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I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback".
I don't understand why this might be the case.
In particular, consider a field $k$ with a map $\varphi:k\rightarrow A$ for $A$ an associative unital $k$-algebra. Then we consider the induced functor ... |
I'd like to learn formal math. Preferably, though not necessarily, starting with predicate logic/first order logic rather than higher order logic. I am trying to find resources (papers, books etc.) for doing this, but I haven't found anything I really like.
There are lots of resources for predicate and first order log... |
I have problem calculating the following summation:
$$
S = \sum_{j=1}^{k^2-1} \lfloor \sqrt{j}\rfloor.
$$
As far as I understand the mean of that summation it will be something like $$1+1+1+2+2+2+2+2+3+3+3+3+3+3+3+\cdots$$
and I suspect that the last summation number will be $(k-1)^2$, but I really can't find the patte... |
So I found this
$$\prod_{k=3}^{\infty} 1 - \tan( \pi/2^k)^4$$
here.
I have only ever done tests for convergence of infinite sums.
At this link it shows a way to convert but
in this case an is less than one for all k. I can see that $ f(k) = 1 - \tan( \pi/2^k)^4 $ is a strictly increasing function and in fact it is inc... |
Suppose $64$ students are taking an exam. The exam has $16$ questions in multiple choice format. Students must fill an oval for every one of the $16$ questions. They can't revise their answers. The time taken for a students to answer a question is exponentially distributed with mean $8$ minutes. So $f(t) = \frac{1}{8}e... |
Suppose $f(x,y)$ is bivariate normal. You are given $E(X)$, $E(Y)$, $\text{Var}(X)$ and $\text{Var}(Y)$. You are also given $\rho$ the correlation coefficient. How would one find $\text{Var}(Y|X=k)$? Would you use the total law of variance? We know that $\rho = \frac{\text{Cov}(X,Y)}{\sigma_x \sigma_y}$.
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How to prove that the power $b^n$ is an irrational number, where $b$ is an integer greater than $1$ and $n$ is an irrational number?
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Suppose I have a 2-variable function $g(k,n)$ and I know that $g(k,n)=O(n^{f(k)})$, for fixed $k$ as $n \rightarrow \infty$, for some function $f=f(k)$. Suppose I also know that $f(k)=O(\log k)$.
Is it legitimate to write $g(k,n)=O(n^{O(\log k)})$?
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I'm dealing with tables that show connectivity within a network. Can anyone point me toward a technique for ordering the rows and columns so that nodes that are connected tend to be close to each other in the table?
For example, if I had a network like this (connections are directional), the structure isn't immediately... |
I am going to a Computer Science Course in University next year. I heard that Discrete Mathematics is whats required for Comp Sci so, I am looking for resources/books that I can read to get started with. I picked up Schaum's Outline of Discrete Mathematics. I find that some things are alittle hard to understand still. ... |
Abel's identity states that if $X(t)$ and $A(t)$ are $n\times n$ matrix-valued functions such that $X'(t)=A(t)X(t)$, then $\frac{d}{dt}(\det X(t)) = \mathrm{tr}\,A(t) \cdot \det X(t)$.
The question is whether there's a nice high-brow basis-independent way to see this. Given that you can state the problem without ever ... |
The companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules.
It is an important fact that the characteristic polynomial and the minimal polynomial of $C(f)$ are... |
Say I have an equation where the value is $m-4$ when $m \ge 5, n=3,4$; $n-4$ when $n \ge 5, m=3,4$; and $m+n-9$ when $m,n \ge 5$. How can I combine these all into one statement? How can I combine them with further ones, like $m-6$ where $m \ge 7, n \in [3,6]$, etc?
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$\max(f(n), g(n)) = O(f(n) + g(n))$
How do I prove this? Also I'd appreciate the markup being corrected, thanks.
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Which are the parallel lines?
I prove that $a$ and $b$ are parallel but can't prove that $c$ and $d$ are parallel.
