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H: The sum of trivial paths for a finite quiver is 1?
let $Q=(E_0, E_1)$ be a quiver and let $P_Q$ be a path algebra of $Q$. Let $p_i$ be the trivial path associated to each vertex $i$ in $E_0$.
Then why is $\sum_{i\in E_0} p_i=1$ for a finite quiver?
Are the $p_i$'s viewed as self-loops (arrows whose source and ta... |
H: What do you call a group that doesn't have a unique identity?
I have a set $M$ and an associative binary relation $+ : M \times M \to M$.
There exists an inversion operator $-$ , such that if $ m \in M$, then $m+(-m) \in Z$ where $Z$ is the set of all zeros ($Z \subset M$).
Additionally, if $z \in Z$ and $m \in M$,... |
H: Finding the all triples
How to find the all positive integer triples such that :
$$ab+c=\gcd (a^2,b^2)+\gcd(a,bc)+\gcd(b,ac)+\gcd(c,ab)=239^2$$
AI: Let $x=\gcd(a,b,c)$. But $\gcd(a^2,b^2)\ge x^2$ and $x$ divides $239^2$, this is impossible due to the equality and 239 is prime. Hence $x=1$.
Let $y=\gcd(c,ab)$. if $y... |
H: A unitary matrix taking a real matrix to another real matrix, is it an orthogonal matrix?
I tried to prove that a real antisymmetric matrix can be taken by an orthogonal tranformation to a form:
where the eigenvalues are $\pm i\lambda_1, \pm i\lambda_2 ... $
which is a statement I saw on wikipedia in http://en.w... |
H: Constructing an equation using proportionality
The question is:
If the rate of a certain chemical reaction doubles for every $10$ degree rise in temperature then which is greater:
a) Twice the rate at $10$ degrees
b) Half the rate at $30$ degrees
According to my text they are same. Here is how I am doing it
$$\r... |
H: Linear Algebra, eigenvalues and eigenvectors
For any $m \times m$ matrix, I will get a characteristic polynomial of degree $m$
with $m$ eigenvalues.
But for the matrix
$$A = \pmatrix{2 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 ... |
H: Intuition for étale morphisms
Currently working on algebraic surfaces over the complex numbers. I did a course on schemes but at the moment just work in the language of varieties.
Now i encounter the term "étale morphism" every now and then (in the book by Beauville). I know Hartshorne's definition as a smooth morp... |
H: Can I get a bound like $[u]_\alpha \leq C\lVert u\rVert_{C^0(S)}$?
I badly need a bound like
$$[u]_\alpha = \sup_{S}\frac{|u(x) - u(y)|}{|x-y|^\alpha} \leq C\lVert u \rVert_{C^0(S)} = C\sup_S |u|$$
where $S$ is compact and $C$ is a constant not depending on $u$.
Can it be done? Please let me know. Thank you.
AI: No... |
H: Graded Ring - Finite Sum
I've just read that if $R=R_0\oplus R_1 \oplus \dots$ is a graded ring and $f\in R$ then there's a unique decomposition of $f$ as $f=f_0+\dots+f_n$ with $f_i\in R_i$. I can't see immediately why in general this would have to be a finite sum! Could someone possibly enlighten me? I've got a f... |
H: Relation between left and right coset representatives of a subgroup
Let $G$ be a finite group and $H$ a subgroup. Is it true that
a set of right coset representatives of $H$ is also set of left coset
representatives of $H$?
AI: Not every left transversal is also a right transversal. The Group Properties Wiki has a... |
H: Evaluating an Integral makes it zero
Please point me out some useful notes for evaluating the following integral. I really wonder why it is $0$. $$\int_{-0.5}^{0.5}\cos(x)\ln\frac{1+x}{1-x}dx=0$$ I apply the rules which I know about solving the integral but, they have been useless. Any help will be appreciated.
AI:... |
H: Given $\forall x \in \mathbb{R} \: h(p^t(x))=th(p(x))$, how to get $h(p(x)) \propto \ln p(x)$?
The whole question is in the title. $p(x)$ is a probability distribution, and $h$ is continuous and monotonic in $p(x)$.
The purpose is to motivate that the "degree of surpise", or the "amount of information" after observ... |
H: A group of order $108$ has a proper normal subgroup of order $\geq 6$.
