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I need a formula which will fill an array of X slots with values (from $0$ to $1$). The sum of all array values should be $1$. The values should look like this:
^
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0.2
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|. ... |
I am having trouble understanding part of a lecture in class.
Let $G$ be a group, $\phi$ a representation of $G$ into $GL_n(C)$, and let $E^{ij}$ denote the matrix with 1 in the $i$th row and $j$th column and zero everywhere else.
Define the matrix $M = \sum_{g \in G} \phi(g) E^{ij} \phi(g)^{-1}$. We showed that $M$ c... |
This is probably a very stupid question, but I just learned about integrals so I was wondering what happens if we calculate the integral of $\sqrt{1 - x^2}$ from $-1$ to $1$.
We would get the surface of the semi-circle, which would equal to $\pi/2$.
Would it be possible to calculate $\pi$ this way?
|
All functions are smooth and continuous.
I use ' to signfy $d/dt$.
Given:
$q_i'(0)=0$, for i=1,2,3,4.
$q_2(t)q_3(t)>q_1(t)q_4(t)$ for $0<=t<T$. T is finite. At t=T the inequality breaks.
$q_4(0)*q_3''(0)>q_4''(0)*q_3(0)$
$(q_4''(0)+q_2''(0))(q_1(0)+q_3(0)>(q_4(0)+q_2(0))(q_1''(0)+q_3''(0))$
This also holds but I susp... |
I am aware that if there is a bifurcating tree with N leaves, then there are (N-1) internal nodes (branching points) with a single root node. How is this relationship proved?
Best,
|
What does "M" stand for in Weierstrass M-Test?
Just asking...
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Please help in to understand why/how $$|\vec{a} +\vec{b}|^2 = |\vec{a}|^2 +|\vec{b}|^2 + 2 \vec{a}\cdot \vec{b}$$
My precise confusion is why the modulus is not included in $2 \vec{a}\cdot \vec{b}$ ?
Since if we do
$(|\vec{a}| +|\vec{b}|)^2$, we have modulus sign in every $\vec{a}$ and $\vec{b}$.
|
In order to learn about vector bundles, I would like to draw the tautological vector bundle over the complex projective line
$$ E = \{(x,v) \in \mathbb{CP}^1 \times \mathbb{C}^2 : v \in x \} .$$
Identifying the complex projective line with the Riemann sphere, $\mathbb{CP}^1 \cong S^2$, I hope that it might be possible ... |
Let $R$ be a commutative ring with unit. Let $R_p$ be a domain for all $p\in SpecR$ and let $SpecR$ be connected. Is it true that $R$ is a domain or can someone provide a counterexample. Note here that $R$ is not necessarily a Noetherian ring. For a Noetherian ring this is easy.
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Let $V$ be a vector space over a field $k$ (not necessarily finite-dimensional) and $T : V \to V$ a linear operator. Is there an accepted term for the following condition on $T$?
For any $v \in V$ the subspace $\text{span}(v, Tv, T^2 v, ...)$ is finite-dimensional, and $T$ is nilpotent on any such subspace.
For exam... |
Is there a way to estimate the reciprocal of a decimal number? e.g. 1/0.388. Are there any formulas for this sort or thing, or do I just have to calculate it by hand?
|
What is the direct formula for the following variation of Josephus problem?
There are $n$ persons, numbered $1$ to $n$, around a circle.. Starting from $k$-th person, every second person is eliminated. Given the $n$ and $k$, determine the index of the $x$-th person who is eliminated.
ADDED:In the first step the $k+1$... |
I asked a question yesterday regarding numbers defined on graph structures that I call graph numbers. I posted the algorithm I was using to define graph numbers, which are simply a natural extension of linear numbers we use all the time. Nolion's response was close but not quite right. The equations are actually:
$$... |
Von Neumann's Minimax theorem (quoted from Wikipedia):
For every two-person, zero-sum game with finite strategies, there exists a value V and a mixed strategy for each player, such that (a) Given player 2's strategy, the best payoff possible for player 1 is V, and (b) Given player 1's strategy, the best payoff possibl... |
If I have only the upper triangular part of a symmetric matrix $A$, how could I compute the SVD?