The angles, which are $135^o$ and $45^o$ are alternate angles.They aren't equal , so $c$ isn't parallel to $d$ but the answer is that $c$ is parallel to $d$. Am I right that they aren't?
Edit:
They are a ... |
How to solve this manually ?
EDIT: In my module the answer is given as $0$ but when I used mathematica N[Sin[10 Degree] + Sin[20 Degree] + Sin[30 Degree] - Sin[360 Degree],50] gives $1.0156683209925990818958162414 \cdots$ (truncated)
So I guess there is some mistake in the problem statement.
|
Of course, it's best explained with an image:
$L1$, $L2$, $h_1$ & $h_2$ are all given, and I would like to calculate the length of $L3$. Is it possible, and how?
|
I want to motivate the quadratic reciprocity theorem, which at first glance does not look too important to justify it being one of Gauss' favorites. So far I can think of two uses that are basic enough to be shown immediately when presenting the theorem:
1) With the QRT, it is immediate to give a simple, efficient algo... |
The group axioms demand that one has an inverse element. When it comes to multiplication zero has no inverse element because one cannot reconstruct the original number after it was multiplied by zero:
There is an irreversible loss of information!
Therefore all numbers cannot form a group under multiplication as long a... |
A power type $b^{a^{1/n}}$ where $b$, $a$ and $n$ are positive integers greater than $1$ and $a^{1/n}$ is irrational, always represents an irrational number?
|
How the following conversion is justified ?
$$\cos \biggl( \theta + \frac{2\pi}{3} \biggr) + \cos \biggl( \theta + \frac{4\pi}{3} \biggr) = \cos \biggl( \frac{\pi}{2} - \theta \biggr) - \cos \biggl( \frac{\pi}{3} +\theta \biggr) $$
|
Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line?
What is a parametric equation of such curve given the parametric equation of a curve along which the master goes?
|
Let $A$ be a finite dimensional vector space, and let
\begin{equation*}
\cdots \to A_{n+1} \xrightarrow{f_{n+1}} A_n \xrightarrow{f_{n}} A_{n-1}\rightarrow \cdots
\end{equation*}
be an inverse system of finite dimensional vector spaces. Let $\phi_i:A\to A_i$ be linear maps inducing a map $\phi : A \to \varprojlim A_i$... |
I have changed the title (replaced "well-behaved" by "càdlàg"), since it seems that "a well-behaved function" might be interpreted as "a function of bounded variation" (rather than "a càdlàg function", which I actually meant).
Let $f:[0,1] \to {\bf R}$ have a following property: $f$ is continuous except at the points $... |
I feel like finding the inverse of $y=xe^x$ should have an easy answer but can't find it.
|
I'm just reviewing for my exam tomorow looking at old exams, unfortunately I don't have solutions. Here is a question I found : determine if the series converges or diverges. If it converges find it's limit.
$$\displaystyle \sum\limits_{n=1}^{\infty}\dfrac{\sin(n-\sqrt{n^2+n})}{n}$$
I've ruled down possible tests ... |
Sorry if this is a little basic, but my math is pretty rusty. I know the point coordinates of the two blue dots. I also know the distances from the middle blue dot to the edges of the square. How can I calculate the coordinates of the red dot?
|
A ring $R$ is a Boolean ring provided that $a^2=a$ for every $a \in R$. How can we show that every Boolean ring is commutative?
|
Let $x \in \mathbb{R} \backslash \mathbb{Q}, x>0$ and $q \in \mathbb{N}, q>0$, prove that there is an $r \in \mathbb{N}, r>0$ with:
$r \cdot x - \left\lfloor r \cdot x \right\rfloor < \frac{1}{q}$ or $1-(r \cdot x - \left\lfloor r \cdot x \right\rfloor) < \frac{1}{q}.$
I was given the hint to divide the sequence $... |
As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, which I believe are slightly different.
