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Specifically, how to show that an affine variety over complex number is never compact in Euclidean topology unless it is a single point. I got a hint on this qiestion: Given an affine variety X, show that the image of X under the projection map onto the first coordinate is either a point or an open subset (in the Zari... |
I want to learn discrete mathematics by reading book by myself but I find sometime it's very hard to understand what author trying to say.
I want to know, are there any good online video that teach discrete mathematic?
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I hope this isn't off topic - sorry if I'm wrong.
In 1976, Kenneth Appel and Wolfgang Haken proved the claim (conjecture) that a map can always be coloured with four colours, with no adjacent regions given the same colour. This was controversial because the proof process required a computer to evaluate many different c... |
I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two Bernoulli variables $X_1,X_2$ that generate sequences of values for $n$ consecutive Bernoulli trials. Now we have $m$ inde... |
I'm trying to figure out how I can work out the number of possible valid hand combinations of a poker game that a player (opponent) could possibly have when the flop has been dealt on the table.
So I'm guessing that I have to take the probability that each hand can occur and then see which one is the highest one that c... |
I have the following question $f(x) = -x^2 + 4x$ when $x = -3$
Should the structure of the equation be as follows:
$$-3^2 + 4 x -3 = -21$$
The issue I'm having is with the double negative in the variable $x$ and in the function formula.
If anyone can help me, be much appreciated.
Thanks
|
I'm trying to derive a taxonomy from sparse "is-a" relationships and am looking for a linear algebraic solution.
More specifically, the data is noisy and I want to detect relations that violate anti-transitivity, remove the least offenders, and then fill-in the transitive relationships, inheriting all relations from pa... |
While working on the problem $g(x)= \sqrt{x^{2}-4x+4}$, I got stuck and plugged the formula into Wolfram Alpha. It simplified alot of things for me, but there is one section I'm having trouble with, since one of my weaknesses is radicals.
According to the Chain Rule, $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$, an... |
I know how to solve linear homogeneous ordinary differential equations with constant coefficients using the differential operator D, by using this method.
Is it possible to use a similar method (using the differential operator) to solve more advanced ODEs? I'm thinking of both more advanced linear ODEs, such as Euler-C... |
I am interested in the relationship between non-linear differential equations and the Hausdorff, or fractal, dimension of the set of solutions.
For example, the Lorenz Attractor, which is a standard example in the theory of dynamical systems, apparently has a "correlation dimension" of 2.05, whatever that means. Accor... |
I have the following function
$$f(x) = 3x^3 - 5x^2 - 4x + 4.$$
I would like to find the value of $f(x)$ when $x = -3$.
I have ordered the equation in the following way.
$$f(-3) = (3 \times -3^3) - (5 \times -3^2) - (4 \times -3) + 4 = -28.$$
Can anyone tell me if the following syntax is correct and if not, where hav... |
How do I solve the following?
$$\lim_{x\to 0} \int_0^1 \cos\left(\frac{1}{xt}\right)\, dt$$
|
I'm starting a mini-course on Combinatorics and, although I can "see" the results, I'm having difficulties proving them.
For instance,
Being $\#A = n \in \mathbb{N}$, prove that $\#A^{k} = n^{k}$, for $k \in \mathbb{N}$.
