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Does the void set ($\phi$) is the proper subset of every set ? |
I am a bit confused about the concept of proper subsets,precisely whether to include one or both of the **void set** and **the set itself**.
An extract from my module goes like this :
***Obviously,every set is the subset of itself and the void set $\phi$ is the subset of every set. These two subsets are called im... |
Working on probability theory in an economics class. This was on of the practice problems that I'm struggling to back up with math. Any help would be appreciated! :)
- An accident insurance company has reserves to cover claims on no more than 3 accidents. If all accidents have a probability of 0.01 of occurring, if ... |
Is the void set (∅) a proper subset of every set ? |
I am a bit confused about the concept of proper subsets,precisely whether to include one or both of the **void set** and **the set itself**.
An extract from my module goes like this :
***Obviously,every set is the subset of itself and the void set $\phi$ is the subset of every set. These two subsets are called im... |
As usual denote L^p the quotient space where two integrable functions are identified if they are equal almost everywhere. So I'm using the definition written here:
http://en.wikipedia.org/wiki/Lp_space
Then we have the following result: for each $p \geq 1$ we have $L^{\infty}(X) \leq L^{p}(X)$ where X is a finite... |
Suppose $X$ is a Banach space and $T: X \to X$ is a bounded linear operator. If $T$ is onto, is it necessarily 1-1? |
As usual denote $L^p$ the quotient space where two integrable functions are identified if they are equal almost everywhere. So I'm using the definition written here:
http://en.wikipedia.org/wiki/Lp_space
Then we have the following result: for each $p \geq 1$ we have $L^{\infty}(X) \subseteq L^{p}(X)$ where X is a... |
How do I show $\frac{d^2 \Gamma}{dz^2} = \displaystyle \sum_{n = 0}^{\infty} \frac{1}{(z+n)^2}$? |
How do I show $\frac{d^2 log(\Gamma (z))}{dz^2} = \displaystyle \sum_{n = 0}^{\infty} \frac{1}{(z+n)^2}$? $\Gamma(z)$ is the gamma function |
How do I show $\frac{d^2 log(\Gamma (z))}{dz^2} = \displaystyle \sum_{n = 0}^{\infty} \frac{1}{(z+n)^2}$?
$\Gamma(z)$ is the gamma function |
I am doing some pre study/revision in maths before i take my next education steps and am working through a [hefty engineering maths book][1] from the beginning. The first chapter is a review of algebraic techniques using indices.
The question is to simplify this
\begin{align}
\frac{2xy^2}{(2xy)^2}
\end{align}
... |
is it possible to calculate shadow areas of buildings or simulate shadows of buildings in a city, using the heights of these buildings and the sun angle and azimuth? the basic light tracing concept using trigonometry is known. But the Problem is how to find away to come up with simulated shadow of group of buildings in... |
Suppose we are given two real numbers: $a, b \in R$, $b > a$. Find $m \in [1, 2, 2.5, 5]$ and $k \in Z$, satisfying the following condition: $\frac{b-a}{m10^k} \in [5; 10]$.
How to solve this problem without using brute force? |
How to solve this problem programmatically? |
When f is a Lebesgue measurable function, is $\Vert f \Vert_p$ is continuous in p, when p is in a set such that f belongs to $L_p$? If yes, how to show that? Thanks!
|
$\Vert f \Vert_p$ is continuous in p ? |
When f is a Lebesgue measurable function, is $\Vert f \Vert_p$ continuous in p, when p is in a set such that f belongs to $L_p$? If yes, how to show that? Thanks!
Yes. It is an exercise taken from Rudin's book. I have spent quite a while thinking but got no idea.
|
So you're asking about $p \mapsto \left( \int_X |f|^p d\mu \right)^{1/p}$. Isn't this clearly a composition of continuous functions?
First $g \mapsto g^p$, then the Lebesgue integral, and finally $x \mapsto x^{1/p}$. |
For a Lebesgue measurable function f, $\Vert f \Vert_p$ converges to $\Vert f \Vert_\inf$ as $p$ goes to infinity.
I was wondering if $\Vert f \Vert_p$ converges to $exp(\int \log |f|)$ as $p$ goes to 0? How to prove this?
Thanks! |
So you're asking about $p \mapsto \left( \int_X |f|^p d\mu \right)^{1/p}$. Isn't this clearly a composition of continuous functions?
