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Smith Company produces desk lamps. The information for June indicated that the selling price was $\$25$ per unit, variable costs were $\$15$ per unit, fixed costs totaled $\$6,000$, and the margin of safety in dollars was $\$12,500$. Smith currently sells 1,100 lamps and earns $\$5,000$ of profit. How much is Smith’s... |
A univariate function $f$ is periodic with period $p_1,\ldots,p_k$ if
$$f(z) = f(z + \sum_{i=1}^k n_i \cdot p_i)$$
for all complex $z$ and integers $n_i$. Elliptic function is an example of doubly periodic function. It is claimed that a triply periodic univariate function cannot exist, but why? Can't we follow the co... |
Take Boolean Algebra for instance, the underlying finite field/ring $0, 1, \{AND, OR\}$ is equivalent to $ 0, 1, \{NAND\} $ or $ 0, 1, \{ NOR \}$ where NAND and NOR are considered as universal gates. Does this property, that AND ('multiplication') and OR ('addition') can be written in terms of a single universal binary... |
I was in a discussion today with a philosopher about the merit of the technique of "proof by contradiction." He mentioned the Law of Excluded Middle, wherein we (typically as mathematicians) assume that we either have P or Not P. That is, if we show not "Not P", then we must have P.
Further along in the conversation,... |
How can I reversibly combing two very large numbers, such as 9,828,485,546,536,174,656,640,115,183 and 43,044,700,185? I need them to be combined into a single number, and the order of the numbers must be remained intact, so being required to flip them would be unfavorable, but I could just add a 0 or 1 at the front or... |
I just read a paper in which an ODE was solved using a step I don't understand.
$${dI \over \kappa (x) dx} = I$$
Let
$$\tau = \int_0^x \kappa(x)dx$$
Rewrite equation as
$${dI \over d\tau} = I$$
and solve.
How does the final expression follow from the definition of $\tau$?
|
Why does $\mu(k,n)=\mu(1,\frac{n}{k})$, where $\mu$ is the Möbius function and $k$ and $n$ are integers.
I'm reading about the Möbius function on a poset of integers ordered by divisibility. The author uses this fact several times, but I don't see why it's so obvious. What exactly is this equality saying? Thanks.
Edit:... |
What are some ways to solve this recurrence relation:
$a(n+1)=2 a(n) - a(n-1) -1, \text{ with }a(0)=0, a(10)=0?$
I tried to first convert this inhomogeneous equation into a homogeneous one following Wikipedia:
$b_{n}=Ab_{n-1}+Bb_{n-2}+K \,$ can be
converted into homogeneous form as
follows: The steady state is fou... |
I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square roots you can get arbitrarily close to any number greater than or equal to $1$. But we are having trouble proving or dis... |
How do I find an orthonormal basis for a plane defined as $x_1+x_2+x_3=0$? I don't know where to start -- not even given dimensions. Thank you.
|
Suppose $\phi$ is a bundle map from $E$ to $F$, where $E$ and $F$ are two vector bundles over a manifold $M$. Now for each $x\in M$, $\phi$ restricted to $E_x:E_x\to F_x$ is a isomorphism, how to show that $\phi$ indeed a diffeomorphism?
|
how can i translate this sentence to a quantifier formula, when the universe - { o|o is a set }
Any master has as elements all and only sets which are not elements of themselves.
i know it starts of with an existential quantifier since its dealing with a specific object "master"
any feedback would be great.
|
Let V , W be finite dimensional vector spaces over R. Let A : V->W be a linear map.
Choose bases of V and W and the corresponding bases of $\Lambda^k$(V ) and of $\Lambda^k$(W). How to show that the entries of the matrix representing $\Lambda^k$(A) are polynomial in the entries of the matrix representing A.
|
I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, 2, each time instead of one.
In Python it would be n += 2 -> I'm looking for an equivalent...
|
For a fixed positive integer n, if
$D = \left|\begin{array}{ccc}
n! & (n + 1)! & (n + 2)! \\
(n + 1)! & (n + 2)! & (n + 3)! \\
(n + 2)! & (n + 3)! & (n + 4)!
\end{array} \right|$
show that $\left(\dfrac{D}{(n!)^{3}} - 4 \right)$ is divisible by $n$.
Any ideas on how to go about solving this??
