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Suppose that $(X, \rho)$ is a metric space, $|X| > 1$. Is it possible to prove that $X$ is an open set without assuming the axiom of choice?
As I understand it, the challenge is to find a way to assign an open ball to each point of the space (assign a radius, basically). The way I found is to select some point $y \in X... |
Using the implicit function theorem one can prove the following:
Let $X,Y$ be Banach-spaces, $U\subset X$ open, $f\colon U\to \mathbf{R}$, $g\colon U\to Y$ continuously differentiable function. If $f|_{g^{-1}(0)}$ has a local extremum at $x$, and $g'(x)\in L(X;Y)$ has a right inverse in $L(Y;X)$, then there is a unique... |
Let $q\in (0,1)$. Is there a way of computing the series
$$
\sum_{n=0}^\infty q^{n^2}
$$
explicitly? Is there at least a nice accurate estimate?
All I could get is the estimate
$$\sqrt{\frac{\pi}{4\cdot\mathrm{ln}\frac{1}{q}}}\leq\sum_{n=0}^\infty q^{n^2}\leq 1+\sqrt{\frac{\pi}{4\cdot\mathrm{ln}\frac{1}{q}}}$$
via in... |
how can one show (hopefully in an elementary manner) that there exist irreducible polynomials of arbitrary degree and number of variables over arbitrary field?
thank you
P.S. induction?
EDIT: ehm, sorry, I meant for $n\geq2$, i.e. in at least two variables.
|
I was recently reading about classifying the extrema of a continuous function of a single variable. I came across the information that if the second derivative is zero then we can examine the higher derivatives such as $f^{\prime\prime\prime}$,$f^{(4)}$ and so on. Suppose $f^{(n)}$ is the first non-zero derivative. ... |
For natural numbers, I would like to prove that if $m < n$, then $k^m < k^n$ for $k$ not $0$ or $1$. Induction seems viable, but I don't know which variable to induct on. Any suggestions?
|
Again, I really just want a nudge in the right direction. Possibly a large nudge, but not the straight forward answer.
I am trying to figure out how to solve Project Euler Problem 191.
I believe I have most of it figured out, but I cannot come up with an algorithm or pattern for the 3-consecutive-Absent days part. I ca... |
Adding to the for dummies.
The real spherical harmonics are orthonormal basis functions on the surface of a sphere.
I'd like to fully understand that sentence and what it means.
Still grappling with
Orthonormal basis functions (I believe this is like Fourier Transform's basis functions are sines and cosines, and sin ... |
Suppose $C$ is an algebraic curve (which has singular points) over an algebraically closed field $k$, and that $f$ is a rational function on $C$. How does one defines the Weil divisor of $f$?
The problem is that the local rings of $C$ at singular points are not DVR's, so I do not have an obvious candidate for an order... |
Suppose $H$ is embedded in $G$ and $H'$ is isomorphic to $H$ and embedded in $G'$. Then we can simultaneously embed $H$, $H'$, $G$ and $G'$ into a single object (the amalgamated free product) such that $H$ and $H'$ become identifiable. This seems similar to the Isomorphism Extension Theorem for fields which is importan... |
How can I prove that a convex function ƒ defined on some open interval C is continuous on C?
Thank you.
|
I have the following exercise:
"Show that if a measure-preserving system $(X, \mathcal B, \mu, T)$ has the property that for any $A,B \in \mathcal B$ there exists $N$ such that
$$\mu(A \cap T^{-n} B) = \mu(A)\mu(B)$$
for all $n \geq N$, then $\mu(A) = 0$ or $1$ for all $A \in \mathcal B$"
Now the back of the book state... |
This is a problem from Hungerford's book:
Let $E$ be an intermediate field of extension $K\subset F$ and assume that $E=K(u_1, \cdots ,u_r)$ where $u_i$ are (some of the) roots of $f\in K[x]$. Then $F$ is a splitting field of $f$ over $K$ if and only if $F$ is a splitting field of $f$ over E.
This is my attempt.
Le... |
We know from Calculus what a series is, and you might have seen infinite products as well. But the Elementary Symmetric Polynomials give an entire spectrum of operators between a sum and product over a finite set.
