields
I'm still stuck on the proof of the following theorem. I've asked two questions so far to get to where I am even at this point.
Theorem: Let $p$ be a prime and let $n\in\mathbb{Z}^{+}$. If $E$ and $E'$ are fields of order $p^{n}$, then $E\cong E'$.
Proof: Both $E$ And $E'$ have ... |
H: Understanding why the roots of homogeneous difference equation must be eigenvalues
There is some obvious relationship between the root solutions to a homogeneous difference equation (as a recurrence relation) and eigenvalues which I'm trying to see. I have read over the wiki article 3.2, 3.4 and the eigenvalues ($\... |
H: show that $x_n$ converges and find the limit
Let $\left\{ x_n \right\}_{n\geq0}$ be a sequence of real numbers such that $$x_{n+1}=\lambda x_n+(1-\lambda)x_{n-1},\ n\geq 1,$$for some $0<\lambda<1$
(a) Show that $x_n=x_0+(x_1-x_0)\sum_{k=0}^{n-1}(\lambda -1)^k$
(b) Hence, or otherwise, show that $x_n$ converges an... |
H: Simplifying Equivalent Functions
Given two functions in closed form such that f(x) is the same for all x for both functions, is there always a way to manipulate either function to make it so they are written exactly the same or can you have two functions that can be proven equivalent yet neither can be simplified t... |
H: What is the chance of an event happening a set number of times or more after a number of trials?
Assuming every trial is independent from all the others and the probability of a successful run is the same every trial, how can you determine the chance of a successful trial a set number of times or more?
For example,... |
H: On the definition of limit of a sequence
I apologize in advance if this turns out to be a trivial question, but when we define the limit of a sequence $x_n$ (it it exists) as the number $a$ such that $\forall \delta > 0\ \exists r : n \ge r \Rightarrow | x_n - a | < \delta$, is there a particular reason why we use ... |
H: Area of a polygon
Possible Duplicate:
How quickly we forget - basic trig. Calculate the area of a polygon
Calculate area of a figure based on vertices
I saw a formula in a book,
$$\mathrm{area}=\frac{1}{2}\left|\sum_{i}(x_iy_{i-1}-x_{i-1}y_i)\right|.$$
Where $x_iy_i$ are the vertices of the polygon. Since it was... |
H: Classifing groups of order 56: problems with the semidirect product
While I was doing an exercise about the classification of groups of order 56, I had some problems concerning the semidirect product.
Let $G$ a group of order 56 and let us suppose that the 7-Sylow is normal (let's call it $H$). Then we want to cons... |
H: Does $A^nx=\lambda ^n x$ apply for $n$ smaller than $-1$ (assuming $A$ is invertible)?
I am able to show (by proving 3 separate cases) that $Ax=\lambda x$ for nonzero $x$ and invertible $A$ implies that $A^nx=\lambda ^n x$ for all integers $n$ greater than or equal to $-1$. I was trying to extend this theorem to th... |
H: Proof: $X\ge 0, r>0\Rightarrow E(X^r)=r\int_0^{\infty}x^{r-1}P(X>x)dx$
As the title states, the problem at hand is proving the following:
$X\ge 0, r>0\Rightarrow E(X^r)=r\int_0^{\infty}x^{r-1}P(X>x)dx$
Attempt/thoughts on a solution
I am guessing this is an application of Fubini's Theorem, but wouldn't that requir... |
H: Proving a relation between 2 sets as antisymmetric
Let $U = \{1,...,n\}$
And let $A$ and $B$ be partitions of the set $U$ such that $$\bigcup A = \bigcup B = U$$
and $|A|=s, |B|=t$
Let's define a relation between the sets $A$ and $B$ as follows: $$B \succ A \iff \forall_{1 \leq i \leq t}, \exists_{1 \leq j \leq s}:... |
H: Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$
Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below:
$\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$
Can it be proven that the result is also a characteristic fu... |
H: Calculate BPM from history of hand positions
I calculate the beats per minute (BPM) out of Kinect hand positions tracked from a conductor. I do that by finding the last and the second last minimum in my history data. I then calculate the time difference between these two minimums and extrapolate this difference to ... |
H: Newton's method - finding suitable starting point
I have some trouble solving a problem in my textbook:
Given the following function: $$f(x) = x^{-1} - R$$
Assume $R > 0$. Write a short algorithm to find $1/R$ by Newton's method applied to $f$. Do not use division or exponentiation in the algorithm. For posit... |
H: Example of a meromorphic function in $\mathbb{C}$ but not in $\mathbb{C}_{\infty}$
I need to produce an example of a meromorphic function on $\mathbb{C}$ but not meromorphic on the Riemann sphere $\mathbb{C}_{\infty}$.
