instruction stringlengths 22 15.8k |
|---|
I want to show that $G$ is open in $X$ iff $\overline{G \cap \overline{A}} = \overline{G \cap A}$ for every $A \subset X$.
This direction $\rightarrow$ is easy.
I'm stuck in showing that $\overline{G \cap \overline{A}} = \overline{G \cap A}$ for every $A \subset X$ implies $G$ is open in $X$.
So I guess the game is to... |
A question from Dugundji's book which I don't even understand the statement.
Describe the relative topology of $\{z: |z|=1\}$ as a subspace of ${\mathbb{R}}^{2}$.
What do they mean by "describe"? I don't understand what they are asking for. Can you please clarify?
|
(1/8)(3m - 5) = (6/5)(6m - 2) + 5...Type your answer as a fraction. Do not use decimals!
**i got -129/273 is this correct ?
|
Two questions from Dugundji's book (not hw, just practice).
1) Let $Y_{1}, Y_{2}$ be subspaces of $X$ and $A \subset Y_{1} \cap Y_{2}$. Assume that $A$ is open in $Y_{1}$ and open in $Y_{2}$. Prove A is open in $Y_{1} \cup Y_{2}$.
Can you please give a hint for this one?
2) a. Let $D$ be dense in $X$. Give an example t... |
First off: I barely have any set theoretic knowledge, but I read a bit about cardinal arithmetic today and the following idea came to me, and since I found it kind of funny, I wanted to know a bit more about it.
If $A$ is the set of all real positive sequences that either converge to $0$ or diverge to $\infty$, we put ... |
This is a similar question to this one, but slightly different.
The question is given two edges ($e_1$ and $e_2$, with the vertex coordinates known), how to find the angles from $e_1$ to $e_2$, with the angles measured in anti clock wise direction?
A diagram is shown below:
One way I can think of is to compute the cr... |
Could you a give a intuitive interpretation of curl, geometrical interpretation, real-world example or physical interpretation would be ok.
EDIT: Consider a specific vector field
$$\mathbf{v} = -y\mathbf{\hat{x}} + x\mathbf{\hat{y}}$$
$$\nabla\times\mathbf{v} = 2\mathbf{\hat{z}}$$
Could you give a intuitive interpreta... |
How do I find generators of group of rotations of d-dimensional cube into itself when viewed as a subgroup of symmetric group of order $2^d$?
This is related to my previous question but not specializing to $d=3$
|
I have two numbers $x$ and $y$ which are $\log_2$ of two other numbers $P$ and $V$. I'm trying to calculate $\log_2 (P-V)$ without transforming $x$ and $y$ back to the linear world.
If $P = AV$, then $P-V = V(A-1)$, and $x-y = \log_2 A$. So, if I could calculate $\log_2 (A-1)$ I could just add it to $y$ and be done. ... |
I'm currently creating a program to plot simple graphs in 2D. In order to draw those properly I have to know if a function graph is continuous between 2 points on the graph.
Assuming the graph for f(x)=tan(x) and two points on the graph P1(1,tan(1)) and P2(2,tan(2)). Is there an algorithmic way to tell if the graph is ... |
For example, I have set of points in 3D. Points lie on straight line. Transformed set of points lies on straight line too. How to check if transformation is affine?
|
Let $A,B \text{ and } C$ are three sets then if $ A \subset B, B \subset C, C \subset A \Rightarrow B = C $
How could we prove this ?
|
I have two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ on a line, $L$,
and another point $P_0(x_0, y_0, z_0)$.
I want to find the distance between $P_0$ and $L$. Could someone help?
|
let $X$ be some Banach space. Let $L(X)$ be the set of continuous operators on $X$ into $X$.
Let $(\tau_i)_i$ be a set of topologies on $L(X)$ s.t. $L(X)$ is topological vector space (i.e. addition and scalar multiplication continuous, not necessarily Hausdorff).
Let $\tau$ be the initial topology on $L(X)$ induced by ... |
Apologies if the title is unclear, I couldn't think of a good way to express what I want.
Given the usual 2D representation of the complete graph $K_n$ as a regular $n$-gon with each pair of vertices joined, let $U_n$ be the set of the disjoint regions of space delimited by the edges of the graph (so $|U_3|=1, |U_4|=4,... |
A while back, I wanted to see if the notion of the arithmetic-geometric mean could be extended to a pair of symmetric positive definite matrices. (I considered positive definite matrices only since the notion of the matrix square root is a bit intricate for other kinds of matrices.)