First $p \mapsto |f|^p$ (no hints, but this should be pretty easy), then the Lebesgue integral (hint: it's a linear map between the Banach space $L^p(X,\mu)$ and $\mathbb{R}$. Which li... |
For a Lebesgue measurable function f, $\Vert f \Vert_p$ converges to $\Vert f \Vert_\inf$ as $p$ goes to infinity.
I was wondering if $\Vert f \Vert_p$ converges to $exp(\int \log |f|)$ as $p$ goes to 0 for probability measure? How to prove this?
Thanks!
|
As we know that $L_p \subseteq L_q$ when $0 < p < q$ for probability measure, I was wondering when $L_p = L_q$ is true and why. Is it to impose some restriction on the domain space? Thanks! |
When $L_p = L_q$ ? |
A Steinmetz solid is the intersection of two cylinders. My question is how to find the surface area of one with both cylinders of radius 1, without parameterizing the solid.
I parameterized it; that is, used $r(u,v)=\langle \sqrt{1-v^2}, u \sqrt{1-v^2}, v \rangle$, $u,v \in [-1,1]$, and integrated the magnitude of t... |
My daughter is stuck on the concept that 2<sup>0</sup> = 1, having the intuitive expectation that it be equal to zero. I have tried explaining it, but I guess not well enough. How would you explain the concept to a child, other than the teachers "that is just the rule" approach? |
This is a "normalization" type operation. For simplicity we can replace the two parameters a,b with the single one c = b - a > 0.
Assuming the interpretation proposed by Eric above, you want c*(10^-k) to fall into an interval m*[5,10) where m is in {1,2,2.5,5}. In other words we want to normalize c by a power of 1... |
How many subgroups with index 2 are there of a free group on two generators? What are their generators?
All I know is that the subgroups should have 2*2 + 1 - 2 = 3 generators. |
Let $F$ be free on $a$ and $b$, and let $N$ be a subgroup of index $2$. Then $N$ is normal, and there is a map $\phi:F\to\mathbb Z/2\mathbb Z$ with kernel precisely $N$. What are the possible images of $a$ and $b$? |
Let $F$ be free on $a$ and $b$, and let $N$ be a subgroup of index $2$. Then $N$ is normal, and there is a map $\phi:F\to\mathbb Z/2\mathbb Z$ with kernel precisely $N$. What are the possible images of $a$ and $b$? For each possible choice you can describe the corresponding generators of $N$ as a normal subgroup, and w... |
I'm teaching first semester calculus and trying to find a way to explain why each hypothesis in l'Hostpial's rule is needed.
If $f$ and $g$ are real differentiable functions on an interval containing a point $c$ then there are three hypotheses one needs to check in order to apply l'Hospitals rule to compute $\lim_{x... |
Mariano's answer is quite right, but relies on the 'accident' that all subgroups of index two are normal. If you wanted to find all the subgroups of index three, say, you would need to try another approach. Here's one idea.
The free group of rank two, $F$, is the fundamental group of the figure-eight graph $\Gamma$... |
I will give a different answer than the answer I gave in the other thread which tries to appeal to intuition. I am sure your daughter has no problem accepting that $2*0 = 0$. Intuitively this is because if you add $2$ to itself zero times, you get zero. Or, to be concrete, if someone gives you two apples zero times,... |
Mariano's answer is quite right, but relies on the 'accident' that all subgroups of index two are normal. If you wanted to find all the subgroups of index three, say, you would need to try another approach. Here's one idea.
The free group of rank two, $F$, is the fundamental group of the figure-eight graph $\Gamma$... |
I have proved this using Venn diagram but when I am trying to prove this using the rule that ***"If $ A \subset B \text{ and } B \subset A $ then $ A = B $"***, I am having some problems with my understanding of the same,here is how I did so far :
Let $x \in A \cap (B-c) \Rightarrow x \in A \text{ and } x \in (... |
If f and g are both constant functions, then it's easy to see that the claim isn't true in general. Let's assume that g is not constant, and since this problem came from a calculus class, let's assume that f and g are differentiable. Compare the derivative of f(x) * g(x) with the derivative of f(x) + g(x). What can you... |
Mariano's answer is quite right, but relies on the 'accident' that all subgroups of index two are normal. If you wanted to find all the subgroups of index three, say, you would need to try another approach. Here's one idea.
