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So I have to either prove or disprove this inequality:
$$ \left\lVert x\right\rVert^2 - \left\lVert y\right\rVert^2 \le \left\lVert x-y\right\rVert \left\lVert x+y\right\rVert$$
I know this to be true in Hilbert space. But i cant figure out what to do in Banach space.
|
I was wondering right that since the notion of a topology is much more general than that of a metric, and that "neighborhodness", if you will, and the concept of continuity, is generalized by the notion of a topology. so is the set of all topological spaces actually bigger than that of metric spaces? in other words if ... |
How would I solve the following Differential Equation
$\frac{dy}{dx}= \sqrt{x+y} $
Clearly, it is nonlinear and non homogeneous, I could not find the way to solve it with Bernoulli or to make it an exact differential equation.
|
If I have a linear programming problem e.g.
$$\max 2x_1 + x_2$$
with these constraints
$$x_1-2x_2 \leq 14$$
$$2x_1-x_2\leq 10$$
$$x_1-x_2 \leq 3$$
And I want to solve the problem starting from a specific point e.g. $A=(x1=5, x2=0)$ and $B=(x1=0, x2=5)$ .
How should I configure the initial tableau to allow to iterate... |
Find all the solutions for $x$ in the following equality.
$$
\sin\left( x - \frac{\pi}{4} \right)
= \sin\left( 3x + \frac{\pi}{4} \right)
$$
I tried using the following formulas for both sides but I got stuck.
$$ \sin(a+b) = \sin(a)\cos(b)+\sin(b)\cos(a) $$
$$ \sin(a-b) = \sin(a)\cos(b)-\sin(b)\cos(a) $$
Is there... |
I am doing an $\epsilon-\delta$ proof ($z \rightarrow i, f(z) \rightarrow \infty$) and currently have the absolute value $$|f(z)|=\left|\frac{z-1}{z^2+1}\right|$$ and I wish to make a statement about $|z-i|$ I have made the fraction above into $$\frac{|z-1|}{|z-i||z+i|}$$ but I cannot proceed. Any tips.
|
Tomorrow I have an exam about graphics and log operations.
Our teacher gave us a paper with exercises to practice and one of the exercises is:
$\log_2 \sqrt[4]4$
I couldn't find the solution. My private teacher helped me but he didn't find the solution.
We don't think this log has a solution.
Does a solution exist ... |
Hello fellow stackexchangers, this is my first post, so sorry if this is too vague or violates guidelines.
I am studying Physics and this problem came up, I will type it verbatim
A small object collides with a large object and sticks. Which object experiences the larger magnitude of momentum change -the small object -... |
What are the methods for solving the following class of problems:
$$
\frac{df(x)}{dx}=a f(x-\xi),
$$
or
$$\begin{cases}
\frac{\partial F(x_1,x_2)}{\partial x_1}=a_1F(x_1-\xi_{11},x_2-\xi_{12})\\
\frac{\partial F(x_1,x_2)}{\partial x_2}=a_2F(x_1-\xi_{21},x_2-\xi_{22})
\end{cases}
$$
and so on, where $a$ and $\xi$ are co... |
I have been trying to find a method to translate the pitch and roll angles of one object to those of another connected object at a different yaw - i.e I have an IMU mounted on a quadcopter frame and a 3 axis gimbal mounted below.
If the gimbal is pointed straight ahead its pitch and roll values equal that of the IMU - ... |
I'm the TA in a discrete math course and there was a problem in this weeks problem set which I had troubles solving. It goes like this:
Find the coefficients of $v^2w^4xz$ in the expansion of $(3v + 2w + x + y + z)^8$.
The multinomial theorem (which I had to look up and have never seen before) solves this very easily, ... |
It is given that
$$\left\lVert x-y\right\rVert =\left\lVert y-z\right\rVert = \left\lVert z-x\right\rVert \qquad (1) $$
where $x,y,z \in \Bbb R^2$ and $ \left\lVert x\right\rVert=\sqrt {x_1^2+x_2^2} $.
