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Prove that if $\displaystyle \sum_{n=1}^ \infty na_n$ converges, then $\displaystyle\sum_{n=1}^ \infty a_n$ converges.
No, $a_n$ are not necessarily positive numbers.
I've been trying summation by parts.
|
Let $f\in C^4[0,1]$ and $p$ a polynomial of degree $3$. Suppose:
$$f(0)=p(0),\quad f'(0)=p'(0),\quad f(1)=p(1),\quad f'(1)=p'(1)$$
Show that for each $x\in [0,1]$ there exists $\xi\in [0,1]$:
$$f(x)-p(x)=x^2(1-x)^2\cdot\frac{f^{(4)}(\xi)}{4!}$$
Work
It was suggested that I use Rolle's theorem. I tried defining... |
Let $V$ be a subspace of $R^n$. Let $V^ \bot$ be a subset of $R^n$ defined by:
$V ^ \bot$ = {$\vec x \in R^n$: $\vec x * \vec v = 0$ for all $\vec v \in V$}
Prove that $V^ \bot$ is a subspace of $R^n$
I really have no idea about it. Can anyone help please?
Or can I just show that:
$\vec 0$ $\in V^\bot$ since every $\v... |
First of all from last question I can obtain some discrete random variable which described below. In this page, we denote this random variable as $Z$.
$P(Z=0) = e^{-\lambda}+(1-e^{-\lambda})(1-\beta)$
$P(Z=n\beta) = \frac{\lambda^{n}}{n!}e^{-\lambda}\beta, (n>0)$
$\beta$ is constant which is lower than 1.
And there i... |
$A$ = the group algebra of the symmetric group $S_2$ over $\mathbb{Q}$
Find a nontrivial idempotent within $A$
So:
I let $S_2 = (e,s)$. I understand that for $a \in A$ I need to find
$a^2 = a$ for $a = \alpha x_e + \beta x_s$. Clearly $\alpha = \beta = 0$ works. I am confused on how to find a $nontrivial$ idempotent
|
How can I show that $x'=hx(1-x)$ can be transformed to $y'=r-y^2$ using a linear transformation (i.e. $y=mx+b$)? I tackled the problem by substituting $x'$ with $y'/m$ and after algebraic manipulation, I cant get it to simplify down to a constant term minus a square term.
|
Region bounded by $y=\sqrt{x}$, $y=2$ and $x=0$.
I got the answer as $8\pi$, is that correct?
This was a quiz question and I was marked wrong but I am thinking it was because of how I derived the answer. Is the radius supposed to be $r=\pi(\sqrt{x})^2$ or $4\pi - \pi(\sqrt{x})^2$?
|
Somehow I learnt that the limit of $ne^{-an}$ (for $a > 0$) is $0$ as $n$ goes to infinity. I was trying to convert the expression to the form $\frac{e^{-an}}{\frac{1}{n}}$ and then use the L'Hôpital's rule but then I am getting the symbol $0\cdot \infty$ over and over again. How can prove it?
|
So far I have that
Since we known that (a,b) = 1 by definition we can write it as ax+by=1. We also know that (c,d) = 1, so cx+dy=1. Since they are both equal to 1 I can set them equal to each other where ax+by=cx+dy.
Now I am unsure what my next steps would be to prove this. It seems that it would only work if the gcd ... |
I know that a non conditional converse of Abel's theorem is not true, but is there a proof for the converse given certain conditions. So if $f(x)=\sum_{k=0}^\infty a_kx^k$ converges when $|x|<1$ and if $a_k\ge0\forall k$ and $\lim_{x\rightarrow1^-} f(x) = l \in\mathbb{R},$ prove that $\sum_{k=0}^\infty a_k=l$.
Is there... |
This question is most likely extremely trivial, but I'm having some difficulty obtaining the least squares equation from the following data points:
{{1.08, 0}, {1.07, 0.0659232}, {0.97, 0.1695168}, {0.77, 0.188352}, {0.84, 0.0847584}}
In particular, I'm trying to obtain a quadratic equation using least squares, so I wa... |
The question asked me to find the harmonic conjugate of $u$ and then express $f(x,y)$ in terms of $z$. I found the conjugate. I also reasoned that it must be the expanded form of $z^2$. However, they give the answer and it includes a $c$. Why is this?
