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I have a bus that does A-------X------------------------B
it goes at 10 hours from A, having:
Speed at A sA = 10 m/s
constant acceleration a = -0,5 m/s2
At what hour it will be at X, knowing the distance AX dAX = 50m ?
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I've read the section "Serre duality" in Hartshorne's book and have several questions.
1) In Remark 7.1.1 it is claimed that on $X = \mathbb{P}^n$
$\alpha = \frac{x_0^n}{x_1 \cdot ... \cdot x_n} d(\frac{x_1}{x_0}) \wedge ... \wedge d(\frac{x_n}{x_0})$
is a Cech cocycle which generates $H^n(X,\omega_X)$ (I have checked ... |
Here's the question, from my $10 mathbook that everyone is growing to love and hate:
A saltwater solution initially contains 5lb of salt in 10 gal of fluid. If water flows in at 0.5 gal/min and the mixture flows out at the same rate, how much salt is present after 20 mins?
I get that this is a differential equation ... |
I don't have much of any awareness about the representation theory of non-compact Lie groups but I bumped into it for my work.
Is there some idea that the representations of a non-compact group are labeled by those of its maximal compact subgroup? If yes then I would like to know of explanations for the above and of r... |
3 colors of balls, blue,red and yellow, there are a large number of each color of balls which are distributed evenly. Ask what's the probability to get 3 balls which are all red color? What is probability to get 3 balls which are red, blue and yellow?
Thanks
|
Let $A$ be $n\times n$ matrix. Prove that if $A^2=\mathbf{0}$ then $A$ is not invertible.
|
What are some examples of functions on the Power Set of the Reals? Is this an abuse of terminology - functions on the reals can be thought of as functions on the power set of the naturals with a specific ordering. I was hoping someone would kindly refer me to a text or article where explicit (not necessarily 'useful') ... |
$(X_n), n \in \mathbb{N}$ is a stochastic process.
I saw in one definition of Martingale that
$$E [X_{n+1} |X_0 , X_1 , . . . , X_n ] = X_n \quad a.s., \forall n \geq 0.$$
I understand what "almost surely" itself means. But I was wondering how to interpret the above usage of "almost surely" in the definition of
Martin... |
1) Let $p$ be an odd prime, $n$ an integer such that $n \geq 3p$ and $a$ a $2p$-cycle in $S_n$.
Let $b$ be a $p$-cycle in $S_n$ and assume that $a$ and $b$ are disjoint. Suppose $K$ is a subgroup of $S_n$ that contains $a$ and $b$. Prove that $K$ is not cyclic.
(we may assume $a = (1\;\: 2\;\: 3\;\: \dotsb \;\:2p)$ and... |
I have an ODE of the form $y''(x)=F(x,y')$ which has the initial conditions, $y(\mu)=\mu$ and $y'(\mu)=1$. Now I have seen that an equation which is equidimensional in x, can be made autonomous by the change of variable $x=e^t$, ie now we must solve an ODE for the unknown $z(t)$ ($=y(x)$). But now what happens to the i... |
Apologies if this is too basic, but given a permutation matrix $M$, is there any parameter or formula based on $M$ that gives the disjoint cycle decomposition, or at least the conjugacy class, of the corresponding permutation?
|
I've been struggling with this for a bit
and was wondering if anyone can give me a hint:
Suppose $\{ T_n\}_1^\infty\subset \mathcal{L}\{ X,Y\}$ is a sequence of bounded operators from a banach space $X$ to a normed space $Y$, such that $\forall x\in X:\lim_n T_n x=0$.
Prove that $\lim_n(\sup\{\Vert T_n x\Vert:x\in K\}... |
I have to find the domain of
$\frac{z^2-1}{z^2+z+1}$
So, there can't be any $z$ that makes null the denominator,
$z^2+z+1$ = 0
and after decomposing $z$,
$x^2-y^2+x+2xyi+yi=0$
How to solve that quadratic complex equation?
|
I'm having some troubles computing this limit.
$$\lim_{M\to\infty}M\left(\frac{1}{s} - \frac{e^{\frac{-s}{M}}}{s}\right)$$
I know that the answer should be 1, but I can't seem to figure out the steps to get there.
Here are the steps I've tried taking:
The exponential approaches 1 as $M\to\infty$,
$$\lim_{M\to\infty}M\l... |
Find the matrix of rotation in $\mathbb{R}^3$ by the angle $\theta$ around the column vector $(1, 2, 3)^T$. We assume that rotation is counterclockwise if we sit at the tip of the vector and look at the origin.