Thank You in ... |
Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Usually when proving a theorem where $f$ is assumed to be continuous, differentiable, $C^1$ or smooth, it is enough to draw intuition by assuming that $f$ is piecewise smooth (something that one could perhaps draw on a paper without lifting your pencil). What I'm saying... |
(1)In set theory, what is the purpose for defining the concept of cofinality?is it that important?
(2)The concept of cofinality finally leads to 2 types of infinite cardinal, for which the first one is regular cardinal and another one is singular cardinal defined this way:
An infinite cardinal $\aleph_{\alpha}$ is ... |
I would like to show the following claim:
Let $l_1, l_2 \in \mathbb{R}^{n+1}$ be two distinct lines then $\exists$ a linear isomorphism $A : \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{n+1}$ with
$Al_1 = (\lambda, 0, 0, \dots , 0)$ and $Al_2 = (0, \lambda, 0, \dots , 0)$
I thought this $A$ needed to prove the claim is th... |
So the Tower of Hanoi numbers are given by the recurrence $h_n=2h_{n-1}+1$ and $h_1=1$. I let my generating function be
$$
g(x)=\sum h_nx^n
$$
Then
$$
g(x)=\sum h_n x^n=\sum (2h_{n-1}+1)x^n=\sum 2h_{n-1}x^n+\sum x^n=2xg(x)+\frac{1}{1-x}.
$$
Solving for $g(x)$ I find
$$
g(x)=\frac{1}{(1-2x)(1-x)}=\frac{2}{1-2x}-... |
I'm doing some programming for a video game in which the player characters and monster enemies can move at different speeds on 2-dimensional, cell-based tile map. Examples of these speeds are as follows:
1600 ms = 0.625 tiles/s
1000 ms = 1 tile/s
800 ms = 1.25 tiles/s
400 ms = 2.5 tiles/s
100 ms = 10 tiles/s
25 ms = 40... |
I am trying to prove young's inequality for integrals
$$
ab \leq \int\nolimits_0^a \! f(x) \, \mathrm{d}x + \int_0^b \! f^{-1}(x) \, \mathrm{d}x.
$$
Can you help me please?
|
I know that all finite subgroups of $\operatorname{SL}_2(\mathbb R)$ are cyclic by standard averaging argument. They are all conjugate to some finite subgroup of $\operatorname{SO}_2(\mathbb R)$ and therefore cyclic. My question is how to classify all finite subgroups of $\operatorname{GL}_2(\mathbb R)$.
Thanking you.
|
Prove by induction. Every partial order on a nonempty finite set has at least one minimal element.
How can I solve that question ?
|
In linear programming, an LP can have multiple optimal solutions if it contains degenerate vertices, i.e. where one of the base-variables is 0.
Can an LP also have multiple optimal solutions if it does not contain degenerate solutions? If so, what is an example LP?
|
Let $X$ an irreducible singular curve with a singular point $x$. Consider $A$, the normalization of $\mathcal{O}_{X,x}$, and $x_1,\dots,x_r$ the points over $x$ in the normalization of $X$. Why $A\cong \mathcal{O}_{x_1}\cap\dots\cap\mathcal{O}_{x_r}$?
Moreover assume that the curve is over a field $k$. Why do the dimen... |
I could really use some step-by-step help on these two problems please. Thank You in advance.
1.) Let $V = \{{\bf{A|A}}$ is an $n \times n$ matrix, $n$ fixed,
det$({\bf{A}}) = 0$ }. Is $V$, with the usual addition and
scalar multiplication, a vector space? Give reason. If yes, find the
dimension and a basis for ... |
I'm a Phd student who teaches part time at a high school and I noticed something when teaching sequences today. I asked my students to find the nth term (the general term) for some sequences. They observed:
If $a_n=n$ then the first differences will be $1$.
If $a_n=n^2$ then the second differences will be $2$.
If ... |
I have 2 polynomials $p_1(x_1,\ldots,x_n)$ and $p_2(x_1,\ldots,x_n)$, of which I have to compute the product, with a special property: The exponent of each variable is always either $0$ or $1$, where every exponent greater than $1$ gets cut down to $1$. E.g. $x_1 * x_1x_2 = x_1x_2$, and the result of $(2x_1x_2x_4 + 4x_... |
I have two points $P_1$ and $P_2$ in the plane. For each of the points, I have two vectors $v_1$ and $v_2$. I want to find a quadratic Bézier curve from $P_1$ to $P_2$ of length $l$ leaving $P_1$ in the direction of $v_1$ and entering $P_2$ in the direction of $v_2$.