I understand that $A^{k} = \{(x_1, \ldots, x_k) : x_i \in A\}$, and that we have $n \times n \ldo... |
I've a $m \times n$ column matrix $A$ that can contain values {0,0.5,1}. i.e $\Omega = \{0,0.5,1\}$
$A$ =
$\begin{bmatrix}
0 & 0.5 & 0.5 & 1 & 1 & \cdots & 0 \\
0.5 & 0 & 1 & 1 & 0 & \cdots & 1 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0.5 & 0.5 & 1 & \cdots & 0
\end{bmatrix}$
let th... |
I found an example of a function $f: \mathbb{R}\to\mathbb{R}$ that is continuous and bounded, but is not uniformly continuous. It is $\sin(x^2)$. I think it's not uniformly continuous because the derivative is bigger and bigger as $x$ increases. But I don't know how to prove this is uniformly continuous. Is $\sin(x^2)$... |
To solve for the $\sin^{-1} z$ ($z$ element of $\mathbb{C}$), the book reads that
$w = \sin^{-1} z$ when $z = \sin w$ implies:
$w = \sin^{-1} z$ when $z = (e^{iw} - e^{-iw})/2i$ implies:
$(e^{iw})^2 - 2iz(e^{iw}) - 1 = 0$ *
I don't see (2) implies (3). I mean it looks like the first step they used was mutiply out t... |
I have the following inequality:
$$\frac{2x-3}{x+1}\leq1$$
so, considering $x \neq -1$, I started multiplying $x+1$ both sides:
$$2x-3\leq x+1$$
then I subtracted $x$ both sides:
$$x-3\leq1$$
and then sum $3$ both sides:
$$x\leq4$$
Therefore, my solution for $x\neq-1$ is:
$$(-\infty,4]$$
But the book solution is:
$$(-1... |
In an exercise asking to mark true or false, it shows:
$$\frac{1}{a/x-b/x}=\frac{1}{a-b}$$
It really look like false to me. But the answer is true! How can it be?
|
My attempt:
Since $a^{k} \equiv b^{k}( \text{mod}\ \ m ) \implies m|( a^{k} - b^{k} )$ and $a^{k+1} \equiv b^{k+1}( \text{mod}\ \ m ) \implies m|( a^{k+1} - b^{k+1} ) $
Using binomial identity, we have:
$$a^{k} - b^{k} = ( a - b )( a^{k - 1} + a^{k - 2}b + a^{k - 3}b + ... ab^{k - 2} + b^{k - 1} )$$
$$a^{k + 1} - b^{k ... |
I am doing calculus integration, and need to show my work for Horizontal slicing (even though Vertical slicing is far easier).
The equation is $$y= x/\sqrt{2x^2+1}$$
I need to rewrite the equation so that it is $x=\;...$ in order to horizontally slice it (in other words, it should be rewritten so that it is dependent ... |
I am interested in a good comprehensive resource on realification and complexification of vector spaces over the reals or complexes (and the interplay of these structures on the 'same' space in general).
In particular, understanding of the basic theory is necessary and useful for a more intuitive approach towards funct... |
What is the difference between a field and a group in algebra?
Could someone briefly explain it?
|
I'm trying to prove divergence of the integral $\int_1^{\infty}\frac{\left|\cos{x^2}\right|}{x^q}dx$, where $0<q<1$, by applying the mean-value theorem to $\int_{n\pi}^{(n+1)\pi}\frac{\left|\cos{x^2}\right|}{x^q}dx$: there exists $c_n\in [n\pi,(n+1)\pi]$ such that $\int_{n\pi}^{(n+1)\pi}\frac{\left|\cos{x^2}\right|}{x^... |
$\displaystyle\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}\,\mathrm{d}x}$
I used partial fractions, solved $A = 2, C = 3$.
$$\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} +\frac{(Dx+E)}{(x^2+2)}$$
\begin{align*}
&8x^4+15x^3+16x^2+22x+4\\
&\quad = A(x+1)^2(x^2+2)+B(x)(x+1)(x^2+2)+C(x)(x^2+2)+(Dx+E)(x)(x+1)^2
... |
Suppose you're given a circle with center $O$, I'm curious, how can one construct with ruler and compass three circles inside the larger circle such that each is tangent to the larger circle as well as to the other two?
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Figured you math dudes (and dudettes) could knock this one out of the park!
I've got a website that stores users' game scenario results. Each scenario record can have one of three possible results: Side 1 wins, Draw, or Side 2 wins.