$$\begin{pmatrix} 1 & 22 & 13 & 14 \\ & 1 & 45 & 24 \\ & & 1 & 34 \\ & & & 1\end{pmatrix}$$
Does having this upper triangular make the computing easier?
|
The symmetric group $S_n$ acts naturally on the set $\{1,2,\dots, n\}$. Is this a left or a right action?
|
here is a pair of quick questions that I am wondering about, I would appreciate any help:
Is it always the case that any matrix $A$ can be represented as $E_{1}...E_{n}$ * rref($A$) (and in special case when the matrix is invertible, the representation becomes simply $E_{1}...E_{n}$) where $E_{i}$ is an elementary matr... |
For a real-valued function $f$ defined on $\mathbb{R}$ or its subset,
is it possible that it is
differentiable at one point and not
in one of its neighbourhoods except
the point itself?
is it possible that it is
differentiable over an interval, but
its derivative over the interval is
not continuous?
I found on this li... |
(Context: polynomial multiplication using DFT/FFT)
Let $f = \sum\limits_{i=0}^{n-1} f_i x^i$ and $g = \sum\limits_{j=0}^{n-1} g_j x^j$ be polynomials in $F[x]$ for some field $F.$ The convolution of $f$ and $g$ is given by $$f \ast g = \sum\limits_{k=0}^{n-1} c_k x^k$$ where $$c_k = \sum\limits_{i+j \equiv k \bmod n} f... |
The answer in the book is $e-1$, and but I can't figure out how to go from $x(\ln(x) - 1)$ to that answer...
|
I believe the following is true, but I do not know how to prove it.
Given two sets of distinct non-zero complex numbers $z_1, \dots , z_n$ and $w_1, \dots , w_n$, if
$$ \begin{bmatrix}
z_{1} & \cdots & z_{n} \\
z_{1}^{2} & \cdots & z_{n}^{2} \\
\vdots & \ddots & \vdots \\
z_{1}^n & \cdots & z_{n}^n
\... |
I'm trying to learn by myself some algebraic geometry, since I will probably be taking an elementary course next semester.
I have some trouble understanding the following:
Let $Y=Z(y-x^{2})$ then by definition $Y$ is algebraic set in $\mathbb{A}^{2}$. The author finds the coordinate ring $K[x,y]/I(Y)$ where $I(Y)=\{f \... |
Find the area bounded by these two functions:
$$y = \frac{\ln x}{x}\quad\mbox{and}\quad y = \frac{1}{e} + \frac{(e^2+1)(x-e)}{e^2-1}.$$
|
Question: Given a twist of the projective space, how do I find unit quaternions that represent it?
Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every pair of unitary quaternions $l,r$ gives a map $[l,r]:x \mapsto \bar{l}xr$, and a map $\ast[l,r]:x \mapsto \bar{l} \bar{x} r$, an... |
Second derivative test in
Wikipedia says that:
For a real function of one variable:
If the function f is twice
differentiable at a stationary point
x, meaning that $\ f^{\prime}(x) = 0$
, then:
If $ f^{\prime\prime}(x) < 0$ then $f$
has a local maximum at $x$. If $f^{\prime\prime}(x) > 0$ then $f$
has
a local minimum... |
How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours?
(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)
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I'm trying to go through the section of Newer Results in Hartshorne's Geometry: Euclid and Beyond.
This particular exercise has been bugging me for a good while:
My first approach was to show that the perpendicular bisectors of $A'B'$ and $A'D'$ intersect at the same point as the perpendicular bisectors of $A'B'$ and ... |
I know there is an easy way to sample standard mixture models if the mixture components can be easily sampled. But, I have a slightly different scenario and I'm wondering if there is an analogous method (or any method at all that is nice). I'll first explain the standard scenario so there is no confusion, and then expl... |
The proof in my text is as such:
Let $a_1,a_2,a_3,\ldots$ and $b_1, b_2, b_3,\ldots$ be the labels of the left- and right- hand endpoints respectively.
