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H: Prove that if a function $f: X\to Y$ continuous then its graph is closed
The graph of $f$ is $G(f) = \{(x,f(x)) : x\in X\} \subseteq X\times Y$
$X$ and $Y$ are metric spaces.
a) Suppose $f$ is continuous and prove that $G(f)$ is a closed set.
b) Suppose that $G(f)$ is compact and prove that $f$ is continous
For a),... |
H: Structure constants for $\mathbb{F}_q$ as an $\mathbb{F}_p$-algebra
Let $k'=\mathbb{F}_q$ and $k=\mathbb{F}_p$, where $q=p^d$. We may regard $k'$ as a $d$-dimensional $k$-algebra via $k\hookrightarrow k'$. For any choice of $k$-basis $\{e_1,\ldots,e_d\}$ we obtain structure constants $a_{ij}^l$ giving the multipl... |
H: Is it possible to remove a variable from an expression?
Given the Expression:
$$\dfrac{N(N+1)(2N+1)}{6} + \dfrac{6N}{6}− \dfrac{3N(N+1)}{6}$$
Is it possible to remove N from the third expression to give -3 / 6, giving a final expression of:
$$\dfrac{N(N+1)(2N+1-3)+6N}{6}$$
If so, what is this called. And how can I ... |
H: Solving a PDE - do you have an idea?
Do you have an idea how to solve
$$
v_{\xi\eta}=\frac{1}{2} v_{\xi}\cdot\xi?
$$
First I thought of using
$$
v_{\xi\eta}=v_{\eta\xi},
$$
substituting $z:=v_{\xi}$ and then getting
$$
\frac{dz}{d\eta}=\frac{1}{2}z\cdot\xi.
$$
First I thought that I can solve this by separa... |
H: Showing that $A(p)\circ A(q)=p \circ q$ for every $p,q\in \mathbb{R}^3$
Let $\{x_A,y_A,z_A\}$ be an orthonormal set of vectors in $\mathbb{R}^3$. We call the coordinate system by these vectors $A$. Let $v\in \mathbb{R}^3$. We define $ A(v)$ to be the unique vector with components $a,b,c$ such that $v=ax_A+by_A+cz_A... |
H: Proving Wilson's theorem for $n=11$
Wilson's theorem establishes that if a $n$ number is prime then:
\begin{align}
(n-1)! &\equiv -1\ \textrm{mod}\ (n)
\end{align}
I have probed the theorem for the particular case where $n = 7$ like this:
I first consider the set $\{2,..,n-2\}$, in my case, $\{2,3,4,5\}$ and then I... |
H: "Nested" recursion preserves primitive recursive functions
Problem: Assume the functions $f$, $\pi$, and $g$ are given. They take one, two, and three arguments respectively. Prove a unique function $h$ exists such that:
$$h(0,y)\cong f(y)$$
$$h(x+1,y)\cong g(x,y,h(x,\pi(x,y)))$$
Furthermore, prove that if the funct... |
H: Integral of quartic function in denominator
I'm sorry, I've really tried to use MathJaX but I can't get integrals to work properly.
indefinite integral
$$\int {x\over x^4 +x^2 +1}$$
I set it up to equal
$$x\int {x\over x^4 +x^2 +1} - \int {x\over x^4 +x^2 +1}$$
$$\text{so } (x-1)\int {1\over x^4 +x^2 +1}$$
OKAY, no... |
H: Trigonometric problem in triangles.
I need your help. I'm studying physics, but I have a trigonometric problem. I attached a figure where depicts the angles and the unknown $x$. The idea that I want to understand is how to express $x$ in terms of $m$, $n$, $a$ and $b$. Because the solution is $x=m\cdot\sin a + n\cd... |
H: From $ \sum^\infty_{\lfloor \log n \rfloor + 1}n/{2^r} $ to $ \sum^\infty_{r=0}1/2^r $?