Will this work: $f(z)=e^z-1/z$?
Other examples are welcome. Thank you.
AI: I just wanted to note ... |
H: Local solutions of a Diophantine equation
I am trying to prove that the equation
$$3x^3 + 4y^3 +5z^3 \equiv 0 \pmod{p}$$
has a non-trivial solution for all primes $p$.
I am sure that this is a standard exercise, and I have done the easy parts: treating $p=2, 3, 5$ as special cases (very simple), and then for $p\geq... |
H: Finding Probability Generating function for $P\left\{ X > n+1\right\} $
I am trying to find probability generating function for $P\left\{ X > n+1\right\} $.
Let X be a random variable assuming the values $0, 1, 2, ...$. The notation both for the distribution of $X$ and for it's tails are
$P\left\{ X = j\right\} = p... |
H: What does the following notation mean: $\Bbb Q[x,y]$?
I came across this notation while at this MathOverflow thread and I could not find its meaning. It makes the biggest sense that $f(x,y) \in \Bbb Q[x,y] $ represents any continuous function on the interval $[x,y] \in \Bbb Q$ and thus $\Bbb Q[x,y]$ is a set of suc... |
H: complex contour line Integral
I am trying to calculate this Integral where Z is an element of the Complex number. I
think I need to find the residue because there is a singularity (the center of the contour is actually the singularity $z = -1 + i$) So I Believe I should find the residue. Is this all correct? and ... |
H: number of zeros of a complex polynomial's leading term
Let $p(z)=a_n z^n + a_{n-1} z^{n+1}+...$ be a polynomial of degree $n$. Prove that in a disc of sufficiently large radius, $p(z)$ and $r(z)=a_n z^n$ have the same number of zeros.
AI: The number of zeros of $p$ is finite, equal to the degree. Consider a disc of... |
H: Is equality proof valid if terms are moved across equality?
For any two events A and B with $Pr(B) > 0$, prove that $Pr(A^c|B) = 1− Pr(A|B)$.
Is there a valid way to finish a proof if a step moves a term across the equality as follows?
Show:
$Pr(A^c|B) = 1− Pr(A|B)$
$Pr(A^c|B) + Pr(A|B) = 1 - Pr(A|B) + Pr(A|B)$
... |
H: $f(x)$ absolutely continuous $\implies e^{f(x)}$ absolutely continuous, for $x \in [a,b]$?
If $f(x)$ is absolutely continuous (a.c.) on [a,b], is the function $e^{f(x)}$ also absolutely continuous on [a,b] ?
thanks
AI: It is true. Suppose $g$ is Lipschitz with rank $L$ on $[a,b]$, and $f$ is AC. Then $g \circ f$ is... |
H: When can one use logarithms to multiply matrices
If $a,b \in \mathbb{Z}_{+}$, then $\exp(\log(a)+\log(b))=ab$. If $A$ and $B$ are square matrices, when can we multiply $A$ and $B$ using logarithms? If $A \neq B^{-1}$, should $A$ and $B$ be symmetric?
AI: If $A$ and $B$ commute, i.e. $AB=BA$, then the same identity ... |
H: Showing that a set of trigonometric functions is linearly independent over $\mathbb{R}$
I would like to determine under what conditions on $k$ the set $$ \begin{align}
A = &\{1,\cos(t),\sin(t), \\
&\quad \cos(t(1+k)),\sin(t(1+k)),\cos(t(1−k)),\sin(t(1−k)), \\
&\quad \cos(t(1+2k)),\sin(t(1+2k)),\cos(t(1−2k)),\sin(t(... |
H: What is the units digit of $13^4\cdot17^2\cdot29^3$?
