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For the level set method, $\phi(\vec{x},t)$ is the level set function in 3D and the level set $\phi(\vec{x},t) = 0$ forms the interface. For evolving $\phi$ the derivation says to imagine a particle $\textbf{x}(t)$ on the surface, then we differentiate with respect to t
$\frac{d}{dt}(\phi(\vec{x}, t)=0)$
Using chain ru... |
On mathoverflow, Terry Tao says the following:
For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom. A megapixel image, for instance, is a point in a million-dimensional vector space; by varying the image, one can explore the space, and various subsets o... |
I have a truncated cone with $\text {smaller perimeter} = 70\space cm$ and $\text{greater perimeter} = 64\space cm$. The edge length is $42\space cm$
How can I now calculate the angle $\alpha$?
|
I'm note sure if this is mathematical, however I know that there's an area of maths about folding. And regardless, this seems like an interesting question that I have no idea how to solve.
I am designing a 5 day paper timetable for university that I will keep in my pocket so I know where and when my classes are. My cur... |
I'm looking for a calculator that shows a full answer regardless of it's length.
For example the assignment 2^1000 will output: 1,07150860718626732094842504906e+301.
Windows calculator shows me that amount because else it won't fit the display.
So the question is:
Is there a calculator out there that shows the full ans... |
Given a topological space that is T3 (i.e. regular) and has a sigma-discrete base (where these conditions allow the topological space to be biconditionally metrizable), is there a way to understand what all the possible admittable metrics are? And do all these possibly admittable metrics form a space?
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I initially asked this question here, and someone suggested this may be a better place to get an answer.
I have a question about a rotation matrix, which can be represented in 2 dimensions as:
$$R_{2}(\theta)=\begin{bmatrix} cos\theta & sin\theta \\ -sin\theta & cos\theta\end{bmatrix}$$
For some arbitrary angle $\theta... |
I need to be able to find the eigenvalues and eigenvectors of a matrix (sticking to $2 \times 2$ matrices at the moment).
I'm fine with finding the eigenvalues, but my answers for the eigenvectors don't match up with those on my problem sheet. However the proportion between my values seem to be the same, but I'm wonder... |
I dread mathematics, and I believe it's because I have come to associate mathematics with the experience of terrible teachers. All of my math teachers have been grumpy, but one in particular was the epitome of evil. She would take any opportunity to yell and scream at me when I struggled to comprehend the problems give... |
I'm looking for a way to mathematically combine two concepts: LFSRs, and Barrel Shifters
I'm looking for a way, in O(1) time, to shift an LFSR loop a given number of shifts.
What I'm hoping to find is a simple process where I have the current state of the LFSR and the number of times I want to shift from that state as ... |
It is known that a string $s$ is actually made up of repetitions of another string $s_1$ of length $L_1$.
Also $s$ can be thought of as made up of repetitions of another string $s_2$ of length $L_2$.
For example the string $s = abababab$ is made by repeating the substring $s_1$ "$ab$" of length $L_1= 2$ or by repeati... |
I have a $N\times N$ symmetric positive semidefinite matrix $Q$, and am considering a class of symmetric positive definite matrices having all eigenvalues in a given bounded interval $[a, b]$.
Is it possible to find a matrix in this class such that the sum of the smallest $k$ elements of $\text{diag}(XQ)$ is minimized... |
I have stumbled upon a remarkable similarity between the proof of Baer's criterion and an extension theorem in field theory. Here are the statements:
Baer's criterion: Let $R$ be a ring. A left $R$-module $E$ is injective iff every $R$-map $f:I \to E$, where $I$ is a left ideal of $R$, can be extended to a map $R\to E... |
Does anybody know of a good way to figure out if the variety defined by the ideal $I=(a_1y_2-a_2y_1,b_1y_2-b_2y_1)$ in $\mathbb A^2_{y_1,y_2}\times\mathbb P^1_{a_1,a_2}\times \mathbb P^1_{b_1,b_2}$ is Toric?
|
The following is possibly true but I can't find a corresponding theorem:
If $E/F$ is the splitting field of some polynomial in $F$ and $F \subset K \subset E$ then:
$Gal(E/K)$ normal subgroup of $Gal(E/F)$ $\Leftrightarrow$ $K/F$ is a normal field extension
Is this true? I think saying that $E/F$ is the splitting fiel... |
If you have an odd number of data values {39, 40, 42, 44, 47, 48, 49, 51, 53}, what is the score that value that corresponds to Quartile 1? In general, I would like to know if you include the median's value (47) in figuring the position of quartile 1.
|
I am taking a probability course and we have been of course learning the classic problems of men with hats or variations of that in some other form. The problems being of the type if $n$ men have $n$ hats and mix them all together, whats the probability $0, 1, 2,$ or $k$ men choose their hat.