Problem: Let $G$ be a group of order $108 = 2^23^3$. Prove that $G$ has a proper normal subgroup of order $n \geq 6$.
My attempt: From the Sylow theorems, if $n_3$ and $n_2$ denote the number of subgroups of order $27$ and $4$, respectively, in ... |
H: Why is $\frac{y}{x}$ greater than $\frac{q}{p}$ -Figure
From the figure why is $\frac{y}{x}$ greater than $\frac{q}{p}$
AI: Let $\ell_1$ be the line through the origin $O$ and the point $P_1=(x,y)$. Let $\ell_2$ be the line through $O$ and the point $P_2=(p,q)$. Let $B=(0,p)$ and let $C=(p,c_2)$ be the point of... |
H: Proving that the set of algebraic numbers is countable without AC
A complex number $z$ is said to be algebraic if there is a finite collection of integers $\{a_i\}_{i\in n+1}$, not all zero, such that $a_0z^n + … + a_n = 0$.
Then can I prove the set of all algebraic numbers is countable without AC? I can only prove... |
H: CW complex structure on a disk with 2 smaller disks removed
For some reason, I'm having trouble visualizing how to put a CW structure on a disk with 2 smaller disks removed. What I'd like to do is have three 0-cells, five 1-cells, and a single 2-cell. Three of the one cells would be glued to the three vertices to g... |
H: Euler's theorem for powers of 2
According to Euler's theorem,
$$x^{\varphi({2^k})} \equiv 1 \mod 2^k$$
for each $k>0$ and each odd $x$. Obviously, number of positive integers less than or equal to $2^k$ that are relatively prime to $2^k$ is
$$\varphi({2^k}) = 2^{k-1}$$
so it follows that
$$x^{{2^{k-1}}} \equiv 1 \... |
H: About fractional differentiation under the integral sign
$1.$ Does $\dfrac{d^n}{dx^n}\int_a^bf(x,t)~dt=\int_a^b\dfrac{\partial^n}{\partial x^n}f(x,t)~dt$ correct when $n$ is a positive real number?
$2.$ How about $\dfrac{d^n}{dx^n}\int_{a(x)}^{b(x)}f(x,t)~dt$ when $n$ is a positive real number?
AI: The first rule o... |
H: What exactly is a manifold?
Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth.
Does this mean that in mathematics a manifold is essentially a representation of something that otherwise would difficult to "model in another way" in order to use it for some other pur... |
H: Why is $\int 1/(t ~\log^2 t) ~dt$ convergent?
$\displaystyle \int \frac{1}{t} dt = \ln t$ diverges.
How do I show that $\displaystyle \int_2^\infty \frac{1}{t ~\log^2 t} dt$ is convergent?
AI: To help yourself figure out the appropriate substitution, reorganize the integral into:
$$ \int \frac{1}{\log^2 t} \frac{dt... |
H: Sum of mutliples b/w $2$ and $10$ . What is wrong with this method
I am trying to find the sum of multiples b/w $2$ and $10$ end points non-inclusive using the following mechanism and I don't know why I am getting the wrong answer. (I selected this small range intentionally just to check the result)
First I find th... |
H: Extending continuous and uniformly continuous functions
What do you say about the following statement:
Let $f\colon(a,b)\rightarrow \mathbb{R}$ be a uniformly continuous function. Then $f$ can be extended to a uniformly continuous function with domain $[a,b]$.
Let $f\colon(a,b)\rightarrow \mathbb{R}$ be a continuo... |
H: Find a bijection from $(A^B)^C$ into $A^{B \times C}$
Possible Duplicate:
How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
Notation: Let A and B be sets. The set of all functions $f:A \rightarrow B$ is denoted by $B^A$.
Problem: Let A, B, and C be sets. Show that there exists a bijection from $(A^B)^... |
H: Show inclusion of a sum of subspaces in another subspace
Let $V$ be a vector space, and let $V',V'',W$ be subspaces of $V$. I want to show that
$$
(V' \cap W) + (V'' \cap W)
\subseteq
(V' + V'') \cap W
$$
So I take an element $a \in (V' \cap W) + (V'' \cap W)$ and show that it is also an element of $(V' + V'') \cap... |
H: Calculating area between circle and line - where have I gone wrong here
I think the initial set up is wrong below I should just integrate over the area as a multiple integral here. correct? the 1step below seems wrong.
