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Let's say we have two functions $f(x)$ and $g(z)$ where $x$ and $z$ are two different, unrelated variables.
Could anyone tell me why the following equality holds (if if does)?
$$\frac{d}{dx}\int f(x)dz=\int \frac{d}{dx}f(x)dz$$
|
What is an infinite collection of maximal ideals containing the ideal $I = (x^2 - y^3) \subset \mathbb{C}[x,y]$?
|
Below is the text of the theorem:
$\mathcal{F}_{i,j}$ are sigma algebras indexed by $i$ and $j$.
I'm having some difficulties in understanding this proof.
Do the $\mathcal{A}_i$ contain $\Omega$ or is $\Omega \in \mathcal{A}_i$? Also, why would $\bigcup_j \mathcal{F_{i,j}} \subset\mathcal{A}_i$?
Any help would be appr... |
This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid.
Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine $\text{dom}(\phi)$, determine if $\phi$ is birational, and if so describe the inverse map.
My answer:
It is rational since $\... |
now here's how I did proceed.
By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| < δ$ and $x,a$ are elements of $E$ implies $|f(x) - f(a)| < ε.$
Then suppose $x, a$ are elements of $\Bbb R. $
Now
\begin{align}
|f(x) - f(a)|
&= \left|\frac{\cos(x... |
I need to calculate an integral over the set:
$$D \colon 0\leq x\leq \pi\text{ and }|y|\leq x$$
from the set (definite integral) $D \int \cos(y)dA$
I don't understand what $|y| \leq x$ means.
Can someone give me a tip how to calculate this?
|
Four accidents occur independently, with each accident following an exponential distribution with a mean of 22.5. What is the expected value of the minimum of the four accidents?
Attempt:
$min(x_1,x_2,x_3,x_4) = n * f(x)* (1-F(x))^{n-1}=4(\frac{1}{22.5}exp(-x/22.5))(exp(-3x/22.5))$
Expectation =$ \int_{0}^{\infty}\frac... |
I have tried to solve the integral
$${\int_0^1 U_n (x) U_m (x)x dx },$$
where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind.
Solving the integration and checking the result, I noticed that it does not match the ones I did analytically for $n=0$, $m=1$; and some other values, so, there must be somethin... |
Again a question about integration. Consider the integral $$\int_{\gamma}\frac{2}{(z+2)^2}dz,$$where $\gamma:[0,1]\to\mathbb{R}$ such that $\gamma(0)=0$ and $\gamma(t)=t+it\sin(\frac{\pi}{t})$ if $t>0$. I first tried to compute this integral by using the formula $$\int_{\gamma}f(z)dz=\int_0^1f(\gamma(t))\gamma'(t)dt,$$... |
Let
$$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{a}{b}$$
and let $a$ and $b$ are coprime, $h_{n}=a$.
$h_{n}$ is prime for
$$n=2,3,5,8,9,21,26,41,56,62,69,79,89,91,122,127,143,167,201,230,247,252,290,349,376,459,489,492,516,662,687,714,771,932,944,1061,1281,1352,1489,1730, 1969,2012,2116,2457,2663,29... |
Say you are given $n$ points such that no three are colinear. Show the number of ways to separate them into two subsets by drawing a straight line depends on $n$ but not the position of the points.
|
Sorry, I know it's not really related to Math itself, but I still need a native English Mathematician to help me through that and don't know where else to ask... So please don't close it because it might be subjective - I really just want to know how to say it in a correct way;
It would be embarassing to have 5 words i... |
I just need some guidance to prove a portion of the following theorem.
Let $V_1, V_2, ... , V_n$ be subspaces of a vector space $V$. Then the following statements are equivalent.
$W = \sum V_i$ is a direct sum.
Decomposition of the zero vector is unique.