To this end, I'm currently representing balance by showing the simple percentage of tot... |
Today I came across two possible canonical maps in the context of representation theory from reading stackexchange and a friend's email. However, I don't really know what they are (and have thus far not found satisfactory information about them):
1) $hom_k(V \otimes V,k ) \rightarrow hom_k(V, V^*)$
2) $(V^*)^* \otimes ... |
I'm having trouble figuring out how to evaluate the integral $\int_{|z|=\rho} \frac{|dz|}{|z-a|^2}$ where $|a| \neq \rho$. This is a problem in Ahlfors in the section on Cauchy's Integral Formula, and I think by convention when he says $|z|=\rho$, he means the parametrization $z=\rho e^{it}, \; 0 \leq t \leq \pi$ (so t... |
is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must be unique(or with very high probability unique)
Thanks a lot
EDIT: calculation is for computer program,so computational ... |
All rings and algebra in this question are commutative and contains unity.
Suppose $M$ is an $A$ module and $A$ a $R$ algebra. If $pd_R(M) < \infty$, then will that imply $pd_A(M) < \infty$?
In particular if $M = \frac{R[X_1, X_2, \ldots,X_n]_N}{I}$, $A = R[X_1, X_2, \ldots,X_n]_N$ where $R[X_1, X_2, \ldots,X_n]$ is... |
Last time I did any maths was A-Level (some time ago!).
This is a programming/layout problem where I need to display 7 items across a page, with 6 gaps between them. I have a fixed width, but need to determine values of x and y that are integers where x > y.
This 'feels' like something I could plot on a graph, hence so... |
I wanted to know if it was true that if we are given a one-dimensional Feller process taking values in $\mathbb{R}$ and a hitting time $\tau_A=\inf\{t>0 s.t. X_t\in A\}$ with $A$ a open set (this to avoid measurability complexities but I would be also interested in the general case)
then if $X_0\in \bar{A}^c$ we have ... |
If anyone here is familiar with the Lowest Common Subsequence problem, they probably know that the number of posibble subsequences in a sequence is $2^n$; $n$ being the length of the sequence.
Another way of phrasing the statement is: given the sample space $\{a,b,c,d\}$, there are $2^4 = 16$ possible events (each even... |
When a mathematical notion is said to exist, it seems that there is a lot of freedom in what this particularly means, a freedom which accounts I think for why some people are finitists or ultra-finitists. How would you define mathematical existence?
|
I had seen a video lecture of Gilbert Strang online, which was about chaos and Newton's method. I could not find it again online, could someone help me locate it? Thanks in advance.
|
How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
|
I have been looking on the net for a textbook that only is about "Generating Functions" but did not found much. Is there any textbook just about "Generating Functions"?
|
In many places I see the entropy definition as:
$H(X) = \sum_{i=1}^n {p(x_i)\,I(x_i)} = -\sum_{i=1}^n {p(x_i) \log_b p(x_i)}$
In Wikipedia I saw: $H(X) = \operatorname{E}(I(X))$ where E is expected value and I is information content.
Today I was discussing an issue with my friends and couldn't decide if chessboard's ... |
I want to show that the product of smooth quasi-projective varieties is smooth.
Call my varieties $A$ and $B$. Since both are smooth, at every point the dimension of the tangent space equals the dimension of the variety. Will the dimension of $A \times B$ be the sum of the dimensions of $A$ and $B$? If so, how can I sh... |
I'm reviewing past assignments and am still having trouble formulating a proof for this:
Consider the sequence $(x_n)$, where $x_n = (1,\frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, 0, 0, \ldots)$. Determine whether $(x_n)$ converges in $l_1$.
It's simple to show that the coordinate-wise limit does not converge, but h... |
I know about algebraic numbers and transcendental numbers.
How the roots of a polynomial with irrational coefficients are classified.
Are they transcendental?
|
Recent news piqued my interest in integer partitions again. I'm working my way back through an old text and I'm completely hung up on this problem:
Recall that $p_k(n)$ is the number of partitions of $n$ into exactly $k$ parts. Prove that for all positive integers $k\leq n$, the inequality $p_k(n) \leq (n-k+1)^{k-1}... |
Let $R$ be a commutative ring. The Cartesian square $A=R\times R$ is endowed with the operation
$(a_1,b_1)\circ(a_2,b_2)=(a_1+a_2,b_1+b_2+a_1a_2^2+a_1^2a_2)$
which turns $A$ into a commutative group. I have two questions concerning this group.
Question 1: For what $R$ is $(A,\circ)$ isomorphic to $R\oplus R$?