Could I use the product of three 3x3 matrices to rotate vectors about the x,y,and z points individually? If s... |
Consider the the following problem.
Define $u'$ and $v'$ to be the projections of $u$ and $v$ onto the plane perpendicular to the axis of $\xi$. If $\omega \in \mathbb{R}^{3 \times 1}$ is a unit vector in the direction of the axis of $\xi$, then
$u' = u -\omega \omega ^{T} u$ and $v' = v -\omega \omega ^{T} v$
Is some... |
Suppose $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$ is a diffusion. Is there a sense in which the dynamics are "dominated" locally by the diffusion term, and dominated globally by the drift term?
If $\mu$ and $\sigma$ are constants, then the law of the iterated logarithm says that the contribution from the diffusion term is ... |
It is easy to see that
$p:\mathbb{R} \rightarrow S^1: p(t)=(\cos t, \sin t)$ is a covering map of $S^1$
(Indeed take a point $x_0$ in $S^1$, take $U=S^1\setminus \{-x_0\}$ as an open neighborhood and then $p^{-1}(U)=\cup_{n \in Z} J_n$ where $J_n=\{t\in\mathbb{R} : t_0-n-1/2<t<t+0+n+1/2\}$. $p$ then maps $J_n$ homeomor... |
$y$ is a function of $t$. Simple differential equations are written to make
$ \frac{dy}{dt} = ky(t) $
The function $y(t)$ that fits this is
$y(t) = y(0) e^{kt} $
Where $y(0)$ is some initial condition.
$ \frac{d}{dt} y(t) = k y(0) e^{kt} $
Achieving our constraint $ \frac{dy}{dt} = ky(t) $
This is all fine and dandy, ... |
I need to confirm that i am approaching this problem correctly.
Problem:
Let $f$ and $g$ be functions $f,g:\mathbb{R}\to \mathbb{R}$
Prove that if $f^2 + g^2$ is bounded then $f$ is bounded.
Attempted Solution:
Definition of bounded: $|f(x)| \leq M$ for all $x \in \mathbb{R}$
Using that definition we can say
$|f^2 + g^... |
I'm lost on how to solve this for z. Can anyone help?
$\left(7^\left(-2z\right)\right)^\left(7-z\right) = 49^\left(-12\right)$
|
Let $X_i$ be a sequence of probability spaces and define $\displaystyle X=\prod_{i=1}^\infty X_i$
Let $A$ be the algebra on $X$ generated by the sets of the form $$\displaystyle \prod_{i=1}^{n-1} X_i \times E_n \times \prod_{i=n+1}^{\infty} X_i, E_n \in P(X_n)$$
Show that every element in $A$ can be written as $$a= \di... |
How to find a differentiable map $T : \mathbb{R} \rightarrow \mathbb{R}$ whose fixed points are exactly integers?
I have to
a)Find points $|T'(x)| > 1$ (if any)
and prove that
b)There must exist at least one $x$ such that $|T'(x)| > 1$.
I think the functions
$$x - Tx = x(x^2 - 1)(x^2 - 4)(x^2 - 9)...$$
and
$$x - Tx = ... |
I have defined this simple sheaf. Take $E:set, B:set, p:E\to B$. Let $P(B)$ be a set of subsets of $B$. Let $S$ be the set of sections of $p$. Let $F$ be a contravariant functor from $P(B)$ as a poset category to $Set$ which sends:
objects: $B'$ to ($S$ with elements restricted to $B'$);
morphisms: an inclusion to the... |
According to Wikipedia, Strassen's Algorithm runs in $O(N^{2.807})$ time. Has anyone seen a more rigorous analysis displaying constants, possibly in a specific language such as C or Java?
I realize this will vary from language to language, machine to machine etc, but does anyone know of an approximate input size whe... |
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a symmetric matrix, and so it has a basis of eigenvectors, and $n$ eigenvalues (not necessarily distinct). We define a funct... |
Starting with a given simple, directed Graph G, I define a two-edge swap as:
select two edges u->v and x->y such that (u!=x) and (v!=y) and (u!=y) and (x!=v)
delete the two edges u->v and x->y
add edges u->y and x->v
Is it guaranteed that I can reach any simple directed graph with the original (in- and out-) degree s... |
How can I compute this limit:
$$\lim_{z\rightarrow i} (z\cdot \overline{z})$$
(where $\overline{z}$ is the conjugate of $z$)?
|
I asked this on mathoverflow, and was suggested to ask it here.