$V_i\cap\sum_{j\neq i}V_i = \{0\}$ for $i = 1, 2, ..., n$
di... |
Let us say I have a function, $g$ that I want to find the extrema of subject to the condition $h=0$ via the method of Lagrange multipliers. i.e. so that I find the extrema of the function:
$$f=g-\lambda h$$
If $f$ is a maximum (say), can we say anything about the nature of the stationary point of $g$ (e.g. can we say t... |
Given the Leibniz rule:
$$\frac{d}{dy}\int^a_b f(x,y) dx = \int_b^a \frac{\partial f}{\partial y} (x,y) dx$$
How do I prove a more general case using the chain rule and the above:
$$\frac{d}{dy} \int_{g_1(y)}^{g_2(y)} f(x,y) dx =?$$
From the fundamental theorem of calculus we have that: $$\frac{d}{dt} \int_{f_2(t)}^{f_... |
How do I integrate the following.I tried by substituting y=sin t but I did not get anything proper.
$\int \sqrt{\frac}{a-2y^2}{3} dy$
|
I have almost always seen the determinant expanded by using the first row:
$$ A =
\begin{vmatrix}
a & b & c\\
d & e & f\\
g & h & i
\end{vmatrix}
$$
Such as:
$ |A| = a(ei−fh)−b(di−fg)+c(dh−eg) $
How does the expansion by 2nd and 3rd rows work?
Here's what I have tried to guess:
Expanding by 2nd row:
$ |A| = -d(bi−ch)+e... |
All explanations and proofs I find about the Pumping Lemma are ambiguous. So if I understand this correctly, if we can find some $p>0$, then for any string $|w| \ge p$, we should be able to split it up into $xyz$ according to some conditions which are satisfied as shown below.
Here's the problem. Say $L$ is the langua... |
Solve the system of inequalities and indicate all the integers which are in the solution set:
\begin{align*}
3−2a&<13 \\
5a&<17
\end{align*}
I solved it then i realized it said indicate all the integer. How do i write this?
I got $-5<a<3.4$ Please help.
|
Let $a \in \mathbb{Z}[i]$ be a prime element. Show that $N(a)$ is a prime or a square of a prime.
I know that the converse is true. That is, if the norm of $a$ is prime then $a$ is a Gaussian prime. But how do I prove the claim above? Thanks!
|
When you draw a planar cubic bipartite graph $\Gamma$ and 3-color its edges you can use this as an orientation $\mathcal O$.
Definition A left-hand turn path on $(\Gamma, \mathcal O)$ is a closed path on $\Gamma$ such that, at each vertex, the path turns left in the orientation $\mathcal O$.
I want to calculate the ... |
In Geometry of Schemes by Eisenbud and Harris, it is claimed in Exercise II-19 that:
There are infinitely many isomorphism types of degree 7 subschemes
supported at the origin in 3-space, and infinitely many types of
degree 8 subschemes supported at the origin in the plane.
However, this exercise is marked "For t... |
I would like to calculate the number of choices of teams I can make in the following scenario.
Suppose a team is comprised of 3 characters (1 leader and 2 support members) and suppose there are 108 characters in total that can play. How many different team combinations would there be?
My attempt at the problem is that ... |
Problem: Show that a $8 \times 8$ chessboard cannot be perfectly covered by $1$ square tetramino, and 15 other tetraminoes chosen from straight tetraminoes and Z-tetraminoes.
My attempt: I tried to color the board with black and white in several different ways but none of them yielded a contradiction. I tried coloring ... |
Here I'm defining my Fourier transform according to the convention
$$
FT\left\{f(x)\right\} = \hat{f}(q) = \int_{-\infty}^{+\infty}dx\, f(x)\, e^{-i q x}\, .
$$
My intuition - which may be off - tells me that this special unit-modulus form for $f(x)$ must result in some special conditions on $\hat{f}(q)$. For example, ... |
In village there are two kinds of people: pessimistics and optimistics.
optimistics- answering "YES" for $70\%$ from the questions and "NO" for $30\%$ from the questions .
pessimistics- answering "NO" for $80\%$ from the questions and "YES" for $20\%$ from the questions
In a survey that made in the village they found t... |
Could you help me to show that
(1) $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$
(2) $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$
where $o(\cdot)$ is little $o$ notation described here and $\omega(\cdot)$ is little $\omega$ notation described here
|
Let the random variables $X_1,X_2,\ldots,X_n$ iid $U[\theta-1\,,\theta+1]$.
