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H: Is the unit ball on the set of continuous functions of a space $X$ strictly convex?
I have been trying to show that $C(X)$ is not strictly convex but I have been having a tough time, any help would be appreciated.
AI: I assume that your $X$ is compact (so that $\|f\|_\infty$ is defined for all continous $f\colon X\... |
H: Show that each vector in an n-dimensional vector space can be represented as the summation of its components along the orthonormal basis.
Show that in an n-dimensional vector space V over the universal set with orthogonal basis {$a_1, a_2,..., a_n$}, each vector B can be expressed as:
B = $\frac{<B,a_1>a_1}{||a_1|... |
H: Find a limit for a sequence of functions with the domain in $(0, \infty)$
How to find limits for function $f_n = \sqrt{n}\left(\sqrt{x - \frac{1}{n}}- \sqrt{x}\right)$ if $Df \in (0,\infty)$
I think as $n \rightarrow \infty$ it would be $\infty$ and then no matter the $x$ the limit would be $\infty$. But in case wh... |
H: Let $R$ be the region enclosed by $x^2+4y^2\geq 1$ and $x^2+y^2\leq 1$. Then the value of $\int \int_R |xy|dx dy$
Let $R$ be the region enclosed by $x^2+4y^2\geq 1$ and $x^2+y^2\leq 1$. Then the value of $$\int \int_R |xy|dx dy$$ is _________
My attempt
I am getting $5/24$. But answer given is $.375$ Where is... |
H: How do you prove this asymptotic relations between $n!$ and $\frac1en+\frac1{2e}\ln(2 \pi n)$?
Let $\varepsilon(n):=\frac1en+\frac1{2e}\ln(2 \pi n)$. Toying around in Wolfram, I found the following results:
$$\lim_{n \rightarrow \infty} \frac{\varepsilon(n)^n}{n!}=1 \tag{1}$$
$$\lim_{n \rightarrow \infty}\varepsilo... |
H: every non-finite set intersects non-void open set
From General Topology by Kelley.
A separable space may fail to satisfy the $2^{nd}$ axiom of countability.
For example, let $X$ be an uncountable set with the topology consisting of the void set and the complements of finite sets. Then every non-finite set is den... |
H: If $z+\frac{1}{z}=2\cos\theta,$ where $z\in\Bbb C$, show that $\left|\frac{z^{2 n}-1}{z^{2n}+1}\right|=|\tan n\theta|$
If $z+\frac{1}{z}=2 \cos \theta,$ where $z$ is a complex number, show that
$$
\left|\frac{z^{2 n}-1}{z^{2 n}+1}\right|=|\tan n \theta|
$$
My Approach:
$$
\begin{array}{l}|\sin \theta|=\left|\sqrt{1... |
H: Find the two complex numbers whose sum is $5 - i$ and product is $8+i$
I have tried to solve this problem, but when I write the equations I end up with:
$$a + c = 5$$
$$b + d = -1$$
$$ab - cd = 8$$
$$ad + cb = 1$$
But substituting $a$ and $b$ get 2 equations I can't be able to solve
AI: Hint: the numbers are the so... |
H: A doubt regarding proof of values of trigonometric functions at allied angles
There are certain identities that help us to determine the values of trigonometric functions at $\dfrac{\pi}{2}+x \text{, } \pi-x$ etc. given the values of $\sin x, \cos x$.
Now, when we prove such identities, we usually take the value of... |
H: sum of :$\sum_{k=1}^\infty\frac{(-1)^k}{2k-1} \cos(2k-1)$
How can I find the sum of :$$\sum_{k=1}^\infty\frac{(-1)^k}{2k-1} \cos(2k-1)$$
I don't fully understand the parseval identity so I am asking if we can use it to find the sum, and if so, how I should use it.
Is there a Fourier series we know the convergence t... |
H: Substitution problem in an integral
Say I have:
$\int_0^\pi\cos^2\theta\sin\theta d\theta$ and I choose to make the substitution $t=\sin\theta$. I then get an integral $\int_0 ^0 t\sqrt{1-t^2}dt$ and the result is zero. What's the fallacy here? I know I can substitute $\cos\theta$, I was just wondering.
AI: The pr... |
H: Fourier Series Expansion, getting an undefined coefficient
I'm asked to find the Fourier series of the following function as a sine series with period $2\pi$.
$f(x)= cosx \ on \ [0,\pi]$
Since we wish to get a sine series we need to make $a_n = 0 \ for \ all \ n\geq 0$. Hence, we need an odd extension. Then I did t... |
H: Diffeomorphism and local isometry
I'm asked to show that $F(x,y,z)=(3x,2y,5z)$ is an diffeomorphism between $\mathbb{S}^{2}(1)$ and the ellipsoid $(\frac{x}{3})^{2}+(\frac{y}{2})^{2}+(\frac{z}{5})^{2}=1$. For this aim what I have seen is that (I) its class is infinity (each component is) (II) its inverse is given b... |
H: Can we find a closed-form formula for $a_{n+1}=\frac{a_n}{a_n+1}$, given $a_1=a$?
