instruction stringlengths 12 30k |
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Two questions:
(1.) Construct a subset of $[0,1]$ in the same manner as the Cantor set, except that at the $k$-th stage, each interval removed has length $\delta 3^{-k}$, $0<\delta <1$. Show that the resulting set is perfect, has measure $1-\delta$, and contains no intervals.
Showing that it's perfect is not dif... |
Define a function $F(A, B, C)$ as the number of ways you can roll $B$ $C$-sided dice to sum up to $A$, counting different orderings (rolling a $2$, $2$, and $3$ with three dice is different from rolling a $2$, $3$, and $2$).
Example:
With three $5$-sided dice, the list of $F(A, B, C)$ values in the domain of the ... |
Combinatorics and Rolling Dice Similarity? |
Define a function $F(A, B, C)$ as the number of ways you can roll $B$ $C$-sided dice to sum up to $A$, counting different orderings (rolling a $2$, $2$, and $3$ with three dice is different from rolling a $2$, $3$, and $2$).
Example:
With three $5$-sided dice, the list of $F(A, B, C)$ values in the domain of the ... |
I have an expected value say 5(minimum) or 6(minimum).
Then I have 4 values [3, 10, 6, 8]
I would like to compute the percentage on how far the 4 values are from the expected value in terms of percentage. Given 100% means they are all within the minimum and maximum and the farther they are from the expected the l... |
What is the degree of the polynomial of the following expression ?
$$ [x + (x^3 – 1)^{1/2}]^5 + [x – (x^3 – 1)^{1/2}]^5 $$
If I am not very wrong the highest power of x is $\frac {15}{2} $ ?! So the degree is floor(15/2) = 7 ?! The answer is 7 though I am not sure of this approach.
|
What is the degree of the polynomial of the following expression ?
$$ [x + (x^3 – 1)^{1/2}]^5 + [x – (x^3 – 1)^{1/2}]^5 $$
If I am not very wrong the highest power of x is $\frac {15}{2} $ ?! So the degree is floor(15/2) = 7 ?! The answer is 7 but I am not sure of this approach.
Please comment.
|
Do you mean 5 (minimum) to 6 (maximum)? You could calculate how many ranges away each item is. So 3 would get a score of 2 (from being 2 away from 5). 10 would get a score of 4 (10-6). 6 would get a score of 0 for being in range and 8 would get a score of 2. This doesn't tell you what percentage are in range, but ... |
I'm solving this question, but I don't know what it is.
============================================================
Consider a plane $x - y + z = 0$, and a point $b = (1, 2, 0)^T$
a) Find a basis of this plane, and a basis of orthogonal subspace to this plane.
b) Find the closest point $\hat{b}$ on the pla... |
Is this a linear algebra problem? |
If in the expansion of $(1 + x)^m \cdot (1 – x)^n $, the coefficients of $ x $ and $ x^2 $are 3 and -6 respectively, then m is ?
I solved it in the following way :
Expanding we get, the coefficient of $ x $ as $ (m-n) = 3 \cdots (1)$ and coefficient of $ x^2 $ as $ \frac{n(n-1)}{2} + \frac{m(m-1)}{2} +- m \cdot n... |
How to calculate this efficiently ? |
If in the expansion of $(1 + x)^m \cdot (1 – x)^n $, the coefficients of $ x $ and $ x^2 $are 3 and -6 respectively, then m is ?
I solved it in the following way :
Expanding we get, the coefficient of $ x $ as $ (m-n) = 3 \cdots (1)$ and coefficient of $ x^2 $ as $ \frac{n(n-1)}{2} + \frac{m(m-1)}{2} +- m \cdot n... |
The coordinate transformation (due to Beukers, Calabi and Kolk)
$$x=\frac{\sin u}{\cos v}$$
$$y=\frac{\sin v}{\cos u}$$
transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the triangle domain $u,v>0,u+v<\pi /2$ (in *Proofs from the Book* by M. Aigner and G. Ziegler).