Edit: It is not needed that I restrict myself to Béz... |
Let $U$ is an ultrafilter on a set $X$, and $V$ an ultrafilter on a set $Y$.
Wikipedia says: Ultrafilters $U$ and $V$ are Rudin–Keisler equivalent, $U\equiv_{RK}V$, if there exist sets $A\in U$, $B\in V$, and a bijection $f: A → B$ which satisfies the condition above. (If $X$ and $Y$ have the same cardinality, the defi... |
By definition of the Riemann Zeta Function, $$\zeta\left(\frac{1}{2}\right) = \sum_{n=1}^\infty \frac{1}{\sqrt{n}}.$$ Since $\forall n \geq 1 : \frac{1}{\sqrt{n}} \geq \frac{1}{n}$, we have that for all $N \geq 1$, $$\sum_{n=1}^N\frac{1}{\sqrt{n}} \geq \sum_{n=1}^N \frac{1}{n},$$ but it is well known that $$\lim_{N\rig... |
How can I solve this cubic equation?
$$H^{3} − 3\left[(1 + A\cos(T) )^{2} + \frac{2r \cdot A \sin(T)}{B}\right]H + 2(1 + A \cos(T))^{3} = 0$$
Solution in terms of H.
Edited in order to give more insight to my problem:
It was an equation which comes as a part of a derivation in Computational Fluid Dynamics. My motivatio... |
Given two polynomials $A = \sum_{0\le k<n} a_k x^k$ and $B =\sum_{0\le k<n} b_k x^k$ of the same degree $n$, which are equal for all $x$, is it always true that $\ a_k = b_k\ $ for all $0\le k<n?$. All Coefficients and $x$ are complex numbers.
Edit:
Sorry, formulated the question wrong.
|
Consider:
if $\log u$ and $\log v$ are both subharmonic function on an open set $U$, then $\log(u+v)$ is also subharmonic on $U$.
How can I prove this?
|
Let $U\subseteq \mathbb{C}$ be an open set and let $u(z)=-\log(\mathrm{dist}(z,\partial U))$.
I need to show that $u$ is subharmonic on $U$?
$\partial U$ it means the boundary of $U$.
|
I am currently a math major in college and my main problem is that it feels directionless. My college offers little in term of variety in undergraduate math so I moved on into taking graduate courses and I am actually loving it. But at this point I am just randomly taking courses that looks interesting and was wonderin... |
I understand how to do differentiation when the elements are individual functions, but am having trouble applying the concepts with nested functions.
For example, given $(\cos \pi x + \sin \pi y)^5= 24$
$$\frac{d}{dx}\left[\frac{d}{dx}(\cos \pi x)+\frac{d}{dx} (\sin \pi y)\right]^5=\frac{d}{dx}24$$
Following the chain ... |
I have a question.
Could you please help me to solve this?
Thanks in advance
Let $\mathfrak a$ be a finitely generated ideal of $A$, commutative ring with identity, such that $\mathfrak a^2 = \mathfrak a$. Show that $\mathfrak a \subseteq Ae$ with $e^2 = e$, and $e$ is not identity.
|
For the following argument, how can I produce an adequate translation, including interpretations function and formal sentence where universe $\mathcal{U} = \{\bigcirc\mid\bigcirc\text{ is a set}\}$.
And we can probably use:
M1: and o is a master set
e: elements
1.) Any master set has as elements all and only set whic... |
This is a programming problem that I'm trying to solve at TopCoder arena. However, I feel like it need Mathematics to prove its correctness.
Problem
Given a list $L$ of $n$ elements, and a number $k, k \leq n$, where $k$ defined as a number consecutive elements can be reversed. For example:
$L = \{ 4, 3, 2, 1 \}$ and... |
Alice released a new version of her software for removing red-eye from pictures. However, she wants to protect her secret algorithm from disassemblers and such while still letting her customers remove red-eye from pictures.
She looks at standard encryption:
A message is encrypted with a key and sent across the wire
I... |
Is there a name associated to rectangular matrices $M \times N$ that have exactly one entry equal to $1$ in each row and $0$ everywhere else?
|
We have a random generator that generates independent random bits with probability $P(x=1) = P$ and $P(x=0)=1-P$.