But from what I can gathe... |
I already know the harmonic series ($1/n$, which diverges) and following the link, $1/n^2$ that converges.
What other useful series could you teach me, or perhaps some general advice? Any website/resource is very welcome also, since most of my search for them reveals only the harmonic one and not much more. (I know how... |
Let $Q$ denote the group of quaternions. Why is every subgroup of $Q \times \mathbb{Z}_{2}$ normal?
$Q= \{I,A,A^{2},A^{3},B,BA,BA^{2},BA^{3}\}$, I the identity matrix and $A=\begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}$ and $B=\begin{bmatrix} 0 & i\\ i & 0\end{bmatrix}$ where $i=\sqrt{-1}$.
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... Or perhaps, what are some interesting examples of simple graphs that are not known to be planar or non-planar?
|
I've been reading about Riemannian manifolds, and have come across a comment that says that for a metric $g$ on an $N$-dimensional manifold $M$, considered as a bilinear map
$$
g:\Omega^1(M) \times \Omega^1(M) \to C^{\infty}(M),
$$
there exists a canonically induced bilinear map
$$
g_k:\Omega^k(M) \times \Omega^k(M)... |
Is there a better way to deal with the series expansion, in particular? I'm interested in attempting to use the series expansion to estimate the value, which I will eventually combine with more complicated expressions.
So, to begin, I calculated the following expansion for this series (note that $n$ is an integer, $i$... |
Can anyone show some hints to this? If $\gcd(a,b)=1$, then $a^{\phi(b)}+b^{\phi(a)}=1 \pmod{ab}$.
I know that $a^{\phi(b)}=1 \pmod{b}$, and similarly, $b^{\phi(a)}=1 \pmod{a}$, but then how do I combine the work so that I get the result I need?
Thanks!
|
So from the fourier series, we can simplify it further and use trig identities to get the following:
$$ f(t) = \frac{a_0}{2} + \sum^{\infty}_{n=1}
\left(\frac{a_n}{2}+\frac{b_n}{2i}\right)e^{i n \omega t} +
\left(\frac{a_n}{2}-\frac{b_n}{2i}\right)e^{-i n \omega t} $$
So how do you go from the above line to
$$ f(t) =... |
Given an expression, for example,
$$f(x,y)=\frac{x}{(x-1)(1-y)(xy+x-1)}$$
How can we find the general expression involving $m$ and $n$ for $$\lim_{x\to 0, y\to 0}{\frac{\partial^{n+m}{f}}{\partial{x}^{m}\partial{y}^{n}}}$$
|
I was reading the proof that $A_{4}$ is the unique subgroup of order $12$.
So the author counts the number of conjugates:
$1$ cycle of type $()$ , $6$ cycles of type $(1 \ 2)$, 8 cycles of type $(1 \ 2 \ 3)$, $6$ cycles of type $(1 \ 2 \ 3 \ 4)$ and $3$ cycles of type $(1 \ 2)(3 \ 4)$.
Now it says, the only possible wa... |
This is the equation that I use to calculate a percentage margin between cost and sales prices, where $x$ = sales price and $y$ = cost price:
\begin{equation}
z=\frac{x-y}{x}*100
\end{equation}
This can be solved for $x$ to give the following equation, which calculates sales price based on cost price and margin percent... |
Does a local minimum of a function always satify the Armijo rule?
|
I am trying to learn differential geometry (i.e., teach myself!)
So here is a question that came up.