I currently think of them in the following ways, but have a tough time reconcil... |
$10 \%$ of people in a population suffer from high blood pressure. Calculate the probability that the twelfth person tested will be the second one with high blood pressure. So this is a negative binomial distribution. Now $P(X=k) = \binom{k+r-1}{k} (1-p)^{r}p^{k}$. So this is the probability of $k$ successes after $r$ ... |
Say we have a set of $12$ numbers - $(0,1,2,0,3,0,4,0,5,6,0).$
We go thru this set one number at a time, and choose a number with a probability $1/3$; we sample. What is the probability we end up with $3$ $0$'s out of the $4$ numbers we will likely choose(on average)?
Is there a closed-form solution for this?
|
so studying for my midterm on Tuesday (intro to abstract algebra). The topics on the exam are Division Algorithm, Divisibility, Prime Numbers, FTA, Congruency, Congruent Classes and very brief introduction to rings.
I was reading a few theorems about Congruency and have a couple of questions.
I want to know what a "c... |
Suppose Jill and Jane are individually painting separate rooms. Each of the rooms has $4$ walls. The time taken to paint each wall is exponentially distributed with mean $1$ hour. What is the probability that Jill will finish painting $3$ walls before Jane finishes $2$ walls?
This seems to look somewhat similar to a n... |
Suppose we have an infinite sequence of intervals $[a_n,b_n]$, where for all n, $a_n<b_n$. Then is there some $x\in \mathbb{R}$ such that for each n there is some $k\in\mathbb{Z}$ such that $kx\in [a_n,b_n]$?
It seems unlikely, but I'm not sure where to begin with the proof. Thank you.
|
The answers are already given after the $=$ sign of each question. But I don't know how they arrived at these answers. What does it mean to say $f(A)=1$ and so on? I can't find the connection.
|
Suppose $P(A) = 0.6$ and $P(B) = 0.7$. Find the smallest possible value of $P(A \cap B)$. Instead of deducing this from $P(A \cup B) = P(A)+P(B)-P(A \cap B)$, could we just use the fact that $P(A|B)P(B) = P(A \cap B)$ and $P(B|A)P(A) = P(A \cap B)$? I notice that $(0.6)(0.7)^2 = 0.294$ which is close to $0.3$, the smal... |
Let X and Y be independent random variables,
$ P(X = k) = P(Y = k) = p(1 - p)^{k-1} $
How do you show that the pmf of $ Z = X + Y $, is negative binomial, and how do you find
$ P(X = Y) $?
|
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
In trying to find this limit, all the (x,y) paths I check approaching (0,0) produce the same result, namely, the function approaches 0. consequently, I came up with the simplest proof I could that the limit indeed exists and is zero. However wolframalpha tells me that the limit doesnt exist. Assuming I'm wrong, can any... |
Let $X_{i} = 1$ if ith person gets his or her own hat back and $0$ otherwise. Let $S_{n} = \sum_{i=1}^{n}X_{i}$, then $S_{n}$ is the total number of people who get their own hats back. Show that:
a) $E(X_{i}^{2}) = \frac{1}{n}$
b) $E(X_{i}.X_{j}) = \frac{1}{n(n - 1)}$ for $i\neq j$
c) $E(S_{n}^{2}) = 2$ (using (a) an... |
Is it algebraic the number 0.2468101214 ...? (After point, the natural numbers are juxtaposed pairs).
|
Let $X$ be an infinite-dimensional Banach space, and $T$ a compact operator from $X$ to $X$. Why must $0$ then be a spectral value for $T$?
I believe this is equivalent to saying that $T$ is not bijective, but I am not sure how to show that injectivity implies the absence of surjectivity and the other way around (or i... |
Does it make sense to say that there are more irrational numbers than rational?
|
The question states:
If $a \in \mathbb{Z}$, prove that $a^2$ is not congruent to 2 modulo 4 or 3 modulo 4.
Our work:
It's an if then proof, so I am thinking we can do a contradiction proof.