Problem is asking for the area inside a region of a circle of radius 2 (centered on origin) and w... |
H: integration by parts question
$\frac {2}{L}(-i\hbar)\int_{0}^{n\pi}\sin(u)\cos(u) \, du$
I tried to solve this using integration by parts
and I got
$\frac {2}{L}(-i\hbar)(\sin^2 u +\cos u)|_{0}^{n\pi}$.
But the answer is zero, and according to the above equation, it is zero or some number.
What did I do wrong here... |
H: Are Cumulative Distribution Functions measurable?
It is well-known that CDFs (Cumulative Distribution Functions) of one-dimensional random variables are Borel measurable. But does the same apply to CDFs of multi-dimensional random variables (rvecs)? It suffices, for my purposes, to consider finite dimensional rvecs... |
H: Polynomial coefficients in exponential-series: how can I convert this into a composite of $\exp(x)$?
Assume we have the exponential-series $ \small \exp(x) = 1+ {x \over 1!} + {x^2 \over 2! } + \cdots = \sum\limits_{k=0 }^\infty {x^k \over k!} $ modified with a polynomial in the coefficients
say $ \qquad \displays... |
H: Geometric intuition for the inequality $(f(y) - c) ( y - d ) \geq (f(d) - c) ( f^{-1}(c) - d )$
Good day to everyone. I am interested in the geometric intuition for the following statement:
Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and $c,d \in \mathbb{R}$. Then for a... |
H: A Time Calculation Problem: 3yrs or 3yrs and 1 day?
Suppose, a person A has his birthday on 30 Sept 1994 and another person B has his birthday on 30 Sept 1997, then which of the following statement would be correct and why?
The difference between the ages of the two persons is
Three years correct to the number of... |
H: Best constant in an integral inequality
Which is the smallest constant $B_d$ such that the following inequality
$$\left|\int_{0}^{1}t^d(1-t)\psi(t) dt\right|^2\le
B_d\int_{0}^{1}t^d|\psi(t)|^2dt$$ holds, provided that $\psi(t)$ is a polynomial?
AI: A hint: Consider the scalar product
$$\langle u,v\rangle:=\int_0... |
H: $\sigma$-algebra of $\theta$-invariant sets in ergodicity theorem for stationary processes
Applying Birkhoff's ergodic theorem to a stationary process (a stochastic process with invariant transformation $\theta$, the shift-operator - $\theta(x_1, x_2, x_3,\dotsc) = (x_2, x_3, \dotsc)$) one has a result of the form
... |
H: Number Theory and combinatorial
Today, I took this observation from my note book. I am looking the strategy to deal this statement. The difference between $$\binom{n}{p}$$ and $$\left\lfloor\frac{n}{p}\right\rfloor\,$$ is divisible by p for a positive integer n and p is prime with >1. Here $$\binom{n}{p}$$ is th... |
H: divergence of $(2^n-n)$
Can anyone give me a satisfactory proof that the real sequence $(x_n)$ defined by $x_n = 2^n - n$ diverges to $+\infty$?
The heuristic reason is that
$$
\lim_{n\to\infty} \frac{n}{2^n} = 0,
$$
but I can't seem to turn this into a rigorous proof.
More generally is there a theorem which says t... |
H: Prove that $(n+\sqrt{n^2 -1})^k$ will always be of the form$ (t+\sqrt{t^2 -1})$ where $n$, $k$, $t$ are natural numbers
Show that $(n+\sqrt{n^2 -1})^k$ will always be of the form$ (t+\sqrt{t^2 -1})$ where $n$, $k$, $t$ are natural numbers
AI: I enjoyed working this out-cool question man. Here's a straightforward in... |
H: 'Linux' math program with interactive terminal?
Are there any open source math programs out there that have an interactive terminal and that work on linux?
So for example you could enter two matrices and specify an operation such as multiply and it would then return the answer or a error message specifying why an a... |
H: Checking efficiency of randomness with entropy
I was going through random numbers and found that the randomness of certain observations is measured by the entropy as given in here. Here, $p(x_i)$ is the probability that $x_i$ will take place. But if I have fair dice then $p(x_i)$ is $\frac{1}{6}$. So, I am assuming... |
H: Atiyah-Macdonald, Exercise 2.17 (direct limit)
I have solved the following exercise, can you tell me if this is correct? Thanks.