Given a sequence $s_n$ and an $a \in \mathbb{N}$(representing which operator we're picking), define $S_n$ ... |
Let $k = \lceil \frac{3 \ln n}{\ln \ln n}\rceil$. How does one show that
$$ \left(\frac{e}{k}\right)^k \frac{1}{1-\frac{e}{k}} \le n^{-2} ? $$
This is from p. 44 of Motwani and Raghavan, Randomized Algorithms, where they're talking about ball/bin probabilities. The $\left(\frac{e}{x}\right)^x$ is motivated but the $k$... |
I would like to learn some differential geometry: basically manifolds, differentiable manifolds, smooth manifolds, De Rham cohomology and everything else that is pretty much part of a course in differential geometry. I do however know some deal of category theory and algebraic geometry, and I would therefore like to le... |
A while back when I was taking a first year cs course, our professor had us write the algorithm for insertion sort in The Scheme programming language. There were also several other similar recursion type programs for us to write, and one of the question in the assignment was to prove that the programs are correct using... |
I am stuck proving the theorem that there exists $x$, $x^4 \equiv 2 \pmod p$ iff $p$ is of the form $A^2 + 64B^2$.
So far I have got this (and I am not sure if it's correct)
Let $p = a^2 + b^2$ be an odd prime,
$\left(\frac{a}{p}\right) = \left(\frac{p}{a}\right) = \left(\frac{a^2 + b^2}{a}\right) = \left(\frac{b^2}{... |
Darwinia features an intro which represents a modified version of Conway's Game of Life.
You can see it in action here.
The game developers added one more rule about the game: no cell may live longer than 50 generations. This is supposed to kill all the cells eventually, which is what does happens in the game.
However... |
I need an NPDA for the following language if it is context-free, and if it isn't I need a proof using the pumping lemma that it is not a CFL:
$$L_1=\{w_1w_2 \in \{a,b\}^* : |w_1| = |w_2|,w_1\neq w_2\}$$
|
A submersion between smooth manifolds is an open map. Is the converse true? That is, is a smooth open map $f:M\to N$ between smooth manifolds a submersion? We can additionally assume that it is surjective, if necessary, because that is the only case I am interested in.
I can see how it is almost true, taking a chart $U... |
Given a published mathematics article, is there a way to find its tree of subsequent citations starting from the date of publication? Does such a (reverse bibliography) database exist?
I'm at a student in a relatively well-known university with access to virtually every mathematics journal imaginable, but I can't seem... |
It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least $n_p(p^n-1)$ elements, where $n_p$ is the number of Sylow $p$-subgroups.
It is explained that the reason this is the case is because distinct Sylow $p$-subgroups interse... |
(sorry, low level so I can't attach the image in the post)
a is given.
b is given.
c is given.
For X, I don't have a clue where he
is.
I was thinking about rect / vector instersection, projection, and other things but still I couldn't come with a solution. If it's computer light, the better.
Sorry for not making cle... |
Suppose there is a second-order real-state stochastic process $X: \Omega \times T \rightarrow \mathbb{R}$ with $T= \mathbb{R}$ and probability space $(\Omega, \mathcal{F}, P)$. I was wondering if the following inequality holds for integrals over an interval $[a,b] \subset \mathbb{R}$:
$$ \Vert \int_a^b X_t dt \Vert_2 \... |
In general, it is not too hard to find isomorphisms between two groups when their order is relatively low. However, as their orders grow, it becomes increasingly irritating to write down their entire Cayley tables and such. Is there a set of tricks that is generally useful when trying to prove that two groups are actua... |
I am trying to prove the recurrence $2T(n-1) + 1$ has the bound $\theta(2^{n})$. $T(1) = \theta (1)$
My attempted solution:
\begin{align*}
T(n) &= 2T(n-1) + 1 \\
&= 2 \{ 2T(n-2) + 1 \} + 1 \\
&= 2 \{ 2 \{ 2T( n - 3 ) + 1 \} + 1 \} + 1 \\
& \ldots \\
&= 2^{i}T(n-i)+\sum_{i=0}^{k} 2^{i} + 1.
\end{align*}
I then proceed ... |
Let $\pi$ be an arbitrary permutation of the set $\lbrace 1,\ldots,n,n+1,\ldots,2n \rbrace$ for some $n \in \mathbb{N}$.
We call a swap local if you swap two neighboring positions in $\pi$, i.e. if you change the positions $i$ and $i-1$ or $i$ and $i+1$ for some $i$.