The free group of rank two, $F$, is the fundamental group of the figure-eight graph $\Gamma$... |
If N stands for the set of natural numbers, the unbounded set is -
1. $ X = \biggl( x|x = \frac{1}{n},n \in N \biggr) $
2. $ Y = \biggl( x|x = \frac{1}{2^n},n \in N \biggr) $
3. $ Z = \biggl( x|x = 2^n,n \in N \biggr) $
4. $ W = \biggl( x| x \in N, x \lt 4532 \biggr) $
After reading the definiti... |
If N stands for the set of natural numbers, the unbounded set is -
1. $ X = \biggl( x|x = \frac{1}{n},n \in N \biggr) $
2. $ Y = \biggl( x|x = \frac{1}{2^n},n \in N \biggr) $
3. $ Z = \biggl( x|x = 2^n,n \in N \biggr) $
4. $ W = \biggl( x| x \in N, x \lt 4532 \biggr) $
After reading the definiti... |
For a Lebesgue measurable function f, $\Vert f \Vert_p$ converges to $\Vert f \Vert_\inf$ as $p$ goes to infinity.
I was wondering if $\Vert f \Vert_p$ converges to $\exp(\int \log |f|)$ as $p$ goes to 0 for probability measure? How to prove this?
Thanks!
|
How do I show
$$\frac{\mathrm{d}^2 \log(\Gamma(z))}{\mathrm{d}z^2} = \displaystyle \sum_{n = 0}^{\infty} \frac{1}{(z+n)^2}$$?
$\Gamma(z)$ is the gamma function. |
A set in $\mathbb{R}$ is bounded if you can find two numbers, $a$ and $b$ such that all elements of the set are between $a$ and $b$. If you want to claim a set is unbounded, whatever $a$ and $b$ I give you, you should be able to find at least one element outside the interval $(a,b)$. Which of your sets satisfies that... |
Imagine I had two sets of samples. The only information I have about them is their
* Size
* Average
* Standard deviation
Is it possible to calculate the standard for a third set that is composed by the sum of all samples in both sets?
And what if the original sets had the same size? |
Is it possible to calculate the standard deviation without the samples? |
Imagine I had two sets of samples. The only information I have about them is their
* Size
* Average
* Standard deviation
Is it possible to calculate the standard deviation of a third set that is composed by the sum of all samples in both sets?
And what if the original sets had the same size? |
Imagine I had two sets of samples. The only information I have about them is their
* Size
* Average
* Standard deviation
Is it possible to calculate the standard deviation of a third set that is composed by the union of all samples in both sets?
And what if the original sets had the same size? |
In my attempts to derive the closed form for the arclength of the hyperbola, I wound up with the following integral:
$$\int\frac{\sqrt{1-m\;\sin^2 u}}{\sin^2 u}\mathrm{d}u$$
I am aware that such integrals are expressible as combinations of elliptic integrals and trigonometric functions, and *Mathematica* does ret... |
**HINT:** When do you say that a set is bounded.
- *A set $X \subset \mathbb{R}$ is said to be bounded if you can find a $M \in \mathbb{R}$ such that $x \leq M$ for all $x \in X$.*
Now consider your cases:
- $X = \\{ x | x = \frac{1}{n} ; \ n \in \mathbb{N}\\}$ **NOTE:** $\frac{1}{n} \leq 1$ for all $n \i... |
If $Y$ is a Hausdorff topological space, then the value of a continuous function $f\colon X\to Y$ is completely determined by the value of $f$ on a dense subset $E$ of $X$; to see this, suppose $f$ and $g$ are two functions that agree on a dense subset $E$ of $X$, and let $u\in X\setminus E$. If $f(u)\neq g(u)$, then t... |
How to solve the recursion:
$$p[n,m] = p[n-1,m-1] + p[n+1,m-1] + p[n,m-1]$$
Ideally in general, but if you need base cases:
$$p[n,0] = 0 \text{ (for } n \neq 0),$$
$$p[0,0] = 1$$
I've asked a similar question previously and believe $n^{m-2}$ is
somehow involved.