2.17. Let $(M_i)_{i \in I}$ be a family of submodules of an $A$-module such that for each pair $i,j$ in $I$ there exists $k$ in $I$ such that $M_i + M_j \subset M_k$. Define $i \leq j$ to... |
H: why do we need the fourth derivative of the function to check the error bound in the simspon's rule?
I understand that trapezoidal or midpoint rule's error bound needs the second derivative,
but I just don't get why the fourth derivative in simpson's rule
please help me :)
AI: Simpson's Rule is exact for polynomial... |
H: convergence of a particular series
Let be $ \Lambda\subseteq \mathbb C$ a lattice, I don't understand why the series $$\sum_{\lambda\in\Lambda\setminus\{0\}} \frac{1}{|\lambda|^s}$$ converges for $s>2$. Can someone help me?
AI: Start by looking initially, the the first 8 terms (with $\lambda$ nearest to $0$), in ... |
H: Usage of Addition principle or Combination.
The question is, " How many ways are there to draw a heart or a club from an ordinary deck of card ? "
The correct method is to use Addition Principle to solve it .
But can I use Combination to solve it as well ? Technically, it is selecting 13 cards out of the 52 cards w... |
H: Randomly selecting a natural number
In the answer to these questions:
Probability of picking a random natural number,
Given two randomly chosen natural numbers, what is the probability that the second is greater than the first?
it is stated that one cannot pick a natural number randomly.
However, in this questi... |
H: Modulus Distributing Over Multiplication?
Given positive integers a,b,c and k:
Define a function $M: \mathbb{Z^2} \rightarrow \mathbb{Z}$ as
$$M(x,y) = (x \bmod y)$$
i.e. the remainder of integer division
The following is always true:
$$a+b=c \implies M(M(a,k) + M(b,k), k) = M(c,k)$$
Under which values of k is the ... |
H: Proving the subset $C$ is a left coset of a certain subgroup iff $x,y,z \in C \Rightarrow xy^{-1}z\in C$
Revising for Group Theory at the moment and I'm sort of stumped on this exercise:
Given a subset $C$ of $G$, prove that $C$ is a left coset of a certain subgroup of $G$ iff $$xy^{-1}z \in C $$ when $$ x,y,z \in... |
H: diagonalizable matrix
$A$ is $n\times n$ matrix over $\mathbb C$.
must exist that:
$A^*A$ is diagonalizable over C
$AA^*$ is unitary matrix
if $A$ is not diagonalizable over $\mathbb C$ so $AA^*$ is not diagonalizable over $\mathbb C$
$i+1$ is not eigenvalue of $A$
I know the answer is 1+4 but I really dont unde... |
H: How do I calculated probabilities for cards?
I am trying to gain basic understanding on how to calculate probabilities. Below are a few examples of what I am trying to calculate. I would prefer (if possible) for formulas to be given on how to solve these using Microsoft Excel. Also, educating me on Probability T... |
H: How do I calculate the probability distribution of the percentage of a binary random variable?
I have an urn containing balls that are all either black or red. I'm interested in discovering the percentage of balls that are red. But I can only sample from the urn (without replacement), so the best I can do is calcul... |
H: Technical question on integral ring extensions
Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and
let $L$ be a finite Galois extension of $K$ with group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $p$ be a maximal
ideal of $A$ and let $\beta$ be a maximal ideal of $B$ suc... |
H: Is an automorphism of a normal extension determined by its image of the maximal separable sub extension?
Let $L / K$ be a normal, algebraic field extension. Suppose that the maximal separable sub- extension $M/K$ is finite, $K \subseteq M \subseteq L$. By the primitive element theorem, $M=K(x)$ for some $x \in L$. ... |
H: Expectation and proofs on $(\Omega,\mathcal{B},P)$ involving moments and MGF
Ok, suppose we have a random variable, X, on $(\Omega,\mathcal{B},P)$ and $r>0$.
I am trying to prove the following 4 things:
1- If $E(|X|^r)<\infty$ then $E(|X|^s)<\infty \;\;\;\forall s\in(0,r]$
2- If $E(e^{t|X|})<\infty$ for some $t>0$,... |
H: Complex variety with Zariski dense set of algebraic points
Let $V$ be an irreducible algebraic variety in $\mathbb{C}^n$ containing a Zariski dense set of points such that every coordinate is algebraic. Then is $V$ a product of one dimensional components?
