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Suppose $E$ and $F$ are given Banach spaces. Let $A$ be a continuous surjective map. Why is there a small ball around $A$ in the operator topology, such that all elements in this ball are surjective?
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In the book "A Course of modern analysis", examples of expanding functions in terms of inverse factorials was given, I am not sure in today's math what subject would that come under but besides the followings : power series ( Taylor Series, Laurent Series ), expansions in terms of theta functions, expanding a function ... |
I learned in different math courses, that $\mathbb{C}$ is a clopen set. I'm very uncertain if I understand this right, please tell me if these two examples are true:
$\mathbb{C} \setminus \{ 0 \}$ is open,
$\mathbb{C} \setminus \{ z \in \mathbb{C} : |z| < 1 \}$ is closed.
Thanks.
|
What does this mean: ($f$ a function, $I$ an interval and $R$ the real numbers)
$f \in C(I,R)$
Does it mean $f$ is an element of the collection of continuous functions with domain $I$ and range $R$ ? (literal translation so the terms might be off)
Also, what would this mean:
$f \in C^\infty(I,R)$
?
|
I have a $3\times 3$ square matrix. I want to find out how many unique matrices I can create if each of the elements can be either $1$ or $0$.
How does the equation change if have $N\times N$ matrix?
|
I was wondering about what is the
general definition of a recursive
mapping between any two sets.
Is there a condition for a mapping
to be able to be written in
recursive form? Is the following
claim true:
A mapping can be represented
recursively if and only if its domain
is a set that can be defined recursively?... |
Hi I have these equations that I wish to know how to reach the variables
for example
${\begin{cases}A=6\\D=A-C\\B=2C\end{cases}}$
All I have is the value of $A$, and $A$ changes from time to time so its not always $6$ it can be $8$, $20$ or anything else.
How can I get the values of each $B$, $C$, and $D$?
Note that, a... |
I am looking for an operator $U$, that can do this to a function:
$$Uf(x)=f(2x).$$
In particular I am happy if there is an $U$ for the general case:
$Uf(x)=f(kx)$.
Does such an operator exist for all $f(x)$?
What is the operator for known classes of $f(x)$?
I have so far 1 condition on $f(x)$:
$$\int_{-\infty}^{\infty}... |
Calculate all complex roots of the polynomial: $8t^{4} -20t^{3} -10t^{2}-5t-3$.
So thanks to matlab, I can easily find out that the roots are $t = 3, -0.5, \pm 0.5i$.
Unfortunately, achieving this answer by hand has been more difficult.
Apparently, one valid method is to try to guess one of the roots and then use it to... |
Is there an epimorphism $f\colon \mathrm{GL}(2,\mathbb{Z})\to \mathrm{GL}(2,\mathbb{Z})$ which is not injective? Here, $\mathrm{GL}(2,\mathbb{Z})$ is the group of invertible $2\times 2$ matrices with integer entries.
Added. The answer is no: this group is Hopfian. I know a proof in which it is shown that this group is ... |
Let $ \chi $ be a character on the field $ F_p $, and fix some $a \in F_p $.
We define a Gauss sum to be:
$g_a (\chi) = \sum_{t\in F_p}\chi(t)\zeta^{at}$ where $\zeta$ is a primitive $p^{th}$ root of unity.
What is the intuition behind this definition?
|
How can I calculate the probability of shaking $7$ of a kind using $10$ six-sided dice?
|
Let $S=\{S_1,S_2,\ldots,S_n\}$ be a set of $n$ distinct subsets with $S_i \subseteq \{1,\ldots,n\}$ for $i=1,\ldots, n$ then $k \in \{1,\ldots,n\}$ exists with $S_i \cup \{k\}$ is distinct for $i=1,\ldots,n$.
I found this in a old problem sheet of mine about sets and graph theory. Is there a elegant solution to this pr... |
Is $\sin\frac{1}{x} \lt \frac{1}{x},\ \forall x\geq 1$?
I tried "copying" the proof of $\ln x \lt x, \forall x\geq 1$ but it didn't quite work.