For some $h > 0$, consider the cone
$C_h = \{ (x,y,z) \; : \; 0 \le z = \sqrt{x^2 + y^2} < h \} \subset \mathbb{R}^3$
endowed with subspace topology. It seems that we can cover this with a single chart $(U,\phi)$ wher... |
The following is a step in a proof: $f$ is a Lipschitz function from $E$ to $F$ where $E$ is a finite-dimensional Banach space and $F$ an arbitrary Banach space. $\phi\geq 0$ is a $C^\infty$ function with compact support, $\int\phi=1$, $\phi(x)=\phi(-x)$. The function $g(z)=\int f(z+x)\phi(x)dx$ is defined. The claim... |
Let $f$ and $g$ be functions such that $f\colon A\to B$ and $g\colon B\to C$. Prove or disprove the following
a) If $g\circ f$ is injective, then $g$ is injective
Here's my proof that this is true.
Let $A = \{4,5\}$, $B = \{3,9\}$, $C = \{1,2\}$
$f(4) = 3$; $f(5) = 9$
$g\circ f(4) = 1$ and $g\circ f(5) = 2$
Since no ... |
Given a measure space $(\Omega, \mathbb{F},\mu)$ and any two measurable real-valued functions $f,g$ defined on $\Omega$, I was wondering if there is an inequality like
$$
\int_\Omega |f*g| d\mu \leq \int_\Omega |f| d\mu * \int_\Omega |g| d\mu?
$$
Or what are the correct relations between product and integral?
What i... |
Given a integer $n$, we want to know if $n=m^k$ for some $m$ and $k>1$. What is the fastest method? (Suppose the factors of $n$ are not given).
The best method I can think of is to try to calculate $n^\frac{1}{k}$ to a certain precision, for $k$ from $2$ to $\log_2 n$. Determine if $n^\frac{1}{k}$ is a integer by test ... |
This is a little exercise I've been fiddling with for a while now.
Let $f\colon X\to Y$ and $g\colon Y\to X$ be functions. I want to show that there are subsets $A\subseteq X$ and $B\subseteq Y$ such that $f(A)=B$ and $g(Y-B)=X-A$.
Of course, if $f$ is surjective, then taking $A=X$, one has $f(A)=B=Y$ and $g(Y-Y)=g(... |
It is well known that the solution to the classical Dido problem is a semicircle, and that the solution to the classical isoperimetric problem is a circle. It's also reasonably obvious that the solution to the following variant is a circular arc:
Let $A$ and $B$ be fixed points on a plane, and let $l$ be a length grea... |
I want to find parametric and symmetric equations for the line of intersection of the planes $x+y+z=1$ and $x+z$=0. Here's what I have so far:
I need to find a point on this line of intersection. So by setting z=0 in the equation of both planes, I get $x=0, y=1, z=0$. So $(0,1,0)$ is on the line. Now since this line is... |
I first saw this on the Missouri State problem page and it has never been solved there.
Consider the group generated by a,b,c, and d subject to the relations
ab = c, bc = d, cd = a, and da = b
Using the first relation, the second relation becomes bab = d. Using this expression and the first relation, we obtain
abbab = ... |
I am trying to refresh my knowledge (and hopefully learn more) about Algorithm Analysis. I took a course on this two years ago but I am trying to catch up on what I had learned back then.
The way I am going about it is by doing exercises from the classic text Introduction to Algorithms, 2nd Edition - CLRS.
The proble... |
Sophie Germain proved Fermat's Last Theorem $x^p+y^p \neq z^p$ for the special case where p is a Sophie Germain prime and $p\not|xyz$. Does any one know of a proof for the other case, where $p|xyz$? Note: I am looking for a proof restricted to the Sophie Germain primes, as of course, Wiles proved this generally.
|
How do I prove this by induction?
Prove that for every natural number n, $ 2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$
Here is my attempt.
Base Case: let $ n = 0$ Then, $2^{0+1} - 1 = 1$ Which is true.
Inductive Step to prove is: $ 2^{n+1} = 2^{n+2} - 1$
Our hypothesis is: $2^n = 2^{n+1} -1$
Here is where I'm getting off tra... |
This time I'm having trouble with the following exercise:
Being $|G| = 20$ and $H$ and $K$ subgroups of $G$ whose order is 5, prove that $K = H$.
I'm also recommended to start by proving that $|H \cap K| = 5$.