I managed... |
Let $x_{1},x_{2},...,x_{m}$ be elements of $\mathbb{A}^{n}$, where $\mathbb{A}^{n}$ is the n-affine space over an algebraically closed field $k$. Now define $X=\{x_{1},x_{2},...,x_{m}\}$. Why is the coordinate ring $A(X)$, isomorphic to $\oplus_{j=1}^{m} k = k^{m}$?
|
Possible Duplicate:
How to find the sum of the following series
Hey there,
How do I (generally) calculate values of series? What tricks and theorems are there to make my life easier? E.g.:
$$\sum_{n=1}^\infty \frac{n^2}{2^n} = ?$$
I know perfectly well how to show whether a series converges or diverges, but (at leas... |
I am writing a scientific article with a few mathematical equations. Can I assume that my audience will know what lhs and rhs mean?
|
Background
As background, I have found that taylor expansion provides poor estimates of a function at extreme parameter values. Indeed, the approximation at extreme values can get worse (more rapid exponential increase) as the order of the taylor series increases.
This seems intuitive, but I don't know that it is a rul... |
How to prove the Energy function of a Hopfield net is monotonically decreasing?
$E = -1/2 \sum_{i,j} {w_{ij}}{s_i}{s_j} + \sum_{i}^{}s_{i} \theta_i$
I'll assume a proof involves the standard definition(s) of the activation function:
(1) $a_i \leftarrow \left\{ \begin{matrix} 1 & \mbox {if }\sum_{j}{w_{ij}s_j}>\thet... |
In my question about the convergence/divergence of
$$
\sum_{n=2}^\infty \frac{1\cdot 3\cdot 5\cdot 7\cdots (2n-3)}{2^nn!}.
$$
here: Why Doesn't This Series Converge?
Zarrax gave the answer:
"You can use Taylor approximations here. Note that the ratio between consecutive terms is ${2n - 3 \over 2n} = \exp(\ln(1 - 3/2n... |
I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.
As such I used... |
That is to ask, is $e^{\ln(q_0)}$ = $\ln(e^{q_0})$?
|
$$\frac{\ln(T+1)-\ln(V-S+1)}{\ln(V-S+1)}=\frac{1}{K}-1$$
What are the steps if I want to solve for $V$?
|
Here is the beginning of a proof that
Every abelian finite group $(B,+)$ with $|B|=p^r$ ($p$ prime, $r\in\mathbb{N}$) is isomorphic to a direct product of cyclic groups.
By induction on $r$. If $r=1$, then $B$ is cyclic.
Else let $b\in B$ of maximal order and $B_1$ the cyclic group generated by $b$. We use the induc... |
The following theorem is in Rudin's Principles of Mathematical Analysis, 8.2:
Suppose $c_n \geq 0$ and $S = \sum c_n$ converges. Put $f(x) = \sum_{n=0}^{\infty} c_nx^n$. Show that
$$
\lim_{x \rightarrow 1} f(x) = \sum_{n=0}^{\infty} c_n.
$$
Rudin goes on to give a somewhat involved proof using Abel Summation. I'm w... |
I am trying to write an equation to solve my problem of figuring out which post is most popular. While one would think the one with the highest stars would be, if the stars are equal then the one with 61 votes should rise higher than the one with 5 for example (since more people are downloading that)
Star system is 1-5... |
$$ \sum_{i=0}^{n-2}\left(\sum_{j=i+1}^{n-1} i\right) $$
Formulas in my book give me equations to memorize and solve simple questions like $$ \sum_{i=0}^{n} i $$ ... However, For the question on top, how would I go about solving it by hand without a calculator? WolfRamAlpha seems to give the equation of 1/6[(n-2)(n-1)n]... |
Is it true that, in a Dedekind domain, all maximal ideals are prime?
|
I know that the recurrence $\displaystyle a(n+1)=a(n)(n-1/2)$ can be represented like $\displaystyle \frac{(2n-1)!!}{2^n}$
Actually the initial recurrence is slightly different:
$$\displaystyle a(n+1)=a(n)(n-1/2)+o(1/n)$$
I am looking for the new representation of it in terms of $\displaystyle \frac{(2n-1)!!}{2^n}$.