A $c$-separation of the pairs $(1,n+1),\ldots,(n,... |
On MO, Daniel Moskovich has this to say about the Hauptvermutung:
The Hauptvermutung is so obvious that it gets taken for granted everywhere, and most of us learn algebraic topology without ever noticing this huge gap in its foundations (of the text-book standard simplicial approach). It is implicit every time one sta... |
People in StackOverflow seems not so into this theme, so I thought I could have better luck in here.
I had the idea of an spaceship game where the world is confined in the surface of an 4-D hypersphere (also called a 3-sphere). Thus, in seeing it from inside, it would look like a 3-D world, but by navigating in every d... |
Suppose I have a graph like this
and a list of its automorphisms. How do I go about getting a set of generators for this group?
|
Which of the following is more surprising?
In a group of 100 people, the tallest person is one inch taller than the second tallest person.
In a group of one billion people, the tallest person is one inch taller than the second tallest person.
Put more precisely, suppose we have a normal distribution with given mean $... |
What is the explicit form of the inverse of the function $f:\mathbb{Z}^+\times\mathbb{Z}^+\rightarrow\mathbb{Z}^+$ where $$f(i,j)=\frac{(i+j-2)(i+j-1)}{2}+i?$$
|
Two permutations are conjugate of each other if and
only if they have the same cycle structure.
If I want to find the number of elemnts in $S_{13}$ that are both conjugate to (123)(45) and (12)(34), then is the answer 0? because (12)(34) and (123)(45) have not got the same cycle structure.
|
Assuming that you start out with a root node, and decide with 50% probability whether or not to add two children nodes. If they do, repeat this process for them. How can you find the average length of this random binary tree?
I'm thinking along the lines of
$\displaystyle\lim_{n\to\infty}\sum\limits_{i=1}^n (\frac{n}{... |
The solid angle spanned by a disc of unit radius, as observed from a point $(r,z)$ at a distance $z>0$ above a point in the disc plane with at distance $r>0$ to the center, can be expressed as $\Omega_1+\Omega_2$ where
$$\Omega_1 =2\pi\Theta[-(r+1)(r-1)] +\frac{2 z}{\sqrt{(r+1)^2
+ z^2}}\frac{r-1}{r+1}\Pi\left[\alph... |
If you have a group, (say you have group table or any other information), is there an algorithm to find the group presentation? What is the general way of finding presentation of a group?
|
There are plenty of theorems out there as well as laws within mathematics. For example, in Boolean algebra:
Theorems
Idempotent
Involution
Theorem of Complementarity
Laws
Commutative
Associative
Distributive
There are countless other examples out there, but my real question is this: What makes a theorem a theorem... |
I often get confused about
when a Markov chain has an equilibrium distribution;
when this equilibrium distribution is unique;
which starting states converge to the equilibrium distribution; and
how finite and countably infinite Markov chains different with respect to the above.
(Google isn't quite clearing up my co... |
I am trying to prove that given a subspace $\mathbf{W}$ in $\mathbb{R^{n}}$, the subspace and its orthogonal complement 'cover' whole of $\mathbb{R^{n}}$ through '+' where we define $\mathbf{W}$+$\mathbf{W^{\perp}}$ as linear combinations of vectors both in the subspace and in its orthogonal complement. It seems intuit... |
It is a well known fact that the geometric series
$$1+x+x^2+x^3+\ldots$$
has the following form
$$\frac{1}{1-x}$$
Another possible representation is
$$\prod_{k=0}^{\infty}\left(1+x^{2^{k}}\right)$$
This comes from the identity
$$1+x+x^2+x^3+\ldots+x^{2^{k}}=\frac{1-x^{2^{k}+1}}{1-x}$$
now taking the numerator of the... |
I have the following generalized eigenvalue problem
$\det[P_i^TLP_j+zP_i^TP_j]=0$
$L$ is a positive-semidefinite matrix with 1 eigenvalue at 0. More precisely, it is the combinatorial Laplacian matrix for a connected graph.
$P_i$ ($P_j$) is the identity matrix with the $i$-th ($j$-th) column removed.