My draft:
Let $x \in (H \cap K)\setminus{\{e\}}$. Then $|x|$ divides $|H| = |K| = 5$. Because $x \neq e$, ... |
On p.3 of the first volume of Spivak's Comprehensive Introduction to Differential Geometry, he says that it is an "easy exercise" to show that the invariance of domain theorem (if $f:U\subset\mathbb{R}^n\rightarrow\mathbb{R}^n$ is one-to-one and continuous and $U$ is open then $f(U)$ is open) implies that in his defini... |
I know how to handle the 2d case: http://www.proofwiki.org/wiki/Set_Difference_of_Cartesian_Products
But I am having trouble simplifying the following:
Let $X=\prod_{1}^\infty X_i, A_i \subset X_i$
How can I simplify/rewrite $X - (A_1 \times A_2 \times \cdots A_n \times X_{n+1} \times X_{n+2} \cdots)$ with unions/inter... |
I have got a proof of $det(AB)$=$(detA)(detB)$ in my book.
It goes as follows (for invertible A): we know that rref[A|AB]=[$I_{n}$|B]
We also know that det(A)=$(-1)^{s}k_{1}k_{2}...k_{r}$ where s is the number of row swaps needed to get to the rrefA, and $k_{i}$ are coefficients by which we divide rows of A to get to r... |
I'm a very, very logical person -- but I've never taken any math classes; really, I know.
So, on that note -- how do I teach myself math? As my question might suggest, I see the answer as being an online reference that chains mathematical knowledge from simple to complex -- but I'm open to other suggestions. One thing ... |
The BNF is defined as followed:
S -> aAb | bBA
A -> ab | aAB
B -> bB | b
The sentence is:
aaAbBb
And this is the parse tree:
Phrases: aaAbBb, aAbB, bB
Simple Phrases: bB
Handle: ?
From the book, handle is defined as followed:
B is the handle of the right sentential from y = aBw if and only if:
$S ->_{rm} \cdot ... |
Let's say we have
$${\bf A} = {\bf B} {\bf X}$$
where $\bf A$ and $\bf B$ are known matrices, $\bf X$ is unknown. In case $\bf B$ was square, a solution can be found by ${\bf B}^{-1} {\bf A} = {\bf X}$. But how do you attempt to solve for $\bf X$ when $\bf B$ is not square, i.e., $n \neq m$?
|
This is for a graduate level course on computational linear systems. I want to do it myself, without help, but it's been ten years since I took linear algebra and I'm not sure I understand what the first part is asking for. The problem states:
Find a solution $x = x^* = (x_1^*, x_2^*, x_3^*, x_4^*)^T$ to the LLS proble... |
I understand what it means for a prime number to ramify in a ring of integers of a number field. However, an infinite prime is an archimedean valuation, what does it mean for an archimedean valuation to ramify in a number field?
|
I'm having trouble with this real analysis/optimization problem. The result seems intuitively obvious, but I don't know how I could possibly formalize it. Any advice is appreciated:
Let $X \subset \mathbb{R}^n$, $Y \subset \mathbb{R}$, $f:x \rightarrow Y$ and $g:: Y \rightarrow \mathbb{R}$.
Suppose $X$ is a compact s... |
Let $f$ be a degree $n$ univariate polynomial in $Z[x]$, defined as $f = \sum\limits_{i=0}^{i=n} f_i x^i$. Let $$g = f^2 = f \cdot f = \sum\limits_{k=0}^{k=2n} g_k x^k$$ be the square of $f$. I am trying to find an expression for the coefficients $g_k$. So far, I have the standard expression: $$g_k = \sum\limits_{i+j=k... |
Is there any way to solve the inequation: $${e^{2x - 1}} \ge 2x$$ without plotting the graph?
|
Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal?
Thanks for any clarification, and this is an exercise from the notes of Commutative Algebra by Pete L Clark, of which I thought as simple but I cannot figure it o... |
If two matrices from GL(2,Z) have the same determinant, will there always be a matrix from SL(2,Z) which transforms one matrix to the other?
|
The problem is: write 101! in canonical form.
I solved it by try to find the highest power of each factor prime factor less than 101. When running factor() function using TI-89, I realized that the power of each prime factor decrease.