So the likelihood function therefore has the form:
$$L(\theta\mid X)=\prod_{i=1}^nf(X_i\mid \theta)=\frac{1}{2^n}I(X_1, \ldots , X_n \in [\theta-1\,,\theta+1])\text{ ?}$$
It is correct ?
|
How can I prove that
$ \lim_{ n\to \infty} \int_{R}^{} \cos(nt)f(t)dt = 0 $
for any $f \in L_{1}(R) $?
I believe that I should use a fact that cosinus is a cyclic function and divide this integral into a series of integrals $ \int_{[\frac{-\pi}{2n} +\frac{2k\pi}{n}] }^{[\frac{-\pi}{2n} +\frac{2k\pi + \pi}{n}]} \cos(nt)... |
If we showed that $\mu(F_n)<\infty$ for all $n\in \mathbb{N}$, can we get $\cup_{n \in \mathbb{N}}F_n<\infty$?
The problem is the following:
In the solution of Folland chapter 1 exercise 14,
Suppose $F^*=${$F:F\subset E, 0<\mu(F)<\infty$} , $\alpha:=sup_{F\in F^*}${ $ \mu(F)$} $<\infty$. Then for every $n$ there exist... |
I'm trying to do a exersie on page 16 of this paper. It says:
Exercise. Show, using the rules of Godel-Lob modal logic (GL), that $\square⊥ ↔ \square \diamond p$; recall that $\diamond p = ¬\square¬p$. Informally, that is, no statement can be proven consistent with Peano Arithmetic unless Peano Arithmetic is inconsist... |
Consider two maps $f, g: X\to Y$, where $X=Y=\{ 0, 1 \}$ with discrete topology, $f$ is the identity and $g$ maps everything to 0. Then it's clear that $\widetilde{H}_0(X;\mathbb{Z})\cong \mathbb{Z}$ and all the other groups are zero. The maps $f$ and $g$ cannot be homotopic, as they are not equal. I want to calculate ... |
Find the zeroes of $\cos z=2$.
Attempt:
$\cos z=\cos(x+iy)=\cos(x)\cos(iy)-\sin(x)\sin(iy)=\cos(x)\cosh(y)+\sin(x)\sinh(y)=2$
I don't know how to proceed form here...
|
Let $C$, $D$ and $E$ be cocomplete categories.
Is there a construction $C \otimes D$ such that there is a correspondence between functors $C \otimes D \to E$ preserving colimits and functors $C \times D \to E$ preserving colimits in each of its compontents ("bilinear")? How can I construct it? Why not?
|
I'm trying to determine the perimeter of a teardrop shape formed by two adjacent circles (non-intersecting) with mutually tangent lines drawn on both sides of the circles. I've attached a sample diagram.
Known variables:
$$
\begin{array}{r|l}
b = r_1 & \text{radius of the larger circle}\\
r_2 & \text{radius of the sm... |
I'm working through Dummit and Foote right now. The lattice isomorphism theorem is stated as follows: "Let I be an ideal of a ring R. The correspondence $A \leftrightarrow A/I $ is an inclusion preserving bijection between the subrings A of R that contain I and the set of subrings of R/I."
First off, to make sure I und... |
Does anyone know if there are any papers/results on functionals of the type :
$$\int_0^tp(X_s)ds$$
where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an object known when $p(x) = x^2$ (in general, or for some special models) ?
|
We have the PDE
$\frac{\partial u}{\partial t}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}=0$. What would be conditions on $a$ and $b$ for the equation to constitute a conservation law? I think that $a=b$ would be sufficient, since then the flux of $u$ into a small control volume in a Cartes... |
application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$ where $1/p + 1/q = 1$
I know this is a straight application of the inequality, but how am I assured that the integral of $(1+|t|)^2|f(t)|^2$ and $(1+|t|)^{-2}$ are finite? I understand the first one I think, but I am not sure for the second.
|
In a binary code of length n
p(exactly i errors in specified positions)=$t^i (1-t)^{n-i}$.
