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Let $G$ represent an undirected graph, let $a$ and $b$ represent vertices and let $k$ represent a non-negative integer.
I have two languages:
$L_1 = \left \{ \left \langle G, a, b, k \right \rangle \mid G \text{ contains a path of at most } k \text{ from } a \text{ to } b \right \}$, and
$L_2 = \left \{ \left \lan... |
I am trying to prove that the sequence $I_n(p)=\int_1^n \frac{dx}{x^p}$ does or does not converge uniformly to the function $I(p)=\int_1^\infty \frac{dx}{x^p}$ for $p>1$.
It seems to me that $\sup |I_n(p)-I(p)|= \lim_{p \to 1} (I(p)-I_n(p))$. Is it enough, then, to show that for all $\epsilon>0$ there exists a natural ... |
...and if it's important, do those ideas have any generalization to more "exotic" number systems?
The motivation for my question comes from reading some of the excellent answers posted to other questions, such as a recent one asking whether $\sqrt{1 + 24n}$ always yields primes. In particular, I've been struck by comme... |
If I understand correctly, the definition of the degree of a field extension $L/K$ is the dimension of $L$ over $K$ interpreted as a vector space. Now if the degree is $n < \infty$, the basis looks like $\{ 1, a_1, \dots , a_n\}$ where $a_i \in L - K$.
My question is:
Is it right to write $\{ 1, a, a^2, \dots , a^n\}$ ... |
I am in a biological field (medicine) but I have genuine passion for mathematics. I want to learn it on my own , in my spare time. Mathematics , as I gather, is learned best when you have grasped the prerequisite concepts for the area you are currently interested in. Kindly suggest a sequence of study for the follow... |
What's the usual method for proving the criticality of a graph?
I've been trying out different methods and theorems but I can't find a decent method that's really convincing.
Thanks a lot in advance!
|
Is there a way to construct a linear feedback shift register (LFSR) which outputs no more than k consecutive 1s or 0s? (It would have to be not a maximal-length LFSR, of course, if k < the degree of the characteristic polynomial)
edit: I am also looking for one where the period is at least 1/16 the maximum period of th... |
We want to reparametrize the curve
$$\displaystyle \vec{r}(t)=<t^3+1, t^2-1, \frac{\sqrt{5}}{2}t^2>$$
in terms of the arc length measured from the point t=0 in the direction of increasing t.
Here is what I tried, but I've hit a snag:
$$\displaystyle \vec{v}(t)=<3t^2, 2t, \sqrt{5}t>$$
$$\displaystyle |\vec{v}(t)|=3t\s... |
From Hatcher, "Algebraic Topology," Chapter 2, "Singular Homology" section (p. 108-109 in my copy):
Cycles in singular homology are defined algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from finite $\Delta$-complexes. To see this, note that a singular $n$-chain $\xi$ ... |
So I have been studying the multi-variable chain rule. Most importantly, and this is what I must have overlooked, is it's not always clear to me how to see which variables are functions of other variables, so that you know when to use the chain rule. For example, if you have:
$$ x^2+y^2-z^2+2xy=1 $$
$$ x^3+y^3-5y=8 $... |
For some reason I cannot figure out how the book is finding the solution to this problem.
find the inverse function of $f(x)=3x\sqrt{x}$.
My steps seem to lead to a dead end:
step 1. switch $f(x)$ with $y$: $y = 3x\sqrt{x}$
step 2. swap $x$ and $y$: $x = 3y\sqrt{y}$
step 3. solve for $y$:
step 3.1: $\displaystyle \fra... |
How would you compute for the definite integral of
$$\int_0^{\infty}\frac{dx}{(1+x^2)^4}$$
I know that integral of $\displaystyle \frac1{(1+x^2)}$ equals $\tan^{-1}x$. I tried using integration by parts without much luck. My teacher pointed me to special functions by which I found out about the hypergeometric distrib... |
Let $x_i$ be iid nonnegative discrete random variables $E[x_i]=N/M$ for some integers $N, M$, variance $\sigma^2$ and higher moments known (finite).
Then, the sum $\displaystyle S = \sum_{i=1}^M x_i$ will have $E[S]=N$.
I'm interested in the probability that
$S$ takes that precise value: $A=P\left(S=E[S]\right)$.
