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How do you transform isoperimetric inequalities of a group to the of Riemann integrals of functions of the form $f\colon \mathbb{R}\rightarrow G$ where $G$ is a metric group so that being $\delta-$hyperbolic in the sense of Gromov is expressible via Riemann integration?
In other words, how do you define "being $\delt... |
I am a total newbie to the world of math and was interesting in learning. I just finished my degree(non-math) and am going to study a few math books to see if it interests me to apply for something more quantitative but I want to study something interesting with interesting problems that won't bore me.
I thought abo... |
If $m$ is an outer measure on a set $X$, a subset $E$ of $X$ is called $m$-measurable iff
$$
m(A) = m(A \cap E) + m(A \cap E^c)
$$
for all subsets $A$ of $X$.
The collection $M$ of all $m$-measurable subsets of $X$ forms a $\sigma$-algebra and $m$ is a complete measure when restricted to $M$.
Is $M$ the largest $\si... |
One reason why the iteration
$$x_{n+1}=x_n-\tan\;x_n$$
converges quickly for appropriate starting values is that this is nothing more than the Newton-Raphson iteration for $\sin\;x$.
This got me thinking: given some arbitrary function $g(x)$, is there always a function $f(x)$ such that
$$\frac{f(x)}{f^\prime(x)}=g(x)$$... |
In this question, The Chaz asks whether $G\times G\cong H\times H$ implies that $G\cong H$, where $G$ and $H$ are finite abelian groups. The answer is to his question is yes, by the structure theorem for finite abelian groups, as noted in the answer by Anjan Gupta.
Even though I don't know the first thing about categor... |
I am reading a section in a book that talks about normal matrices, and I see the following:
A normal matrix is a matrix that commutes with its adjoint,
Eh? What do the brackets and the comma inside mean?
|
I am unable to see how to eliminate $t$. Wolfram Alpha fails at it too.
$$x=2t-4t^3$$ $$y=t^2-3t^4$$
I can guess that the curve is a polynomial equation so in principle I can write this as
$$w_1 x^4 +w_2 x^3 +w_3 x^2 +w_4 x +w_5 y^3 +w_6 y^2 +w_7 y +w_8=0$$
for the powers of $t$ to cancel when plugged in I am determi... |
This is a question I had on a test, we were told not to use brute-force and figure out a smart way to solve the problem.
We have a matrix $A =$
$\displaystyle\begin{bmatrix}
2 & 3\\
3 & 2
\end{bmatrix}$. Find a matrix $C$ such that $C^{3}$=$A$.
What is the 'smart not brute-force' way to solve this, without picking num... |
Suppose $N$ is an RSA modulus (ie, it's the product of two distinct primes), 256 bits long. What is the best method to factor it?
Trial division is out of the question, Pollard's Rho is probably out as well (without significant parallelization). I doubt there are any online tools or math libraries that can handle thi... |
While studying the use of the trapezoidal rule for numerically evaluating the complementary error function $\mathrm{erfc}(z)$, the following integrals showed up when I was trying to derive expressions for the truncation error:
$$\int_0^\pi \exp\left(-z^2\tan^2\frac{u}{2}\right)\cos(2mu) \mathrm du$$
where $z$ is positi... |
Greetings All
I have a function that creates an affine matrix that works for
2D rotation around an arbitrary point but now I would like
it to work for 3D rotations also. The working function example is below
with it's example function:
%affine matrix rotation about a point rtaffine[theta,rpx,rpy,sigx,sigy,sigz]
%crx ... |
How can one show the following identity
$ne^n = \sum_{k=0}^{\infty}n^k(n - k)^2/k!$ ?
|
I want to show, Image of $unit~ disc$ under a discontinuous linear functional from a Normed Linear Space is $\mathbb{C}$ . I know the Image is Unbounded
|
Is this true that: "The order of wreath product of $\mathbf{Z}_{p^n}$ and $\mathbf{Z}_{p^m}$ is $p^{m+n}$" ?
|
I'd like to understand a step in a counterexample of Reichardt and Lind to the Hasse principle. The example is given by the equation $2y^2=x^4-17z^4$ (1).