I have an example where $C=\{000,111\}$, the binary repetition code of length 3.
Suppose 111 is transmitted.
Then the received words will be decoded as 111 are
${111,011,101,110}$
Why does P$($decoding as $111$ $)$= $(1-t)^3+3t... |
Can we deduce that $F$ is a convex function (i.e $F''(t)>0, \forall t>0$) from the following conditions:
$F(t)=\int_{0}^t f(\xi) d\xi$, $0\leq \theta F(t)< t f(t), \forall t>0$
The function $\frac{f(t)}{t^{p-1}}$ is increasing in $(0,+\infty)$ and satisfy $$\limsup_{|t|\rightarrow\infty}\frac{f(t)}{|t|^s}=0~\text{an... |
The isosurfaces I'm reading about are defined by a constant value v in a scalar field.
The scalar field is defined by placing n vectors in k-space such that
$iso(\vec{x})=\sum_{i=1}^{n}\sum_k(x_k-p_{i_k})^2$
So the field value at each point in space is the sum of all squared distances.
I'm interested to find a funct... |
Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$.
The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is defined as the largest nonnegative real eigenvalue of a matrix $N_i$, where the $(k,j)$ entry is $(N_i)_{k j}=N_{i,j}^k$.... |
Talking about colored graphs there is a definition given for graphs with bounded color valence. This definition is as follows:
A vertex-colored graph $G=(V,E)$ has bounded color valence, if there is a constant $D$, such that for every color class $C$ every vertex $v$ (possibly in $C$) has at most $D$ neighbors or at m... |
Consider the system
$$
\dot{x}=x+ay-y^3,\quad \dot{y}=b-2y+x.
$$
The task is to give the bifurcation curve for the equilibria.
First of all, equilibria are determined by
$$
x+ay-y^3=0\quad b-2y+x=0.
$$
This gives $x=2y-b$. Hence
$$
y^3-(a+2)y+b=0.
$$
Now, it is said, we need to determine the multiple roots ... |
Let $f:\mathbb{R}\to \mathbb{R} $ be an uniformly continuous functions, how to prove that
there are $a, b $ such that $|f(x)|\leq a|x|+b$
thanks for any suggestions
|
Given u $\in \mathbb{R}^{n}$ and v $\in \mathbb{R}^{m}$ with unit $L^{2}$ norm, i.e. $\|u\|_{2}$ = $\|v\|_{2}$ = 1. Construct a rank-one matrix B $\in \mathbb{R}^{mxn}$ such that $Bu = v$ and $\|B\|_{2}$ = 1.
|
My apologies for being very brief with this question, the reason for this is because I don't know where to start.
The question is as follows:
A curve has the equation $\lambda \cosh(x) + \sinh(x)$, where $\lambda$ is a constant.
Determine the coordinates of the turning point of the curve in the case when . Is this a ... |
Calculate the are enclosed by ${y = 2x - 1}$ and ${y= x^2 + 6x + 2}$
First of all I combine the equations into:
${x^2 + 4x + 3 = 0}$
${(x + 3)(x + 1), x = -3, x = -1}$
They intersect at ${(-3 -7) (-1, -3)}$
I would say that ${y = 2x -1}$ is the top equation so to work out the area, I would use:
${\int_{-3}^{-1} (2x ... |
If I had four microphones, located on poles that were at corners of a square that's 300 feet on a side (or some other specific configuration that would make the math easier), would I be able to use the difference in time between each mic to determine where the gun shot occurred? Let's presume 2 dimensions, and that th... |
I’m trying to find a closed form for this integral.Any help is appreciated.Thanks
$$I=\int_0^{1}\frac{{ \arcsin}({x^2})}{\sqrt{1-x^2}}dx$$
|
Let $ G=
\begin{pmatrix}
a & a\\
a & a\\
\end{pmatrix}
$ where $a\in \Bbb R, a \neq0$. I need to show that $G$ is a group under matrix multiplication. The problem is determinant of the matrix is $0$, and if I am to find the inverse and conclude that the inverse is within the set, I come ... |
For some values of $x$ the limit of infinite tetration converges. For example when $x=\sqrt{2}$ this is fixed point
$$\lim_{n\rightarrow \infty} \sqrt{2} \uparrow \uparrow n = \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}} = \sqrt{2}$$
This limit converges for all $e^{-e} \le x \le e^{1/e}$. What is the minimum... |
i am currently studying a course in homology theory and have done a basic introductory course in algebraic topology which deals with the idea of the fundamental groups and their topological invariance.I am really stuck in getting through the basic idea of homology ,espacially , what properties of the space does it reve... |
As you can see in the title, I want to know how the number before a root is called.