We have a sequence:
$$a_{n+1}=\frac{a_n}{a_n+1}$$
$$
a_1 = a
$$
$$
a_2 = \frac{a}{a+1}
$$
$$
a_3 = \text{here things get really messy and for all the following }
$$
I can't get the formula since the expressions just get more and mor... |
H: Let $X=\mathbb R$ with cofinite topology and $A=[0,1]$ with subspace topology - show $A$ is compact
Let $X=\mathbb R$ with the cofinite topology and $A=[0,1]$ with the subspace topology. I've just proven that every closed subspace of a compact space is compact, and now I'm asked to show that $A$ is compact, but not... |
H: How does Grinberg's theorem work?
Grinberg's theorem is a condition used to prove the existence of an Hamilton cycle on a planar graph. It is formulated in this way:
Let $G$ be a finite planar graph with a Hamiltonian cycle $C$, with a fixed planar embedding. Denote by $ƒ_k$ and $g_k$ the number of $k$-gonal fac... |
H: How can Complex numbers be written in this way?
In quantum optics, we assume that the eigenvalues of coherent states are complex numbers. So, when we determine the overlap of two different coherent states we have to deal with complex numbers all the time. In an academic book, I encountered an equation and I want to... |
H: Need to find maximum in order to prove that the function converges uniformly.
So, here is the function: $f_n = \sqrt{n}\left(\sqrt{x+\frac{1}{n}}-\sqrt{x}\right)$
I found that the limit for it is $0$. Now I want to show whether it converges uniformly. So first I tried to find a maximum.
$$f'_n(x) = \frac{1}{2}\sqrt... |
H: what does this notation $Z1\{condition\}$ mean?
I am reading that you can decompose a random variable like this:
$$
Z = Z1\{Z\leq\theta\mathbb{E}[Z]\} + Z1\{Z>\theta\mathbb{E}[Z]\}
$$
What does $$Z1\{condition\} $$ mean in this case?
Thanks for your help.
AI: $1_{\{condition\}}$ is an indicator function: $1$ when c... |
H: Integral of Stochastic Integral $ x(t) = \int_0^te^{as}dW_s $
I'm working through an exercise where I have got the following object:
$$ x(t) = \int_0^te^{as}dW_s $$
for some constant $a$. Now, I need to find the distribution of $\int_0^t x(s) ds$. I'm struggling to do this and I don't know where to look for help.
S... |
H: Units of $R[X]/(aX-1)$
In the most upvoted answer to another question here, the author states that:
$R\to R[x]$ followed by the quotient map $R[x]\to R[x]/(ax-1)$. Call this $f$. Note that $f(a)$ is a unit in $R[x]/(ax-1)$
If I understand correctly, the result is $f(a)=ax^0\mod{ax-1}=ax^0$, and I fail to see why ... |
H: Number of Elliptic Curves over Fp
I am a beginner/amateur in the topic
according to https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf on page 45,
There are approximately 2p different elliptic curves defined over $F_p$.
On SAGE, I tried to enumerate elliptic curves over $F_5$ looking at all curv... |
H: Which integral Steifel-Whitney classes are universally $0$?
Let $BO(n)$ denote the classifying space of the orthogonal group $O(n)$. Then there is the well-known ring isomorphism
$$H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}/2[w_1,\dots,w_n] $$
where $w_i \in H^i(BO(n);\mathbb{Z}/2)$ is the $i$-th universal Steifel-W... |
H: Relation between variance and a type of expectation
I'm asking if there are some relations/inequalities between those two:
$$
\mathbb{E}[f(X)(X - \mathbb{E}[X])] \tag{1},
$$
and
$$
\mathrm{Var}[X] = \mathbb{E}[(X - \mathbb{E}[X])^2] \tag{2}.
$$
Let us assume $f$ is some regular bounded continuous funciton.
My gues... |
H: Question similar to Collatz game
A problem is this:
Something I'm thinking about.
Let $f(n)$ be the number of $1$ bits in $n$, i.e
$f(3)=2,f(8)=1$
consider the trajectory of a natural $k$ under the map $n -> n+f(n)$. does it always collapse into the trajectory of $1$ under the same map?
PLEASE Help!!!
do you need... |
H: For $T(t)$ strongly continuous, check that $T(t)x - x = tAx + \int_0^t (t-s) T(s)A^2 x ds$.
I am reading Lemma 2.8 of "Semigroups of Linear Operators and Applications to
Partial Differential Equations" by Pazy:
Let $A$ be the infinitesimal generator of a strongly continuous semigroup $T(t)$ satisfying $\|T(t)\| \l... |
H: Estimated Time of Completion given Progress and Elapsed Time
Say I have some task that I have spent an hour on, and I am 33% done. It should take me 3 hours to complete this task. Another task may be 50% done, and I have spent 3 hours on it. That should take a total of 6 hours. What is that equation?