Here's what I did: Let $f(x)=\sin\frac{1}{x}-\frac{1}{x}$, I'd like to show that $f(x)\lt 0$. $f'(x)=\frac{1}{x^2}(1-\cos\frac{1}{x}) \gt 0$. So $f(x)$ is increasing, then $f... |
Are Riemann integrals special cases of Haar integrals? Why do we need the invariant property under some actions of groups in the definition of Haar integrals? For example, if we have a group of real matrices $G$ and we have an inner product $\langle\>\ ,\ \rangle: G \times G \to \mathbb{R}$, for any $A, B$ in $G$, $C$ ... |
Suppose $p$ is irreducible and $M$ is a tosion module over a PID $R$ that can be
written as a direct sum of cyclic submodules with annihilators of the form $p^{a_1} | \cdots | p^{a_s}$ and $p^{a_i}|p^{a_i+1}$. Let now $N$ be a submodule
of $M$. How can i prove that $N$ can be written a direct sum of cyclic modules wi... |
As Wikipedia says, there are two ways to describe a particular set: intensional definition and extensional definition.
For a set able to be described by
extensional definition, is it necessary and sufficient to say the set has
to be countable? Do there exist sets that are not describable by extensional definition?
For... |
For no real reason, I did a classification of (associative, with a 1) rings with less than 8 elements (they all happen to be commutative). Most of the rings I got were of a type I knew - namely: cyclic, field, or Boolean - but one wasn't.
$R \hspace{.04 in} := \hspace{.04 in} \Bigg\langle \bigg\{\begin{pmatrix} 0 & 0 ... |
After solving a system of equations I get this:
\begin{cases} x + 0.03125z = 0.242813 \\ y + 0.40625z = 0.456563\end{cases}
How can I filter solutions that are only positive for $x$, $y$ and $z$
Ted
|
In a finite dimensional (think Euclidean) ambient space, let $S$ a compact, convex set and $x$ not in $S$. The two sets can be (weakly) separated, i.e. there exists a vector (normal) that defines a hyperplane separating the point and the set. In fact, there exists an entire (open) set of normals that separate the point... |
I have a line graph, Given some certain points on the graph, I wish to add some explanation on them, such as this:
Is there any mathematical algorithm that allows me to overlay my explanation textbox "nicely" on the graphs, avoiding self overlapping as far as possible?
I understand that "nicely" is quite a subjective ... |
Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet."
I have a (not very mathematically intelligent) friend who presented it as if it were a mathematical fact, which got me thinking... Is this really true? Of course, I've learned that deal... |
Does anyone know of a nice example of a non-singular complete intersection $X$ (say in $\mathbb{P}^n_k$, maybe even $k=\overline{k}$, char($k$)=0) such that it cannot be written as $E\cap H$ where $E$ is a non-singular complete intersection and $H$ is a hypersurface (or maybe this can always be done)?
For instance, it ... |
We have skew normal distribution with location $=0$, scale $=1$ and shape $=0$ then it is same as standard normal distribution with mean $0$ and variance $1$. But if we change the shape parameter say shape $=5$ then mean and variance also changes. How can we fix mean and variance with different values of shape paramete... |
Let $L/K$ a finite extension and $f(x)\in K[x]$ a non-linear irreducible polynomial. Prove that if $\mathrm{gcd}\left( \mathrm{deg}(f) , \left[ L:K \right] \right)=1$ then $f(x)$ has no roots in $L$.
Added: (Solution based on the answer below)
Suppose $f(x)$ has a root in $L$, namely $\alpha$ and consider the exten... |
What is the property where f(g(x)) = g(f(x))?
|
According to my textbook:
$$\int_{-L}^{L} \cos\frac{n \pi x}{L} \cos\frac{m \pi x}{L} dx =
\begin{cases} 0 & \mbox{if } n \neq m \\
L & \mbox{if } n = m \neq 0 \\
2L& \mbox{if } n = m = 0
\end{cases}
$$
According to the trig identity given on this cheat sheet:
$$
\cos{\alpha}\cos{\bet... |
Let $a<b$ be real numbers. Define a cube $Q$ in $R^n$ to be the $n$-fold Cartesian product of $[a,b]$ with itself. Find a formula of the angle between the longest diagonal of $Q$ and any of its edges.
