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Helping a buddy with his "intro to math" course for CompSci. I'm afraid my 'leet math skills are already giving out, so any help would be appreciated!
If f and g are both surjective functions, then so is the composition f o g. (so far so good).
I would expect the converse to be true too: If f o g is a surjective functi... |
I think this is really weird.
I want to solve the following equation:
$$(2x^2+3x+5)^3+(2x^2+3x+5)^2=0$$
This is a polynomial equation of 6 degrees. It should have 6 roots.
But wolframalpha only returns 4 roots! Anything I miss?
This is the command I use:
solve ((2x^2+3x+5)^3)+((2x^2+3x+5)^2)=0
Edit: Ah, Wolframalpha ... |
I'm trying to convey to some folks some logical and bitwise operators such as AND, OR, XOR, &c in programming languages such as C &| Java. Though these I guess could be considered "boolean" operators there is no "Boolean Algebra" in here, just the results of various operation, yes?
I've never heard of Boolean Arithmet... |
A simple question here, which likely demands a somewhat complex answer... Or rather, a set of related questions.
What are the advantages of typed lambda calculus over untyped lambda calculus in terms of proof theory?
Specifically, Church's original Lambda Calculus was untyped and allows arbitrarily high-order function... |
I'm going to ask a very silly question, so I'm begging you to be understanding if it is absolutely trivial, or if it's an exercise in some Bourbaki. I'm afraid of asking you, because the question entails this particular case: is the addition of (a finite, but variable quantity of) real numbers a continuous function?
Ok... |
I want to show that there is a finite conjunction $\phi$ of axioms of $ZF$, such that every transitive proper class $M$, which satisfies $\phi$, is already a model of $ZF$.
This is an exercise in Kunen's set theory. There is a hint, it seems to be useful to apply the Reflection principle to the union $M = \cup_{\alpha}... |
I've been trying to understand how
${x^3-12x+9}$
factors to
$(x-3) (x^2+3 x-3)$
What factoring rule does this follow? The net result seems to be similar to what is attained through the sum/difference of cubes factoring pattern, but the signs are different.
Additionally, what type of problem is this, so I can make b... |
How would you find the image of ${ x+yi : -\pi < x < \pi, y > 0 }$ under $\sin(z)$? I see that $\text{Re } \sin(x+yi) = \sin x \cosh y$ and $\text{Im } \sin(x+yi) = \cos x \sinh y$, but I'm not sure what to do next since $\sinh$ and $\cosh$ are unbounded.
|
If $h(x,t)$ is measurable and the measures involved are $\sigma$-finite, does there exist a sequences of functions
$$h_n(x,t) = \sum_{j=1}^{N_n} f_{j,n}(x) 1_{F_{j,n}}(t)$$
where the sets are pairwise disjoint (in $j$) such that $h_n \to h$ pointwise almost everywhere?
I know that if we fix $x$ we can get a sequence ... |
Right, this is an exercise in Apostol, which I am not being able to solve. I was able to prove this result for a small case, that is the case when $n=2$, $[x] + \Bigl[x + \frac{1}{2}\Bigr]=[2x]$, but I am struggling with the generalization.
Prove that $$\sum\limits_{k=0}^{n-1} \Biggl[x + \frac{k}{n}\Biggr] = [nx],$$ wh... |
Looking at a 2nd-order Taylor series approximation of the function $f$, I have this:
$$f(t_1) = f(t_0) + hf'(t_0) + {h^2\over 2}f''(t_0) + O(h^3)$$
Now say I approximate $f''(t0)$ with a $2$nd-order central difference method:
$$f''(t) = {1\over 2}{f'(t+h) - f'(t-h)\over h} + O(h^2)$$
What's the resulting error of this ... |
I made a picture of the problem here:
If the link does not work, read this:
Let $a$ be a non-zero vector in $\mathbb{R}^n$. Let S be the set of all orthogonal vectors to $a$ in $\mathbb{R}^n$. I.e., for all $x \in \mathbb{R}^n$, $a\cdot x = 0$
Prove that the interior of S is empty.
How can I show that for every point ... |
I was reading about Euler's totient function on wikipedia, and it eventually led me to this book on google:
Page 74 of the book, Prime numbers: the most mysterious figures in math
By David G. Wells.