AI: No. If you take any (say projective, irreducible ) vari... |
H: Calculating the average speed using only velocities
For the following question:
Pedro travels by bus to school at an average speed of $40$ km/hr. He is driven home by the same route by a friend's car at an average speed of $50$ km/hr. Which of the following is greatest:
(a) Average speed of both legs of the journ... |
H: Taylor polynomial of $f(x) = 1/(1+\cos x)$
I'm trying to solve a problem from a previous exam. Unfortunately there is no solution for this problem.
So, the problem is:
Calculate the Taylor polynommial (degree $4$) in $x_0 = 0$ of the function:
$$f(x) = \frac{1}{1+\cos(x)}$$
What I tried so far:
calculate all $... |
H: Why does Strassen's algorithm work for $2\times 2$ matrices only when the number of multiplications is $7$?
I have been reading Introduction to Algorithms by Cormen. Before explaining Strassen algorithm the book says this:
Strassen’s algorithm is not at all obvious. (This might be the biggest understatement in thi... |
H: Solving for unknown, trouble with $\ln$ and $\exp$
Having some trouble understanding $\ln$ and $\exp$ rules and what to do in this situation. Perhaps it has just been a very long day...
$$\hat{Y} = \exp \left[\left(\hat{\beta_0} + \sum_i \hat{\beta_i}{x_i}\right)\space A' \right] $$
Solving for $A'$.
AI: Hint:
$\ln... |
H: Characterization of uniform continuity via sequence
For $f(x)$, defined on the interval $X$, $f(x)$ is uniformly continuous in X if and only if for every sequences $x_{n}$, $y_{n}\in X$, when we have $\lim_{n\rightarrow\infty}(x_{n}-y_{n})=0$, then $\lim_{n\rightarrow\infty}[f(x_{n})-f(y_{n})]=0$.
For this characte... |
H: Unsolved arithmetic exercise for printing numbers in book
Here is the exercise from the "Arithmetic" book (ISBN: 5-02-013764-2).
There are 1989 numbers had printed to enumerate book's pages. How many
pages in book?
Honestly I decided to refresh my arithmetic knowledge at all but stuck with that problem.
Here's ... |
H: Closed set on Euclidean space that is not compact
I have read that a subset of Euclidean space may be called compact if it is both closed and bounded. I was wondering what a good example of a closed but unbounded set would be?
Would a closed ball inside a sphere with an infinite radius do the trick? If that examp... |
H: Question about Lie superalgebra.
What are the generators and relations for the Lie superalgebra $\mathfrak{psu}(2, 2 | 4)$? Thank you very much.
AI: The article arXiv:0505234, "Non-linear Realization of $\operatorname{PSU}(2,2|4)$ on the Light-Cone" by Pierre Ramond et al. has details and further references.
The al... |
H: How can I construct a $2^{63}$-gon with a straightedge and compass?
I entered $2^{63}$ as a stand alone value at WolframAlpha. Among
the responses was a factoid that 'A regular 9223372036854775808-gon is constructible with a straightedge and compass.'
What is such a shape and how can I construct one?
AI: Start with... |
H: What is the difference between following approaches to line integrals?
What is the difference between following approaches concerning line integrals:
first approach is for complex function with parametrization $\gamma(t) = \cos t+i\sin t$ (Line_integral :: Example from Wikipedia).
Second approach is for $f(x,y)$... |
H: Distributing an item equally.
For the following question
A $10$ foot plank of wood is cut to give three equal lengths with a shorter length left over. Which is more
a)The length of one of equal pieces
b) $3$ feet
Now here is how I am solving it --> Each piece gets =$\frac{10}{3}$ and remaining is $1$ foot.
So... |
H: How do I show that $f(z) \equiv \sum_{n \in \mathbb{Z}} \frac{1}{(z-n)^2}$ is a meromorphic function?
Let
$$f(z)=\sum_{n=-\infty}^\infty \frac{1}{(z-n)^2}.$$
Show $f$ is meromorphic on $\mathbb{C}$ with double poles at each integer.