(1) has no rational solutions ($\neq (0,0,0)$), but it has non-zero solutions in $\mathbb{Q}_{p}$ for all $p$. The part that I don't get is the justification that th... |
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we restrict our definition of Lebesgue Integrability to absolutely integrable? Wouldn't it be better to extend our definition ... |
I'm currently working on some graph theory, and I'm checking the robustness of graphs, by removing random vertices from a graph and then check how many vertices are outside the giant component. Then I'll do that for a lot of vertices, and plot them using ListPlot.
This takes along time to calculate, is there a way I ca... |
I have two linear, skew, 3D lines, and I was wondering how I could find a point on each of the lines whereby the distance between the two points are a particular distance apart, and that the vector formed between these points is perpendicular to both original lines?
I'm not after the points where the lines are the clos... |
I'm trying to get a value in a web application, using some informations. I succeeded on creating this system (I need $x$):
$
\begin{equation}
\left\{
\begin{matrix}
x & = & \dot a + b - c - (\dot d \cdot \dot e)\\
b & = & f \cdot \dot g\\
f & = & \frac{x - \dot d - h}{1 + \dot i}\\
c & = & \dot \alpha(b)\\
h... |
An exercise says: if the real sequence $(x_n)$ is defined by $x_{n+1}=F(x_n)$ for some $F$, $x_n\to x$ and $F'(x)=0$, prove that $x_{n+2}-x_{n+1}=o(x_{n+1}-x_n)$.
The textbook says assume $F$ is continuously differentiable and apply the mean-value theorem. It's straightforward: $$\frac{x_{n+2}-x_{n+1}}{x_{n+1}-x_n}=\fr... |
According to Wikipedia with a little rephrasing:
A state $i$ is transient if and only
if $P(T_i < \infty) <1$, recurrent if
and only if $P(T_i < \infty) =1$,
where $T_i$ is the first hitting time
to i, i.e. $T_i=\inf\{n \in \mathbb{N} \cup \{ \infty \}: X_n=i \mid X_0=i \}.$
If I understand correctly, this ca... |
There are three circles, all of them touching each other. The bottom two circles are laying on an imaginary floor, such that they touch the line g=-r as well.
Given are all three radii, r1 (A), r2 (B) and r3 (C).
Assuming circle A has its center on (0/0), B has its center on (2 sqrt(r1 * r2), r2 - r1). I am now suppose... |
Can you use noninteger powers Like is $x^{8.3} / x^{2.2} = x^{6.1}$?
|
Let $$f(x)=\sum_{s=-\infty}^{\infty}e^{-2\pi ixsk}$$
$k$ integer.
Does this integral exist?
$$\int_{0}^{1}(f(x))^{2}dx$$
|
We recently went through the wreath product in my group theory class, but the definition still seems a bit unmotivated to me. The two reasons I can see for it are 1) it allows us to construct new groups, and 2) we can use it to reconstruct imprimitive group actions. Are there any applications of the wreath product outs... |
Is it possible to simplify this expression?
$\frac{2^{2x-1} - 2^{x-1}}{2^{2x-1}}$
|
The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study.
We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S \in {\cal L}(V), TS = ST$. Here, ${\cal L}(V)$ denotes the set of all line... |
Given integers $a, b, c$ construct a single-tape Turing machine recognizing the language
$\{w \in \{0,1\}^{*}: a*\#_{0}w+b*\#_{1}w+c=0\}$ in time $O(n*logn)$, where $n=|w|$.
$\#_{x}w$ denotes the number of occurrences of the symbol $x$ in $w$.
Updating counters somewhere on the tape would result in $O(n^2)$ complexity... |
I've kind of forgotten the name of the following statements:
9x = 11
10x = 9y
What are they called? And how do you solve them?
|
Is the set of KKT conditions
necessary for local min, only under
some regularity conditions (ie
constraint qualifications)?
So does this mean that when the cost
functions and the functions in
constraints are continuously
differentiable at a point, it is
possible that the point is a local
min but is not regular?
I am m... |
I'm in a class titled "Puzzle Based Learning" and we were given this problem:
There is a new game show and you are
the participant. There are two doors,
each has a suitcase with gold coins
behind it. You know that these
suitcases contain amounts from the
set: 25, 50, 100, 200, 400, 800,
1,600, 3,200, and 6... |
Let $W_t$ be a standard one dimension Brownian Motion with $W_0=0$ and $X_t=\int_0^t{W_sds}$.