Ap... |
Consider the following diagram:
What does it mean precisely to say "$f$ factors through $G/\text{ker}(f)$"?
Does it mean $f = \tilde{f} \circ \pi$, for some $\tilde{f}$?
I've seen texts use the phrase, but never a definition of this notion.
|
I recently found out that there is no algorithm which, given an arbitrary group presentation, will determine in finite time if it represents the trivial group*. Additionally, in a lecture I recently attended, it was portrayed that given any desired property P of a group, there is no algorithm that will determine if an... |
It is well-known that any set $E \subseteq \mathbb{R}$ with positive outer measure contains a nonmeasurable subset $V$. I know that $0 < m^*(V) \le m^*(E)$. Nevertheless, my question is the following: given $r \in \mathbb{R}$ such that $r>0$, is there a nonmeasurable subset of $\mathbb{R}$ whose outer measure is exactl... |
I need to simplify the following equation, by completing the square:
$$\frac{\left(x-z\right)^{2}}{2(u-s)}+\frac{(z-y)^{2}}{2(t-u)}$$
As $\displaystyle\frac{\left(x-y\right)^{2}}{2(t-s)}+C$.
How can I do this? I can't seem to be able to deal with the fractions effectively when completing this square...
|
Suppose there is a family (can be
infinite) of measurable spaces. What
are the usual ways to define a sigma
algebra on their Cartesian product?
There is one way in the context of
defining product measure on
planetmath. Let $(E_i, B_i)$ be
measurable spaces, where $i \in I$
is an index set, possibly infinite.
We defin... |
If for each $t\in I=[0,1]$ I have a measurable space $(X_t,\Sigma_t)$, is there a standard notion which will give a measurable space deserving to be called the integral $\int_I X_t\,\mathrm d t$?
Motivated by this question and curiosity...
|
Let $C_1, C_2$ be two concentric circles in the extended complex plane. Is it true that if another "circle" $C$ is orthogonal to both $C_1$ and $C_2$, then $C$ must be a line?
I think that this should be true, because at the point of intersection between $C_1$ and $C$, the tangent of $C$ must intersect the center of $C... |
I don't know the proper terms to type into Google, so please pardon me for asking here first.
While jingling around a few coins, I realized that one nice puzzle might be to figure out which $n$ or so coins of a currency (let us, say, use the coins of the American dollar as an example) can be used to total to a given am... |
I've been trying to wrap my head around the basic concepts of category theory, and I thought I would attempt to illustrate what I understand with the category of sets, probably the easiest example. Particularly, I've been trying to prove that $id_A$ (the identity morphism on $A$, for all $A \in Obj(\mathbf{Set})$) is $... |
Let $x$ and $y$ be continuously differentiable real-valued functions defined on $\mathbf{R}$ and suppose that $x'(t) = f\,(x(t),y(t))$ and $y'(t) = g(x(t),y(t))$ where $f$ and $g$ are continuously differentiable on $\mathbf{R}^2$. According to the ODE book I am reading, if $f$ is never 0, then
\[
\frac{dy}{dx} = \frac... |
I have a quite small data set (on the order of 8-20) from an essentially unknown system and would like to predict a value that will be higher than the next number generated by the same system 90% of the time. Both underestimation and overestimation are problematic.
What is the mathematically "correct" way to do this?
I... |
A flat two-torus, $T^2$ that is the torus with Euclidean metric needs to be embedded at least in $\mathbb{R}^4$. If we puncture the torus and leave the Euclidean metric on it as inherited (ignoring the issues of completeness), what ambient space could you embed the punctured torus with Euclidean metric?
|
I'm learning about sufficient statistics and I understand basic stuff that it is the minimum information we need to represent a statistic (very vague definition, I know. I'm learning). I'm trying to solve problems in that. The problem is suppose a sample of size $n$ is taken from the geometric distribution with paramet... |
I want to evaluate the following:
$$\lim_{r \rightarrow 0}\frac{-r^2}{2 \left(\sqrt{1-\frac{r^2}{4}}-1 \right)}$$
I look at the graph and see that it seems to be going to zero. This makes sense to me because if I replace r with zero this function is defined and continuous near zero and the value of the function is zero... |
I'm primarily a programmer, so forgive me if I don't know the proper nomenclature or notation.