Suppose $a^2 \equiv 2 \pmod{4}$ and $a^2 \equiv 3 \pmod{4}$.
Then you can write by definition
\begin{align*}
a^2 &= 4k + 2 \\... |
I figured out this order on $\omega^2$: elements of different columns are ordered by their column number; within column $m$, the order is $0 \gt 1 \gt 2 \gt \cdots \gt n$, where $n$ is the first such that $P(m,n)$ is true; then $0 \lt n+1 \lt n+2 \lt \cdots$. The column is an infinite descending sequence if there is no... |
The title is somewhat deceptive: I know that $|\mathbb{Q}|=|\mathbb{Z}|.$
But suppose I wanted to compare the sets, knowing that they are of the same cardinality but still wondering if there was some sense in which Q could be larger. There are two usual ways of thinking about this:
$\mathbb{Z}\subset\mathbb{Q}$ but $... |
Consider a $G$-equivariant map $\pi:X\to Y$ for $G$ an affine algebraic group, such that $\pi$ is a good categorical quotient. Is there any relationship between $H^*_G(X)$ and $H^*(Y)$? Is there if $\pi$ is a good geometric quotient, or if the quotient space is smooth?
EDIT: A categorical quotient is an equivariant m... |
Combining the answers given by me and Ralth to the probability question at Probability of getting three zeros when sampling a set, we get the following identity:
$$
\sum\limits_{k = m}^n {{n \choose k}p^k (1 - p)^{n - k} {k \choose m} p_j^m (1 - p_j )^{k - m} } = {n \choose m} (p p_j)^m (1 - p p_j)^{n-m},
$$
where $m \... |
How would I go about creating a polynomial $f$ such that
\begin{align*}
f(x) &= A_1\\
f(y) &= A_2\\
f'(x) &= B_1\\
f'(y) &= B_2
\end{align*}
With $x$ not equal to $y$ and $A_1$, $A_2$, $B_1$, $B_2$ real numbers?
|
I have a finite alphabet A and a set S of $n$ ordered pairs $(a, b)$ with $a,b\in A.$ I want to find out how many sequences $(a_1, b_1),\ldots,(a_n,b_n)$ there are where adjacent elements have either $a_{k-1}=a_k$ or $b_{k-1}=b_k.$ (Since S is a set, they can't have both.)
Any ideas on how to go about finding this? F... |
I have trouble understanding an argument in the proof of the Kummer-Dedekind theorem.
I am referring to a proof given in Peter Stevenhagen's notes.
http://websites.math.leidenuniv.nl/algebra/ant.pdf
This is theorem 3.1 on page 27. For the proof of the second part, the first statement says that since $p_i$ contains $pR$... |
Two vectors $\mathbf{x}=(x_1,x_2)$ and $\mathbf{y}=(y_1,y_2)$ are orthogonal if their scalar product is zero, i.e. $x_1 y_1 + x_2 y_2=0$. What surface does this equation describe in the four-dimensional $(x_1,x_2,y_1,y_2)$ space? How about the orthogonality condition for $3$-dimensional (or higher dimensional) vectors?... |
There are two equivalent definitions of a locally convex topological vector space. Note that since the vector space is topological, addition by a fixed vector x and multiplication by a fixed scalar r are both homeomophisms, which means it is enough to give a neighborhood basis at the vector 0.
The first definition is t... |
I'm going through an examples section (on improper integrals) but I got lost at this bit:
$$\lim_{t\to-\infty} \frac{t}{e^{-t}} = \lim_{t\to-\infty}\frac{1}{-e^{-t}}.$$
I think it's a simple algebra trick but I don't see how the right hand side came to be. How did it become $1/-e^{-t}$?
|
How to describe all ring homomorphism from $\mathbb Z \times \mathbb Z \to \mathbb Z$ ?
|
Where is the well-pointedeness assumption of the Elementary theory of the category of sets (Lawvere's category-theoretic axiomatization of set theory) used in everyday math?
