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H: $(n-1)$-th derivative of a complex polynomial
I cannot wrap my head around the $(n-1)$-th derivative of the polynomial $(z-2)^{n+1}$.
$$ \frac{d^{n-1}}{dz^{n-1}}(z-2)^{n+1} = \frac{(n+1)!}{2}(z-2)^2, \quad z \in \mathbb C.$$
I get why the term $(z-2)^2$ is there, the problem is with $(n+1)!/2$. Why divide by $2$?... |
H: If sets $A, B$ in Euclidean space are closed sets, they have the same boundary and their interior's intersection is non-empty, can we say $A=B$?
If sets $A, B$ in Euclidean space are closed sets, they have the same boundary and their interiors intersection is non-empty, can we say $A=B$? Any suggestions and comment... |
H: Evaluation of a limit.
Find $\lim_{n\to \infty}\frac{1}{2^n} \Bigg\{ \dfrac{1}{\sqrt{1-\frac{1}{2^n}}}+\dfrac{1}{\sqrt{1-\frac{2}{2^n}}}+\cdots+\dfrac{1}{\sqrt{1-\frac{2^n-1}{2^n}}}\Bigg\}$
Evaluation of this limit using integration as the limit of a sum doesn't work here. Is there any other way of doing this probl... |
H: Extreme points of a function at domain ends
Consider the following function: $f(x) = x\sqrt{9-x^2}$
$\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad\quad \quad $
$f'(x) = \frac{-2x^2+9}{\sqrt{9-x^2}}$ and $D(f) = [-3,3]$ therefore the critical points of the function are $x_{c_i} = \left\{ -3, -\frac{3\sqrt2... |
H: $\mathbb{Z}_p=\varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$ is uncountable.
The ring of $p$-adic integers is given by $\mathbb{Z}_p=\varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$. From this description how can we conclude that $\mathbb{Z}_p$ is uncountable ?
It follows from the description that each nonzero element of $\mathb... |
H: $[X,Y]=0 \implies \exp(X+Y)=\exp(X)\exp(Y)$
I am trying to show that if $[X,Y]=0$ then the exponential map $\exp : Lie(G)\to G$ is such that
$$\exp(X+Y)=\exp(X)\exp(Y), \forall X,Y\in Lie(G).$$
The hint is to show that $\gamma : t\mapsto \exp(tX)\exp(tY)$ is a one-parameter subgroup, which I can show :
\begin{align... |
H: Prove binomial coefficient
Equality of Binomial coefficients.
I was wondering why these two Binomial coefficients (x is just a place holder):
$\binom{x}{ k!(n+1-k)!}$ = $\binom{x}{k!(n-k)!}$.
Both lead to $\binom{x}{k}$.
Does the answer come from Pascal's triangle?
I don't get it. Why is it equal?
AI: This is not a... |
H: If $a^3=b^3$ then $a=b$ for all $a,b\in \mathbb R$ direct proof
While I was working through the first chapter of Spivak's Calculus, "Basic Properties of Numbers", where he introduces the Field Axioms and Order Axioms (P1 - P12), I thought about a problem which is not stated in the book, but could be formulated like... |
H: Inverse of sum of inverses of matrices
Is it somehow possible to reformulate the following exuation into something easier to calculate:
$$(A^{-1}+ B^{-1})^{-1}$$
A and B are both square real matrices: $A, B \in \mathbb{R}^{n \times n}$,
and are positive definite and therefore invertible.
AI: Note that
$$
A^{-1}(A +... |
H: Proving that if $\forall n\in\mathbb N,\exists x_n \in \mathbb R: |x_n - a| < \frac{1}{n}$, then $a \in \bar S$
This exercise is in my general topology textbook:
Let $S$ be a non-empty subset of $\mathbb R$ and $a \in \mathbb R$. Prove that $a \in \bar S$ if and only if $\forall n\in\mathbb N,\exists x_n \in S: |x... |
H: Find $\min \{ x+y: x+2y \ge 5, 4x+y\ge6\}$
Find $\min \{ x+y: x+2y \ge 5, 4x+y\ge6\}$ Could anyone tell me what is the answer? Is it zero?
I drew all the lines: $x+y=0$ which intersect the 2nd line at $(2,-2)$.
and with the first line at $(-5,5)$.