Since the invert transform... |
For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. I want to prove:
$$f^{-1}(\sigma(C))=\sigma(f^{-1}(C))$$
I could prove that
$$\sigma(f^{-1}(C)) \subset f^{-1}... |
For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. I want to prove:
$$f^{-1}(\sigma(C))=\sigma(f^{-1}(C))$$
I could prove that
$$\sigma(f^{-1}(C)) \subset f^{-1}... |
For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. I want to prove:
$$f^{-1}(\sigma(C))=\sigma(f^{-1}(C))$$
I could prove that
$$\sigma(f^{-1}(C)) \subset f^{-1}... |
According to Mathworld, hyperbolic geometry is well understood in 2 dimensions but not in 3 dimensions.
http://mathworld.wolfram.com/HyperbolicGeometry.html
What isn't well understood about hyperbolic geometry in 3D, that is in 2D or that is in 3D-Euclidean geometry? |
Hyperbolic geometry. 3 dimensions. What is not well understood ? |
How to calculate without calculator or something like this the values of
π^e and e^π
in oder to compare them? |
Let \psi be a wavelet. Can its Fourier transform \hat{\psi} be also wavelet? produce an example or prove that it is not possible. A wavelet is a function \psi:R->R such that (i) \psi \in L^1(R) \cap L^2(R), (ii) \int_{-infty}^{infty} \psi(t) dt = 0, fourier transform is \hat{f(\omega)} =\int_{-infty}^{infty} f(t)e^{-i*... |
Let \psi be a wavelet. Can its Fourier transform \hat{\psi} be also wavelet? |
This is an old chestnut. As a hint, it's easier to consider the more
general problem: for which positive $x$ is $e^x>x^e$? |
How to calculate without calculator or something like this the values of
$\pi^{e}$ and $e^{\pi}$
in oder to compare them? |
Let $\psi$ be a wavelet. Can its Fourier transform $\hat{\psi}$ be also wavelet? produce an example or prove that it is not possible. A wavelet is a function $\psi:R->R$ such that (i) $\psi \in L^1(R) \cap L^2(R)$, (ii) $\int_{-infty}^{infty} \psi(t) dt = 0$, fourier transform is $\hat{f(\omega)} =\int_{-infty}^{infty}... |
I was wondering this today, and my algebra professor didn't know the answer.
Are subgroups of finitely generated groups finitely generated?
I suppose it is necessarily true for finitely generated abelian groups, but is it true in general?
And if not, is there a simple example of a finitely generated group wi... |
Let $\psi$ be a wavelet. Can its Fourier transform $\hat{\psi}$ be also wavelet? produce an example or prove that it is not possible. A wavelet is a function $\psi:R->R$ such that (i) $\psi \in L^1(R) \cap L^2(R)$, (ii) $\int_{-infty}^{infty} \psi(t) dt = 0$, fourier transform is $\hat{f(\omega)} =\int_{-infty}^{infty}... |
Let $\psi$ be a wavelet. Can its Fourier transform $\hat{\psi}$ be also wavelet? produce an example or prove that it is not possible. A wavelet is a function $\psi:\mathbb R\to\mathbb R$ such that (i) $\psi \in L^1(R) \cap L^2(R)$, (ii) $\int_{-\infty}^{\infty} \psi(t) dt = 0$, fourier transform is $\hat f(\omega) =\in... |
Let $\psi$ be a wavelet. Can its Fourier transform $\hat{\psi}$ be also wavelet? |
I want to show that there is a finite conjunction $\phi$ of axioms of $ZF$, such that every transitive proper class $M$, which satisfies $\phi$, is already a model of $ZF$.