Something like... |
H: First isomorphism theorem for groups proof that function is well-defined
Theorem: If $\alpha:G \to H$ is a homomorphism, then $G/ \ker{(\alpha)}$ is isomorphic to $\alpha(G)$.
Denote $\ker(\alpha)$ as $K$, and recall that $K$ is a normal subgroup of group $G$ for homormorphism $\alpha$. Then the function $\beta:G/... |
H: What is meant by “how an element in the domain is mapped to its image”.
In the following lecture given by Fredric Schuller, he mentions this during the lecture which is on multilinear algebra (know that $P$ is a set of polynomials such that $p$ $\in$ $P$):
“consider the map $I$: $P$ $\to$ $\mathbb{R}$, now I need ... |
H: Why do we use the method of matrix exponential?
I have a linear system of homogeneous ordinary differential equations, i.e.:
$$ \dot{x}=Ax $$
where $A$ is an $n\times n$ real matrix.
The matrix exponential method (described for example here) tells me that
$$ e^{At} C $$
where $C=(C_1,C_2,C_3)$ are arbitrary consta... |
H: Entire function that is a bijection on the unit disk is a rotation
I'm working on this problem
"Let $f$ be an entire function. Suppose $f$ restricted to the unit disk is a bijection. Prove that $f$ is a rotation."
My attempt: It is tempting to use Schwarz lemma. Let $T$ be the linear transformation that maps the un... |
H: Does $f'(x) > 0$ and $f''(x) < 0$ imply that $f'(x)$ is converging to $0$ as $x$ is tending to infinity?
I have a question related to another question which I asked today, where some people gave me two nice counterexamples.
Suppose you have a differentiable function $f:\mathbb{R}_+ \rightarrow \mathbb{R}_+ $, where... |
H: Find a convergence interval for the series of functions.
I tried to use the ratio test to determine the convergence interval for the series of functions given as $\Sigma^{\infty}_{k=0}\frac{k!(x-2)^k}{k^k}$. But the ratio test was infinity no matter the x. Here is the working:
$$\Sigma^{\infty}_{k=0}\frac{k!(x-2)^k... |
H: Find irreducible factors without factorizing
I have an exercise from my course notes that states:
Find how many irreducible factors has $f(x) = x^{26}-1$ over $\mathbb{F}_3$ and their degrees. (don't factorize it)
I see immediately that the $1$ is a root of $f$. So I have $f(x)=(x-1) g(x)$ where $g$ has degree $2... |
H: Help with exercise in complex analysis on the existence of a mapping
I have encountered the following exercise from a practice exam:
Does there exist an analytic function mapping the annulus:
$ A = \{ z | 1 \leq |z| \leq 4 \} $
onto the annulus:
$ B = \{ z | 1 \leq |z| \leq 2 \} $
And which takes $ C_1 \to C_1 $ a... |
H: Difference of two increasing functions.
Let $f$, and $g$, both be an increasing function w.r.t $x$, s.t $x\in(0,1)$. What can we comment on the nature of $f-g$?
AI: The information provided on $f$ and $g$ is not enough to deduce how $f-g$ behaves. Depending on which function increases faster affects whether or not... |
H: Uniform convergence for series
Given the series:$\Sigma_{k=1}^{\infty} \frac{x^2 \cos (2x)}{1+k^4x^6} $ show that the series converge uniformly.
I tried to do it like this:
$$\frac{x^2 \cos (2x)}{1+k^4x^6} \le \frac{x^2 }{1+k^4x^6} \\ \text{Atvasina pēc } x \text{ un pielīdzina nullei: } \frac{2x(1+k^4x^6) -x^2 (... |
H: Determine $I_n=\int_{-\infty}^\infty t^ne^{-\frac{t^2}{a}}dt$
if $\Large \int_0^\infty t^ne^{-t}dt =n!$
What's about $\Large I_n=\int_{-\infty}^\infty t^ne^{-\frac{t^2}{a}}dt$ ; $a\in \mathbb{R}$
AI: Let $t=\frac{x^2}a$ in $\int_0^\infty t^m e^{-t}dt =m!$ to get
$$\frac2{a^{m+1}} \int_0^\infty x^{2m+1}e^{-\frac{x... |
H: If $f(x)$ is continuous on $[0,1], \text{ and } 0\leq f(x)\leq1, \forall x \in [0,1], \text{ prove } \exists t \in [0,1] \text{ s.t. } f(t) = t$
My thinking is that $f(x)$ has to intersect with function $g(x) = x$ at some point, but I don't know how to prove this.
AI: Guide:
$f$ is continuous, $g$ is continuous, he... |
H: Use eigenvectors and eigenvalues for transformation matrix of quadractic form.