What is the units digit of $13^4\cdot17^2\cdot29^3$?
I saw this on a GMAT practice test and was wondering how to approach it without using a calculator. Thanks.
AI: If you compute modulo $10,$ then you'll get $$13^4 17^2 29^3 \equiv 3^4 7^2 (-1)^3\equiv -81\cdo... |
H: Show that $A$ is non-singular
(a) Let $A$ be an $n × n$ real matrix such that $(A + I)^4 = 0$ where $I$
denotes the identity matrix. Show that $A$ is non-singular.
(b) Give an example of a nonzero $2×2$ real matrix $A$ such that $x′Ax = 0$
for all real vectors $x$.
(a) Note that $(A + I)^4 = 0=>A^4+4A^3+6A^2+4A+I=0... |
H: Am I right in thinking $\frac{x^{2}}{ax+b}$ is an improper rational expression?
Am I right in thinking $\dfrac{x^{2}}{ax+b}$ is an improper rational expression? If so, can someone help me figure out how to write it
as the sum of a polynomial and proper rational expression?
I have not a clue.
AI: I'll do the first... |
H: which of the following statements are true and which are false
Let $f$ and $g$ be continuous functions such that $f(x) ≤ g(x)$ for all
$x ∈ [0, 1]$. Determine which of the following statements are true and
which are false:
$$
\begin{align}
(a) & {}\quad \int_0^x |f(t)|~dt \leq\int_0^x |g(t)|~dt ~\forall~x ∈ [0, 1]\... |
H: Is the result always n+1?
I'm reading a book on algorithms by Kurt Mehlhorn and Peter Sanders.
On page 2, the following Theorem is stated:
The addition of two n-digit integers requires exactly n primitive
operations. The result is an n+1-digit integer.
Is the result always n+1. What if you have 2 2-digit intege... |
H: Show that $g$ is differentiable on $(0,\infty)$
Let $f:\mathbb{R}\to \mathbb{R}$ be a bounded continuous function. Define $g:[0,\infty)\to \mathbb{R}$ by $$g(x)=\int_{-x}^{x}(2xt+1)f(t) ~dt.$$ Show that $g$ is differentiable on $(0,\infty)$ and find the derivative of $g$.
I can find the derivative. I can check diff... |
H: Bézier approximation of archimedes spiral?
As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic and quadratic Bézier curves. What is a good method of approximating an Arch... |
H: A question about probability
I have met a interesting question:
If today rains, the probability that tomorrow rains is $0.6.$ If today doesn't rain, the probability that tomorrow rains is $0.2.$ Given Tuesday rained, what's the probability that Monday rained?
I have no idea how to solve this.
If I make the quest... |
H: Trivial solution when solving in integers
Suppose we want to solve $4(x+y)^{2}-3xy-6(x+y)=0$ where $x$ and $y$ are both integers. Why we only get the trivial solution?
AI: The equation can be arranged as a quadratic equation of x: $4x^2+x(5y-6)+(4y^2-6y)=0$.
The discriminant(D)= $(5y-6)^2-4.4.(4y^2-6y)=36+36y-29y^... |
H: Why does A=$\{\langle M_1,M_2,M_3 \rangle : L(M_1) \cap L(M_2) \ne L(M_3)\}$ isn't in $RE$.
I'm trying to figure out what's wrong with this following Turing machine which determinate that the following language
A=$\{\langle M_1,M_2,M_3 \rangle : L(M_1) \cap L(M_2) \ne L(M_3)\}$ is in $RE$.
I said that we can build... |
H: Compute $\lim_{x\to\infty} \frac{{(x!)}^{\frac{1}{x}-1} (x\Gamma(x+1) \psi^{(0)}(x+1)-x! \log(x!))}{x^2}$
What's the strategy one may use when facing a limit like this one? I think it's more important to know the possible ways to go than the answer itself. It's a problem that came to my mind again when I was workin... |
H: Calculating the extinction probability
I am trying to solve the following problem. In a branching process the number offspring per individual has a binomial distribution with parameters 2, p. Starting with a single individual, calculate the extinction probability.
I believe the solution to such a problem is evalua... |
H: $\alpha < \beta$ implies that $\gamma+\alpha<\gamma+\beta$ and $\alpha+\gamma\le\beta+\gamma$ for ordinals
This is an exercise from Kunen's book.