What, if anything... |
Consider the inner product by $\langle f,g \rangle_{H^1} = \langle f, g \rangle_{L^2} + \sum_{|\alpha|=1} \langle D^\alpha f, D^\alpha g \rangle_{L^2}$ where $\alpha$ is a multi-index and $D$ denotes the weak derivative. Define $H^1(\Omega)$ as the space of functions that are finite under the norm induced from this inn... |
Please suggest a reference for a proof of Gibbs phenomenon at jump discontinuities of a function.
|
I would like to rewrite this two-level program as a Non Linear Program using KKT conditions
$\min_{x \in X, \space y \in Y} \space f(x)$
$\text{s.t. } \space g(x, y) \leq 0$
$y \in \text{argmin}_{z \in Y} \space F(x,z)$
$\space \space \space \space \space \space \text{s.t.} \space Ax + Bz - b \leq 0$
Here $A$ and $B$ a... |
This is one of those perhaps rare occasions when someone takes the advice of the FAQ and asks a question to which they already know the answer. This puzzle took me a while, but I found it both simple and satisfying. It's also great because the proof doesn't use anything fancy at all but it's still a very nice little ... |
Define $D^n = \{ x \in \mathbb{R}^n : |x| \leq 1 \}$. By identifying all the points of $S^{n-1}$ we get a topological space which is intuitively homeomorphic to $S^n$.
If $n = 2$, this can be visualised by pushing the centre of the disc $D^2$ down so you have a sack, then shrinking the boundry of the sack to a point wh... |
If you form the Euler-Lagrange equation for some calculus of variations problem in x, f(x) and f'(x), and the resulting expression is always non-negative over the domain of x (because the expression is independent of f(x) and f'(x)), does that mean any perturbation that increases f(x) increases the functional? This see... |
When $a_2/a_1 = b_2/b_1$, $a_1 \neq b_1$, we have
$$\frac{a_{1}}{a_{2}/a_{1}}+\dfrac{b_{1}}{b_{2}/b_{1}}=
\frac{a_{1}+b_{1}}{1+\dfrac{(a_{2}+b_{2})-(a_{1}+b_{1})}{a_{1}+b_{1}}}.$$
So why when $a_2/a_1 \neq b_2/b_1 , a_1 \neq b_1$ we don't have a similar equality?
$$\frac{a_{1}}{a_{2}/a_{1}}+\dfrac{b_{1}}{b_{2}/b_{1... |
symbolize the following in Lg using the the interruption function below.
U: {o|o is a person}
L2: { , m loves n}
H1: { p|o, o is exaited}
C1: {o|o, o is a kid
b: Bob
"if any loves Bob, then Bob loves everyone." if theres a x (Lxb --> Lcx)
right?
"happy people and kids love Bob." For any x[Hx ^ If therex a x Hbx) ..... |
The Fokker-Planck equation for several variables is :
$$\frac{\partial W}{\partial t} = L_{FP}W\qquad(1)$$
where
$$L_{FP} = -\frac{\partial}{\partial x_i}D_i(\{x\})+\frac{\partial^2}{\partial x_i \partial x_j}D_{ij}(\{x\}).\qquad(2)$$
The summation convention for Latin indices is used here. The drift vector $D_i$ and ... |
I have solutions to Exercises 5.16–5.19 in Atiyah–Macdonald's Introduction to Commutative Algebra, but not in the order desired; I find myself needing later exercises to do earlier ones, and it's been frustrating me. Online solution sets (I count five, in various stages of completion) seem to either not notice this pro... |
I have a set of points $(x, y)$ where each one comes from either one of two linear functions:
\begin{align*}
y &= m_1 x + b_1\\
y &= m_2 x + b_2
\end{align*}
Is there a fitting method to find such functions, without knowing from which function each of the points come from?
PS. can somebody add fit (or fitting) ... |
Suppose we have a $n\times m$ rectangular grid (namely: $nm$ points disposed as a matrix with $n$ rows and $m$ columns).
We randomly pick $h$ different points in the grid, where every point is equally likely.
If only horizontal or vertical movements between two points are allowed, what is the probability that the point... |
Sorry for my English.
Here is the question:
Graph (V,E).
Definition: Legal's up-path in a graph from s to t is existent if and only if for every Vi, Vi+1 (for each i) fulfill w(Vi)<=w(Vi+1), when w(v) is a weight of the vertex.