... |
If I have a covariance matrix for a data set and I multiply it times one of it's eigenvectors. Let's say the eigenvector with the highest eigenvalue. The result is the eigenvector or a scaled version of the eigenvector.
What does this really tell me? Why is this the principal component? What property makes it a p... |
Studying some integral table, I came across the following definite integral
$$\int_0^{\pi} \log [ a^2 + b^2 -2 a b \cos \phi ]\,d\phi$$ for $a,b \in \mathbb{R}$. Does somebody know a nice way to get the results?
|
I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know.
Let me get more specific. Let $\left(\Omega,\mathcal{A},P\right)$ be a probability space and $X$ an ... |
I am trying to upper bound the following sum:
$$\sum_{k=1}^{n/2} \frac{\binom{n-2}{k-1}k^{k-2}(n-k)^{n-k-2}}{n^{n-3}}.$$
Based on numerical computations, it seems like the upper bound is a constant (there is also another complicated proof that suggested the upper bound should be a constant). Any idea how to prove this?... |
So the problem was to integrate $\overline{z}$ over the half circle of radius 2, $ z = 2e^{i \theta}$ from $z = -2i$ to $z = 2i$. After going through the math, they say
this integral is equal to the integral over the curve of $\frac{dz}{z}$ which is equal to $\pi i $. I don't follow how this equivalence was establishe... |
Suppose $(\Omega, \mathcal{F}, P)$ is a probability space and $(U, \mathcal{\Sigma})$ is a measurable space. There seem to be two ways of defining the conditional expectation of a r.v. $X: \Omega \rightarrow \mathbb{R}$ given another r.v. $Y: \Omega \rightarrow U$, denoted as $E(X\vert Y)$:
As a $\sigma(Y)$-measurable... |
Consider the collection $$\left\{e^{2\pi ikx/(b-a)}\,:\, k=0,\pm 1, \pm2,\ldots \right\}\,,$$ on $(a,b)$. Verify that this system is orthogonal in $L^2$.
I have attempted the two most obvious things: (1) integrating with exponentials, (2) integrating with trig functions. Both are messy, and neither are working out. I d... |
How can we show that the projective dimension of the $\mathbb{Z}/p^2 \mathbb{Z}$-module $\mathbb{Z}/p \mathbb{Z}$ is infinite?
|
Let's say we have a real, continuous,
positive function f(x) for which we
define the quantity:
$$\pi(f,a) = \frac{\int_0^a f(x)
dx}{\int_0^a \sqrt{1+\left(\frac{df(x)}{dx}
\right)^2 }dx}$$
we want to find the function f that
maximizes $\pi$ for a given $a$.
In general how do we attack problems of this kind:... |
I do not have enough complexity theory background, but I was wondering about the kind of reductions that we normally do to show NP-Completeness. I think all of the reductions that I have seen are one-to-one; in that, given an instance $I_1$ of problem $P_1$, they map it to instance $I'_1$ of problem $P'_1$ such that $I... |
EDIT:Let $S$ be any finite set and suppose $x \notin S$. Let $K=S \cup \{x\}$.
1.Prove that $P(K)$ is the disjoint union of $P(S)$ and $X=\{T \subset K : x \in T\}$. That is, show that $P(K) = P(S) \cup X$.
2.Prove that every element of $X$ is the union of a subset of $S$ with $\{x\}$, and that if you take different su... |
I have $n$ coins and each of them shows head with probability $p$ if I throw them separately. Now I throw them one-by-one, and each current throw has a correlation $\rho$ with the previous result. (I hope this is rigorously defined through the definition of correlation coefficient.) After one round of tossing, what is ... |
Prove that for every positive integer $x$ of exactly four digits, if the sum of digits is divisible by $3$, then $x$ itself is divisible by 3 (i.e., consider $x = 6132$, the sum of digits of $x$ is $6+1+3+2 = 12$, which is divisible by 3, so $x$ is also divisible by $3$.)
