instruction stringlengths 12 30k |
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What makes writing mathematics difficult is lack of confidence. Someone confident who happens to be wrong will not face difficulties writing up their misapprehensions.
Willie's argument, getting familiar with lots of proof techniques that together solve most problems in some area, builds confidence: when you see su... |
What makes writing mathematics difficult is lack of confidence. Someone confident who happens to be wrong will not face difficulties writing up their misapprehensions.
Willie's argument, getting familiar with lots of proof techniques that together solve most problems in some area, builds confidence: when you see su... |
**Statement:**
There exist a polynomial P such that |P(x) - cos(x)| <= 10^-6 for all (real) x.
**My answer**: False. All polynomials of a degree >= 1 are unbound as x tends to infinity. A polynomial of degree n = 0 is bound only when it is in the form y = a (horizontal line) but this will not help because cos(x) va... |
Statement about existence of a polynomial - true or false? |
**Statement:**
There exist a polynomial P such that |P(x) - cos(x)| <= 10^-6 for all (real) x.
**My answer**: False. All polynomials of a degree n >= 1 are unbound as x tends to infinity. A polynomial of degree n = 0 is bound only when it is in the form y = a (horizontal line) but this will not help because cos(x) ... |
Matrix Image http://img714.imageshack.us/img714/3540/selection002od.png
Is it possible to derive formula from above matrix that converts YCbCr to RGB? I am newbie to maths.
I have YCbCr value already available, I want to convert it to RGB. |
Do somebody knows anything about the Dirac's identity?
\begin{equation} \label{Dirac}
\frac{\partial^2}{\partial x_{\mu}\partial x^{\mu}} \delta(xb_{\mu}xb^{\mu}) =
-4\pi \delta(xb_0)\delta(xb_1)\delta(xb_2)\delta(xb_3)
\end{equation}
here
- $xb$, is the 4-vector $x-b$ in Minkowsky spacetime
- $\delta... |
I have a problem understanding this equation (I encountered it in a mechanics problem).
Given:
$v = \partial x/\partial t$
$w = \partial q/ \partial t$
Then:
$\frac {\partial v}{ \partial w} = \frac{\partial x}{\partial q}$
Why is that true? How can I proof that? |
You don't even need to look at the coordinate entries of your matrices. If E and F are vector spaces, and A and B are linear maps from E to F such that for all x in E: Ax=Bx, then by linearity (A-B)x = 0 for all x. Now what can you say about the linear map A-B, and what does that imply about A and B? |
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 $\mathbf{M}\mathbf{s} = \mathbf{d}$ and $\mathbf{s} \geq 0$, how can it be shown $\mathbf{s}$ is op... |
How can I prove that for every positive integer $n$ we have
$\frac{n\pi}{4}−\frac{1}{\sqrt{8n}}<\frac{1}{2}+\sum_{k=1}^{n−1}\sqrt{1−\frac{k^2}{n^2}}$? |
I have a problem understanding this equation (I encountered it in a mechanics problem).
Given:
$x = x(t)$
$q = q(t)$
$v = \partial x/\partial t$
$w = \partial q/ \partial t$
Then:
$\frac {\partial v}{ \partial w} = \frac{\partial x}{\partial q}$
Is that true? If yes, why is that true? How can I proo... |
I recommend chapters 3 and 5 of [Numerical Recipes][1]. It sounds like you have a function f which is hard/expensive to evaluate. You can evaluate it wherever you like, just not too many times. You then use those values to model f in some way that is much easier to compute and use that model to get values at other p... |
$$
\begin{pmatrix}
Y \\\\ Cb \\\\ Cr
\end{pmatrix}
=
\mathrm{Round}
\left\\{
\begin{pmatrix}
0.2126\times\frac{219}{255} &
0.7152\times\frac{219}{255} &
0.0722\times\frac{219}{255} \\\\
-\frac{0.2126}{1.8556}\times\frac{224}{255} &
-\frac{0.7152}{1.8556}\times\frac{224}{255} &
0.5\times\frac{224}{255} \\\... |
I have a problem understanding this equation (I encountered it in a mechanics problem).
Given:
$x = x(t)$
$q = q(t)$
$v = dx/dt$
$w = dq/dt$
Then:
$\frac {\partial v}{ \partial w} = \frac{\partial x}{\partial q}$
Is that true? If yes, why is that true? How can I proof that?
EDIT:
Actually it's... |
I have made up this "fun" problem.