I expected that some complications w... |
In the definition of the complexity class NP, we allow the certificate to be polynomial in the input size. But do we ever need a certificate that is longer than the input?
In all the NP problems I could think of, we don't. The "worst" problem was factorization, where the obvious incoding is $n+O(\lg(n))$ (where n is l... |
Three fellows accused of stealing CDs make the following statements:
(1) Ed: “Fred did it, and Ted is innocent.”
(2) Fred: “If Ed is guilty, then so is Ted.”
(3) Ted: “I’m innocent, but at least one of the others is guilty.”
If the innocent told the truth and the guilty lied, who is guilty?
(Remember that false stateme... |
What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a first-year-student in college. Do you know a good book on the subject?
Thanks... |
Please forgive me if this is not the right Stack Exchange (I also posted it at Cross Validated and Theoretical Computer Science). Please also forgive me for inventing terms.
For discrete random variables X and Y, the mutual information of X and Y can be defined as follows:
$I(X;Y) = \sum_{y \in Y} \sum_{x \in X}
... |
I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
|
First some context, I'm not a mathematician, not even close (as you will soon see) I do grasp some things about it but in a need to know basis, so plain english answers are appreciated (too).
I can't figure out this one even though some have tried very hard to explain/convince me.
So let's say I'm on a roulette (let's ... |
Greetings,
I'm performing research that will help determine the size of observed space and the time elapsed since the big bang. Hopefully you can help!
I have data conforming to a piecewise linear function on which I want to perform two linear regressions. There is a point at which the slope and intercept change, and I... |
I was trying to write a poker program yesterday, and trying to figure out a way to tell the computer how to detect a straight. I realized that if you add $5$ consecutive numbers and divide by $5$, you'll get no remainder. I tried for $6$ consecutive numbers, but it didn't work. It turns out it only works for odd number... |
I have asked this question on cstheory.
Let $\Pi$ be NP-complete problem.
Can we partition the set of instances of $\Pi$ into finite number of subsets (subproblems) each of which is polynomially solvable (and not necessarily polynomially recognizable)?
On cstheory I have obtained two answers on my question: "trivially ... |
Suppose we have a square real $n\times n$ matrix $X=[x_1,...,x_n]$, where $x_i$ is $i$-th column of the matrix.
Now define $X_k=[x_1,..,x_k]$, i.e. matrix $X_k$ columns are the first $k$ columns of the matrix $X$. Define $\lambda_k$ as the maximal eigenvalue of $(X_k^TX_k)^{-1}$.
Is it possible to prove that $\lambd... |
Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \infty}\,\lim_{n\to \infty}{a_{nm}}$? Bonus points for necessary and sufficient conditions.
For an example of a sequence w... |
Wikipedia says the midpoint formula for numerical integration has error of order $h^3 f''(\xi)$. I am trying to replicate this result.
I'm guessing that I want to use Lagrange's formulation of the remainder for Taylor series. Let $x_0=\frac{a+b}{2}$ (i.e. the midpoint).
The midpoint method says $\int_a^b f(x)dx \appro... |
Suppose $w(z)$ is an analytic function in a domain $G$ which is symmetric with respect to the real axis. Show that $f(z) = \overline{w(\bar{z})}$ is then an analytic function of $z$ in $G$.
So this is interesting because we are looking at the conjugate of a function of a conjugate. Let $w(z) = u(x,y)+iv(x,y)$. Then $f... |
Suppose $\Omega$ is a bounded domain in the plane whose boundary consist of $m+1$ disjoint analytic simple closed curves.
Let $f$ be holomorphic and nonconstant on a neighborhood of the closure of $\Omega$ such that
$$|f(z)|=1$$ for all $z$ in the boundary of $\Omega$.
If $m=0$, then the maximum principle applied to $f... |
I read somewhere that a set is infinite if and only if it has a proper infinite subset. I also remember seeing someones name attached to this theorem on Wikipedia once, but I can't even find that now. I haven't been able to find a proof of this theorem, nor been able to generate one myself.