Consider the set A of the left-hand endpoints of the intervals, and let x = sup A. Since x is an upper bound for A, we have $a_n \leq x$. Since each $b_n$ is an upper b... |
I'm talking about these specific relations, where g is the determinant of the metric tensor (so it's symmetric spscific), which is a function of $x^k$:
$\frac{1}{2g}\frac{\partial g}{\partial x^k}=\frac{1}{2}\frac{\partial \ln g}{\partial x^k}$
I've tried making discrete variations of the matrix, but I can't reach this... |
If I have 5 people classify an image as offensive or not, how can I calculate my confidence that the answer that 3 or more people agree on is the correct answer?
I have past accuracy rates, but am not sure how to incorporate my confidence in those accuracy rates. For example, Person A may be 90% accurate, but with only... |
This question comes from computer security, but I'll distill it into a probability question:
I have a biased die with 96 sides. 95 sides are equiprobable, each having a 1% chance of landing up. The remaining side, side X, has a 5% chance.
All sides look identical; I want to identify side X. My best strategy is obvio... |
Why is it that for quaternions, $u*v = \mathrm{cross}(u,v)-\mathrm{dot}(u,v)$?
I wonder for what reason they are equal to each others.
|
I have an exercise scribbled down, and I am not sure what it is asking. It is somewhat similar to Burnside's lemma.
We have a finite group $G$ acting on a set $X$. For each $g \in G$, let $X^g$ denote the set of elements in $X$ fixed by $g$.
$$\sum_g |X^g|^2 = |G| \cdot \text{(number of orbits of a stabilizer)}$$
I am ... |
My brother has a 30 year fixed mortgage. He pays monthly. Every month my brother doubles his principal payment (so every month, he pays a little bit more, according to how much more principal he's paying).
He told me he'd pay his mortgage off in 15 years this way. I told him I though it'd take more than 15 years. Who's... |
I'm reading a statistics textbook which defines the mean of a random variable $X$ with CDF $F$ as a statistical function $t(\centerdot)$, where
$$ t(F) = \int x \, dF(x).$$
Can someone explain this definition? I'm familiar with the definition of the mean as an integral over the PDF $f$:
$$ \int x \, f(x) \, dx.$$
But w... |
Let's say I have a 2D coordinate space defined by $-5 \leq x,y \leq 5$.
Then let's say I have coordinates $(x_1,y_1)$ and $(x_2,y_2)$ for a line that will run from $(4,4)$ to $(6,7)$.
How do I truncate that line so that it ends at the edge of the coordinate space (but maintains the same slope)? In other words, how d... |
this is a problem from my textbook I am working on. I am not sure how to approach it -- I would like to solve it using a 'matrix' way if possible -- that is, to set up a matrix with coefficients etc. and solve for the basis of the kernel, which would be the basis of what I am looking for (not sure if that's possible).
... |
This problem came up in my conversation with a friend—not sure how basic it is, but it seems quite interesting:
Prove that if $f_n$ where $n>3$ is prime, then $n$ is prime for a Fibonacci sequence where $f_1=f_2=1.$
How can this be tackled? Thanks a bunch for any ideas.
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I want to solve transcendental equations by Mathematica 7. Solve does not help me, reduce - as well. Seams that I shoulkd use FindRoot, but it does not give all solutions. It gives only one solution. So how can I solve transcendental equations?
Here is the equation I want to solve:
So for a fixed value of L, A1, A2 (L... |
A series arising in perturbation theory in quantum mechanics:
$\sum_{m\neq n} \frac{1}{n^2 - m^2}$, where $n$ is a given positive odd integer and $m$ runs through all odd positive integers different from $n$. I have a hunch that residue methods are applicable here, but I don't know complex analysis.
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Let R be a ring. If a direct sum of R-modules is projective, can we imply that the direct sum is also a free module over R ?
|
This is given in my module as a part of a problem's solution:
$$m^4 + 4 = 0 $$ $$\Rightarrow m = 1 \pm i,-1\pm i$$
I am not getting how this conversion is taking place,could somebody explain?
|
Find
$$ \lim_{x \to \infty} \sqrt[3]{1 + x^{2} + x^{3}} -x$$
|
In reading this article about updated estimates for the number of exoplanets in the Milky Way, I am curious how to get an estimate of the mean distance between them. The Milky Way is ~50,000 light years in radius and an average of ~1000 ly in thickness, and the article estimates there to be ~500 million exoplanets in ... |
Let $R$ be a commutative ring with identity. Let $I$ be an ideal of $R$. Suppose, we give a topology on $R$ where a set is open if and only if it is a union of cosets of powers of $I$. Then, is $R$ a topological ring?