Last night, an old teacher of mine told me about a question that had caused some noodle-scratching for him:
For any two sequences of consecutive integers, can the sums of their reciprocals be equal?
Now, I gather that these s... |
Is there any reason that this generalization of the mean value theorem would fail?
Let A be a subset of Rn that is differentiably connected, and let f : A --> R be continuously differentiable at every point in A. If x and y are two points in A then there exists a point c in A such that
f(y)-f(x) = (grad(f(c)))*(y-x)
It... |
This question was inspired by the question on examples of classes that are not sets.
From the discussion in the comments there, it seems there does not exist a set of all sets of a given cardinality.
To me, it seems easy to see that this is true for any nonzero finite cardinality $n$. Given any set $x$, you can always ... |
Let $\mathbb{Q}^{alg}$ be the algebraic closure of the rationals. Given a point $P\in \mathbb{A}^n(\mathbb{Q}^{alg})$, $P = (a_1,\dots,a_n)$, we define the degree of $P$ to be the degree of the minimal field extension of $\mathbb{Q}$ over which $P$ is defined: $\text{deg}(P) = [\mathbb{Q}(a_1,\dots,a_n):\mathbb{Q}]$.
I... |
I'm having trouble keeping the definitions straight, so to help me remember the definitions: is it correct to say that a function is:
Injective (one-to-one) iff it is not many-to-one.
Surjective iff its codomain and range are equal.
?
|
I saw definitions and theorem about power series are in the form of $\sum_{k=0}^n a_k (x-x_0)^k$. And it definitely doesn't include negative or noninteger powers. Nevertheless, I saw the theorems like
If the two series converge to the same value on some interval, then the corresponding coefficients are the same.
The ... |
The bound formula is:
$$H(M,N,n,k) \leq (( \frac{p}{p+t} )^{p+t} (\frac{1-p}{1-p-t})^{1-p-t})^n$$
So how to proof it?
Some papers discuss it:
this one,
that one.
They say the proof is in the following paper:
The tail of the hypergeometric distribution. Discrete Mathematics
But I can not find the online version.
|
Could someone help me with the proof that a square matrix has maximal rank if and only if it is invertible?
Thanks to everybody
|
I have a matrix of the following form:
$
\begin{matrix}
a_1 & 0 & \ldots & 0 \\
a_2 & a_1 & \ldots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
a_n & a_{n-1} & \ldots & a_1
\end{matrix}
$
I'd like to know if this matrix has a special name (or even if it's a sum/product of special matrices).
The thing is, I need t... |
I was following an example in my text, and there was one step I got stuck on.
Given $\frac{2x(\Delta x) + (\Delta x)^{2}}{\Delta x}$, how does the denominator cancel out to produce $2c+\Delta x$?
My thinking is that the formula expands to $\frac{2x(\Delta x) + (\Delta x * \Delta x)}{\Delta x}$, but how does the denomin... |
Let $X$ be the following topological space (with the subspace topology): Connect the rational points of $([0,1]\cap \mathbb{Q})\times \{0\}$ with the point $(0,1)$ and connect the points of $([-1,0]\cap \mathbb{Q})\times \{1\}$ with $(0,0)$, as shown in the figure. Is $X$ contractible?
|
On Planetmath, product measure
is roughly defined as follows:
Let $(E_i, \mathbb{B}_i, u_i)$ be
measure spaces, where $i\in I$ an
index set, possibly infinite.
When each $u_i$ is totally finite,
there is a unique measure on the
product measurable space $(E,
\mathbb{B})$ of $(E_i, \mathbb{B}_i,
u_i)$. such that "taking... |
I'm trying to find the root space decomposition of a lie algebra wrt a toral subalgebra h. Both a matrix lie algebras. I'm confused about how do I find the linear forms $\lambda \in \mathfrak{h}^*$ on $\mathfrak{h}$.