For example, if you have the cubic root of 8, I want to know how the 3 before the roof is called.
Actually, I already know it - but in German... In German, it's called "Wurzelexponent".
If I translate that german eord into English, it i... |
I'm trying to learn about topological groups but I can't seem to google anything that provides a simple and clear example of how the set of elements of a group correspond to open sets of a topology.
Also, what does it mean for a group binary operation to be continuous. I would think that is always true since if $g_1\i... |
Prove that $\aleph_0^{\aleph_0} = \mathbb{c}$
without using Cantor's $2^{\aleph_0} = \mathbb{c}$
Card $\mathbb{N}^\mathbb{N} = \aleph_0^{\aleph_0}$
Card $(0, 1) = \mathbb{c}$
Define: $f: (0, 1) \rightarrow \mathbb{N}^\mathbb{N}\:\:$, by
$f(0.a_1a_2...a_n...) = \langle a_1,a_2...a_n...\rangle$
$f$ is a injective $\Rig... |
Been racking my brain on this one..
I've tried some things but not sure if it flows logically:
$\forall x \in \mathbb{Z}: n \ge 1$
$n+2 \ge 1$
$2n+2 \ge n+1$
$\frac{2n+1}{2n+2} \ge \frac{n+1}{2n+1}$
Does this make sense? I'm not sure where to go..
EDIT: Thanks everyone for the answers. I'm trying to learn how to flow... |
In what contexts is it conventional to write $ ^a B$, and in those contexts what does it mean?
I remembering seeing this but I can't remember where. I believe we would expect $a$ to be a function here and $B$ to be a set, possibly with additional structure. I feel like I may have seen this in field theory to mean the e... |
I recently come across the following statement mentioned in a proof:
Let $X,Y$ be normed linear spaces and $T:X \rightarrow Y$ be a linear operator. if for every bounded linear functional $U: Y \rightarrow \mathbb{R}$ , $UT$ is bounded, then $T$ is also bounded.
How can we justify this statement?
|
I am kinda lost here. All I did until now was finding eigenvalues and vectors for a matrix but as far as I can understand the question it asks me to find the eigenvalues of a Linear Transformation?
Let V be the vector space of all real polynomials p(x) of degree ≤ n. Define
T : V → V, T (p) = q, q(t) = p(t+1). Determin... |
How to show that the only elimination ordering on the ring $k[x,y]$ is the lexicographic ordering? (Ene and Herzog, Gröbner Bases in Commutative Algebra, Problem 3.1.)
Definition (Elimination Order): A monomial ordering on $R = k[x_1,\dots, x_n,y_1,\dots,y_m]$ is an elimination ordering for $x_1,\dots, x_n$ if whenev... |
Given points P1, P2, and P3, I need to calculate the center point of 2 tangent circles, C1 and C2, with radius R. Line P1P2 is tangent to circle C1 at P2, line P2P3 is tangent to C2, and C1 and C2 are tangent to each other.