I am not sure where to start. Any help would be appreciated.
|
A child walks due east on the deck of a ship at 1 miles per hour.
The ship is moving north at a speed of 9 miles per hour.
Find the speed and direction of the child relative to the surface of the water.
Speed = __ mph
The angle of the direction from the north = ___ (radians)
The last time I solved a problem like this... |
The modified Euclidean Algorithm has you pick an $a,b\in\mathbb{Z}^+$. Perform the usual division algorithm with the dividend being $b$ and the divisor being $a$. This produces $b=aq+r$. The next step is $bq=rq_2+r_2$. The algorithm terminates and $\sum_{i=0}^n\frac{(-1)^i}{q_i}=\frac{a}{b}$.
Results like this come... |
Find the positive real number(s) $c$ such that the graphs of $y=x^c$ and $x=y^c$ intersect (somewhere) at an angle $\frac{\pi}{4}$.
A problem from Mark Krusemeyer. Should one be using the symmetry of the graphs along $x=y$ due the functions being inverses of one another? Just a thought.
|
We are trying to implement transformations to evaluate the incomplete integral of the third kind $\Pi(n;\phi|m)$ for arbitrary inputs, and I can't find any references for how to calculate this function with phase $\phi$ less than zero or greater than $\pi/2$.
Are there periodic identities for this function for various ... |
Please forgive me a really simple question: Am I calculating the relative error reduction correctly?
Example: Model 1 gets 80/100 examples wrong, model 2 gets only 20/100 examples wrong, so the 80 errors are reduced by 60, which is an error reduction of 75%.
So the relative error reduction formula seems to be
(m1_erro... |
In Tom Apostol's Analytic Number Theory book there is a problem which states:
That there do not exists polynomials $P(x)$ and $Q(x)$ such that $$\pi(x) = \frac{P(x)}{Q(x)}$$ for all $x \in \mathbb{N}$.
I tried this problem, but couldn't find a solution. Here is what i attempted. Since $\pi(x) \sim x\log{x}$ as $x \to... |
This is my first question here.
As I'm not a matematician I thought I'd ask here for advice how to approach something I'm working on as a hobby project.
A bit of context
Let's say there is a collection of items with a description of features and a price. Imagine a list of cars and prices. All cars have a list of featu... |
Given a divisor $D$ on an algebraic curve $X$, there is a corresponding map $\phi_D$ from $X$ to the projective space (of dimension $\dim L(D)-1$). In particular, we know that if $D$ is a very ample divisor, then this is a holomorphic embedding into projective space. When $D$ is not very ample, how does one read off in... |
The following is a problem in Miranda's Algebraic Curves and Riemann Surfaces.
Given any algebraic curve $X$ and a point $p \in X$, show that there is a meromorphic $1$-form $\omega$ on $X$ whose Laurent series at $p$ looks like $dz/z^n$ for $n > 1$, and which has no other poles on $X$.
The point of this is as a step ... |
This question was inspired by this question dealing with finding the probability of rolling a $7$ of a kind with $10$ dice.
Is there a general method for finding the chance of rolling a $k$ of a kind with $n$ $m$-sided dice when $k\leq n/2$? When $k>n/2$ the probability is found relatively easily with the Bernoulli dis... |
The Wikipedia article on sprouts, seems to suggest that it does not, although I can't see how to prove it.
|
We know there are statements that are undecidable/independent of ZFC. Can there be a statement S, such that (ZFC $\not\vdash$ S and ZFC $\not\vdash$ ~S) is undecidable?
|
I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague.
I would like to have a reference where I can go look at images of functions and say: that looks like a behavior/character I want. Or maybe even ... |
Consider $\mathbb{R}^n$ equipped with the standard scalar product. Let $f(n)$ denote the maximal cardinality of a set of vectors in $\mathbb{R}^n$ with a pairwise negative scalar product.
What is $f(n)$?