The first row of a 3x3 matrix is $(a_{11} a_{12} a_{13})$. The next row consists of the variables $h, g$ and $c$ in any order. The last row are distinct non-zero fixed scalars, such as $2, 1, 7$ or $-4$ etc. They could be any integer.
Given an expression for $det(A)$ for example... |
*Wondering if my following question is an application of information theory:*
Lets say we have a factory and ship boxes of stuff outside. If a competitor stands outside my factory, observes the stream of boxes coming out of a factory and note their sizes, he can get valuable information about my manufacturing proc... |
In this formula for Euler numbers:
$\begin{align*}A_n = i^{n+1}\sum _{k=1}^{n+1} \sum _{j=0}^k{k\choose{j}} \frac{(-1)^j(k-2j)^{n+1}}{2^ki^kk}\end{align*}$
what is $i$? I have to generate the $n-th$ Euler number, and I don't have $i$.
Isn't it the imaginary unit, is it? |
I have made up this "fun" problem.
The first row of a 3x3 matrix is $(a_{11} a_{12} a_{13})$. The next row consists of the variables $h, g$ and $c$ in any order. The last row are distinct non-zero fixed scalars, such as $2, 1, 7$ or $-4$ etc. They could be any integer.
Given an expression for $det(A)$ for example... |
I know it is a very general question, but i'm not sure how to get started and improve number theory related skills related to ACM ICPC and other programming competitions.Any good way or book recommendations ? |
How to improve number theory related skills ? |
What meaning of "()" in this formula?
[Image_click here][1]
(That is Catalan number formula (http://en.wikipedia.org/wiki/Catalan_number))
Is it a division?
[1]: http://upload.wikimedia.org/math/3/b/d/3bd4ac77a4af3f894d8e88ed7e1ba418.png |
What meaning of () in Catalan number formula? |
What meaning of "()" in this formula?
![alt text][1]
(That is Catalan number formula (http://en.wikipedia.org/wiki/Catalan_number))
Is it a division?
[1]: https://i.stack.imgur.com/wBRDD.png |
What meaning of "()" in this formula:
$$C_n=\frac1{n+1}\binom{2n}{n}=\frac{(2n)!}{n!(n+1)!}\qquad\mathrm{for}\;n\geq 0$$
(That is Catalan number formula (http://en.wikipedia.org/wiki/Catalan_number))
Is it a division? |
If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$ ?
I would be very grateful if somebody show me how to factor this polynomial by hand, as of now I have used to mathematica to get $(x-2) (2x - 1) (1 + 3x + x^2)$ but I am having trouble to factor this manually.
Best regards, |
How to solve this ? |
You need practice in formulating your intutions. This is a skill that can be learned through practice, but like all skills, you can't begin just by jumping into the deep-end: you need to work consistently at extending your comfort zone, one small step at a time; over time you will then make substantial improvements. ... |
If a,b,c are in arithmetic progression., p,q,r in harmonic progression and ap,bq,cr are in geometric progression., then $\frac{p}{r}+\frac{r}{p} = $ ?
I need some idea to proceed on this problem. |
If a,b,c are in arithmetic progression., p,q,r in harmonic progression and ap,bq,cr are in geometric progression., then $\frac{p}{r}+\frac{r}{p} = $ ?
**EDIT:** I have tried to use the basic/standard properties of the respective progressions to get the desired result, but I am not yet successful. |
*Added:* I interpreted the question as follows: suppose that we use the integration by parts formula,
$$\int u(x)v'(x)\\,dx = u(x)v(x) - \int v(x)u'(x)\\,dx$$
and $\int v(x)u'(x)\\,dx = \alpha \int u(x)v'(x)\\,dx$ for some constant $\alpha$, so that we can "solve" for the original integral as
$$\int u(x)v'(x)\\,dx =... |
And if not, what are the most common hypotheses (on open sets it induces, for example) under which distance is a continuous function?
|
Is the function distance continuous? |
As Qiaochu points out $d(x,y)$ is continuous for fixed $x$. You may like to see this as well, as this is a familiar result in Topology.
- If $A$ is a non empty closed subset of a metric space $(X,d)$ then show that the function f on X given by $f(x)=d(x,A)$ is continuous.
Because $$| f(x) - f(y) | = | d(x,A) - d(... |
At the Blizzcon 2010, StarCraft II multiplayer panel, this stuff was supposed to explain the ladder matchmaking system. I look at this and go eh? what!?