$$
E[h] = E[\sum^\infty_{r=1}I_r]
= \sum^\infty_{r=1}E[I_r]
$$
$$
= \sum^{ \lfloor \log n \rfloor}_{r=1}E[I_r] + \sum^\infty_{\lfloor \log n \rfloor + 1}E[I_r]
$$
$$
\leq
\sum^{ \lfloor \log n \rfloor}_{r=1}1 + \sum^\infty_{\lfl... |
H: Prove that if $A$ is compact and $B$ is closed and $A\cap B = \emptyset$ then $\text{dist}(A,B) > 0$
Let $X$ be a metric space. For nonempty subsets $A,B\subseteq X$. Define $\text{dist}(A,B) := \inf\{d(x,y) : x\in A, y\in B\}$
a) Prove that if $A$ is compact and $B$ is closed and $A\cap B = \emptyset$ then $\text{... |
H: nitpicking the definition of a polynomial function
A textbook I'm reading says that $f(x)=0$ is NOT a polynomial function, yet $g(x)=8$ IS a polynomial function since $g(x)=8=8x^0$ which satisfies the non-negative integer degree requirement. Yet, it's still a monomial! Or, can it be considered $g(x)=8=8x^0=8x^0+0... |
H: Uniform Convergence of Sequence of Functions Doubt
For the sequence of functions $$f_n={nx\over 1+n^3x^3},x\in[0,1]$$ If we wish to check the Uniform Convergence. I tried doing it in the following two ways
Method 1 :
$\lim f_n=0=f(x)$ , for a given $\epsilon>0$
$|f_n(x)-f(x)|=|{nx\over 1+n^3x^3}-0|={nx\over 1+n^3x... |
H: Nonzero subspace that is invariant under any operator cannot be proper?
This is not for homework, and I would just a like a hint please. The question asks
Prove or give a counterexample: If $U$ is a subspace of $V$ that is invariant under every operator on $V$, then $U = \{ 0 \}$ or $U = V$.
In the question, $V$... |
H: Is it a Transitive Set?
Is $\{\{0\},\{\{0\}\}\}$ a transitive set? Or only $\{0,\{0\},\{\{0\}\}\}$ is transitive?
if the first isnt a transitive set, can someone give me an example of a transitive set which does not contain urelements?
Thanks!
AI: We have $0\in\{0\}$ and $\{0\}\in\{\{0\},\{\{0\}\}\}$, but $0\notin\... |
H: Verify that an ellipse has four vertices.
Verify that an ellipse has four vertices.
The ellipse is given by $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$
Please can someone help me how to verify this?
AI: Hint:
Consider the values $t=\frac {k\pi}2$ for $k\... |
H: Deciding whether a subset of a regular language is regular
So let's have $2$ regular languages $R_1$ and $R_2$. Then we have language $L$ where $R_1 \subseteq L \subseteq R_2$. Decide whether $L$ is always regular language or not.
So my approach was this. Let's have
$$R_1=\{a^mb^l \;|\;m,l\ge0\}$$
$$L=\{a^nb^n\;|\;... |
H: Prove that $\mathbb{F}_5[X]/(X^2+3)$ is a field with 25 elements
Prove that the ring shown above is a field containing 25 elements.
Research effort:
$$\mathbb{F}_5[X]/(X^2+3)\cong (\mathbb{Z}[X]/5\mathbb{Z}[X])/ \overline{(X^2+3)} \cong \mathbb{Z}[X]/(5,X^2+3)$$
Modifying the term $3$, I tried to find $k$ such that... |
H: Solving a recurrence relation using repeated substitution
So, basically I am having a big issue with this recurrence relationship:
$$T(n) = T(n-1)+n, T(1) = 0$$
using repeated substitution I get down to:
$$i=1, T(n-1) + n$$
$$i=2, T(n-2) + 2n - 2$$
$$i=3, T(n-3) + 3n - 3$$
$$T(n) = T(n-i) + i*n - i$$
Base case: $n-... |
H: Problem of color painting on a circle.
Is it possible to color each point on a circle either red, yellow, or blue in such a way that no three points of the same color or totally different colors from the vertices of an isosceles triangle?
I think we must be able to find at least one of such an isosceles triangle. M... |
H: Find the next four largest 4-combinations
Find the next four largest 4-combinations of the set 1,2,3,4,5,6,7,8 after 1,2,3,5.
Not sure how to do this, need some help~!
AI: If you want the next ones, increment the last digit as far as you can go, then go back and increment the next to last, bringing the last up as c... |
H: Let $I$ be a prime ideal. Show that $f^{-1}(I)$ is a prime ideal of $R$. Is this also true for maximal ideals?
I'm solving some exercises to prepare for my ring theory exam.
Let $f:R→R'$ be a ring homomorphism, with $f(1)=1$, and $R,R'$
commutative rings with $1$. Let $I$ be a prime ideal. Show that
$f^{-1}(I)... |
H: For any given set of 13 distinct real numbers, prove we can always find two numbers $x$ and $y$ that $0<\frac{x-y}{1+xy}\leq 2-\sqrt{3}$.
For any given set of 13 distinct real numbers, prove we can always find two numbers x and y that $0<\dfrac{x-y}{1+xy}\leq 2-\sqrt{3}$.
I knew we can always make $0<\dfrac{x-y}{1+... |
H: Finite field extension
Let $K\supset F$ be a finite field extension of degree 2 and the characteristic of $F$ is not 2. Show that there exists an isomorphism of rings $\sigma:K\to K$ such that
$$F=\{x\in K:\sigma(x)=x\}.$$
In this case, I didn't see why we need the condition char $F\neq 2$. The extension has degree... |
H: How does this factor to 6(1-ln6)
-6(ln6)+6
I got....