Given $N$ random independent bits, we estimate $P$ by $\hat{P} = N_1/(N_0+N_1)$.
where $N_0$ is the number of $0$'s and $N_1$ is number of $1$'s.
The expected value for $\hat{P}$ can be simply shown to be $... |
Consider the following definition of closed maps, defined in the book Nonlinear Programming by Bazaraa et al.:
Let $X$ and $Y$ be nonempty closed sets in $\mathbb{R}^p$ and $\mathbb{R}^q$, respectively. Let $\mathbf{A} \colon X \to Y$ be a point-to-set map. The map $\mathbf{A}$ is said to be closed at $\mathbf{x} \in ... |
This is an old exercise I wrote up, but I'm unhappy with my solution. I assume only the basic properties of addition and multiplication for natural numbers as sets.
Assume that $m$ and $n$ are natural numbers with $m$ less than $n$. Show that there is some $p$ in $\omega$ for which $m+p^+=n$.
First see that $n\neq 0$,... |
Given two natural numbers I am supposed to reverse each of them and then sum them up and reverse the sum to get the final answer.
For example if the numbers are $4358$ and $754$ then the answer should be $1998$
I always thought that there should be a mathematical trick avoid brute force (i.e reversing each numbers and ... |
I've found in Wikipedia the formula for calculating an individual row in Pascal's Triangle:
$$v_c = v_{c-1}\left(\frac{r-c}{c}\right).$$
where $r = \mathrm{row}+1$, $c$ is the column starting from $0$ on left and $v_0 = 1$.
Now, I've tried to do by hand and it works, but I don't understand how to find this magic formu... |
Suppose I chose n uniformly distributed random points in a 3D cube. What is the expected value for the percentage of points lying on the convex hull as a function of n?
Just as a reference, I made the following experiment in Mathematica 8:
Needs["TetGenLink`"]; Show[
DiscretePlot[
1/k Mean[Length /@ Union /@ (Fl... |
What is the isomorphism between dual vector bundle $E^*$ and $\mathrm{Hom}(E,M\times \mathbb R)$? There is a natural isomorphism on bundle that is $\mathrm{Hom}(E,E')=E^*\otimes E$, therefore I am wondering if I can use this isomorphism to get the result that $E^*$ is isomorphic to $E^*\otimes (M\times\mathbb R)$ and t... |
As I understand it, many of the ideas that were introduced into algebraic geometry in the mid 20th century by french mathematicians were done by transporting over ideas from the theory of manifolds (for example defining things locally). Instead of there being an analogy between these 2 areas, is it possible to do more ... |
Can manifolds be uniformly approximated by varieties in the way that continuous functions can be uniformly approximated by polynomials?
I got the idea from reading the Princeton companion to mathematics when it gave:
Theorem (Nash) Let $M$ be a manifold in $\mathbb{R}^n$. Fix any large number $R$. Then there is a pol... |
Given $(5x+5y~)^3= 125x^3+125y^3$, find the derivative.
Using the chain rule and power rule, I came up with
$3(5x+5y)^2 \cdot (\frac{d}{dx}5x+\frac{dy}{dx}5y)= 3 \cdot 125x^2 +3 \cdot 125y^2$
Now, the derivative of $5x~$ is 5, but what about the derivative of $5y~$?
I know that $\frac{dy}{dx}5y~$ turns to $5(\frac{dy... |
I've just sat through several lectures that proved most of the results in Tate's thesis: the self-duality of the adeles, the construction of "zeta functions" by integration, and the proof of the functional equation. However, while I was able to follow at least some of the arguments in the individual steps, I understan... |
Having some difficulty doing translations for complicated neither...nor sentences.
With these characters:
~: Negation; $\vee$: Disjunction; &: Conjunction.
I'm trying to translate and understand, for example:
"Neither John nor Mary are standing in front of either Jim or Cary"
I have been told that a successful translat... |
suppose $v$ is a signed measure on $(X,M)$ and $E\in M$
how do i go about showing that $|v|(E)=sup\{\sum_1^n |v(E_j)|: E_j\cap E_i=0 \forall i\neq j, \cup_1^n E_j=E\}$
sorry it took me awhile to fix the latex
|
I have one proof of the following statement, and I would like to know if there is a simpler proof. I am not sure if “simpler” is the right word or not but, for the purpose of this question, I prefer an elementary proof rather than a short proof depending on a powerful and difficult-to-prove theorem.