Some thoughts:
WLOG one can assume, that all vectors have length 1. This turns this problem in a space packaging pr... |
I've been struggling with the following problem for a couple of days and I don't seem to get any further:
Let $R$ be a commutative ring. I would like to get (something like) a classification of all finitely generated $R$-modules $M$ that satisfy the following condition:
When we look at $M \otimes_R M \otimes_R M$, the ... |
I've been referred to this website, hopefully you have the
background in set theory to help me out here.
Got two questions, the first is on number systems arising out of set theory
and the second is a small one about functions. I've tried to be as clear as
possible but please bear with me; the only reason I'm postin... |
How many possible combinations are there in Hua Rong Dao?
Hua Rong Dao is a Chinese sliding puzzle game, also called Daughter in a Box in Japan. You can see a picture here and an explanation here .
The puzzle is a $4 \times 5$ grid with these pieces
$2 \times 2$ square ($1$ piece)
$1\times 2$ vertical ($4$ pieces)... |
We have a set of numbers, of size $m$. We are going to pick $a$ numbers with uniform probability from that set, with replacement. Let X be the random variable denoting the probability of having X of those picks distinct (exactly X distinct values are picked).
Motivation: I need to calculate this probability in order to... |
I encountered a programming problem that need a little math. Since I took number-theory long time ago, from the top of my head, I could not think of a way to approach this problem.
Give a base 'b' ( 2, 3, 4, 5, 6 ... -> 26 ). So if I'm at base 10, I will have 10 digits: 0, 1, 2, 3...9. The digit of a base 'b' is calle... |
Assuming I have the graph of a function $f(x)$ is there function $f_1(f(x))$ that will give me a rotated version of the graph of that function?
For example if I plot $\sin(x)$ I will get a sine wave which straddles the $x$-axis, can I apply a function to $\sin(x)$ to yield a wave that straddles the line that would resu... |
If you divide an number by another numbers reciprocal
i.e
x / (1 / y)
Can this be expressed in another way
is dividing by the reciprocal equivelant to another operation?
|
Is the subspace of $\mathbb C^2$ of those vectors with real coordinates a complex submanifold?
|
I vaguely remember the Frobenius matrix norm
( ${||A||}_F = \sqrt{\sum_{i,j} a_{i,j}^2}$ ) was somehow considered unsuitable for numerical analysis applications. I only remember, however, that it was not a subordinate matrix norm, but only because it did not take the identity matrix to $1$. It seems this latter proble... |
A common characterization of
M-matrices are non-singular square
matrices with non-positive
off-diagonal entries, positive
diagonal entries, non-negative row
sums, and at least one positive row
sum
It seems that this characterization depends on the use of rows or columns. I don't understand how that is pos... |
In this question on Math.SE, I asked about Ramanujan's (ridiculously close) approximation for counting the number of 3-smooth integers less than or equal to a given positive integer $N$, namely,
\begin{eqnarray}
\frac{\log 2 N \ \log 3 N}{2 \log 2 \ \log 3}.
\end{eqnarray}
In his posthumously published notebook, Raman... |
In Maple the command NullSpace turns a given matrix into a set of vectors -- a basis for its null space. This is precisely what it should in a mathematical sense. However, for computations using the obtained basis, it would be more convenient to have as output a matrix whose columns are the basis vectors.
I'm having a ... |
Let $\displaystyle f: [0,1] \rightarrow \mathbb{R}$ given by
$$f(x) = \begin{cases}
0 & x \notin \mathbb{Q} \\
\\
0 & x = 0 \\
\\
\frac{1}{q_x} & x = \frac{p_x}{q_x} \in \mathbb{Q} \backslash \{0\}, \ p_x \in \mathbb{Z}, \ q_x \in \mathbb{N}, \ \text{gcd}(|p_x|, q_x) = 1
\end{cases}$$
Is $\displaystyle f$ Riema... |
Cantor's Diagonal Argument is the standard proof of this theorem. However there must be other proofs, what are some of these proofs?
I am asking this because whenever I think of this question, I immediately think of the Cantor's Argument, which kills the possibility of other interesting finds.
|
Let's consider linear operators on the set of complex-valued functions to the same set. I wonder to which categories such operators can be classified. All linear operators I encountered so far fall into the following categories:
Differintegral of complex or variable order $L:f\to\mathbb{D}^{g}f$
Multiplication by a va... |
I have problems in understanding few concepts of elementary set theory. I've choosen a couple of problems (from my problems set) which would help me understand this concepts. To be clear: it's not a homework, I'm just trying to understand elementary set's theory concepts by reading solutions.