EDIT: The question has been edited in the light of comments below.
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Suppose $w \in L^{1}(G)$ and $w \geq 0$. The geometric mean of $w$ is defined by $$ \Delta(w) = \exp \int_{G} \log w(x) \ dx$$
where $\Delta(w) = 0$ if the integral is $-\infty$. What is the intuition behind this definition?
Source: Fourier Analysis on Groups by Walter Rudin
|
Snell's law from geometrical optics states that the ratio of the angles of incidence $\theta_1$ and of the angle of refraction $\theta_2$ as shown in figure1, is the same as the opposite ratio of the indices of refraction $n_1$ and $n_2$.
$$
\frac{\sin\theta_1}{\sin \theta_2} = \frac{n_2}{n_1}
$$
(figure originally f... |
Define the rank of a group $G$ as the minimal size of a subset $S$ such that $S$ generates $G$. I want to prove that, given $G$ and $H$ two groups, if there exists a surjective homomorphism $f$ between $G$ and $H$ then rank$(G)$ $\geq$ rank$(H)$.
I thought of this: let $S$ be a minimal generating set of $G$. Let us con... |
Given an integer N and a smooth base B; what is the (approximate) probability that N is completely divisible by primes <= B.
I assume there is some nice connection to the de Bruijn or Dickman function but I can't see it.
|
I am to prove that being $x$ a string and $|x|$ its length, one should have the following property hold true for any two strings $x$ and $y$:
$$ |xy| = |x| + |y| $$
with $x, y \in \Sigma^*$.
To prove this, I am expected to make use of the following two definitions:
for $x=\epsilon$ we have $|x|=0$
for $x=au$, being $u... |
If $X$ is inductive, then the set $\{x\in X\mid x\text{ is transitive}\}$ is inductive. Hence
every $n\in\mathbb{N}$ is transitive.
If $X$ is inductive, then the set $\{x\in X\mid x\text{ is transitive and }x\notin x\}$ is inductive. Hence, $n\notin n$ and $n\neq n+1$ for each $n\in\mathbb{N}$.
|
What are the open sets in a topology specified by a fundamental system of neighbourhoods of $0$ of a topolgical group? Also, how is this topology unique. I searched this online, but the books I found only mention these statements, without specifying what the open sets are and why the topology is unique. Any help will b... |
I am using quaternions to rotate my camera.
I only rotating the camera around it's x and y axises.
But for some reason, after some rotation, y axis also gets rotated(looks like around z axis).
I could avoid the problem with fixing y to (0,1,0), but I am curious where I am wrong.
|
I'm having trouble getting my head arround the concept.
Can someone explain it to me?
|
Suppose I have to solve the following boundary value problem:
$$-u''=f\quad\text{and}\quad u(0)=0,\ u(1)=0.\qquad \mbox{Where $f \colon [0,1] \to \mathbb{R}$.}$$
In the solution to the exercise, the professor integrates as follows to solve the equation:
$$u(x)=-\int_{0}^{x}\int_{1}^{y}f(s)dsdy+c_{1}x+c_{2}$$
Could so... |
A Fourier series arising in perturbation theory in quantum mechanics is
$$\sum_{m\neq n} \frac{1}{n^2 - m^2} \cos \frac{m\pi x}{2a} \, .$$
where $n$ is an odd positive integer and $n$ runs through all odd positive integers other than $n$. (The numbers are odd so that the Fourier terms are zero at $\pm a$.)
I have no id... |
I'm an undergrad student fairly keen on algebra. Over the different algebra courses I've taken, I've often encountered the so-called $p$-Prüfer group on exercises but somehow never got around to them. Now I'm trying to take care of that, but there are some statements I've seen about this group which I don't know how to... |
Question 5.33 from "Topology and its Applications" by Baesner is to compute the fundamental group of the torus ($T^2$) with $n$ points removed. I can "see" in my mind that if we remove one point we get a bouquet of two circles. Less clear is what happens when we remove two (or more) points. Any hints?
|
I have been looking for some good material covering Markov Chains but everything seems so difficult to me...