Would turning this into a DF... |
If N stands for the set of natural numbers, the unbounded set is -
1. $ X = \\{ x|x = \frac{1}{n},n \in \mathbb{N} \\} $
2. $ Y = \\{ x|x = \frac{1}{2^n},n \in \mathbb{N} \\} $
3. $ Z = \\{ x|x = 2^n,n \in N \mathbb{N} \\} $
4. $ W = \\{ x| x \in N, x \lt 4532 \\} $
After reading the definitio... |
Let $f \in L^{1}(\mathbb{R})$ such that $f(x) >0$ for almost all $x \in \mathbb{R}$. Let A be a Lebesgue measurable set such that $\int_{A} f =0$. Prove $m(A) = 0$.
wlog we can assume $f(x)>0$ everywhere define $A_{n}=\{x \in A: f(x) > \frac{1}{n}\}$
so since $f(x) > 0$ everywhere, $\int_{A_n} f dm \leq \int_{A... |
Which of the following is the unbounded set ?
1. $ X = \\{ x|x = \frac{1}{n},n \in \mathbb{N} \\} $
2. $ Y = \\{ x|x = \frac{1}{2^n},n \in \mathbb{N} \\} $
3. $ Z = \\{ x|x = 2^n,n \in N \mathbb{N} \\} $
4. $ W = \\{ x| x \in N, x \lt 4532 \\} $
After reading the definition from [Wikipedia][1]... |
It is claimed in some logic books that the compactness theorem of first-order logic can be proven using Tychnoff's theorem from topology.
Now, to me this feels very strange because I consider logic as something "lower" in the sense that the theorems of topology follow from the axioms and theorems in logic. How can w... |
Let X be a sigma finite measure space with $p > 0$. The book states the following:
There exists a map $f \in L^{p}(X)$ such that $f>0$ and f is bounded by $1$.
Proof: Let $C_{n}$ be a sequence of disjoint sets of finite measure such that X is the union
of these sets. Now just put $f(x) = (\frac{2^{-n}}{1+\mu(X... |
What are some interesting applications of the Mean Value Theorem for derivatives (both the 'extended' or 'non-extended' versions as seen [here][1] are of interest)?
So far I've seen some trivial applications like finding the number of roots of a polynomial equation. What are some more interesting applications of it?... |
It is a rather surprising fact (to me, at least) that $\binom{14}{4} = 1001$; $\binom{14}{5} = 2002$; $\binom{14}{6} = 3003$. Actually, this is the only instance where three consecutive binomial coefficients are in the ratio 1 : 2 : 3. I found it quite interesting to investigate under what conditions consecutive member... |
It is a rather surprising fact (to me, at least) that $\displaystyle \binom{14}{4} = 1001$; $\displaystyle \binom{14}{5} = 2002$; $\displaystyle \binom{14}{6} = 3003$.
Actually, this is the only instance where three consecutive binomial coefficients are in the ratio $\displaystyle 1 : 2 : 3$. I found it quite inter... |
Let X be a sigma finite measure space with $p > 0$. The book states the following:
There exists a map $f \in L^{p}(X)$ such that $f>0$ and f is bounded by $1$.
Proof: Let $C_{n}$ be a sequence of disjoint sets of finite measure such that X is the union
of these sets. Now just put $f(x) = (\frac{2^{-n}}{1+\mu(C_{... |
What is an intuitive explanation of a positive-semidefinite matrix?
|
We have three counters, $i, j, k$, all initialized to zero. Each step consists of adding or subtracting one from one of the counters, so $(\Delta i, \Delta j, \Delta k)$ is selected among $(\pm1, 0, 0), (0, \pm1, 0), (0, 0, \pm1)$, each with probability 1/6. What is the expected number of steps before all the counters ... |
What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition.
The definition is:
A n × n Hermitian matrix M is called positive-semidefinite if
$$x^{*} M x \geq 0$$
for all $x \in \mathbb{C}^n$ (or, all $x \in \mathb... |
Injective functions are not the same thing as injective *correspondences*, correct? Injective functions are a subset of the injective *correspondences*.
For example in $y^2 = x$, y is not a function of x, but this is still a correspondence between $\mathbb{R}$ and $\mathbb{R}$. Can we say that the correspondence, $... |
What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.
The definition is:
A $n$ × $n$ Hermitian matrix M is called *positive-semidefinite* if
... |
Injective functions are not the same thing as injective *correspondences*, correct? Injective functions are a subset of the injective *correspondences*.
For example in $y^2 = x$, y is not a function of x, but this is still a correspondence between $\mathbb{R} \rightarrow \mathbb{R}$. Can we say that the corresponde... |
I am writing a program to solve a rubiks cube, and would like to know the answer to this question.