I think I got it to be meromorphic. I fixed an integer $m$ and considered... |
H: Riemann Sum question
limit as $n$ goes to infinity of $$\sum_{k=1}^n \left(\frac k {n^2}-\frac{k^2}{n^3}\right)$$ I know I need to make this an integral but I cannot figure out how to acquire the limits of integration. Any help would be great.
AI: We actually don't need to perform an integral here, we can simply t... |
H: What does $\lim_{n \to \infty} \sum_{m=n}^{2n} \frac{1}{m}$ equal?
Possible Duplicate:
Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?
I encountered the following sum in Boros & Moll's "Irresistible Integrals" question 5.2.11 (pg. 78):
$$\lim_{n \to \infty} \sum_{m=n}^{2n} \frac{1}{m}$$
H... |
H: Example that $u\in W^{1,2}$, but $u \notin W^{1,3}$
I'm doing the calculations about the following assertion
Let $\Omega$ be $\{(x,y):0<y<x^2, 0<x<1\}$. The function $u(x,y)=\log (x^2+y^2)$ belongs to $W^{1,2}(\Omega)$, which you can check by integrating $|\nabla u|^2\approx 1/x^2$ within $\Omega$. We have $\Delta ... |
H: OEIS A000255 recursion.
I encountered the sequence A000255. $a(n)$ counts permutations of $[1,...,n+1]$ having no substring $[k,k+1]$
I am finding difficulty in proving it.
Can you please give any clues or hints on how to attack the problem?
AI: The recurrence is $a(n)=na(n-1)+(n-1)a(n-2)$ The first term is the nu... |
H: Is every monomial over the UNIT OPEN BALL bounded by its L^{2} norm?
Let $m\geq 2$ and $B^{m}\subset \mathbb{R}^{m}$ be the unit OPEN ball . For any fixed multi-index $\alpha\in\mathbb{N}^{m}$ with $|\alpha|=n$ large and $x\in B^{m}$
$$|x^{\alpha}|^{2}\leq \int_{B^{m}}|y^{\alpha}|^{2}dy\,??$$
AI: No. For a count... |
H: Relationship between real inverses of analytic functions
Take some analytic function, $f(x)$, that goes from $-\infty$ to $\infty$, with a finite number of points such that $\frac{df}{dx}=0$. You can divide the y axis into intervals, where the boundary between each interval is the y value at a critical points (see ... |
H: Number of bit strings with 3 consecutive zeros or 4 consecutive 1s
I am trying to count the number of bit-strings of length 8 with 3 consecutive zeros or 4 consecutive ones. I was able to calculate it, but I am overcounting. The correct answer is $147$, I got $148$.
I calculated it as follows:
Number of strings wit... |
H: is division by zero automatically irrational?
I know that division by zero is undefined and is also not rational, but I am not sure whether this means it's irrational because it is undefined.
Can anyone clarify?
AI: An irrational number is a real number that is not rational. Dividing by zero doesn't give you a num... |
H: Can't follow a proof involving Prime-Power Fields
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 $\mathbb{Z}_{p}$ as prime fields (up to isomorphism). By Corollary 33.6, $E$ is a simple extension of $\mathbb{... |
H: How to prove that the language of a DFA is some $L$
Consider the following DFA:
It is quite clear that the language of this FDA is all the words that don't have the word $aa$ as a subword.
My question is: How can I formally prove that this is the language of this FDA ?
My efforts: I tried to determine $L(q_0)$ and... |
H: Perfect set in $\mathbb{R}$ which contains no rational number
Possible Duplicate:
Perfect set without rationals
Does there exist a nonempty perfect set in $\mathbb{R}$ which contains no rational number?
This problem is on p.44 PMA - Rudin
I found a proof of this on google but the proof is not 'suitable' for me, ... |
H: Question about algebraic field extensions
If I have a subfield $F$ of a field $E$, and an algebraic (over $F$) $\alpha\in E$, I can form $F(\alpha)$ which is isomorphic to $F[x]/\langle f(x)\rangle$ for $f(x) = irr(\alpha, F)$. That is, $f(x)$ is the minimal degree and monic element of $F[x]$ such that $f(\alpha) ... |
H: Is There a Continuous Analogue of the Hypergeometric Distribution?
As the title states, is there a continuous analogue of a Hypergeometric distribution?