AI: The constraints can be written (with $s:=x+y$)$$s\ge5-y,\\s\ge\f... |
H: $f: A \rightarrow R^{n} .$ Show if $f^{\prime}(a, u)$ exists, then $f^{\prime}(a ; c u)$ exists and equals $c f^{\prime}(\mathrm{a} ; \mathrm{u}).$
Definition. Let $A \subset R^{m} ;$ let $f: A \rightarrow R^{n} .$ Suppose $A$ contains a neighborhood of a. Given $\mathbf{u} \in \mathbf{R}^{m}$ with $\mathbf{u} \neq... |
H: Uncontrolable subset and stabilizability of a linear dynamical system
I'm reading "Control System Design" by G. Goodwin, and I can't wrap my head around his definitions of uncontrolable subspace and stabilizability of a controlled dynamical system.
Consider a linear dynamical system of state $X\in\mathbb{R}^n$, con... |
H: Vector Calculus Gradient - are my answers correct?
Could someone please clarify if I have the correct answers for these following questions?
[Assume $r=x$i +yj+zk and $a=a_1$i +$a_2$j+$a_3$k for some constants $a_1,a_2,a_3$]
$\nabla f forf=cos(x)+3y^2sin^3z $
Answer: $\nabla f = -sin(x)i+6ysin^3(z)j+9y^2sin^2(z... |
H: 'Fake' identity regarding the closure in the subspace topology
I have the following argument which I encountered, and can't seem to find why it's not true:
Let $X$ be a topological space and let $A$ and $B$ be nonempty subsets of $X$. Then $\overline{A\cap B}^A=\overline{A\cap B}^X\cap A$.
"Proof"
$\overline{A\cap ... |
H: Find poles and residues of \begin{equation} f(z)=z^2/(\cosh z-1) \end{equation}
I understand that the point
\begin{equation}
z=0
\end{equation}
is an obvious singularity, but since it's also a root of multiplicity 2 of the nominator the residue of the given function at 0 will be 0.
As for the other singularities we... |
H: Integrate $\Omega=\int_{-\infty}^{\infty}\frac{\operatorname{arccot}(x)}{x^4+x^2+1}dx$
A friend of mine got me the problem proposed by Vasile Mircea Popa from Romania, which was published in the Romanian mathematical Magazine. The problem is to find:
$$\Omega=\int_{-\infty}^{\infty}\frac{\operatorname{arccot}(x)}{... |
H: Show that at least one of $x(t),y(t),z(t)$ must be the constant solution at the origin.
Let $A$ be an invertible $3\times3$ matrix, and consider the equation
$\dot{x}(t)=Ax(t)$ Suppose there are three solutions
$x(t),y(t),z(t)$ with the properties:
$\lim\limits_{t\to\infty}x(t)=0$
$\lim\limits_{t\to-\infty}y(t)... |
H: Expected value of the number of white balls in a box
In a box there are $k$ white and $l$ black balls, and next to the box there are $m$ white and $m$ black balls. Out of the box one ball is randomly taken out and then returned to the box also with $m$ more balls of the same color.
a) Find expected value of the num... |
H: If $2\tan^{-1}x + \sin^{-1} \frac{2x}{1+x^2}$, find the values for $x$ for which the function is independent of $x$
For $x>1$, then
$$2\tan^{-1} x = \pi -2\sin^{-1} \frac{2x}{1+x^2}$$
For $x<-1$
$$2\tan^{-1} x =-\pi -2\sin^{-1} \frac{2x}{1+x^2}$$
So $x\in (-\infty, -1) \cup (1, \infty)$
But the given answer is only... |
H: Integrate: $\int \frac{x}{\left(x^2-4x-13\right)^2}dx$.
Integrate:
$$\int \frac{x}{\left(x^2-4x-13\right)^2}dx$$
Here's my attempt:
I first completed the squares for the denominator:
$$\left(x^2-4x-13\right)^2=(x-2)^2-17 \implies \int \frac{x}{\left(\left(x-2\right)^2-17\right)^2}dx$$
I then used $u$-subsituition... |
H: Conditional Probability - how to find $P(A'|B')$
I have been trying to answer this problem but it appears I have the incorrect approach.
Given events $A,B$ with $P(A)=0.5$, $P(B)=0.7$, and $P(A\cap B)=0.3$, find: $P(A'|B')$
I know $P((A\cup B)') = 1 - P(A\cup B)$
I also know that $P(A\cup B) = P(A) + P(B) - P(... |
H: Find the greatest integer less than $3^\sqrt{3}$ without using a calculator and prove the answer is correct.
Find the greatest integer less than $3^\sqrt{3}$ without using a calculator and prove the answer is correct.
I'm puzzled on how to solve this problem, any help is appreciated. There was hints about turning t... |
H: Is there a simple proof that a non-invertible matrix reduces to give a zero row?