I can prove that if a se... |
Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result:
(A-S 1.6.3) Prove that for every two independent identically distributed real random variables $X$ and $Y$,
$$Pr[|X-Y| \leq 2... |
Are there any analogues of Galois Theory for complex numbers (with non-zero imaginary part)? This is motivated by Schwarz Principle. It says that if $f$ is analytic, $f$ is defined in the upper-half disk, and $f$ extends to a continuous function on the real axis, then $f$ can be extended to an analytic function on the ... |
I am under the impression that the standard convention for the homology (singular) of the empty set is 0 in all non negative degrees and $\mathbb{Z}$ in degree $-1$. I have no problem with this convention, I am just curious what role it plays. Most conventions helps something sensible remain true in a particular case, ... |
For an university assignment I have to visualize some curves in 3 dimensional space.
Until now I've implemented Bézier, helix and conical spiral.
Could you give me some advice about some famous curves in geometry history?
|
For improper single integrals with positive integrands of the first, second or mixed type there are the comparison and the limit tests to determine their convergence or divergence. There is also the absolute convergence theorem.
For multiple integrals I know that the comparison test can be used as well.
Question: But c... |
How to prove that the Sorgenfrey line is hereditarily separable?
|
I'm going through Enderton's Elements of Set Theory, as I heard it is a gentle introduction to set theory. I'm a little confused on how the subset axioms are used. In the text, the axiom is given as:
Subset Axioms For each formula $\underline{\hspace{1cm}}$ not containing $B$, the following is an axiom:
$$
\forall ... |
Almost everyone is familiar with the famous Taylor Series:
$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n $
which, if it converges at more than one point, will converge in some interval about $a$. Has anyone considered the "Reverse" Taylor Series:
$ g(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x-a)}{n!} a^n $
I c... |
I've revised my question a bit in response to the (very helpful) advice so far--
I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather unstructured; the Fourier transform sparked my interest in mathematics, and I've been following since I first learned of it as... |
I became interested in how numbers actually work recently. I want a source that covers number theory. Explaining how and why numbers react the way they do. A friend recommended Mathematical Proofs: A Transition to Advanced Mathematics.
I want to learn about the concept of numbers, how they interact etc. I am a high sch... |
It's famously unknown whether the natural log of 2 is rational or not.
How about the natural log of other numbers. Is it known/unknown whether these are rational?
Obviously ln(1) is 0, and ln(2^n) is n*ln(2) (and is thus rational iff ln(2) is rational), but how about other cases?
|
Let $T: E \rightarrow E$ be an endomorphism of a finite-dimensional vector space, and let $S$ be a circle in the complex plane that does not intersect any eigenvalues of $T$. Now let $Q = \frac{1}{2\pi i} \int_S (z-T)^{-1} \, dz$.
Why is $Q$ a projection operator?
The motivation behind this question is that the above ... |
I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.
However when it comes to the area under a curve for some reason when you break it up into an infinite amount of rectangles, magically it turns into the anti-d... |
Suppose $\kappa$ is an uncountable cardinal. Then $S\subseteq\kappa$ is club (CLosed UnBounded) if $S$ is unbounded in $\kappa$ and is a closed subset of $\kappa$ under the order topology. My question is the following: suppose $S\subseteq\kappa$ is club, and $S=S_1\cup S_2$. Is there necessarily some club set $C$ such ... |
This is part of an assignment that I need to get a good mark for - I'd appreciate it if you guys could look over it and give some pointers where I've gone wrong.
(apologies for the italics)
$$\prod_{n=1}^\infty\left(1+\frac{2}{n}\right)\; \text{ converges when }
\sum_1^\infty \ln\left(1+\frac{2}{n}\right)\; \text{ co... |
One of the early applications of fractional calculus is the tautochrone problem set up by Abel in the integral form or its fractional derivative one. i wish to know its solution.
|
OK, I know that Matiyasevich's solution to Hilbert's 10th problem shows that there is no algorithm to decide whether or not a polynomial $p(x_1,\ldots,p_n)$ with integer coefficients has a solution over the integers. I have no doubts about the truth of that. However, I have been unable to figure out why the following... |
I got an assignment to prove certain things about right triangles in Lobachevsky geometry, but so far I don't know where to start. What model is the best for studying these objects? What is the general approach to this kind of task?
|
Everything is in $\mathbb{Z}$. Let $v_1 < v_2 < ... < v_n = k$, and $v_1 = 1$ for $k >> n$. Let $ P = \Pi_{i < j} (v_j - v_i)$. How can I show that $P \le k^{n^2}$?