With the help of ito formula, we could get
$$E[(X_t)^2]=\frac{1}{3}t^3$$
$$E[(X_t)^3]=0$$
When I try to employ the same method to calculate the general case $E[(X_t)^n]$, I got stuck.
I guess $X_t$ should be normal distributi... |
I am reading a document it says
A group realization is a map from elements of G to transformations of a space M that is a group homomorphism, i.e. it preserves the group multiplication law. Thus if $T:G \rightarrow T(M): g \mapsto T(g)$ where $T(g)$ is some transformation on M, then T is a homomorphism if $T(g_1 g_2) ... |
Does there exist a quaternion $q$ on the unit sphere such that, given the vanilla complex plane $\mathbb{C}$, $q\mathbb{C}q^{-1} = \bar{\mathbb{C}}$?
Motivation: ordinarily, the plane is rotated by multiplying it by a complex number in the unit circle. Taking the conjugate amounts to flipping the plane along the real a... |
Triggered by previous question, can one prove GCD(0,8)≠1 purely by lattice laws?
Brute force Prover9/Mace4 assertions
x ^ y = y ^ x.
(x ^ y) ^ z = x ^ (y ^ z).
x ^ (x v y) = x.
x v y = y v x.
(x v y) v z = x v (y v z).
x v (x ^ y) = x.
1 v x = x.
1 ^ x = 1.
0 ^ x = 1.
exhibit no [finite] model, which is indicati... |
I'm trying to work on the following problems:
Prove that if a set S of sentences axiomatizes a finitely axiomatizable class K of structures then K can be axiomatized by a finite subset of S.
I have an idea for this: Suppose not, i.e. for all finite subset $S_0$ of S and all $M \in K, M \nvDash S_0$. Suppose K is finit... |
How does one find the arc length of the Cantor function? Wikipedia says that the length is $2$. I can "see" that the length is at most $2$ by a simple triangle inequality argument. I am struggling to come up with a partition $P$ such that the arc length is at least 2. I tried a partition of the form $\{ 1/ 3^n : 0 \le... |
How would you find $x$ in a modulo arithmetic expression $x^e \bmod p$ knowing only $e$ and $p$?
$e$ is an integer, $0 \leq e \lt p$, that is relatively prime to $p-1$; and $x$ is an integer, $0 \leq x < p$.
|
Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a harmonic (or loop-based) decomposition.
Are there deep connections between these two transforms? The formulaic connection... |
Possible Duplicate:
Modulo Arithmetic
How would you find x in a modulo arithmetic expression x^e mod p knowing only e and p?
e is an integer, 0 ≤ e < p, that is relatively prime to p-1; and x is an integer, 0 ≤ x < p.
|
Using the tools of linear-algebra it seems like $\mathbb{R^{2}}$ or $\mathbb{R}$ $\times$ $\mathbb{R}$ is isomorphic to $\mathbb{C}$, since both of the spaces are of dimension 2. Does this mean that the these sets are equinumerous? $\mathbb{R}$ is a subspace of $\mathbb{C}$, but I am wondering how could one prove that... |
Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the polynomial in terms of its coefficients assuming the galois group of the polynomial is solvable?
|
I was solving an exercise on Isosceles trapezoid whose diagonal was given, and I note that If I draw a diagonal in the isosceles trapezoid I got two triangles
To determine the area of the triangles I draw their heights, which are perpendicular to the diagonal. The problem arises when I suppose that the sum of the heigh... |
I was reading the Wikipedia article on senary numbers (base 6), which states that:
all primes, when expressed in
base-six, other than 2 and 3 have 1 or
5 as the final digit
Unless I am converting to senary incorrectly, I find this not to be true. For example, the senary representation of the decimal number 2047 i... |
Let $e^{2 \pi i k s} = f(s)$, $f \colon [0,1] \to S^1$ subset of Complex numbers ($S^1$ = unit circle at origin). So $f$ is a loop at the basepoint $1$ in $S^1$. Show that it is not homotopic to the constant loop $c(s) = 1$. This is obvious, but what is a more rigorous way of showing it.
