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H: Is notational compactness in tensors (compared to linear algebra) relevant?
In this post you can read:
A matrix is a special case of a second rank tensor with 1 index up and 1 index down. It takes vectors to vectors, (by contracting the upper index of the vector with the lower index of the tensor), covectors to co... |
H: Prove or disprove that the given group is abelian
Let $K=K_1 \cup K_2 \cup \dots \cup K_n$ be a finite union of tori in 3-space (each $K_i$, $i=1,2,\dots, n$ is a torus). For $i=1,2,\dots, n$, let $a_i$ be the meridian and let $b_i$ be the longitude of $K_i$. Define the group $G$ that is generated by the set
$ X=\... |
H: Example where $\operatorname{Spec} S^{-1}B$ is neither open nor closed in $\operatorname{Spec} B$
I know that $\operatorname{Spec} S^{-1}B$ is open in $\operatorname{Spec} B$ with respect to the Zariski topology when $S=\{1,f,f^2,\ldots\}$ for $f\in B$.
However, is this true for every multiplicative subset $S$ of ... |
H: Prove complex numbers $a$ and $b$ are antipodal under stereographic projection $\iff a \overline{b} = -1$
I'm trying to prove the following statement:
Given $a, b \in \mathbb{C}$, prove that $a$ and $b$ correspond to antipodal points on the Riemann sphere under stereographic projection if and only if $a \overline{... |
H: interiors and closures of sets where the interior of the boundary is empty
Let $A, B\subseteq \mathbb{R}^n, int(\partial A) = int(\partial B) = \emptyset.$If $A\cap B\neq \emptyset,$ is it necessarily true that $\overline{A\cap B} = \overline{A}\cap \overline{B}$? Is it true that if $A\cap B = \emptyset,$ then $in... |
H: Question about cluster point and subsequence on topological space.
First, take some definition: Given a topological space and net, defined on : . We say that x is cluster point of net, if for every open set and for every , there exist , such that .
We know that sequence is a special kind of net. Also, every subs... |
H: Confusion about the definition of a multifunction/multi-valued function
I had this definition of a multi-function (for the case of complex-valued functions)
A multi-function on a open subset $U$ is $f:U\to \mathcal{P}(\mathbb{C}).$
Or at least how I interpret this definition is that $f(z)\subset \mathbb{C}$ for e... |
H: Image of the multiplication of a function by a scalar
$f$ is a linear function that maps from a vector space $E$ to $E'$. How can I prove that $\operatorname{Im}f = \operatorname{Im} \alpha f$?
Lets say that $f(a,b,c) = (2a, 2b, 2c)$, the image is $(2a, 2b, 2c)$.
Now $2\cdot f(a,b,c)$ has image $(4a, 4b, 4c)$ which... |
H: Prove that a function between metric spaces is continuous iff the preimage of any open set in the codomain space is an open set.
So, here's the full question:
Let $f:(X,d) \to (Y,d')$ be a function between two metric spaces. $f$ is continuous iff for each open set $O \subseteq Y$, $f^{-1}(O)$ is an open subset of $... |
H: Using the interpolation to find some data
For part d of this question i want to know what does the sentence "one standard deviation of the means" mean because it is not clear for me
AI: Within one standard deviation of the mean is interval centered on the mean with length equal to two standard deviations. So if t... |
H: How do I create an offset shape that is a specific distance from a given circle, in the direction of the origin?
I'm an amateur engineer, working on a CAD design - but sadly, I'm not a mathematician.
In other words, this question might sound like homework, but it's not, I promise.
I have an existing circle, which h... |
H: Little o notation question with the prime number theory?
I don't understand what is being used in the little o notation for a description of the Prime Number Theorem. Specifically I do not understand what is f(x) or g(x) for their little o notation which they state as:
"The Prime Number Theorem states the number o... |
H: Combinatorics problem from Introduction to Probability
Problem
A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. Thus, the superdeck has 52 · 10 = 520 cards, with 10 copies of each card. How many different 10-card hands can be dealt from the superdeck... |
H: How to find $f(777)$ for a particular function $f$?
A function $f$, defined on the set of positive integers, has $f(1) = 2$ and $f(2) = 3$.
Also $f(f(f(n))) = n + 2$ if $n$ is even and $f(f(f(n))) = n + 4$ if $n$ is odd.
What is $f(777)$?
I'm not sure where to start. I've tried looking for a pattern by finding th... |
H: Prove $\cos^2(\theta)+\sin^2(\theta) = 1$
$$\cos^2(\theta) + \sin^2(\theta) = 1$$
I solved this by using right triangle,
$$\sin(\theta) = \frac{a}{c}, \quad \cos(\theta) = \frac{b}{c}$$
$$\cos^2(\theta) + \sin^2(\theta) = 1$$
$$\Bigl(\frac{b}{c}\Bigr)^2 + \Bigl(\frac{a}{c}\Bigr)^2 = 1 $$
$$\frac {a^2 + b^2} {c^2} =... |
H: In a cyclic pentagon $ABCDE$, prove $\frac{a}{\sin(B+E)}=\frac{d}{\sin(C+E)}$
I've found this question online without a solution, I have tried myself and I couldn't go further than what I've mentioned in the picture.