I'm really confused by a task I have been given:
We are looking at a quadratic form $Q(\vec{x})=\vec{x}^TA\vec{x}$ where the symmetric matrix $A = A^T$ has normalized eigenvectors $\vec{v_i}$ with corresponding eigenvalues $\alpha_i$. So... |
H: $\sum_{n=0}^\infty\frac{H_n(x)H_n(y)t^n}{2^nn!}$=$\frac{\exp\left[\frac{2xyt-(x^2+y^2)t^2}{1-t^2}\right]}{\sqrt{1-t^2}}$
I am told to prove that :
$$\sum_{n=0}^\infty\frac{H_n(x)H_n(y)t^n}{2^nn!} = \frac{\exp\left[\frac{2xyt-(x^2+y^2)t^2}{1-t^2}\right]}{\sqrt{1-t^2}}$$
where $H_n(x)$ is Hermite polynomial.I am wo... |
H: Let $x$ be a non zero vector in the complex vector space $\mathbb C^n$ and $A=xx^H$.Find all the Eigenvalues and their Eigen spaces.
Let $x$ be a non zero vector in the complex vector space $\mathbb C^n$ and $A=xx^H$.Find all the Eigenvalues and their Eigen spaces.[where $x^H=$ conjugate transpose of $x$]
Here ra... |
H: Question About the Definition of a $\mathbb{C}-$algebra
In the 'Quick review of commutative algebra' chapter a $\mathbb{C}-$ algebra is defined as a commutative ring that contains $\mathbb{C}$ as a subset. I don't understand if these coordinate rings are vector spaces nor if a variety is supposed to be a vector spa... |
H: Investigate convergence of a series using comparison test
The series with $a_n$:
$a_n = (n^{1/3}-(n-1)^{1/3})/n^{1/2} $
I tried comparing it to the $1/n^2$ and $1/n^{3/2}$ because those definitely converge, but proving the inequality gives rise to pretty complicated polynomials with high degree, and I cant seem to... |
H: An unbounded function whose square is uniformly continuous on $\mathbb R$
I want an example of a continuous function $f:\mathbb R \to [0, \infty)$ which is unbounded but such that $f^2$ is uniformly continuous.
However much I try I am able to get only bounded functions like $\sin x$. We have $\sin^2 x$ is unifor... |
H: Find if series converge ( series containing complex numbers)
Here I try to determine whether series converge:
$$\sum^{\infty}_{k=1} \frac{(z-2i)^{3k}}{k^3 2^k}$$
$$\left|\frac{a_{k+1}}{a_k}\right| =\left| \frac{(z-2i)^{3(k+1)} 2^{k} k^3}{(k+1)^32^{k+1}(z-2i)^{3k}}\right| = \frac{1}{2} \left|\frac{z-2i}{k+1}\right|^... |
H: Why any integer $n$ can only have one prime factor greater than $\sqrt{n}$?
I know the proof that for a composite number $n$, there is at least one prime factor less than or equal to $\sqrt{n}$ but I don't know how to prove this following statement:
Any number $n$ can have only one prime factor greater than $\sqrt... |
H: How does $H$ act on $G^t$ in the wreath product $G^t \wr H$?
I'm reading this expository paper about group theory in the Rubik's cube.
I'm a little confused by the definition of the wreath product in this paper.
Example 3.12 on page 12 states that the elements of the wreath product $(\mathbb{Z}/2\mathbb{Z})^3 \wr S... |
H: Defining a complex polynomial with certain roots
How can i define a complex polynomial, with real coefficents $a_{0}$ and the form $p(z)=a_{2}z^2 + a_{1}z + a_{0}$ with the roots being at the same time $z_{1}=1-i$ and
$z_{2}=-1+i$,
AI: In a quadratic polynomial $p(z)=a_2z^2+a_1z+a_0$ with roots $z_1,z_2$, we h... |
H: Lebesgue measure/integral problem - where do I go wrong?
Let $f\geq0$ be a bounded function supported in a measurable set $E$ with $m(E)<\infty$. Show that if $\int_E f=0$, then $m(E)=0$.
My proof: Let $E_\epsilon=\{x\in E:f(x)\geq\epsilon\}$. Then for any $\epsilon>0$,
$$
0=\int_Ef\geq\int_{E_\epsilon}f\geq\int... |
H: If $f(x)$ is continuous on $(a,b), \text{ and } x_1,x_2 \in (a,b), m_1,m_2 > 0$
$$\text{If} f(x) \text{ is continuous on } (a,b), \text{ and } x_1,x_2 \in (a,b), m_1,m_2 > 0$$
$$ \text{ prove }\exists c \in (a,b) \text{ s.t. } f(c) = \frac{m_1f(x_1)+m_2f(x_2)}{m_1 + m_2}$$
I need a hint on how to prove this
AI: $f... |
H: Proof that there exists a unique $x^* \in X$ such that $T(x^* ) = x^*$ .