Specifically, if you have a topos with natural numbers object (assume choice if you want to), what familiar theorems don't hold? I've heard that s... |
For a sequence defined by a formula normally the usual limit rules allows
one to find its limit. But for a sequence defined by a recurrence, up to now
I have only seen some refined ad hoc methods, mostly in Problems.
The Trick explains that "There is one trick that is (...) first to prove
that a limit exists, and then ... |
I'm in a little bit of a pickle here, can someone tell me if it is actually possible to solve these? From the general rule of thumb, 4 variables, 4 equations should be able to right?
|
It is known that, in the universe of complex numbers, the only root of the equation $x^2 - 2x + 1 = 0$ is $1$. Could we say that the equation has two equal real roots? Or should we say that the equation has one real root with multiplicity 2?
|
I'm stuck with this problem I'm trying to solve from about an hour. Here's the question.
What is the remainder when (3^202)+137 is divided by 101?
There are 4 options -> 36, 45, 56, 11
I want to know the answer of the question with proper and possibly easiest method to solve the problem.
Thanks in advance, waiting for ... |
I'm stuck with the dual of an exercise in Atiyah-MacDonald.
It's easily seen that tensoring over $R$ with $-\otimes M$ the sequence
$$
0\to \mathfrak a\to R\to R/\mathfrak a\to 0
$$
one gets the isomorphism $R/\mathfrak a\otimes M\cong M/\mathfrak aM$.
What can I deduce from "Hom(-,M)ing" the same sequence? I have, ... |
I have to solve the following recurrence :$\displaystyle T(n)=2T(n/2)+T(n/3)+\theta(n^2)$
I have done the whole tree analyses and now I have to prove that $\displaystyle T(n) \leq dn^{2}\log_{2}(n)$ but I can not find mathematical way to prove that, is there any trick with logarithms that I can use?
|
Let's say I have a truth table like this:
X Y A
t t f
t f t
f t t
f f f
Now, I have to find the formula for A. This case is rather easy because I can immediately see that this looks like the inverted values of X~B, thus ¬(X~B). Now, for a more complicated problem I do not know the method.
X Y Z A
t t t... |
I believe I've found all Amicable Pairs $(n, m)$ such that n is in the closed interval $[1, 34000000]$. Here's my list.
By inspection I believe that for all $n$ in the list, $n$ is odd iff its last digit is $5$. Can it be proved that this is true for all Amicable Pairs?
|
Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal?
|
I want to describe the polynomials with integer coefficients and the property that $f'(x) \mid f(x)$ (the derivative divides the polynomial).
So I know that $f(x)$ divides $g(x)$ if all of $f(x)$'s roots are roots of $g(x)$. I also know that $f'(x)$ has degree one less than $f(x)$. Finally, $f'(x)$ detects multiple roo... |
Given a $1$ dimensional domain $R$ with fraction field $K$. Suppose $\mathfrak{m}$ is a maximal ideal of $R$. Then, I need to show that $R_m$ is a maximal subring of $K$ in the sense that if $x\in K-R_\mathfrak{m}$, then, $R_m[x]=K$.
1)Is this true? How would you go about showing this?
2)If this is true, can any of the... |
Imagine we have a wizard that knows a few spells. Each spell has 3 attributes:
Damage, cooldown time, and a cast time.
Cooldown time: the amount of time (t) it takes before being able to cast that spell again. A spell goes on "cooldown" the moment it begins casting.
Cast time: the amount of time (t) it takes to use a s... |
If you have a function, implicitly defined, how would understand the following;
Assume $y=y(x)$, find $y'(0)$ and $y''(0)$.
After I implicitly derivate it the first time, I have an expression with $dy/dx$. Do I just fill in zero for $x$, remove the parts that gets canceled out? I'm not sure if the answer is a number ... |
Given a finite abelian group $G$. $H$ and $K$ subgroups of $G$ such that $G/H$ and $G/K$ are primary cyclic. $H$ and $K$ intersect trivially ($H\cap K=1$), and $H\neq 1$, $K\neq 1$.