Oh yeah, $k$ is an an integ... |
well I'm having a hard time understanding the tensor product. Here's a problem from Atiyah and Macdonald's book:
Let $A$ be a non-trivial ring and let $m,n$ be positive integers. Let $f: A^{n} \rightarrow A^{m}$ be an isomorphism of $A$-modules. Show this implies that $n=m$.
Well the solution is at follows:
Let $m$ be... |
Tim hortons has a roll-up the rim to win contents. your chances of winning in a single play is 1 in 6. so you have a 5/6 (83% chance) of winning nothing.
How do I calculate the odds of losing multiple times in a row? In 2 plays, what are my chances of losing both times? and what are my chances of losing 41 times in... |
How would I solve these differential equations? Thanks so much for the help!
$$P'_0(t) = \alpha P_1(t) - \beta P_0(t)$$
$$P'_1(t) = \beta P_0(t) - \alpha P_1(t)$$
We also know $P_0(t)+P_1(t)=1$
|
If $f(z) = (g(z),h(z))$ is continuous then $g$ and $h$ are as well.
The converse is easy for me to prove, but I'm not seeing how to prove it using the terminology of open sets and not metric spaces.
|
I would like to know how to prove the convergence (or not) of the following serie:
$\sum\limits_{n=1}^{\infty }(-1)^{n+1} \frac{\sin^2(n)}{n}$
Thank you in advance for any suggestion.
|
I am interested in exam questions that are "backwards" from how they are usually asked. For example:
Brian and Megan have the following question on their exam:
Find the volume of the solid obtained by rotating the region bounded between $y=x^2$ and $x=y^2$ about the $x$-axis.
Megan's integral looks like this: $2\pi \... |
I am working through a bank of previous exams and couldn't figure a problem out to my satisfaction.
Let $f(x) : \mathbb{R} \to \mathbb{R}\,$ be a continuous function.
Show that $f$ can have at most countably many strict local maxima.
Assume that $f$ is not monotone on any interval. Then show that the local
maxima o... |
What does the "^" represent in the statement below? It definitely doesn't mean to the power of as that makes no sense in this context. Thanks for the help!
Since X(t) is a pure death process with constant death rate , Y(t) can be viewed as a
truncated Poisson process below N, namely, Y(t) has the same distribution as Z... |
I understand the arc length formula is derived from adding the distances between a series of points on the curve, and using the mean value theorem to get:
$ L = \int_a^b \sqrt{ 1 + (f'(x))^2 } dx $
But is there an intuition here I'm missing? Something about taking the integral of the derivative seems like it should me... |
I've got the following finite sum:
$s_{n}=\sum\limits_{k=0}^{n}\binom{n}{k}^2p^k$ (esp. if $p$ is a function of $n$, like $p=\frac1{n}$), which can be rewritten as $s_{n}=\sum\limits_{k=0}^{n}\binom{n}{k}\sqrt p^k\binom{n}{k}\sqrt p^k$. Using the generating function approach (from Graham, Knuth and Patashnik) with each... |
I'm given the vectors $v_1=(4,1)$, $v_2=(-7,-8)$, and I'm trying to figure out see if they form a basis for $\mathbb{R}^2$.
I think that it is a basis because $v_1$ and $v_2$ are independent of each other but I'm not sure if it's that easy. Am I on the right track?
|
My approach to this SE question uses the following joint moments of
Brownian motion. For $n=1,2$ they are obvious and well-known, the others
are not terribly hard to work out. Is there a reference where these
formulas are given, or/and is there a pattern to the coefficients?
Fix $t_1\leq t_2\leq t_3\leq\cdots \leq ... |
My question comes from Spivak's "Comprehensive Introduction to Differential Geometry Vol 1" (It's Chapter 1, Problem 24).
Background: Let $X$ be a connected, locally connected, locally compact, and hemicompact Hausdorff space.