![StarCraft II: Ladder math][1]
Is any of this real? or are they just messing with me, because I can't make heads or tails of this humongous equation. Can anyone... |
*Added:* I interpreted the question as follows: suppose that we use the integration by parts formula,
$$\int u(x)v'(x)\\,dx = u(x)v(x) - \int v(x)u'(x)\\,dx$$
and $\int v(x)u'(x)\\,dx = \alpha \int u(x)v'(x)\\,dx$ for some constant $\alpha$, so that we can "solve" for the original integral as
$$\int u(x)v'(x)\\,dx =... |
As promised in my response to Byron, here is my very partial progress to
date on this problem. Though long-winded, it doesn't amount to much.
I'm still looking for an answer to the general case, or to other special
cases (like $n=2$ or $c=2$) or even just better estimates. I'm also
interested to know about any ap... |
It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total pro... |
It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total pro... |
Suppose $B \subset X$ where $X$ is a vector space. $B$ is called balanced if $\alpha B \subset B$ for every $\alpha \in \Phi$ with $|\alpha| \leq 1$. Note that $\Phi = \textbf{R}$ or $\Phi = \textbf{C}$. What is the intuition behind balanced sets? Why do we define them? Is it "good" for a vector space to have more bala... |
I'm working on a game which relies on "flinging" an object. That is, click and hold on the object, and then drag and release it, and it continues on the path you were dragging it. Of course, the most well-known example of flinging is with iPhone and Android devices, where you can quickly scroll down a list by quickly s... |
It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total pro... |
For the ACM you should probably focus on their problem sets and could also look into problems from discrete mathematics. I really don't think they ask questions that rely on a graduate level understanding of number theory so reading books could be overkill for your purposes and would only confuse you if you rush it. It... |
Does any compact subset belong to Borel Sigma Algebra?
Why?
Here Borel Sigma algebra is defined to be the sigma algebra generated by all open subsets in the topological space.
Thanks and regards! |
Does any compact subset belong to Borel Sigma Algebra ? |
Suppose $A$ is a set of finite measure. Is it possible that $A$ can be an **uncountable** union of disjoint subsets of $A$, each of which has positive measure? |
How is a Borel measure on Euclidean space defined?
I mean the Borel measure on the Borel sigma algebra generated by usual toplogy on an Euclidean space, which coincides with the normal definition of volume of subsets in the Euclidean space.
For Lebesgue measure on an Euclidean space, it is defined in terms of th... |
How is a Borel measure on Euclidean space defined? |
It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total pro... |
It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total pro... |
Does any compact subset belong to Borel Sigma Algebra?
Why?
Here Borel Sigma algebra is defined to be the sigma algebra generated by all open subsets in the topological space.
Thanks and regards!
****************
UPDATE:
I found that there are two kinds of definitions for a Borel sigma algebra: one gen... |
It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total pro... |
Does any compact subset belong to Borel Sigma Algebra?
Why?
Here Borel Sigma algebra is defined to be the sigma algebra generated by all open subsets in the topological space.
Thanks and regards!
****************
UPDATE:
I found that there are two kinds of definitions for a Borel sigma algebra: one gen... |
It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total pro... |
I am not a math expert nor a native English speaker, so maybe I am using wrong terminology... In this case, help me to improve my question :)
**Part 1:**
This is my set of equations, I have to calculate values xn, yn and zn. I know the values of A, B, C, Total and sums of parameters (for example I know the sum o... |
I am not a math expert nor a native English speaker, so maybe I am using wrong terminology... In this case, help me to improve my question :)
**Part 1:**
This is my set of equations, I have to calculate values $x_n$, $y_n$ and $z_n$. I know the values of A, B, C, Total and sums of parameters (for example I know ... |
Let $X$ be integral scheme and $\mathcal K$ sheaf of rationnal functions on $X$. For any
point $y\in X$ different of generic point we know that fiber of $\mathcal K$ (defined as usual as $\mathcal K _y / \mathcal m_y \mathcal K_y$) is zero. I'll be very gratefull if you explain intuitively why this is so, in language... |
Let $X$ be integral scheme and $\mathcal K$ sheaf of rational functions on $X$. For any
point $y\in X$ different of generic point we know that fiber of $\mathcal K$ (defined as usual as $\mathcal K _y / \mathcal m_y \mathcal K_y$) is zero. I'll be very gratefull if you explain intuitively why this is so, in language ... |
It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total pro... |
I'm quite positive that I saw before an encyclopedia (more accurately, dictionary/glossary) on Elsevier website. I remember I looked up the definition of a hyperplane there, few months ago. But I can't find it through Google anymore.