Is it then true that $HK=G$?
My attempt so far: It's easy to show that $[G:HK] | ([G:H], [G:K])$. So if $G/H$ and $G/K$ are primary cycl... |
What is $x$ in closed form if $2x-\sin2x=\pi/2$, $x$ in the first quadrant?
|
All power sets contain an empty set, so does this make the power set itself an empty set? Or am I mistaken?
|
Let a statement P is "X is true if and only if Y is true". What is the negation of P? I am little confused. It seems that digital equivalent of this statement is P = X and Y. Hence negation of P is (not X) or (not Y) i.e. Either X or Y is false. Am I right guys?
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I'm working through an exercise in Hartschorne concerning direct limits of sheaves on noetherian topological spaces. It seems natural to use the the fact that every open subset of a noetherian topological space is the quasi-compact in its induced topology. I'm some trouble proving what I need when I only use this prope... |
In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts.
I learned that both are connected in some way, i.e. there is an eigenvector/eigenvalue approach to solve differential equations. Unfort... |
I was reviewing quantum factorization and am slightly unclear on a classical detail of order-finding.
Given a (suitably nice) periodic function $f$ with unknown period $r$ and a power of two $N > r^2$, the quantum subroutine yields (with bounded probability) a result of the form $\lfloor jr/N \rceil$ for $1 \le j \le ... |
If $a_1,\ldots,a_n$ are in arithmetic progression and $a_i\gt 0$ for all $i$, then how to prove the following two identities:
$ (1)\large \frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + \cdots + \frac{1}{\sqrt{a_{n-1}} + \sqrt{a_n}} = \frac{n-1}{\sqrt{a_1} + \sqrt{a_n}}$
$(2) \large\frac{1}{a_... |
This is a newbie question, but I would be grateful for any reference that you could give.
let $f(x) \in \mathbb{A}$, where $x \in \mathbb{A}$
Is there a symbol to indicate the repeated application of $f$ unto its results, in a manner similar to $\sum$ and $\prod$?
Thank you!
|
In Classical Fourier Analysis by Loukas Grafakos we have in Proposition 2.3.25 the following definition for $\mathcal{S}_\infty(\mathbf{R}^n)$, namely that these are all the Schwartz functions $\phi$ such that for all multi-indices $\alpha$ we have that
$$\int_{\mathbf{R}^n} x^\alpha \phi(x) \, dx = 0.$$
Now I'm tryin... |
I'm a CompSci university student and i want some help for my math questions.
Therefore i am looking for a good discussion board for math homework questions.
I don't know any (good), because I come from an non-native-english-speaking country.
any ideas ?
note : I know that I could post my questions here, but:
1. this... |
I'm reading Hatcher's notes on spectral sequences and he mentions that steenrod squares commute with the coboundary operator for pairs (X,A) which would then explain why these operations commute with the transgression. It says it's because
that coboundary operator can be defined in terms of suspension and we know stee... |
While reading about Moufang loops in the book "An introduction to Quasigroups and their Representations" by Smith, I've encountered the following statement:
The set $ S^7 $ of nonzero octonions of norm 1 forms a Moufang loop under multiplication. Geometrically, this set is a 7-sphere.
While I understand why this set ... |
From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not ... |
so I dont know if this is really what the website is meant for so let me know. I have a midterm on abstract algebra (intro) and I've been doing practice problems and I just need to know whether my answers are correct and if my way of doing them is right.
1) Does the following have a solution: $x^2 \equiv 1 \pmod{3}$
... |
Background
Recall that an integral scheme $X$ is a scheme which is both irreducible and reduced; equivalently, its ring of functions is an integral domain on every open subset.
Given any point $p$, there is a local ring $R_p$ at $p$, which is given by localizing the ring of functions $R_U$ on any affine neighborhood $U... |
So some of you may have noticed my algebra exam is tomorrow!
