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Find $A^8$ using Cayley Hamilton Therorem, when
$$A = \begin{pmatrix}
0 & 0 & 0 & -1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & 0
\end{pmatrix}$$
I found that $P(t) =t^4-2x^2+1 = (t-1)^2(t+1)^2$.
But how can I use Cayley-Hamilton to find $A^8$?
|
The title resume all the point I'll try to make now. Reading this post, I realize that is possible to have another type of PDF (probability density function).
Usually, we have a probability space $(\Omega, \mathcal{F}, P)$, a random variable $X:\Omega\to\mathbb{R}$ and the PDF of $X$, $f_X:\mathbb{R}\to[0, \infty)$. T... |
I am given the problem: suppose for measurable, real-valued functions $f$ and $g$, and an open set $A \subset \mathbb R ^2$, prove that $\{x \in \mathbb R : (f(x),g(x)) \in A\}$ is a measurable set.
I know that if I could reformulate it such that $\{x \in \mathbb R : (f(x)) \in C\}$, and $\{x \in \mathbb R : (g(x)) ... |
We are given a system of equations:
$2w+3x-2y+z=-1$
$6w+10x+6z=14$
$3w+2.5x-15y-4.5z=-35.5$
and a particular solution to that system of equations, $\begin{bmatrix}0\\2\\3\\-1\end{bmatrix}$
A) Find the solution set to the corresponding homogeneous system of equations, with all variables written in terms of free variable... |
Given $A_{n\times n} \in \mathbb R$ such that $A^4=4A^2$ then
if $A$ is invertible and isn't of the form $cI, c\in \mathbb R$ then $m_A(x)=x^2-4$.
if $A$ isn't diagonalizable over $\mathbb R$ then $0$ is an eigenvalue of $A$.
From the given we get $x^2(x-2)(x+2)=0$ zero $A$ so the eigenvalues of $A$ are from $\{... |
I have a below method which is called every one minute from background thread and that background thread keeps running always.
public void testMethod() {
// some other code
int value = new Random().nextInt(20 - 1 + 1) + 1;
if(value > 19) {
// this will print hello line
System.out.print... |
I try to estimate the supremum of $|\cot(\pi z)|$ and where $z=(n+1/2) e^{i t}$, $n\in\mathbb N$ and $t\in[0,2\pi)$. I should be a constant.
So far I did by wiriting it in exponential form and expanding it
$|\cot(\pi z)|=|1+\frac{2}{e^{2\pi i z}-1}|\le 1+\frac{2}{|e^{2\pi i z}-1|}$
I focus on $|e^{2\pi i z}-1|$ and put... |
The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$,
$Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle $\theta$ in
clockwise direction. If $sin(\theta) = 0.6$ and $cos(\theta) = 0.8$, the new coordinates of the vertex $Q$ are
... |
If $R$ is a ring with unity $1$, then $S \subseteq R$ is called a subring if it is itself a ring with $1 \in S$. A subset $I \subseteq R$ is called an ideal if it is a group with respect to addition and $rx, xs \in I$ for each $r,s \in $ and $x \in I$. With these definitions an ideal $I$ is a subring iff $1 \in I$ iff ... |
I have learnt that a fraction with a surd in its most simplest form should have the surd in the numerator and not the denominator? Why is it convention not to leave the surd on the denominator? Is it just arbitrary convention or does it have an actual purpose?
|
Let $(X,d)$ be a metric space, with $Y$ a subset of $X$.
How do I prove that a subset $U\subseteq Y$ is open in the metric space $(Y,d|_{Y\times Y})$ iff there exists an open subset $V$ of $X$ such that $U=Y\cap V$?
So, I have to prove both sides. For the $"\implies"$-side, can I just take $V=U$? And the other side, ... |
Let there be two independent events $A$ and $B$. To calculate the probability (for a particular condition) we have two relations.
$P(A \cup B)=P(A)+P(B)-P(A \cap B)$.
$P(A/B)P(B)=P(A \cap B)$, i.e., $P(A)P(B)=P(A \cap B)$.