This is an exercise in Kunen's set theory. There is a hint, it seems to be useful to apply the Reflection principle to the union $M = \cup_{\al... |
This is NOT a homework, I am over homework stage ages ago :-)
What function draws semicircule?
y=f(x) = something
and I end up with a unit circle. |
This is NOT a homework, I am over homework stage ages ago :-)
What function draws semicircle?
$$y=f(x) = \text{something}$$
and I end up with a unit circle. |
i have 2 perpendicular vectors in space . How can i determine the plane determined by the 2 vectors?
Regards,
Alexandru Badescu |
Is this 2d five-fold Venn Diagram original and valid? |
Are there known sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (The factor of +1 in the denominator and the constraint of the sum of inverse "shifted prime divisors" is bounded by 1 are both necessary.)
A related ques... |
are continuous functions always differentiable? |
I am learning about various signal processing methods in my university course, and I can't seem to grasp what 'energy' in signals represent. I mean, I know that it is the integral of the absolute value of energy squared (side question: why is absolute value needed?), but I don't know the meaning of what this energy is.... |
How can I interpret "energy" in signals? |
Notation question (I believe the notes that I'm reading uses pretty common notation):
Let $V $ be a vector space over a field $K$. What is $V^\mathbb{N}$? Is it a vector space of infinite dimensions? What are the elements of $V^\mathbb{N}$? Are they infinite length tuples? |
If $V$ is a vector space, that what is $V^\mathbb{N}$? |
In my opinion, this notation is ambiguous and should be avoided. It could mean two things: it could mean the space of functions $\mathbb{N} \to V$, which is a vector space in the obvious way. This is the countable product in the category of vector spaces. This vector space is very large; in particular, it has uncoun... |
Are Continuous Functions Always Differentiable? |
It would be the vector space consisting of an infinite product V × V × ... of copies of V.
More generally: for two sets A and B, A<sup>B</sup> often denotes the set of functions *f* : B → A. (The only exception I'm aware of is when the sets A and B happen ... |
in R3 (3d) , having a vector perpendicular with a plane ( so we know where is 'up') , how do we determine if a certain point is below our plane ?
Regards,
Alexandru Badescu |
Isn't this the same as your question [vector question][1]? Take the dot product of the perpendicular and the coordinates of the point. If it is less than zero, the point is below the plane.
[1]: http://math.stackexchange.com/questions/7780/vector-question |
In my opinion, this notation is ambiguous (edit: but maybe I'm wrong). It could mean two things: it could mean the space of functions $\mathbb{N} \to V$, which is a vector space in the obvious way. This is the countable product in the category of vector spaces. This vector space is very large; in particular, it has ... |
One observes that $4!+1 =25=5^{2}$, $5!+1=121=11^{2}$ is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of this question:
- Is $n!+1$ a perfect square for infinitely many $n$? If yes, then how to prove. |
UPDATE: I feel like my original question was too vague and didn't provide enough information (as others have mentioned). So I'm going to restate it.
<strong>Purpose:</strong> I want to find an equation in order to provide a smooth slope for an animation that I am creating. My animation consists of a lever and a set ... |
The expression on the right just screams "*Binomial Theorem! Binomial Theorem!*". It's a quick matter to figure out *what* to apply the Binomial Theorem to...
**Added:** So, how can we find out to what to apply the Binomial Theorem? Well, I can tell you what *my* thought process was; I'm sure there are better or mor... |
I think the easiest way may be piecewise—not piecewise-linear, but a piecewise-defined function that is relatively smooth, so there isn't a point where the speed suddenly changes. Let's say that you want slope $m$ for $-a\le x\le a$ (and $y=0$ when $x=0$), then curving up to the point $(b,c)$.
$$y=\begin{cases}
\fra... |
The expression on the right just screams "*Binomial Theorem! Binomial Theorem!*". It's a quick matter to figure out *what* to apply the Binomial Theorem to...