I have prove or disprove $$ (1) \Rightarrow \left\lVert x\right\rVert \le \left\lVert y\right\rVert+\left\lVert z\r... |
Let $\zeta = e^\frac{2 \pi i}{p}$, with $p$ prime. Show that the elements of the form $1+\zeta +\zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$.
I know that $f(\zeta) \in \mathbb{Z}[\zeta]$ is invertible if and only if $N(f(\zeta)) \in \{1, -1\}$.
Definition : $N(f\zeta)) =... |
I am having trouble understanding exercise 1.25 from the picture below.
I know what order means, but the second sentence puzzles me. I have included the exercises before it in the case that 1.25 makes sense in their context, but I doubt it.
|
I was trying to figure out the curve of intersection of these
2 planes:
$$3x - y + z = 4 $$
$$ y + z = 2.$$
I realize it will be a straight and not curved, and feel like I should be able to do the math but I just can't arrive at the posted answer. I was confused how they parametrized this and would just like to see t... |
If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a Groebner Basis for $I$ w.r.t. some monomial order $>$, then $\forall i$ $\exists g_i \in G$ such that $LT(g_i) = x_i^{d_i}$ ... |
Let's say I have the following decomposition:
$$\{100,10011,00110\}^*$$
How would I determine if the decomposition is ambiguous or unambiguous?
|
Problem: Let $0<a_1<b_1$ and
$$a_{n+1}=\sqrt{a_n\cdot b_n},b_{n+1}=\frac{a_n+b_n}{2}.$$ Prove that $\{a_n\}$ and $\{b_n\}$ converge to some limit.
Attempt: By induction and AM-GM, I can show that $0<a_n<b_n$ for any $n\in\mathbb{N}$.I am thinking maybe I can use Cantor intersection theorem at some point? Provided tha... |
Let $A = A_0[X_1, \dots , X_s]$ be a polynomial ring in $s$ variables over
an Artin ring $A_0$. This is a graded ring, and can be regarded as a
graded module over itself.
1. What are the Poincaré series $P(A, t)$ and the Hilbert polynomial of $A$?
2. Let $M ⊆ A$ be a maximal ideal containing $Q = (X_1, \dots ,... |
How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t?
$$
\int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt
$$
I know the answer is
$$
e^x\sqrt{1+e^{2x}} + e^{-x}\sqrt{1+e^{-2x}}
$$
but I'm not entirely sure how to get there. I know it involves using FTC part two to get F(b)-F(a), and I ... |
Two workers, if they were working together, could finish a certain job in 12 days. If one of the workers does the first half of the job and then the other one does the second half, the job will take them 25 days. How long would it take each worker to do the entire job by himself?
I'm really bad at these type. Can som... |
I had one curiosity regarding maxwell equations in 3-D
From the curl equations, you get 6 unknowns, with 6 equations. The divergence equations add 2 additional equations. When these are combined, we have 6 unknowns and 8 equations. I was curious if someone could give insight for why this is the case? More equations tha... |
I have a simple question: in differential equations, it has been common in several of my homework problems to raise a base $e$ to the power of both sides of an equation to get variables out of natural log functions.
My question is what happens if you have a zero term? For example:
$$
ln(x) + ln(y) = z
$$
is equivalent ... |
Given $f(x;θ_{1},θ_{2})=\frac{1}{(θ2-θ1)}$ when $θ_{1}\leq x\leq θ_{2}$ and $0$ otherwise.
How would I find the MLE? I know you're supposed to take the $log$ of the likelihood function and then the derivative, but I honestly have no idea where to start. It looks like a foreign language to me... Can anyone help me with ... |
In Hartshorne's AG, he discusses the classification of curves by birational equivalence class says 'based on the idea that a nonsingular projective variety is the best kind..'. What exactly makes a projective variety better than an affine one? I know for each class there is a unique nonsingular projective curve. While... |
Proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space.