Anyway, the books lists many assertions without proof, only references which I can't find. One I could not solve for mys... |
I was in the middle of writing the same old geographic distance calculation using the Haversine formula when it occurred to me: shouldn't there be simpler way to do this? Haversine is of course derived from the Law of Cosines. But in thinking about this problem, I ran across Napier's Rules for right-angled spherical tr... |
There is a lemma in my introduction to analysis book for which the proof eludes me. I was hoping to get some clarification. Here goes:
Lemma: If $\lbrace b_n \rbrace_{n=1}^\infty$ converges to B and B $\not=$ 0, then there is a positive real number $M$ and a positive integer $N$ such that if $n \geq N$, then $\mid b_n ... |
Me and a team of programmers are programming a robot to play a game in a competition. We did not create the hardware, and all teams will use the same type of robot.
GAME DESCRIPTION
The game has two players opposing each other and are both trying to achieve an objective. This objective is to move to a position to pick... |
By looking at an integral and bounding the error?
|
An exercise from Herstein asks to prove that the number of elements of order $p$, $p$ a prime in $S_{p}$, is $(p-1)!+1$. I would like somebody to help me out on this, and also I would like to know whether we can prove Wilson's theorem which says $(p-1)! \equiv -1 \ (\text{mod} \ p)$ using this result.
|
What is the general solution to the differential equation:
$$\frac{\partial^2 u}{\partial y^2}=\frac{\partial u}{\partial x}$$
I'm a little stuck because all the techniques I know are unable to solve it.
|
The goal is to show that the product of two Riemann integrable functions is integrable.
First step is to use the identity $f\cdot g = \frac{1}{4} \left[(f+g)^2 - (f-g)^2\right]$ so that we only need to consider squares of functions.
The second step is to reduce to positive valued functions because $f(x)^2=\left|f(x)... |
I think Cauchy's integral formula and the Hilbert transform can be used to prove one direction, but is this an equivalence or only an implication?
edit for clarification:
Is a function $f : \mathbb C \to \mathbb C, z\mapsto f(z)$ analytical $\Leftrightarrow$ The Fourier-Transform $\mathcal F\{f\}(\omega) = N \int_{\mat... |
I will restate the 3-D version of the problem. In how many ways can you color a regular cube with 2 colors up to a rotational isometry. The answer is of course a special case of Burnsides Lemma which can be used to show that the number of distinct face permutations is $\frac{1}{24}(N^6 + 3N^4 + 12N^3 + 8N^2)$ where $N$... |
Let $G$ be a (not necessarily finite) group with the property that for each subgroup $H$ of $G$, there exists a `retraction' of $G$ to $H$ (that is, a group homomorphism from $G$ to $H$ which is identity on $H$). Then, we claim :
$G$ is abelian.
Each element of $G$ has finite order.
Each element of $G$ has square-free... |
If I have an equation
$p(x)\frac{\partial^2u}{\partial x^2} + r(x)\frac{\partial^2u}{\partial x\partial y} + q(x)\frac{\partial^2 u}{\partial y^2}=f(x,y,u)$
Where $f$ maybe contains first partial derivatives for $u$.
Can anyone give me a worked example of how to solve this using the method of characteristics.
Thanks... |
I've recently spent some time learning (the very basics of) K-theory for $C^*$-algebras and topological K-theory. Actually, my main fields of interest are PDEs and related topics, in particular functional calculus for unbounded operators, Sobolev/Bessel potential/Besov spaces, interpolation theory, semigroups and so on... |
This is (a slight paraphrase) of question 1.3.14 in Chang and Keisler's Model Theory book.
"Show that for each natural number $n$, there is a language $L_n$ and finite model $M_n$ of $L$ such that $M_n$ has precisely $n$ undefinable elements."
Here, an element $x\in M$ is definable if there is a (first order) formula i... |
Let $f : \mathbb{R}^n \to \mathbb{R}$ be a convex function and let $c$ be some constant. Show that the following set $$s= \{x \in \mathbb{R}^n \mid f(x) \le c \}$$ is convex.