Suppose $(X, \rho)$ is a complete metric space, and suppose
the function $T : (X, ρ) \rightarrow (X, ρ)$ is such that $T_n = T ◦ T ◦ · · · ◦ T$ (n times) is
a contraction map for some $n \ge 2$. Prove that there exists a unique $x^* \in X$ suc... |
H: Unit circle coordinates to pi
If there are two coordinates of a unit circle, e.g. $x=0$, $ y=1$, I know this is $\frac{\pi}{2}$.
How can I calculate pi for any two coordinates, even if they are not places on the unit circle, like $x=1.23$, $y=-0.1$?
AI: If I am understanding correctly, you can take the $\text{atan2... |
H: Cauchy product of two formal power series
I am thinking if I could get help for the following question:
Given a formal power series
$$g(z)=\sum_{i=0}^\infty a_i z^{-i}$$
does there always exists another (non-trivial) formal power series $y(z)$, such that the Cauchy product between $y$ and $g$
$$
y \times g= \sum_{... |
H: Showing that $y=P\cos(\ln(t))+Q\sin(\ln(t))$ satisfies $t^2\frac{d^2y}{dt^2}+t\frac{dy}{dt}+y=0$
Show that, if $P$ and $Q$ are constants and $$y = P\cos(\ln(t)) + Q\sin(\ln(t))$$
then
$$t^2\frac{d^2y}{dt^2}+t\frac{dy}{dt}+y=0$$
AI: $P$ and $ Q $ are constant, thus by chaine rule differentiation,
$$y'=-P\sin(\l... |
H: "Inverse" moment generating function of standard normal distributed random variable
This is just a trivial question maybe but, is the Moment generating function for $X$ the same as for $-X$ for a normally distributed random variable, so $E(e^{tX})=E(e^{-tX})$? If not, what is the difference between them?
AI: If $$X... |
H: Is $\mathbb{R}$ a vector space over $\mathbb{R}$?
It might be an interesting question before studying the concept of orientation on $\mathbb{R}$ as it is studied on $\mathbb{R^2}$ & $\mathbb{R^3}$
AI: Any ring $R$ is a an $R$-module via its intrinsic multiplication. So in the case when $R$ is a field, $R$ is a vect... |
H: Quotienting $\Bbb{Q}$ by a fractional ideal, what happens?
Consider arithmetic $\pmod N, \ N \in \Bbb{N}$.
Now suppose there is $a \in \Bbb{Z}/N$ such that $(a,N) = 1$. Then Can we consider arithmetic in $R:=\Bbb{Q}/(N/a)\Bbb{Z}$ equivalent in some way to arithmetic in $\Bbb{Z}/N$?
If we're in $R$ then equality $x... |
H: eigen pair for sin(A)
original image
I don't know how to get the eigen pair for $\sin (A)$
AI: Let's prove that :Eigen pair of A and sin A are the same in the sense that if $(\lambda, v) $is an eigen pair of A then (sin$\lambda, v) $ is an eigen pair of sin A. $\tag{1}$ Since A is a symmetric matrix, it is diagona... |
H: Bound probability of deviation of norm
For $X, Y$ iid, show that:
$$P(||X|| > t) \leq 3 P(||X + Y|| >2t/3)$$
for any norm $||\cdot||$ and $t >0$
The proof I've seen is just two steps, but I am not understanding the argument leading to the inequality:
$$P(||X|| > t) = P(||(X + Y) + (X + Z) - (Y + Z)|| > 2t) \leq 3 P... |
H: number of ideals in a set and determine the maximal ideals
Let $f(X)=(X^2-2)(X^4-X)$ and $g(X)=(X^2-1)X\in \mathbb{Q}[X]$.
Let $I=(f,g)$ the ideal generated by $f$ and $g$.
1) How many ideals does $\mathbb{Q}[X]/I$ has?
2) What are the maximal ideals?
I have already computed that $I=X(X-1)$ and proved that $\math... |
H: If $A \in L(R^n,R^m)$, then $||A|| < \infty$ and A is a uniformly continuous mapping of $R^n$ into $R^m$.
Definition: for $A \in L(R^n,R^m)$, define the norm $||A||$ of A be the sup of all numbers $|Ax|$, where x ranges over all vectors in $R^n$ with $|x| \leq 1$. Observe that $|Ax| \leq ||A|| |x|$ holds for all $... |
H: Prove inequality $\|Z\|_2\le \|Z\|_1^{1/4} \|Z\|_3^{3/4}$
Prove $\|Z\|_2\le \|Z\|_1^{1/4} \|Z\|_3^{3/4}$ for random variable $Z$.