After reading about the subject, I figured out that there is basically three kinds of processes: discrete-time, continuous-time and decision Markov Processes.
I started reading Introduction to Probability Models... |
In Lebesgue integration we usually approximate the function we want to integrate with step-functions on measurable sets. How much "power" do we take away if we require that the step functions are on intervals instead? What functions are left that are integrable?
I'm asking this because I want some integral to converge ... |
Let $F$ be a local field, $G=GL_2(F)$, $\chi_1,\chi_2$ quasicharacters, and $B(\chi_1,\chi_2)$ be the principal series (we are assuming $\chi_1,\chi_2$ to be such that the representation is irreducible).
Is $B(\chi_1,\chi_2)$ a constituent of $L^2(Z\backslash G)$?
I'm pretty sure it never is, but can't prove it. Am fam... |
I'm trying to prove that any group $G$ of infinite order has an infinite number of subgroups.
I think that if the group has an element of infinite order, then it's easy because I can take the groups generated by the powers of this element.
What if it doesn't? Every element generates a cyclic subgroup. Every element bel... |
My interest in combinatorics was recently sparked when I read about the many things that the Catalan numbers count, as found by Richard Stanley. I picked up a copy of Brualdi's Combinatorics, and while browsing the section on counting sequences I found a nice little puzzle that has definitely puzzled me.
Let $m$ and $... |
Suppose there are three measurable spaces
$(\Omega, \mathbb{F})$, $(S_i,
\mathbb{S}_i), i=1,2$, and two
measurable mappings $f_i: \Omega
\rightarrow S_i, i=1,2$. Is the
mapping $f$ defined as
$f(\omega):=(f_1(\omega),
f_2(\omega))$ a measurable mapping
from $(\Omega, \mathbb{F})$ to
$(\prod_{i=1}^2 S_i... |
One learns in a probability course that a (real) random variable is a measurable mapping of some probability space $(\Omega,\mathcal{A},\mathbf{P})$ into $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. But as soon as one gets into topics that are a little advanced, the space $(\Omega,\mathcal{A},\mathbf{P})$ is not mentioned u... |
I recently learned that the only parallelizable spheres are $\mathbb{S}^1$, $\mathbb{S}^3$, and $\mathbb{S}^7$. This led me to wonder:
What is $T\mathbb{S}^2$? Is it diffeomorphic to a more familiar space? What about $T\mathbb{S}^n$ for $n \neq 1, 3, 7$?
EDIT (for precision): Is $T\mathbb{S}^2$ diffeomorphic to some... |
Given a $\sigma$-algebra on a Cartesian product of a collection of sets, do there always exist a $\sigma$-algebra on each set, so that their product $\sigma$-algebra on their Cartesian product will be the same as the given $\sigma$-algebra?
If yes, how to construct the $\sigma$-algebras on the sets from the $\sigma$-... |
This question is based on a homework exercise:
"Let $m$ be a negative, square-free integer with at least two prime factors. Show that $\mathbb{Z}[\sqrt{m}]$ is not a PID."
In an aside comment in the book we're using (Milne), it was noted that a lot was known about quadratic extension with $m$ negative, but very little ... |
I got this problem within others as homework, and I don't know how to do it. Does anyone know how to start solving it?
Question : Let $X_1$ and $X_2$ be independent exponentially distributed random variables with parameters $\lambda_1$ and $\lambda_2$ so that $Pr\{X_i>t\}=e^{-\lambda_it}$ for $t\ge0$.
Let
$N=1$ if $X_... |
I am working on this practice problem, and I was wondering if I could get some help.
I have a $T$:$\mathbb{R^{2x2}}\to \mathbf{P_{2}}$, that is, from 2x2 matrices to polynomials of degree at most 2. The transformation is given as following:
$$T\left(\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}\right)
= a-c+2d+(b+2... |
I am in the process of learning about Mapping class groups. At this point it seems like most of what I've read involves very low dimensional (surfaces and 3-manifolds) applications.