Looking for a hint.
|
Let $K$ be a number field and $L$ an infinite algebraic extension of $K$. Fix a non-trivial absolute value $v$ on $K$ (so $v$ is induced either by an embedding into the complex numbers or by a prime ideal in the ring of integers of $K$). If $K_v$ is the corresponding completion and $\overline{K}_v$ a choice of algebrai... |
What is the column space of a matrix? What equation do they satisfy? Why are they useful in linear algebra?
|
What is the homogeneous problem? What is the purpose of null space of a vector in this context?
|
Suppose $G$ is a compact topological group. We can construct the profinite completion of $G$; let's call this $\Gamma$.
My questions are:
1) Assuming that we know nothing about the (original) topology of $G$ other than that it is compact, is there anything we can say linking the topology of $G$ to the topology of $\Ga... |
Can someone shed a light on how to solve something like this?
I've been looking for permutations but so far I found it very confusing.
As I'm not at home atm, I can't reach for my math books either for an answer.
Maybe you can help?
Given:
a 5x5 grid
each cell can have 2 possible values: 0 or 1
each resulting grid "l... |
This question is from the book How to Prove It and I'm having trouble getting started with it. The book provides the hint "first prove $a \lt 0$". However, I can't figure out how to get that far with the provided givens.
The question is:
Given $a\ne 0$ and $b\ne 0$ and $a \lt \frac{1}{a} \lt b \lt \frac{1}{b}$. Prove ... |
In the literature it is stated that to each quadratic irrational $\gamma=\frac{P+\sqrt{D}}{Q}$ there is a corresponding ideal $I=[|Q|/\sigma , (P+\sqrt{D})/\sigma]$, where $\sigma=1$, if $\Delta \equiv0$ mod $4$ and $\sigma=2$, otherwise.
Thus, in the case of $\frac{2+\sqrt{13}}{3}$ the associated ideal must be $I=[3/2... |
How do I use second order Taylor method to solve a system of non-linear equations? Is there a good reference that gives details? I found mentions of it in a dozen of numerical analysis books, but no examples
Specifically, $f:\mathbb{R}^n \to \mathbb{R}^m$, solve $f(\mathbf{x})=\mathbf{0}$ using second order Taylor expa... |
A variation of the Root of Unity problem.
I want to find all possible answers to this:
$$z^n = i$$
Where $$i^2 = -1$$
|
I'm looking for a convenient upper bound on the integral
\begin{equation*}
\int_y^\infty x^k \exp(-(x-\mu)^2/2) dx
\end{equation*}
for (possibly large) positive integer $k.$ This is equivalent to finding higher moments of a truncated normal distribution. A bound that works for non-integer $k$ as well would be even b... |
What is the largest transitively reduced acyclic connected graph on $n$ vertices, for every $n$?
How many edges $e$ does it have? How does $e$ grow as a function of $n$? Faster than $n^2/4$?
|
If you start with an infinite set, you can have a sequence of nested sets which converge to a single point. (ie Intersection of $\left(\large\frac{-1}{n}, \frac{1}{n}\right)$ as $n\to \infty$)
However, at no time during the sequence is there a first element with only one point. In fact, for any finite (but unbounded) $... |
Suppose $x_1, x_2, y_1, y_2$ are known integers which satisfy
$y_1 \equiv \gamma x_1 \pmod c$
and
$y_2 \equiv \gamma x_2 \pmod c$
where $\gamma$ and $c$ are unknown integers. Is there a way to determine the values of $\gamma$ and $c$?
|
I had asked this question: Characterising Continuous functions some time back, and this question is more or less related to that question.
Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ and suppose the set $G = \\{ (x,f(x) : x \in \mathbb{R}\\}$ is connected and closed in $\mathbb{R}^{2}$, then does it impl... |
I've proven that $\neg \neg \beta \rightarrow \beta$ is a theorem, but I can't figure out a way to do the same for $\beta \rightarrow \neg \neg \beta$.
It seems the proof would use Axiom 2 and the deduction theorem (which allows $\beta$ to be an axiom)--but I've endlessy tried values to no avail.
Axiom 1: $A \rightarr... |
I have recently been working out the full game tree for tic-tac-toe, just for fun. I am using the well known equivalence relation of rotations/reflections to simplify this tree in the standard way (which begins by noting that there are only 3 opening moves: edge, corner, and center). I actually could not find an imag... |
Given a tuple $\left(a_1,a_2,\ldots,a_n\right)$ over the alphabet $\\\{1,2,\ldots,m\}$ chosen uniformly at random among the $m^n$ possibilities. What is the expected size of the set $\{a_1,a_2,\ldots,a_n\}$?