$\mathfrak{g}_{\lambda } {\text{:=} \{ X\in \mathfrak{g}|\text{ad}_H(X).=\lambda (H)X \forall H\in \m... |
Given a random vector, what are the
domain, range and sigma algebras on
them for each of its components to
be a random variable i.e. measurable
mapping? Specifically:
is the domain of each component random variable same as the domain
of the random vector, and are the
sigma algebras on their domains
also the... |
Why does a smooth surjective morphism of schemes admit a section étale-locally?
|
For $A=\mathbb{Z}[x]/(f)$ with quotient field $K$ and ring of integers $B$, does $U(B)/U(A)$ have a name?
For instance $u = \tfrac{1+\sqrt{5}}{2}$ is a unit in $\mathbb{Q}[\sqrt{5}]$, but neither $u$ nor $u^2$ has integer coefficients in the basis $\{ 1, \sqrt{5} \}$. Of course $u^3$ has integer coefficients (spooky... |
I'm stuck on this question from a past paper.
A parachutist jumps out of a hot air balloon. The resistance is proportional to the speed of the parachutist. The parachutist has a terminal velocity of 4 ms-1
How far did he fall after 3 seconds.
I'm really not sure how to go about this question, any help much appreciated.... |
Markov's inequality states that $Pr(X \geq tE(X)) \leq 1/t$. This is great for asking "What is the probability that we get more than t times our expected value". However, if rather than more, we want to ask what is the probability that we get LESS than our expected value, how is this handled?
Say our expected value i... |
I have $m$ continues integer points on a line, randomly uniform select $n$ points from the $m$ point without replacement. Order the points ascendingly.
Let the random variable $A_i$ is the position (coordination on the line) of the $i$th point. So, $$P(A_i=k)=\frac{{k-1\choose i-1} {m-k \choose n-i}}{{m \choose n}} $$... |
How do you describe the normalization of $\mathrm{Spec}(k[[x,y]]/(xy))$? In particular, call this $N$. Why can $N$ be written as the intersection of 2 local rings and the same holds for the maximal ideal (with 2 maximal ideals)?
|
I'm trying to solve coordinate related problem.
I have a large set of X,Y coordinates (logged by the GPS device during circuit racing, the driver makes 10-20 laps during one racing session) and a vector (a line that marks start/finish of the circuit/lap).
I need to divide large set of coordinates into sub-sets using ... |
I have two real vectors $v = (v_1,\ldots,v_n)$ and $u = (u_1,\ldots,u_n)$.
I know that the dot product of $u$ and $v$ is larger than $\delta > 0$:
$\langle u,v \rangle \ge \delta$.
What would be an interesting condition on $u$, $v$ or both such that I have $u_i v_i \ge f(\delta)$ for each coordinate $i$ with some real... |
This sort of question is probably in bad taste for math.stackexchange, but is probably in high demand. (I tried to start a site on Area 51 to house questions like this, but my request was closed due to the existence of math.stackexchange.)
I am advising talented students about graduate school. I believe that the most i... |
Let $X$ be a topological space, $Y$ and $Z$ subspaces of $X$. Let $C$ be a connected subset of $Y\cap Z$ such that $C$ is a component of $Y$ and a component of $Z$. Does it follow that $C$ is a component of $Y\cup Z$?
Intuitively, I would say yes, but I don't know how to prove it.
In case further assumptions are neces... |
What exactly does this notation mean when using it to define a function?
$$f(x) = f(\text{ sum of the digits of }x) \{
\mathrm{if}( x \ge 10) \} = x \{ \mathrm{if}( x \lt 10) \} $$
Thanks
|
My apologies if this comes out as stupid.
Problem (Domain : stock exchanges):
I have been given the sum of orders placed per instrument(security or company) for about 25000+ instruments. So my dataset looks something like this:
[Instrument] [Order count] [Average Order Size(AOS)] [Orders above 10*AOS]
AAA... |
I have a question more about terminology than anything else: If $f:X\rightarrow Y$ is an isomorphism of schemes, then what does it mean to say "the invertible sheaf $\mathscr{L}$ on $X$ corresponds to the invertible sheaf $\mathscr{M}$ on $Y$?" Naturally, it makes sense to assume that this statement means $f_{\ast}\m... |
How do you prove that $\sin(n)$ with $n=0,1,2...$ is not uniformly distributed mod 1?