I was wondering if they were studied (or arise naturally) in higher dimensional settings?
In particular, any references to their uses in h... |
Is $(\mathbf{V} \cap \mathbf{W})^{\bot}=(\mathbf{V}^{\bot} \cap \mathbf{W}^{\bot})$? I tried element-chasing, but I am getting confused when trying to determine mutual containment.
|
I used computer program to generate numbers of this form, then verify if they are amicable. However, it took too long to finish, is there any special technique to verify if a pair is amicable?
One pair that I found is $m = 284 = 2^{2}.71, n = 220 = 2^{2}.5.11$
Thanks,
Chan
|
This is a continuation of my previous question. I admit I still don't fully understand the concept of infinite product of $\sigma$-algebras.
Let $(E_i, \mathbb{B}_i)$ be measurable spaces, where $i \in I$ is an index set, possibly infinite.
I understood the definition of product sigma algebra. However, I am not sure af... |
At the moment, I'm studying Ergodic Theory and I find myself a little stuck.
The Weyl Equidistribution Theorem states that the following are equivalent:
1. For any $f \in L^{1}([0,1])$ and sequence $\{ u_{k} \}_{k \geq 1}$ on $[0,1]$,
\begin{eqnarray}
\lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} f( u_{k} ) = \int_... |
Next question:
Let $X$ be a wedge of two circles, $X=A \vee B$ and let $x_0$ denote the point of tangency.
Let $q:S^1 \to X$ be the identification map that identifies -1 and 1 $\left(q\left(-1\right)=x_0=q\left(1\right)\right)$
The question is to show that $(S^1,q)$ is not a covering space of $X$
My question: Does this... |
In Wikipedia, bilinear mapping to the underlying field $f \in \operatorname{Hom}(V \otimes V, k)$ is defined as non-degenerate iff $X \mapsto f(-,X)$ is an isomorphism to the continuous (?) dual.
I tried to prove the proposition that $X \mapsto f(-,X)$ is an iso iff $X \mapsto f(X,-)$ is an iso, but only managed to pro... |
How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals?
|
Does knowing the exact number of $\mathbb{Z}^{n}$-lattice points in an $n$-sphere of arbitrary radius help one compute $n$-sphere packing density and kissing number?
Thanks!
|
Suppose we divide the the interval $[0,1]$ into $t$ equal intervals labeled $i_1$ upto $i_t$, then we make a function $f(t,x)$ that returns $1$ if $x$ is in $i_n$ and $n$ is odd, and $0$ if $n$ is even.
What is $\lim_{t \rightarrow \infty} f(t,1/3)$?
What is $\lim_{t \rightarrow \infty} f(t,1/2)$?
What is $\lim_{t \rig... |
How can I show for some transcendental number $\alpha \in \mathbb{C}$, that two distinct polynomials of the form $b_{n}\alpha^n + \cdot \cdot \cdot + b_{2}\alpha^2 + b_{1}\alpha$ and $a_{m}\alpha^m + \cdot \cdot \cdot + a_{2}\alpha^2 + a_{1}\alpha$ can never equal each other?
|
How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below.
In the above illustration, the points A, B and X lies on a spherical surface, I need to find the distance between points (A,B) and X. I am ... |
I need to express the value of $\pi$ as numerator/denominator. What is the best pair considering that the numerator should be less than or equal to $2^{62}$? Or how to get this pair?
|
I am trying to calculate the touching point of two circles. I have the following information.
Circle $1$:
Centre $(h,k)$
Radius $r_1$
Circle $2$:
Point on Circumference $(x_1,y_1)$
Radius $r_2$
From this I would like to calculate the centre point of circle $2$: $(s,r)$ and the point where they touch: $(x,y)$.
I h... |
Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks!
|
Today we had a probational exam in analysis. I wasn't able to solve one of the exercises and I have no idea what theorem to apply in order to solve it:
Let $I=[0,1]$ and $f: I \rightarrow \mathbb{R}$ be continuously differentiable. Assuming that $f$ has a root. Show: $$\max_{x \in I} |f(x)| \leq \max_{x \in I} |f'(x)|$... |
recently I've come across a few cases where I had to evaluate $e^{At}$ (differential equations) where A is in Frobenius canonical form. The algorithm looks like:
$$e^{At} = Pe^{Jt}P^{-1}$$ so I need to do Jordan decomposition. From Frobenius canonical form I can easily write the characteristic equation => compute the r... |
As we all know, Sturm's axioms have completely solved the problem for finding the number of roots in an arbitrary interval $[a,b]$, using the derivative and forms a Sturm set.