There are $n + (n-1) + ... + 1 = \frac{n(n+1)}{2}$ terms in the product. Starting from $v_n - v_{n-1} = 1$, etc. Clearly, $P = 1(1*2)(1*2*3) ...(k-1)! = \P... |
I have n points that determine a polygon. I need to color the interior of this polygon with a color . How can I do this?
|
The eigenvalues of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same.
My question
What is the intuition that the eigenvalues are complex? Why do they exist at all for rotation matrices? I mean it is not so that every time a... |
Given the Right ascension in h m s, Declination in deg ' " and the Trigonometric parallax How can I get the cartesian (x,y,z) coordinates of a star? I'm guessing I need 3 separate formulas to get each x, y and z values.
|
there is a prove that there is no $2^n$ that is divisible by 5?
|
I'm trying to develop an architecture hardware to make an implementation of an algorithm that can be descomposed in terms of sums, multiplications, subtractions and exponential functions. I'm trying to model $\exp(-x)$ through Taylor series. The domain of my function is bounded between $0$ and $1500$, but I want to use... |
It is well known how to use algebraic geometry (differentials, divisors, and Riemann-Roch) in order to prove the finite dimensionality of the vector space of modular forms of some fixed weight and level.
Is it possible to prove the finite dimensionality of modular forms using representation theory? If so, any reference... |
If we start with $\int\frac{v}{g-2v} dv$ how would we go about integrating it? (g is a constant)
The answers to the past exam paper I have tell me to rearrange it to $\frac{1}{2}\int ( -1 + \frac{g}{g-2v} ) dv$ which integrates to $\frac{1}{2}v-\frac{1}{4}g\ln(g-2v) + c$
I can see that the rearrangement works 'in rever... |
My question is ; How can I solve the following integral question?
$$\int{(1 + \cos x)^3\mathrm dx}$$
Thanks in advance,
|
Is $$\left |\int\limits_1^N e^{ix^\alpha} \ \text{dx} \right| = \mathcal{O}(N^\alpha)?$$
where $\displaystyle 0\lt \alpha \lt 1$ and $\displaystyle i=\sqrt{-1}$.
${\bf Edit.}$ As Willie Wong shows below, the correct equation is
$$\left |\int\limits_1^N e^{ix^\alpha} \ \text{dx} \right| = \mathcal{O}(N^{1-\alpha}).$$... |
Let $A \subset X$ be closed and $U \subset A$ open in $A$. Let $V$ be any open
set in $X$ with $U \subset V$.
Prove $U \cup (V \setminus A)$ is open in $X$.
|
Given a directed graph $G\langle V,E\rangle$, and weight $w[u][v]$ of each $\langle u,v \rangle$ representing an edge of $E$ the question is, try to divide the whole graph into several "rings", making each vertex $u$ belong to a certain "ring" and a "ring" contains at least two vertices, it's guaranteed that the given ... |
(I'll keep this one short)
Given a Dirichlet series
$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$
where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero real numbers), and assuming that $g(s)$ can be analytically continued, does it follow that $g(s)$ possesses a critical strip ... |
How may one go about proving
$\displaystyle\frac{\Gamma'(s)}{\Gamma(s)}=O(\log|s|)$,
(away from the poles) directly? By a direct proof, I mean not to go through the usual Stirling formula with the exact error term. The use of a rough form of Stirling's formula is welcome.
|
I'm working with percentage splits and I'm trying to add extra players into the split but I'm coming unstuck...
Scenario:
I have agreements with a number of parties that set out what type of split we get from a pot of cash. With some it's 50/50, with others it may be 60/40 in favour of either party or perhaps even 70/... |
The integral test of convergence states that, if $f:[1,+\infty)\to[0,+\infty)$ is a monotonically decreasing nonnegative function, then the series $\sum_1^\infty f(n)$ converges iff $\int_1^\infty f(n) dn$ is finite.
Is the high-dimensional generalization also true? That is, given $f:[1,+\infty)^N \to[0,+\infty)$, and ... |
I'm doing a physics course in my spare time and the problem set is:
Ball A is dropped from rest from a building of height H exactly as ball B is thrown up vertically from the ground. When they collide A has twice the speed of B. If the collision occurs at height h, what is h/H?