(This is an exercise in Uniform Distribution of Sequences by Kuipers and Niederreiter.)
|
I have a module M over some commutative ring R. A square matrix $\Omega$ of size $m$ with coefficients in $R$ defines a $R$-linear map from $M^m$ to itself.
Proposition. If $x = \left( \begin{matrix} x_1 \\ \vdots \\ x_m \end{matrix} \right) \in M^m$ is in the kernel of $\Omega$, then $\det \Omega$ annihilates all the... |
i am developing an application for iPhone, and i need some help to solve this difficult problem.
I am working with the device compass, from where i get the angle where the iPhone is oriented.
I need to check what points are on a angle view(oriented with the compass) from the current location.
I have no idea how to mak... |
On the Wikipedia article for codomain, in the third paragraph, it roughly says:
When the domain of a function is a
proper class X, in which case there is
formally no such thing as a triple (X,
Y, F). (?) With such a definition
functions do not have a codomain.
As a proper class is a class that cannot be a member of ... |
I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following
$$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$
Now I'm wondering what this limit really says, it must be stronger than pointwise convergence since we can pick $M = 0$. ... |
I need to find a point (A on this diagram) given the center point of the ellipse as well as an angle. I've been melting my brain all day (as well as searching through questions here) testing out different equations. What's the best way to do this?
I intend to grab point A at $225^o$ as well as another point at approxi... |
How can set theory and category theory both be plausible theories for the foundations of mathematics? If these two theories are not mathematically equivalent, does it not mean that the rest of mathematics, when taken exclusively with either of these two theories as foundation, will be distinct; that is, unless these tw... |
I am still studying the topics in forcing and did not yet study much about forcing with a class of conditions.
I know from Jech's Set Theory that you can force that the class of ordinals in the world will be countable in the generic extension, which means that you can take a proper class and turn it into a set.
I was w... |
I was wondering if there is a name for a function that satisfies the conditions
$f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$?
Thanks and regards!
|
Came across this conundrum while going over the proof that
$$A \cdot \sin(bx) + B \cdot \cos(bx) = C \cdot \sin(bx + k)$$
for some numbers $C$ and $k$. ($A$, $B$ and $b$ are known.)
The usual method is to expand the RHS using the compound angle identity
\begin{align}
C \cdot \sin(bx + k) &= C \cdot \bigl( \sin(bx)\cos(... |
In the link http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse, it talks about solving $Ax=b$ by $x = A^+b + [I − A^+A]w$ for any vector $w$. Let's say $A$ is $m\times n$, and $b$ and $x\in \mathbb{R}^n$. My question is: since $w$'s $n$ components are in general more than needed given the kernel of $A$ mi... |
Let $F\subseteq\mathbb{C}$ be a Galois extension of $\mathbb{Q}$ such that $[F:\mathbb{Q}]=2^n$; then all elements in $F$ are constructible.
Added.
Here is what I have so far.
Since there exist the finite normal extension $F\subseteq\mathbb{C}$ over $\mathbb{Q}$ which contains an element say, s, then s has a degree ... |
So we have an equation $x^2 + By^2=z^2$, and we're supposed to find for what values of $B$ this graph would show two intersecting planes. That value is $0$. Now I need to find the equation of these two planes. I'm not really sure how to start this one. I'm very shakey on this stuff. Any hints?
|
For a homework assignment, I was asked to find the derivative of the function $$f(x)= \frac{x^{3}-6x^{2}+6}{x^{2}}.$$
Using the product rule, I worked through the following:
$$f'(x)=\frac{\frac{d}{dx}(x^{3}-6x^{2}+6)}{x^{2}}+(x^{3}-6x^{2}+6)\left[\frac{d}{dx}\frac{1}{x^{2}}\right]$$
Since the derivative of $\frac{1}{x{... |
How to solve this limit?
$$\lim_{x \to \infty}{\sqrt[5]{x^5 - 3x^4 + 17} - x}$$
|
Consider the plane defined by $2x+3y+5z=30$. Is the line parametrized by $l(s):=(-1,0,2)+s(-4,-12,18)$ for all s orthogonal to the plane?