But since the two events are independent,they can be expressed by a Venn diagram as two disjoin... |
Working through foundations of mathematical anlysis by johnsonbaugh per suggestion and wondering if the following proof works? (no solutions to book)
Problem:
Let X and Y be nonempty subsets of real numbers such that $X \subset Y$ and Y is bounded above. Prove that $\mathop{\mathrm{sup}} X \le \mathop{\mathrm{sup}} Y$.... |
Can somebody help me ,please? I have the following function:
$$\phi:c_{o}\to \mathbb{R}$$, $\phi(x)=\sum_{n\ge 1}(-2)^{1-n}x_{n}$, where $c_{0}$ is the set of the sequences of real numbers convergent to $0$. I was asked to prove that this function is well defined. So I have to show that, for every sequence in $c_{0}$, ... |
I am reading Royden and Fitzpatrick's book on Real Analysis, and I have a question about the length function on an interval in $\mathbb{R}$. Is it true that given $\{I_n \}_{n=1}^{\infty}$ of countable collection of open intervals in $\mathbb{R}$:
$$ \ell( \bigcup_{n=1}^{\infty}I_{n}) \leq \sum_{n=1}^{\infty} \ell (I_... |
Show that any two different functions in $C(\Omega)$ differ as generalized functions as well.
Let $f,g \in C(\Omega)$. Since $f\ne g$, there is a $x_0 \in \Omega$ such that $f(x_0)\ne g(x_0)$. Hence there exists a neighborhood say ($U_{x_0}$). Let's say $h(x)=f(x)-g(x) \gt 0$ for all $x$ in $U_{x_0}$. Let $K$ be a com... |
So, I have to prove that the sequence defined as
$a_{n+1}=\frac{6(1+a_n)}{7+a_n}$ converges and then find the limit.
I have few questions; Do i have to assume that $a_n \geq 0$ or $a_n \leq0$. Because if i assume that $a_n \geq 0$, I can prove that
$\left|a_{n+1}-2\right| \leq \frac{4}{7}\left|a_{n}-2\right|$, and by... |
I was wondering about the derivation for the area of a triangle and the area of a rectangle. Of course, we all know them to be $\dfrac{1}{2}bh$ and $bh$ respectively, but where is the derivation of these? If we derive the rectangle one then the triangle one obviously follows, but I don't recall ever seeing a derivation... |
Let $F$ denote a finite field and $A$ a square matrix with coefficients in $F$. The set of all matrices $B$ such that $BA=AB$ is called the centraliser of $A$.
Now consider the set $C(a,A)$ of all matrices $B$ satisfying $AB=aBA$ where $a$ is in $F$. We call this set the twisted centraliser of $A$. It is an $F$-vector... |
There is a weird Theorem that comes about when considering whether a function is homogeneous (in the sense of the title definition).
I was unable to prove it, or to find a proof to it.
Can any one help?
If an equation is of the form $$f(x,y) = \frac{x^{a_1}y^{b_1} + x^{a_2}y^{b_2} \cdots}{x^{c_1}y^{d_1} + x^{c_2}y^{d_2... |
I came across this problem yesterday: Let $R$ be a ring and $M$ an $R-$module. $\varphi:R^n\to M$ is a surjective $R-$module homomorphism if and only if $M$ is finitely generated.
Given the set of generators of $M$ as $\{a_i\}_{i=1}^n$, I was able to show that the map $\varphi(r_1,\ldots,r_n)=\sum_{i=1}^nr_ia_i$ define... |
I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises.
If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $c$ and graded for 7%, so it's supposed to be pretty easy.
Find the range of $x$ for which the sequence $\dfrac{n!} {k!(n-k)!}x^n$ conver... |
What is the probability that a randomly chosen positive
three-digit integer is a multiple of $7$?
Is my answer right?:
$$\frac{100}{7} = 14 , \qquad \frac{999}{7} = 142$$
Then there are $142 - 14 = 128$ numbers that are multiples of $7$.
Then the probability is: $\frac{128}{900}$.
|
Let $\Omega$ be a bounded domain with smooth boundary in $R^n$. Fix $u_0 \in H^1(\Omega) - \{ 0\}$.