If $m=n$ it seems the answer tends to $(1-1/e)n$ as $n\to\infty$, but I don't know why.
I bumped into this while... |
I know I'm supposed to do $\sin(3x) - \sin x = 0$ but beyond that I'm stuck.. I tried expanding $\sin(3x)$ but that didn't help.
I want the value of $x$ in the interval $[0, 2\pi)$
|
This is probably a very simple question to all of you, but I have never taken a physics class, so I am lost for what I need to do for a programming problem. I will need to specify a force by inputting the newtons x, newtons y, and newtons z.
How can I calculate these 3 forces based on the desired angle and desired forc... |
Please help - in my notes, it is the group $V$ itself. I just want to confirm this. Can you also explain and give an example if that is possible?
|
it has been years since I have done logs, I remember something like this:
$$x^{\log_z(y)} = y^{\log_z(x)}$$
(where $z$ is the base)
Is that correct? It doesn't seem so, since
$$3^{\log_2(4)} \neq 4^{\log_2(3)}$$
Am I right in that assumption?
If I am, does
$$x^{\log_z(y)}$$
convert to anything easily?
|
Here is the dot
And here is how I arrived at it »
Pretty simple IFS using a 2 x 2 grid as the base for the iterations. Is there a way to describe this point as well as its siblings, all of which lie on the dotted red line(pic 2), using an equation?
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Suppose I have a weakly sequentially continuous linear operator T between two normed linear spaces X and Y (i.e. $x_n \stackrel {w}{\rightharpoonup} x$ in $X$ $\Rightarrow$ $T(x_n) \stackrel {w}{\rightharpoonup} T(x)$ in $Y$). Does this imply that my operator T must be bounded?
|
Consider a canonical elliptical cone $C$ with its vertex at the origin, with height $h,$ and with a base given by:
\begin{equation*}
\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1;~z=h
\end{equation*}
for $a, b$ not equal to $0$. Given a point $p=(p_x,p_y,p_z)$, how would you determine if $p$ is interior to $C... |
This is an old exam problem I found online. For any positive integers $a,b$, one has $a^4|b^3$ implies $a|b$. Clearly if $a^4|b^3$, then $a|b^3$. It seemed simple on first reading, but I can't figure out how to show that $a|b$ follows, since $a$ is not necessarily prime. Is there some observation I'm missing?
|
I think this is asked as a standard exercise in books about wavelets (e.g. exercise 7.2 in Mallat's book), but I couldn't find a proof. Let $\phi$ be a scaling function (see definition below). I would like to learn why
$$\sum_{k\in\mathbb Z} \phi(x-k) = 1 $$
almost everywhere.
Definition. A sequence of subspaces $... |
Suppose I have a $8 \times 6$ rectangle, with its lower left corner at the origin $\left(0, 0\right)$. I want to scale this rectangle by $\frac{1}{2}$ at an anchor point $\left(3, 3\right)$. So the resulting rectangle is $4 \times 3$, but I cannot figure out how to compute the distance from the origin to the lower left... |
I hope I'm at right place, as I asked this question at stackoverflow.com where the question was closed and suggested I go ask here. I'll try to be more comprehensive also, looking at couple of replies I got there.
The question is simple for those who know how algorithms work, and as I'm physics student I'm not sure abo... |
Given a differential equation of the form
$\epsilon \frac{d^ny}{dx^n} + \sum_{k=0}^{n-1} a_k(x)\frac{d^ky}{dx^k}=0$
Then the WKB method says to choose the ansatz $y\sim exp({\frac{i\phi(x)}{\epsilon}})A(x,\epsilon)$
where $A(x,\epsilon) = \sum_{k=0}^{\infty} A_k(x)\epsilon^n$
I wondered what is the motivation for this... |
This is an exercise from Apostol, which i have been struggling for a while.
Given a prime $p$, how does one show that $${n \choose p} \equiv \biggl[\frac{n}{p}\biggr] \ (\text{mod} \ p)$$ Note that $\Bigl[\frac{n}{p}\Bigr]$ denotes the integral part of $\frac{n}{p}$.