Part of Proof:
Let $x$ and $y$ be distinct points of $G$. Then $x^{-1}y \neq e$ (identity element).
The set $G$ \ $\{x^{-1}y\}$ is an open neighborhood of $e$ and so there exists an open symmetric neighborhood $V$ of $e$ ... |
Let's say I have a matrix
\begin{bmatrix}a&b\\c&d\end{bmatrix}
What would I have the multiply the matrix above by to obtain the following?
**\begin{bmatrix}c&d\\a&b\end{bmatrix}
|
Given the fixed point iteration $$x_{n+1}=\frac{-x_n^2-c}{2b}$$ where $b$ and $c$ are fixed, $x_n\longrightarrow x$, what does $x$ solve? Additionally, what is the region for $(b,c)$ values where our iteration converges at a rate of $O(2^{-n})$ or better from an interval of starting values $x_0$ near $x$?
|
Singmaster (1981) writes, on page 32 of his Notes on Rubik’s Magic Cube:
Frank Barnes observes that the group of the cube is generated by two moves:
\begin{align*}
\alpha &= L^2 B R D^{-1} L^{-1} &=(RF,RU,RB,UB,LD,LB,LU,BD,DF,FL,RD)& \\
&&\cdot (FUR,UBR,LDB,LBU,DLF,BDR,DFR)\\
\beta &= UFRUR^{-1}U^{-1}F^{-1} &=(UF,UL... |
I'm trying to determine whether $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$, but I'm confused since $\mathbb{Z}/9\mathbb{Z}$ is not a field, but $x^2 + x + 1$ is irreducible in $\mathbb{Z}/9\mathbb{Z}$. Thus the quotient in the title is a field, but which field?
|
NY State Lotto works by "$6$ winning numbers and a bonus number are randomly selected from $59$ numbered balls". The $3rd$ Prize level states that you have matched $5$ numbers, and not the bonus number. Your chances of winning the 3rd Prize level is $1:144,414.98$. (NY State Lotto Chances Page) It has higher odds f... |
Let $1_R$ be the identity of a ring $R$ and let $\mathbb{Z}1_R=\{k1_R\mid k\in\mathbb{Z}\}$.
If char$R=n$, show that $\mathbb{Z}1_R\cong \mathbb{Z}_n$.
So my thought is I just have to think of some rule for $\sigma:\mathbb{Z}1_R\rightarrow\mathbb{Z}_n$ and show that $\sigma$ is an isomorphism I will be done.
So clearly... |
Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.
No, $a_n, b_n$ are not necessarily positive numbers.
I've been trying to use the Dirichlet's Test, but I have no way to show that $b_n$ goes to zero. If I switc... |
This is an extremely simple problem, but I can't find an example anywhere for some reason.
I know that $30^k$ is not correct
But I have no idea what else makes sense.
|
(*Constraint) Kinds of number should be limited in 3
(ex. {1,1,1,1} o/ {1,2,3} o / {1,1,2,2,5,5,4} x)
And I want to find series of Integer.
For example, if n = 4 (n is length of numbers)
{2, 7, 2, 7}
the whole combinations of these numbers are..
2, 7, 2, 7
2, 2, 7, 7
7, 2, 2, 7
7, 7, 2, 2
2, 7, 7, 2
7, 2, 7, 2
six.
It... |
For example we want to find a and b such that av+bw=0 (bold text means vector, otherwise scalar)
Usually we would just solve the equation. But before solving that equation we need one assumption: that the solution exists. How can we know that the solution exists?
|
I'm asked to prove that:
$n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$
I know that for Big O, I need to show: $f(n) <= c*g(n)$
But I'm not sure how to show this, since it involves theta.
Any help would be greatly appreciated.
|
Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as
$$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta \ \right\},$$
whereas the sphere $S(a; \delta)$ is defined as
$$S(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) = ... |
An edge coloring of a graph is an assignment of colors to the edges of the graph. I have $K_{18}$ colored with blue and red and I want to show that it contains a $K_4$ colored with just one color. Also I want to prove that it is the best possible.