Define
$$ J(u):= \int_{\Omega} \langle A(x) \nabla u(x), \nabla u(x)\rangle dx,$$
$$u \in K:=\{ v \in H^1(\Omega); v-u_0 \in H^{1}_{0}(\Omega)\}$$
where $A(x) = a_{ij}(x), i,j=1,...n , x \in \Omega$ is a matrix with smoo... |
I'm studying about linear algebra and came across with the following:
Let $A\in \mathcal{M}_{n\times n}(K)$ for some field $K$. If
$\text{dim}\,K^n = n$ is finite then
$\text{Im}(A^i)=\text{Im}(A^{i+1})$ for some $i\leq n.$
Why is this true? I get it that $\text{Im}(A^{i+1})\subseteq \text{Im}(A^{i})$, but why d... |
I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it:
Let $p$ be a probability measure on the symmetric group $S_n$. Consider the composition:
$$\Sigma_N = \sigma_1 \cdot \ldots... |
The spectrum-functor
$$
\operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set}
$$
sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a prime ideal of R}\}$ and a morpshim $f:S\to R$ to the map $\operatorname{Spec}(R)\to \operatorname{Spec}(S)$ with $\math... |
Problem: Find all integer solutions of $x^2 \equiv -x \pmod{2015}$.
I proceeded this way: first, I realized that $2015 = 5 \times 13 \times 31$. I rewrote $x^2 \equiv -x$ as $x^2 + x \equiv 0$.
Then, for each factor $d$, I have that $d\ |\ x(x+1)$, therefore $d\ |\ x$ or $d\ |\ x + 1$.
This gives eight cases:
$3,13,... |
I am learning real analysis now but I really dislike the notion of limit, infinity... They seems to generate lots of paradoxes and unreasonable results.
For example, when I am reading the uniform continuous notion, there is an example I find quite annoying to accept:
Consider the sequence of function: (check the pictu... |
My problem is that I have multiple complex number in trigonometric form and I want to add those and get the magnitude of the result.
I am aware that the normal route would be to calculate the rectangular form, add the numbers and then I can easily calculate the magnitude.
I also know the trick, where you factor out the... |
Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are orthogonal to $\dot{\gamma}(0)$ for each differentiable curve $\gamma$ with $\gamma(0) = x$.
We define $\nu$ as a unit normal fie... |
A topological space is a set $X$ together with a topology $\tau$ (a collection of open subsets) such that.
$\emptyset\in \tau$ and $X\in \tau$.
The intersection of a finite number of sets in $\tau$ is also in $\tau$.
The union of an arbitrary number of sets in $\tau$ is also in $\tau$. (it could be the union of inf... |
Let $T: V \to V$ linear transformation, and let $W$ to be an invariant subspace of $V$. we mark $T_w: W \to W$ the from $T$ to $W$.
Prove that if T is diagonalizable, then $T_w$ is diagonalizable.
How do I start to prove it?
I know that if T is diaognizeable then it's factors are different from each other. but how does... |
I'm trying to show that this sequence $$a_n = \frac{3^n-7}{4^n+5}$$ is decreasing for all $n$ greater than some $N\in \Bbb N$.
All I can see to do is something like
$$a_{n+1} = \frac{3^{n+1}-7}{4^{n+1}+5} = \frac{3\cdot3^n-7}{4\cdot 4^n+5}\le \frac{3\cdot3^n-7}{4^n+5}$$
But that last expression is not less than $a_n$ f... |
So i completed an year ago my schooling and i am pretty good at maths well at my level and i am very interested in maths and want to learn as much maths as possible and i like stuff like number theory. what got me really fueled up is the fact that i watched a video on numberphile (its a youtube channel on math stuff) a... |
Conditonal Jensen's Inequality says that for a convex function $\varphi$, a random variable $X$, and a sub-sigma-field $\mathcal{F}$, $E[\varphi(X)\mid \mathcal{F}] \geq \varphi(E[X\mid \mathcal{F}])$. In ordinary Jensen's Inequality, $E[\varphi(X)]\geq \varphi(E[X])$, and we have equality if and only if $X$ is degener... |
I have a basic school academic question which despite of trying very hard I was unable to find a solution -
Average score of girls is an exam is $67$ and that of boys is $62$. Find the percentage of girls and boys?