Problem 1
(I don't unders... |
I'm trying to understand a proof in "Groups with a maximal irredundant 6-cover"
by A. Abdollahi et al, page 3231, Lemma 4.1 itens (2) and (3).
Here's my problem:
Lemma 4.1
(2) The following are $\mathfrak{G}_6$-groups:
(d) $G \cong (C_5 \times C_5) \rtimes C_4$ with $Z(G) = 1$. In this case, the intersection of a maxi... |
I am having trouble with this:
If $4^x + 4^{-x} = 34$, then what is $2^x + 2^{-x}$ equal to?
I managed to find $4^x$ and it is:
$$4^x = 17 \pm 12\sqrt{2}$$
so that means that $2^x$ is:
$$2^x = \pm \sqrt{17 \pm 12\sqrt{2}}.$$
Correct answer is 6 and I am not getting it :(. What am I doing wrong?
|
I have a problem in vector algebra.
In this Wolfram-page the last two formulas (9) and (10) are:
$$
\frac{ | (x_2 - x_1) \times (x_1 - x_0) | }{|x_2 - x_1 |} =
\frac{ | (x_0 - x_1) \times (x_0 - x_2) | }{|x_2 - x_1 |}
$$
I've tried to apply properties founded in this other Wolfram-crossproduct-page but I still don'... |
I would like to hear some intuition about difference between subtraction and division. For binary subtraction operator the standard development is introduction of unary operation of taking negative number. For division however it depends on the domain. For "nice" domains (such as rational and complex numbers) we have u... |
How do we show that if $g \geq 2$ is an integer, then the two series $$\sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}} \quad \ \text{and} \ \sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}}$$ both converge to irrational numbers.
Well, i tried to see what happens, if they converge to a rational but couldn't get anything out i... |
Is there a categorification of $\pi$?
I have to admit that this is a very vague question. Somehow it is motivated by this recent MO question, which made me stare at some digits and somehow forgot my animosity about this branch of mathematics, wondering if there is a connection to the branches I love so much.
$\pi$ is t... |
OK, so here's the question (not my homework - which is line segments -_-)
According to Chebyshev's theorem, how
many standard deviations from the mean
would make up the central 60% of
scores for this class? [What are the
corresponding grades? Answer the same
questions for central 80%. Do these
values capt... |
I am confused when reading Wikipedia's article on limsup and liminf of a sequence of subsets of a set $X$.
It says there are two different ways
to define them, but first gives what is common for the two.
Quoted:
There are two common ways to define
the limit of sequences of set. In both
cases:
The sequence accum... |
I'm looking for the method by which the partial derangement formula $D_{n,k}$ was derived. I can determine the values for small values of N empirically, but how the general case formula arose still alludes me.
Any links/books or an explanation will be appreciated.
The formula is:
$D_{n,k} = {n \choose k}!(n-k)$
Links:
... |
Consider a simple random walk on the integers with a reflecting barrier at $0$. What is the expected time the walk spends in the set $\{0, \ldots, k\}$ in the first $n$ steps?
I'm curious about how the answer scales with $n,k$, and don't really care about the constants.
|
Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies
$$ f(x+1) - f(x-1) = g(x) $$
|
[1)
I'm wondering if there are two angles in an
'angle.'
For example, the image below would typically be
identified as a right angle.
But couldn't you also say that it's a right angle
and a $270^o$ reflex angle? Are there two
angles?
http://tinyurl.com/4fs85kg
[2)
Or-- An equilateral triangle is defined (?) as a
trian... |
How would I go about solving a problem that goes roughly as such?