So I guess I have two questions:
How can write this idea using Math notation? I'm t... |
I need to find a counterexample to disprove the following statement:
for upper semi-continuous function $g: \mathbb{R}^n \rightarrow \mathbb{R}$, the set $\{x \in \mathbb{R}^n: g(x) \leq 0\}$ is closed.
|
I've found some time to read a little more on set theory, and I've come across the following question.
Suppose I have four sets $X$, $Y$, $Z$, and $W$ such that $Y\subseteq W$ and $Z\subseteq X$. Suppose also that $X\cup Y\sim Y$, where by $\sim$ I mean that the two sets $X\cup Y$ and $Y$ are equinumerous. How can I s... |
I know about the structure theorem for finitely generated abelian groups.
I'm wondering whether there exists a similar structure theorem for abelian groups that are not finitely generated. In particular, I'm interested in torsion groups. Maybe having a finite exponent helps?
|
Premise: I'm a bit unsure about the terminology most frequently used, so I'll try to be as clear as possible, please tell me if something isn't clear.
Talking about regular smooth manifolds, by which I mean function $P: E \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}^3$ such that $P_u \times P_v \not\equiv 0 \; \forall... |
Perhaps this is a very basic question, but I can't find a fast solution.
I have a cyclic interval [0,n] (distance from n to 0 is 1). I need a function that for a given value x returns the point in the interval where it maximizes the distance to all previous points (x-1, x-2...)
For instance, if n=100 and f(0) = 0
f(1) ... |
I wonder how $qvq^{-1}$ gives the rotated vector of $v$.
Is there any easy-to-understand proof for it?
I was on Wikipedia, but I could not understand the proof there
because of the conversions.
Why is $uv-vu$ the same as $2(u \times v)$, and why is $uvu$ the same as $v(uu)-2(uv)u$?
|
Suppose $h \in \mathbb{N}$, is there a known non trivial upper bound for
$$\left| \frac{1}{n} \sum_{m=1}^n e^{2 \pi i h (2 \pi m)} \right|?$$
|
A matrix $A \in R^{n\times n}$ is said to be orthogonally equivalent to $B\in R^{n\times n}$ if there is an orthogonal matrix $U\in R^{n\times n}$, $U^T U=I$, such that $A=U^T B U$. My question is what kind of matrices are orthogonally equivalent to themselves? i.e.,
$A=U^T A U$
A similar interesting question is: if
$... |
Transport equation:
$$
\begin{cases}
\frac{d}{dt}u +cu = 0\qquad \mbox{ in } \mathbb{R}^n\times (0,\infty) \\
u(x,0) = 0 \qquad \mbox{ on }\mathbb{R}^n\times \{t=0\}
\end{cases}
$$
b-const
|
Is it possible to create an "antigroup"? What I mean by this is, given some group G, and some "antigroup" H, then the "free product" of G and H, G*H will equal the "group" (vacuously a group) of no generators and no relations. Can I construct such a thing? It makes me think that I need "anti-generators" and "anti-relat... |
Given a function series, what are
the ways often used to find the
function that the series converges
to?
For example, how do you construct
the function on the right hand side
from the series on the left hand
side: $$\sum_{n \in \mathbb{N} \cup \{ 0\}} (\begin{array}{c} 2n \\ n
\end{array}) x^{n}=(1-4x)^{-1/2}$$ I... |
Matsumura mentions this as if it is obvious, and I can't find this result anywhere. Am I missing something obvious here?
|
I am working through L.C.Evans' Partial Differential Equations -- the chapter on second-order elliptic equations.
I have got a general question on symmetric vs. non-symmetric elliptic operators. Consider an operator of the form
$\displaystyle Lu = \sum_{i,j}a^{ij} u_{,ij} + \sum_i b^i u_{,i} + cu$
In his book, Evans ma... |
using sum of geometric progression and inverse DFT we can show:
$${\sum_{j=0}^{k-1}\frac{\exp(2\pi ij\frac{u+1}{k})}{\exp(2\pi i\frac{j}{k})-a}=k\frac{a^{u(mod\ k)}}{1-a^{k}}}$$
My question is how to compute
$${\sum_{j=0}^{k-1}\frac{\exp(2\pi ij\frac{u+1}{k})}{(\exp(2\pi i\frac{j}{k})-a)^m}}$$
here $m,u,k$ are natural ... |
Let $g$ a Lie algebra and $V$ a finite-dimensional irreducible $g$-module, then each generalized eigenspace of $V$ is actually an eigenspace? If not, what is a condition to guarantee this fact?