Line/Circle Image
I am plotting points along a route as a vehicle travels from P1 to P2 and P3. ... |
I need to find number of permutations $p$ of set $\lbrace 1,2,3, \ldots, n \rbrace$ such for all $i$ $p_{i+1} \neq p_i + 1$.
I think that inclusion-exclusion principle would be useful. Let $A_k$ be set of all permutation that for every permutation $a$ in this set $a_{k+1} \neq a_k + 1$. So our answer would be $| A_1 \c... |
Let $X=\{(x_1,x_2,x_3) \in \mathbb C^3 : x_1^2=x_2^2+x_3^2\}$, an algebraic variety. How do i prove that $X$ is normal, but not smooth? I guess the non-smoothness appears at the point $(0,0,0)$, but I don't know how to put that rigorously.
|
I am trying to evaluate the following limit:
$$
\lim_{x \to 0} \frac{e^x}{\sum_{n = 1}^\infty n^k e^{-nx}},
$$
where $k$ is a large (but fixed) positive integer. I am unsure how to proceed. Can this be done using L'Hopital's rule? Just started learning calculus, thanks guys!!
|
According to the researchers who cracked it, RSA-768 took an equivalent 2000 years to factor on a 2.2GHz single-core computer.
Using the complexity equation for the General Number Field Sieve with n=2^768 yielded 8.7*10^31 operations (calculated using Maxima). Let's naively assume a computer can perform 1 billion opera... |
The question asked here is, suppose we are given a a conjecture to prove in number theory (with numerical evidence showing its true). Say an important well studied conjecture that most will believe is true-though it may have remained unproven for more than 200 years or more (a long time).
How do we know it is true tha... |
I am trying to prove that when solving numerically diff. eq.:
$$
y'(t)=f(t,y(t)), \hspace{0.5cm} y(t_{0})=y_{0}
$$
using trapezoidal rule, namely:
$$
y_{n+1}=y_{n} + \frac{h}{2} \left( f(t_{n},y_{n}) + f(t_{n+1},y_{n+1}) \right)
$$
we get LTE (local truncation error) proportional to $C h^{3}$, $C=const$.
Let's denote ... |
Any ideas on evaluating the definite integral
$$\int_0^\frac{\pi}{2}x\ln^2(\sin x)\ dx$$
The best numerical approximation I could get is $0.2796245358$.
Is there even a closed form solution?
|
So I am having trouble with finding the generalized solution and I am not sure why my answer is interpreted as incorrect and I wanted to double check.
$$ \overrightarrow{y'} = \begin{pmatrix} -6 & 1 \\ 0 & -6 \end{pmatrix} \overrightarrow{y} $$
I found the eigenvalue of $$ \lambda = -6 $$
Then I found the eigenvectors... |
Suppose that $X_1, \ldots, X_n$ are independent, but not necessarily identically distributed. If we know that $E(X_n) = 0$, it is possible to show that $\frac{X_1 + \ldots + X_n}{n} \to_{a.s.} 0$ using only the assumption that the first moment, $E|X_n|$ is finite?
|
I know there are some posts about the same thing but I am unable to do my specific question or at least, I don't think I'm doing it the right way.
The question asks me to use the Trapezoidal Error bound formula to determine N so that if $\int^{10}_0e^{-2x}dx$ is approximated using the trapezoidal rule with N subinterva... |
Let $A$, $B$, and $C$ be disjoint events where
\begin{align*}
P(A) & = 0.50\\
P(B) & = 0.30\\
P(C) & = 0.20
\end{align*}
If $D = B \cup C$, what is the value of $P(C \mid D)$?
I know that
$$P(C \mid D) = \frac{P(C \cap D)}{P(D)}$$
But how do I get $P(C \cap D)$ with only this info given?
|
Write out a minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$, and determine the primes belonging to $I$. Determine the dimension of the ring $\mathbb C[X, Y, Z]/(XY, Y Z, XZ)$.
If the decomposition of ideal $I$ is given then the proof is straightforward. However, without any pre-inf... |
The nature of infinity and irrational numbers (among other things) always gave me trouble. But recently I learned to think about infinite sequences in terms of recursive algorithms and their time evolution - and it turned out to be much easier to accept.