We know that there is an algorithm like BFS that solve the problem of finding the shorter p... |
One of my professors wrote the following open question on the blackboard:
If $M$ is a compact, connected smooth $4$-manifold such that $\pi_1(M) = 0$, $\pi_2(M) = 0$ (first two homotopy groups are trivial), does it follow that $M$ is diffeomorphic to the $4$-sphere?
and warned us, that if we managed to solve it, we wou... |
How do I prove the following:
For every irrational number $q$, given $\varepsilon>0$, there exist natural numbers $N$ and $M$ such that $|Nq-M|<\varepsilon$.
|
I've been presented with a function expansion which I'm told is correct but I can't figure out where the sign in the second term might be coming from.
$$
e^{i\alpha(x_\mu + \epsilon \, n_\mu)} = e^{i\alpha(x_\mu)} ( 1 - i\,\epsilon \,n^\nu\, \partial_\nu \alpha(x_\mu) )
$$
$x_\mu$, $n_\mu$ are four vectors, the... |
I've long dreamt of an occasion where hundreds or thousands of mathematicians work on a single problem in a truly coordinated way. Well, now that I'm nearing my doctoral defense in mathematics, I've decided to consider making this dream a reality.
But I can't do it alone....
What are some of your thoughts on how best ... |
I checked most of the posts about Taylor expansion with scalar functions. Could anyone tell me what is the multivariate version of Taylor's Theorem, and how I can use it?
|
Given continuous and Periodic function,
How can I prove that it is Uniformly continuous?
Thank you!
|
Show that deleting at most (m-s)(n-t)/s edges from a K_{m,n} will never destroy all its K_{s,t} subgraphs. I'm completly stuck here and any help would be welcome!
|
Vector $\vec{a}$ can be broken down into its components $\vec{a}_\parallel$ and $\vec{a}_\perp$ relative to $\vec{e}$.
$\vec{a}_\parallel = (\vec{a}\vec{e})\vec{e}$
and
$\vec{a}_\perp = \vec{e} \times (\vec{a} \times \vec{e})$ (f1)
The orthogonal part can be found via application of the triple product:
$\vec{a}_... |
Given two connected open sets $U,V \subset X$ such that $U \cap V$ is path connected and $U \cup V = X$, then $\pi_1(U) \ast_{\pi_1(U \cap V)} \pi_1(V) \cong \pi_1(X)$. This is of course the Seifert Van Kampen theorem, a question in Munkres asks if the homomorphism induced by the inclusion map of $i: V \rightarrow X$ i... |
I wonder is there an elementary proof for Fermat's last theorem. Why it's so difficult to prove this theorem by elementary method?
Thanks,
|
There are few well-know styles of logical systems (LS): Hilbert-style, Sequent-style, Natural deducton style. And such lesser-know styles as Gottlob-Frege two-demensional notation, systems with graphical syntax etc. (In wikipedia)
And I have some questions:
For example, is the Resolution method (language of clauses an... |
I'm doing one of the exercises of Stewart and Tall's book on Algebraic Number Theory. The problem concerns finding an expression for the norm in the cyclotomic field $K = \mathbb{Q}(e^{2\pi i / 5})$. The exact problem is the following:
If $\zeta = e^{2 \pi i / 5}$, $K = \mathbb{Q}(e^{2\pi i / 5})$, prove that the nor... |
I've been reading the following papers in access control.
http://faculty.nps.edu/dedennin/publications/lattice76.pdf
http://www.winlab.rutgers.edu/~trappe/Courses/AdvSec05/access_control_lattice.pdf
The papers ultimately try to reduce the access control mechanisms to lattice based access control. But I do not understan... |
NOTATION: Let $\mathcal{C}=\mathcal{V}(F)\subseteq\mathbb{P}^2$ be a curve of degree $3\!=\!deg(F)$ with no singularities and let $A_0\!\!\in\!\mathcal{C}$ be fixed. Let $Div(\mathcal{C})$ denote the group of divisors on $\mathcal{C}$, i.e. the set of all formal sums $$\{\sum\limits_{P\in\mathcal{C}} n_PP\,|\; n_P\!\in... |
Let $f \in m(\Omega,\mathcal{F})$, i.e. $f \mapsto [-\infty,\infty]$ and let $A \in \mathcal{F}$ be an atom. Prove that $f$ is almost everywhere constant on A: there exists $k \in [-\infty,\infty]$ such that
$\mu (\{\omega \in A : f(\omega) \neq k \} )=0$.