I think the question is incomplete - cause average is nothing but addition of each student marks divided ... |
Let $(X,\tau)$ be a topological space that admits a one-point compatification $(X^{*},\tau)$ (Alexandroff compatification). I know that if the space $X$ is connected, then $X^*$ is connected as well. However, the converse is not neccesarly true, that is, the Alexandroff compatification can connect a disconnected space.... |
ZFC doesn't allow for classes like the Russell Set (aptly named as a set I suppose...) to be sets, but my question is the following:
In general, if X is a set in the ZFC universe, then for some sentence, A(X), can it be the case that A(X) iff not A(X)?
|
If I want to do a hypothesis test of a sample that contain $49$ women and $51$ men. The hypothesis test is only regarding the women which has a given sample standard deviation of $12,032$ and sample mean of $13,633$.
Do I just ignore the fact that my sample includes men and just go with the number of women and their da... |
If $X$ is a Banach space, $X'$ its dual and $x,x_n\in X$ and $x',x_n'\in X'$, then the following implication holds:
$x_n\to_w x$ (weak convergence) and $x_n'\to x'$ in $X'$ $\implies$ $x_n'(x_n)\to x'(x)$
I tried the following:
$$\left|x_n'(x_n)-x'(x)\right|\leq \left|x_n'(x)-x'(x)+x_n'(x_n-x) \right|\leq\|x_n'-x\|\|... |
Given $A_{n\times n},B_{n\times n} \in \mathbb R$ such that $A$ is diagonalizable then:
if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable over $ \mathbb R$.
if $A,B$ have then same characteristic polynomial, then $B$ is also diagonalizable over $ \mathbb R$.
Those two state basically the same t... |
Let $A \in M_{3\times 3}(\mathbb{R})$. Its characteristic polynomial is $P_A(t) = t^3+t^2+t-3$. Find the coefficient of the characteristic polynomial of $A^2$.
I tried to solve it by finding the factors of A, but it looks like it doesn't have factors in the real numbers $\mathbb{R}$ only above $\mathbb{C}$.
|
Let $G$ be a profinite group and consider a continuous surjective homomorphism:
$$\phi:G\rightarrow \widehat{\mathbb Z}$$
where $\widehat{\mathbb Z}:=\varprojlim \mathbb Z/n\mathbb Z$. Moreover Let $H<G$ be a (closed) subgroups such that $f=|\widehat{\mathbb Z}:\phi(H)|<\infty$.
I Have two questions:
Why the map $\f... |
The problem is: $\exists x \in \mathbb{R} (x^3 = -1)$
I understand the following:
$\exists x$ = There exists an $x$
$\in$ = shows the element before it is a member of a set after it
$\mathbb{R}$ = Real number
This question doesn't make sense to me. If I try to translate it into English I get something like:
There ex... |
I'm trying to think about exercise 4.5.O in Vakil's notes on Algebraic Geometry.
Before we defined the scheme $\mathbb{P}^n_k := \operatorname{Proj}(k[x_0,...,x_n])$ and showed that that for $k$ algebraically closed the closed points of $\mathbb{P}^n_k$ correspond to the points of "classical" projective space.
Now the ... |
It's been awhile since I've worked problems like these and I am a bit hazy on some of the rules. I was hoping someone could show me how these are solved so that I can make sure I'm on the right path:
Equations:
$$Y=\dfrac a x + \dfrac 1 b\\
Y=\dfrac a x + \dfrac x b\\
Y=axe^{-b/x^2}$$
I need to convert those equation... |
$$\lim_{n\to\infty}\left(1+\frac{1}{3}\right)\left(1+\frac{1}{3^2}\right)\left(1+\frac{1}{3^4}\right)\cdots\left(1+\frac{1}{3^{2^n}}\right)$$
You have to find the given limit when $n$ tends to infinity. I can't think of a way I tried taking $\log$ and proceeding but nothing helped please help.
|
would appreciate any hints with this question:
Assume $x$, $y$ are both $n \times 1$ vectors, and that $B$ is $n\times n$.