**Added:** So, how can we find out to what to apply the Binomial Theorem? Well, I can tell you what *my* thought process was; I'm sure there are better or mor... |
I was trying to teach myself some basic Lie theory, and I came across this statement [on Mathworld](http://mathworld.wolfram.com/Commutator.html), relating the commutator of a group, $\alpha\beta\alpha^{-1}\beta^{-1}$, to the commutator of its Lie algebra, $[A,B] = AB-BA$:
> For instance, let $A$ and $B$ be square m... |
Let $\mathbf C$ be an abelian category admitting projective limits. Let's consider the category whose objects are those of the form
$$
0\to A\to B\to C\to 0
$$
and whose morphisms are triples of morphisms of $\mathbf C$ such that the diagram
$$
\begin{array}{ccccc}
A&\hookrightarrow & B & \to & C & \newline
\do... |
If I have 5 students (A-E) that score 80%, 70%, 70%, 60% and 50% on a test what percentiles do they fall in?
- A - 20th percentile (80%)
- B - 40th percentile (70%)
- C - 40th percentile (70%)
- D - 80th percentile (60%)
- E - 100th percentile (50%)
Is this correct?
(PS can someone with 150+ rep add a... |
How do percentiles work? |
If I have 5 students (A-E) that score 80%, 70%, 70%, 60% and 50% on a test what percentiles do they fall in?
- A - 20th percentile (80%)
- B - 40th percentile (70%)
- C - 40th percentile (70%)
- D - 80th percentile (60%)
- E - 100th percentile (50%)
Is this correct?
|
Can the Nil-Radical and the Jacobson Radical be equal in a non-trivial case (i.e. not every prime ideal in said ring is maximal)?
Are there any interesting examples of this case? |
Nil-Radical equals Jacobson Radical even though not every prime ideal is maximal? |
Let's talk about some general methods. This is a special case of the following problem: given that you understand a sequence $a_n$, how can you understand the sequence $b_n = \sum_{k=0}^n {n \choose k} a_k$? Generating function techniques will come to your rescue. If you let $A(x) = \sum_{n \ge 0} a_n x^n$, then
... |
From $e_n(x)+D_n=\prod(x+b)$
it follows that $\prod b=D_n$ and the $x^1$ coefficient of $e_n(x)$
is $D_n\sum 1/b$. But the $x^1$ coefficient is equal to $\prod'a$
where the product is over all nonzero polynomials $a$ of degree $ < n$.
All you have to do is to prove this product equals $(-1)^nD_n/L_n$.
The leading ... |
Can the Nil-Radical and the Jacobson Radical be equal in a non-trivial case (i.e. not every nonzero prime ideal in said ring is maximal)?
Are there any interesting examples of this case? |
Let's assume we have a commutative ring with identity. Can the Nil-Radical and the Jacobson Radical be equal in a non-trivial case (i.e. not every nonzero prime ideal in said ring is maximal)?
Are there any interesting examples of this case? |
Suppose for company A, there is a $60 \%$ chance that no claim is made in the coming year. If one or more claims are made, then the total amount is normally distributed with mean $1000$ and standard deviation $2000$.
Suppose for company B, there is a $70 \%$ chance that no claim is made during the coming year. If ... |
You don't state the question explicitly, but presumably it is what is the chance that A's claims exceed B's. As the expected difference in claims is 2.8 standard deviations, there is little (how little?) chance that A's claims will exceed B's claims assuming they each get one. If we ignore it, and assume that the cha... |
Suppose for company A, there is a $60 \%$ chance that no claim is made in the coming year. If one or more claims are made, then the total amount is normally distributed with mean $1000$ and standard deviation $2000$.