Solution A: .1%
Solution B: .15%
How many milliliters of each A & B are required to create 260 milliliters of a solution, C, that is .14%?
|
There's a $30$-year home loan for \$$100000$ at $7$%. After $15$ years the loan is paid off in order to refinance at a lower rate. The loan has a prepayment penalty of six months interest of $80$% of the remaining balance of the loan.
a) How much is the remaining balance of the loan?
b) If the loan can be refinanced wi... |
The problem is: Suppose $\phi_1(x)$ and $\phi_2(x)$ are solutions of the differential equation (1):
$$ y''(x)+p(x)y'(x)+q(x)y(x)=0 $$
on the interval $I$. Show that $\lambda(x)=c_1\phi_1(x)+c_2\phi_2(x)$, where $c_1$ and $c_2$ denote arbitrary constants, is also a solution of (1) on the interval $I$.
What I did was to... |
How to characterize the speed of a divergent series ? I have a divergent series with a parameter $x$ in it. How can i characterize the speed of divergence for different $x$ ?
|
I know that for any surface, the Gaussian curvature $K$ and mean curvature $H$ satisfy the inequality $H^2 \geq K$ , and the sphere is a surface where that inequality becomes an equation. Thus, the sphere has both constant Gaussian and mean curvature.
Are there other surfaces whose Gaussian and mean curvatures are cons... |
My ways to define/write limsup and liminf of a sequence of points in a set $X$:
They come from what I have understood. If you have other ways of understanding, really appreciate if you can reply here.
When $X$ is a complete lattice, then
$$\limsup \, x_n := \inf_{n \geq 0}
\left(\sup_{m \geq n} \, x_m\right... |
Let $X$ and $Y$ be Riemann surfaces, and $\mathscr{O}_X, \mathscr{O}_Y$ be the sheaves of holomorphic functions on $X$ and $Y$ respectively.
It is obvious that a holomorphic map $f:X \to Y$ gives rise to a morphism of locally ringed spaces $(X, \mathscr{O}_X) \to (Y, \mathscr{O}_Y)$.
I was always under the impression t... |
I am experiencing what I think is really simple confusion.
Take $y(t) = \sin(2 \cdot \pi \cdot t \cdot\omega(t))$
and $\omega(t) = a \cdot t+b$ for $t \in [0,p)$ and let $\omega(t)$ have a periodic extension with period $p$. The values $a,b,p$ are parameters
$\omega(t)$ looks like the blue function here
with $\omega$ a... |
Recently, I have been programming a simple game. Very simple:
There is a tank, and the cannon will aim at whatever position the mouse is at.
Now lets talk about the cannon graphic. The cannon graphic points to the north, with 0 rotation.
Here are the variables I have for my game, and that might be important factors for... |
I'm confused as to how far to go in terms of getting $s_n$ in a telescoping series. I cannot find an explanation for this at all in my textbook. But from the problems I've seen, perhaps it's just coincidence, does it have anything to do with the numerator?
For example, the section example gives:
$(\sum_{n=1}^\infty) \f... |
Wait before you dismiss this as a crank question :)
A friend of mine teaches school kids, and the book she uses states something to the following effect:
If you divide the circumference of any circle by its diameter, you get the same number, and this number is an irrational number which starts off as $3.14159... .$
... |
I have trouble understanding this question and its supposed answer.
Let $A$ be a subgroup of $G$. Prove that $A$ is a direct summand of $G$ if and only if there is a homomorphism $p:G \to A$ with $p(a)=a$ for all $a \in A$. Here we talk about internal direct sum.
When I see this, I was thinking, why do homomorphism o... |
$$ D = \{ z \in \mathbb{C} \ : \ |z| < \frac{1}{2} \}$$
It seems a disc is open or closed whether it has a < or <= but I don't know how to prove this. All examples have slightly different proofs so I've got no clarity.
Thanks for your help.
|
I am thinking about following problem, but am not able to find answer.
In how many ways can a number be formed only using $3$ digits $(0, 1, 2)$ given following constraints with $K$ and $D$:-
1) Each digit appears exactly K times. The total length of the number is $3*K$.
2) At any given position (staring from left), t... |
Let $X,Y$ be connected sets and $A,B$ proper subsets of $X$ and $Y$ respectively. Then it can be shown that the space $ (X \times Y) \setminus (A \times B)$ is connected. This is an problem from Munkres book. My question is:
Is the above result still true when we replace connectedness by path-connectedness instead?
|
Please answer as many questions as you can.