Show that $\alpha < \beta$ implies that $\gamma+\alpha<\gamma+\beta$ and $\alpha+\gamma\le\beta+\gamma$. Given an example to show that the "$\le$" cannot be replaced by ... |
H: Why is $\int_{-\infty}^{\infty} f(x)\sin(tx)dx$ continuous?
This is embarrassing.
I asked this question several months ago:
Let $f$, a Lebesgue integrable function in $\mathbb{R}$
($\int_{\mathbb{R}}|f| < \infty$). Let: $$g(t) :=
\int_{-\infty}^{\infty} f(x)\sin(tx)dx.$$ Show that $g$ is continuous
and that:... |
H: Prove that: $\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $
I conjecture that in some specific conditions a differentiating function gives the following equality:
$$\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $$
However, I'm not sure yet what exactly those conditions are in order to p... |
H: Factoring for extremely large numbers that are a power of 2.
This is a variation of this question. I want to find the number of factors for a given large integer that I already know to be a power of 2.
Given that the number is a power of 2, does that help by eliminating most scenarios e.g.
factors cannot be odd.
a... |
H: What is the cokernel of this map?
Consider the matrix
$$\begin{pmatrix} -12 & 6 &0\\ 58 & 34 & 18 \\ 18 & 12 & 6\end{pmatrix}$$
The problem ask me to decompose the kernel and cokernel of this matrix (regarded as a linear map) into cyclic $\mathbb{Z}$ modules (abelian groups). I immediately found the one dimensional... |
H: What is the relationship between $\pi_{2}\overline{X}$ and $\pi_{2}(X)$?
From Harvard qualification exam, 1990. Consider the space $X=\mathbb{S}^{1}\wedge \mathbb{S}^{2}$, alternatively viewing it as a sphere with north and south poles connected. I was asked to:
1): the relationship between $\pi_{2}X$ and $\pi_{2}\... |
H: How do I prove the existence of infinite union in ZFC?
Given an infinite set of sets A - how can I prove in ZFC that the union of all the elements of A exists?
AI: It is the axiom of union which assert this.
The axiom states that if $A$ is a set, then there exists a set $B$ such that $B=\bigcup A$, that is to say
... |
H: Hölder continuous and uniformly convergence subsequence
Let $\alpha \in (0,1]$. A function $f: [0,1]\rightarrow \mathbb{R}$ is defined to be $\alpha$-Hölder continuous if
$$
N_{\alpha}(f)=\sup\{ \frac{|f(x)-f(y)|}{|x-y|^\alpha} : x,y\in[0,1] \ \ \ \ x\neq y \} < \infty
$$
(a) Suppose $\{f_n\}$ is a sequence of fu... |
H: Transformation under rotation of Riemann sphere
Suppose the Riemann sphere $S$ is rotated by the angle $\phi$ round the diameter whose end points have $a,-1/\bar{a} $ (which have antipodal preimages) as stereographic projections. Suppose moreover, $z$ and $\zeta$ are stereographic projections of points correspondin... |
H: Does $L=\{(\langle M \rangle,k)| M$ is a TM and $\exists w\in \sum^*$ s.t when $M$ runs on $w$, $M$ visits some state at least $k$ times$\} \in R$?
I'd like your help with understanding , how come the following language is decidable (in $R$):
$L=\{(\langle M \rangle,k)| M$ is a TM and $\exists w\in \sum^*$ such tha... |
H: Is this CRC calculation correct?
I am currently studying for an exam and trying to check a message (binary) for errors using a polynomial, I would like if somebody could verify that my results below are (in)valid.
Thanks.
Message: 11110101 11110101
Polynomial: X4 + x2 + 1
Divisor (Derived from polynomial): 10101
Re... |
H: the measure of a set..
$C(n)$ is defined as the set that remains after removing from $[0, 1]$ an open interval of length 1/n centered at 1/2, then an open interval of length $1/n^2$ from the center of each of the two remaining intervals, then open intervals of length $1/n^3$ from the centers of each of the remaini... |
H: How to integrate $\frac{1}{\sqrt{1+x^2}}$ using substitution?