I would also like to know as to how does one try to ... |
Let's say we have some topological space.
Axiom $T_1$ states that for any two points $y \neq x$, there is an open neighborhood $U_y$ of $y$ such that $x \notin U_y$.
Then we say that a topological space is $T_4$ if it is $T_1$ and also satisfies that for any two closed, non-intersecting sets $A,B$, there are open neigh... |
Let us say that a Rubik's cube in a particular configuration is in a particular "state". All other configurations of this cube (other "states"), which can be achieved by rotations of the cube can be thought to be connected to each other... rotations are like walking the edges of the graph where the different states are... |
How to prove connect sum of two manifolds doesn't depend on the choices of balls(which would be cutted) and different gluing of boundary spheres? Is it still true in differential category? Thanks!
|
This is a problem which appeared in one of my tests, which i wasn't able to solve.
Let $\Omega$ be a uncountable set. Let $S$ be the collection of subsets of $\Omega$ given by: $A \in S$ if and only if $A$ is countable or $A^{c}$ is countable. Suppose $f: \Omega \to \mathbb{R}$ is a real measurable function. Prove that... |
(Apologies for the vague way the question is stated.)
We define the function like this:
(1) $f(x) = x$ for $0\leq x \leq a$ (for some $a \geq 15 \in \mathbb{R}$) $\land$ $f(x) = a$ if $ x > a$ . For $x<0$, it doesn't really matter what $f(x)$ looks like.
Has someone studied this type of function? Does it have a 'nice... |
I'm looking for good online resources for 2nd order system dynamics. Any recommendations?
I'm looking for stuff that ideally includes discussion of Q, damping ratio, overshoot, bode plots, for systems with transfer functions of
$$ H_1(s) = \frac{1}{\tau_2^2 s^2 + \tau_1 s + 1} $$
$$ H_2(s) = \frac{\tau_1 s + 1}{\tau_... |
A student of mine has cooked up a new graphical notation for computing with knots on surfaces. The trouble is, writing up his results is difficult due to his new notation. Is there a good "drawing tool" for mathematics that anyone can suggest?
|
Given that I know the point coordinates of A and B on segment AB and the expected length of a perpendicular segment CD crossing the middle of AB, how do I calculate the point coordinates of segment CD?
|
I have read a few weeks ago that some mathematical researchers have discover that the minimum number of movements to solve any initial configuration of a Rubik's cube has been downsized to 20.
How do they calculate such a number ?
Thanks.
|
I'd like to know if the function $ y = f(x) : [0,1] \rightarrow [0,1]$ defined implicitly by the transcendental equation $$\displaystyle y = e^{(y-1)/x}$$ is "well known" (name, properties) or is related to some other well known function.
|
This is the question which i am attempting to solve and it seems to be difficult to get rid of the exponents.
Show that a the two cubic curves $Y^3 = X^2 + X^3$ and $X^3 = Y^2 + Y^3$ intersect each other at nine points.
Any help would be appreciated. By the way i would also like to know whether there is any general m... |
I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are:
\begin{equation*}
\frac{dP}{dt}=a\cdot P-b\cdot PZ=P\left(a-bZ\right), \\
\frac{dZ}{dt}=c\cdot PZ-d\cdot Z=Z\left(cP-d\right).
\end{equation*}
Multiplying both sides by $\frac{1}{PZ}$ giv... |
Let $(\Omega, \mathcal{F}, P)$ be a probability space, $X$ an
integrable random variable, $\mathcal{G} \subset \mathcal{F}$ a
$\sigma$-field. The conditional expectation of $X$ given
$\mathcal{G}$ is by definition the unique random variable $Y$ which is
$\mathcal{G}$-measurable and satisfies $E[Y;A] = E[X;A]$ for all ... |
If you are given of a ring, how do you find its ideals?
|
Given lines $\mathbb{L}_1 : \lambda(1,3,2)+(-1,3,1)$, $\mathbb{L}_2 : \lambda(-1,2,3)+(0,0,-1)$ and $\mathbb{L}_3 : \lambda(1,1,-2)+(2,0,1)$, find a line $\mathbb{L}$ such that $\mathbb{L}$ is parallel to $\mathbb{L}_1$, $\mathbb{L}\cap\mathbb{L}_2 \neq \emptyset$ and $\mathbb{L}\cap\mathbb{L}_3 \neq \emptyset$.
