instruction stringlengths 22 27.2k |
|---|
Suppose you have two distinct large primes $p$ and $q$.
Explain how you can find an integer $x$ such that $x^2 \equiv 49$
(mod $pq$), $x \not\equiv\pm 7$ (mod $pq$).
|
Suppose I have two machines, $A$ and $B$. $A$ encrypts a message $m$ and outputs the ciphertext $m^e \pmod n$. $B$ outputs $c$ such that $c = m^e \pmod p$ and $c = m^e + 1 \pmod q$. How can I use $A$ and $B$ to find $p$ and $q$? I am allowed to choose $m$ and $n$.
|
Geometric multiplicity of an eigenvalue of a matrix is the dimension of the corresponding eigenspace. The algebraic multiplicity is its multiplicity as a root of the characteristic polynomial.
It is known that the geometric multiplicity of an eigenvalue cannot be greater than the algebraic multiplicity. This fact ca... |
I am trying to derive the following equation for an analytic function, $f$, defined on the unit disc, $D$
\begin{equation*}
f(z)=\frac{1}{\pi}\int_{D} \frac{f(w)}{(1- \bar{w}z)^{2}}dA(w).
\end{equation*}
Question: Anyone have any ideas on how to derive this formula?
I've been able to verify that the above is true for $... |
How does one prove that if $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $ \displaystyle n \mid \frac{a^{n}-b^{n}}{a-b}$ where $a,b, n \in \mathbb{N}$.
What i thought of is to consider $$(a-b)^{n} \equiv a^{n} + (-1)^{n}b^{n} \ (\text{mod} \ n)$$ and if we suppose that $n$ is odd then we have, $$(a-b)^{n} \equiv a^{n} -b^{n... |
Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
|
It seems to me not, since this would seem to imply that for all functors $F$ and all objects $A$ in $C$ there exists a morphism $F(A) \to A$ (making all functors co-pointed?). However, intuitively it seems like the identity functor acts like a terminal object; a monad $M$ on $C$ is a monoid on $[C, C]$ where the "unit"... |
Any number less than 1 can be expressed in base g as $\sum _{k=1}^\infty {\frac {D_k}{g^k}}$, where $D_k$ is the value of the $k^{th}$ digit. If we were interested in only the non-zero digits of this number, we could equivalently express it as $\sum _{k=1}^\infty {\frac {C_k}{g^{Z(k)}}}$, where $Z(k)$ is the position o... |
I've been self studying abstract algebra from these notes. I've ran across the following Lemma:
The confusing part is that $X$ is a "any subset" not any "subgroup", and that the term "normal set" appeared in place where I expected "normal subgroup". The definition of a "normal set" has not been provided and I couldn't... |
I understand in mathematics there are many "quotienting " proceduce, is this the only reason that we consider equivariant theory for different "unequivariant" theory? Are there any more applications for equivariant theory?Thanks!
|
This came up in proving non-regularity of a certain language (powers of 2 over the ternary alphabet). Any clue to the above equation could help me move forward.
Edit:
Of course, $x = 1, y = 1$ is a solution. I am looking for non-trivial solutions.
|
Suppose $P(x,y)$ gives transition probabilities of a random walk. I've seen $Pf=f$ being called the discrete version of Laplace's equation. In what sense are they analogous?
|
Convex polyhedron $P$ is a subset of $\mathbb{R}^n$ that satisfies system of linear inequalities
\begin{align}
a_{11}x_1 + \cdots + a_{1n}x_n & \sim_1\, c_1 \\
& \vdots \\
a_{p1}x_1 + \cdots + a_{pn}x_n & \sim_p\, c_p,
\end{align}
where $\sim_i \in \{\leq,\geq\}$. It can be alternatively represented by two finite sets ... |
"Gromov Witten potential", when does "potential" mean here? What does the whole thing mean in physics? Thanks!
|
Using generating functions one can see that the $n^{th}$ Bell number, i.e., the number of all possible partitions of a set of $n$ elements, is equal to $E(X^n)$ where $X$ is a Poisson random variable with mean 1. Is there a way to explain this connection intuitively?
|
If not, is there only a limited rangle of polynomials for which the root can be found?