Question:
My attempt
So I simplified the question into proving $\frac{a}{\sin \beta} = \frac{d}{\s... |
H: Does proving the equivalence between a function being analytic and being holomorphic imply that a holomorphic function is infinitely differentiable?
In my book (Serge Lang's Introduction to Complex Analysis at a graduate level) we prove that a function is holomorphic i.f.f it is analytic. My question is, since a ho... |
H: Proof verification: Baby Rudin Chapter 6 Exercise 4
I am trying to prove
If $f(x) = 0$ for all irrational $x, f(x) = 1$ for all rational $x$, prove that $f \notin \mathscr{R}$ on $[a, b]$ for any $a<b$.
My attempt:
Suppose $f(x) = 0$ for all irrational $x$ and $f(x) = 1$ for all rational $x$. Let $[a, b]$ be an ... |
H: What can we say if the gradient at the boundary has constant norm?
Let $(M^n,g)$ be a Riemannian manifold and consider $\Omega$ a smooth and bounded domain in $M$. Let $u : \overline{\Omega} \to \mathbb{R}$ be a smooth function that satisfies both $u = 0$ and $\Vert \nabla u \Vert = 1$ along $\partial \Omega$. Is i... |
H: Liminf of union of two sequences
Let $A_n$ and $B_n$ be two sequences of sets. How $(\liminf_n A_n \cup \liminf_n B_n)$ and $\liminf_n (A_n\cup B_n)$ are related?
Def. Given a sequence of sets $E_n$, the limit inferior of $E_n$ is defined as $$\liminf_{n\to\infty} E_n=\bigcup_{n=1}^\infty \bigcap_{k=n}^\infty E_k$$... |
H: Group of units of $C[0,1]$
Is the group of units of $C[0,1]$ cyclic?
I think it is not cyclic. The first argument that came to my mind is that if it is cyclic then its generator must be a constant function, but not all units of $C[0,1] $ are constant.
Is my argument correct or do we have some better argument then t... |
H: Is it possible to pull out a diagonal matrix from a trace operation
Suppose I have a real $m \times m$ symmetric matrix $A$, and a real $m \times m$ diagonal matrix $D$. I'm interested in finding $\operatorname{tr}(DA)$. Suppose $d$ were a scalar constant, then I know the following holds: $$\operatorname{tr}(dA) = ... |
H: Inductive step of $2^2 + 4^2 + \cdots + (2n)^2$
I am trying to prove that $\sum_{k=1}^n (2k)^2 = \frac{2n(n+1)(2n+1)}{3}$. I managed to do the base case but I am stuck at the inductive step. Am I doing it wrong?
This is what I need to prove:
$$2^2 + 4^2 + \cdots + (2n)^2 = \frac{2n(n+1)(2n+1)}{3}$$
Here is the work... |
H: Find conditional probability
For two events A and B, the probability that A occurs is 0.6, the probability that B
occurs is 0.5, and the probability that both occur is 0.3. Given that B occurred, what
is the probability that A also occurred?
My work: $P(A|B) = \frac{P(A\cap B)}{P(B)} = \frac{0.3}{0.5} = 0.6$ whic... |
H: EGMO 2015/P5: Let $m, n$ be positive integers with $m > 1$. Anastasia partitions the integers $1, 2, \dots , 2m$ into $m$ pairs.
Let $m, n$ be positive integers with $m > 1$. Anastasia partitions the integers $1, 2, \dots , 2m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these... |
H: Why is $6$ the multiplicative identity of the ring $2 \Bbb Z_{10}$?
Just wondering how you are able to determine that the multiplicative identity of the ring $2 \Bbb Z_{10}$ which is $\{ 0,2,4,6,8 \}$ is 6. I tried multiplying every element in this ring by 6, but I never got the original element for an answer with ... |
H: Prove $\lim_{x \to a} \Re f(x) = \Re A$ and $\lim_{x \to a} \Im f(x) = \Im A$
I'm reading Ahlfors' complex analysis. In the book, he states that
From $\lim_{x \to a} f(x) = A$ we obtain that \begin{align*}\lim_{x \to a} \Re f(x) = \Re A\\
\lim_{x \to a} \Im f(x) = \Im A\end{align*}
where $x$ and $f(x)$ can be re... |
H: A quiz question related to contour integration i am unable to solve
I am trying quiz questions of last year in Complex analysis by myself and I was unable to solve this particular problem .