I know that in principle, general Cauchy sequences can't be defin... |
I am trying to evaluate this integral:
$$\int_{0}^{1}\int_{x}^{1} y^2 \sin(2\pi \frac{x}{y})dydx$$
$$=\int_{0}^{1}\int_{0}^{1} \chi_{[x,1]}(y) y^2 \sin(2\pi \frac{x}{y})dydx$$
$$=\int_{0}^{1}\int_{0}^{1} \chi_{[x,1]}(y) y^2 \sin(2\pi \frac{x}{y})dxdy$$
$$=\int_{0}^{1}\int_{0}^{1} \chi_{[0,y]}(x) y^2 \sin(2\pi \frac{x}{... |
As in the title I'm trying to work out what the chances of drawing $m$ copies of a specific card in $n$ draws are given a deck size of $d$ containing $a$ copies of $A$.
I've tried using permutations for this but must be doing something wrong as the solution i get:
\begin{equation}
\frac{n!(d-n)!}{m!(n-m)!d!}
\end{equa... |
I have shown a group of order 10 has a cyclic subgroup of order 5. But now I am stuck. Any hints?
|
Let be $X,Y$ independent random variables with $X\sim\mathcal{U}(\{0,1\})$ and $Y\sim\mathcal{U}(\{0,1,2\})$. Compute the CDF of $Z:=\min\{X,Y\}$.
My idea:
$$
\begin{align*}
P(\min\{X,Y\}\leq x) &= 1-P(\min\{X,Y\}>x)\\
&= 1-P(\{X>x\}\cap\{Y>x\})\\
&=1-P(X>x)\cdot P(Y>x)\\
&= 1-(1-P(X\leq x))(1-P(Y\leq x))\\
&=1-(1-F_X... |
Is $x\sin(\frac{1}{x^2})$ uniformly continuous on $(0,1]$? I am really unsure how to start this one off.
I have done checks for "similar" functions like:
$f=\sin(\frac{1}{x}), x\in(0,1]$, by using $|f(x_1-f(x_2)|=|\sin(\frac{1}{x_1})$-$\sin(\frac{1}{x_2})|=2$ when $\frac{1}{x_1}=\frac{\pi}{2}+2\pi n$ and $\frac{1}{x_2}... |
Say we have a standard deck of 52 cards. Probability of drawing the King of Hearts OR the Queen of Diamonds is $\frac{2}{52}$ obviously (or $\frac{1}{26}$). If we were to make 30 draws with replacement (so each time the card is drawn, it is put back in the deck and and the deck is shuffled), the odds that the King of ... |
Let we have a forest $F_n(P)$ with $n$ nodes defined by set $P$ of all pairs $\{\text{father}, \text{son}\}$. For instance $P=\{\{1, 2\}, \{3, 4 \}, \{1, 3 \}\}$ defines a forest $F_5(P).$
Let $k_i$ is the number of descendants the node $i$. For the forest $F_5$ we have $k_1=3, k_2=0, k_3=1,k_4=0.$
Let $K_n(P)$ ... |
How do I go about converting programming logic to mathematic notation?
For example, I read a question that asks:
A piece of software is set to run every 35 minutes. The time is now 2.03pm. The software last ran at 1.25pm. Write a solution that checks to see if the software needs to be ran.
Easy enough. First convert th... |
I am trying to prove this using mathematical induction, but I'm lost once I get to comparing the two sides of the equation.
Proposition:
For all integers n such that n ≥ 3,
$ 4^3 + 4^4 + 4^5 … 4^n = \frac{4(4^n - 16)}{3}$
Proof:
Let the property P(n) be the equation
$P(n) = 4^3 + 4^4 + 4^5 … 4^n = \frac{4(4^n - 1... |
I am having difficulty understanding the solution to this problem.
Since the solution is in the form of Bayes theorem I expected something along the lines that looked similar to Bayes theorem.