Also, if $u=x^{\frac{3}{2}}+x$, is $x$ expressible in terms of $u$?
|
Suppose that $A \in {\cal S}_+^n$ is a symmetric positive semidefinite matrix. Let $B = {\rm sign}(A)$, where the sign is taken elementwise. Is the resulting matrix $B$ always positive semidefinite?
If not, under what conditions can we say that $B \in {\cal S}_+^n$ ?
|
Just to clarify, I want to show that:
If $f$ is entire and $\int_{\mathbb{C}} |f|^p dxdy <\infty$, then $f=0$.
I think I can show that this is the case for $p=2$, but I'm not sure about other values of $p$...
|
I am looking for a good reference that could clearly explain finite field arithemtic. Specifically I want to understand the importance and utility of the field generator polynomial and why it is important in multiplication.
|
I know this question is easy, but for the life of me, I cannot remember what we call this thing. Googling for this has offered no help.
Consider an object $A$ and a second object $B$(let them be groups if you so choose). We wish to consider and action of $A$ on $B$. Moreover there is a subobject $C \hookrightarrow B$(s... |
Suppose $U$ and $V$ are $2$ dimensional subspaces of $\mathbb{R}^4$. How do you determine the dimension of $U \cap V$? I know that
\begin{equation*}
\text{dim}(U + V) = \text{dim}(U)+ \text{dim}(V) - \text{dim}(U \cap V).
\end{equation*}
So it seems that $\text{dim}(U \cap V) = 0$.
|
I'm currently studying the notion of a quotient topology. The one thing I'm having trouble with understanding is what we're actually doing to the points as we're identifying them. Say we have a $[0,1] \times [0,1]$ in $E^2$ (euclidean $2$-space) with the subspace topology and we partition $X$ into:
The set $\{(0,0),(1... |
In particular for two real numbers $a$ and $b$, I'd like to know if there are formulas for $\zeta (a+b)$ and $\zeta (a-b)$ as a function of $\zeta (a)$ and $\zeta (b)$.
The closest I could find online is a paper by Harry Yosh "General Addition Formula for Meromorphic Functions Derived from Residue Theorem" in some litt... |
I can't seem to solve this one. I've tried Dirichlet's test but to no avail.
Any help is greatly appreciated.
|
How does one find the no of positive integers such that find all possible numbers such that $$\lfloor{\sqrt{n}\rfloor} \mid n$$
What i did was to subsitute $n=t^{2}$ so that the equation becomes $\lfloor{t\rfloor} \mid t^{2}$ But this means that we want $t^{2} = k \lfloor{t\rfloor}$, where $k \in \mathbb{N}$. I don't r... |
A map $s : \mathbb{N} \to X$ is a computable sequence in $(X,\nu_X)$ when there exists a computable map $f : \mathbb{N} \to \mathbb{N}$ such that $s(n) = \nu_X(f(n))$ for all $n \in \mathrm{dom}(\nu_X)$.
My best guess would be, "A map s taking N onto X is a computable sequence in the ??? (X, nu??) when there exists a c... |
In the UK the lottery uses numbers $1$ to $49$ and a total of six numbers are picked. It has been said may times that there is as much chance of numbers $1, 2 ,3 ,4 ,5 ,6$ to be picked as any other random combination.
My question is this:
Let's say that the first $3$ numbers to come out are $1,2$ and $3$. What are the ... |
For instance, say $G = \langle x , y \ | \ x^{12}y=yx^{18} \rangle$. I want to know what is the normal closure of $y$ in $G$.
In general, what are the standard approaches to compute the normal closure of a subset of a finitely presented group? Are there algorithms?
|
Let $X$ be a Hausdorff space and $A$ a closed subspace. Suppose the inclusion $A \hookrightarrow X$ is a cofibration. Let $f, g: X \to Y$ be maps that agree on $A$ and which are homotopic. Are they homotopic relative to $A$?