Now my question follows naturally that is there any other use of this axiom.
When I learned the Sturm set, I thought it will be difficult to f... |
Possible Duplicate:
On sort-of-linear functions
It is an interesting exercise to show that $\operatorname{Gal}(\mathbb R ~\colon \mathbb Q)$ is trivial. The only solution I know hinges on the fact that the automorphism is order-preserving, which in turn depends on the fact that $\theta(xy)=\theta(x)\theta(y)$ for $\... |
Given a random vector $X: (\Omega,
\mathbb{F}, P) \rightarrow (\prod_{i
\in I} S_i, \prod_{i \in I}
\mathbb{S}_i)$, is each component
variable $X_i, \forall i \in I$ of
the random vector $X$ always a
random variable from $(\Omega,
\mathbb{F}, P)$ to $(S_i,
\mathbb{S}_i)$?
If yes, I guess there... |
Can one state and prove the Pythagorean theorem using Hilbert's axioms of geometry, without any reference to arithmetic?
Edit: Here is a possible motivation for this question (and in particular for the "state" part of this question). It is known that the theory of Euclidean geometry is complete. Every true statement in... |
I have an equation:
$(1-\epsilon)x^2 -2x +1=0$ (regularly perturbed problem, we anticipate all roots to remain bounded when $\epsilon$ goes to $0$)
I substitute $x=\displaystyle \sum_{n\geq0}C_n \epsilon^n$ and have to work out the equations for $C_0,C_1,C_2$.
So I get: $C_0^2-2C_0+1=0$ so $C_0=1$
and there a problem s... |
Suppose $T_1(X)$ is a sufficient statistic for $\theta$ for a family of distributions indexed by $\theta$ and $T_2(X)$ is another statistic that induces a “finer”(what is meant by a finer ? ) partition than $T_1$. How do I know that $T_2$ is also sufficient ?
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For some formulae $\phi(x)$ it can be proved from the axioms of ZFC, that there is no set $X$ with $(x)x\in X \equiv \phi(x)$. Thus the collection $\lbrace x\ |\ \phi(x)\rbrace$ is a proper class.
Furthermore, there is no set $Y$ that is in bijection with such a collection, right? Thus, every proper class must be in so... |
At time $t_{0}$ I have $N$ server instances running. Each instance is at $100\%$ CPU utilization. Collectively, the instances are attempting to respond to $R_{0}$ web requests. Some unknown percent of $R_{0}$ requests are being ignored because all the instances are under too much load.
At time $t_{1}$, the load lessens... |
this will probably be an easy-to-answer and a not-well-posed question, since I'm a total beginner in the field, but here goes:
Let $V(F)$ and $V(G)$ be two projective curves in $\mathbb{P}^2$ ($F,G\in\mathbb{F}[x_0,x_1,x_2]$ homogenous polynomials) and $P\in V(F)\cap V(G)$ a point. Let $\mu_P(F,G)$ denote the intersec... |
How does one prove that the class number of $\mathbb{Z}[\zeta_3]$ is $1$?
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If P=NP, is there a polynomial-time algorithm $A$ that can decide the $\text{Independent Set}$ decision problem?
That is, with an undirected graph $G = (V, E)$ and a positive integer $k$, does $G$ contain a set of $k$ independent vertices (i.e. a set of vertices no two of which are joined by an edge)?
I am unsure of wh... |
I'm having trouble comprehending the following.
http://www.stat.berkeley.edu/users/pitman/s205s03/lecture28.pdf in the proof of theorem 28.3.
It says that $H_t(x,y)=H_t\delta_y(x)$, where $\delta_y(x)=1$ if $x=y$ and $0$ if $x\neq y$.
I'm stuck with understanding this. Please help?
Also why is $E(H_t f)=Ef$ ? I particu... |
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