For any integer $... |
function $f$ on $R$ can be made periodic with period 1 by wrapping
$$g(x)=\sum_{k=-\infty}^{\infty}f(x+k)$$
Given $g$ with period 1. How to find continuous $f$ in general case?
|
In Leibniz notation, the 2nd derivative is written as
$$\dfrac{\mathrm d^2y}{\mathrm dx^2}\ ?$$
Why is the location of the $2$ in different places in the $\mathrm dy/\mathrm dx$ terms?
|
I have been going through the RSA cipher and have been wondering if there is a way other than the Extended Euclid Method to find
$a^{-1} \mod n$
where a,n $\in$ Z
P.S : n is not necessarily prime
|
I have a problem in solving my number theory homework. My question is as follows:
Let $p$ be an odd prime. Prove that $a$ is a quadratic residue mod $p$ if and only if the $I_{g}(x)$ (index with respect to any primitive root of $p$) is even.
Please edit my writing. Thanks. Does anyone know where to start? Thank you ver... |
I'm reading a tutorial on expectation maximization which gives an example of a coin flipping experiment (the description is at http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html?pagewanted=all). Could you please help me understand where the probabilities in step 2 of the process (i.e. in the middle of part b in... |
Can anybody help me out with this number theory question?
My question is as follows:
If $n$ is a positive integer and if an integer $x$ exists such that
$x^{n-1}\equiv 1 \pmod n$ and $x^{\frac{n-1}{q}} \neq 1 \pmod n$ for all prime divisors $q$ of $n-1$, then $n$ is prime.
I believe we have to use some reasoning on t... |
Let $\{X(t), t\ge 0\}$ and $\{Y(t),t\ge 0\}$ be independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$, respectively. Define $Z_1(t)=X(t)+Y(t)$, $Z_2(t)=X(t)-Y(t)$, $Z_3(t)=X(t)+k$, $k$ a positive integer. Determine which of the above processes are Poisson and find $\lambda$.
Any help is appreciated!... |
Let $x_1,\ldots,x_n$ be any numbers and $\bar{x} = (x_1+ \ldots + x_n)/n$. Then I need to prove that $$min_a \sum_{i=1}^n (x_i - a)^2 = \sum_{i=1}^n (x_i - \bar{x})^2$$
I do not need the answer but I need an visual/intuitive understanding of the question. I do not understand what min$_a$ mean. Actually I'm kind of str... |
I'm trying to get insight into the eigenvalue spectrum of a square matrix (large N, symmetric, positive semi-definite matrix) using Fourier transforms (I've tried transforming a bunch of things: the diagonal, the skew diagonal, the entire matrix, each eigenvector, etc. . Is there anything I should be looking at specifi... |
Prove that $x^{2} \equiv 1 \pmod{2^k}$ has exactly four incongruent solutions.
My attempt:
We have,
$x^2 - 1 = (x - 1) \times (x + 1)$, then
$(x - 1)(x + 1) \equiv 0 \pmod{2^k}$
which implies,
$2^k|(x - 1)$ or $2^k|(x + 1) \implies x \equiv \pm 1 \pmod{2^k} (1)$
Furthermore, $2^{k-1} \equiv 0 \pmod{2^k} \Leftrighta... |
I apologize in advance for how basic this question is...
Let $j:V\rightarrow V/U$ be the ultrapower map where U is an ultrafilter on a set S, and $j(x)=[c_x]$. Now, let $f\in j(0)$. Then $f$ is undefined almost everywhere on S, but we're assuming $f$ is a function defined on all of S, so we conclude no such $f$ exist... |
We have a forest of rooted trees. Two players makes alternating moves according to the following rule: one move is to cut vertex and all its children. Player which makes last move (no vertices remain) wins.
How can we compute Grundy function for the positions in the game?
Suppose we have a trees and we need to say whet... |
I would like to determine the cardinality of the sets specified bellow. Nevertheless, I don't know how to approach or how to start such a proof. Any help will be appreciated.