My motivation for asking this question comes from the following result:
Let $i: A \to X, j: A ... |
Is there any relation between the homology of a space with local coefficients (in $\mathbb Q$ vector space) and the homology with coefficients in $\mathbb Q$? Thanks!
|
Let $n$ be a positive integer, $n!$ denotes the factorial of $n$. Let $d = \gcd(n! + 1, (n + 1)! + 1)$. Show that $d$ divides $n$.
(Hint: notice that $(n+1)(n!+1) = (n+1)!+n+1$)
|
Let's say I am distributing a bunch of encryption keys, where if two users have the same key, they can communicate. I have $20$ different keys, and I am giving each user $2.$ Since encryption keys don't disappear when I hand them out, it can be considered 'with replacement', but I'm not giving the same user the same ... |
So I was bored and decided to figure out the indefinite integral of the absolute value function, $|x|$. Using integration by parts ($u=|x|, dv=dx$, $dx = \text{sgn}(x)=\frac{|x|}{x}$), it can be shown that $\displaystyle\int |x| dx = \frac{x |x|}{2}+C$.
Now I decided to take the integral again, finding that $\displayst... |
I'm looking for a fast algorithm to perform division of large numbers (by hand). Traditional long division just isn't fast enough for my needs.
In most of these cases, I'm only looking for the modulus/remainder of the division, so if there is a fast algorithm specifically for this purpose, it would suffice. (But, if th... |
I am trying to understand left/right cosets in group theory.
Here is the example in my text:
Let $G = \lbrace 1, a, b, c, d ,e \rbrace$
Lets define the group operation $.$ by the following table, where the entry at row $x$ and column $y$ gives $x.y$
ex. $d.e = b$
1 a b c d e
1 1 a b c d e
a a b 1 d e c
b b 1 a e c d
c... |
Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\mathcal{F}$ is an $\mathcal{O}_X$-module and $\mathcal{G}$ an $\mathcal{O}_Y$-module (and the Homs are in the category of ... |
Suppose you would like to find the two tangent lines that support two given disks in the plane
to the same side. Parameterizing the circles using $( \cos \theta, \sin \theta )$,
I find that ultimately I am computing the roots of a 4th-degree polynomial in $x = \cos \theta$
(two of whose roots are imaginary, and two re... |
I know that a universal quantifier can be distributed over conjunction and not disjunction, but I'm having a hard time wrap my head around it. Why is this the case? Is there an example of a statement that would demonstrate this principle?
|
Let $(X,||\cdot||_X)$, $(Y,||\cdot||_Y)$ be a pair of normed linear spaces, and $(X \times Y, ||\cdot||_{X \times Y})$ the induced product space and norm.
If $(x,y)$ is an element in $X \times Y$, is it true that $||(x,y)||_{X \times Y} \lt \delta$ implies that $||x||_{X \times Y} \lt \delta$, where $(||x||_{X\times Y}... |
I'm guessing that this is a standard enough problem, but can't seem to find an intuitive solution.
I have an undirected weighted graph $G = (V,E,f)$ with vertices $V$, edges $E$ and $f$ the weighting function for an edge.
I have a set $\mathbf{I} = \{ I_1, ..., I_n \}$ such that for $I \in \mathbf{I}$, $I \subset V$.
I... |
...subject to the conditions that (i) the projection be smooth and that (ii) smooth sections correspond to smooth vector fields.
This homework problem is really bugging the hell out of me. Of course it can be checked locally, so we'll look at an open neighborhood $U\in \mathbb{R}^n$ and check that $TU\cong U\times \ma... |
First question: Let $a$, $b$, $c$ be distinct primes larger than $3$, and let $x = abc$. Show that if $p$ is a positive integer and $p^2 \equiv 9 \pmod {x}$, then $p \equiv \pm 3 \pmod {a}$, $p \equiv \pm 3 \pmod{b}$, and $p \equiv \pm 3 \pmod{c}$.
Second question: Using the fact that $1001 = (7)(11)(13)$ find the eig... |
As I'm trying to improve my maths I've now reached the subject of differentiation of more general exponential functions.