My answer: $$f(x,y)=x^2-y^2+2xyi$$
$$f(z)=z^2$$
Given answer:
$$f(z)... |
So I'm supposed to figure out how many names an alphabet consisting of A,L,I,S and T can generate. Each name must consist of $2$ vowels and $5$ consonants, begin and end with a consonant, two vowels can't be adjacent and two adjacent consonants must be different. I calculated that there are $2,592$ names possible by lo... |
Define
$$\begin{align*}
f(t) = \sum_{j=0}^\infty (-1)^j\binom{\frac{1}{2}}{j}
t^j
&= \sum_{j=0}^\infty (-1)^j \binom{2j}{j} \frac{(-1)^{j+1}}
{2^{2j} (2j-1)} t^j \\
&= -\sum_{j=0}^\infty \frac{(2j)!}{(j!)^2} \frac{1}
{2^{2j} (2j-1)} t^j \\
&= 1 - \sum_{j=1}^\infty c_j t^j
\end{align*... |
Mathematical induction Follows Thus:
$1.$ The basis (base case): prove that the statement holds for the first natural number $n$. Usually, $n = 0$ or $n = 1$.
$2.$ The inductive step: prove that, if the statement holds for some natural number $n$, then the statement holds for $n + 1$.
For example, when $F_{n}$ is the... |
Evaluate:
$$\lim_{x \rightarrow 1^+} (\ln\ x)^{x-1}$$
|
$$\int x^3f''(x^2)\,\mathrm{d}x$$ Solve using Integration by Parts.
\begin{align}
u&=x^3\qquad\mathrm{d}v=f''(x^2) \\
\mathrm{d}u&=3x^2\qquad v=f'(x^2) \\
&=x^3f'(x)-\int f'(x^2)3x^2 \\
u&=3x^2\quad\mathrm{d}v=f'(x^2) \\
\mathrm{d}u&=6x \qquad v=f(x^2) \\
&=x^3f'(x^2)-[3x^2f(x^2)-\int f(x^2)6x]
\end{align}
No clue w... |
Can this be expressed in a closed form?
$$f(x)=\sum_{n=0}^{\infty}\dfrac{\sin(2^n x)}{2^n}$$
I believe it converges as $-1\le\sin x\le 1$, but I am not sure whether it is even possible to evaluate it in a closed form. I tried to evaluate $f'(x)$, which I failed. I tried to transform the series into a telescoping one, w... |
$$f(x,y)=e^y\sin x+ie^y\cos x$$
The problem requires me to express in terms of $z$ only.
My attempt:
$$=e^y(\sin x+i\cos x)$$
$$=ie^y(\cos x-i\sin x)$$
If $e^{-i\theta}=(\cos\theta - i\sin \theta)$, then:
$$=ie^ye^{-i\theta}$$
But then I get stuck. I'm not sure what to do with the $e^y$.
Note: - the final answer is gi... |
The question is as follows:
Suppose that the conditional distribution of $N$, given that $Y = y$, is Poisson with
mean $y$. Further suppose that $Y$ is a gamma random variable with parameters
$(r, λ)$, where $r$ is a positive integer.
I want to get Var(N)
I know that $$\text{Var}(N) = E(N^2) - (E(N))^2 = E(\text{... |
I was wondering, why are we not allowed to take arbitrary unions (likewise intersections) in the definition of a sigma algebra?; I am looking for a more or less intuitive reason. It seems to me that most of the motivation in defining sigma algebras lies in Measure Theory. So, does allowing arbitrary unions/intersection... |
Based on my previous question: (somewhat related to it)
$$\int f''(x^2)~dx$$
How would you go about and find the integral in an abstract sense as you can do the following with derivatives using chain rule:
$$\frac{d}{dx}[f'(x^2)]$$
$$=f''(x^2)2x$$
I kind of call the above abstract as I think of it because we're not giv... |
If $\{E_\alpha\}_{\alpha\in A}$ is connected in $\mathbb{R}^n$, $\bigcap\limits_{\alpha\in A}E_\alpha \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E_\alpha$ is connected.
I have zero intuition on how to do a proof of this statement. Please help.