Since ... |
Given a set $E$ which is bounded, measurable and $m^{\ast}(E)=x >0$, then for each $y \in (0,x)$ we can find a measurable set $A \subset E$, such that $m^{\ast}(A)=y$. To see this one considers measure as a continuous function and applies the Intermediate Value property.
Now we extend the question in the following mann... |
The web is littered with any number of pages (example) giving an existence and uniqueness proof that a pair of squares can be found summing to primes congruent to 1 mod 4 (and also that there are no such pairs for primes congruent to 3 mod 4).
However, none of the stuff I've read on the topic offers any help with actua... |
I am having a little trouble understanding some results from a program i wrote. I think i have the correct results but i need to understand the theory behind it a little more (i am a programmer, not mathematician).
My program generates symmetric k-regular graphs (single undirected edges). It generates an adjacency matr... |
I have the following assertion in my notes from last year that I'm trying hard to digest, but I think it isn't true:
If $p$ is prime $\Leftrightarrow$ if $p | ab$ then either $p | a$ or $p | b$ or both.
A valid proof must prove both directions, so:
if $p$ is prime, let $a_p$ be the prime factorisation of $a$ and lik... |
Let $G = A *_C B$ be an amalgamated free product of groups.
My question is: suppose $C$ and $G$ are finitely generated, can we prove that so is $A$?
I've been trying to prove it by contradiction. Any suggestion?
Thanks...
|
As it is required for most students who wish to do a Ph.D in maths in the US to sit the GRE subject specific mathematics exam, I hope this question will be of interest to the mathematical community and will not be closed.
Essentially, the exam was "rescaled" (made more difficult) in 2001 and I have only been able to fi... |
$\sum_{n=1}^{\infty} \frac{\varphi(n)}{n}$ where $\varphi(n)$ is 1 if the variable $\text n$ has the number $\text 7$ in its typical base-$\text10$ representation, and $\text0$ otherwise.
I am supposed to find out if this series converges or diverges. I think it diverges, and here is why.
We can see that there is a ser... |
If the order of a group is pqr with p q r primes, then there exist three elements A B C with order p q r. Is it possible that the order of the subgroup generated by A and B has the order of pqr ?
|
Suppose $V, V'$ are subspaces of dimension $d$ of a vector space $X$. Then there is a subspace $W$ of $X$ of codimension $d$ such that $W \cap V = W \cap V' = { 0 }$.
This can be proved by choosing an explicit basis for $X$ which contains a basis for $V$ and a basis for $V'$ and a basis for $V \cap V'$. On the other ... |
I was wondering what are the differences and relations:
between geometry and topology;
between differential geometry and differential topology;
between algebraic geometry and algebraic topology?
For example:
Are they studying different objects? Such as different mathematical structures/spaces?
if they study the same... |
Does the basis of an Euclidean space have to be ordered by definition? Or can be left unordered?
I was also wondering about what is the morphism (i.e. the mapping that can preserve all the structures) on Euclidean spaces? Is it Euclidean transformation (rigid transformation), consisting of rotation, translation and ref... |
Is a triangle (its sides and the region enclosed by its sides) in a 2D Euclidean space $\mathbb{E}^2$ a manifold? I was thinking to use the identity mapping as its charts, but for each point on the sides of the triangle, there is no neighborhood of it can be mapped to an open subset in $\mathbb{E}^2$.
Thanks!
|
Consider a simple random walk on a undirected, connected graph. This is the random walk which, at every time step, moves to a random neighbor, with all neighbors being equally likely. Lets assume every node has a self-loop to avoid issues associated with periodicity. Then, the following (surprising to me) fact is true:... |
How to prove that the only integral solutions to the equation $$y^{2}=x^{3}-1$$ is $x=1, y=0$.
I rewrote the equation as $y^{2}+1=x^{3}$ and then we can factorize $y^{2}+1$ as $$y^{2}+1 = (y+i) \cdot (y-i)$$ in $\mathbb{Z}[i]$. Next i claim that the factor's $y+i$ and $y-i$ are co-prime. But i am not able to show this... |
So if I'm given $H$, when I can conclude there is a group $G$ such that $H\cong G/Z(G)$? It's easy to show that non-trivial cyclic groups are not of this form. More generally, any group with the property that all finitely generated subgroups are cyclic (e.g. $\mathbb{Q},\mathbb{Q}/\mathbb{Z}$) cannot be of this form. A... |
F y(w) >= F x(w), for all w. Prove P{y<x} >= 0.5
I had this question on the exam and boy did it stump me. It was simple to understand why it was true, but I tried several different paths and got nowhere.