And end of $X$ is defined to be a function $e$ that assigns to each compact set $C$ of $X$ a ... |
I don't understand how to get from the first line to the last. What happens to the summation? Please help! An important piece of information given in the question is that the transition probability from state 0 to state 1 is equal to 1. N.B. o(h) is just an error term that goes to 0 when divided by h and a limit as h-... |
Find the value of '$x$' if, $$\large \left(\frac{1}{2^{\log_x 4}} \right) \cdot \left( \frac{1}{2^{\log_x 16}} \right) \cdot \left(\frac{1}{2^{\log_x 256}} \right) \cdots = 2 $$
I tried to make it simple by resulting series is not converging, the suggested answers in my module is either ($2,\frac{1}{2},4,\frac{1}{4}$) ... |
How to find $f(x)$ if $\frac{df(x)}{dx}=f(x)$?
I know $c e^x$ is a solution, but how does one find it and how to prove it is the complete solution?
|
From what I understand, the question is asking me to find P(X(t) = n | X(0) = N). I know that with a linear death rate this probability is (N choose n) * [e^(-alpha*t)]^n * [1 - e^(-alpha*t)]^N-n but I don't think this is true for a constant death rate. Any help on how to approach this would be great! Also, I see in t... |
Let's say we have two exams, each out of 50 points. The correlation rate between them is 0.75. If the teacher decides to add 10 points to the results of the first test, what will happen to the corr. rate?
The way I see it, the correlation should decrease, but by how much? Would it decrease by 1/5=20%? And the result wo... |
I'm a uni student doing a real analysis course and am finding it very interesting, but at the same time very confusing.
One question that has me stumped is how to get a countable set of functions from a set $S$, where $S$ is a set of all functions $u: \mathbb{N} → \{ 0,1,2 \}$. They also tell us that a function $v(n)$ ... |
I wanted to give some of the new undergrad analysis students the following problem: given the real numbers (with the standard topology, as they'd expect) one cannot have an uncountable set such that every point is an isolated point. A few of my fellow grad students attempted solutions first.
A sketch of a potential pr... |
In my analysis lecture I am given a topology on the space of distributions as follows:
Let $u_k$ be a sequence in $\mathcal D'(u)$, $u \in \mathcal D'(u)$. We say $u_k \rightarrow u$, if $\forall \phi \in \mathcal D(u) : u_k(\phi) \rightarrow u(\phi)$.
This is the weak-$*$-topology on $\mathcal D'(u)$. It seems lecture... |
I'm trying to modify a function to create fisheye effect on image (i use a software that permit to modify every single pixel of an image).
At the moment i use this function:
float kk=width*0.5;
float ll=height*0.5;
float dx=(x-kk);
float dy=(y-ll);
float aa=atan2(dy,dx);
float rr=hypot(dy,dx);
float rs=rr*rr/hypot(... |
I would like to show that the diagram
$$\begin{array}{}
A & \stackrel{f}{\longrightarrow} & Y \\
i \downarrow & & \downarrow {\phi_2} \\
X & \stackrel{\phi_1}{\longrightarrow} & X \coprod_f Y \end{array}$$
where $i:A \to X$ is an inclusion is a pushout.
Here $A$ is a closed subset of $X$, all maps given are contin... |
Does anyone know a way to do a quick sort with trivalued logic?
The problem I’m trying to solve is this: I’m trying to display a view of a complex 3d object from a given viewing angle. I’ve broken the object into many 2d surfaces that I can draw separately, but to display the image properly, I need to determine the z-o... |
Suppose we have an Abelian category $\mathfrak A$ and a ring $R$.
From this data we can form a new Abelian category $\mathfrak A[R]$ whose objects are objects $A\in\mathfrak A$ together with a ring homomorphism $R\to\mathfrak A(A,A)$ and whose morphisms are morphisms in $\mathfrak A$ commuting with the $R$-actions.
Let... |
I have some confusion when understanding the concept of conditional probability.
Given any two random variables $X$ and $Y$ and any two constants $m$ and $k$, Is it true that $$P(Y-X=m | Y > X) = P(Y = m+k | Y > k)? $$
My guess is it is not always true,
because
$$\begin{align*}
P(Y-X=m | Y > X) & = \sum_{k} P(Y-X=... |
I am trying to understand how the curvature equation
$$\kappa = -\frac{f_{xx} f_y^2-2f_{xy} f_x f_y + f_x^2 f_{yy}}{(f_x^2+f_y^2)^{3/2}}$$
for implicit curves is derived. These curves arise from equalities such as $f(x,y)=0$. I found this on the net:
http://www.cad.zju.edu.cn/home/zhx/GM/001/00-rep_dg.pdf
I can follow... |
What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think
$$ \mathbb R^+ $$
but I believe that that is usually used to denote "all real numbers including infinity".