I started it by contradiction like this: suppose that $\bigcup\limit... |
$$(z+1)^2=u(r,\theta)+iv(r,\theta)$$
We are learning how to apply the polar form of the Cauchy-Riemann equations. I understand how to do this using the C-R equations the non-polar way, but I'm at a total loss when doing it in polar form. My guess would be to first convert $(z+1)^2$ into polar form?
Update - I still nee... |
Show that $$\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))dt=\ln(b/a),\,a,b>0.$$
Thanks to wikipedia I know that
$$\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))\,dt \overset{?}{=}\int\limits_{0}^\infty\left[\frac{p}{p^2+a^2}-\frac{p}{p^2+b^2}\right]=\left.\frac{1}{2}\ln\left[\frac{p^2+a^2}{p^2+b^2}\right]\r... |
The question is:
Consider the linear system $\left( {\begin{array}{*{20}{c}}
1&\alpha \\
a&1
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
x \\
y
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1 \\
0
\end{array}} \right)$, what is the condition number for computing $x$.
I know the con... |
Assume you have a strictly decreasing convex differentiable function $f(x)$, $x \in \Bbb R^+$, I am wondering if the increment of the first derivative is also convex; i.e.,
$$g(x) = f'(x+\delta) - f'(x)$$ where $\delta$ is any positive number.
What I concluded :
I can say that $f'(x)$ is a strictly increasing functi... |
I've been thinking about the distribution of the coprimes of n, and in particular, about one particular symmetry that they have. Here's a problem that I have formulated for myself:
Given natural numbers $n > 0$ and $m$ prove that there exists $0 < k < ub(n)$ such that $(n, m-k) = 1$ and $(n, m+k) = 1$.
The theorem stat... |
Fix positive integers $n$ and $k$. Find the number of $k$-tuples $(S_1, S_2,\dots, S_k)$ of subsets $S_i$ of $\{1, 2, \dots , n\}$ with the $S_i$’s pairwise disjoint.
I am looking for a way to find this combinatorially, instead of generating functions/recursion. The balls and boxes idea seems like the way to go, but ... |
How to prove that there a constant $C$ such that $\arcsin \frac{1-x}{1+x}+2\arctan\sqrt{x}=C$?
I have no idea using which theorem to prove. Could someone show me how to start the problem?
|
I am really stuck on a homework problem, which boils down to the following:
We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions I know that satisfy such properties take their zeroes at integer values, but I have no idea how to adjust the picture to... |
In the proof of Lemma $15$, the author claimed that if there exists constants $C>0$ and $D>0$ such that
$$\begin{array}{lcl}C \sup \{ | \sum_{i=1}^k{\alpha(i) f(x_i) | \colon \| f \|_{\infty}^1 \leqslant 1, f(0)=0} \} &\leqslant &\sup \{ |\sum_{i=1}^k{\alpha(i) f(x_i)| : \| f \|_{Lip} \leq 1, f(0)=0} \} \\
&\leqslant ... |
Is the operator $\frac{1}{x} \frac{\partial }{\partial x}$ Hermitian?
I think it is because I found $f(x)=Ce^{\frac{1}{2}x^{2}\lambda}$ to be the set of eigenfuctions with ${\lambda}$ as the eigenvalue, and it works for ${\lambda} = i$. I thought Hermitian operators couldn't have complex eigenvalues, so this made the ... |
I am really struggling when proving injectivity and surjectivity for mult variable function. Please if someone can guide the way.
|
In $\mathbb{R} ^3$, $v$ is a fixed vector and $\alpha:I\rightarrow \mathbb{R}$ is a regular curve. We have: $\alpha'(t)$ is orthogonal to $v$ for all $t\in I$ and $\alpha (t_0)$ is orthogonal to $v$. How can I show that $\alpha(t)$ is orthogonal to $v$ for all $t\in I$?
|
I have enclosed one image of two textbook pages.
There is a system of equation (see frame) on page 2.
I do not understand why both terms can be set equal to 0 (zero)to solve it.
Thank you for the assistance.
|
In complex analysis how to prove that if $S$ is closed in $\mathbb{C}$ then it contain all of its accumulation points.