If $X$ and $Y$ are well-ordered sets, then determine the cardinality of:
$\{f : f$ is a function from $X$ to $Y\}$
$\{f : f$ is an order-preserv... |
How do you prove $$\sum_{j=0}^n (x)^j (-1)^{n-j} \left\{{n \atop j}\right\} = x^n,$$
where $(x)^j=x(x+1)...x(x+j-1)$ and $\left\{{n \atop j}\right\}$ is a Stirling number of the second kind?
|
Can someone help me to construct a linear functional in $\mathcal{C}([0,1])$ that does not attain its norm?
Actually, I want to prove that $\mathcal{C}([0,1])$ is not reflexive Banach space. Is it sufficient to construct that kind of functional?
|
I'm looking for some basic information about vector before going through a chapter in book. Can someone provide links to some good tutorials?
|
Question that seems pretty easy, but I can't formalize it:
Let $L \subset C$ be a lattice, and $f(z)$ be an elliptic function for $L$, that is a meromorphic function so that $f(z+w) = f(z)$ for all $\omega \in L$. Assume that $f$ is analytic except for double poles at each point of the lattice $L$. Show that $f = a\wp... |
$$ y'= \log(ax+by+c)~ ;\qquad a,b,c \in R$$ So is this how I am supoosed to ask questions here?
|
I'm just getting started with Markov chains and have a few simple questions:
Is it possible to define a period for a reducible Markov chain? If so, how?
Can we define balance equations and a stationary distribution for reducible Markov chains?
Thanks.
|
I have a assignment for Minitab 15, which about varoius statistics concepts (basic and foundamental topics) but have a big problem with understanding of of questions! Also I do not know how it's possible to use Minitab® to answer this question, I would be glad some one get me clue. (- I don't need some one to solve the... |
I want to prove the following theorem (no idea whether it has a name):
Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all $x \in V$:
$$C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$$
I first let $x \neq 0$ (otherwise it would be trivial). Th... |
I'm trying to derive the LTE for CN applied to the linear heat equation;
$u_t = u_{xx}$.
The problem is that I end up with terms of the form $\frac{{\Delta t}^k}{{\Delta x}^2}$
when using a two dimensional Taylor expansion around $(x,t)$ for the term:
${\Delta x}^2 {\delta^2_x} = (u_{i+1}^{n+1} - 2 u_{i}^{n+1} + u_{i-... |
Suppose matrix $A\in R^{3 \times 3} $ and
$$A=
\left[ \begin{array}{ccc} b_1 & b_2 & b_3 \end{array} \right]
\left[ \begin{array}{ccc} c_1 & 0 & 0 \\\ 0 & c_2 & 0 \\\ 0 & 0 & c_3 \end{array} \right]
\left[ \begin{array}{ccc} b_1^T \\\ b_2^T \\\ b_3^T \end{array} \right]
$$
where $b_i\in R^3$ and $B=(b_1,b_2,b_3)$ ... |
Here is an interesting infinite series. It would be great to see a method to evaluate it, if possible. I know it converges to a little less than 11/40
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{4^{k}+k!}$
I could not think of any good identities to start this.
Thanks a million to those who can show how to evaluate it.
... |
I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow\mathbb{C}P^3$ is given explicitly by $(z,w)\mapsto(z^3,z^2w,zw^2,w^3)$ in homogenous coordinates.
I was wondering if the same could be d... |
). To find the Point where 2 lines intersect, is there any simpler method than this?
http://en.wikipedia.org/wiki/Line-line_intersection
For example could you use the equation of the line? (y = mx + b)
Another question: what would happen if I use Determinants for 2 lines that never intersect? (parallel lines). I thin... |
). I'm working on a Flash game and I need to find when a shooting line intersects an enemy. The enemy's shape is rectangular. So far, I was thinking to test this like so:
Test if the shooting line intersects ANY of the lines composing the rectangle. BUT while the shooting line can be endless, I need to only validate a... |
I'm unable to prove the following two statements:
$ \displaystyle x^n+x^{\frac{1}{n}}+n = 0$ has no real zeros for all $n\geqslant 2.$
$ \displaystyle x^n+x^{\frac{1}{n}}-n = 0$ has exactly one real zero for $n \geqslant 1$.
(where $n$ is an integer and $x$ is a positive real number).
|
I would like to know how the MSU - Video Quality measurement tool:
here
calculates the blur metric.
What could be the mathematical computation to calculate blur?
I want to do something similar in my Image pipeline.
EDIT: If I have a blurry image, I upsize it / downsize it, and then compute the blur again for this resa... |
Does anyone know how to derive a formula for the coefficients.