How you integrate $\frac{1}{\sqrt{1+x^2}}$ using following substitution? $1+x^2=t$ $\Rightarrow$ $x=\sqrt{t-1} \Rightarrow dx = \frac{dt}{2\sqrt{t-1}}dt$... Now I'm stuck. I don't know how to proceed using substitution rule.
AI: By the substitution you s... |
H: $HAM-NO-HAM$ problem: How do I show reduction from $HAMcircuit$ problem to the following problem?
consider the following language: $HAM-NO-HAM=\{(G_1,G_2)|G_1, G_2\}$ are undirected graphs, $G_1$ has a Hamilton and $G_2$ doesn't have one$\}$.
i need to decide wheter it's in $P, NP, CONP$ or none of the above.
If I... |
H: Floating point binary arithmetic question
I'm doing a basic class on computer architecture and we dwell into Floating Point Arithmetic, I'm not looking for someone to solve my homework, I'm actually just going through old exams and I'm kinda stuck on one exercise here.
So here it goes:
1,010010*2^(-9) - 1,000101*2^... |
H: is there a nice way to find the fourier transform of...
I am looking for a nice way to calculate the FT of the following function
$f(x)=\biggl(\sum_{n=1}^{c}~a_n~e^{-\frac{i}{2}~x~b_n}\biggr)^d$,
where $d,c>0$, $a_n$ and $b_n$ are real coefficients, strictly monotonously rising in $n$ and $x$ is the free variable a... |
H: Calculating powers of 2 on a 2D grid without factoring.
Consider the following 2D infinitely large grid where the dots represent infinity:
1 2 3 4 5 6 7 8 9 10 ...
2 4 6 8 10 12 14 16 18 20 ...
3 6 9 12 15 18 21 24 27 30 ...
4 8 12 16 20 24 28 32 36 40 ...
5 1... |
H: Can we descend field extensions of prime degree of number fields to number fields of the same degree
Let $K$ be a number field and let $p$ be a prime number.
Let $L$ be a degree $p$ field extension of $K$.
Does there exist a degree $p$ field extension $M$ of $\mathbf{Q}$ such that $$M\otimes_{\mathbf{Q}} K = L?$$
I... |
H: How to solve the inequality $2^x\ge a+bx$?
Let $a$ and $b$ be real constants where $b$ is positive. What is the small real number $x_0>0$ such that
$$2^x \ge a+bx?$$
Here is what I have tried. Suppose $a=0$. Fix a positive integer $n$. Then $2^x=e^{(\log 2)x}=\left(e^{\frac{\log 2}{n}x}\right)^n\ge \left(1+\frac{\l... |
H: What rule do I use to differentiate this function?
I am new to calculus, and thought I had my head round the product, quotient and chain rules, but I can't work out how to tackle this:
$$
\frac 1{x(x+1)^2}
$$
Apparently, the first step of the solution is
$$
f'(x) = - \frac {(x+1)^2 + 2x(x+1)} { x^2(x+1)^4}
$$
but... |
H: Cauchy's Integral Theorem
I am trying to understand Cauchy's Integral Theorem which states
$$
\int_\gamma f(z)\,dz = 0.
$$
If function $f(z)$ is holomorphic (has no singularities) within the area contained by the contour $\gamma$. I understand the proof comes from Green's theorem, but I don't understand conceptuall... |
H: Submodules of direct sums of simple modules
I'm reading these online notes on representation theory, and I don't fully understand this:
Isn't $V\oplus(\bigoplus_{i\in I}S_i)$ a direct sum by definition, so we'd get $I=\{1,\cdots,n\}$? Can we assume the $S_i$ are pairwise disjoint submodules of $U$ based on how the... |
H: Determine $\alpha >0$ for which $\iint_Af(x,y)^\alpha dx \, dy < +\infty$
Let $A=\{(x,y)\in \mathbb R^2: 0<x<1, 0< y < \sqrt{x}\}$ and $f \colon A \to\mathbb R$ a continuous function s.t.
$$
\frac{1}{x^2+y^2} \le f(x,y) \le\frac{2}{x^2+y^2}
$$
for every $(x,y) \in A$. Determine the set of value $\alpha >0$ su... |
H: Why does this function converges almost everywhere and not pointwise?