Suppose that in each individual of a large population there is a pair of genes that determine eye color. Each of these genes ... |
Components $1$ and $2$ are connected in parallel, so that subsystem works iff either $1$ or $2$ works;since $3$ and $4$ are connected in series, that subsystem works
iff both 3 and 4 work. If components work independently of of one another and $P(\text{component works})=0.9$, calculate $P(\text{system works})$.
My try... |
I am wondering why I get an incorrect answer when trying to find the volume of a rotated function about the x-axis when using the Average Value Theorem.
I want to find the volume of $y=\sqrt{x-2}$ as it is revolved around the $x-axis$ on $ x\in[2,6]$
Classic Calculus (area with disks)
$\int_2^6\left(\pi x - 2\pi\right)... |
We are told to use the definitions of Big-Oh and Big-Omega to prove that a given function is in $O(f(n))$ or $\Omega(f(n))$. It requires being able to use $c$ and $n_0$.
Use the definitions to show that $6n^2 + 20n \in O(n^3)$ but $6n^2 + 20n \not\in \Omega(n^3)$
The only way I know that these are true are by looking a... |
Let $N = \{1,(12)(34),(13)(24),(14)(23)\}$. Determine if the quotient group $S_4/N$ is isomophic to $S_3$.
I computed the cosets: $N, (12)N, (13)N, (14)N, (123)N, (234)N$, and the others are equivalent to one of these. This took me a while to do. Is there a faster way and what do I do next?
|
A teacher put this problem up the other day and I'm confused about how he got to the answer. Can you explain it to me?
Job $X$ provides $20$ vacation days and $143,000$ euro annual salary.
Job $Y$ provides $16$ vacation days and $145,400$ euro annual salary.
The question asks: what salary would $Y$ need to proved to ma... |
Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by
$$
D_v f(x) = \lim_{h\to 0} \frac{f(x + hv) - f(x)}{h} .
$$
What bothers me about this definition is that we're calling $x$ a "point" ... |
Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$.
Show that if $V^0 \subset W^0$ then $W \subset V$
This is an exercise problem in Linear Algebra Done Right, 3rd Edition, in Chapter 3.F . For a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ ... |
I was watching these videos from MIT's series: Green, Stokes.
And I didn't understand the justification: their "extended version" of the theorems.
I looked up on google and couldn't find many references (or they explained things with differential forms, which haven't yet studied).
Does anyone know some references about... |
A cruise ship left port $A$ and traveled towards port $B$ $225$ km away. After $1.5$ hours of travel, the cruise ship was stopped for a half an hour and then it had to increase its speed by $10$ km/hour in order to reach port $B$ on time.
Find the original speed of the cruise ship.
|
If $S$ and $T$ are subrings of $R$, is $S+T=\{s+t\mid s\in S, t\in T\}$ a subring of $R$?
So I think that $S+T$ is a subring, but I am getting stuck trying to prove it.
Clearly since $S$ and $T$ are rings, $0,1\in S$ and $0,1\in T$. So, $0+0=0\in S+T$ and $0+1=1\in S+T$.
Then if $a,b\in S+T$ then there exist $s_1,s... |
I was given a recursive formula and I need to convert it into a $3\times3$ matrix. What is a general formula to do this?
My recursion is in the form: $$R_{n+2} = 4R_{n+1} + 5R_n + 2R_{n-1}$$
Just want a general equation to help me solve this!! Thanks
|
How to express
$$\int^{\pi}_{-\pi}e^{c \cdot e^{j\omega}}d\omega$$ in closed form, where $c$ is a constant?
Should it be some Bessel function?
|
If, let $f: \left[ 0, 1\right] \to \mathbb{R}$ continuous with only one discontinuity point is integrable on $\left[ 0, 1 \right]$.
I think this is false but I can't find a example that contradicts the statement.
|
Let $L(x,y)$ be "$x$ loves $y$". Then is the statement: "Nobody loves everybody" equivalent to $$∀x ∀y \overline{L(x,y)} $$
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