6(-1(ln6)
shouldn't the 1 be negative not the ln6
AI: You forgot the part of the expression when we add $6$; when you factor out $6$ from each term, you're left with:
$$-6\cdot (\ln (6)) + 6 = 6\cdot\Big ((-1)\ln(6) + 1\Big) = 6\cdot \Big(-\ln(6) + 1\Big) = 6\cd... |
H: Poisson Process: probability
Find the chance that the fifth waiting time of the length at least T for an arrival, occurs at the time of tenth arrival.
{N(t), t>0} with rate lampda
AI: The probability that the waiting time between two arrivals is $\ge T$ is $e^{-\lambda T}$. Call this number $p$.
We want the probab... |
H: Matrix representation of a co-domain restriction of a linear operator
Consider the finite-dimensional linear operator:
$\mathcal{A}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3},$ with $Ax=y,$
$A=\left[\begin{array}{ccc}
1 & 0 & 1\\
1 & -2 & -1\\
0 & 1 & 1
\end{array}\right].$
Let $\mathcal{A}_{1}:\mathbb{R}^{3}\rightarr... |
H: Calculus position/velocity question.
Disclaimer: I am not a student trying to get free internet homework help. I am an adult who is learning Calculus. I am deeply grateful to the members of this community for their time.
Here is the question (Not sure of the best way to format this)
$$s(t)=t^3-9t^2+24t-6$$
Q: Th... |
H: Limit as $x$ approaches $0$ with constant $a$
Find the limit where $a$ is a constant
$$ \lim_{x \to 0}\frac{\left [ \cos(a+x)-\cos(a-x) \right ]^2}{\tan^2(3x)} $$
I don't know what to do. At first I thought I could replace $a$ with an arbitrary number and then solve the limit but then I got stuck as to how to use ... |
H: Partial derivative with respect to a function?
I have $f(x, y)$ is a class $C^2$ function where $x = u + v$, and $y = u - v$
How do I get $$\frac{\partial^2 f}{\partial x^2}$$
Well before we even talk about that one, I don't even know how to get the first partial. I'm confused about how to go about solving it when ... |
H: How to solve for $n$?
e.g.
$$\int_1^n \frac{1}{x} dx = \pi$$
I want to know the method used to solve it for any constant $c$ of any integral.
AI: We have $\int_1^n \frac{1}{x} dx = \ln (n) = \pi$ therefore $e^{\ln n} = e^{\pi} = n.$ |
H: Meta Theory when studying Set Theory
What exactly serves as the meta theory in the study of set theory (for example in Kunen's text). It seems to involve a certain amount of number theory and a certain concept of what it means to be finite.
Also: We usually use $\bigcup_{\alpha\in{ON}}R_{\alpha}$ as our set theort... |
H: How to obtain $y$
The question was written with dark-blue pen. And I tried to solve this question. I obtained $x$ as it is below. But I cannot obtain $y$ Please show me how to do this.
By the way, $\gamma (t)$ may not be clearly readable. So, I wrote again.
$$\gamma (t)=( \cos ^2 (t)-1/2, \sin(t)\cos (t), \sin... |
H: Stuck because of possible error in exercise of "How to Prove It" by Velleman
I am self-studying "How to Prove It" by Velleman, and I believe there must be an error on exercise 3.3 #14.
I'll show the question here, and where I think the error is, and then I'd love to find out if you believe I'm correct that there is... |
H: How to calculate the minimal and maximal distance between these two objects?
Suppose that an object $\mathcal{O}$ travels on the $xy$ plane following a path with respect to the time $t$ of the form $\mathcal{O}(t)=(2\cos(t), 2\sin (t))$ and another object $\tilde{\mathcal{O}}$ at the same time follows the path $\ti... |
H: Prove that $I= \{a+bi \in ℤ[i] : a≡b \pmod{2}\}$ is an maximal ideal of $ℤ[i]$.
I'm making some exercises to prepare for my ring theory exam:
Prove that $I= \{a+bi \in ℤ[i] : a≡b \pmod{2}\}$ is an maximal ideal
of $ℤ[i]$.
I know that $a+2l=b$ with $l\in ℤ$ (or should I say $l \in ℤ[i]$)? Therefore I can write $... |
H: How can I optimize this? Finding someone using several factors
I have 100 students, and they all need colored pencils.
Each of them needs the same colors of pencils, however, they can have different shades of the color.