I've already studied the so called "chain rule".
Now, my book tells me:
$$y=a^x \rightarrow y^{\prime}=a^x\ln(a)$$
Then, there's the following function:
$$f(x) = 10^{-x}$$
Applying the above rule, th... |
Let's say I have a bag of coins, which contains $1$ quarter, $2$ dimes, $3$ nickles and $4$ pennies. If I were to randomly pull out $3$ coins, on average, how much money would I get?
|
What I currently meet are all fiber bundles.
|
The vector space $V(n,d)$ has dimension $n$ and the vector coordinates are from set
$\Sigma$.
$d:=|\Sigma|$
How many vectors can the vector space contain? Why?
|
While searching a question about fibre bundles, which was asked here, i got directed to Vector bundles. I noticed this word "Hairy Ball" which sounded eccentric and made a search at Wikipedia.
How is the hairy ball theorem related to this statement: You can't comb the hair on a coconut.
|
What number of positive integers satisfying the given inequality :
C(n + 1,n – 2) – C(n + 1,n – 1)<= 100
What I did so far :
C(n + 1,n – 2) – C(n + 1,n – 1) <= 100
=> C(n + 1,3) – C(n + 1,2) <= 100
=> n(n+1)(n-4) <= 600
Now to derive after this ?
PS: This is not my homework, It originated my test p... |
Given finite-dimensional vector spaces $V,W$, there is an isomorphism $\text{Hom}(V,W) \rightarrow V^* \otimes W$. In particular, any linear map $\phi : V \rightarrow W$ has a tensor expansion $\sum v^*_i \otimes w_i$ where $v^*_i \in V^*, w_i \in W$.
For example, if one chooses dual bases $\{x_i\}, \{e_i\}$ of $V^*$ a... |
Let $T_f g = f \cdot g$ where $f, g, f \cdot g$ are in $H^2(\mathbb{D})$ (where $H^2$ is the Hardy space on the open unit disk). Now $T_f$ is a bounded operator.
I want to show this by showing that $f \in H^\infty$. So I try to write $f = G_1 h_1$ and $g = G_2 h_2$ where $G_i$ are outer functions and $h_i$ inner functi... |
I'm trying to show that the Möbius strip with boundary circle identified to a point is homeomorphic to $P^2$ (real projective space). I get geometrically why this is so, but, how does one generally construct maps between these spaces to show that they're homeomorphic?
|
Given the binary alphabet {a,b}, I'm trying to find the generating function that distinguishes, for all words of fixed length $n$, the count of blocks of a's and the number of a's. Let $x^p$ count the number of blocks of size $p$ and $y^q$ count the number of a's in a word. As a first example, take the word:
$$abbaaaba... |
I need to write a piece of software that analyzes packet switched computer networks and calculates certain characteristics based on packet loss (the application is Voice over IP quality monitoring). One approach is to use the following $4$-state Markov model (as described here):
State 1 – packet received successfully... |
I'm studiying for an exam I have on combinatorics next week and I'm stuck with the solution to this question:
Imagine you have a pentagon and you want to color the corners of the pentagon so that no two adjacent corners have the same color. If you have 'q' colors, how many ways of coloring the pentagon are possible.
I ... |
Let $A$ and $B$ be compact subsets of a metric space. Denote by $\mathcal{P}(A)$ the set of (Borel) probability measures in $A$, and similarly define $\mathcal{P}(A \cup B)$. They are compact metric. Define $C$ as the set of all elements of $\mathcal{P}(A \cup B)$ which assign probability 1 to $A$.
Question: Is there a... |
In physics there are lots of identities like:
$$\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - (\nabla \cdot \nabla) A$$
I'm wondering if there is an algorithmic algebraic method to prove and/or derive these identities (something like using $e^{i\theta}$ to prove trigonometric identities)?
|
Let $G$ be the fundamental group of the Klein bottle,
$G = \langle x,y \ ; \ yxy^{-1}=x^{-1} \rangle = {\mathbb Z} \rtimes {\mathbb Z} \ .$
What are the nilpotent subgroups of $G$?