Suppose for company B, there is a $70 \%$ chance that no claim is made during the coming year. If ... |
How to represent logical operators as functions? |
As the expected difference in claims is 2.8 standard deviations, there is little (how little?) chance that A's claims will exceed B's claims assuming they each get one. If we ignore it, and assume that the chance of a claim at A is independent of the chance of a claim at B (can we?) We have probability of no claim ei... |
If $x, y \in [0, 1]$ you can take $f(x, y) = xy$. This comes from replacing "true" and "false" with probabilities and from assuming that $x, y$ are probabilities of independent events. Similarly for NOT take $f(x) = 1 - x$, and for OR take $f(x, y) = 1 - (1 - x)(1 - y)$.
If this isn't what you're looking for then ... |
I'm aware that one can use a Fast Fourier Transform (FFT) to take the cost of multiplication of two polynomials of degree N from O$(N^2)$ to O$(N \ln N)$ (which is an amazing reduction when dealing with large polynomials!). Does a similar transformation procedure exist for multinationals?
I'm interested in the speci... |
I'm aware that one can use a Fast Fourier Transform (FFT) to take the cost of multiplication of two polynomials of degree N from O$(N^2)$ to O$(N \ln N)$ (which is an amazing reduction when dealing with large polynomials!). Does a similar transformation procedure exist for multinomials?
I'm interested in the special... |
I think the easiest way may be piecewise—not piecewise-linear, but a piecewise-defined function that is relatively smooth, so there isn't a point where the speed suddenly changes. Let's say that you want slope $m$ for $-a\le x\le a$ (and $y=0$ when $x=0$), then curving up to the point $(b,c)$.
$$y=\begin{cases}
\fra... |
If I have 5 students (A-E) that score 80%, 70%, 70%, 60% and 50% on a test what percentiles do they fall in?
- A - 20th percentile (80%)
- B - 40th percentile (70%)
- C - 40th percentile (70%)
- D - 80th percentile (60%)
- E - 100th percentile (50%)
Is this correct?
Let me expand the question as I was... |
Is there a succinct proof for the fact that the rank of a non-zero skew-symmetric matrix ($A = -A^T$) is at least 2? I can think of a proof by contradiction: Assume rank is 1. Then you express all other rows as multiple of the first row. Using skew-symmetric property, this matrix has to be a zero matrix.
Why does ... |
The question I've heard on many levels (including the grad level): what is the square root of $a^2$? And everyone says: it's $a$!
In fact it is $|a|$. |
Suppose I have an optimization problem of the form:
min d'y
Subject to My \geq d and y \geq 0
If a solution s which satisfies As = c and s \geq 0,m how can it be shown s is optimal for the above linear programming problem? |
Suppose I have an optimization problem of the form:
$$\min \mathbf{d}^{T}\mathbf{y}$$
subject to
$$\mathbf{M}\mathbf{y} \geq \mathbf{d}, \qquad \mathbf{y} \geq 0$$
If a solution $\mathbf{s}$ which satisfies $\mathbf{A}\mathbf{s} = \mathbf{c}$ and $\mathbf{s} \geq 0$, how can it be shown $\mathbf{s}$ is op... |
> "It is a profoundly erroneous truism, repeated by copybooks and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thi... |
So I feel a bit strange asking a Calculus question, but this came up today while teaching.
One can check that if you start with some integral, which can be see as an "obvious u-substitution problem", that you can instead use integration by parts, and wind up with the scenario where you have have the original integra... |
Integration by Parts implies U-substitution? |
Let $R$ be a ring and $f(X) \in R[x]$ be a non-constant polynomial. We know that the number of roots, of $f(X)$ in $R$ has no relation, to its degree if $R$ is not *commutative*, or *commutative but not a domain*. But,
- *The number of roots, of a non-zero polynomial over commutative integral domain, is at most it... |
Is the [lemniscate][1] what you want? I don't know what you need as a jump at the crossing point, but maybe you can get that with a trigonometric parameterization.
[1]: http://mathworld.wolfram.com/Lemniscate.html |
For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. I want to prove:
$$f^{-1}(\sigma(C))=\sigma(f^{-1}(C))$$
I could prove that
$$\sigma(f^{-1}(C)) \subset f^{-1}... |
I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it? |
Can someone please explain the Riemann Hypothesis to me... in English? |
I have an expected value say 5(minimum) or 6(minimum).