So I know this is easy, but I'm missing one key piece. If the normal vector of the plane cross the line is equal to zero, then this confirms the line is indeed orthogonal. But now ho... |
I'm looking for a module $E$ of a commutative ring $A$ with two submodules $E_1$ and $E_2$ such that the associated primes of $E_1 + E_2$ strictly contain the union of the associated primes of $E_1$ and $E_2$. There's a short exact sequence
$$
0 \to E_1 \to E_1 + E_2 \to (E_1 + E_2)/E_1 \to 0
$$
where the last term i... |
For some reason I can't seem to see why
$$\frac{az+b}{cz+d} = \frac{bc-ad}{c^2\left(z+\frac{d}{c}\right)} + \frac{a}{c}.$$
Its been so many years since I've taken Algebra I, so when this was stated in class, without even the motions of manipulation, I felt lost. Sorry if this question is too dumb to belong on the site.... |
The following has come up during an undergraduate research project. It's rather elementary, but we are having trouble finding anywhere that mentions it. It might have something to do with rigidity.
Consider two tetrahedrons that share a base. What is the relationship between the lengths of its sides and the diagonal? W... |
Given a subset of the hyperbolic upper half plane, say an ideal triangle (so with vertices on the boundary), what is the cardinality of all points contained in the interior?
|
Is there a central wiki or something where open questions (and relevant research on them) takes place?
|
Suppose you have a set of twelve distinct objects, and you were interested in ways of arranging them into a 3 by 4 rectangle. You wonder: does it matter if I arrange them in 3 bunches of 4 or 4 bunches of 3?
Label them by primes. (I could, and probably should write $a_{2}$ in the rest of this post, but I don't feel lik... |
What is the definition of an envelope of a function ? For example if we multiply a sinusoid function of certain frequency ($1/f$ < support of bump) with a bump function we get a function whose envelope looks like bump function. The points where the envelope (bump here) and the given function (the product of sine and bu... |
Suppose I show that:
$$x^{f(z)/g(z)} = y \pmod{4}$$
is impossible for some given positive integers $x$ and $y$, where,
\begin{align*}
f(z) &= \phi(4) k_1(z) + 1 \\
&= 2 k_1(z) + 1\\
g(z) &= \phi(4) k_2(z) + 1 \\
&= 2 k_2(z) + 1
\end{align*}
and $k_1(z)$ and $k_2(z)$ are integer functions, that approach infinity, such... |
This is a very basic question, and its funny that I'm able to solve more advanced problems like this, but I was presented with a basic one and got stumped. I have the equation
$$y=-\frac{3}{4}x+6.$$
In $\mathbb{R}^2$, this is a line. I want to find a parametrization of this line. I guess the problem here is I never r... |
I've been reading about Hilbert's Axioms of geometry, and I have two models on $\mathbb{R}^2$ which satisfy the usual axioms of incidence, betweenness, and congruence.
The first model has a distance function given by
$$
d(A,B)=|a_1-b_1|+|a_2-b_2|
$$
where $A=(a_1,a_2)$, $B=(b_1,b_2)$, and two segments $AB\cong CD$ ... |
I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property.
|
Suppose I have a discrete sequence (or would it be a set?) of numbers labeled by two integer indices, $\{a_{ik}|i,k\in 1,\ldots,n\}$. Also, I know that these numbers satisfy the properties
$$a_{ij} + a_{jk} = a_{ik}$$
and
$$a_{ik} = -a_{ki}$$
Are these conditions sufficient for me to conclude that there exists a sequen... |
I'm confused with these definitions. The metric of certain space and the second fundamental form seem to be the same object.
I don't know what else to say, this is a pretty straight forward question.
|
I have been looking around for this question, but all results I found only describe the definition and not the answer I seek.
Is "kernel" basically a synonym of "function"? When should be the time we should use the word "kernel" instead of "function"?
|
In the book I'm reading there is a question,
Prove that given a rotation $R(\phi \hat{n})$ and a reflection $\sigma$, they commute iff $\sigma \perp \hat{n}$ and $\phi=\pi$. (Binary rotation normal to the reflection plane.)
The answer in the back of the book is,
Take $R(\phi z)$. For $\phi=\pi$, the eigenvectors, wi... |
I'm using Macdonald's "Symmetric Functions and Hall Polynomials" as a reference and did not find what I was looking for -- apologies if I only missed it.