I was only able to find a normal series of abelian subgroups with cyclic quotients in $G$, namely
$1\leq \langle y^2 \ ; \ \ \rangle\le... |
I am wondering about the quickest way to prove the following from scratch: the lower shift is not similar to a direct sum of other linear transformations.
Here are the definitions of the terms used in the question. The lower shift is a mapping from $R^n$ to $R^n$ which maps $(x_1, x_2, \ldots, x_{n-1}, x_n)$ to $(0, x... |
Given representations $\rho_1 : G \to \mathrm{GL}(V)$ and $\rho_2 : G \to \mathrm{GL}(W)$, how can we define explicitly the representation of $G$ over $\mathrm{Hom}(V, W) \cong V^* \otimes W$?
|
Suppose $p$ and $q$ are two prime numbers. How can one quickly calculate how many numbers $x$ there are such that $\gcd(x, (q-1)\cdot(p-1)) = 1$, without using brute force?
|
I wish to prove the following inequality. I have done a plot and it looks ok. $\mu \geq 1$ is an integer and so is $h$, where $0 \leq h \leq \mu-1$.
I started like this:
$\frac{e^{-\mu}\mu^{(\mu+h)}}{(\mu+h)!} \geq \frac{e^{-\mu}\mu^{(\mu-h-1)}}{(\mu-h-1)!}$.
I have rewritten this to
$(\mu-h-1)!\mu^{(\mu+h)} \geq (\mu+... |
How would I prove that if $f: \mathbb R^2 \to \mathbb R$ is a function such that
$$\lim_{(x,y)\to(a,b)} f(x,y) = L$$
and for every $y_0 \in \mathbb R$
$$ \lim_{x\to a} f(x,y_0) = L'_{y_0}$$
and for every $x_0 \in \mathbb R$
$$ \lim_{y\to b} f(x_0,y) = L''_{x_0}$$
then
$$
\lim_{x\to a}\left(\lim_{y\to b} f(x,y)\right)... |
Why in homological mirror symmetry, we restrict us to a projective variety (Calabi-Yau manifold)? Because in physics we don't need this condition.
What's the general picture for general Calabi-Yau manifold?
|
Consider the following optimization problem.
$$ \min_{x,y} \quad cx+dy \quad \text{subject to}
\quad e^x + e^y = 1 \tag{OP} \label{OP}$$
Given the lack of inequality constraints in \eqref{OP}, does Slater's condition hold trivially? Can I conclude there is a zero duality gap here?
|
As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get.
I went to Wikipedia to see what the Axiom of Choice is, but as often happens with things like this, the Wikipedia entry is not in plain, simple, understandable language. C... |
Reffered here by https://mathoverflow.net/questions/41750/determining-n-in-sigma-x0n
I'm not entirely sure if this question falls under MathOverflow but neither of my Calculus AP teachers in high school could help me with this:
Given $\Sigma_{x=0}^n {f(x)\over2}$ and the output of the summation, how would you find $n$... |
Can we have a measurable function $f$, whose inverse is not measurable?
|
Think of $t$ and $r$ as two independent variables.
Suppose $E$ be a function of $r$ and $V~$ be a function of $(t,r)$ such that both go to $0$ at $r=0$.
There exists a positive function $M(r)$ such that $M(0)=0$ and $V(t,r) = -\dfrac{M(r)}{R(t,r)}$ where $R$ is another positive function such that $R(0,r)=r$.
Let $p... |
So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My question is: Has anyone constructed a well ordering on the reals?
First, I was going to ask this question about the rationals,... |
One of the definitions of stochastic continuity is: $\forall a > 0$ and $\forall s \geq 0$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$ What does it mean intuitively? I know that it implies the jumps but am not sure how and why.
|
Online, I have read contradicting opinions on whether intersect should take precedence over union (by analogy to logical and and or), or whether all set operators should have equal precedence.
Which way makes more sense and why?