What's the least amount of color and shade combinations I need to purchase so that each student ... |
H: Prove the limit related to a recurrence
For the following sequence $\{a_n\}_{n=1}^{\infty}$, we define $a_1=\alpha\in(0,1)$, and for any $n\geq 2$, $a_{n+1}=a_n(1-a_n)$.
Prove: $\lim_{n\rightarrow\infty}{na_n} = 1$.
AI: Hint
Prove that $(a_n)$ is decreasing sequence bounded below by $0$ so it's convergent to $0$
P... |
H: Largest box fitting inside an ellipsoid
Find the volume of the largest box with sides parallel to the $xy$, $xz$, and $yz$ planes that can fit inside the ellipsoid $(x/a)^2 + (y/b)^2 + (z/c)^2 = 1$.
My answer: We want to maximize $f(x,y,z) = xyz$ subject to $(x/a)^2 + (y/b)^2 + (z/c)^2 \leq 1$. So $\nabla f $ paral... |
H: Binary representation of 2-adic integers
I would like some examples of the binary representation of 2-adic integers that are not standard integers. What is the 2-adic expansion of $1/3$? Of $-1/3$? What number does $...010101$ represent?
AI: To answer your last question first, $$...010101 = 1 + 4 + 16 + \cdots + 4... |
H: Compact Subsets of $C[a,b]$
Consider the set $G = \lbrace f \in C\left[a,b\right] : |f(x)| \le |g(x)|,\ \forall x \in [a,b] \rbrace$
Find all values of $g$'s for which $G$ is a compact subset of $C[a,b]$ with the max norm.
Attempts:
I understand the fact that it suffices to just show that the set $G$ is closed bec... |
H: Reference for Deligne-Mumford
What is a good reference for someone new to the theory of Deligne-Mumford stacks, other than the original Deligne-Mumford paper itself? The paper itself seems readable with some effort; but the fear is that the reader will miss out on whatever happened in the later years. Is there a go... |
H: Kind of a silly question regarding compact operators
If $T \in \mathcal{B}(X)$ is a compact operator, how about $-T$, i.e. it's additive inverse? Does this notation even make any sense with regard to the minus sign? Is it trivial?
Further, suppose we have another compact operator $S \in \mathcal{B}(X)$. I know that... |
H: Prove that a function is irreducible
Let $F$ be a field. Let $\varphi : F[x] \rightarrow F[x]$ be an isomorphism such that $\varphi(a)=a$ for every $a \in F$.
Prove that $f \in F[x]$ is irreducible if and only if $\varphi(f)$ is.
How will I be able to start this proof? Any help will be greatly appreciated. Thanks ... |
H: Improper integral divergence
Use the graph of 1/x and the sum of areas of rectangles to show that $\int _{ 1 }^{ \infty }{ \frac { 1 }{ x } dx }$ = +$\infty$.
Would the sum of rectangles just be:
1 + 1/2 + 1/3 + 1/4 +....+1/n + = +$\infty$.
AI: You will need to draw a picture. Then maybe use:
First rectangle: ba... |
H: Let $V$ be $n$ dimensional real vector space. Show that of $T[V]=\ker(T)$, then $n$ is even
Let $V$ be an $n$-dimensional real vector space and $T:V\rightarrow V$ a linear transformation from $V$ to itself. Suppose that $T[V]=\ker(T)$. Show that $n$ is even.
I'm really lost. Since $\operatorname{Im}[T]=\ker(T)$, do... |
H: Find the volume of the solid obtained by rotating the region bounded by the curves $y=x^2$, $x=4$ and $y=0$ about the $x$-axis
I do not understand how they want the volume between $y=x^2$ and $y=0$? I don't understand how to do the problem. Please help. I have other similar homework problems and would like to LEARN... |
H: Differentiation/Integration continuous function (metric spaces)
I have the following two questions:
Is differentiation, $f(x) \mapsto f'(x)$ a continuous function from
$C^1[a,b] \longrightarrow C[a,b]$
and
Is integration, $f(x) \mapsto \int_a^x f(t) \ dt$, a continuous
function from $C[a,b] \longrightarrow C[... |
H: Decomposition of $S^2(\wedge^2 E)$
Consider bianchi map $$ b(T)(x,y,z,t) = \frac{1}{3}(T(x,y,z,t)+T(y,z,x,t) + T(z,x,y,t))$$ where $T\in S^2(\wedge^2 E)$
I already checked that $b(b(T))=b(T)\in S^2(\wedge^2 E)$
But how can we derive the following ?
$$ S^2(\wedge^2 E) = {\rm Ker}\ b \oplus {\rm Im}\ b$$ and
$${\rm... |
H: Is sum of tail probability always less than integral of tail probability?