What are the topics one should know before delving into probability theory? (Please recommend any books you know on those topics too.) I think there is set theory, but set theory is a large topic in itself. Does probability need only a little bit of set theory? Also, there is... |
What sorts of mathematical tools, models and methods and theoretical frameworks do people use to simulate the function of the brain's neural networks? What mathematical properties do different brains have?
|
I'm posting this as more of a theory/mathematic how-to followup to a stackoverflow question.
The non-iterative method for calculating graycode depends on Log2N bytes, to store position information for the next bit in the iteration sequence.
Specifically, the goal is to know the next bit to change without having to look... |
Fubini's theorem states that if you have a double integral over a function $f(x,y)$, then you can compute the integral as an iterated integral, if $f(x,y)$ is in $\mathcal{L}^1$. But to find out if $f$ is in $\mathcal{L}^1$ you need to compute the double integral.
What am I missing? The examples I found all apply Fubin... |
I'm trying to re-write $v (u x)$ where $v$ and $u$ are row vectors and $x$ is a column vector as some expression $M x$ (or $\bar{v}x$, etc.).
The motivation is because I'm trying to compute the gradient (and $\nabla^{'}_x [ v (u x) ] \cdot x$ should yield a row vector).
Intuitively, since the expression $v (ux)$ can ... |
The coordinates of an arc of a circle of length $\frac{2pi}{p}$ are an algebraic number, and when $p$ is a Fermat prime you can find it in terms of square roots.
Gauss said that the method applied to a lot more curves than the circle. Will you please tell if you know any worked examples of this (finding the algebraic p... |
A linear system of form $A\vec{x}=\vec{0}$ is called homogeneous. Why are all homogenous systems consistent?
|
I ran across a group with presentation $<a,b | a^2b=ba^2, b^2a=ab^2, a^2=b^4>$. Does this group have a name?
|
How does someone mathematically round a number to its nearest integer?
For example 1.2 would round down to 1 and 1.7 would round up to 2
|
I am trying to make a formula where a "table lookup" is not required. Given these two tables:
Days Discount Refs Rent_Cost
------------------ ------------------------
15 0.00 0-250 $0.200
30 0.05 251-500 $0.210
60 0.10 5... |
The formula for the Gordon growth model is:
$\hspace{1in}P= \sum_{t=1}^{\infty} D\times\frac{(1+g)^t}{(1+k)^t}$
So summing the infinite series we get:
$\hspace{1in}P=\frac{D(1+g)}{k-g} \hspace{2in}\mbox{(1)}$
Here's my attempt to arrive at equation (1):
The sum of a geometric series is:
$\hspace{1in}\frac{a}{1-r}$
So, ... |
I'm reading a text on ray tracing. There is this section about radiometric quantities where radiance is defined as
$L = \frac{d^2\Phi}{dA cos\Theta d\omega}$
$\Phi$ is the radiant flux
$\Theta$ is the solid angle (sr) subtended by the observation or measurement
$\omega$ is the incidence angle measured from the surface ... |
Let $X$ be a set and consider the collection $\mathcal{A}(X)$ of countable or cocountable subsets of $X$, that is, $E \in \mathcal{A}(X)$ if $E$ is countable or $X-E$ is countable. If $X$ is countable, then $\mathcal{A}(X)$ coincides with the power set $\mathcal{P}(X)$ of $X$. Now suppose that $X$ is uncountable. Assum... |
Let $M$ be the positive borel measures on a hausdorff topological space $X$, which are finite on compacts sets $--$ i.e. the real cone of radon measures.
I am given a definition of a derivative of two radon measures, say $\mu, \eta \in M$, as follows.
Let
$D^+_\mu \eta (x)
:= \left\{ \begin{array}{1 1}
\limsup \limit... |
I'm not so sure if my "solution" to the following problem is correct. Can you please let me know if is? or what is wrong? I'm new with this stuff so sometimes I get confused with it.
Anyway here it is:
Suppose $Y$ is an arbitrary compactification of $X$, if $\beta X$ is the Stone-Cech compactification, prove there is ... |
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