There are 12 ways to make one move on a rubik's cube. How many ways are there to make a sequence of six moves?
From my project's specification: up to six moves may be used to scramble the cube. My job is to write a ... |
i have to solve the following problem
find an f(k) where Sk=θ(f(k))
where Sk is sum from 1 to K^2-1 of sqrt(n)
I tried first of all to calculate or "limit" my sum using integral rule so i came up with
2/3(K^2-1)^(3/2)<=Sk<=2/3(k^3-1)
but after that i am in a dead end as i do not know SK so i can not simplify anyth... |
i have to solve the following problem
find an $\displaystyle f(k)$ where $\displaystyle S_k=\theta(f(k))$
where $\displaystyle S_k =\sum_{n=1}^{k^2-1} \sqrt{n}$
I tried first of all to calculate or "limit" my sum using integral rule so i came up with
$\displaystyle \frac{2(k^2-1)^{3/2}}{3} \leq S_k \leq \frac{2(k... |
This may be embarassingly simple, but I can't see it.
Let $M$ be a Riemannian manifold of dimension $n$; fix $x \in M$, and let $B(x,r)$ denote the geodesic ball in $M$ of radius $r$ centered at $x$. Let $V(r) = \operatorname{Vol}(B(x,r))$ be the Riemannian volume of $B(x,r)$. It seems to be the case that for smal... |
Greetings!
I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean?
I've read a <a href="http://mathforum.org/library/drmath/view/52251.html">few</a> <a href="http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH057.HTM">pages</a> on this... |
A Google search for the title of the question finds [this article](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.mmj/1029001150) where the first five coefficients in the series expansion of the volume (in powers of $r$) are computed. The first term is the same as the Euclidean volume ... |
It is a way to extend your real variable function into the complex domain in a "nice" (read nice as holomorphic) way. You might ask the question why do we need to even extend in the first place? The need for extension comes from the fact that a $n^{th}$ order polynomial with real co-efficients need not have all its roo... |
How do I calculate the outcome of taking one complex number to the power of another, ie (a + bi)^(c + di)? |
It is a way to extend your real variable function into the complex domain in a "nice" (read nice as holomorphic) way. You might ask the question why do we need to even extend in the first place? The need for extension comes from the fact that a $n^{th}$ order polynomial with real co-efficients need not have all its roo... |
If A and B are two points on the earth, how could I find any arbitrary point between them along the shortest distance side of their great circle path?
Points are in radians
longitude = 0 to 2pi
latitude = 0 to pi, 0 being at north pole
Points are not antipodal
I desire something where I specify a range 0... |
Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; \mathbb{R}^{n})$ by the points $(x_{1}, \dots, x_{n}) \subset \mathbb{R}^{n}$ which satisfy the inequality
\begin{eqnarray}
\omega_{1} x... |
Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; \mathbb{R}^{n})$ by the points $(x_{1}, \dots, x_{n}) \subset \mathbb{R}^{n}$ which satisfy the inequality
\begin{eqnarray}
\omega_{1} x... |
Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; \mathbb{R}^{n})$ by the points $(x_{1}, \dots, x_{n}) \subset \mathbb{R}^{n}$ which satisfy the inequality
\begin{eqnarray}
\omega_{1} x... |
Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; \mathbb{R}^{n})$ by the points $(x_{1}, \dots, x_{n}) \subset \mathbb{R}^{n}$ which satisfy the inequality
\begin{eqnarray}
\omega_{1} x... |
If $\omega_1 = \cdots = \omega_n = 1/k$ then the equation translates to
$\displaystyle x_1 + \cdots + x_n \leq k$
and so the number of non-negative integer solutions is $\binom{n+k-1}{k}$, which is roughly $n^k/k!$, compared to your estimate $k^n/n!$ (that's assuming $k$ is much smaller; if we assume $n$ is much ... |
The definition of small is that they have O(lg n) bits. One way is just to test the integers 2,3,... for primality and keep the first n primes, but this takes at least O(n log n) time (times the cost of primality testing) using a naive algorithm. Is it possible to do it in linear time? |
How can one efficiently generate n small relatively prime integers? |
Are they isomorphic? |
Ok this is a really silly question and I should know this, but I cant seem to figure something out:![alt text][1]
[1]: https://i.stack.imgur.com/F0U3t.png