I'm working through some Probability and Measure Theory, and frequently we have been using the fact that for $X_i$ iid
$\sum\limits_{k = 1}^{\infty} P(|X_1| > k) \leq \int\limits_0^{\infty}P(|X_1| >t)dt$
Intuitively this makes sense but othe... |
H: What's wrong with my answer? $\int \frac{3x^2-2}{x^2-4x-12} \mathrm dx$
Here is the original problem: $\int \frac{3x^2-2}{x^2-4x-12}\ \mathrm dx$
After doing polynomial division and factoring the denominator I got this:
$$\int 3 + \frac{12x+36}{(x-6)(x+2)}\ \mathrm dx$$
Then using partial fraction decomposition I g... |
H: Inner Product Space on linear transformation on itself
So $V$ is an inner product space and $T : V \to V$ is a linear map such that $$||T(v)|| = ||v||$$ for all $v \in V$. Prove that $$\langle T(v), T(w)\rangle = \langle v, w\rangle$$ for all $v,w \in V$.
However, I missed the class where they talked about this, an... |
H: $S_n = \sum_{k=1}^n \frac{(-1)^{k+1}}{k !}$ for each $S_n$ in terms of $n$?
How can we compute
$S_n = \sum_{k=1}^n \frac{(-1)^{k+1}}{k !}$ for each $S_n$ in terms of $n$?
I tried to compute 2 items together but that didn't work for me. I also tried to find a recurrence relation between $n$ and $n-1$ but that also ... |
H: question on Brownian Motion stopping time and end state
I came across this equation in my lecture notes, which states:
$P(T_a < t , W_t \ge a) = P(W_t \ge a)$ where $T_a = \min\{t \ge 0, W_t \ge a\}$.
I'm really confused by this equation: as far as I understand it, the R.H.S $P(W_t \ge a)$ is the probability that ... |
H: How do I prove that the limit as one function goes to infinity is equal to another function?
I was playing around with the integral $\int x^ndx$ and noticed that it is always $\frac{x^{n-1}}{n-1}+c$ except for the singular case $n=-1$. So I could pick $n$ arbitrarily close to $-1$, and the formula works, but as soo... |
H: Boundary of a set
Can boundary of any subset of $\mathbb R$ under the usual topology be $[0,1]$? I think it is not possible. If $\partial A=[0,1]$, then $[0,1]\subset \overline{A}$ and $[0,1]\cap A^{0}=\emptyset$. Then what to do? Please give some hint.
AI: Well, the boundary of a dense set with empty interior is i... |
H: Factoring with 5 terms
I'm doing some factoring and have arrived at the point where the book says that:
$$(k+1)/30(6k^4 + 39k^3 + 91k^2 + 89k + 30)$$
factors to:
$$(k+1)/30(k+2)(2k+3)(3(k+1)^2+3(k+1)-1)$$
I cannot derive the latter using the former. Can someone help? This must be a tough cookie to crack because eve... |
H: Cylindrical Shell Volumes Problem
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by $y=3+2x−x^2$ and $x+y=3$ about the y-axis.
I have already turned $x+y=3$ into $y=3-x$. However I don't know what to do with the polynomial to continue into graphing them and using th... |
H: Limit of a multivariable function approaching to $(0,0)$
So I have
$$\lim_{(x,y) \to (0,0)}\frac{x\sin(xy)}{y}$$
And I was wondering if I'm allowed to do this:
$$= \lim_{x \to 0} x * \lim_{(x,y)\to(0,0)} \frac{\sin(xy)}{y} \\=0 * \lim_{(x,y)\to(0,0)} \frac{\sin(xy)}{y} \\ = 0$$
AI: \begin{align*}\lim_{(x,y) \to (0,... |
H: Showing a set of functions $F$ is bounded
I have a set of functions given by;
$$F = \{f:[0,1]\rightarrow\mathbb{R}|\int_0^1 f(x)dx = 0, |f(x)-f(y)|\leq|x-y|, x,y\in[0,1]\}.$$
I have a solution for the question so my questions are about the solution.
Let $f\in F \implies \int_0^1 f(x)dx = 0$.
$\implies$ there exist... |
H: Integrate $\int \cos x\cos^5(\sin x)dx$
I set $u$ to $\sin x, du$ to $\cos x$, then had $\cos^5udu$, then replaced $(\cos^2u)^2$ with $(1-\sin^2u)^2$ to get $\cos u(1-\sin^2u)^2$, then set $w$ to $\sin u$ and $dw$ to $\cos u$ and got $(1-w^2)^2dw$ then integrated to get $w- 2/3w + w^5/5$ then plugged back in $\sin ... |
H: Detailed proof that no essential singularity at infinity implies polynomial
Suppose $f(z)$ is holomorphic in the whole plane, and that $f(z)$ does not have an essential singularity at $\infty$. Prove that $f(z)$ is a polynomial.