The question is: If $[a]_{n} = [1]_{n}$ prove that $\mathrm{gcd}(a, n) = 1$.
Well I tried it one way. We know the above means the following: $a \equiv 1 \pmod{n}$. This is also written as $1 \equiv a \pmod{n}$
Then the gcd of these are as follows:
$$\mathrm{... |
If $X,Y$ and $Z$ have identical distributions and are independent, and can assume $0$ or $1$, what is $\text{Var}(XYZ)$? So this is either $0$ or $1$. It's $1$ with probability $1/8$ and $0$ with probability $7/8$. This means that $\text{Var}(XYZ) = 7/16$? Or is this not right? How does this use the definition of varia... |
Given the natural numbers $bbb\dots b$ ($n$ digits equal to $b$) and $bbb\dots b$ ($m$ digits equal to $b$), show that the greatest common divisor of these numbers is $bbb\dots b$ ($M$ digits equal to $b$, where $M$ is the greatest common divisor of $n$ and $m$).
|
Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular cohomology of $K(G,1)$, and it is a theorem that this is isomorphic to the group cohomologies $H^*(G, \mathbb{Z})$. Acc... |
Let $T$ be a distribution, $f$ be smooth. It is known that
$T(f(x - \cdot) ) \in C ^ \infty$ .
On the other hand, for $\phi$ a test function
$( T \ast f ) (\phi) = T( f( - \cdot ) \ast \phi ) = T( \int f( x - \cdot ) \phi( \cdot ) )$,
because the convolution is still a distribution. However, with regard to the first re... |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the origin and the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with pe... |
I'm approximating an integral with only exponentials. i.e., it is equal to
$$\int_{-\pi}^\pi{
\left({\sum\limits_{j=a}^b{c_j e^{i\cdot d_j \cdot t}}
}\right)\left(
{\sum\limits_{k=a}^b{r_k e^{i \cdot s_k \cdot t}}
}\right)^{-1} dt},$$
where $c_j, d_j, r_k, s_k \in \mathbb{Z}$
In other words, it's a fraction of ex... |
We have points A, B & C in 2D plane. How having point coordinates $(x, y)$ to calculate area of triangle formed by them?
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Suppose $r_i$, $1 \le i \le n$, and $c_j$, $1 \le j \le m$, are nonnegative integers. When does there exist an $n \times m$ matrix in $\text{Mat}_{n \times m} (\mathbb{Z}^+)$, i.e. nonnegative entries, such that $r_i$ is the sum of the entries in its $i$th row and $c_j$ is the sum of the entries in its $j$th columns?
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what is the sigma field created by X(t,w) where t belongs to [0,1] and X(t,w) =1 if t=w and zero otherwise.
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in a game of contract bridge, partner and me together have 11 cards in a suite. The remaining 2 cards in the same suite can be distributed amongst our opponents as either 1-1 or 2-0.
What is the probability that it will be distributed as 1-1 and what is the probability it will be distributed as 2-0?
Once the above is ... |
The area of a pyramid with a square base with side length $L$ and height $h$ is calculated as follows:
In $\mathbb{R}^3$ place the pyramid upright with one side flush with the z-axis so that the corners of the base are at $\left(0,0,\frac{1}{L}\right)$, $\left(0,0,-\frac{1}{L}\right)$, $\left(L,0,\frac{1}{L}\right)$ ... |
Suppose that $\mathcal{B}$ is a base for a topology on a space $X$. Is there a nice way of thinking about how we can modify $\mathcal{B}$ (for instance, to simplify computations) without changing the topology it generates? It seems non-trivial to compute the topology generated by a base, but maybe some "small enough" c... |
I am currently reading Logic an Structure by Dirk van Dalen (2008).
However, I am missing some basics I try to find related lectures on online / on youtube. I frequently watch MIT, Stanford, and University of Nottinham lectures on youtube.
However, I have a hard time to find lectures on topics like predicate logic, hig... |
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