If $ X \sim H(m,n,N)$ is a common Hypergeometric distribution, where $N$ is the population size, $n$ is the number of draws, and $m$ is the number of success. In m... |
H: Determining limits on variable change
Ok, I've seen some questions similar to mine but it didn't really get me what I want so I figured I'd ask. I was given the following problem to solve by making the change of variables $ u = x-y, v = x+y$ in the following integral
$ I = \int_0^1dy\int_0^{1-y}e^\frac{x-y}{x+y}dx$... |
H: Consider $x = (2+\sqrt[]{3})^6$, $x=[x]+t$, where $[x]$ is the integer part of $x$, and $t$ is the 'non integer' part of $x$. find $x(1-t)$
consider $x = (2+\sqrt[]{3})^6$, $x=[x]+t$, where $[x]$ is the integer part of $x$, and $t$ is the 'non integer' part of $x$. find the value of $x(1-t)$
AI: Note that $(2+\sqrt... |
H: Integral dependence and rings of fractions
I have a question on a Proposition in Atiyah and MacDonald's text. It concerns Proposition 5.12 ($A$ and $B$ are commutative rings with an identity) pictured here:
Here's my concern: After multiplying the equation of integral dependence in the ring of fractions through by... |
H: $\sum\limits_{i=1}^n z_i=0\Longleftrightarrow z_1,\ldots ,z_n$ are the vertices of ...
Suppose for $n\geq 3$ we have, $z_1,\ldots ,z_n\in\mathbb{C}$ and $|z_1|=|z_2|=\cdots=|z_n|=1$. Now I need to determine a property $P$ such that the following is true :
$$\sum_{i=1}^n z_i=0\Longleftrightarrow z_1,\ldots ,z_n\mbox... |
H: What does String mean in context of counting theory?
What does "String" mean in context of counting theory ? For example :
There are 8 binary strings of length 3 .
What does binary and length signify here ?
Edit : I want to specifically why it is called a String ?
AI: binary string means $\textit{a sequence of 0'... |
H: Maximum Profit at which combination
A dealer deals only in colour TVs and VCRs . He wants to spend up to Rs.12 lakhs to buy 100 pieces. He can purchase colour TV at Rs. 10,000 and a VCR at Rs. 15,000. He can sell a colour TV at Rs. 12,000 and a VCR at Rs. 17,500. His objective is to maximiz profits
Fpr the maximum ... |
H: Analysis of how-many-squares and rectangles are are there on a chess board?
I know the formula $$f(n) = \frac{n \cdot (n + 1) \cdot (2n + 1) } 6$$
Since chess board consists $8\times 8$, hence here $n=8$.
But I want to know how it has been concluded? Also how to tackle the number of rectangles?
AI: Here is an easy ... |
H: Can $(\Bbb{R}^2,+)$ be given the structure of a matrix Lie group?
I have an assignment problem that is coming from Brian Hall's book Lie Groups, Lie Algebras and Representations: An Elementary Introduction.
Suppose $G \subseteq GL(n_1;\Bbb{C})$ and $H \subset GL(n_2;\Bbb{C})$ are matrix Lie group and that $\Phi:G... |
H: Find the probability of the the sum
A new drug has been released and produces some minor side effects. 8% of users suffer
only a loss of sleep and 12% of users suffer only bouts of nausea. 75% of users will have
no side effects at all. What percentage of users will suffer from both loss of sleep and
nausea?
... |
H: What's the proof of correctness for Robert Floyd's algorithm for selecting a single, random combination of values?
I read about it in a SO answer:
Algorithm to select a single, random combination of values?
initialize set S to empty
for J := N-M + 1 to N do
T := RandInt(1, J)
if T is not in S then
... |
H: Question about proof that $C(X)$ is separable
In my notes we prove Stone-Weierstrass which tells us that if we have
a subalgebra $A$ of $C(X)$ such that it separates points and contains
the constants then its closure (w.r.t. $\|\cdot\|_\infty$) is $C(X)$.
A few chapters later there is a lemma that if $X$ is a compa... |
H: Evaluating $I(x)=\int_{0}^{\infty}\frac{e^{-xy}}{y^2+a^2}dy$
I examine currently this integral:
$$I(x)=\int_{0}^{\infty}\frac{e^{-xy}}{y^2+a^2}dy;x\geqslant0$$ where $x$ and $a$ are real.