I've tried following the hint given in this question. Since $f(z)$ has a nonessential... |
H: sketching the graph of functions
Good morning!
How to draw the graph of the following function and discuss the increasing and decreasing intervals of y.
$y = $x$ + \sqrt[]{x^2}$
Also, I would like to know free online graphing calculator
Thanks in advance.
AI: Hint: Recall that $\sqrt{x^2} = |x|$, so
$$f(x) = x + ... |
H: Is $\max(f)$ well defined if $f$ is finite?
Say I have a function that's finite almost everywhere within an interval $[a,b]$. Does that mean that it has an upper bound if I ignore those points on which it is infinite? i.e. is:
$$\text{max}_{[a,b]}(f)$$
well defined?
AI: No, for one of two possible reasons:
1) The ... |
H: How should I grade my students?
Anyone who is interested in my experience as a grader in the past can check this thread.
This semester I was assigned a grader for a certain class, which runs two sections. There are two professors, Professor $A$ and Professor $B$. Both of them using Professor $A$'s personal written... |
H: combinations possible of distributing 8 apples to 4 people
I saw a problem where you need to distribute 8 apples to 4 different people where there is also a possibility of one or more of them getting 0 apples too. How do you compute in combinatorics the number of total combinations. The answer is given as 165. I am... |
H: Find a closed expression for the sum of the entries of the Pascal triangle inside the upper n x n rhombus.
Find a closed expression for the sum of the entries of
the Pascal triangle inside the upper $n \times n$ rhombus.
For example, for $n = 3$, you need to sum the entries:
\begin{array}{cccccccc}
&&&&1\\
&&&1&&1... |
H: Can a complex function be complex-differentiable at a point and not in a neighborhood?
Is it possible for a function $f:\mathbb{C} \to \mathbb{C}$ to be complex-differentiable at a point $z_0\in \mathbb{C}$ without being analytic in a neighborhood of $z_0$? How can we prove this?
AI: Yes; try $f(z)=|z|^2$ ; then Ca... |
H: The Hahn-Banach theorem for Hilbert spaces follows from Riesz's theorem
How does the Hahn-Banach theorem for Hilbert spaces follow from Riesz's representation theorem?
AI: Yes. Given a subspace $M$ of a Hilbert space $H$, and a continuous linear functional $f : M \to \mathbb{C}$, you can extend $f$ to a unique cont... |
H: A particular version of Gronwall's inequality
We have this theorem (Gronwall's inequality):
Let $f$, $g$ and $h$ be continuous nonnegative functions defined for $t\ge t_0$. If$$f(t)\le h(t)+\int_{t_0}^{t}g(s)f(s)\,ds\>,$$then$$f(t)\le h(t)+\int_{t_0}^{t}g(s)h(s)e^{\int_s^tg(u)\,du}\,ds$$
How do I prove another ver... |
H: Generating all solutions for a negative Pell equation
How to get all solutions for a negative Pell equation? For example, the equation $x^2 - 2 y^2 = -1$ has two solutions - $(7, 5)$ and $(41, 29)$, and the $(7, 5)$ is the fundamental one, right? How to get the $(41, 29)$ solution from the fundamental one?
AI: The ... |
H: integrate $(t\sin^2(t))dt$... why is my answer coming out wrong?
I tried setting $u = t, dv = \sin^2t\mathrm{d}t$, then converting $dv$ to $1/2(1-\cos 2t)\mathrm{d}t$
with half angle identity,
Then $du = 1dt$ and $v = 1/2 - 1/4(\sin2t)$, then $$t\sin^2tdt = t(1/2 - 1/4 \sin2t) - (1/2 - 1/4 \sin2t)dt$$ then after ... |
H: Why is the $0$th power mean defined to be the geometric mean?
Mentioned in the wikipedia article, the $0$th power mean is defined to be the geometric mean. Why is this? I understand that a convenient consequence is that the means are ordered by their exponent. But is there an intuitive reason why the $0$th mean sho... |
H: Vector space and vector subspace
We know that a vector space V is a set of vectors on which two operations are defined, sum, which satisfies the following properties:
\begin{equation}
u+v=v+u
\end{equation}
\begin{equation}
u+(v+w)=(v+u)+w
\end{equation}
\begin{equation}
u+0=0+u=0
\end{equation}
\begin{equation}
u+... |
H: A problem of minimizing distance.
A power house, P, is on one bank of a straight river $200$ m wide,
and a factory, F, is on the opposite bank $400m$ downstream from P.
The cable has to be taken across the river,
under water at a cost of Rs $6/m$.