I know this question is related to Ramsey-theory, but I don't want to u... |
All numbers are real, WLOG positive.
$A + B + ... + N = T$ and $A' + B' + ... + N' = T$
I'm trying to figure out some function, f, such that if
$f(A,B,... ,N) > f(A',B',...,N')$
then,
$max(A,B,..,N) > max(A',B',...,N')$
I've been going at this for a couple weeks and starting to think no such function exists. Thanks i... |
The cost for $1$ car is $\$10.00$.
Every time you buy $1$, the cost increases by $\$50$.
What is the cost for $1000$ units.
If you have $10$ million dollars, how many units can you buy.
Thanks
Peter
|
I have two different images and with them an estimation of two planes ( defined in the same system). I would like to get the rotation matrix, quaternion or euler angles of a surface within this planes.
What I have is:
an image of an object ( let's say a cuboid to simplify things a bit) taken from a fixed point.
an es... |
Suppose that $X_1, \ldots$ were a sequence of random variables with $E(X_i)=0$ and $E(X_i^2)= \sigma_i^2$ and that $S_n := X_1 + \cdots + X_n$ converged in $L^2$. Does this imply $\sum \sigma_i^2 < \infty$?
Without some sort of bound on the $X_iX_j$ I'm not sure how to proceed.
|
$$\int\frac{dx}{x-\sqrt[4]{x}}$$
given the substitution $x=u^{4}, dx=4u^{3}du$
$$=\int\frac{4u^{3}du}{u^{4}-u}=\int\frac{4u^{3}du}{u(u^{3}-1)}=\int\frac{4u^{2}du}{(u^{3}-1)}$$
At this point I believe I can factor the denominator and use partial fractions
$$\int\frac{4u^{2}du}{(u-1)(u^{2}+u+1)}=\int\bigg(\frac{A}{u-1}+... |
if $r,s$ are rational numbers, Prove $r+s\sqrt2$ is irrational unless $s=0$?
I need to prove this simple question, but not sure if my method is acceptable
I'm trying to prove it by contradiction, Suppose $r+s\sqrt2 = a$, where $a$ is a rational number. Then by given $s$ not equal to $0$, we can say $\sqrt2 = \frac{a-... |
$x^2 \tan x \mid _{x=\pi}$
What does this little line after the $\tan x$ mean? Does it mean I should take the derivative and then substitue in $\pi$?
I need to find the second derivative.
|
Let $I \subset K[x_1,\dots,x_n]$ be a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety).
I know that $|V(I)| \leq \dim_K(R/I)$, but can the gap $\dim_K(R/I) - |V(I)|$ be made arbitrary large? That is, $\forall \alpha > 0$, is t... |
The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible.
But reducible quintic polynomials may be invertible in a case by case scenario:
$$f(x)=x^5-1$$
$$f^{-1}(x)=\sqrt[5]{x+1}$$
But are all reducible quintic polynomials inverti... |
In my Digital Logics class, we are doing boolean algebra.
In the case where $ a * b * c $ (a and b and c)
can we assume either of those values to be 1?
so can we say that $a * b * c = 1 * b * c$,
in which case that would be $(a + a') * b * c$?
|
Prove that the altitudes of an acute triangle intersect inside the triangle.
I can pretty easily see that this is true by a pythagorean theorem argument. Given any two sides, the smaller length to the side will be closer on the base so thus the orthocenter must lie inside the triangle. What would be a more mathematic... |
Let $\Omega$ a limited domain in $\mathbb{R}^{n}$, the space $L^{\infty}(\Omega)=\{f: \Omega\to\mathbb{R} $ measurable $; ||f||_{L^{\infty}(\Omega)}<\infty\}$. Then if a function $f \in L^{\infty}(\Omega)$, by definition of $L^{\infty}$ norm, $f$ is bounded a.e x, we can say that $\sup\limits_{\Omega} |f(x)| <\infty$... |
If I have that $X_n \to_p X$ (convergence in probability), and if $\limsup_n E|X_n|^r \leq E|X|^r$ for all $r \geq 1$, how can I show that $X_n \to_r X$ (this means $L^{r}$ convergence)? My goal is to show that $\limsup_{n} E|X_n|^r \leq E|X|^r$ implies uniform integrability, then I can use a standard result to show th... |
Let S be a non-empty set, and Q be a set of non-empty subsets of S such that $\bigcup Q=S$. Let $P'$ be the set of all non-empty subsets x of S such that:
$\forall q\in Q. x\subseteq q \lor x\cap q=\emptyset$
and $P$ be the set of all maximal elements of $P'$.