AI: By Holder's inequality,
$$\mathbb{E}[Z^2]=\mathbb{E}[Z^{1/2}Z^{3/2}]\le (\mathbb{E}|Z|)^{1/2}(\mathbb{E}|Z|^3)^{1/2}$$
Rearrange and then it is proved. |
H: Derivative of $\text{Tr}[B X^T A X^{-1}]$
Let $A, B, X \in \mathbb{R}^{n \times n}$ and assume that $X^{-1}$ exists. Derive $\frac{\partial K}{\partial X}$ where
$K(X)= \text{Tr}[B X^T A X^{-1}]$
I have tried the following so far ($U = B X^T A X^{-1}, K = \text{Tr}[U]$):
$$
\frac{\partial K}{\partial X} = \frac{\p... |
H: Is this set in $\mathbb{R}^2$ closed?
Is $A=\{(x,y)\in \mathbb{R}^2 : x=\frac1n\}$ $, n \in \mathbb{N}, 0\leq y\leq 1$ a closed set? I think it is but I don't have any demonstration yet.
AI: Following the hints given in the comments: notice that for all $n \in \mathbb{N}$, $a_n \doteq \left(\frac{1}{n}, 0 \right) ... |
H: Is it possible to solve this probability question without knowing if these two events are independent or not?
The question gives
P(A) = 1/3, P(B) = 1/4 and P(A⋂B) = 1/6 and asks for P(A⋃BC)
The solution provided is the following:
P(A∪Bc) = P(A) +P(Bc)−P(A∩Bc)
=P(A) +P(Bc)−(P(A)−P(A∩B))
=P(Bc) +P(A∩B) = 11/12
My q... |
H: How to calculate the following Probability given the combination of discrete and continuous events? Is it even possible?
This is a problem i came up with myself, but i don't even know how to start to solve it? So maybe somebody here knows or sees a way to approach it:
You are given a dice with $n \in \mathbb{N}$ ... |
H: Radius of convergence $\sum_{n=0}^{\infty} (n+a^{n})z^{n}$
hi could you please help me find the radius of convergence for the next series
$\sum_{n=0}^{\infty} (n+a^{n})z^{n}$
please i have tried by the quotient criterion and using the hardy formula but i did not come to any conclusion and put it into wolfram but i... |
H: An entire function whose only zeros are positive integer
This is a problem from my past Qual: "Prove or disapprove. There is an entire function $f$ s.t. $f(n)=0$ for all $n\in \mathbb{N}$ and nonzero elsewhere."
For this type of problems, I think of Identity Theorem. $f(n)=0$ for all $n$ and hence the set $\{z|f(z)... |
H: Prove that all singularity of $\frac{1}{e^z+3z}$ is of order 1
This is a problem from my past QUal: "Prove that all singularity of $$\frac{1}{e^z+3z}$$ is of order 1. You don't need to find the singularities."
Usually this kind of problem is easy to me. My procedure is to find the singularities, and use Taylor seri... |
H: Finding $P(X+Z>Y)$ where $X,Y,Z$ are exponential random variables
Let $X$,$Y$,$Z$ be independent random variables with exponential distribution of parameter $\lambda$, then $X,Y,Z$ ~ $\xi(\lambda)$.
The task is to calculate $P(X+Z>Y)$.
Comment:
In previous excersices, by finding the joint density function of $X,Y$... |
H: Understanding $(n \cdot 1)(m \cdot 1) = (nm) \cdot 1$
I read that $(n \cdot 1)(m \cdot 1) = (nm) \cdot 1$ in a ring with unity $1$ because $\left(\underbrace{1+\cdots+1}_{n \textrm{ times}}\right)\left(\underbrace{1+\cdots+1}_{m \textrm{ times}}\right) = \underbrace{1+\cdots+1}_{nm \textrm{ times}}$ by the distribu... |
H: Why this conditional variation and expectation equality holds?
Assume $(X_i)_{i\ge 0}$ are random variables (not necessarily martingale) adapted to the filtration $(\mathcal{F}_i)_{i\ge 0}$. I found a statement that says
$$\text{Var}(X_{i+1}-X_i\mid\mathcal{F}_i)=E((X_{i+1}-X_i)^2\mid \mathcal{F}_i).$$
I am wonderi... |
H: How to find the intergral $I_{A}=\int_{0}^{2\pi}\frac{\sin^2{x}}{(1+A\cos{x})^2}dx$
Let $A\in (0,1)$be give real number ,find the closed form intergral
$$I_{A}=\int_{0}^{2\pi}\dfrac{\sin^2{x}}{(1+A\cos{x})^2}dx$$
This integral comes from a physical problem，following is my try:
since
$$I_{A}=\int_{0}^{2\pi}\dfrac{... |
H: Sequence of terms in a series is unbounded out side its region of convergence
Let $\sum_{n=0}^\infty a_nz^n$ be a series with radius of convergence $R$, then show that the sequence $\{a_nz_0^n\}$ is unbounded for $|z_0|>R$.