On land the cost is Rs $ 3/m$.
What path should be chosen so tha... |
H: distribution of chocolate problem?
I have 6 packet where , 10 20 30 35 50 60 are chocolate in that 6 packet.I have to distribute one packet to k children. where k=3 .Then unfair is calculated by
find=min(|sum(Xi-Xj)|) where Xi and Xj are set of number chocolate .ex:-
10 20 30 we get unfair distribution as
|30-20|... |
H: Is there any weighted graph which smallest eigenvalue of its adjacency matrix is greater than 1?
Is there any weighted or unweighted graph which smallest eigenvalue of its adjacency matrix is greater than 1?
AI: If the graph has no loops, then the answer is "no" since the sum of the eigenvalues is equal to the trac... |
H: Using counting formula to get |G| = |kernel φ||image φ|
The counting formula I am saying :
Let S be a finite set on which a group G operates, and let Gs and Os be the stabilizer and orbit of an element s of S. Then
|G|=|Gs||Os|
or
(order of G)=(order of stabilizer)(order of orbit)
then I have th... |
H: Finding the characteristic polynomial of a linear transformation
Let $T: M_n(\Bbb{R}) \to M_n(\Bbb{R})$ be a linear transformation defined by $T(A)=A^t+A$. What is the characteristic polynomial of $T$?
If I use the basis $E_{ij}$, I get $T(E_{ij})=E_{ij}+E_{ji}$, but don't know how to compute the characteristic po... |
H: Calculate unique Integer representing a pair of integers
I have a pair of positive integers $(x, y)$, with $(x, y)$ being different from $(y, x)$, and I'd like to calculate an integer "key" representing them in order that for a unique $(x, y)$ there is an unique key and no other pair $(w, z)$ could generate the sam... |
H: How to evaluate this using algebra?
We have $$1.001^6 - 1.001^5$$
How do I evaluate this? Normally I would use algebra to rewrite it, but I don't know how to cleverly rewrite $a^6 - a^5$ to a simpler form.
AI: How about (let $x = 0.001$ for simpler writing)
\begin{align*}
(1+x)^6 - (1+x)^5 &= (1+x)^5 \cdot (1+x -... |
H: Nonnegative measurable function on measure zero set
Let $f$ be a nonnegative measurable function from $X$ to $R$.
Then, for $\mu$$(A)=0$, $\int_A f d\mu$=0?
I think by using increasing simple function to $f$ and Monotone convergence theorem,
it's true but i'm not sure...
AI: Let $s=\sum_{i=1}^na_i\chi_{A_i}$ be a s... |
H: Fact about PSD matrices
I'm reading something where $C$ is a positive semi-definite matrix. Also, the centring matrix is $H = (I - (1/n)11^{T})$.
My notes say that $C$ is p.s.d. $\implies HCH$ is p.s.d. I'm just not seeing seeing why.
AI: A matrix $C$ is called positive semidefinite if $y^\ast Cy\ge0$ for all vecto... |
H: Unbounded function with uniform bounds on Integrals
I'm looking for examples of unbounded functions $f\colon\Bbb R\rightarrow\Bbb R$ with the property that for all $L$ exists some $C(L)$ with
$$\int_x^{x+L} f(t)dt \le C$$
for all $x$.
AI: We can choose $f$ non-negative, continuous, integrable and unbounded, for ex... |
H: Area between curve $x^n$ and other lines
Find the area between the curve $y = |x^n|$, where $n$ is a positive constant, the line defined by $y=-2$, and the lines defined by $|x| = 2$.
I have answered this question, but I don't know if my answer is correct, so that's why I'm posting it here.
We want $\displaystyl... |
H: A notation problem of partial derivatives
I have a notational problem of partial derivatives:
Let $z=f(x,\phi(x))$,and let $\phi(x),f$ be a differentiable functions.What is the difference between $\frac{\partial z}{\partial x}$ and $\frac{\partial f}{\partial x}$?In my eye,they ought to have no difference,but in my... |
H: Probability with average of measures
I am trying to solve Problem 2 from this problem set.
Let $\mathbf{Y}$ be the avreage of $5$ independent measurements.
For a single measurement one have $\sigma^2 = 0.060^2$ and $\mu = 6.8$.
$\textbf{b})$ What is the probability that $\mathbf{Y}$ deviates more than $0.06$ fr... |
H: Proving sequential compactness from open cover compactness.
Let $(\mathcal M,d)$ be a metric space and $A\subset\mathcal M$. The following types of compactness are equivalent:
(i) Each open cover of $A$ contains a finite subcover.