Thanks!
|
Let $f(t) = \sum_{i = 0}^{n} c_i \ t^i$ be a degree $n$ polynomial. In the binomial basis, we can write $f(t) = \sum_{k = 0}^{n} h_k \ \binom{t + n - k}{n}$. Using the calculus of finite differences, I can prove that the coefficient $c_i$ can be computed from $\{ h_0, \dots, h_n \}$ by the equation
$$c_i = \frac{1}{n!... |
When solving an equation with irrational (or algebraic in the case) powers, are the roots likely to be transcendental or algebraic, or does it vary?
As an example, I was trying to figure out if $(x + 1)^{\sqrt{2}}=x^2 - 2 x + 2$ had an exact solution. I tried it myself and then let Wolfram Alpha work on it, only gettin... |
How do you determine the different odds in Blackjack? For example, what would be the difference in odds from using 1, 2, or 3 decks? Also, what would be the difference in odds if you shuffle the deck before each hand instead of after the cards are used up?
|
The most basic example of a topologically non-trivial real line bundle is the well-known Möbius strip. Everyone who learns about vector bundles will be confronted by it, if only because it has the distinguished advantage that we can draw a picture of it.
I would like to draw pictures of other line bundles, too. In par... |
I need some help understanding why $\frac{x}{x+1}=\frac{1}{x^{-1}+1}$.
I would be grateful if someone could explain. Be explicit.
Thank you!
|
The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school geometry classes even today.
What I'm getting at is this: Are the rules of construction just arbitrarily imposed restrictions,... |
My question is the following
How can I get point $(x_3, y_3)$ from points $(x_1, y_1)$ and $(x_2, y_2)$ ?
The distance of point $(x_3, y_3)$ from $(x_1, y_1)$ is $300$.
|
Assume that I have a large, non-symmetric matrix $\bf{A}$ of zeros and ones. If I want the exact diagonal elements of $\bf{A}^{k}$ (not the trace, but the elements themselves) where $k$ is some large whole number, I can
Multiply the matrices out using with exact integers, possibly speeding the computation by using som... |
Let $\pi: X \times Y \to X$ be the projection map where $Y$ is compact. Prove that $\pi$ is a closed map.
First I would like to see a proof of this claim.
I want to know that here why compactness is necessary or do we have any other weaker condition other than compactness for the same result to hold.
|
$\begin{align*}
\lim_{x\to 0}\frac{\frac{1}{\sqrt{4+x}}-\frac{1}{2}}{x}
&=\lim_{x\to 0}\frac{\frac{2}{2\sqrt{4+x}}-\frac{\sqrt{4+x}}{2\sqrt{4+x}}}{x}\\
&=\lim_{x\to 0}\frac{\frac{2-\sqrt{4+x}}{2\sqrt{4+x}}}{x}\\
&=\lim_{x\to 0}\frac{2-\sqrt{4+x}}{2x\sqrt{4+x}}\\
&=\lim_{x\to 0}\frac{(2-\sqrt{4-x})(2+\sqrt{4-x... |
Let $A$ and $B$ be two groups and let $G=A*B$ be their free product. Suppose both $A$ and $B$ are nontrivial. I want now to prove that $G$ is infinite and that $Z(G)$ is trivial.
I think I can prove that $G$ is infinite using this argument: two reduced words of $A*B$ are equal if and only if they have the same length a... |
Let $\mathfrak {so}_{n}$ denote the skew-symmetric complex $n \times n$-matrices and let $M$ denote the symmetric $n \times n$-matrices of trace 0.
As I understand, $M$ is a module over $\mathfrak {so}_n$. What then is its decomposition into irreducibles?
The standard representation of $\mathfrak {so}_n$ has dimension... |
We usually say that a graph is planar if it can be embedded into 2-space s.t. no edges intersect. Here's a different way to describe the same situation: a graph is planar if it can be embedded into 2-space s.t. the edges form the boundaries of cells which nowhere overlap.