Someone mentioned, in passing, to me that $u \mapsto \nabla \cdot ( c^2 \nabla u)$ is a Laplace-Beltrami operator. Does anyone have some insight into this? From my understanding, the Laplace-Beltrami operator generalizes the Laplacian to Riemannian manifol... |
Statement: If $E$ is a bounded set of real numbers, there exists a $G_\delta$ set $G$ such that $G$ contains E and has the same Lebesgue outer measure with $E$.
I completed the proof for the case that $E$ is countable, but how should I prove this for the case that $E$ is uncountable?
|
$$\int (2x^2+4x-2)^{-\frac{3}{2}} \ dx$$
Complete the square?
$$\int \frac{1}{(2(x+1)^2-4)^\frac{3}{2}} \ dx$$
Not sure what do do next, or if I should try something else?
Big help if you can show as "step-by-step" possible as you can.
Thanks in advance!
@Chandru1:
I'm confused at what happens to the 2 that gets facto... |
As we all know, the ring $\mathbb{Z}_p$ can be constructed as the projective limit of the rings $\mathbb{Z}/p^{n}\mathbb{Z}$.
Now is there any generalization such as the $p$-adic completions of a Dedekind Domain?
This might be said to be inspired by the general treatment of extensions of Dedekind Domains from the treat... |
The table on the left gives the birth and death rates (per million) by region. The table on the right gives the populations (in millions) in each region for a number of years. Use matrix multiplication to get a matrix showing the total number of births and deaths in each year.
Births Deaths
Afr... |
Let $f\colon\mathbb{R}\to\mathbb{R}$. Prove that the set
$$\{x \mid \mbox{if $y$ converges to $x$, then $f(y)$ converges to $\infty$}\}$$ is countable.
My book told me to consider $g(x)=\arctan(f(x))$, then it said "it is easy to see the set is countable." But I still can't understand what it mean.
|
There's a way to represent proper colorings as graph homomorphisms, is there a similar construction for spanning trees of a graph?
For proper $m$-colorings of graph $G$, the relevant homomorphism is the mapping that takes adjacent vertices of $G$ to adjacent vertices of $K_m$
|
We have a function $f$ of $N$ variables which is the product of $M$ polynomials:
$$f(x_1,x_2,\ldots, x_N) = P_1 \cdot P_2 \cdots P_M.$$
Each $P_i$ is a polynomial of at most three variables ($x_j$s) with degree 1, and each variable appears in at most two of the $P_i$ 's, so the highest degree of each variable in $f$ i... |
Let $I$ be an uncountable set of indices and $U_\alpha\subset\mathbb{R}^3$ be an open subset for each $\alpha\in I$. We know that $U=\bigcup_{\alpha\in I} U_\alpha$ is an open subset.
My question is: does there exist a countable subset $J \subset I$ such that $U=\bigcup_{\alpha\in J} U_\alpha$?
In general it is not tr... |
Simple but Stuck: How do I find the point of intersection of two lines in Vector Calculus? Given symmetric equations and deriving parametric equations.
Find the point of intersection of the lines L1: $\frac{x}{2} =\frac{y+1}{3} = \frac{z-1}{2}$ and L2: $\frac{x}{4} = \frac{y}{5} =\frac{z}{5}$
The parametric forms shoul... |
$\forall x \in Z \exists y \in Z(x *y = x + y)$
If I'm understanding this correctly this is FALSE.
|
Let S be the collection of vectors $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$ in $\mathbb{R}^3$ such that $|x-y|=|y-z|$. Either prove that S forms a subspace of $\mathbb{R}^3$ or give a counterexample to show that it does not.
(from Poole, Linear Algebra - A Modern Introduction 2nd ed., exercise 3.5.8)
I'm working on... |
$\displaystyle\int \left(\frac {x-1}{3-x}\right)^\frac{1}{2}\,\rm dx$
I am stuck on this part:
Let $u=\dfrac{x-1}{3-x}~\longrightarrow$ $~~\rm du=\dfrac {2}{(x-3)^2}\,\rm dx,$ which can be represented as
$\rm du=\dfrac{1}{3-x} - \dfrac{1-x}{(3-x)^2}\,\rm dx$
I cannot "see" how to get to this $2$ $\displaystyle\int \: ... |
I'm hating these variable resistance questions.