for the last step, how do they know that 0<=x<=4? If we use the minimum value of theta, which is $-pi/2$, and plug that into $x=cos(theta)$, then we get 0,... |
I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five pieces. My question is :
"Is it possible to construct (mathematically) these five sets or is the proof more of an existe... |
This question in its full generality is probably best tackled by some generating function approach as Qiaochu points out. I toyed around with the $N=1$ case and got some interesting results that I am sharing below. For the $N=1$ case, we don't have to distinguish between the $+1$ and the $-1$s. In fact, a nice visualiz... |
This question in its full generality is probably best tackled by some generating function approach as Qiaochu points out. I toyed around with the $N=1$ case and got some interesting results that I am sharing below. For the $N=1$ case, we don't have to distinguish between the $+1$ and the $-1$s. In fact, a nice visualiz... |
If | | denotes the absolute value then is it always true $|x^{s}|=|x|^{s}$ whenever s is any _positive_ real number? I know this holds for any positive integer. What about positive reals? |
Let V be a vector space with a non-trivial subspaces U, W. $ U \neq W$. Consider $u_0 \in U$, with $u_0 \neq 0$ and $w_0 \in W$ with $w_0 \neq 0$ then we have $u_0 + U \in V/U$ and $w_0 + W \in V/W$. Is there any sensible way to define:
$(u_0 + U) + (w_0 + W)$?
I'm thinking "no" $(2,4) + (2,6,7)$ makes no sense.... |
Bad ideas: Adding vectors from different vector spaces? |
I think that it is worth pointing out that, strictly speaking, explaining “2<sup>0</sup> equals 1” is not really what people do in this situation. The best we can do is to convince a child of the following facts:
1. If 2<sup>0</sup> is any number, it makes more sense to consider that 2<sup>0</sup>=1 than considerin... |
Jonas: What you need to show is that if $Y$ is well-ordered, then any subset $Z$ of $Y$ is isomorphic to an initial segment of $Y$. How would you define such an isomorphism $f$? You do not have many possibilities: The least element of $Z$ must be mapped by $f$ to the smallest element of $Y$. The next element of $Z$ mus... |
The question is: Given a licence plate that can have either 2 or 3 letters followed by either 2 or 3 numbers, how many different license plates can be printed.
My math is as such $(26^2 + 26^3)*(9^2+9^3)$ which comes to 14,784,120 but the book says the answer is 20,077,200. What am I doing wrong here? |
Check out Chapter 1 of Paul Pollack's book "[Not Always Buried Deep][1]". He lists several elementary proofs for the infinitude of primes, some of which can be adapted to form simple iterative algorithms.
(Sorry, I can't get Latex to work here. Isn't it like MathOverflow? Anyway..)
Let S = {p1...pk} be the se... |
Assuming you're talking about real $x$, the only time you would be concerned with whether the equality holds is if $x$ is negative. (Otherwise, $|x|=x$ and $|x^s|=x^s$, so there's nothing to show.) Now, how do you define $x^s$ if $x$ is negative and $s$ is irrational?
One answer is given in more generality by use... |
This is a homework question so specifically I am looking for a direction (help). Not an absolute answer.
I understand how to solve 2nd order nonhomogeneous (and I think 3rd order is similiar) recurrence relations. Whats kinking me up with this one is a term with n.
h[n] = 4h[n- 1] - 4h[n-2] + 3n + 1
Here I... |
How to solve 3rd order nonhomogeneous recurrence relation with n in equation? |
A triangle on a sphere is comprised of points A, B and C. How to determine its area?
I know the formula:
A = E * R^2, where R is radius of sphere, and E is the excess angle of (a + b + c - pi), but how to determine the angles between ABC, ACB and BAC?
Thanks! :-) |
According to the book the following is true:
Let $f$ be a measurable function and suppose $|f| \leq h$ where $h$ is a real-valued function
and $h$ is integrable. Then f is integrable.
My question is: suppose instead we have the inequality $ |f| \leq h$ but almost everywhere and of course $h$ assumed to be integr... |
You have three bags, A, B, and C. Bags A and B hold x red and y blue marbles each; bag C holds 2x red marbles and 2y blue marbles.
Let us say that we pull out z marbles from bag A, z marbles from bag B, and 2z marbles from bag C. Will the expected number of red marbles pulled from bag A and bag B together be the sam... |
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