If $S$ is closed then $S$ contain all its boundary points.(If $z_{0} $ is a boundary point then every neighbourhood of it will contain both the points in $S$ and $S'$.) Suppose $z_{0}$ is an acccumul... |
Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$?
To start this question, we need to integrate $\arctan(s+2s^2)$ but how do you do that?
Then, what do we do with the $\int_0^{G(x)}$?
|
Suppose that $f:(a,b)\to\mathbb{R}$ is a differentiable function. Does it follow that $f$ has bounded variation on some subinterval $[c,d]\subset (a,b)$?
Details and ideas
Being differentiable means only that $f'(x)$ exists for all $x\in (a,b)$. Continuity of $f'$ is not assumed.
$f$ need not be of bounded variation ... |
So, because this is honestly homework for a course, I'm primarily looking for a hint from where I've gotten so far. The question is very quick.
$X, Y$ are independently distributed Poisson $(\lambda)$, find the distribution of $$Y^* = Y\mid X_1 + X_2 = t$$
So, the logic I've gotten to so far, but not really further i... |
Consider the local base over the space of complex continuous functions over $[0,1]$ (denoted by $\mathcal{C}[0,1]$) defined for each fixed $x\in [0,1]$ and $\epsilon>0$ by
$$\mathcal{B}_{\epsilon,x} = \{f\in \mathcal{C}[0,1] : |f(x)| < \epsilon\}.$$
In an effort to better understand what is going on with the topology i... |
An $n\times n$ real symmetric matrix is a correlation matrix, if it is positive-semidefinite and all its diagonal entries equal 1. According to most references it is easy to see that the space of correlation matrices is both convex and compact. How?
|
Yes, I know the title is bizarre. I was urinating and forgot to lift the seat up. That made me wonder: assuming I maintain my current position, is it possible for the toilet seat (assume it is a closed, but otherwise freely deformable curve) to be moved/deformed such that stream does not pass through the hole anymore w... |
A Banach space $X$ is called weakly sequentially complete if all weakly Cauchy sequences are weakly convergent.
Question: If $Y$ is a subspace of a Banach space $X$, must $Y$ be weakly sequentially complete?
My guess is yes, but I don't know how to prove it.
Any hint is much appreciated.
|
Let $T:\mathbb R^{4}\to \mathbb R^{4}$ be a linear transformation satisfying $T^{3}+3T^{2}=4I$, where $I$ is the identity transformation. Then the linear transformations $S=T^{4}+3T^{3}-4I$ is:
One-one but not onto
onto but not one-one
Invertible
Not invertible
Given $T^{3}+3T^{2}=4I$, let $S=T^{4}+3T^{3}-4I$. Th... |
Let $A \subset \mathbb{R}^{2}$ be the annulus
$A = \{(x,y) \in \mathbb{R}^{2} \colon 1 \leq x^{2} + y^{2} \leq 4 \}$.
Prove that $A$ is homeomorphic to $S^{1} \times I$, where $I = [0,1]$ is the unit interval.
Define $f \colon S^{1} \times I \rightarrow A$, and $g \colon A \rightarrow S^{1} \times I$ by the following ... |
How can I compute the below integral?
$$
\int e^{-1/2*tr[(\mu\mu^T-\mu m^T-m\mu^T)\Sigma]}d\mu
$$
in which $\mu \in R^{n},m \in R^{n},\Sigma \in R^{n*n}$ and $tr(.)$ is trace of matrix.
I have problem with integral join with trace. how can I solve it?
|
A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category of fields. But I cannot formally prove why categorical product does not exist in these categories.
I can see that categ... |
Take for example $Q ∩ [ - \sqrt(2) , \sqrt (2)]$? Would it be $[ - \sqrt(2) , \sqrt (2)]$ or is this untrue since they are not in $Q$?
|
Find the volume of the cylinder with base as disk of unit radius in the $xy$ plane centered at $(1,1,0)$ and top being the surface $z=[(x-1)^2+(y-1)^2]^{3/2}$
According to me it should be
$$\int_0^{2\pi}\int_0^1\int_0^{r^3}r dzdrd\theta$$
Am I right?