And where does difference fit in? (I'd say it should be equal precedence to intersect beca... |
I am of, and I would like to retain, a mindset that mathematics does not have to have numbers as the central object of interest. With that in mind, I have done a fair amount of self-study on topics in undergrad modern algebra, looking for examples to show this is the case.
For objects like groups, it is often stated ex... |
I want to prove the estimate
\begin{equation*}
P(|A+B|>c)\leq P(|A|>c/2)+P(|B|>c/2),
\end{equation*}
where $A$ and $B$ are random variables and $c>0$. Can anyone help?
|
Let $M$ be an $n \times n$ matrix whose elements are random reals in [0,1].
Two questions.
What is the growth rate of the magnitude of the elements of $M^k$ as a function of $k$? It is definitely
exponential, but maybe the exponent is known?
Is it the case that eventually one element of $M^k$ dominates, as $k \righ... |
Given an $n >0$ is it possible to partition the set $\mathcal{P} = \{1,2, \cdots, 2n\}$ into $n$ pairs $(a_{i},b_{i})$ such that $a_{i} + b_{i}$ is a prime?
|
Given n segments of line (into the $X$ axis) with coordinates $[li; ri]$. You are to choose the
minimum number of segments that cover the segment $[0;M]$. All segment are within $[0,M]$.
|
Let $k$ be a field, $V$ a finite-dimensional $k$-vectorspace and $M \in End(V)$. How can I determine $Z$, the centralizer of $M \otimes M$ in $End(V) \otimes End(V)$?
For example, if
$$M=[[1,0],[0,2]],$$ then $M$ is 6-dimensional, consisting of block matrices of shape 1,2,1.
I was confused at first, because this see... |
So i am reading up on some interesting network topologies for a CS course and what i am reading is $r$-dimensional meshes of tree's.
An $r$-dimensional mesh of trees is formed by adding complete binary tree's to an $N$-sided $r$-dimensional grid where the grid vertices act as the leafs of each binary tree.
Something l... |
I need help understanding how to derivate this function:
$$f(x) = x^{\arctan(x)}$$.
Any suggestions?
|
It is clear that a finte group of order $105$ is not simple since it contains a normal Sylow $7-$subgroup or a normal Sylow $5-$subgroup. I wonder if it is possible that there exists a Sylow $7-$subgroup and a Sylow $5-$subgroup ?
|
It can be shown that
$$ n \int_0^1 \frac{x^n - (1-x)^n}{2x-1} \mathrm dx = \sum_{k=0}^{n-1} {n-1 \choose k}^{-1}$$
(For instance see my answer here.)
It can also be shown that $$\lim_{n \to \infty} \ \sum_{k=0}^{n-1} {n-1 \choose k}^{-1} = 2$$
(For instance see Qiaochu's answer here.)
Combining those two shows that
$$ ... |
The motivating factor behind this question is the comments given for this question asked some hours ago Group of order $105$
So, given a group $G$ of order $n$, what are different methods for constructing a non abelian group of order $n$? Well, I have seen a method which Herstein uses in his book. He takes a cyclic gr... |
I have multiple objects in my lab which are fully described in two coordinate systems. Both systems are Cartesian and in meters. If they had a common origin but were rotated relative to each other I could easily reconcile the two. Additionally, if it was just a displacement it would be even easier. However the two are ... |
I'm looking to brush up on SAT math. (That's basic algebra, probability and trigonometry.)
I'm interested in books and websites.
|
I've been trying to find a way to do this: given a point $P(\alpha,\beta)$, a function $f(x)$, and an angle $\theta$, find the area of the sector determined by extending a horizontal line from $P$ to $f$, and then another one from $P$ to $f$ at the angle $\theta$ from the first line. If you look at this picture:
it's ... |
Can you help me understand how
$$g^{-1}(x) = f^{-1}(x + 2)$$
is the inverse of
$$g(x) = f(x) - 2$$
I cannot get my head around how the 2 ends up inside $f$ :/
thanks
|
Consider I have two sets that neither one is a subset of the other (for example, the set of prime numbers and the set of odd integers). Is there a specific notation that combines the meanings of both ⊄ and ⊅ ?