I am working on the following question. I believe I am almost done, but I still have a hole in my solution:
Let $\{r_k\}_{k=1}^{\infty}$, a counting of $\mathbb{Q}$. For every $k \in \mathbb{N}$, let: $$ f_k(x):= \begin{cases} (x - r_k)^{-1/2} ... |
H: Definition of Grothendieck group
I'm reading the Wiki article about the Grothendieck group.
What's the reason we define $[A] - [B] + [C] = 0 $ rather than $[A] + [B] - [C] = 0 $ (or something else) for every exact sequence $0 \to A \to B \to C \to 0$? What is the property we obtain if we define it this way? I supp... |
H: Is concave quadratic + linear a concave function?
Basic question about convexity/concavity:
Is the difference of a concave quadratic function of a matrix $X$ given by f(X) and a linear function l(X), a concave function?
i.e, is f(X)-l(X) concave?
If so/not what are the required conditions to be checked for?
AI: A l... |
H: Evaluate $\lim\limits_{n\to \infty}\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{6n}$
Show that $$\lim_{n\to \infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{6n}\right)=\log 6$$ Here I need to use the definition of integral but I faced problem in range . Please help.
AI: Maybe it is intended that you mention... |
H: Proof strategy for Pointwise converging sequence of Riemann integrable functions to not uniformaly converge
I am wondering of a proof strategy to show. That a sequence of Riemann integrable functions which converges point wise to a function may not actually uniformly converge to it. If it makes the argument simpler... |
H: Show that $k\ln k \in \Theta(n)$ implies $k \in \Theta(n/\ln(n))$?
It is exercise (3.2-8) from Introduction to Algorithms book.
I need help to solve it.
I am confused by the fact that there are two parameters. Because usually one parameter is used.
There is related exercise
which can be helpful.
Thanks.
AI: ... |
H: How to show x and y are equal?
I'm working on a proof to show that f: $\mathbb{R} \to \mathbb{R}$ for an $f$ defined as $f(x) = x^3 - 6x^2 + 12x - 7$ is injective. Here is the general outline of the proof as I have it right now:
Proof: For a function to be injective, whenever $x,y \in A$ and $x\neq y$, then $f(x) \... |
H: Exponential Distribution Maximum Likelihood
I found the following question in a past exam paper and I would like to ask how to solve it as I can't find anything in the notes related to it:
If three samples taken from Exponential(lambda) are 0.1, 0.5
and 0.9, what's the MLE for lambda?
I don't really understand ho... |
H: Some naive questions about embeddings
First, if I may, I would like to ask for help in getting an intuitive understanding regarding embeddings.
Wikipedia gives examples such as $\mathbb N$ in $\mathbb Z$. My first question is: were they not there already? And my second question is what advantage (or advantageous co... |
H: For regular expression $E$ and a context free grammar $G$- why deciding if $L(G)\subseteq L(E)?$ is a recursive problem?
I'd love your help with understanding why does the following language is recursive:
Input: a regular expression $E$ and a context free grammar $G$
question: $L(G)\subseteq L(E)?$
I tried to thin... |
H: If $\sum \limits_{i=0}^{\infty}{|x_i|}<\infty$, prove that $\sum \limits_ {i=0}^{\infty}{x_i^2}<\infty$
I have been trying to prove this for the last three days. It's for my Time Series Econometrics homework. I think you'll notice I'm not good at math as I can't even express my solution very well, and I'm sorry fo... |
H: Newton's method - determine accuracy in calculation
I have almost managed to solve a problem (I think), but I am a bit unsure if my procedure is correct, and my answer is not quite the correct one. Would appreciate any input! The problem is as follows:
If Newton's method is used with $f(x) = x^2 - 1$ and $x_0 = 1... |
H: Am I talking right?
I'm trying to describe expected value. My paragraph goes:
From probability theory we have
$E[f(x)] = \int{f(x)p(x)dx}$.
That is, the expected value of $f(x)$ is
equal to the sum of infinitesimals $f(x)dx$ weighted by
the probability that $f$ should take on
those values $f(x)$ at each $x$.