How could I approach this proof? I'm not sure w... |
Given a lie algebra $g$, how does one approach finding the Gel'fand models? For clarity, by this I mean
$\bigoplus_{\lambda\in P^+}V(\lambda)$ where $P^+$ are the dominant weights, and $V(\lambda)$ is the highest weight representation of weight $\lambda$.
One can calculate the weight modules and just take their sum... |
Given a Möbius transformation that maps one pair of concentric circles to another pair of concentric circles, why is the ratio of the radii preserved through the map?
I thought about how Möbius transformations are compositions of rotations, scaling, inversion, and translation, and that intuitively, these types of maps... |
I know this is a simple question for most of you, but I am currently studying for a Calculus exam and was just wondering why an online calculator I am using to double-check my work was disagreeing with me on this question:
$$\lim_{x\to 0} \cot(x)\sec(x)$$
I reduce this down to $\frac{1}{\sin(x)}$, and in that case $x\t... |
This is about problem $5$ in Section IV.$5$ of Hungerford's Algebra book. The question is the following:
If $A'$ is a submodule of the right $R$-module $A$ and $B'$ is a submodule of the left $R$-module $B$, then $$\frac{A}{A'} \otimes _R \frac{B}{B'} \cong \frac{A \otimes _R B}{C}$$ where $C$ is the subgroup of $A \... |
So I am trying to understand what a coset is and what a quotient ring is.
So I am going to tell you guys what I know. And please let me know if my thinking is right or wrong, and if I am missing something. For the rest of this post, assume $R$ is a ring and $I$ is an ideal of that ring.
So $I = (m)$ is a principal id... |
What are the various sets of postulates that can used to derive Euclidean geometry?
It might be nice to have several different approaches together for comparison purposes and for ready reference.
It might also be interesting to include an axiomatization (or two) of elliptical geometry.
|
Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?
|
I'm looking for a way to project a vector (in this case a function on the real line) onto a basis for that space (in this case the set of N-degree polynomials over the domain of a closed interval) with respect to minimizing the L-infinity norm.
Another way you could phrase this is that I'm looking to find the polynomia... |
In the Wikipedia article for conditional expectation, conditional probability is defined in terms of conditional expectation.
Given a sub sigma algebra of the one
on a probability space.
Given a probability space $(\Omega,
\mathcal{F}, P)$, a conditional
probability $P(A \mid \mathcal{B})$
of a measurable subset $A \... |
In Introduction to Set Theory by J. Donald Monk, he defines ordinal as follows.
Definition (1): $A$ is an ordinal iff $A$ is $\in$-transitive and each member of $A$ is $\in$-transitive. $A$ is $\in$-transitive iff for all $x$ and $y$ , $x \in y \in A \implies x \in A$.
And using definition (1), I have a problem of sh... |
I have a question.
Let $f,g$ be continuous functions from $X$ to $Y$, $X$ is a topological space and $Y$ a topological space under ordered topology. Then prove that the set $\{x \in X \ | \ f(x) < g(x)\}$ is open.
I want to know that what intrisic property of order makes it possible.
|
When there is no hash function used, $s = k-x \cdot m \mod {q}$ instead of $s = k-x \cdot H(m||r) \mod{q}$?
When a hash function is defined as $H(m)$ instead of $H(m||r)$?
Ref: Schnorr Signature
|
I saw this expression in a book and I cannot understand how did he get this expression.
Suppose $Z_t$ and $D_t$ are some stochastic processes and we have these expressions,
$Z_{t_k} - Z_{t_{k-1}} = (t_k - t_{k-1})(D_1 + ... + D_{n-k+1}) - t_{k-1}D_{n-k+2}$
$Z_{t_i} - Z_{t_{i-1}} = (t_i - t_{i-1})(D_1 + ... + D_{n-i+1})... |
Let $v$ be the 3-adic valuation on $\mathbb{Q}$ and consider the subring $\mathbb{Z}_{(3)}$ of $\mathbb{Q}$ defined by
$$
\mathbb{Z}_{(3)} = \{ x \in \mathbb{Q} : v(x) \geq 0 \}.
$$
That is, $\mathbb{Z}_{(3)}$ is the ring of rational numbers that are integral with respect to $v$. $\mathbb{Z}_{(3)}$ is also the loca... |
Given random variables $X, Y, Z$, is
joint distribution of $X$ and "$Y|Z$"
same as distribution of $(X, Y)|Z$?