|
I am studying probabilites and the notion of poisson random variable was introduced in the class. But it seems to me that the introduction of poisson random variable is to provide a easy approximation of the binomial random variable conditioned that n is large and p is small. Besides, the preconditions in the poisson d... |
While reading through a statistics book earlier today I came across a notation I'm unfamiliar with and can't find a way to search for it. It is not expected value $E[\,]$, but instead the following. What does the '$E$' Denote here?:
$$E\{RSS/(n-p-1)\} = \sigma^2$$
and that, provided $H_0$ is true,
$$E\{(TSS-RSS)/p... |
Let $A$ and $A'$ be two annuli in $\mathbb{R}^3$. Suppose $A$ has $n$ half twists and $A'$ is with $m$ half twists, where $m$ and $n$ are even and $m\ne n$. It is clear that the surface resulting by cutting along the two boundary circles of an annulus is a 2-sphere. Let $S$ and $S'$ denote the two spheres resulting fro... |
I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: $$\frac {\partial \phi}{\partial x}= yf(u)$$ $$\frac {\partial \phi}{\partial y}= xg(u)$$ Calculating the mixed partial f... |
Let $p_i$ denote the $i$th prime number, and let $p_k\#$ denote the $k$th primorial, $p_k\# \overset{\textrm{def}}= \prod_{i \le k} p_i$.
I am interested in asymptotic upper bounds for the function $S(N) = \sum_{i =1}^{k(N)} p_i$, where $k(N)$ is the smallest $k$ such that $N \le p_k\#$.
As a concrete example, if $N... |
The Von Neumann Problem is as such:
$\Delta u = f(x,y,z)$ in $\ D$
$\frac {\partial u} {\partial n} = 0$ on bdy $\ D$.
Using this you can prove that for there to be a solution to this Von Neumann problem, $\int\int\int f(x,y,z) dx dy dz=0$
.
So I found a partial answer to this question.
Help with the Neumann Problem
Ho... |
The Axiom of Choice is as follows:
Given a collection $\mathcal{A}$ of disjoint non-empty sets, there exists a set $C$ consisting of exactly one element from each element of $\mathcal{A}$; that is, a set $C$ such that $C$ is contained in the union of the elements of $\mathcal{A}$, and for each $A \in \mathcal{A}$, th... |
Having taken some courses in higher algebra, I realized that what I truly appreciate in mathematics is abstract algebra. But it also appears that I'm not a big fan of real analysis [at least I don't seem to have a good feel of topology, and I don't seem to enjoy some other things in analysis either, and I'm not even ha... |
If $\mathscr{A}$ is an unital c*-algebra, $I$ is a closed ideal of $\mathscr{A}$ ,and $\mathscr{B}$ is a unital c*-subalgebra of $\mathscr{A}$ .
Show that the c*-algebra generated by $I\cup\mathscr{B}$ is $I+\mathscr{B}$.
I know that it only need to prove that $I+\mathscr{B}$ is closed, however I do not know how.
Any ... |
I am working on a problem and want to verify that my logic and reasoning is correct. This is my first time working with metric spaces.
Show that the following define equivalent metrics on $\mathbb{R}^n$.
\begin{eqnarray}
\varphi^*(x,y)&=&|x_1-y_1|+\cdots+|x_n-y_n|\\
\varphi^+(x,y)&=&\text{max}\{|x_1-y_1|,\cdots,|x_... |
Suppose a real-valued function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by
$$f(x) =
\begin{cases}
\hfill e^{-\frac{1}{\delta^2 - x^2} + \frac{1}{\delta^2}} \hfill & \text{ $|x| < \delta$} \\
\hfill 0 \hfill & \text{ otherwise} \\
\end{cases}$$
where $\delta>0$ is any fixed real number.
Questi... |
How do you show a set is dense? For example, is The set of all ration numbers ${p \over q}$ with $q ≤ 10, p \in \Bbb Z, q \in \Bbb N$ a dense set?
I know that a set is considered dense in $\Bbb R$ if an element of the set can be found between any two real numbers ($a<b$), but I am not sure how to prove this for any g... |
We have that $(a_n)$ is a bounded sequence of real numbers that satisfy $2a_{n+1}\leq a_n+a_{n+2}, \forall n\in \mathbb{N}$.
I want to show that the sequence $b_n=a_{n+1}-a_n$ converges and that the limit is $0$.
$$$$
I have shown that $b_n$ is increasing and bounded.