I was thinking let $k=\frac{1}{\mu(A)}\int \limits_{A} f\,d\mu... |
I'm trying to prove that, given a principal fibration $\Omega B \rightarrow F \stackrel{p}{\rightarrow} E$ such that $p$ is a retraction of $F$ onto $E$, the total space $F$ is homotopy equivalent to the product $\Omega B \times E$. This is essentialy a problem from Hatcher (4.3, Q22).
Anyway, since $E$ is a retract of... |
D. My first post here ::- >. I got a rather simple question. But please, allow me to introduce myself a bit first. I think it's polite for a first post ::- D.
I'm a game developer (free Flash games) with a few ahem big holes in my math (read: I haven't done any math since high school - thanks to the school, I was total... |
This is spurred by the comments to my answer here. I'm unfamiliar with set theory beyond Cohen's proof of the independence of the continuum hypothesis from ZFC. In particular, I haven't witnessed any real interaction between set-theoretic issues and the more conventional math I've studied, the sort of place where you... |
I want to show that there are no simple groups of order $p^{k}(p+1)$ where $k>0$ and $p$ is a prime number.
So suppose there is such a group. Then if we let $n_{p}$ denote the number of $p$-Sylow subgroups of $G$ we have that $n_{p}=p+1$. Now by letting $G$ act on $Sylow_{P}(G)$ by conjugation we obtain a group homomor... |
Let $a ,$ $b ,$ $\alpha ,$ and $\beta$ be non-zero binary strings of length exactly $n$ where $$ \beta = a \alpha + b. $$
Now, consider the following scenario. Fix $a$ and $b.$ I give you $\alpha$ and $\beta,$ but keep $a$ and $b$ as secrets from you. What is the probability that you can modify both strings (same lengt... |
I have run across the following multinomial series:
$$ \sum_{j = a}^{N}
\binom{N}{j} \binom{j}{a} d^{-j}
$$
Here, $d>1$. This seems like a formula which has either a well-known identity, or which has no further closed form or simplification. Can anyone shed any light on this formula, or possibly point me to a... |
I mean if there is any construction that satisfies all the conditions for an vectorial space except it lacks an identity element? This questions was posed to me by a classmate last semester and I have been puzzling over it since then.
|
According to the Wikipedia article, the Simplex algorithm depends on constraining all the unknowns to be >= 0. I have a problem where one of my variables is highly likely to be negative in many cases. How can I adapt the problem so I can use Simplex?
EDIT: example: say I have x, y and z and I want to find values (bot... |
It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.
Along similar lines, you can show that $\mathbb{R^2}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>2$ by subtracting a point and checking if t... |
In general, are papers published by mathematicians the same thing as PhD theses? In other words, if a mathematician submitted a paper for a PhD would it get accepted? Or are papers of higher quality than PhD theses?
|
The first few important algebraic number fields I have read about are:
$\mathbb{Q}$ The integers
$\mathbb{Q}[\sqrt{d}]$ quadratic
$\mathbb{Q}[e^{\frac{2 i \pi}{p}}]$ cyclotomic
What could be read about next?
|
I have to compute this limit:
$$\lim_{x \to\infty} 3(\sqrt{\strut x}\sqrt{\strut x-3}-x+2)$$
wolfram alpha says that answer is $\frac{3}{2}$, but I can't get why. Does anyone know how to get this limit?
|
$$\frac{(z x^{-1} y)^5 y^5}{x^{-4} z^{-4}}$$
How would I evaluate this if $x = 10$, $y = -3$ and $z = 3$ ? I would like a step-by-step solution to help me fully understand it.
|
I have the following summation:
$$\sum_{i=0}^n\left(6in + n(8n+2)\right).$$
Evaluating I get
\begin{align*}
6n\left(\frac{n(n+1)}{2}\right) + n(8n+2) = 3n(n(n+1)) + n(8n+2)\\
&=3n^2(n+1) + n(8n+2)\\
&=3n^3 + 3n^2 + 8n^2 + 2n\\
&=3n^3 + 11n^2 + 2n.