Given that $x^{T}By = 1$, should $\operatorname{Tr}(Byx^{T}) = 1$ ?
Thank you very much.
|
Can weak convergence of a sequence $f_n\in L^p(\Omega, \mu)$ to some $f\in L^p(\Omega, \mu)$ be characterised as almost everywhere pointwise convergence?
Let us also assume the measure space is $\sigma$-finite.
I'm asking because for continuous functions on the unit interval, weak convergence is equivalent to pointwise... |
I wish to find the eigenvalues and eigenfunctions of the following, but am unable to and further don't know what I am doing wrong at all
$y''+\lambda y =0$ where $y(0)=0$, $y'(1)+y(1)=0$
My attempt
Let $y=e^{\mu x}$, then $y''=\mu ^2 e^{\mu x}$. By substitution into the original equation we find that $\mu = \pm i \s... |
Here's the question which I'm struggling with -
In how many ways can we pick a group of 3 different numbers from the group
$1, 2, 3, ..., 500$
such that one number is the average of the other two? (The order doesn't matter)
Here's what I've done -
Choose any 3 number $x$, $y$ and $\frac{x+y}{2}$. If $x + y$ is even t... |
First of all, I'm sure this question has been answered somewhere on the web, but I am just starting probability and I don't have the vocabulary to know what to look for, which is why I am asking here.
I have $5$ white tokens and $5$ black tokens and I want to find the the number of possible orderings.
I know how to do ... |
As a prospective grad student, I would like to get an entry level introduction to classical (i.e. non-quantum) gauge theory. Please direct me to resources suitable for a novice.
|
In Hartshorne chapter I.6 is discussed the construction of the abstract nonsingular curve as part of the proof for the well known correspondence between complete regular irreducible algrebaic curves and the function fields in one variable, in the case of algebraically closed ground field.
It seems that Milne (in his no... |
Consider two sequences of real numbers $\{a_n\}_n, \{b_n\}_n$.
I know that if $a_n\geq b_n$ $\forall n$ then $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$.
Suppose instead with $a_n\geq b_n$ for $n>\bar{n}$. Can I still say $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b... |
I have two questions:
If $X$ is a countable product of spaces having countable dense subsets then does $X$ have a countable dense subset?
Let $X$ $=\prod_{i=1}^\infty X_i$. Let $D_i$ denote the countable dense subset of the space $X_i$. Then $\scr D$ $=\prod_{i=1}^\infty D_i$ is a countable set as Cartesian product... |
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. Fix $u_0 \in H^1(\Omega) - \{ 0\}$.
Define
$$ J(u):= \int_{\Omega} \langle A(x) \nabla u(x), \nabla u(x)\rangle dx,$$
$$u \in K:=\{ v \in H^1(\Omega); v-u_0 \in H^{1}_{0}(\Omega)\}$$
where $A(x) = a_{ij}(x), i,j=1,...n , x \in \Omega$ is a matrix ... |
Prove that
$$\det\bigg(\frac{1-s^{i+j-1}}{1-t^{i+j-1}}\bigg)^n_{i,j=1}=t^{n^3/3-n^2/2+n/6}\prod_{1\leq i<j\leq n}(1-t^{j-i})^2\prod_{i,j=1}^n\frac{1-st^{j-i}}{1-t^{i+j-1}}$$
In his book, D. Bressoud also gives some hints on how to prove this. Below are the hints (and my progress so far in brackets):
Show that the det... |
Recently, when fiddling around with integration by parts, I noticed that it is possible to define infinite series that led to an integral. My calculus teacher noticed this, and told me to find
$$
\int \frac{e^x}{x}dx
$$
which I already knew didn't have a elementary function definition. After integrating by parts a few ... |
Let $E$ and $F$ two normed vector spaces, $A \subset E$ compact, $B
\subset F$ and $f: A \to B$ is a bijective continuous function. As $f$
is bijective, we can defining the inverse function $f^{-1} : B \to A$
by $$f^{-1}(y)=x \iff f(x)=y.$$ Show that $f^{-1}$ is continuous.