So, $b_n$ converges.
How do we find the limit?... |
For a parabola, let Focus: $(a_1,b_1)$ Equation of directrix: $y-mx-c=0$
The equation of parabola is,
$\sqrt{(x-a_1)^2+(y-b_1)^2}= \frac{|y-mx-c|}{\sqrt{1+m^2}}$
There are 4 parameters $m,c,a_1,b_1$. To determine these 4 parameters, we need 4 equations. Therefore, if 4 points are given, the four parameters wi... |
I understand in order to find number of divisors, you need to follow following method, But I don't seem to find why it works.
In order to find number of divisors a number has, you find the prime factorization, and add one to exponents and multiply them.
Eg:
The number 48 has how many positive integral divisors?
a. Fi... |
Let $A\subset \mathbb{R}^n$ be an open set, such that the boundary $\partial A$ has only finitely many connected components. Is it true, that $A$ can only have finitely many connected components as well?
If we fix a connected component $C\subset A$, I have shown that $\partial C\subset \partial A$ has to hold, but the ... |
I am wondering the equivalence of the following problem.
When we computing
$$
P(\mathbf{x}_1 | \mathbf{x}_2, \mathbf{x}_3)
$$
is it equivalent as following, at first define $\mathbf{y} = \mathbf{x}_1 | \mathbf{x}_2$, as new random variable, and
$$ P(\mathbf{y} | \mathbf{x}_3) $$
In particular when $\mathbf{x}_i \si... |
Compute the following integrals:
$I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$
$J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$
I do not know where to begin. I know I am supposed to use the substitution $z\rightarrow \frac{1}{t}$. I can see that $I$ has 13 poles inside the circle of radius 2.
Using a hint provided by Winther I was... |
I do not even know how to start.
Find a symmetric matrix $A \in \mathbb R^{3\times 3}$ with the following properties. Both $[1,2,2]^T$ and $[2,1,-2]^T$ are eigenvectors. It has three distinct eigenvalues. Furthermore, $A^n=A$ for some $n>1$. How many different solutions are there?
|
I am looking at the co-unital property of a coalgebra.
This is my work:
Let $\epsilon: C\to \Bbb C$ be the counit, and let $\Delta$ be the coproduct of $C$, a coalgebra. So $(C,\Delta,\epsilon)$ is a coalgebra.
To satisfy counit requirements, we need:
$$(\epsilon \otimes \text{Id})(\Delta(c))=(\text{Id}\otimes \epsilon... |
My initial reaction is to say that the answer is $6^4$, since 4 dice can have 6 outcomes. In my train of thought, the first dice can have 6 outcomes, same as the second, third and fourth, thus $6*6*6*6$ would seem fitting. However, my professor claims that answer is wrong because it overcounts, and says that the answer... |
In Second-Order ODEs ,There is a problem which I haven't solved.
Method of Variation of Parameters;
In derivation of the method , there is a part which is following.
$$k'[2Pu'_1 + Qu_1] + k''[Pu_1]=0 \\
k'=\frac{e^{-\int [Q/P]dx}}{u_1^2}$$
But I find $$k=\frac{e^{-\int [Q/P]dx}}{u_1^2}$$
I could't understand how they ... |
I am reading "A First Course in Partial Differential Equations with Complex Variables and Transform Methods" (Weinberger, p. 168).
if we have the eigenvalue problem
$$ (pu')'- qu + \lambda \rho u = 0 $$
$$ u(0) = 0 $$
$$ p(1)u'(1) + au(1) = 0, a \ge 0 $$
we find that the eigenvalues are defined by the minimum principle... |
I have to evaluate the integral $\int_{0}^{\infty}t^3e^{-3t}dt$ using complex analysis techniques (the laplace transform). Can you check my steps, please?
$$\int_{0}^{\infty}t^3e^{-3t}dt =\Rightarrow L(t^{3}e^{-3t})=\frac{6}{(s-3)^{4}}$$ evaluated from 0 to $\infty$ gives $\frac{-6}{3^{4}}$ even if the exact result is ... |
How do I set up this problem ?