So is there a standard way to denote the set
$$ \{x \in \mathbb R : x \geq 0\} \; ?$$
|
Suppose we choose randomly 230 different numbers from 1 to 25,000.
What is the probability that at least 9 of them will be less than or equal to 22?
What is the probability that exactly 9 of them will be less than or equal to 22?
Thanks!
|
Can someone please explain what this sentence means "a Bayes net specifies a canonical factorization of a probability distribution into the product of the conditional distribution for each variable conditioned on its parents"? Thank you.
|
Supose we have a vector field $E:R^3\rightarrow R^3$ with the property $\nabla\times E=0 \Longleftrightarrow E=-\nabla \phi$ where $\phi:R^3\rightarrow R$
How does the boundedness of $E$ imply the continuity of $\phi$
I can solve this physically for a certain case by assuming a rectangular curve through a surface acro... |
This a reformulation of a practical problem I encountered.
Say we have an infinite sequence of random, i.i.d bits. For each bit $X_i$, $P(X_i=1)=p$.
What is the expected time until we get a sequence of $n$ 1 bits?
Thanks!
|
Let $B_{t}$ be a Brownian motion relative to a filtration $F_{t}$, is $(B_{t}+t)^{2}$ a Markov process? Thanks!
|
What is the symbol for demonstrating syntactic conversion (transformation or translation)? For example, I want to show a calculation sequence, from $ \neg ( A \wedge B ) $ to $ \neg A \vee \neg B $. Now I just use $ \vdash $: $ \neg ( A \wedge B ) \vdash \neg A \vee \neg B $. Is there a suitable symbol to replace $ \... |
I don't have much expertise in this area but I am confused by a remark I overheard regarding Teichmüller spaces.
I was always under the impression that for a surface $S$ (say genus $\geq 2$) that the Teichmuller space of $S$ was give by $\mathcal{T}(S) = \{\text{Hyperbolic structures for S} \} / \text{homotopy}$.
I was... |
Given $f: \mathbb{R}^n \backslash \{0\} \rightarrow \mathbb{R}$, a twice differentiable, rotationally symmetric function, that is to say $\exists \varphi: \mathbb{R}_{>0} \rightarrow \mathbb{R}$ with $f(x) = \varphi(\|x\|)$, where $\|.\|$ is the Euclidean norm. I want to express $\Delta f$ in terms of derivatives of $\... |
Let $C, D, E$ be three non-degenerate triangles in $\mathbb R^2$. Let $c, a, b$ be the vertices of $C$, let $d, a, b$ be the vertices of $D$, and let $e, a, b$ be the vertices of $E$. I want to show that there is one point contained in the interior of at least two of the given triangles.
Here are my thoughts: If two of... |
Mathworld notes that "The Fibonacci and Lucas numbers have no common terms except 1 and 3," where the Fibonacci and Lucas numbers are defined by the recurrence relation $a_n=a_{n-1}+a_{n-2}$. For Fibonacci numbers, $a_1=a_2=1$; for Lucas numbers, $a_1=1$, $a_2=3$. How do you prove mathworld's statement?
|
I need to make a constraint for the following condition:
Among students 1, 2, 3, and 4, at least two of them must be on the team, if there are any on the team at all.
I have defined Y1, Y2, Y3, and Y4 to be 1 if student i is on the team and 0 if they are not.
My constraint looks as follows:
Y1 + Y2 + Y3 + Y4 >= 2 + MZ ... |
Note: Not good at math and my terminology may be very wrong.
I have a uniform random number generator that outputs a number between [0,1]. I'd like a function that returns a random number between 0 and 1 using an exponential or exponential-like distribution - i.e. F(random()) is more likely to be between 0.5-1.0 than ... |
I have an equation and I want to know why it is true. It can be found in Boyd's Convex Optimization book, page 82, line 11-12. It is example 3.9.
\begin{equation}
b^T W A (A^T W A)^{-1} A^T W b = \displaystyle\sum\limits_{i=1}^n w_i^2 b_i^2 a_i^T (\displaystyle\sum\limits_{j=1}^n w_j a_j a_j^T)^{-1} a_i
\end{equation}
... |
Given: a set $S$ of points in $\mathbb{R}^3$.