Am I... |
H: Proving :$ \frac{\textrm{d}}{\textrm{d}x}\int^{g(x)}_{h(x)}f(t)\textrm{d}t =f(g(x))g'(x)-f(h(x))h'(x). $
How to prove that :
$$ \frac{\textrm{d}}{\textrm{d}x}\int^{g(x)}_{h(x)}f(t)\textrm{d}t =f(g(x))g'(x)-f(h(x))h'(x). $$
AI: Let us first assume that $f$ has a primitive, which we shall refer to as $F$. By the fund... |
H: Simultaneous equations, two unknowns
I really should've paid more attention to maths in school...
I have some fairly simple simultaneous equations in the following format.
VMax = DMax + (DMax - DMin) * GMax
VMin = DMin - (DMax - DMin) * GMin
Knowns are VMax, VMin, GMax, GMin
Unknowns are DMax, DMin
All values a... |
H: Compute the limit of $\sqrt{1-a}\sum\limits_{n=0}^{+\infty} a^{n^2}$ when $a\to1^-$
I need some suggestions, hints for the limit when $a \to 1^{-}$ of $$\sqrt{\,1 - a\,}\,\sum_{n = 0}^{\infty}a^{n^{2}}.$$
AI: Note that $\frac1{\sqrt{1-a}}=\sum\limits_{k=0}^{+\infty}c_ka^k$ with $c_k=\frac1{4^k}{2k\choose k}\sim\fra... |
H: Basics of probability - Independent Events.
I read that the probability of two events provided they are independent is obtained by the following formula:
$Probability _ {Independent~ Events} = Probability_{1st Event} \times Probability_{2nd Event} \times ...$
Now it states that the probability of getting a Tails an... |
H: There is no $C^1$ function $f$ mapping an open interval in $\mathbb{R}$ onto open ball in $\mathbb{R}^2$
We know that there are no $C^1$ functions which map open $E\subset \mathbb{R}^2$ INTO $\mathbb{R}$. (Actually we can drop the $C^1$ requirement and just use continuity). A nice related question is that There is ... |
H: How to classify 3-sheeted covering space for $S_{1}\vee S_{1}$?
This might be a duplicate. This question also feels routine (it is also the execrise 10, page 88 in Hatcher). From Harvard qualification exam, 1990.
Let $X$ be figure eight.
1) How many 3-sheeted, connected covering space are there for $X$ up to isom... |
H: Prove $\cos^2 x \,\sin^3 x=\frac{1}{16}(2 \sin x + \sin 3x - \sin 5x)$
How would I prove the following?
$$\cos^2 x \,\sin^3 x=\frac{1}{16}(2 \sin x + \sin 3x - \sin 5x)$$
I do not know how to do do the problem I do know $\sin(3x)$ can be $\sin(2x+x)$ and such yet I am not sure how to commence.
AI: Using $\sin 2\the... |
H: Function writen as two functions having IVP
I heard this problem and I am a bit stuck.
Given a function $f : I \rightarrow \mathbb{R}$ where $I \subset \mathbb{R}$ is an open interval.
Then $f$ can be writen $f=g+h$ where $g,h$ are defined in the same interval and have the Intermediate Value Property. I tried to co... |
H: Boundary of $L^1$ space
Is there any rigorous or heuristic notion of boundary of $L^1$ that is studied? I mean something loosely like the collection of functions or distributions defined by
$$\left\{f\notin L^1: f_n\to f\quad\text{a.e.}\quad \text{as} \quad n\to \infty \quad \text{where} \quad f_n\in L^1\right\}$$
... |
H: Show in the form $A\sin(x+c)=\sin x - \cos x$?
Not sure what identity I should be using here: My gut tells me to use the Sin sum formula: $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$, but can't figure out how to.
AI: Yes, you want to use that form. You have:
$$A\sin(x+y)=A\sin x \cos y + A\sin y \cos x=\sin x - \... |
H: Equivalent characterizations of ordinals of the form $\omega^\delta$
Let $\alpha$ be a limit ordinal. Show the following are equivalent:
$\forall \beta, \gamma<\alpha (\beta+\gamma<\alpha)$
$\forall \beta<\alpha(\beta+\alpha=\alpha)$
$\forall X\subset \alpha(\text{type}(X)=\alpha, \text{ or,} \text{ type}(\alpha-X... |
H: Is there a trick to finding the number of odd numbers b/w two values?