Does anyone know the link? |
Website for Elsevier Encyclopedia (or glossary)? |
Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that will allow me to build widgets for a cost of A/B dollars. Presume that I know that for each C dollars I spend on research ... |
How to analyze risk vs. reward for spending on research and development work? |
`Given three containers of specified volume, how many different volumes can you measure ?`
For example, suppose we have cans with capacities 2, 3, 4 litres.
We can measure :
<code>
1 litre by filling the 3 litre can and pouring it in the 2 litre can. </code>
<code> 2, 3, 4 litres are trivial.</code>
<co... |
Is there a short way to say $f(f(f(f(x))))$?
I know you can use recursion:
$g(x,y)=$
{
$f(g(x,y-1)) \space if \space y \space > \space 0$
$x \space if \space y \space = \space 0$
(That's supposed to be a "piecewise" function; I'm not good at LaTeX. This is the first time I've ever used it.) |
**1.How many proper subset of {1,2,3,4,5,6,7} contain the numbers 1 and 7 ?**
Lets consider {1,7} as a single element then the number of possible subset is $2^6$ and hence the number of proper subset is **62**.
2.A survey show that 63% of the Americans like cheese where as 73% like apples.If x% of Americans like... |
**1.How many proper subset of {1,2,3,4,5,6,7} contain the numbers 1 and 7 ?**
Lets consider {1,7} as a single element then the number of possible subset is $2^6$ and hence the number of proper subset is **62**.
2.A survey show that 63% of the Americans like cheese where as 73% like apples.If x% of Americans like... |
Is there a short way to say $f(f(f(f(x))))$?
I know you can use recursion:
$g(x,y)=\begin{cases}
f(g(x,y-1)) & \text{if } y > 0, \\
\newline x & \text{if } y = 0.
\end{cases}$ |
does $\sum_{n=1}^{\infty} \frac{1}{\zeta(1+\frac{1}{n})}$ diverge or converge? |
A union of finite sets can always be converted to a union of finite disjoint sets.
I was wondering if a union of countable sets can always be converted to a union of countable disjoint sets? If yes how to do that?
Thanks! |
I'm trying to think of an example of a homomorphism of commutative rings $f:A\rightarrow B$ and ideals $I,J$ of $B$ such that $f^{-1}(I)+f^{-1}(J)$ is not a preimage of any ideal of $B$. I can't seem to come up with one... anyone know one? |
What is the easiest way to find the maximum of 9cos(t) - 8cost(3t)? |
What is the easiest way to find the maximum of 9cos(t) - 8cost(3t)? |
A union of finite sets can always be converted to a union of finite disjoint sets.
I was wondering if a union of countable sets can always be converted to a union of countable disjoint sets? If yes how to do that?
Thanks!
**************
Update with Paul's question:
"I wonder about the natural extension o... |
Of course, Euler proved that the Riemann zeta function can be defined as the analytic continuation of a product over all primes.
$\zeta(s) = \displaystyle\prod_{p \in P = primes}\frac{1}{1-p^{-s}}$
It is well known (but not something I understand) that the positions of zeros of the zeta function allows one to m... |
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 $\mathbf{M}\mathbf{s} = \mathbf{d}$ and $\mathbf{s} \geq 0$, how can it be shown $\mathbf{s}$ is op... |
What is the easiest way to find the maximum of 9cos(t) - 8cos(3t)? |
Of course, Euler proved that the Riemann zeta function can be defined as the analytic continuation of a product over all primes.
$$\zeta(s) = \prod_{p \in \mathbb{P}}\frac1{1-p^{-s}}$$
It is well known (but not something I understand) that the positions of zeros of the zeta function allows one to make inference... |
Let $z_1, z_2 \ldots z_m$ be complex numbers, $m \in \mathbb{N}$. Can anybody tell me how to prove the following inequality?
$| z_1 z_2 \ldots z_m - 1 | \leq e^{|z_1 - 1| + \ldots + |z_m - 1|} - 1$
In case you're wondering, this is asserted without proof in a paper by Von Neumann, about infinite tensor products o... |
How to prove this inequality involving complex numbers? |
Define the relation ~ on Z by a ~ b if and only if 5a = 2b mod 3 (= means congruent)
Proove that ~ is an Equivelence relation on Z
Identify the distinct equivalence classes on Z/~
How can I prove something like this? Do I need to prove something for reflexiveness, symmetry, and transitivity?
Thanks! |
Define the relation $\sim$ on $\mathbb{Z}$ by $a \sim b$ if and only if $5a \equiv 2b \pmod{3}$.