Let $A$ be a square matrix that is non-invertible. I was wondering if there is a simple proof that we can apply elementary row operations to get a zero row. (For a matrix $C$ to be invertible, I mean there is $B$ such that $CB = BC = I... |
H: Factor $x^5-5x^3+4x$
I am trying to factor$x^5-5x^3+4x$ so that I can find the roots. I know from the answers section that the roots are where $x = 0, 1, -1, 2$ and $-2$.
I'm stuck, here's as far as I got:
$$
x^5-5x^3+4x =
x(x^4-5x^2+4)
$$
Let $u = x^2$ and just focus on the term on the right (drop the first $x$ f... |
H: find function $f$ such that $f(x)=xf(x-1)$ and $f(1) = 1$
Find function $f$ such that $f(x)=xf(x-1)$ and $f(1) = 1$.
I can prove that there is just one function as $f$ (see Proof1).
I know that there exists a pi function $\Pi(z) = \int_0^\infty e^{-t} t^z\, dt$ that fits as $f$ so it is the only solution.
My pro... |
H: Significance of Codomain of a Function
We know that Range of a function is a set off all values a function will output.
While Codomain is defined as "a set that includes all the possible values of a given function."
By knowing the the range we can gain some insights about the graph and shape of the functions. For e... |
H: An stronger inequality than in AoPS.
For $x,y,z >0.$ Prove$:$
$$\sum {\frac {y+z}{x}}+{\frac {1728 {x}^{ 3}{y}^{3}{z}^{3}}{ \left( x+y \right) ^{2} \left( y+z \right) ^{2} \left( z+x \right) ^{2} \left( x+y+z \right) ^{3}}} \geqslant 4\sum {\frac {x}{y+z }}+1$$
I check when $xyz=0$ and $x=y$ and see it's true. So I... |
H: Inequality on a domain and on compact subsets of the domain
Let $u \in L^1_{loc}(\Omega)$ on a bounded domain $\Omega$. Suppose
$$u > 0 \quad \text{a.e. on compact subsets of $\Omega$}.$$
Does this imply that
$$u > 0 \quad \text{a.e. on $\Omega$}?$$
This is trivially true, isn't it? Because we can exhaust almost ev... |
H: Does $\mathrm{Log}(\zeta)$ extend meromorphically past $\Re(s)=1$?
Let $\zeta(s)=\sum_{n\ge 1}n^{-s}$ be the Riemann zeta function. It is well-known that the infinite sum
$$\mathrm{Log}(\zeta)=-\sum_{p\text{ prime}}\log(1-p^{-s})$$
converges to an analytic function on the right half-plane $\Re(s)>1$.
Question. Is ... |
H: Does $\ker T\cap {\rm Im}\,T=\{0\}$ imply $V=\ker T\oplus{\rm Im}\,T$?
Let $T: V\rightarrow V$ be a linear operator of the vector space $V$.
We write $V=U\oplus W$, for subspaces $U,W$ of $V$, if $U\cap W=\{0\}$ and $V=U+W$.
If we assume $\dim V<\infty$, then by the rank-nullity theorem, $\ker T\cap {\rm Im}\,T=\{0... |
H: $H$ normal iff $Lie(H)$ is an ideal.
I was reading a proof of the following theorem.
Theorem 20.28 (Ideals and Normal Subgroups). Let $G$ be a connected Lie group,
and suppose $H \subset G$ is a connected Lie subgroup. Then $H$ is a normal subgroup of
$G$ if and only if $\operatorname{Lie}(H):=\mathfrak{h}$ is an i... |
H: Iterates of $\frac{\sqrt{2}x}{\sqrt{x^2 +1}}$ converge to $\text{sign}(x)$.
In this post, a comment states that if $f(x):= \dfrac{\sqrt{2}x}{\sqrt{x^2 +1}}$ and $F_n:=\underbrace{f\circ \dots\circ f}_{n\text{ times}}$, then the pointwise limit $\lim\limits_{n \to \infty} F_n$ is equal to the sign function
$$
\text{... |
H: Can the step in gradient descent be negative?
For a single-variable function, gradient descent works just by repeatedly computing the next x that is closer and closer to the minimum of the function.The formula is
$$x_{i+1}:=x_i-t\nabla f\vert_{x= x_i} $$
But can "t" be negative or is it just positive? Does the grad... |
H: Is cardinality a number?