It seems that the integral has no closed form in terms of elementary functions. But perhaps it has a closed form in terms o... |
H: Number Expected of Tests
A medical clinic tests blood for certain disease from which approximately one person in a hundred suffers. People come to the clinic in group of 50. The operator of the clinic wonders whether he can increase the efficiency of the testing procedure by conducting pooled tests. In the pooled t... |
H: All possibilities of seven numbers in ascending order?
I had posted a question about a pretty old permutation cipher here on Stack Overflow.
One @Mark Adler has commented that all possibilities of seven numbers with range $0$-$255$ in ascending order is $9,503,812,464$.
I tried to speculate by saying "Ok you mean... |
H: Is a countable product of compact intervals in $\mathbf R$ compact (without using the AC)?
Let $\{I_n=[a_n,b_n]\}_{n\in\mathbf N}$ be a countable collection of closed, bounded intervals in $\mathbf R$. Is the infinite Cartesian product $$\prod_{n=1}^\infty I_n$$ compact without using the Axiom of Choice?
AI: See He... |
H: What's the supremum of this?
I would like to know how to show that the supremum
$$\sup_S\frac{(x-y)^2 + (t-s)^2}{(|x-y|^2 + |t-s|)^{\epsilon}}$$
is less than infinity, where $0<\epsilon \leq 1$ and
$$S = \{ (x,t), (y,s) \in [0,2\pi] \times[0,T] \mid (x,t) \neq (y,s)\}$$
where $0 < T \leq 1$.
What can I use?
AI: We ... |
H: Prove that the solution tends to $0$ as $t$ goes to infinity
I have to prove that, if $A(t)$ is a real symmetric $N\times N$ matrix whose
eigenvalues are all less than $-1$ for all $t$, then if we consider $u$ to be the solution of
$$\dot u(t)=A(t)u(t),$$
Then
$$\lim_{t\to\infty}|u(t)|^2=0.$$
The only thing I w... |
H: characteristic polynomial of the adjacency matrix of a tree
I have read that if $A$ is the adjacency matrix of a tree $T$, then we have that
$$\det(\lambda I - A) = \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k N_k(T) \lambda^{n-2k} $$
where $N_k(T)$ is the number of matchings of size $k$ of $T$.
However, I couldn't find... |
H: upper bound for derivatives of analytic functions on upper half-plane
Let $f$ be analytic mapping of the upper half-plane into the unit disc. Given $f(i)=\alpha$, how does one rigorously obtain an upper bound for $|f'(i)|$?
AI: We can assume that $f(i)=i$ without loss of generality (scale and rotate). Then we use C... |
H: What is the image of $|z-4i|+|z+4i|=10$?
What is the image of $|z-4i|+|z+4i|=10$?
I tried to simplify this equation but it is too difficult for me. I tried squaring both side but it was too long and I can't get equation like any curve.
Can anyone give me any simple simplification? Or is there any result of comple... |
H: Find the probability of three tosses of a fair coin
Find the probability that, in three tosses of a fair coin, there are three heads, given that
there is at least one head.
I manage to get $\frac{3}{6}$ or $\frac{1}{6}$ but the right answer is
$\frac{1}{7}$
I have no idea, Can you please explain?
thanks! App... |
H: Why this proof $0=1$ is wrong?(breakfast joke)
We have $$e^{2\pi i n}=1$$
So we have $$e^{2\pi in+1}=e$$
which implies $$(e^{2\pi in+1})^{2\pi in+1}=e^{2\pi in+1}=e$$
Thus we have $$e^{-4\pi^{2}n^{2}+4\pi in+1}=e$$
This implies $$e^{-4\pi^{2}n^{2}}=1$$
Taking the limit when $n\rightarrow \infty$ gives $0=1$.
AI: Yo... |
H: Additive quotient group $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the multiplicative group of roots of unity
I would like to prove that the additive quotient group $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the multiplicative group of roots of unity.
Now every $X \in \mathbb{Q}/\mathbb{Z}$ is of the form $\frac{p}{q} ... |
H: Why there may be no single "maximum" element in a partially ordered set?
From Appendix B.2 (relations) of Introduction to Algorithms by Cormen et al:
In a partially ordered set $A$, there may be no single "maximum" element $a$ such that $b R a$ for all $b ∈ A$. Instead, there may several maximal elements a such th... |
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