A product can be made in three sizes, large, medium, and small, which yield a net unit profit of $12, 10$ and $9$ respectively. The company has three centers where this product can be manufactured and these centers have a capacity of turning out $550, 750$ and $275$ units of the product ... |
There are three events: $A$ and $B$ and $C$. We know $P(A|B) = P(B|C) = 0.5$. Then
$P(A|C)$ should be?
Is it $0.5\cdot0.5=0.25$?
The question only provide the above information and the question provide the following choices.
$a. 0.25$
$b. 0.5$
$c. 1$
$d. 0$
$e.$ None of the above is correct
|
Can anyone please help me with:
Find a some base for complementary subspace of
$$M=(p\in P_4 : p(2x)=p(x+1)), $$
|
Find $\text{gcd}(2^{21}-1,2^{27}-1).$
My proof: We know that $2^{21}-1=(2^3)^7-1=8^7-1=(8-1)(8^6+\dots+8+1)=7(8^6+\dots+8+1)=7N_1$ and $2^{27}-1=(2^3)^9-1=8^9-1=(8-1)(8^8+\dots+8+1)=7(8^8+\dots+8+1)=7N_2.$ Also $N_2-N_1=8^8+8^7$. Hence $$\text{gcd}(2^{21}-1,2^{27}-1)=\text{gcd}(7N_1,7N_2)=7\text{gcd}(N_1,N_2).$$
I gues... |
Suppose we have the function $g:\mathbb{R}\rightarrow \mathbb{R}$ and $g(x)=\left\{\begin{matrix}
x\cdot (-1)^{[\frac{1}{x}]} & x\neq 0\\
0 & x=0
\end{matrix}\right.$
I want to show that $g$ is continuous at $0$ and discontinuous at $\frac{1}{2}$.
How can we calculate the limit at $0$ ?
|
The problem
A psychic uses a five-card deck to demonstrate ESP, claiming to be able to guess a card correctly with $0.5$ probability (of course, ordinary guessing is $0.2$). A single experiment consists of making five guesses, reshuffling the deck after each guess. The experiment is executed and the the psychic guess c... |
I have problems to understand a definition from lecture:
Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., a_k$ with degree $|a_1|,..,|a_k|$ and with $a^2=0$ over $R$ is the graded commutative algebra $\Lambda_R[a_1,..,a_k]$ with ... |
compute 3/5+2/7+1/6 in Z11. Please give me a hint on how to go about it. I have created a table for Z11 but unsure of the next step.
|
I know that I have to use the binomial theorem. So, in following the formula of ${(1+x)}^{n} = {n \choose 0}+{n \choose 1}{x}+{n \choose 2}{x}^{2}...+{n \choose k}{x}^{k}+...+{n \choose n}{x}^{n}$, I came up with ${200 \choose 99}$, but I am certain that is not all the way correct, if at all.
Can anyone help me with th... |
Question
We now define the following "ugly" function:
$$ A_c(s,r,n,m) =
\begin{cases}
1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise}
\end{cases}
$$
How does the "ugly" function asymptotically behave?
$$ \sum_{n=1}^c \sum_{m=1}^c \sum_{r=1}^c \sum_{s=1}^c A_c(s,r,n,m) \sim (?)$$
Does this connecti... |
Define $f,g:\Bbb{R}\to \Bbb{R}$ by $$ f(x)=\begin{cases} x, & \text{if} \,\,\, x\in\Bbb{Q},\\ \sin(x), &\text{if}\,\,\, x\in\Bbb{R-Q}\end{cases}$$ and $$g(x)=\begin{cases} x \sin(x)\sin(1/x), & \text{if}\,\,\, x\ne0,\\
0, & \text{if}\,\,\, x=0\end{cases}$$ at $x=0.$
how do I know if they are differentiable or not a... |
I'm thinking along the lines of attempting to explore mathematics by making my own discoveries as if it's the new frontier. As if I was a mathematician from 2000 B.C.
I know this sounds really strange but I'm curious - has anyone tried rediscovering mathematics? (who isn't an advanced mathematician) because I think it ... |
I am given $T z= \frac{z+2}{z+3}$ , where $T_1$ is a linear transformation on the complex number $z$. I need to find its inverse $T^{-1}z$.