It's easy to find definitions such as
If A and B are sets (finite or infinite) A and B have the same cardinality (written $|A|=|B|)$ if there is a bijection between them.
and equally easy to find statements such as
The cardinality of a finite set is equal to the number of elements in it.... |
H: Proof verification: polynomials $\mathbb R[X]$ are a vector space that is not isomorphic to its dual
I have never seen an elementary proof of the fact that $V$ need not be isomorphic to $V^*$ that does not require some set theoretic background. I came up with this [most likely incorrect] argument, which does not se... |
H: Confusion about proving logical implication statements
I've got four statements* which I'm meant to evaluate as being either true or false.
a. If 25 is a multiple of 5, then 30 is divisible by 10.
b. If 25 is a multiple of 4, then 30 is divisible by 10.
c. If 25 is a multiple of 5, then 30 is divisible by 7.
d. If ... |
H: Integrate $\int \frac{\tan ^3\left(\ln \left(x\right)\right)}{x}dx$
Integrate:
$$\int \frac{\tan ^3\left(\ln \left(x\right)\right)}{x}dx$$
My attempt:
$$u=\ln(x) \implies \int \tan ^3\left(u\right)du=\int \tan ^2\left(u\right)\tan \left(u\right)du=\int \left(-1+\sec ^2\left(u\right)\right)\tan \left(u\right)du$$
... |
H: Solution verification: Picking stones consecutively puzzle
Puzzle:Two players pick consecutively 1,2,3 or 4 stones from a stack of 101 stones. The player who picks the last stone wins. Suppose that both players play perfect, does the first or second player win? What about the situation that the one who picks the la... |
H: dot product of direction cosine vector
I have been reading a scientific paper and they defined Di as the direction cosine vector, [ cos(latitude of point i)cos(longitude of point i) , cos(latitude of point i)sin(longitude of point i), sin(latitude of point i)]'. The same was done for another coordinate j. Next the ... |
H: every trail from start node having the same end node
given a directed graph (can have cycles) with:
an arbitrary number of nodes
an arbitrary number of edges
that satisfies the condition that there is (at least) one trail (i.e. a walk where no edge is repeated) that visits all nodes.
Would this be a true statemen... |
H: Why is the answer of $\frac{ab}{a+b}$ always smaller than the smallest number substituted?
If $\frac{ab} {a+b} = y$, where $a$ and $b$ are greater than zero, why is $y$ always smaller than the smallest number substituted?
Say $a=2$ , $b=4$ (smallest number here is $2$. Thus, the answer would be smaller than $2$)
$\... |
H: calculate $ \intop_{a}^{b}\left(x-a\right)^{n}\left(x-b\right)^{n}dx $
I need to calculate $ \intop_{a}^{b}\left(x-a\right)^{n}\left(x-b\right)^{n}dx $ this.
Now, this exercise came with hints. I have followed the hints and proved :
$ \intop_{-1}^{1}\left(1-x^{2}\right)^{n}dx=\prod_{k=2}^{n}\frac{2k}{2k+1}\cdot\fra... |
H: Show that $E = [0,1]$ is not open.
Here is my proof. Not sure if I'm doing it correctly. We recall that $E$ is open provided every point of $E$ is an interior point. We then have $\forall p \in E$, there exists a neighborhood of $p$, $N$, such that $N \subset E$. Suppose that $E$ is open and consider the point $p=1... |
H: prove or disprove: if $\sum_{n=0}^\infty a_n$ converges, then $\sum_{n=0}^\infty (-1)^n a_n^2$ converges
if $\sum_{n=0}^\infty a_n$ converges, then $\sum_{n=0}^\infty (-1)^n a_n^2$ converges
I think I'm supposed to disprove it, but i can't think of anything.
non of the usual stuff disproved it.
I know that becuase ... |
H: Prove that $13\sqrt{2}$ is irrational.
I am currently a beginner at proofs and I am having trouble proving this problem...
I know that the square root of $2$ is irrational because the square root of $2$ can be expressed as $\frac{p}{q}$ and once both sides are squared it is true that both $p$ and $q$ are even which... |
H: Is $S^{-1}R$ a free $R$-module?
As in the title, let $R$ be a (commutative unitary) ring and $S\subset R$ a multiplicatively closed subset. Then since there is a canonical map $\tau:R \to S^{-1}R$, which in general need not be injective or surjective, $S^{-1}R$ is a $R$-module by restriction of scalars. Is it free?... |
H: How to study a function (e.g $\frac{e^x-e^{-x}}{2}$) whose $\{1,2\}$-th derivatives have complex solutions?
I am wondering how one can study a function whose roots exist only in complex plane $\mathbb{C}$.
A similar question have been asked here, but it is quite different from this one.
For example's sake let:
$$f... |
H: Solve for $x$ with exponents
I am trying to solve an equation to find a value of $x$ like this:
$(1.08107)^{98/252}=(1.08804+x)^{23/252}(1.08804+2x)^{37/252}(1.08804+3x)^{38/252}$
That is pretty straightforward using Excel Solver, but I am not quite grasping how to do it by hand.
The result is $-0.00323$.