It also gives a hint: Write equations f... |
This is prompted by question 15312, but moved to the reals. It must be solved already. Pick n points $x_i \in [0,1]$ to maximize $\prod_{i < j} (x_i - x_j)$. A little playing shows you don't want them evenly distributed-they need to push out to the ends. With four points, Alpha says to use $\{0,\frac{1}{2}\pm\frac{... |
The hint for this problem is $a^{n+1} - 1 = (a + 1)(a^n - 1) - a(a^{n-1} - 1)$
I see that the problem is true because if you distribute the $a$ and the $-1$ the terms cancel out to equal the left side. However, since it is telling me to use strong induction I am guessing there is more I am supposed to be doing. On th... |
I was reading a comment on MathOverflow and did not understand what was being suggested. I also don't understand Anton's response:
https://mathoverflow.net/questions/17732/difference-between-measures-and-distributions/18062#18062
I initially thought Regenbogen's comment was being applied to the derivative of the dirac ... |
I have found the following ellipse representation $(x,y)=(x_0\cos(\theta+d/2),y_0\cos(\theta-d/2))$, $\theta \in [0,2\pi]$. This is a contour of bivariate normal distribution with uneven variances and correlation $\rho=\cos(d)$. I know that this is a rotated ellipse with centre $(0,0)$. How to find the lengths of major... |
I just started real analysis. I don't have a background in proofs or logic, simply calculus. So I'm trying to learn more about proofs--so forgive the basic question, please.
How do you go about proving this theorem: If $a$ and $b$ are any numbers, then there is one and only one number $x$ such that $a + x=b$. This n... |
Let $\kappa$ be a measurable cardinal, we say that $\mathcal{D}$ is a normal ultrafilter iff whenever $g\in\kappa^\kappa$ such that $g<_\mathcal{D} Id$, we have some $\alpha<\kappa$ such that $\{\gamma<\kappa | g(\gamma) = \alpha\}\in\mathcal{D}$.
Given $\kappa$ a measurable cardinal and some $M$ which is a transitive ... |
I'm looking for an algorithm for the transitive reduction of a digraph (a DAG in fact).
The wikipedia article displays a formula $R^-$ = $R - (R \times R^+)$ — where $R^-$ is the transitive reduction of $R$ and $R^+$ is the transitive closure of $R$ — but that Cartesian product doesn't form an edge (unless I'm missing ... |
While combing around my notes looking for other possible examples for this question, I chanced upon another one of my unsolved problems:
Cycloidal curves are curves generated by a circle rolling upon a plane or space curve. It's not too hard to derive the required parametric equations if the fixed curve is a plane curv... |
I'm developing an open-ended strategy game. I am using the following formula to calculate damage (In PHP)
$rand = rand($a, $b) + $c;
$damage = $rand * sqrt(($d / 20) * $c));
a, b, c, and d are all values that can be modified by the user over the course of play, either by buying a better item (a and b), investing in t... |
How do we numerically integrate a rapidly decaying exponential function? A simple Gauss quadrature which is based on approximating the function by polynomial, I think will not work, since rapidly decaying exponential function will require more terms in the series. Is there a series approximation for the function?
|
How do I show that $\text{Hom}(\mathbb{Z},G)$ and $G$ are isomorphic?
The method suggested by the book is to define a map $k$: $\text(\mathbb{Z},G) \to G$ by $f \to f(1)$.
I am stuck on how exactly to show that it is an isomorphism. Could anyone shed some light into this? Is $f(1)$ an element of G?
(For those who want... |
Prove that there is a largest integer $n$ such that $n \le x$ for any fixed rational $x$.
What about any fixed real number $x$?
|
If we define $x$ = displacement, $v$ = velocity and $a$ = acceleration then I am used to the ideas that $a= \frac{dv}{dt} = \frac{d^2x}{dt^2}$
However I also understand $a=v \frac{dv}{dx}$. Can someone explain to me why this is?
|
Given set of points in $3D$ ( $X = (x_1, x_2, x_3)$, $Y = (y_1, y_2, y_3)$ ), how can I fit transformation from $X$ to $Y$?
As far as I know this is called projective transformation.
Here is example of $X$ and $Y$.
Blue and red lines in $X$ are parallel, but they are not parallel in $Y$.
|
I have this function, $e^{-x}$ bounded between 0 and 1500 and I have an approximation by Taylor Series of the same function bounded between 0 and 0.5. I would like to express my function $e^{-x}$ bounded between 0 and 15oo in terms of the last approximation I described, what is bounded between 0 and 0.5.
Does anyone kn... |
Say we have $N$ spheres indexed as $1,2,3,\dotsc, N$ such that all of them have identical weight apart from one, and we don't know if that one is heavier or lighter. We have to determine which sphere has the odd weight using just a balance scale.