Prove that $\sim$ is an equivalence relation on $\mathbb{Z}$.
Identify the distinct equivalence classes on $\mathbb{Z}/\sim$.
How can I prove something like this? Do I need to prove something for reflexiveness, sy... |
Yes, you need to "prove something".
To show that $\sim$ is an equivalence relation, you need to prove that it is reflexive, symmetric, and transitive. That is, you need to prove that:
1. For every $a\in\mathbb{Z}$, $a\sim a$ is true.
2. For every $a,b\in\mathbb{Z}$, if $a\sim b$ is true, then $b\sim a$ is tr... |
Yes, you need to "prove something".
To show that $\sim$ is an equivalence relation, you need to prove that it is reflexive, symmetric, and transitive. That is, you need to prove that:
1. For every $a\in\mathbb{Z}$, $a\sim a$ is true.
2. For every $a,b\in\mathbb{Z}$, if $a\sim b$ is true, then $b\sim a$ is tr... |
I have a graph $G$ with a list of edges $E$, all the edges can form cycles, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Ux3b0FtuIMpVGX4e-AbblH05h2gIZII9u8pACRzymPE&w=960&h=720">
All the coordinates for the vertex $V$ **are known**.
Wat is the algorithm to find all the minimum cyc... |
Yes, you need to "prove something".
To show that $\sim$ is an equivalence relation, you need to prove that it is reflexive, symmetric, and transitive. That is, you need to prove that:
1. For every $a\in\mathbb{Z}$, $a\sim a$ is true. Since "$x\sim y$" means "$5x\equiv 2y\pmod{3}$", then that means that you need ... |
How to calculate a bezier curve with only start and end points? |
Let $C$ be a circle of radius $r$ with $n$ points. Prove that there is a point on the circle such that the product of the distances from this point to the other $n$ points is greater than $r^n$. So we seem to be looking at chords that are of length $\leq 2r$. To show the existence of such a point, would you use a const... |
How many distinct integral value of $n$ satisfies the equation $2^{2n} - 3 \cdot (2^{n+2}) + 32 = 0 $ ? |
How many distinct integral value of $n$ satisfies the equation $2^{2n} - 3 \cdot (2^{n+2}) + 32 = 0 $ ? |
We know that the points of discontinuity of a monotone function on an interval $[a,b]$ are countable.Using this can we prov that:
- Any bijection $f: \mathbb{R} \to [0,\infty)$ has infinitely many points of discontinuity.
If yes, how or otherwise how to prove the above result?
|
Is the localization of a reduced ring (no nilpotents) still reduced? |
Does localization preserve reducedness? |
**1.** Say {1,7} to be a single element, then total number of subsets = $2^6$ Now, how many subsets among these don't have {1,7} ? It will be $2^5$.
Hence,number of subsets having {1,7} is $2^6$ - $2^5$ = 32.But since you have asked for proper subset so the answer would be 31. |
**1.** Say {1,7} is a single element, then total number of subsets possible = $2^6$ Now, how many subsets among these **don't have {1,7}** ? It will be $2^5$.
Hence,number of subsets **having {1,7}** is $2^6$ - $2^5$ = 32.But since you have asked for proper subset so the answer would be 31. |
Suppose to the contrary that $f$ has a finite number $n$ of discontinuities, at $x_1, x_2, \ldots x_n$. Then $\mathbb{R} - ( x_1, \ldots x_n )$ is a union of open intervals $I_1, \ldots I_{n + 1}$. $f$ restricted to each $I_m$ is continuous and injective, and therefore monotone, so each image $f|_{I_m}(I_m)$ is an open... |
If a set contain $(2n+1)$ elements and if the number of subsets which contain at most n elements is 4096, then what is the value of $n$? |
If a set contain $(2n+1)$ elements and if the number of subsets which contain at most n elements is 4096, then what is the value of $n$? |
How much do you know about vectors? You can take two points and find the difference between them, as a vector; this is the analogue of the slope of a line in $\mathbb{R}^2$. If the vector is $\mathbf{v}$ and one point is $a$, then points on the line are of the form $a+\lambda\mathbf{v}$, where $\lambda$ is real. (Un... |
I am not a math expert nor a native English speaker, so maybe I am using wrong terminology... In this case, help me to improve my question :)
**Part 1:**
This is my set of equations, I have to calculate values $x_n$, $y_n$ and $z_n$. I know the values of A, B, C, Total and sums of parameters ($S_1$ = $x_1$ + $x_... |
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