I know you could find the number of even numbers (since they are a multiple of two). For example the number of even numbers between $11$ and $30$ will be
$$n= \frac{28-12}{2} + 1 = 9 $$
I wanted to know is there a similar way to find the number ... |
H: Show that $\overline{U\cap \overline{A}}=\overline{U\cap A}.$
Show that:
For every open set $U$ in a topological space $X$ and every $A\subset X$ we have $$\overline{U\cap \overline{A}}=\overline{U\cap A}.$$
The simple and new proof is welcome. Thanks for any help.
AI: Clearly $\overline{ U \cap \overline{A} } \s... |
H: Probability that a man will hit the target
The question is
A man can hit a target once in $4$ shots. If he fires 4 shots in succession, what is the probability that he will hit his target?
Here is how I am solving it:
Since the probability of man hitting the target is $\frac{1}{4}$ so for four consecutive shot... |
H: Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?
When I was in third grade, I was playing with rectangles and diagonal lines, and discovered something very interesting with fractions. I've shown several math teachers and professors over the years, and never g... |
H: Group with an automorphism of order 2 (Jacobson BA1)
I am having trouble with Exercise 11, Section 1.10 of Basic Algebra 1 by Nathan Jacobson (pub. Freeman & Co. 1985). The statement to prove is:
Let $G$ be a finite group and $\phi$ an automorphism of $G$. Let
$$ I = \{ g \in G : \phi g = g^{-1} \} $$
If $|I| >... |
H: Reference request for examples of probabilistic heuristics, help put some examples in a broader context.
I was thinking about how probability is used in heuristic arguments, an example being the argument that there are an infinite number of twin primes: the probability that $n$ is the first of two twin primes is ab... |
H: Dealing with "at least" in Permutation
For the following question (which I pulled of the internet)
A five member committee is to be selected from among four Math teachers and five English teachers. In how many different ways can the committee be formed under the following circumstance?
A) Anyone is eligible to ... |
H: Canonical Isomorphism Between $\mathbf{V}$ and $(\mathbf{V}^*)^*$
For the finite-dimensional case, we have a canonical isomorphism between $\mathbf{V}$, a vector space with the usual addition and scalar multiplication, and $(\mathbf{V}^*)^*$, the "dual of the dual of $\mathbf{V}$." This canonical isomorphism means ... |
H: Is a 2-dimensional subspace always called a plane no matter what the dimensions of the space is?
Is a 2-dimensional subspace in a 7-dimensional space still called a plane? I know that a 6-dimensional space in 7-dimensional space is called a hyperplane because the difference in the number of dimensions of the space ... |
H: Image of commutative diagram is commutative under functor?
The Wikipedia article for Functor ( http://en.wikipedia.org/wiki/Functor ) claims:
Two important consequences of the functor axioms are (where $F \colon C \to D$ is a covariant functor between categories $C$ and $D$)
F transforms each commutative diagram i... |
H: why is $0=0$ not possible?
Hi one of my friend showed me one proof, i.e.,
$2^2 - 2^2 = 10 - 10$
$(2+2) (2-2) = 5 (2-2)$
dividing both sides by $(2-2)$
$(2 + 2) = 5$
I know this is wrong in first line as both LHS and RHS goes to $0$ and you cannot directly make an equation $0=0$
because $\frac{0}{0} \neq 1$,
b... |
H: Regular Polyhedrons
In $\mathbb{R}^3$, there are five regular polyhedrons (up to similarity), and can be parametrized by number of vertices, edges and faces. What is the number of regular polyhedrons in $\mathbb{R}^n$, and their parametrization? Please suggest the reference(s) also. (Thanks in advance.)
AI: In shor... |
H: Partial trace of a system with isolated evolution
Let $\rho_{AB}$ be the state of a composite quantum system with state space $H_A\otimes H_B$ (two finite dimensional Hilbert spaces). Now assume that $A$ and $B$ are isolated and suffer a unitary evolution given by $U_A$ and $U_B$. If we measure the system $A$ then ... |
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