Thanks in... |
H: Solving the functional equation $f(x)f(y)=c\,f(\sqrt{x^{2}+y^{2}})$
Find all probability density function $f:\mathbb{R}\to\mathbb{R}$ such that there exists a constant $c\in\mathbb{R}$ for which $$f(x)f(y)=c\,f(\sqrt{x^{2}+y^{2}})\text{ for all }x,y\in\mathbb{R}\,.$$
The following is part of a derivation of the G... |
H: How to solve for $x$ given $x⇔A$ in a truth table?
How can I solve for $x$ in terms of A, B and C given the truth table below?
$$\begin{array}{ccc|c}
A & B & C & x ⇔ A\\
\hline
0 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
0 & 1 & 1 & 1\\
1 & 0 & 0 & 0\\
1 & 0 & 1 & 1\\
1 & 1 & 0 & 0\\
1 & 1 & 1 & 0
\end{array}$$... |
H: How to tell if a system of ordinary differential equations is homogeneous?
Suppose that I have the following set of 2 equations:
$\frac{d(x(t))}{dt} = 5tx(t) + 2y(t)$
$\frac{d(y(t))}{dt} = 5ty(t)+2x(t)$
I have read that this is a system of "Linear first order non-constant coefficient homogeneous" differential equat... |
H: True or false: Suppose $p$ and $q$ are propositions. Then $\lnot(p\implies q) \equiv p \land q.$
I am not very familiar with truth tables but I think that the $\lnot$ should get distributed among both $p$ and $q$ making the problem $\lnot p \implies \lnot q$ which does is not the same as $p\land q$ making the state... |
H: Sum of Continued Fractions
Let $x$ be a positive integer.
Consider the following sum (maybe there is a better notation with continued fractions, but I am not aware of it):
$\frac{1}{x} + \frac{1}{x- \frac{1}{x}} + \frac{1}{x-\frac{1}{x}-\frac{1}{x-\frac{1}{x}}} +\frac{1}{x-\frac{1}{x}-\frac{1}{x-\frac{1}{x}} - \fra... |
H: Find surface area of part of cylinder.
I ran into trouble when I'm trying to find a surface area of parts of the cylinder $x^2+z^2=4$ bounded by another cylinder $x^2+y^2=4$, I simply used a traditional way of double integral, change into polar coordinate calculate
$$
\iint\limits_{x^2+y^2=4}
\sqrt{\left(\frac{\p... |
H: Solving $n(4n+3)=2^m-1$ in positive integers
Find all positive integers $m$ and $n$ such that $$n(4n+3)=2^m-1\,.$$
This is an interesting equation which was sent to me by a friend (probably found online). I have been scratching my head about whether or not this has a unique solution in positive integers which I h... |
H: Lee's Intro to Topology, generating the same topology
Suppose $M$ is a set and $d, d^\prime$ are two different metrics on
$M$. Prove that $d$, and $d'$ generate the same topology on $M$ if and
only if the following condition is satisfied: for every $x \in M$ and
every $r > 0$, there exists positive real numbers $r... |
H: Why do some partial fractions have x or a variable in the numerator and others don't?
Why do rational expressions like $\left(\frac{1}{(x-2)^3}\right)$ do not have x in the numerator of the partial fraction but a rational expression like $\left(\frac{1}{(x^2+2x+3)^2}\right)$ does have x in the numerator of its part... |
H: All numbers that are less than four units from zero
Looking at a simple algebra question which is to graph "All numbers that are less than four units from zero."
The knee jerk response is to draw a number line with an open circle at -4 and a line to the left pointing at negative infinity. A second interpretation co... |
H: Solution verification: sum at least $4N/7$ times odd
Problem: Let $a_{j}$,$b_{j}$,$c_{j}$ be whole numbers for $ 1 \leq j \leq N$. Suppose that for each $j$ at least one of $a_{j}$,$b_{j}$,$c_{j}$ is odd. Show that there are whole numbers $r$,$s$ and $t$ such that the sum $$r\cdot a_{j} + s\cdot b_{j} + t\cdot c_{... |
H: How can I use the fact that $6=2\cdot 3=(\sqrt{10}-2)(\sqrt{10}+2)$ to prove $Z[\sqrt{10}]$ is not a UFD?
How can I use the fact that $6=2\cdot 3=(\sqrt{10}-2)(\sqrt{10}+2)$ to prove $Z[\sqrt{10}]$ is not a UFD?
They are different ways of factoring 6 into irreducibles?
AI: Hints:
That's exactly the idea, but showi... |
H: Which is the correct integration using an integrating factor?