\end{align*}
Is this correct?
|
A curve has parametric equations
$$\begin{array}{rcl}
x & = & \cos(t)\\\
y & = & \frac{1}{2}\sin(2t)
\end{array}\qquad \text{ for }0\leq t \leq 2\pi.$$
Show that the Cartesian equation of the curve is $y^2=(x^2)(1-x^2).$
What are the steps for showing this?
|
Let $R$ be a ring, $I$ an ideal. According to Atiyah-Macdonald, if $R$ is Noetherian, then, we have $\hat{I}=\hat{R}I$ where hat denotes $I$-adic completion of $R$ and (I presume) $\hat{I}$ denotes the induced completion on $I$. I don't understand how to arrive at this equality and why the Noetherian hypothesis is nece... |
How to show that $\sin(2 \pi nx)$ does not converge as n goes to infinity? $x \in (0,1/2)$
|
Let $f$ be a strictly increasing positive continuous function defined on $[1,\infty)$ with limit $\infty$ as $x$ goes towards $\infty$. Then $\sum_{k=0}^{\infty} \frac{1}{f(k)}$ converges if and only if $\sum_{k=0}^{\infty} \frac{f^{-1}(k)}{k^2}$ converges, where $f^{-1}$ denotes the inverse of $f$.
I saw this claim a... |
I have some LP problem and I'm willing to solve it (this is an exercise from some optimization-related book).
Now, Mathematica tells me that the problem is unbounded and I want to make a generic proof of that.
What would be the easiest way to prove that the selected linear optimization problem is unbounded (and obvious... |
So I am having a lot of trouble proving this. It was for an assignment due tuesday, but the prof said I can have a bit of extra time with this question.
Please do not give me the answer because I am sure my prof is on this website :P
okay so the question is as follows:
Let $R$ be a ring with identity. Show that the m... |
I was curious (even though this is a very amateur question)... what would the multiplication equivalent of sigma (the summation symbol) be?
$$\sum$$
I want to do a series of multiplication of terms.... instead of addition...
|
Question: In Functional Analysis we can note things like: every closed subspace of a Banach space is Banach. In this case, what does "closed subspace" mean?
Does this mean closed under the norm topology?
Or does this mean closed in the sense that multiplication of scalars and addition of vectors is closed?
Or ... |
$$
\begin{align*}
&= (n+1)! − 1 + ( (n+1) · (n+1)! )\\
&= (n+1)! (1+n+1) − 1\\
&= (n+1)! (n+2) − 1\\
&= (n+2)! − 1\\
\end{align*}
$$
I'm confused at how the first step goes to the second step. Can someone please clarify this?
|
Question:
The area covered by a newly-introduced species of invasive weed doubles every 4 years. Suppose the weed currently covers 23 km^2.
a) Write an equation that gives the area, A, covered by the weed, in t years.
b) Estimate how many years it will take for the weed to cover 51 km^2
Equation:
$$A = 23(2)^{t/4}$$
I ... |
I need to calculate the inverse Fourier transform of the following functions:
$\displaystyle f(w) = e^{(-\pmb{i}5w)} * {\rm sinc}(2w) $
$\displaystyle g(w) = \frac{\pmb{i}w}{(3+\pmb{i}w)(1+\pmb{i}w)} $
I'm really stuck and I don't know where to start, any help would be appreciated!
|
While reading through my textbook it says "the most important example of an inner-product space is $F^n$", where $F$ denotes $\mathbb{C}$ or $\mathbb{R}$ .
Our definition of an inner product on a vector space $V$ is as follows:
1) Positive definite: $\langle v,v \rangle \ge 0$ with equality if and only if $v=0$
2) Lin... |
A two dimensional silo shaped figure is formed by placing a semi-circle of diameter 1 on top of a unit square, with the diameter coinciding with the top of the square. How do we find the radius of the smallest circle that contains this silo?
|
I am given a square matrix $A$, and I need to prove that if c is its eigenvalue, then it is also an eigenvalue of its transpose. How should I approach this? Clearly $Av$=$cv$, but I am not sure how to bring transpose into the equation.
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Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to non-commutative groups in such a way that these groups contain more information than the higher homotopy groups?
|
The Nullstellensatz for $\mathbb{C}[x_1, \ldots, x_n]$ gives a dictionary between radical ideals and varieties, which makes the following assertion obvious: a radical ideal in $\mathbb{C}[x_1, \ldots, x_n]$ is the intersection of the maximal ideals containing it. Is there a simple purely algebraic way of seeing this?
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