(ii) $A$ is sequentially compact, i.e. each sequence in $A$ contains a subsequence wh... |
H: Sum of sines $\sum_{k=0}^{n} \sin(\phi +k\alpha)$
I've got the following problem. I'd like to prove that
$$\sum_{k=0}^{n} \sin(\phi +k\alpha) = \frac{\sin\left(\frac{n+1}{2}\right)\alpha + \sin\left(\phi + \frac{n\alpha}{2}\right)}{\sin\frac{\alpha}{2}}$$ Can you help me? I tried writing sines using complex numbers... |
H: Isomorphism between partially ordered sets - What is wrong with my argument?
I'm trying to show
Isomorphism between two partially ordered sets is an equivalence relation.
Suppose $M$ and $M^{\prime}$ are two partially ordered set and $f:M\to M^{\prime}$ is isomorphism between them. To show reflexivity, let $a\in... |
H: Which is bigger $x^{log(x)}$ OR $(log(x))^x$
I'm trying to find out which is bigger $x^{log(x)}$ OR $(log(x))^x$
As $x \to \infty$
I tried to take the log of both but I didnt't reach any where.
AI: You can try to take $\lim\limits_{x\rightarrow +\infty}x^{\log(x)}/\log(x)^x$. To do that, see that:
$$\begin{aligned}... |
H: $2$-dimensional Hyperbolic space with fundamental group ${\bf Z}$ and constant curvature $-1$
$$ d\rho^2 + \cosh^2\rho\ d\theta^2$$
Only one ? Is there any other example ?
AI: You have essentially two choices: either the quotient of the hyperbolic plane by (the cyclic group generated by) a hyperbolic isometry or t... |
H: Solve recurrence relation $ a_{n+1} = (n+1)a_n + 1 $
$a_0 = 1 \\ a_{n+1} = (n+1)a_n + 1 $
Could you help me solve this?
And maybe someone know good source explaining how to solve recurrence relations?
AI: as @Did said in the comments we can put
$$ n!b_n = a_n \Rightarrow (n+1)!b_{n+1} = (n+1)!b_n + 1 $$
$$ \Right... |
H: How to guess that $ \sum_{i=1}^{n}3^i = \frac{3}{2}(3^n - 1)$
As in title how do you guess that $ \sum_{i=1}^{n}3^i = \frac{3}{2}(3^n - 1)$?
I have homework about solving recurrence relations and using iterate method I can find that http://www.wolframalpha.com/input/?i=RSolve%5B%7Ba%5Bn%5D+%3D+a%5Bn-1%5D+%2B+3%5En%... |
H: Continuous irrational function proof
Let $f,g$ be two defined and continuous function on $[a,b]$ suppose that $f(x)=g(x)$ for all rational $x∈[a,b]$. Prove that that $f(x)=g(x)$ for all real $x∈[a,b]$
Since this is true for rational I know that I need to prove it also true for irrational, but I don't know how to do... |
H: Flat space Minkowski metric
I am having some problem understanding the why in Minkowski spacetime, the continuity equation is written as $$\partial_\mu J^\mu=0.....................(*)$$
Physically, I know that $$\partial_t \rho=-\nabla\cdot\vec j$$
but isn't the Minkowski metric diag$\{1,-1,-1,-1\}$, such that $(*)... |
H: $\prod_{k=1}^n (1+ \frac{z}{k})$ converges to $0$ when $\Re (z)<0$
We have a complex number $z$ such that $\Re (z)<0$ and the sequence $ z_n = \prod_{k=1}^n (1+ \frac{z}{k})$. Prove that $\lim_{n \rightarrow \infty} z_n = 0$.
How to do it? I guess it will be easier to prove that $\lim_{n \rightarrow \infty} |z_n... |
H: Working out modulo without a calculator
How does one determine modulo without a calculator in cases like this:
$$15^7 - 13^5(\mod14)$$
Normally I would simply divide what is given by the modulo number and take the decimal output and times it by the modulo number. How can I work out $15^7 - 13^5(mod14)$ without the... |
H: Integrating $\int_\gamma \frac{\cosh z}{2 \ln 2-z} dz$
I need to solve the following integral using Cauchy Integral Formula:
$$\int_\gamma \frac{\cosh z}{2 \ln 2-z} dz$$
with $\gamma$ defined as:
$|z|=1$
$|z|=2$
With $|z|=2$ I've solved already, as it is quite easy. All one needs to do is re-write the denominator... |
H: If $k^2-1$ is divisible by $8$, how can we show that $k^4-1$ is divisible by $16$?
All is in the title:
If $k^2-1$ is divisible by $8$, how can we show that $k^4-1$ is divisible by $16$?
I can't conclude from the fact that $k^2 - 1$ is divisible by $8$, that then $k^4-1$ is divisible by $16$.
AI: Hint: $$k^4 - 1... |
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