Theorem : $f$ is continuous $\iff$ for... |
I have a question regarding Congruent relations:
In Euclidean geometry, two subsets of $\mathbb{R}^{d}$ are said to be congruent if one set can be mapped onto the other by translations and rotations.
Claim. Two congruent measurable sets must have the same Lebesgue measure.
I need to prove two things:
If B is a rotatio... |
Is is true that in any topological vector space, which is $T_1$ there exists bounded neighbourhood of zero ? Is is still true if we omit $T_1$ axiom ?
|
THE RED LINE IS $ \Im(z) = x$
Now, my proof is as follows,
Let $z' \in A$, then take $\epsilon = x - \Im(z')>0$
Now let $w \in D_{\epsilon}(z')$ and suppose $w \notin A$ then
$$\epsilon > |z'-w| \geq|\Im(z)-\Im(w)| = \color{red}{\Im(w) - \Im(z')} \geq x - \Im(z') = \epsilon$$ A contradiction. This seems all good, but ... |
Do you know of a proof which uses homology to demonstrate a property about a topological space which is made easier (or even possible) because they work over $\mathbb{Z}/2\mathbb{Z}$ instead of $\mathbb{Z}$?
I have found that certain computer calculations on a finite set of generators and a explicit boundary function a... |
This will be largely a trivial question.
But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ?
Of course, I can treat it like a quadratic inequality by saying $t=x^2$
So I can solve it like: $ 3t^2 - 4t +1 >0$
The solutions are: $t_1=1; t_2=\frac{1}{3}$
Or in termes of $x$: $x_1=1; x_2=-1; x_3=\frac{\sqrt{... |
I have to solve if $C=C(1,\frac{1}{2})$ $$I=\int_{C} ze^{\frac{1}{z-1}}$$
I know that $I=2\pi i \operatorname{Res}(f(z), 1)$, but I do not know how could I calculate that residue.
What I did:
$$f(z)=ze^{\frac{1}{z-1}}=z\sum_{n=0}^{\infty}\frac{1}{n!}\left( \frac{1}{z-1}\right)^n=\sum_{n=0}^{\infty}\frac{1}{n!} \frac{1... |
I am having problems understanding how the connective not works in natural deduction. We were given the below example but I cannot workout how the lecturer got the values in table. If someone could explain how the result was derived I would really appreciate it.
¬¬p |- p
1 ¬¬p ass 0
2 ¬p |- ¬p ∧ ... |
I have to prove the chain-rule for conditional entropy. I kept getting stuck on one step, so I looked up a proof and found this:
\begin{align}H(Y\mid X)&= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log \frac {p(x)} {p(x,y)}\tag{1}\\&= -\sum_{x\in\mathcal X, y\in\mathcal Y}p(x,y)\log\,p(x,y) + \sum_{x\in\mathcal X, y\... |
Please, can somebody help me? In the vectorial space $\mathbb{R}^{2n}$,is the canonical basis a symplectic one?
|
How do I solve these types of problems? What method or technique do I use? I want to learn how to solve the first one so that I can try to figure out the second part to the question myself.
|
Given an undirected, unweighted graph with |V| = 11 and |E|= 19 and given probability p=0.5 I have to calculate the probability of the graph being generated using the Erdos-Renyi Model. I applied the following formula: $$p^{|E|} * (1-p)^{\binom n 2-|E|}$$ but the answer i get it's not correct (I have a list of possible... |
If $P$ contains a set of conjugate vectors that span Krylov space of matrix $A$, i.e. $\mathcal{K}(A, x)$, and also $P$ span Krylov space of matrix $\mathcal{K}(B, x)$, is it true that the diagonal matrix:
$$
P^T A P = P^T B P = D
$$
And is there a proof for this?
EDIT: Pardon for the lack of clarity, here is my situat... |
Is q-Lipschitz continuity necessary condition for fixed point iteration?
Meaning does it have to be so that $d(f(x),f(y))\leq qd(x,y)$ where $0\leq{q}<1$.
|
An air bubble arises from the bottom of the sea. Find its acceleration if the resistance force is proportional to $\rho$*A*$v$ where $\rho$ is density of water, A is cross section area and $v$ is velocity of the bubble. Density and temperature of water are constant.