As an example, let us consider the partition $\lambda=(3,2,2,1)$ of $8$. Is the notation $2\lambda$ clear / in use / ... for describing the partition $(3,3,2,2,2,2,1... |
In the book "Markov Random Field Modeling In Image Analysis" by Stan. Z Li, 3d Edition. Page 15.
An observation d of a field f (with m nodes) with i.i.d. gaussian noise:
$d_i = f_i + e_i$, there $e_i$ ~ $N(\mu, \sigma^2)$
$p(d|f) = \frac{1}{\prod_1^m\sqrt{2\pi\sigma^2}} e^{-U(d|f)}$
$U(d|f) = \sum_{i=1}^{m} \frac{(f_i ... |
Let $f$ be a holomorphic function on the open unitary disk $\mathbb{D}$ and continuous on $\mathbb{\overline{D}}$. If $f(\frac{z}{2})= \frac{1}{2}f(z)$ for all $z\in \mathbb{\overline{D}}$ and $f(1)=1$, then $f(z)=z$ for all $z\in \mathbb{\overline{D}}$.
Got this as homework. Any hints would be highly appreciated.
|
I have this definition
A field $F$ is perfect if it has characteristic $0$ or if it has characteristic $p$ and $F^p = F$
From Wikipedia, I have this fact about separable polynomials:
Irreducible polynomials over perfect fields are separable.
I can show that if $p(x)$ is irreducible and $char(F) = 0$, then $p$ cannot h... |
I have the following setting: Let (M,g) be a Riemannian manifold and $\iota: M \to R^N$ some isometric embedding. This means especially that the connection $\nabla^{TM}$ of M is given by the ordinary Euklidean connection $\nabla$ on R^N composed with the projection on the subbundle TM of TR^N. I also have the normal bu... |
As this question suggests, I quite like the notion of permuting the coefficients of polynomials.
And, moreover, I have another question on this direction:If L|F is a finite normal field extension, then it must be normal, then my question is: is there a name for the field extension L|F such that L contains all roots of... |
I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations.
If $\sim$ is the equivalence relation on $n\times m$ binary matrices such that $A \sim B$ iff one can obtain B from applying a permutation matrix to A, I'm interested in the number of $\sim$-equ... |
I have a query below, and want to understand the reason of this in detail.
Here is the graph of $x^2+(y-\sqrt{x^2})^2=1$ drawn by Wolfram Alpha:
In this equation if you replace the term, $\sqrt{x^2}$ by $x$, which is a valid substitution, the graph seems to become an ellipse:
Shouldn't the curve remain same? What is... |
Let $A$ be a commutative (unital) ring, and $A[x_1,\ldots,x_n]$ a polynomial ring over it in some finite number of variables. The inclusion $i\colon A \hookrightarrow A[x_1,\ldots,x_n]$ induces (by contraction) a continuous surjection $\mathrm{Spec}(i)\colon \mathrm{Spec}(A[x_1,\ldots,x_n]) \twoheadrightarrow \mathrm{S... |
I have two vectors of matching lengths. They are readings from two different sensors (one is from a smartphone and the other is from a wiimote) of the same hand movement. I am trying to find the time offset between them to synchronise the readings for further processing. The readings I get are of the format (Time(ms) V... |
I'm hoping that such a calculation as below will be very easy for the users of this website:
I have a 3D Carousel on my website and a colleague is going to render out some 3D components to sit on the carousel. However, he needs the angle that the "camera" is at on my 3D carousel. The issue is that my 3D carousel is fau... |
It asks me to graph the region in the complex plane.
$Re(z+iz) \le 1$
|
The expression is this:
$|3z+2| \leq 1$
then, I did
$|3x+2+3iy| \leq 1$
and I got stuck.
|
I am just starting out with vector spaces and I am having a hard time understanding them. One of the requirements states that $1\mathbf{v}=\mathbf{v}$ where $1$ is the multiplicative identity.
Does 1 have to be the identity? Or is that whatever is the multiplicative identity is labelled 1?
I ran across a question where... |
I have a value generated from complex circle geometry. That value is about 1 for the bottom edge of the circle and 0.2 for the top edge. I don't have the luxury of changing that variable. However, I can make an additional variable calculation from it.
I wish to apply an effect to the circle where the top edge has a val... |
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