Then I have 4 values [3, 10, 6, 8]
I would like to compute the percentage on how far the 4 values are from the expected value in terms of percentage. Given 100% means they are all within the minimum and maximum and the farther they are from the expected the l... |
I manually arranged two complete latin squares from ordered pairs of 4 and 6 like so:
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
1 2 3 4 5 6
2 4 1 6 3 5
3 1 5 2 6 4
4 6 2 5 1 3
5 3 6 1 4 2
6 5 4 3 2 1
I like this kind of sequence, but can tell it is going to be more di... |
I'm just starting to learn some basic topology, and I've mostly encountered definitions. For instance, I read that for a topological space $X$, for any $E\subseteq X$, define its closure $\overline{E}$ as the set of points $p\in X$ such that each neighborhood $N$ of $p$ has a nonempty intersection with $E$.
I wanted... |
Just wondering if equations used as props in SG1, atlantis and most recently Universe are just random and cool looking, or real/meaningful and cool looking.
Most recent episode of Stargate Universe S02E04 has a "corridor of equations" where dr nick Rush is trying to decipher something. Has anyone familiar with the s... |
Do the equations used in Stargate make sense or are they gibberish? |
Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. Hence there are at least countably infinite many orders of infinity.
**But do there exist uncountably infinite orders ... |
Are there uncountably infinite orders of infinity? |
I'm looking for lightweight, free, Windows, Math software.
Something I can put an expression and get an answer, or graph it.
I tried Euler, but it is quiet complicated and HUGE.
Basic needs:
1. Expression Based
2. Supports Variables
3. Support Functions, User defined and auto loaded.
4. Supports ... |
Well, in a previous post regarding the explanation of *Riemann Hypothesis* Matt answered that:
- The prime number theorem states that the number of primes less than or equal to $x$ is approximately equal to $\int_2^x \dfrac{dt}{\log t}.$ The Riemann hypothesis gives a precise answer to how good this approximation ... |
Well, in a previous post regarding the explanation of [*Riemann Hypothesis*][1] Matt answered that:
- The prime number theorem states that the number of primes less than or equal to $x$ is approximately equal to $\int_2^x \dfrac{dt}{\log t}.$ The Riemann hypothesis gives a precise answer to how good this approxima... |
As calculator [wxMaxima][1] ([screenshots][2]) with export to png and LaTeX syntax.
For playing with graphs I use [GeoGebra][3] with export to png, eps, pdf...
For some special cases [Wolfram|Alpha][4].
And also MS Excel.
And also a normal calculator - is often faster than PC programs ;)
(And for hard di... |
Lots of people (including myself) face lot of problems in tackling Mathematical Problems, which appear as if we can solve it, but then writing out a solution becomes difficult.
Let us consider some examples:
- I was asked this question, some time back in an exam. Give an example of a continuous function on $(a,b... |
See [this question][1] on applying probability theory principles in software design.
The question is generally the following: you design some system (say software) and rely on some well-known mathematical concept (say hash function). You know that when this concept is used without caution your system can sometimes f... |
How safe is it to ignore low probability events? |
I have a (joke) template file for writing papers, which contains in it
> **Lemma** *(Main technical lemma)*
> Let $D$ be a domain in (INSERT SPACE HERE) such that the following properties hold:
> 1. Technical condition 1
> 2. Technical condition 2
> Then $D$ is both opened and closed in (INSERT SPACE AG... |
I have a (joke) template file for writing papers, which contains in it
> **Lemma** *(Main technical lemma)*
> Let $D$ be a domain in (INSERT SPACE HERE) such that the following properties hold:
> 1. Technical condition 1
> 2. Technical condition 2
> Then $D$ is both opened and closed in (INSERT SPACE AG... |
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