Also, is there a notation to mean that two sets are completely distinct and share no elements in common (for e... |
This is part of a homework question
I'm trying to figure out the perimeter of the black-filled object below (sorry, it's crudely drawn in paint, but I think you get the idea). To find the perimeter, I'm assuming that I should first find the perimeter of the rectangle (12*2+3*2 in this case) then subtract each little tr... |
The other day I came across the following statement:
A polynomial $f(x,y)$ of degree at most $3$ that vanishes at $8$ of the $9$ points $(x,y)$ with $x, y \in \{-1,0,1\}$ must also vanish at the $9$th point.
I am wondering about how this statement generalizes. Specifically, I am looking for a theorem of the form
Sup... |
I was wondering about the relation between complement of a subset, different between two subsets, union and intersection of subsets. Can we reduce the set of the above operations in a minimal way so that other operations can be represented by these "independent" operations? And how many ways to do this?
For example, t... |
(1). According to Wikipedia, a field of subsets of $X$ is defined to be a non-empty subset of the power set of $X$ closed under the intersection and union of pairs of sets and under complements of individual sets. I was wondering if there is any redundancy in this definition?
Can it be defined to be closed only under p... |
If G is a finite group containing an irreducible character of degree 5, deduce its order.
I know that the order of G is divisble by 5
I also know that the order of G is equal to the sum of the squares of the degrees of irreducible characters.
However I am unable to deduce the order of G with just this information.
Coul... |
Since union of sets is defined to be independent of the number of sets as in Wikipedia
If M is a set whose elements are themselves sets, then x is an element of the union of M if and only if for at least one element A of M, x is an element of A.
I was wondering why for a class of subsets of a set, it being closed und... |
prove by structural induction that in any tree T, the number of leaves is 1 more than the number of nodes that have right siblings.
My proof so far:
s(n). in any tree T, the number of leaves(L) is 1 more than the number of nodes(N) that have right siblings.
s(1) for a tree of one node, there are no right siblings so n... |
This is a contest problem. It asks for all the real solutions of
$$
\sqrt{x^2-p}+2\sqrt{x^2-1}=x
$$
for arbitrary real $p$.
From the eq. it's clear that $p$ cannot be negative. By squaring once, splitting the radicals from the other terms and squaring again, I found a formula for $x^2$:
\begin{equation*}
x^2 = 1+\frac{... |
Given a certain coordinate frame, I can compute a new one by applying a set of rotations in a given order (what I call Euler $Z-Y-X$). So I yaw, then pitch then roll.
Now imagine that I want to do exactly the opposite: given two coordinate frames (same origin to simplify), how do I find out roll, pitch and yaw angles t... |
A field is an abelian group under addition and a group under multiplication. But for the power set of a set $S$, under union and intersection operations and with $S$ and the empty set being their identities, the inverse of any set is not unique for either union or intersection. So is the power set a field of sets?
Must... |
I have a couple things I want to prove. I'm pretty sure a proof by induction is the best route for these.
First, I need to show that $5^n < n!$ from some $n_{0} > 0$. I'm choosing $n_{0} = 12$ since that's the smallest positive integer where $n! > 5^n$.
So: $P(k) : 5^k < k!$
Show $P(12)$: $5^{12} < 12!$
Assume $P(k)$... |
I am an amateur mathematician (but I do have degree's in computer science (with mathematics)). Anyways, I have written this paper, where I have proved that for $\zeta(\rho) = 0$ if $\Im(\rho) \to \infty$ then $\Re(\rho) \leq \log_2(3) - 1$.
Now this is a big claim, and I am a cranky fella ;) who has a number of withdra... |
If f is in $L^1(\mathbb{R})$, is it true that $\lim_{h \to 0^+} \frac1{h} \int_x^{x+h} f(t)\mathrm{d}t$ exists and is finite for every x in $\mathbb{R}$?
Would it be possible to use something along the lines of the following argument: The Lebesgue Differentiation Theorem says that this integral is equal to f(x) a.e., w... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.