Is P a partition of S? How can I prove this? Is there some ... |
I know how to take a fraction and get it's binary expansion. For example, $\frac{1}{5}$ would go like so:
$\frac{1}{5} \cdot 2 = \frac{2}{5} \rightarrow 0$
$\frac{2}{5} \cdot 2 = \frac{4}{5} \rightarrow 0$
$\frac{4}{5} \cdot 2 = \frac{8}{5} \rightarrow 1$
$\frac{3}{5} \cdot 2 = \frac{6}{5} \rightarrow 1$
$\frac{1}{5} \... |
Does anyone know if wolfram alpha has step by step solutions for systems of differential equations? When I input them, it comes up with an answer but it does not give me the step by step solution. I would like it just for practicing purposes.
For example, I input the following into wolfram but it does not show me the s... |
I'm trying to follow a video lecture on solving the heat equation.
$I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$
$II) \space u(x,0)=\phi (x), $
$k$ is const, $\phi (x) $ is a known function.
An earlier video in lecture series proved these properties:
If $u(x,t)$ is a solution to $I)$ then for each fixed $y$, t... |
It is given that a,b,c are be of same sign and a,b,c are in Harmonic progression i.e. $\frac{2}{b}=\frac{1}{a}+\frac{1}{c}$ and also
$\frac{a+b}{2a-b}+\frac{c+b}{2c-b}\geq \sqrt{\lambda \sqrt{\lambda \sqrt{\lambda.............\infty}}}$.
Find the greatest possible value of $\lambda$.
How should I begin? I think we so... |
What is bigger?
lim n->infinity n^x
or
lim n->infinity x^n
I have a relationship where I am trying to find the lim n->infinity (2^n + n^20) / 3^n and am having a hard time deciphering it.
|
Let $c$ be prime and $d$ be an associate of $c$.
Claim: $d$ is prime.
$c$ prime $\Rightarrow c\neq 0$, nonunit
Suppose that $c\vert ab$. Since $c$ is prime, $c\vert a$ or $c\vert b$
$c\sim d \Rightarrow c\vert d$ and $d\vert c$
$R$ ID $\Rightarrow d\neq 0$
$d\vert c$ and $c\vert a \Rightarrow c=dx_1$ and $a=cx_2, \;x_... |
Let $(M,g)$ be a connected smooth Riemannian manifold and denote by $(M,d)$ the induced metric space following by taking topological metric to be the infimum over length of curves in the standard way. Suppose that $(M,d)$ is not complete and let $(\hat{M},d)$ denote the metric completion.
What can be said about $(\hat{... |
So, I've been working my way through the following as I'm messing about with programming some helper functions for orbital mechanics: https://downloads.rene-schwarz.com/download/M002-Cartesian_State_Vectors_to_Keplerian_Orbit_Elements.pdf
Given I don't have the greatest understanding of mathematical notation I've run i... |
Consider randomly selecting a single individual and having that person test
drive 3 different vehicles.
Define events $A_1, A_2, A_3$ by:
$A_1$ = likes vehicle #1
$A_2$ = likes vehicle #2
$A_3$ = likes vehicle #3
Suppose that $P(A_1) = .55 \,, P(A_2) = .65 \,, P(A_3) = .7 \,, P(A_1 \cup A_2) = .8 \,, P(A2 \cap A3... |
Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angles of $\frac{\pi}{3}$
Now our required line should be $\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$. But I don't see how we are going to get three equations to obtain values of a,b,c. Any hint?
|
Show that every region of a maximal planar graph is a triangle.