When shown equation $(1)$, I have two different answers for its integration, one mine, one more from a colleague and I am uncertain of which is the correct one.
$$\left( \frac{\partial r}{\partial T}\right)_{E/T}- r\frac{c_0}{T}= - \frac{c_0}{T} \tag{1}$... |
H: What do you call a "circle-ish" polygon with 256 sides?
I would like to find an image of such a geometric shape or generate one myself, but I am not sure what to look for.
AI: It's a dihectapentacontahexagon.
For a more general answer, see List of polygons - Wikipedia.
The late John H. Conway mentions a few other s... |
H: Prove or disprove the following: If $n^3 − 5$ is an odd integer, then $n$ is even.
Prove or disprove the following proposition: If $n^3 − 5$ is an odd integer, then $n$ is even.
I know that $n$ must be even in order for $n^3 - 5$ to be odd which means I have to prove the statement.. possibly with a contradiction? I... |
H: How could a path be homotopic to a point
I have been following Serge Lang's Intoduction to Complex Analysis at a Graduate Level and I met this theorem. I want to ask what does it mean for a function to be homotopic to a point? I am only familiar with paths being homotopic to each other but not to a point. Here is t... |
H: Tricky Schwarz Lemma Type Qualifying Exam Question
I have a question from a past complex analysis qualifying exam that I would like some help on. Let $f(z)$ be an analytic function on the unit disk $\mathbb{D}$ such that $\lvert f(z)\rvert \leq 1$. Suppose that $z_1,z_2,...,z_n\in \mathbb{D}$ are zeroes of $f(z)$.
... |
H: Non constant harmonic function on $\mathbb C$
If $U(z)$ is a non constant real valued harmonic function on $\mathbb C $ then prove that there exists $\{z_n\} \subset\mathbb C $ with $z_n\to \infty $ and $u(z_n) \to 0$ as $n \to\infty$.
Non constant harmonic functions are surjective and unbounded. They do not attai... |
H: Basic Understanding of Construction of Spectral Sequences - Limit Page
I have always wanted to learn spectral sequences, and I finally found some time to do so.
However, I have some problems to understand the very basics of the construction (the answer is probably obvious, but I don't see it). I found the following... |
H: How many strings of six lowercase letters from the English alphabet contain the letter a?
How many strings of six lowercase letters from the English alphabet
contain the letter a?
The answer to this problem is:
$26^6-25^6$ by the principle of inclusion-exclusion
My question is: why we can't calculate it in this... |
H: What's the gradient of a vector field?
Imagine I have the following function
$$ \vec{f}(\vec{x}) = x \vec{x}, x = | \vec{x} |, \vec{x} \in R^3 $$
That is, the function is essentially a quadratic function, but contains a vector direction as well. Intuitively from single variable calculus I would expect the gradient ... |
H: If $(a+bi)$ is a unit in $\mathbb{Z} [i]$, then $N(a+bi) = 1$?
How does one prove that, if some $a+bi \in \mathbb{Z}[i]$ is a unit, then the norm of $a+bi$, $N(a+bi) = 1$ without simply checking each unit? I've tried a few different things (applying the Well-Ordering Principle considering only the norms, trying to ... |
H: Well-definition of the quotient norm
Consider $X$ a normed space with norm $\|\cdot\|$ and $M$ a closed subspace of $X$. In the quotient space $X/M$ we define the quotient norm:
$$|||\hat{x}||| = \inf_{y\in M} \|\hat{x}+y\|, \quad \hat{x}\in X/M.$$
I'm trying to prove the well-definiton of this norm, that is, given... |
H: Link between a particular function and cos(nt)
There was a particular question we were given that I can't work out. The question's fine but the difficulty is the extra bit. So the original question is:
Define the polynomials $f_n$ by
$$ f_0(x) = 1, $$
$$ f_1(x) = x, $$
$$ f_n(x) = 2x f_{n-1}(x) - f_{n-2}(x) \quad ... |
H: Joint probability for 2 uniform distributions
X and Y are independent and uniformly distributed on the interval (0, 1). If U = X + Y , and V =X/Y
find the joint density for U and V and the marginal densities for U and V.
Given that $$f_{UV}(uv) = f_X(x)f_Y(y)*|J|$$
Where i get the $|J| =\frac{vu+1}{(1+v)^{3}} $
I ... |
H: Combinatorics Analysis of Game "Olaf Hits The Dragon With His Sword"
I recently encountered the short tabletop roleplaying game Olaf Hits The Dragon With His Sword and I've been trying to do a combinatorics analysis of its dice mechanics. The full rules are available behind the link I've provided but I'll summarize... |
H: How is the ideal $(x,y)$ isomorphic to $k[x, y]$ as $k[x, y]$-modules?