A body of mass $m$ falls from rest in a medium that produces a resistance of magnitude $m\cdot k \cdot v$. where $k$ is a constant, where the speed of the particle is $v$. Show that when the body has reached a speed $V$ it will have fallen for a time
$$\frac{1}{k} \ln \le... |
Calculate $lim_{x \to (\frac{9}{8})} \frac{12x}{9-8x}$ and $lim_{x\to(-\frac{9}{8})} \frac{12x}{9-8x}$.
I am confused how to do this. L'hospital, substitution and conjugates will not solve this as they result in the division of zero. I tried factoring out the largest degree:
(x/x)(12/[9/x]-8)
integer over variable resu... |
Find two different $5$-Sylow subgroups in $S_{16}$.
Hint: use group multiplication.
Any hints?
|
Possible Duplicate:
Computing GCD of all permutations (of the digits) of a given number.
How can we find the greatest common divisor (GCD) of all numbers that can be obtained by permuting the digits in the given number?
Please suggest a feasible algorithm.
|
Sorry for this whole bunch of questions. Please note, that I know what a commutative diagram is, and that I can somehow read them, at least the simpler ones. But often enough the diagrams are labelled and/or explained verbally, thus my question:
How are commutative diagrams to be
read in general? Can they always be
... |
I'm super rusty on my real analysis and I'm stuck on this easy question.
Show that for a semi-continuous function $ g: \mathbb{R}^n \rightarrow \mathbb{R}$, the set $\{x \in \mathbb{R}^n: g(x) \leq 0\}$ is closed
I have found some proofs online though I am working with somewhat different definitions of semi-continuous ... |
In my book, Galois group is defined to mean the set of automorphisms on $E/F$ that "leave alone" the elements in $F$.
On Wikipedia it says:
"If $E/F$ is a Galois extension, then $Aut(E/F)$ is called the Galois group of (the extension) $E$ over $F$, $\dots$"
And Wikipedia's definition of Galois:
"An algebraic field exte... |
How to find this sum?
$$\sum\limits_{k=1}^{2n} (-1)^{k} \cdot k^{2}$$
|
I know this is fairly basic stuff compared to most of the questions here, but I have to start somewhere.
How would I find out whether the following lines intersect the same point, and if so, where is that?
$y - 2x + 3=0$
$2x + y - 53=0$
$y - x - 11=0$
I've read the answers to "How to find where 3 lines intersect.", b... |
Let $X$ and $Y$ be two Poisson variables with different mean.
Is there a better (as in more concise or numerically faster) way to compute $P(X\leq Y)$ than using
$$ P(X\leq Y) = \sum_{y=0}^\infty P(Y=y) P(X\leq y) $$
|
Prove that for all natural numbers statement n, statement is dividable by 7
$$15^n+6$$
Base. We prove the statement for $n = 1$
15 + 6 = 21 it is true
Inductive step.
Induction Hypothesis. We assume the result holds for $k$. That is, we assume that
$15^k+6$
is divisible by 7
To prove: We need to show that the result h... |
Say I have 100 numbers that are averaged:
number of values = 100
total sum of values = 2000
mean = 2000 / 100 => 20
If I want to add a value and find out the new average:
total sum of values = 2000 + 100
mean = 2100 / 101 => 20.79
If I want to subtract a value and find out the new average:
total sum of values = 2100 ... |
I'm having trouble seeing why this (it's stated as obvious in a proof of Sylow's first theorem): if $n=mp^r$ ($p$ is prime), and $p^t\mid m$ but $p^{t+1}\nmid m$, then $p^t\mid\binom{n}{p^r}$ but $p^{t+1}\nmid\binom{n}{p^r}$.
I also wonder if a combinatorial proof of this fact can be given.
Thank you in advance for any... |
Can there be such a thing as a classification of classification theorems?
|
Does the principle of mathematical induction extend to a cardinality larger than that of the countably infinite?
|
I have an algorithm and I would like to know how many times each line is called.