A planar graph G is called maximal planar if the addition of any edge to G creates a nonplanar graph.
Will this proof use the theorem that states that if G is planar with $n \ge 3$ and $e$ edges, then $e \le 3n-6$. Furthermore, if the equality holds th... |
I'm trying to prove the following theorem using the axioms quoted below.
Theorem: If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$
Axiom 1: For every event $A$ in the class $C$,
$$P(A) \ge 0$$
Axiom 2: For the sure or certain event $S$ in the class $C$,
$$P(S) = 1$$
Axiom 3: For any number of mutua... |
I have this rather complicated loop:
sum=0
for i=1 to n do
for j=1 to i^2 do
if(j (mod i) = 0) then
for k=1 to j do
sum++
How the heck do I analyze this? I would try to use a sum but the if statement throws me off. How can I determine what time this nested for loop runs in?
|
If $a \ne b$ in a ring $R$ satisfy $a^3 = b^3$ and $a^2b = b^2a$, show that $a^2 + b^2$ is not a unit.
So I am thinking that I should be able to do this by contradiction. So if I assume there is some element $z\in R$ such that $(a^2 + b^2)z=1$ then
$a^2z + b^2z=1$, but I'm not sure this helps in any way.
I also tried... |
Prove or disprove: The set Y of numbers in (0,1) with a decimal expansion that contains only 0s and 1s, and only finitely many 0s, is an uncountable set.
Uncountable means it's not finite or not denumerable. It seems that this would not be finite, since there may be unlimited 1s in the decimal expansion. Denumerable me... |
I recently learned that $\sin(z)$ has an extension into the complex plane, namely:
$$\frac{e^{iz}-e^{-iz}}{2i}$$
Is there any complex number $z=a+bi$, with $b≠0$ such that $\sin(z)=0$ ?
I am already aware of the 'trivial' zeros when $a=πk$ with $k\in\Bbb Z$ and $b=0$.
Is there any way to prove this? Either complete so... |
In class we encountered this function
$$f_n(x)=\begin{cases}
n^2x, & 0 \leq x \leq 1/n\\
2n - n^2x, & 1/n \leq x \leq 2/n\\
0, & 2/n \leq x \leq1
\end{cases}$$
The prof said that this function is not uniformly convergent because the integral of the limiting function $f(x) = 0$ is $0$, but the integral of th... |
For a finite case we have $\lim\limits_{n\rightarrow\infty}f(n)\cdot g(n) =\lim\limits_{n\rightarrow\infty}f(n)\cdot\lim\limits_{n\rightarrow\infty}g(n)$ however when is it possible to interchange the folowwing?
$\lim\limits_{n\rightarrow\infty}\prod\limits_{i=1}^n f(i)$
A specific problem I have is $\lim\limits_{n\rig... |
Let $X$ have a uniform distribution on the interval $(0,1)$.
a) Find the c.d.f. and p.d.f. of $Y=\dfrac{X}{1-X}$.
b) Find the c.d.f. and p.d.f. of $W=\ln Y$.
I am extremely confused on part A, and part B also. I get to this very early step and am stuck:
$$Y= \dfrac{X}{1-X} \\ \boxed{ F_Y(y)=\Pr(Y\le y)=\Pr\left(\fra... |
I understand that "If every finite subset of $S$ is linearly independent, then $S$ is independent" is true. In fact, every textbook proves this statement. But why not "If every subset(either finite or infinite) of $S$ is linearly independent, then $S$ is independent"
To me, there seems to be no problem at all.
|
Solve $$e^{4z} +e^{3z} + e^{2z} + e^z + 1 = 0.$$
I have attempted this problem with the usual definition by writing $z=x+iy$ and using $e^z = e^x(\cos y + i \sin y)$ but got a large unsolvable mess. Is there something I'm missing?
|
I have a question about this statement in the Cauchy's Integral Formula in Conway text.