Let $R = k[x, y]$ where $k$ is a field and consider the ideal $I = (x, y)$ as a $R$-module.
Consider the $R$-module homomorphism $\varphi : R^2 \to I$ given by $\varphi(a, b) = ax + by$.
Prove that the kernel of $\varphi$ is the set $\{(−cy... |
H: yes/ No Is $T$ is linear transformation?
Given $T : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be transformation defined by $T(x,y)= (x+1,y+1)$.
Now my question is that Is $T$ is linear transformation ?
My attempt : The main concept of linear transformation is that $T= 0$, so if we put $x=y=-1$ then $T=0$
So i think ... |
H: Coefficient Estimators of $\frac{1}{x^{2}}$ Weighted Least Squares Linear Regression
I have a feeling there should be a mathematical formular for determining the estimators of the coefficients of a $\frac{1}{x^{2}}$ Weighted Linear Regression.
I was able to derive the estimators ($a$ and $b$) for the non-weighted ... |
H: 3D Parametric Equation changing over time
I can create a 3D Parametric Equation of a spiral but I'm having trouble getting the angle of "decent" to also change over time.
$$x=u\sin(u)\cos(v)$$
$$y=u\cos(u)\cos(v)$$
$$z=-u\sin(v)$$
The Octave code I have so far seems close, I'm just not sure how to "tweak" it.
The i... |
H: Arc length of the Inverse Function
I am looking to prove that the arc length of the inverse function $f^{-1}(x)$ on the interval $[f(0),f(a)]$ is the same as the arc length of the function $f(x)$ on the interval $[0,a]$. One may see that this is true visually, as it comes from flipping the function across the line ... |
H: limits and L'Hospital's rule
$$\lim_{x\to 0^{+}} \frac{1+x\ln x}{x}$$
I think it is an indeterminate form so I applied the L'H rule and did the derivative of numerator and denominator and got limit as x tends to 0 from right [ $\ln (x) + 1$ ] which clearly shows that it depends on $\ln$ function. we know that $\ln$... |
H: Can we say that $\text {tr}\ (A) = 0\ $?
Let $A$ be an $n \times n$ real matrix with $A^3 + A = 0.$ Can we say that $\text {tr}\ (A) = 0\ $?
I think it's true but can't prove it. Any help will be highly appreciated.
Thanks in advance.
AI: The minimal polynomial of $A$ must divide $x^3+x$.
Then the real Jordan form ... |
H: Result of $\int_{-\infty}^{\infty}\frac{\cos(ax)}{e^x+e^{-x}}$
I'm trying to validate if result of this integral is equal to:
$$
\frac{\pi*ch(a\frac{\pi}{2})}{ch(\pi))+1})
$$
I'm trying to resolve it using the reside in $\frac{\pi}{2}$ but couldn't find a resolution to compare.
Any help is most appreciated.
AI: Not... |
H: Rudin proof for compact subsets $\{ K_{\alpha} \}$ (theorem 2.36) — Contrapositive or contradiction?
I am having doubts about Theorem 2.36 pasted below. I was able to follow all the steps individually, but I don’t see how this is a proof by contradiction. It seems to be it is a proof by contrapositive.
That is bec... |
H: Image of morphism of sheaves
Suppose I have a morphism of sheaves $f : E^{\oplus4} \to I_p$ on $X$ a degree $3$ Fano threefold, where $E$ is a rank $2$ vector bundle on $X$ and $I_p$ is an ideal sheaf of a point $p \in X$. I'm interested in knowing the image of $f$.
For each of the direct summands, I have a morphis... |
H: Rudin's PMA: Theorem 3.29 Proof
Theorem 3.29: If $p>1$,
$$
\sum_{n=2}^{\infty}\frac{1}{n(\log\ n)^p}
$$
converges; if $p\leq1$, the series diverges.
Proof: The monotonicity of the logarithmic function implies that $\{log\ n\}$ increases. Hence $\{1/n\ \log\ n\}$ decreases, and we can apply Theorem 3.27 to the serie... |
H: Does $\mathbb{Q}$ have the finite-closed topology?
Let $\mathbb{Q}$ be the set of all rational numbers with the usual topology
Does $\mathbb{Q}$ have the finite-closed topology?
My attempt : I think yes
Finite - closed topology mean cofinite topology .we know that in the cofinite topology-$ (\mathbb{R} , T)$, the ... |
H: How to convert from a flattened 3D index to a set of coordinates?