There I wrote which lines I understand and some lines is left.
j := 1 ---------------------- 1 time
while j < n do --------------- n times
x := B[j] ---------------- n-1 times
k := j ------------------- n-1 times
for i := j ... |
The definition for the two processes
to be independent is given by
PlanetMath:
Two stochastic processes $\lbrace X(t)\mid t\in T \rbrace$ and $\lbrace Y(t)\mid t\in T \rbrace$ are said to
be independent, if for any positive
integer $n<\infty$, and any sequence
$t_1,\ldots,t_n\in T$, the random
vectors
$\boldsymbol{X}... |
Do there exists permutations $\pi_1,\pi_2$ and polynomial size CFG that describe the finite language {$w \pi_1(w) \pi_2(w)$} over alphabet {0,1}?
Polynomial size in $|w|=n$
|
Here I have an interesting problem on linear algebra. It looks very simple, but not so easy to solve for me.
Let $r_i, i=1,…,n$ be unit vectors in $\mathbb{R}^n$, find a unit vector $x$ to minimize
$\sum \| r_i\times x \|^2$
Remark: if let $\theta$ be the angle between $r_i$ and $x$, then $\sum \| r_i\times x \|^2 = \s... |
First of all, I would like to call a group immaculate provided that the orders of $G$ and $\Sigma$ (the order of $N$) where $N$ varies over all normal subgroups of $G$, are equal. From here it has been said that any abelianizer of a group whose order is less than that of $D(G)$, where $D(G)$ is the latter function defi... |
Given two Martingale processes $(X_t)$ and $(Y_t)$, are their sum $(X_t+Y_t)$ and their product $(X_t \times Y_t)$ also Martingale?
If not, will the two $(X_t)$ and $(Y_t)$ being independent grant their sum and product Martingale? Thanks!
|
The equations for getting the expected value of the binomial distribution in one of my textbooks is:
$$\sum_{x=i} x_i P(X=x_i)$$
and in another is:
$$\sum_x x P(x)$$
and both texts say that this actually equals:
$$\mu = np$$
and from what I can see the 'p' means the probability of success in the binomial equation.
A... |
For a function $f(x,y)$ of two independent variables we have an incomplete specification of its partial derivatives as follows:
$$\frac {\partial f(x,y)} {\partial x} = \frac {1} {g(x,y) \sqrt {1 - (\frac {k y} {x^{(1/3)}})^2}}$$
$$\frac {\partial f(x,y)} {\partial y} = \left(\frac {3 x} {4}\right)\left (\frac {k} {x^... |
Let $B$ and $F$ be compact Hausdorff spaces.
Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$.
I think this induces a fiber bundle $E'$ over $B$ with fiber $C(F,\mathbb C)$, the C*-algebra of continuous functions on $F$, and with structure group $... |
Given a polynomial $y=C_0+C_1 x+C_2 x^2+C_3 x^3 + \ldots$ of some order $N$, I can easily calculate the polynomial of reduced order $M$ by taking only the first $M+1$ terms. This is equivalent to doing a Taylor series expansion with $M<=N$ around $x=0$.
But what if I want to take the Taylor series expansion around a di... |
I am having an NxN matrix . I want to read the elements of that NxN matrix diagonally and need to store it in an array.How?
For example, I am having one 3x3 matrix
$$\begin{bmatrix}
A B C \\
D E F\\
G H I
\end{bmatrix}$$
I want to read this matrix diagonally(starting from A and downwards) and output should be
A... |
I have a question related to this: Projective modules
I'm trying to understand the "philosophy" of the statement, because it seems too similar to the statement "a module is free iff every element can be written uniquely as a finite linear combination of elements of a basis".
Is this "projective basis" property saying t... |
I also have to prove this for $$2^2 \times 4^2 \times 6^2 \cdots \times (p-1)^2 \equiv (-1)^{(p+1)/2} \pmod{p}$$
I made some progress so far and got stuck. I said that since p is odd, $(p+1)/2$ is even. Then we can say that $(-1)^{\mathrm{even}}= 1$, so $(-1)^{(p+1)/2}\pmod{p}$ can be written as $1\pmod{p}$. I know tha... |
I am looking for any reference on Wigner's classification of irreducible representations of the Poincaré group. I know the classification, but is there any reference where the representations are constructed and explained. This classification gives the different spin particles in Quantum mechanics.
Thanks.
Edit (Qiaoch... |
For a family of random variables, I was wondering if independence and conditional independence under any condition among them imply each other?