Find: the smallest oriented bounding box that contains all the points. Note, the bounding box is "oriented" and thus need not be axis-aligned.
Can this be formulated as an optimization problem? Preferably a convex optimization problem!
The optimal box should be smallest by ... |
Let $I \subset \mathbb R$ be open, $u \in \mathcal D'(I)$ be a distribution whose distributional derivatives vanishes (i.e. is zero for all test functions, which we may assume to be complex valued ).
We show $\forall c \in \mathbb C: \forall \phi \in \mathcal D(I) : u(\phi) = \int c\cdot\phi dx$. (EDIT: Correctly, $c$ ... |
This problem has been giving me some troubles. Does anyone have any ideas on how to go about proving this?
Let $X$ and $Y$ be Banach spaces. If $T: X \to Y$ is a linear map such that $f \circ T \in X^*$ for every $f \in Y^*$, then $T$ is bounded.
Thanks in advanced!
|
Where can I read a corrected up to date version of Heegner's solution of the class 1 problem of Gauss?
|
If I can show that $\displaystyle \sum_{k\geq1}a_k^2<\infty$, then is this an acceptable proof that $\displaystyle \sum_{k\geq1}a_k^2$ is convergent?
|
Here we have that $\pi$ is the invariant distribution of a Markov chain $K$. Let's say that $\mu=f\pi$, that is, $f$ is the density of $\mu$ and $\mu$ is a probability measure.
Why does $\mu K$ have density $K^* f$? I don't understand where the adjoint comes from because we're not dealing with any inner products here... |
Let $A \in \mathbb{R}^{n\times n}$ be a real symmetric matrix. Please help me clear up some confusion about the relationship between the singular value decomposition of $A$ and the eigen-decomposition of $A$.
Let $A = U\Sigma V^T$ be the SVD of $A$. Since $A = A^T$, we have $AA^T = A^TA = A^2$ and:
$$A^2 = AA^T = U\Si... |
Expressing the product Ax as a linear combination of the column vectors of
$A$=
$\begin{bmatrix}
4 & 0 & -1\\
3 & 6 & 2\\
0 & -1 & 4
\end{bmatrix}$
$\vec{x}$=$\begin{bmatrix}
-2\\
3\\
5
\end{bmatrix}$
I get it now. They just want me to multiply the two vectors together.
I end up with $\begin{bmatrix}
-13\\\
... |
I've heard this result bandied about many times, and I know that it follows from e.g. the theory of divisors, but I'd like to see some simpler, straightforward ways of proving this fact.
|
If $r : X \to A$ is a deformation retract and $i : A \to X$ is inclusion, then $i(r)$ is homotopic to $id$ on $X$.
I know that since r is a deformation retract, it is homotopic to id on X, but I don't see an obvious construction of the above homotopy.
|
so I have the x,y,z value for 3 points to define a plane in 3d space.
I need to find the z value of an arbitrary point given the x,y.
I can sort of see some ways to calculate this, but they seem like they might be doing a lot of extra steps.
I eventually need to encapsulate the process in a algorithm for a computer pro... |
Three points are placed at independently and at random in a unit square. What is the expected value of the area of the triangle formed by the three points?
|
Suppose that $x_1 = -1$, $x_2 = 2$, $x_3 = 4$, $x_4 = -3$ is a solution of a non-homogeneous linear system $A\mathbf{x} = \mathbf{b}$ and that the solution set of the homogeneous system $A\mathbf{x} =\mathbf{0}$ is given by the formulas:
$$\begin{align*}
x_1 &= -3r + 4s,\\
x_2 &= r - s,\\
x_3 &= r,\\
x_4 &= s.
... |
Let $E$ be an open set in $[0,1]^n$ and $m$ be the Lebesgue measure.
Is it possible that $m(E)\neq m(\bar{E})$, where $\bar{E}$ stands for the closure of $E$?
|
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