My thought: Can we take $|a_n|^\frac{1}{n}\geq \frac{1}{R} $ from Cauchy Hadamard formula? A... |
H: Boundary of union of open subsets
Let $X$ be a topological space and each $V_i \subset X$ be an open subset of $X$, where $i \in I$. Denote $V_I = \{V_i : i \in I\}$. Below I'll show that
$$(*) \quad \quad \partial \left(\bigcup V_I\right) = \overline{\bigcup \partial V_I} \setminus \bigcup V_I $$
provided
$X$ is ... |
H: Recurrent sequence convergence.
I'm trying to prove this but I can't, I tried to use an infinite integral for it but I can't do it.
Let $$f_o(x)$$ continuos in $$0\leq x\leq a$$ Show that the sequence of functions defined by $$f_n(x)-\int_{0}^{x}{x}f_{n-1}(t)dt$$ for $$n=1,2,3,\ldots$$ converges uniformly to $$f(x)... |
H: Unique Solution to 1st Order Autonomous ODE
Take the ODE $y'=F(y)$. Show it has a unique solution with initial condition $y(t_0) = y_0$ in a neighborhood of $t_0$ provided $F$ in continuous and $F(y_0) \neq 0$. I am trying to use the inverse function theorem by solving the ODE the inverse function satisfies but I a... |
H: Show that $\sum_{v=0}^N (-1)^v {N \choose v} \left(1- \frac{v}n\right)^r \to (1-e^{-p})^N$.
(Feller Volume 1, Q.13, p.61) Let
$$u(r,n) = \sum_{v=0}^N (-1)^v {N \choose v} \left(1- \frac{v}n\right)^r.$$
Show that if $n\to \infty$ and $r \to \infty$ so that $r/n \to p$, then $u(r,n) \to (1-e^{-p})^N$.
Althoug... |
H: Proving $\sum_{k=1}^\infty \rightarrow -\infty$ almost surely if $P(X_k=k^2)=\frac{1}{k^2}=p_k, P(X_k=-1)=1-p_k$.
Suppose $\{X_k\}_{k\geq 1}$ are independent with
$$P(X_k=k^2)=\frac{1}{k^2}=p_k, P(X_k=-1)=1-p_k.$$
Show $\sum_{k=1}^n X_k\rightarrow -\infty$ almost surely as $n\rightarrow \infty.$
I can see that each... |
H: Proving that in a graph with 400 vertices, each with a valency of 201, there exists a subgraph isomorphic to $K_3$
I need to prove that a graph $G$ with 400 vertices, each of a valency 201, has a subgraph isomorphic to $K_3$
As far as I understand, I need to prove that there exists a triangle within the graph $G$, ... |
H: How to prove only answer
The question is
Find all prime numbers, $p$ such that $16p+1$ gives a perfect cube.
By trial and error, I have found 307 to be a solution but how do I prove that it is the only solution (if it is)?
I've got that $16p+1$ can only give odd solutions that end in $7,9,1,2,3$ but that's about ... |
H: I want to prove that given a limit point, it is possible to find a sequence that converges to it
Given this definition, Prove carefully that if $E \subset X$ and if
$x$ is a limit point of $E$, then $\exists$ a sequence $(a_n)$ in $E$
that converges to $x$.
Proof: (attempt)
Let x be limit point of $E$. Let $r >... |
H: Solve $(1+i)^z = i$ where $z$ is a complex number.
How can I solve this equation?
$$(1+i)^z = i$$
The only way I can start is to use $e^{z\log(1+i)}$.
AI: Note that $1+i=\sqrt{2}\left(\frac{1+i}{\sqrt{2}}\right)=\sqrt{2}e^{\frac{i\pi}{4}}$. Now let $z=x+iy$. Then
\begin{align*}
(1+i)^z & =\left(\sqrt{2}e^{\frac{i\p... |
H: Newton method exchanging row
suppose to have a function $F(x,y,z) = [ f_1(x,y,z),f_2(x,y,z),f_3(x,y,z)]$ and that $f_1$ depend only by x, $f_2$ depends only by y and $f_3$ depends only by z.
Now if I apply newton method I can write $[d_x,d_y,d_z] = -J^{-1}(x,y,z)*F(x,y,z)$.
The question is if I exchange the rows ... |
H: Evaluating erf(x) using Taylor's series
I tried to evaluate error function using Taylor series by using its definition
$$ erf(z) = \frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}dt$$
I've used Taylor expansion to evaluate this integration and i got this
$$ \frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}dt = \frac{2}{\sqrt{\pi}}\int_0^... |
H: Behaviour of meromorphic functions near poles.