Any ideas?
|
So I wanted to figure out a meaningful probability for the card deck order problem. I chose arbitrary values in an attempt to show the magnitude of permutations:
If you and everyone on the Earth right
now shuffled a million randomized
decks every millisecond that universe
has existed, the probability that any
... |
I have a problem with the binomial coefficient $\binom{5}{7}$. I know that the solution is zero, but I have problems to reproduce that:
${\displaystyle \binom{5}{7}=\frac{5!}{7!\times(5-7)!}=\frac{5!}{7!\times(-2)!}=\frac{120}{5040\times-2}=\frac{120}{-10080}=-\frac{1}{84}}$
Where is my mistake?
|
The definition of the Mandelbrot set that most enthusiasts first encounter is that of the set of all complex numbers $c$ for which the sequence $z_{n+1} = z_n^2 + c$ starting from $z_0 = 0$ does not diverge. For convenience, let us name this familiar quadratic map $P_c(z) = z^2 + c$.
I have read that an equivalent defi... |
What is the usefulness in mathematics, I know it can provide topological invariants, any other things?
|
In this answer to a question about a series, a theorem was stated:
If $A= \{a_i \}$ is a set such that $\sum_{i = 1}^{\infty} \frac{1}{a_i}$ converges, then $d(A) = 0$, where $d(A)$ is the natural density of the set.
My background in number theory is basically zero and all my attempts to prove this have been utterly ... |
Suppose
$P(Z|X) = A$
$P(Z|Y) = B$
where $X$ and $Y$ are independent. How does one determine $P(Z|X,Y)$?
I haven't studied maths in years, so what I post below may be entirely off base, but I don't like to ask a question without some attempt, so:
From what I read, Baye's theorem is used to calculate update a probabili... |
How to prove the following identity?
$$\left(\frac{4}{3-\sqrt{5}}\right)^2 - \left(\frac{6 - 5 \sqrt{6}}{5 - \sqrt{6}}\right)^2 = 2\sqrt{61+24\sqrt{5}}$$
I come up with $\dfrac{16}{14-6\sqrt{5}}-6=2\sqrt{61+24\sqrt{5}}$, but still can't get it to the obvious equality.
Any ideas?
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I need to find the exact solution for the lateral surface area of the solid generated by revolving the region bounded by $y=x^2$, $y=0$, $x=0$, and $x=\sqrt 2$ about the X axis.
I have come up with my own solution, but I'm not sure if it's right.
Before substituting values back in, I got
$$ \frac\pi{16} \left( (1+4x^2)... |
I believe that if we are given two modules $M,N$ over a ring $R$, and a pairing between them
$M \otimes_R N \to R$, we can construct a perfect pairing $M'\otimes_R N' \to R$ by taking kernels.
Is there a name for this construction?
Thanks.
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Are there any positive integer solutions $(n,x)$ to the equation $(x)(x+1)=n!$ except $(2,1)$ and $(3,2)$?
If not (as I suspect is the case), how do you prove that?
In general, is there a way to approach the Diophantine equation $n!=P(x)$ where $P$ is a polynomial?
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If $\phi:X\rightarrow Y$ is a rational map of varieties, why is the graph of $\phi$ well-defined? That is, if $\langle \rho,U\rangle$ and $\langle \psi,V\rangle$ are two representatives of $\phi$ (so that $U,V\subseteq X$ open, and $\rho\vert_{U\cap V}=\psi\vert_{U\cap V}$), why is the closure of the image of $\rho$ eq... |
I suspect this is easy, but that I'm missing something obvious:
If a ring homomorphism $f:R\rightarrow S$ is such that a maximal ideal $\mathfrak m$ in $S$ does not have $f^{-1}(\mathfrak m)$ maximal in $R$, must there be some non-unit element $x\in R$ whose image under $f$ is a unit in $S$?
My intuition is just tha... |
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