Given random variables $X_1, X_2,
Y_1, Y_2$, is joint distribution of
"$X_1 | X_2$" and "$Y_1|Y_2$" same as
distribution of $(X_1, Y_1) | (X_2, Y_2)$?
For two independent random vectors
$X$ and
$Y$... |
I've never seen the following comparison before. Let me start with a specific example:
Given a finite structure with two symmetric binary relations, i.e. a graph $G$ with one vertex set $V$ and two edge sets $E_1$, $E_2$.
Giving an explicit description of $G$ in a formal language can be seen as a special case of defini... |
$\frac{256}{2^S}=64$
How would you solve for S?
|
This question stems from Probability of 3 of a kind with 7 dice as a means to check my understanding. I'm not sure whether this is general practice on this website, so I'm sorry if creating a new topic is frowned upon. I figured the question is large enough on its own to warrant a new topic. (Note: I intend to go ba... |
If I have two exact triangles $X \to Y \to Z \to X[1]$ and $X' \to Y' \to Z' \to X'[1]$ in a triangulated category, and I have morphisms $X \to X'$, $Y \to Y'$ which 'commute' (i.e., such that $X \to Y \to Y' = X \to X' \to Y'$), thene there exists a (not necessarily unique) map $Z \to Z'$ which completes what we've go... |
I was wondering what is the relation between the first and second Borel–Cantelli lemmas and Kolmogorov's zero-one law?
The former is about limsup of a sequence of events, while the latter is about tail event of a sequence of independent sub sigma algebras or of independent random variables. Both have results for limsup... |
I would be glad if someone can give me some (hopefully easy to understand!) references for learning about these groups Osp, USp and PSU and their representations.
I run into these mostly while studying about supersymmetry and I wonder if PSU groups are the same as what are called ``supergroups".
I frequently run in... |
I read about how Hecke groups are a particular generalisation of the modular group, generalising one of the generators of the modular group to $z\mapsto z + \lambda$. Have people studied generalisations of the other generator of the modular group, perhaps generalising to $z \mapsto \dfrac{-1}{z^n}$ ?
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As we know,a hyperplane can seem as a divisor,and a divisor can become a linear bundle,I want to know what the structure of linear bundle is.
For example, the hyperplane is given by $a_0 z_0+a_1 z_1+\ldots+a_n z_n=0$,what is the projective map、transform function and so on?
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I am poking on a proof of the subject theorem. Given sets $A_i$ and injections $f_i:A_i\to A_{1-i}$, $i\in \{0,1\}$, theorem defines a bijection $b$ between $A_0$ and $A_1$. $b$ uses an auxilary function
$degree (x:A_i) := \cases{
1+degree(f_{1-i}^{-1}(x)), x\in f_{1-i}(A_{1-i}) \cr 0}$
“degree” may loop, denote this $... |
Everywhere I look in the literature, Sobolev spaces are defined on an open subset of the real line. What are the technical issues with defining a Sobolev space on a closed subset, i.e. are there problems at the boundary, and does anyone know any good references that cover this?
My main purpose is to prove $H^1([0,T];\m... |
i need a mathematical function F which is defined as F(x) = 0 if x equal to zero and F(x) = 1 if x is not equal to zero. can F(x) be represented in the form of a single mathematical expression in terms of x?
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Can someone please share a link or source where I can find the derivation of FFT(base-2) from the DFT. I need to put this in latex for my thesis and am finding so many different explanations that I don't know which one to use.
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I hit a snag whilst revising some log rules, could anyone confirm my suspicion:
$$\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y ?$$
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Cheers.
So if I don't make sense, I have a value for $y$, I need to know what $x$ is.
$$y = a \exp(bx) + c \exp(dx)$$
$a = 12.85$,
$b = 0.001857$,
$c = -54.24$,
$d = -0.05316$
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