There was also a part a) of this exercise, where ther... |
Let $a,b\in \mathbb{F}_p^*$ with orders $o_p(a)=|\langle a \rangle|=\alpha$ and $o_p(b)=|\langle b \rangle|=\beta$. I have few questions:
1) Is it true in this case ($\mathbb{F}_p^*$ cyclic) that $o_p(a,b)=|\langle a,b \rangle| = \rm{lcm}(\alpha,\beta)$? I tried to prove it (supposing it's true!) using primitive roo... |
A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads
I already know that the exact same question has already been answered here
But I am trying to solve it using another method.
The first case is when at least $m$ consecutive heads occur before getting a tail. That is
$$HHH\cdo... |
Suppose you roll a fair 6-sided dice three times. There are $6^3$ possible outcomes and each is equally likely.
Let $A_1, A_2, A_3, A_4, A_5,$ and $A_6$ be the events that the last value is 1,2,3,4,5, and 6 respectively.
Let B be
the event that the first value is less or equal to the second value and the second value ... |
On a hypothetical computer with a word length of three digits and truncation, compute the solution of
$$
\begin{matrix}
-3x & + & y & = & -2 \\
10x & - & 3y & = & 7 \\
\end{matrix}
$$
a. Without partial pivoting
b. With partial pivoting
c. Exactly
We consider the matrix $$
\beg... |
Calculate $$\int_{\left|z-1\right|=2}z^{n}\sin\left(z\right)dz$$ for $n\in \mathbb{Z}$
My attempt: According to the following result which was presented at my course as Cauchy's integral formula for Disks
Let $U\subseteq \mathbb{C}$ open, $z_{0}\in U$, $f:U\rightarrow\mathbb{C}$ holomorphic, $R>0$ such that $\overli... |
I believe it means the greatest common divisor of $a_i$ and $m$ is $1$, meaning $a_i$ and $m$ are co-prime, but I want to be sure. Here is the context:
A reduced residue system modulo $m$ is a set of integers $a_1,...,a_k$ such that if $i \neq j$ and $(a_i,m)=1$, $a_i \not\equiv a_j \mod m$...
|
In this question, we have five circle that touch each other. we draw their tangents. If we know that smallest circle radius is 8 and biggest circle radius is 18, then what is the length of PF?
Note: Lines MB, HN, PF, DR, AE are perpendicular lines CE.
I'm sorry for bad English. Is it possible to help me?
Note:
If we ... |
I need help with $part$ $c$ of the following problem:
An accountant has a salary scheme as outlined below.
STARTING SALARY £16,000.
ANNUAL INCREASES OF £1,000 GUARANTEED THEREAFTER.
Let £$u_n$ denote the accountant's salary at the beginning of the the $n$ th year, after any increases which are then due have been added.... |
Imagine the following scenario:
You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep score of each deck. Of course, some deck(s) will perform better in that day's meta than others. Your goal, after each match... |
I am trying to solve this exercise from a textbook:
$y = x^4, y = 0, x = 1;$ rotated about $x=2$
This is my attempt at solving the problem:
Shell radius: $2 - x$
Shell height: $x^4$
$a = 1$
$b = 2$
$\int_1^2 2π(2-x)x^4dx$
$= 2π(2x^5/5 - x^6/6)_1^2$
$= 2π(2*2^5/5 - 2^6/6 - 2/5 + 1/6)_1^2$
$= 2π(19/10)$
$= 19π/5$
However... |
It is easy to argue that the Kempner series converges:
$$
\sum\limits_{\substack{n \text{ s.t. 9 is}\\\text{ not a digit} \\\text{ of } n}}
\frac{1}{n} < \infty$$
Let $E \subset \Bbb N_{>0}$ the subset of the positive integers such that $9$ is not a digit of the decimal expansion of $1/n$ (the decimal expansion is... |
I need to generate a random variable with density function:
$$
f(x)=
\begin{cases}
a (1+|x-2|) , & {-1 \le x \le 4} \\
0, & \text{elsewhere}
\end{cases}
$$
For that I need to inverse the probability function and use a uniform variable to generate the variable with this distribution.