In the Integral formula, it states that
" Let $G$ be an open subset of the plane... If $\gamma$ is a closed rectifiable curve in $G$ such that $n(\gamma;w) = 0 $ for all $w \in \mathbb{C}-G $."
Now before this theorem, Conway pre... |
What points in $[0,1)$ will have two binary expansions?
I know that $\frac{1}{2}$ has the two expansions $0.1\bar{0}$ and $0.0\bar{1}$
$0.1\bar{0}$ is found by starting with $\frac{1}{2}$ and finding the binary expansion
$0.0\bar{1}$ is found by working backwards from the expansion to find the fraction.
But how do I ... |
Prove that if $A$ and $B$ are mutually exclusive, $P(A \text{ and }B)=0$, then $A$
and $B$ are dependent.
So I know that $A$ and $B$ are independent if $P(A \cap B)= P(A) P(B)$
How do I use that piece of information to prove the statement?
|
Let $(X,\Sigma,\mu)$ be a $\sigma$-finte measure space and $f \in L^+(X,\Sigma)$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$.
Theorem:
Define the area under the graph of $f$ to be $S_f = \{(x,y) \in X \times [0,\infty]\mid y \leq f(x)\}$. Then $S_f \in \Sigma \otimes \mathcal{B}(\mathbb{R})$, where $\mathca... |
What standard distribution would be suitable for the random phenomenon at hand, and what are the knowns and unknowns?
e) The size of an automobile insurance claim
I'm thinking that the distribution has to be continuous and whose density is non-negative on the entire interval $[0 , ∞)$. Would the gamma distribution be a... |
The standard log function form is $a \log[k(x-d)] + c$
Where
$a$ vertically stretches or compresses
$k$ horizontally stretches or compresses
$d$ translates left or right
$c$ translates up or down
If we have $2 \log(2x)$ does this have a vertical translation when rewriting to $\log 4 + \log x$?? And why? Standard form ... |
Problem Statement: Let $W$ be a subspace of a Hermitian space $V$. Prove that $W^{⊥⊥}=W$
I am trying to figure out a good strategy for this proof.
I know that:
$W$ is a subspace of $V$
$W^{⊥}=\{\mathbf{v}\in V\ \ |\ \ \langle \mathbf{v}, \mathbf{w}\rangle=0\ \ \ \forall\mathbf{w}\in W\}$
$W^{⊥⊥}=\{\mathbf{v}'\in V... |
So here's the problem:
$x+3=3^x$
Obviously, graphing both sides and finding the intersection would reveal the answer, but algebraically, how can this be solved?
|
$\frac{\ln(x^2)}{\ln(x)} = 2$?
Upon trying to evaluate $\frac{\ln(x^2)}{\ln(x)}$, i've found that google plots it as always equal to 2, other than 0 where it is undefined. Why is this the case?
|
I am having difficulty find the centroid of the region that is bound by the surfaces $x^2+y^2+z^2-2az=0$ and $3x^2+3y^2-z^2=0$ (lying above $xy$ plane, consider the inner region). I know the first surface is a sphere, while the second is an infinite cone.
I just dont know how to even approach it as I dont know how to e... |
I know there already are questions like these, but I still don't understand how to prove it.
Question:
Prove that $f$ is Riemann integrable on $[0,1]$ if
$$f(x) = \begin{cases} x^2 \sin (1/x) & \text{if } 0 < x \le 1 \\ 0 &\text{if }x = 0.\end{cases}$$
Do I use an epsilon proof or do I use the method of showing ... |
Okay, I realize this seems like a really stupid question, but on a math contest (without calculators) I got down to this equation:
$$\frac{26}{672-x} + \frac{24}{372-x} = \frac{50}{480-x}$$
whereupon I spent 15 minutes trying to solve for $x$ until I eventually gave up. Especially, on a math contest, this is not an ide... |
One day my mind wandered off and came upon the following.