If not, can these two concepts imply one another under some special cases?
|
I'd like to calculate, or find a reasonable estimate for, the Mertens-like product
$$\prod_{2<p\le n}\frac{p}{p-2}=\left(\prod_{2<p\le n}1-\frac{2}{p}\right)^{-1}$$
Also, how does this behave asymptotically?
Hmm... trying to think this one out, I get
$$\left(\prod_{2<p\le n}1-\frac{2}{p}\right)^{-1}=\exp\log\left(\lef... |
Write the following sequence as a recurrence relation (with sufficient initial values specified):
$$b_n=1-\frac{1}{2^n} \forall n\in\mathbb{N}^*$$
I think I am suppose to use induction (e.g. n-1) to prove the recurrence, but I am confused, how?
|
As often happens I'm not sure of the name of what I'm trying to do, so I'm having difficulty searching it. I know this is basic algebra, but my math is null so I'm struggling with it greatly.
I have a coordinate system that starts from 0,0 in the bottom left corner, and ends in 1,1 in the top right corner. I must "tran... |
Prove that any positive real number $r$ satisfying:
$r - \frac{1}{r} = 5$ must be irrational.
Using the contradiction that the equation must be rational, we set $r= a/b$, where a,b are positive integers and substitute:
$\begin{align*}
&\frac{a}{b} - \frac{1}{a/b}\\
&\frac{a}{b} - \frac{b}{a}\\
&\frac{a^2}{ab} ... |
I know that I can use P(AB)=P(A)P(B) to prove whether sets are independent. But how can I use this to say that P(A)=0.3 and P(B)=0.4 and P(AUB)=0.6 are or aren't independent?
It's fine not to give an outright solution, an explanation would be much more helpful!
|
This is another homework question I can't figure out.
For $|G|$ even, $\forall x\in G\exists b\in G\setminus{\{x^{-1}\}}$ such that $bxb = x^{-1}$
I tried to toy with associativity but to no avail. Also, I can't see the relevance of $|G|$ being even. Any hint (I'm not here for the solution) is appreciated.
Thanks for... |
How would you go about proving the $(p, q, r)$ -pretzel knot is equivalent to the $(p, r, q)$ -pretzel knot?
By "equivalent" I mean you can change one knot into the other by elementary deformations.
I've found this question/example in several books and papers on knot theory where they state the proof as obvious.
|
Every rotation in 3D space can be defined by a rotation axis and an angle.
Now let's say we have two rotations $R_1 (\text{(axis)}_1, \text{(angle)}_1)$, $R_2 (\text{(axis)}_2, \text{(angle)}_2)$.
I remember that Rotation operator is closed under composition, so $R_1 (R_2(\text{object}))$ will be a rotation again.
Does... |
I'm trying to solve the problem of showing that
$$\lim_{x\to6}\left(\frac{x}{4}+3\right) = \frac{9}{2}$$
using the $\epsilon$-$\delta$ definition of a limit.
|
If I understand correctly, the analytic multiplicity of a linear operator say $T:V\to V$ is the amount of times $\lambda$ shows up as a root in the characteristic polynomial (Assuming you have a matrix $A$ representing $T$ with respect to some basis of $V$, then the characteristic equation is $\det(A-\lambda I)$).
I ... |
$\log(a - b) - \log(a - c)$
Does this have a simpler form? Perhaps one where the $a$s have cancelled out? I know it can also be expressed as a log of the fraction: $\log\frac{(a-b)}{(a-c)}$, but the same question applies.
|
I have several BNF defined as follow:
1. <S> -> 0<S> | 1 | 0
2. <S> -> 1<S> | 1 | 0
3. <S> -> 0<S> | 1<S> | 1 | 0
4. <S> -> <S>0 | 1 | 0
5. <S> -> <S>1 | 0
This is the solution from my friend, however, I doubt the correctness of some of them
1. { 0^n: n >= 1 } V { 0^n1: n >= 0 } -> This is fine.
2. { 1^n0^k: n > ... |
What is the formula for the derivative of the product of two matrices when they are different sizes? Further, what is the formula for the derivative of a Hadamard product when the derivatives of the inputs aren't the same size as the inputs?
Here's what I'm thinking: assume an $n\times 1$ vector $x$, a $n\times n$ symm... |
For a p-primary group $G$, if $\langle y\rangle$ is a cyclic group of order $p^i$ for any natural integer $i$, then $ng \in \langle y\rangle$ implies $g \in \langle y\rangle$ for any $g \in G$?