That is if,
$f(x)=\sum _{n=0}^{\infty } a_nx^n$ and $g(x)=\sum _{n=0}^{\infty } b_nx^n$
suppose the composition is an analytic function, $h(x)=f(g(x))=\sum _{n=0}^{\infty } c_nx^n$
Is there an expression we can find for the coefficients $c_n$ in terms of $a... |
Does anybody have a simple proof this inequality
$$e^x\le x+e^{x^2}.$$
Thanks.
|
I'm trying to check if a line hits a rectangle, and for that, I found this nice solution:
Line triangle intersection
The problem is that, having forgot almost all I ever knew about math, I don't know how to obtain that equation of a line which they use there: $4x − 3y + 2 = 0$
I know a line's equation is $y = mx + b$. ... |
Suppose $p$, $k$ and $s$ are integers with $s,k \le p$. Consider the following polynomial in $x$ and $y$,
$$ \sum_{\ell=0}^k \binom{s}{\ell} \binom{p-s}{k-\ell} x^\ell
y^{p-\ell}$$
Does this expression look familiar to anyone? Is there closed form?
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Let $a$ and $b$ be two nonzero elements of a PID $R$. Prove that direct sum $R/aR\oplus R/bR$ is isomorphic to the direct sum $R/cR\oplus R/dR$, where $c=\mathrm{lcm(a,b)}$, $d=\gcd(a,b)$.
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I am looking for all $k$-dimensional subspaces of $(\mathbb{Z}/2\mathbb{Z})^n$ up to permutational equivalence.
Is there a database of all $[n,k]$-codes up to equivalence for reasonable values of $(n,k)$? For instance, $1 \leq k \leq 6$ and $6 \leq n \leq 12$ (giving around 35,000 codes).
When $k=1$, there are $n$ s... |
My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer.
After tiring of memorizing $\pi$, he resolved to discover a new irrational whose expansion is easier to memorize. He invented (probably ... |
How can I prove that $n^8 \lt 8^n-1$ for all $n$?
I'm searching for technical way to show that without using calculus theorems, using well known facts such as $n^2 \lt 2^n$ and so.
Thanks!
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Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills:
$$
s = \sum_{i=1}^k n_i d_i\quad\text{with } n_i > 0, d_i > d_{i-1}
$$
We start with the highest denomination $d$ not exceeding $s$ and subtract it. We repea... |
In an additive category, why is finite products the same as finite coproducts?
This is relatively easy to prove when the category is R-mod, but my intuition/creativity fails to see how the method can be extended to arbritrary additive categories
Specifically, a category (in Weibel, "An introduction to homological algeb... |
What does := mean?
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Attempt:
$\sum_{k=1}^n (p_k + \frac{1}{p_k})^2 = 2n + \sum_k p_k^2 + \sum_k \frac{1}{p_k^2}$
I used the Cauchy inequality to decompose 1 as $\sqrt{p_k}(\frac{1}{\sqrt{p_k}})$ and got
$n^2\leq \sum_k \frac{1}{p_k}$
I could, use the Cauchy inequality again on $\frac{1}{p_k}\cdot 1$ to get
$(\sum\frac{1}{p_k})^2 < n \s... |
I'm finishing up reading about Hilbert's axioms, and this final problem from Hartshorne's Geometry: Euclid and Beyond is throwing me for a loop.
I'm working in a Hilbert plane, so the axioms of incidence, betweenness and congruence apply. I have the additional Archimedes' axiom, which states that for given line segment... |
Let be $n_1$, $n_2$ such natural numbers that $n_1\geq 3$ and $n_2\geq 3$ and let be $G_{n1,n2}$ a graph that takes shape by taking $G_{n_1}$, the cycle of $n_1$ vertices, and $G_{n_2}$ the cycle of $n_2$ vertices, and linking all of $G_{n_1}$ vertices with all of $G_{n_2}$ vertices.
a) For which values of $n_1,n_2$ h... |
Is there a connection between the Hardy Spaces on the unit disk and on $\mathbb R^n$?.
If so, can we use results from the Hardy Spaces on the unit disk to prove $(H^1)^* = \text{BMO}$?
Further, what is the most fruitful way to define $H^1$ on $\mathbb R^n$ but avoiding (bounded) distributions? I was writing something f... |
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