I have a flattened 3D array in row-major format with an index, $I$, defined as $I = x + y D_x + z D_x D_y$, where $x$, $y$, and $z$ are the indices and $D_x$, $D_y$, and $D_z$ are the dimensions of the 3D array. How can I obtain the original set of c... |
H: Holomorphic in an open set containing the closed unit disc $\mathbb D$ and real on the boundary of $\mathbb D$
If f(z) is holomorphic in an open set containing the closed unit disc, and if f($e^{i\theta})$ is real for all $\theta$ in $\mathbb R$, then prove that f(z) is constant.
I came across this problem in the e... |
H: area being finite in $\int_{0}^\infty \alpha\frac{x}{\beta+x}dx<\infty.$
For $x>0$, for what values of $\alpha$ and $\beta$, do we have:
$$\int_{0}^\infty \alpha\frac{x}{\beta+x}dx<\infty.$$
This is known as the saturaion-growth model specification in nonlinear regression.
I would like to know for what values the a... |
H: Integrate: $\int \:x\left(\frac{1-x^2}{1+x^2}\right)^2dx$.
Integrate:
$$\int \:x\left(\frac{1-x^2}{1+x^2}\right)^2dx$$
My attempt:
$$\text{Let} \ u = x, v'=\left(\frac{1-x^2}{1+x^2}\right)^2\\$$
\begin{align}
\int \:x\left(\frac{1-x^2}{1+x^2}\right)^2dx & = x\left(x-2\arctan \left(x\right)+\frac{2x}{1+x^2}\right... |
H: Cyclic Field Extensions of sum of radicals
Given a field $K$ of characteristic 0, which contains a primitive root of order 3,
I would like to show that the extension $K(\sqrt2+3^{\frac{1}{3}})/K$ is cyclic.
My attempt was to look at the "bigger" extension $K(\sqrt2, 3^{\frac{1}{3}})/K$.
The problem is that I don't ... |
H: How to easily identify how many distinct roots a polynomial has?
For example, I have $a(x) = 4 x^4 + 5 x^2 + 7 x + 2 $.
$$$$ Using Descartes rule, I know $a(x)$ has maximum $2$ negative real roots. As imaginary roots come in pairs, $a(x)$ will have either $0$ or $2$ negative real roots. I also imagine the possible ... |
H: Proof that $\frac{d(\sin x)}{dx} = \cos x$ for $\frac{\pi}{2} < x < \pi$
Let us assume that $x$ is an angle that lies in the second quadrant i.e. $\dfrac{\pi}{2} < x < \pi$.
We have to prove that $\dfrac{d(\sin x)}{dx} = \cos x$. I will use the unit circle to prove this. The method will be like the one used by Gra... |
H: Trigonometry Ferris Wheel Question
Question: Suppose you wanted to model a Ferris wheel using a sine function that took $60$ seconds to complete one revolution. The Ferris wheel must start $0.5\,\textrm{m}$ above ground. Provide an equation of such a sine function that will ensure that the Ferris wheel's minimum h... |
H: Problem on ratio wrt time distance
This is a rough translation from a local language so please bear with it,
Say we have two people, police $p1$ and thief $p2$.
$p1$ takes $4$ $steps$ and $p2$ takes $5$ $steps$ in the same amount of time. Also the distance $p1$ covers in $6$ $steps$ is equal to the distance $p2$ co... |
H: Isometry on inner product space
$V$ is an inner product vector space. If a transformation $T\colon V\to V$ satisfies $\langle T(x), T(y)\rangle = \langle x, y\rangle$ for every vector $x, y \in V$, prove or disprove that $T$ is linear.
Seems true, but can't prove it. Tried plugging $x+y$ into $x,y$ and got
$\lang... |
H: Identities in first order logic
When given the statement
(∀)
A()
→
B()
→
C,
can I rewrite it as
(∀)
(A()
^
B())
→
C
?
Also, if so, how can I can prove it please?
Thanks!
AI: Yes, indeed, $(\forall x)~(A(x)\to(B(x)\to C)) \iff (\forall x)~((A(x)\wedge B(x))\to C)$
We may derive $C$ when we assume that $(\forall x)~(... |
H: Having trouble deciphering this function space notation
I am working through some fluid mechanics and I can't seem to find a precise definition anywhere for this function space notation:
$$z(\alpha,t) \in C^1 \left( [-T,T]; C[0,2\pi] \right)$$
I am specifically looking at the bottom of page 332 (Proposition 8.6) in... |
H: norm of a functional attains in real part
I tried to prove that, when $X$ is a Banach space, it holds that $$\sup Ref(B_X)=\|f\|$$, where $f \in X^*$ is a functional and the norm is defined by $\|f\|=\sup\{ |f(x)|: x \in B_X \}$.
One of the inequalities is straightforward, but I do not see the other one.
AI: If $z=... |
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