Let $F(z)$ be a meromorphic function on a domain $\Omega$ with a pole at $z=a .$ Let $C \subset \Omega$ be a simple closed contour with the point $z=a$ enclosed in its interior domain $I,$ and assume that $F(z)$ is continuous on $C .$ Prove that there exists a constant... |
H: Proving functions equality is reflexive, symmetric, and transitive
The reason I am posting this is that these seem too trivial and my "proofs" feel like I am doing nothing other than stating definitions, not even manipulating them.
Here is how functions equality is defined in the book:
Two functions $f : X → Y$ ,... |
H: Find matrix from given minimal polynomial
I have to find a $3$x$3$ matrix with the minimal polynomial: $x^2 -9$.
What I've tried:
If it would be a 2x2 matrix, then :
$$
C:=\begin{pmatrix}
0&9 \\ 1 & 0
\end{pmatrix}
$$
Then i tried adding columns and rows with zeros, but the minimal polynomial is now $x^3-9x$
$$
C:... |
H: Evaluation of limit using $1$ st principle
Express the following limits as $f'(c)$ for some function $f$ and some number $c$
$$\lim_{h\rightarrow 0}\frac{5(1+h)^{20}-6(1+h)^3+1}{h}$$
What i try
I have tried using D L Hopital rule
$$\lim_{h\rightarrow 0}\frac{100(1+h)^{19}-18(1+h)^2}{1}=82$$
But i did not understa... |
H: A three digit number was decreased by the sum of its digits .Then the same operation was carried out with the resulting number,et cetera
A three digit number was decreased by the sum of its digits .Then the same operation was carried out with the resulting number,et cetera ,100 times in all .Prove that the final nu... |
H: Is it enough if $g$ is injective for $g ◦ f$ to be injective?
This question came to mind when I got the following question:
Let $f : X → Y$ and $g : Y → Z$ be functions. Show that if $f$ and $g$ are both injective, then so is $g ◦ f$
But this got me wondering, isn't enough for just $g$ to be injective? We have $g... |
H: ${\lim_{x\rightarrow \infty}}(\sqrt{x^2+2x+3} - \sqrt{x^2+3})^{x}$
$${\lim_{x\rightarrow \infty}}(\sqrt{x^2+2x+3} - \sqrt{x^2+3})^{x}$$
I tried taking log both sides (on paper).
After taking log how do I proceed? You get $\infty$ * $(\infty-\infty$). But $\infty-\infty$ could be any number. How can I take this as ... |
H: Inequality for the expectation of a nonnegative random variable
Let $X$ be a nonnegative random variable with distribution $F$ and mean $\mu=E(X)>0$. Let $A_{\mu}=[\mu, \infty)$. Is it true that
$$
\int_{A_\mu} x dF(x) \geq \mu/2
$$
must then hold?
I'm trying to find counterexamples, both for continuous or discret... |
H: Why Bond(graph theory) is non-empty
According to Diestel, bond is defined as the minimal non-empty element of cut-space at page 25.
He explain the non-empty condition in the difinition of a bond bites only if G is disconnected.
What is the meaning of this?
AI: Recall the definition of bond :
A minimal non-empty c... |
H: Prove that a cycle of length $k\geq 2$ can be written as a product of $k-1$ transpositions.
Prove that a cycle of length $k\geq 2$ can be written as a product of $k-1$ transpositions as follows:
$$
(a_1 ... a_{k-1} a_{k})=(a_1 a_{k})(a_1 a_{k-1})...(a_1 a_2).$$
I found an answer here: Permutations as a product of t... |
H: How to prove that $f(x)=x+\frac{1}{x}$ is not cyclic?
Let $f(x)=x+\frac{1}{x}$ and define a cyclic function as one where $f(f(...f(x)...))=x$.
How do prove that $f(x)$ is not cyclic?
What I tried was to calculate the first composition:
$f(f(x))=x+\frac{1}{x}+\frac{1}{x+\frac{1}{x}}=\frac{x^4+3x^2+1}{x^3+x}$
Intui... |
H: Would the man in the white room have a 100% of crossing the line?
I had this thought in my head for a couple of months now and I really wanted to see what the answer to it is. So here it is:
A man is sitting in a white room, where all he has to do is cross over a purple line to travel back to Earth. However, he has... |
H: Does there exist an infinite set S that is closed under infinite unions but not finite unions?
Does there exist an infinite set $S$ of sets, such that for every infinite subset $I$ of $S$, $\bigcup I \in S$, but $S$ is not closed under finite unions?
AI: Let $$S=\{\,A\subseteq \Bbb N\mid 1\in A\,\}\cup \{\{2\},\{3\... |
H: Find all possible positive integers $x$ and $y$ such that the equation: $(x+y)(x-y)=\frac{(y+1)(y-1)}{24}$ is satisfied.
My approach so far:
The given equation can be rewritten as: $x^2 -y^2=\frac{y^2 -1}{24}.$ This gives
$24x^2 +1=25y^2=(5y)^2.$ So $(24x^2+1)$ must also be a perfect square. This implies $x=0, 1$ i... |
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