$x^2 \equiv 1 \pmod{x+1}~\forall x>0, x \in \mathbb{Z}$.
My markdown might be a little bit broken :)
I tested this out in Python for the first $1000$ values of $x$ and it seems to work out. In case my congruence is hard to understand, I just mean that the remain... |
I was messing around recently with a unit cube.
If you draw vertices on the midpoint of each edge of the cube, then connect those points by new edges, you will form the wireframe of what I figured out is a cuboctohedron. If you repeat this process on the cuboctohedron (drawing vertices on the midpoints of the edges of ... |
Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Is the function $f(r) = \mu(A_r)^{1/n}$ concave? ($\mu$ is the Lebesgue measure.)
Context
This came up in the discussion of George Lowther's answer, where I asserted t... |
Consider the mean combined SAT score for high school seniors is 1500, and the standard deviation is 250. Calculate the percentage of students who scored at the following levels:
Below 1000.
Above 1750.
|
It is a very well known fact that Smith Normal Form has proven useful when dealing with the development of the structure theorem of finitely generated abelian groups. In this context, there is an approach that takes advantage of the next result, which indeed is a very particular case of a much more general theorem rela... |
Notation in my book, Introduction to Probability and Mathematical Statistics by Bain and Engelhardt, is $x_{1:n}$ and $x_{n:n}$. For example, see Example 10.2.3 (page 340). Here is a screenshot.
Can anybody tell me what this denotes?
|
Suppose I am forming a semidirect product of finite groups $G \rtimes_\theta A$. Here, $\theta : A \to \mathrm{Aut}(G)$ is a homomorphism. Now, suppose I notice that the homomorphism $\theta$ factors through a quotient of $A$. That is, suppose $\pi : A \to Q$ is a surjective group homomorphism, and $\phi: Q \to \mathrm... |
Rudin's Principles of Mathematical Analysis(3rd ed) says on page 53 that 'If $\{p_n\}$ is a sequence in $X$ and if $E_N$ consists of the points $p_N,p_{N+1},p_{N+2},\dots$, it is clear from two preceding definitions (about Cauchy sequence and diameter) that $\{p_n\}$ is a Cauchy sequence if and only if $$\lim_{N\right... |
Is it provably difficult to generate a prime larger than a prescribed $N$? For instance, if I want a prime of $1000$ digits, is there a way to do that deterministically, i.e., without resorting to AKS and exhaustive search?
|
Definition: A graph G is bipartite if its vertices can be partitioned into two sets V1 and V2 and every edge joins a vertex in V1 with a vertex in V2. Bipartite graphs can be characterized by all circuits in such graphs having even length (if there are no circuits, the graph is also bipartite), where the length of a ci... |
The question is : $r$ is the solution of equation $x^2+bx+c=0$ and $r$ is a rational number, so there is another solution $s$, how to prove s is a rational number as well? I have no idea about it and I think that I shall prove it through $(x-r)(x-s)$ equals to the equation, but still no idea about it. Please help me a ... |
I am working on an application that integrates with a payment service that charges a fee to use. I'm trying to come up with a formula, so I'll ask my question in a different format so that I don't have to get into unnecessary details.
Let's say there is a 'gatekeeper' whose job is to transfer money from person $A$ to p... |
Let $I$ be an ideal in a polynomial ring $P = K[x,y_1,\dots,y_n]$ and assume that $I \cap K[x]\neq (0)$. Let $>$ be an elimination ordering for $\{y_1, \dots, y_n\}$ and $G$ is a Groebner basis for $I$ with respect to $>$.
Why $G\cap K[x]$ is in fact a single polynomial f?
edit: yes I agree $G \cap K$ is also a Grob... |
The problem is specific as an example from hw. But it is more the concept/process I could use clarification on. Given a complex number
$$\Big(\frac{-2}{1-i\sqrt3}\Big)^{\frac{1}{4}}$$
Find all possible roots. I know the method is to change into the exponential form, solving for magnitude (r) and theta. Which I did and... |
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