Also, in the terms of group theory, what does the symbol $G^p$ means for a group $G$?
I wanted to ask this because I was refe... |
I'm trying to understand a solution I was given in a tutorial regarding a problem with Carmichael numbers and I was wondering if you guys can help clarify things:
A composite number $m$ is called a Carmichael number if the congruence $a^{m-1} \equiv 1 \pmod{m}$ is true for every number a with $\gcd(a,m) = 1$.
Verify t... |
I have a BNF defined as follow:
<S> -> 0
<S> -> 1
<S> -> <S><S>
I think this grammar is not ambiguous, but the solution was 'yes'. Any idea?
The book solution is odd to me, because I think it is the same as the one above.
<S>-> 0|1|<S><S>
Thanks,
Chan
|
I'd like to show that among any $52$ integers, there are two whose sum or difference is divisible by $100$.
Assume all the integers are placed into boxes based on their remainders, then there must be one box $y$ that contains two integers.
If $(a - b) = (100x + y) - (100z + y)$, where $x,z$ are integers.
$= 100(x - z)$... |
Assume R is commutative. Show that an R-module M is irreducible if and only if M is isomorphic (as an R-Module) to R/I where I is a maximal ideal of R.
If I remove the hypothesis that R is communtative, is it result still true?
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This is asserted in Exercise 1.15 of Joe Harris's algebraic geometry book (Algebraic Geometry: A First Course, Pg. 11 in my copy). This result struck my fancy but I'm unable to solve it myself or find references to it elsewhere. The closest reference I've found is the following document, which asserts something weaker:... |
This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help.
(By the way- this question does come from home-work, but I've already solved and handed it, and I'm posting this out of interest, so no HW tag.)
Let $B_n=B(x_n,r_n)$ be a sequence of ... |
Definition: F is a function iff F is a relation and $(x,y) \in F$ and $(x,z) \in F \implies y=z$.
I'm reading Introduction to Set Theory by Monk, J. Donald (James Donald), 1930 and i came across a theorem 4.10.
Theorem 4.10
(ii)$0:0 \to A$, if $F : 0 \to A$, then $F=0$.
(iii) If $F:A\to 0$, then $A=F=0$.
Wher... |
How would I go about calculating the total number of possible triangles where the perimeter is given, and all sides are integers?
Exempli gratia:
$a + b + c = 15$
$d + e + f = 16$
$x + y + z = 100$
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Why a torsion-free flat module over a discrete valuation ring is Cohen-Macaulay?
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I created a certain algorithm and now I try to prove it's convergence.
Here is the essence of the algorithm:
Given are finite sets $A_i$, positive constants $\alpha_a$, and a sequence $ \vec{\delta}^{(k)} \in \mathbb{R}^n$:
$$ A_1,\; \ldots,\; A_n \subset \mathcal{A} $$
$$ B_a = \{i : a \in A_i\}$$
$$ \sum_{a\in \m... |
Philosophical questions concerning the difference between equality, isomorphism, equality upto (unique) isomorphism, undistinguishability, and the like are not very popular among practicing mathematicians: there's a difference between equality upto isomorphism and equality upto unique isomorphism, and that's it (not to... |
I have an equation like this:
$$te^{t} = \int\nolimits_0^t e^\tau u(\tau)d\tau$$
I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how can I do it - is it like differentiate one side and then the other or is it more complex? And if it is possible what conditi... |
I have:
A1: {3, 5, 6} and A2: {3, 4, 6, 8}
Avg A1: 4.(6)
Std.Deviation A1: 1.247
Avg A2: 5.25
Std. Deviation A2: 1.92
Merging A1 and A2, we get
A12: {3, 3, 4, 5, 6, 6, 8}
Avg A12: 5
Std. Deviation A12: 1.69
I want to be able to obtain those values (avg and std. deviation for A12), having only the following informati... |
Suppose I have a singular matrix given by
$$A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} & a_{14}\\
a_{12} & a_{11} & a_{14} & a_{13}\\
a_{31} & a_{32} & a_{33} & a_{34}\\
a_{32} & a_{31} & a_{34} & a_{33}
\end{pmatrix}$$
Which is a homogeneous system in $(x_1, x_2, x_3, x_4)^T$ where there is a variable $\Lambda$... |
I'm searching for a formula to create http://oeis.org/A057537.
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