I need to use the inverse transfom... |
An urn contains $nr$ balls numbered $1,2..,n$ in such a way that $r$ balls bear the same number $i$ for each $i=1,2,...n$. N balls are drawn at random without replacement. Find the probability that each of the $n$ numbers will appear at least once.
My approach differs from the book so I would appreciate some help under... |
What is the ideal structure of group $C^*$-algebras? Do there exist any books or articles in the field ?
If G to be the group of integers $Z$ , then $C^*$($Z$)=C($T$). so because ideal structure of C(X) characterized then it is trivial ! The conditions on the G in mind , is locally compact group or finite group or tota... |
How about over a field (i.e. f.d vector spaces)? Can these categories be considered as a site?
|
Suppose I have $K = \mathbb{Q}(\theta)$ and let $f$ be the minimal polynomial of $f$ over $\mathbb{Q}$. Suppose $f$ has degree $n$ so that the degree of $K$ over $\mathbb{Q}$ is $n$. Suppose further that $K/\mathbb{Q}$ is a Galois extension. Let $G = \{ \sigma_{(j)} : j = 1, ..., n \}$ be the Galois group of $k /\mathb... |
Can I get help solving this DE?
$$ y' = \frac 2xy - 1$$
Doesn't look too hard but i just can't get to the correct result.
Thank you in advance
|
Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| \mid x \in E\}$.
Why does the inequality $|f(x)| \leq \lVert f \rVert \cdot |x|$ stand for all $x \in E$? It's trivial ... |
I'm reading the Conway's complex analysis book and I'm trying to solve theses exercises on page 80:
4.Prove that $e^{z+a}=e^ze^a$
5.Prove that $\cos(a+b)=\cos a\cos b-\sin a\sin b$
The author asks to use the corollary 3.8 which says:
Corollary 3.8: If $f$ and $g$ are analytic on a region $G$ then $f\equiv g$ iff $\{... |
Let $\tau$ be a polynomial involution of affine space $k^n$ for $k$ algebraically closed, so
$\tau(x_1,...,x_n)=(f_1(x_1,...,x_n),...,f_n(x_1,...,x_n))$
where $f_i$ are polynomials, and $\tau^2=id$. I want to prove, that the set of fixed points of $\tau$ is non-empty. It probably follows from the Lefschetz fixed-poin... |
Prove that $\lim_{z \to i} |1-z|^2=2$.
I let $\epsilon>0$ and $\delta$ = (blank, since I don't know what to choose yet)
When $|z - i| < \delta$, $||1 - z|^2 - 2| ≤ |(1 - z)^2 - 2| = |z^2 - 2z - 1|.$
And then I'm stuck. I'm not sure how I can extract out a $|z - i| < \delta$ in order to further the proof.
|
My math teacher is a little bit picky on the order we write our simplified algebraic math problems, and I forgot to take notes on this. Let's say you have the problem $2(y + 3) - 4x$. Would I write that $2y - 4x + 6$ or $6 + 2y - 4x$?
Please help me,
-LtLobster
|
I am trying to construct finite automata for this regular expression:
Every block consisting of 5 characters need to contain at least two zeros. The regular expression would look sth like this:
(00(1|0)(1|0)(1|0)(1|0) + 0(1|0)0(1|0)(1|0) + 0(1|0)(1|0)0(1|0) + 0(1|0)(1|0)(1|0)0 + (1|0)00(1|0)(1|0) + (1|0)0(1|0)0(1|0) + ... |
Let A be a finite set of topological spaces $\ X_\alpha $ set. Now let us consider the product set of this topological spaces P=$\ \prod_\alpha X_\alpha $.
Now a topology in P , with all sets of the form $\ \prod_\alpha U_\alpha $, where U is open in $ X_\alpha $ as its basis is called a box topology and a top... |
Problem:
How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality?
My attempt:
$$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$
$$\implies |z^2 - 2iz+1|\le16$$
However, this does not provide a strict upper bound on the inequality, where the equality holds.
I also tried writing it as:
$$|(z... |
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