I considered a complex number $w =\frac{z+2}{z+3} $. This gives me $z = \frac{2-3w}{w-1} $. So, the inverse linear transformation is $\frac{2-3z}{z-1}$.
Am I correct or is this ... |
I know that $\frac{a^n - 1}{a-1}$ is a geometric series, but I don't know how that can help me solve it.
This is not a duplicate question...
|
I want to check if the series $$\sum_{n=2}^{\infty}\frac{1}{n\log n\log (n+1)}$$ converges.
Let $a_n=\frac{1}{n\log n\log (n+1)}$.
I found that $\frac{a_{n+1}}{a_n}\rightarrow 1$ so we cannot use this criterion, right?
What could we do?
|
I am not sure where to start with this one, any help would be appreciated. I tried using the definition of a span but I couldn't see where to go from there.
|
I saw this post on the eigenvalues of a matrix plus a constant times the identity matrix.
Say $A$ is an $n\times n$ matrix (real and non-singular) with eigenvalues $\lambda_1,\ldots,\lambda_n$, then the eigenvalues of $A+cI$ are $\lambda_1+c,\ldots,\lambda_n+c$.
My question is on the condition number of $A+cI$, can thi... |
Let $a$ be a root of the cubic $x^3-21x+35=0$. Prove that $a^2+2a-14$
is a root of the cubic.
My effort
Working backwards I let $P(x)$ be a polynomial with roots $a,a^2+2a-14$ and $r$.
Thus, $$P(x)=(x-a)(x-r)(x-(a^2+2a-14))$$
Expanding, I get
$$P(x) =(x^2-(a+r)x+ar)(x-(a^2+2a-14)) $$
$$P(x) =x^3+x^2[-(a^2+2a-14)-(... |
To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible?
Just used to help understand a Linear Transformation assignment question, don't necessarily need an explanation as to why, but it would help. Thankyou.
|
$$2^{x}+3^{x}=6$$
How to find the real number x? I mean it's approximately $1.19$ bur can we write $x$ as the form of $a, b, c$ when $a^{x}+b^{x}=c$ in general. Maybe an infinite sum?
|
I have managed to implement both forward and inverse FFT transforms in C#. And, i tested them by taking signal both ways and got the real part after iFFT equals the original signal.
Now , I have taken the same signal in frequency domain and changed both magnitude spectrum as well as phase. Then, i have taken the comple... |
I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some cosets spaces came from.
The setup is as follows:
Consider the group $ISO(m,n)$ of isometries of $\mathbb{R}^{m,n}$, i... |
$\require{cancel}$
I want to see if I have solved this problem appropriately or not.
If we have
$$[C_\beta(2\theta)]=\frac{1}{8}\begin{bmatrix}-2\cos2\theta&\sqrt{2}\sin2\theta&0\\\sqrt{2}\sin2\theta&0&\sqrt{2}\sin2\theta\\0&\sqrt{2}\sin2\theta&2\cos2\theta\end{bmatrix}$$
and
$$p(\theta,\theta_0,n)=\frac{2^{2n-1}}{\bin... |
I followed the binomial theorem and got this:
The Binomial Theorem is: $(a+b)^{n}= \sum_{k=0}^{n} {n \choose k}{a}^{k}{b}^{n-k}$
Then let $a=x, b={1\over x}, n = n, k = k.$
I then get $\sum_{j=0}^{n}{n \choose j}{x}^{j}{1 \over x}^{n-j}$
I'm not sure if I'm doing it right. Can someone help me out with this?
|
I would like to proove the Lemma 3.1. in this book.
My attempt...
I want to split the lemma into several parts.
Part 1:
$$\prod_{j=1}^{n} (1 + \epsilon_j) = 1 + \sum_{j=1}^n \epsilon_j + O(|u|) = 1 + \theta_n,$$
where
$$\theta_n = \sum_{j=1}^n \epsilon_j + O(|u|).$$
For the bound i observe that
$$1 + \theta_n = \prod_{... |
Prove that $\lim_n (-n+\sin(n)) = -\infty$
So I need to show that if $B'$ is any number, then there is a number N' such that
$$n>N' \implies S_n\lt B'$$
I am having trouble feeling confident in my proof so please tell me if my method is right, if my answer is correct, and if there is a more precise way to use the def... |
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