My question: Can you take this second definiti... |
Let the heat kernel on $(0,\infty)\times \mathbb R^n$ be given by
$\Psi(x,t) = (4\pi t )^{-\frac{n}{2}} e^{ -\dfrac{|x|^2}{4t} }$
for $t>0$, otherwise $0$ except at the origin of space-time.
It is clear that the heat kernel is smooth everywhere except at the origin. Let $\delta > 0$. How can you show that ( the absolu... |
I don't need to list all the proper divisors; I just want to compute their sum. While checking and summing up all proper divisors isn't an issue for small numbers, it becomes significantly slower for larger numbers. Any suggestions? Thanks!
|
When considering an infinitesimal distance/interval/in calculus, what is the intuitive interpretation? Is it too small to be measurable but still has some distance on an unattainable scale? Are there different interpretations? If so, what I am considering for the time being is the interpretation in calculus, but I'm st... |
How could I find all the pairs $(n, k)$ for this equation. The most obvious pair solution that I can see is $(1, 1)$.
Using summation identity, I have:
$$\frac{n(n+1)}{2} = \frac{k(k + 1)(2k + 1)}{6}$$
Then I thought of using cubic formula for $k$-equation, but it involved many variables. Any idea?
Thanks,
Chan
|
If there exists such an $n$ then $n$ must be the highest of power of 2 that divides $n!$. But to find out the highest power we need to increase n for each step until it reaches $n$. Any idea?
Thanks,
Chan
|
Let $H$ be the Hawaiian earring and let $H'$ be the reflection of the Hawaiian earring across the $y$-axis (in the Wikipedia picture). There is a canonical homomorphism from the free product $\pi_1(H) * \pi_1(H')$ to $\pi_1(H \cup H')$ (with basepoint their intersection), but it is not an isomorphism.
This was intended... |
I'd like to know why inner products in Reproducing kernel Hilbert spaces are (linear) evaluation functionals.
I understand that inner products are linear functionals, and I know what an evaluation functional is, I just can't explain why an inner product (in a RKHS) is evaluation functional, and vise-versa.
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I am unfamiliar with the general process of converting NFA to DFA. I have general understanding of the theory, but I don't have the method established. Please help explain the process required to transform an NFA to DFA. Thank you.
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Possible Duplicate:
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
Given an odd prime $p$, how does one find the highest power of $p$ that divides
$$\displaystyle\prod_{i=0}^n(2i+1)?$$
I wrote it down all paper and realized that the highest power of $p$ that divides ... |
I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is.
The definition is basically this: "A matrix $A$ is orthogonally diagonalisable if there exists a matrix $P$ which is orthogonal and $D = P^tAP$ where $D$ is diagonal". I d... |
eg. $1000$ has $16$ factors $(1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000)$
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I want to compute the dimension of $\mathbb{A}_{\mathbb{C}}^{1}$, that is the dimension of the affine space in 1 dimension over the field $\mathbb{C}$ but with respect the $\textbf{Euclidean}$ topology.
My definition of dimension of a topological space $X$: is the greatest upper bound of the set of all ascending chains... |
Certainly is not a circle; it looks like an ellipse but I don't think it is.
Thanks
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What does it mean to say a function is differentiable with respect to the lebesgue measure almost everywhere. A definition would be helpfull.
Do I need to learn about the Radon Nikodym derivative to understand this.
Thanks
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Lets say we construct a list as follows.
It has f(n) 1's, and f(2) 2's and f(3) number of three's in it etc.
Let L(n,f) be this list, so if f(n)=n^2 we get L(3,n^2)=(1,2,2,2,2,3,3,3,3,3,3,3,3,3)
If f(n) is such that f(m)>(sum f(n) n=1 to n=m-1). Then the probability to pick the number n from the list L(n,f) is above 1/... |
I've been thinking about this for awhile now (I am trying to find a method of proving the Borsuk-Ulam theorem in 2 dimensions without resorting to the usual, and not so intuitive to non-mathematicians topological arguments). I saw the question Does a continuous scalar field on a sphere have continuous loop of "isother... |
$w$ is a complex number that satisfy:
$$5{w^3} - 3i{\left| w \right|^2} - 2i=0$$
Can we find $|w|$ without finding $w$ or easy root?
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