We could solve this problem by weighing repeatedly, but I am interested i... |
For an abelian variety $A_{/\mathbb{Q}}$, its volume $vol(A(\mathbb{R}))$ appears in the conjectured Birch Swinnerton-Dyer formula for the L-series at 1.
I am having trouble in understanding the size of this volume, as a function of the (minimal) equations defining the variety. Hence the question:
Say we have a hyperel... |
Situation A: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at one of the coins and tells me that at least one of the coins is a head.
Situation B: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at both of the coins... |
Which is a good introductory book for Markov chains and Markov processes?
Thank you.
|
This question is about Schauder bases in Banach spaces.
Suppose $(x_n)$ is a basic sequence in a Banach space $E$. Prove that $0$ is the only possible weak limit point of $\{x_n \in E \;\vert\; n \in \mathbb{N}\}$.
|
(sorry for the draw)
I try to get the blue angle , i thought it was easy but...
the curve is a Bézier curve.
the black dot shows the tangent of D
I know position of A B C D.
I tried to get the angle by addition of angle a and angle b
the point F is found by multiplying AB per the time factor (between 0 and 1)
the poi... |
I want to make an origami of a sphere, so I planned to print some net of a pentakis icosahedron, but I have a image of another sphere with more polygons:
I would like to find the net of such model (I know it will be very fun to cut).
Do you know if it has a name ?
|
This is a very basic question that I feel that I should know the answer to but haven't been able to think through clearly.
In linear algebra, we learn that the basis of a finite dimensional vector space can be thought of as 'co-ordinates' of that space. And, we model what we intuitively understand as the euclidean plan... |
I ran into this expression in a paper I was reading, and I'm confused about part of the meaning. Here $u$ and $v$ are two polynomials.
$$u, v \in \mathbb{R}[x]$$
I'm not really familiar with usage of $[x]$ here, but if it means "nearest integer", then isn't this expression equivalent to simply:
$$u, v \in \mathbb{Z}$$... |
What does the $\ni$ (backwards element of) symbol mean? It doesn't appear in the Wikipedia list of mathematical symbols, and a Google search for "backwards element of" or "backwards epsilon" turns up contradictory (or unreliable) information.
It seems it can mean both "such that", or "contains as an element". Is this... |
I'm stuck on this problem:
I have a "truth-table" (well, I don't know if it can be called truth table, if there aren't true/false values only):
string | a | b
---------------------
x | 1 | 0
z | 1 | 0
xx | 1 | 1
xz | 1 | 1
zx | 1 | -1
zz | 1 | -1
xxx | 0 | 1
xxz | ... |
Does anyone know a combinatorial proof of the following identity, where $F_n$ is the $n$th Fibonacci number?
$$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3$$
It's not in the place I thought it most likely to appear: Benjamin and Quinn's Proofs That Really Count. In fact, this may be a hard problem, as they say th... |
Let $f: X \to Y$ be a proper morphism of noetherian schemes. If the natural map $\mathcal{O}_Y \to f_*(\mathcal{O}_X)$ is an isomorphism, then a version of Zariski's main theorem states that the fibers $X_y, y \in Y$ are all connected. (The case listed as "Zariski's main theorem" in Hartshorne is the case of $f$ birat... |
The sum of a finite number of rational numbers is of course a rational number, but the sum of an infinite number of rational numbers might be an irrational number. Can someone give me some intuition why this sum might be irrational? I just "don't feel it."
|
Let $0<a\le 1$ be fixed. Is the sequence $$a_N=\int_1^N x^{-a}e^{ix} dx$$ bounded?
|
Question: What are some good resources for a student who has taken algebraic and point-set topology and who wishes to learn how filters and ultrafilters are applied in topology?
Motivation: I've seen in a number of papers where ultrafilters are briefly mentioned as tools which can be used to generalize an argument or s... |
This question is vaguely related to: Different formulations of Class Field Theory
As I said there, I'm currently learning class field theory. For some motivation, I've also read a little about Langlands. Everything I've read about that program seems to be a generalization of the global formulation of CFT. Is the local ... |
Can you help me to prove this inequality
\begin{aligned}
|\sqrt{3} - m/n| \geq 1/(5n^2)
\end{aligned}
where m and n are integers.
Hint:$sqrt(3)$ is irrational.
|
How to understand the compactness in topology space in intuitive way?
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.