Adding it's image:
I tried by putting z= 3$e^{it}$ and then changing limits from 0 to 2π but I am not able to simplify the ... |
H: Help with Example 5.18 in Rudin's PMA
Rudin showed a counterexample that L'Hospital's rule fails when it comes to complex-valued functions. My question is how can we get $$ \lim_{x\rightarrow 0} \frac{f(x)}{g(x)}=1$$
using the fact that $|e^{it}|=1$ for all real $t$?
I would appreciate if you could explain in deta... |
H: Proof Verification: Baby Rudin Chapter 4 Exercise 9
I am trying to prove:
Show that the requirement in the definition of uniform continuity can be rephrased as follows: To every $\epsilon > 0, \exists \delta > 0$ such that $\forall E \subset X \textrm{ with } \operatorname{diam} E < \delta, \textrm{ we have that ... |
H: Identify the function represented by $\displaystyle \sum_{k=2}^\infty \frac{x^k}{k(k-1)}$
So first I wrote it out in the terms, and I got
$\displaystyle \sum_{k=2}^\infty \frac{x^k}{k(k-1)} = \frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{12}+\frac{x^5}{20}+...$
I know the power series for $\displaystyle ln(1+x) = x-\frac{... |
H: Find the area bounded by the curve.
I want to find the area under the curve:
$$(2018x+2y-1)^2+ (2018x-3y+2)^2 =1$$
What I tried is:
Firstly, I substituted $2018x=X$ and kept $y=Y$, as this would scale my area by $2018$, so in the end, we have to divide the answer by $2018$. Further, I found that it was not possible... |
H: A quiz question in number theory related to chinese remainder theorem
I am trying quiz questions of previous year in number theory and I was unable to solve this particular problem.
I tried by taking various integers ie fixing a and b and then trying to finding k and also letting that such a k exists and then equ... |
H: What does it mean for an improper integral to exist even though it diverges
I have been working through problems in Spivak Calculus and in Chapter 14, (Fundamental Theorem of Calculus) Problem 26 it asks if the Integral $\displaystyle\int_0^\infty \frac{\mathrm{d}x}{\sqrt{1+x^3}}$ exists.
Now I thought it doesn't b... |
H: Convergence Notation: $L^k$ and $L^k(dQ)$
What is the difference between the following convergence notations: $L^k$ and $L^k(dQ)$?
I am familiar with typical $L^n$-convergence, but what does $dQ$ represent?
AI: Maybe you could give a little more of context (book, exercise?) but it comes to my mind is that $dQ$ is a... |
H: Prime Ideal with 1
I know that it is possible for a prime ideal $P$ to not contain $1$ (the even numbers are a prime ideal of $\mathbb{Z}$), but I can't figure out if every prime ideal does not contain $1$, and I can't find an example of a prime ideal with 1.
AI: One of the defining properties of an ideal $I$ of ... |
H: A high school quadratic problem
"If $a_1,a_2,a_3,\dots,a_n$ are distinct non-zero numbers such that
$$
\left(\sum_{k=1}^{n-1} {{a_k}^2}\right)x^2 +2\left(\sum_{k=1}^{n-1} a_k a_{k+1}\right)x + \left(\sum_{k=2}^{n} {{a_k}^2}\right)\le 0
$$"
We have to tell the kind of sequence that $a_1,a_2,a_3,\dots,a_n$ produces. ... |
H: Prove the zero set of a proper ideal of the ring of continuous complex-valued function on a compact space is nonempty
Prove the zero set of a proper ideal $I$ of the ring of continuous complex-valued function on a compact space $X$ is nonempty.
The above problem is from Lang' s Real and Functional analysis Chapte... |
H: Calculus of $ \lim_{(x,y)\to (0,0)} \frac{8 x^2 y^3 }{x^9+y^3} $
By Wolfram Alpha I know that the limit
$$
\lim_{(x,y)\to (0,0)} \dfrac{8 x^2 y^3 }{x^9+y^3}=0.
$$
I have tried to prove that this limit is $0$, by using polar coordinate, the AM–GM inequality and the change of variable $ x^9= r^2 \cos^2(t) $ and $y... |
H: Open Filter Characterization of Minimal Hausdorff Spaces
For minimal Hausdorff spaces, we have the equivalence -
Hausdorff $X$ is minimal Hausdorff iff every open filter with unique cluster point converges to that point.
I've been able to prove the forward implication, but I've not been able to prove the reverse.... |
H: Can the distance ($r$) or angle ($θ$) of the Polar coordinates contain Complex numbers ($a+bi$)?
Is it possible that distance ($r$) or angle ($θ$) contains Imaginary or Complex number?
If the answer is yes, how can I convert a number like that (Polar with complex argument) to Rectangular number?
For example:
$(r,... |
H: Prove that $\binom{n}{0}\binom{n+1}{n} +\binom{n}{1}\binom{n}{n-1} +\binom{n}{2}\binom{n-1}{n-2} +\cdots +\binom{n}{n}\binom{1}{0} = 2^{n-1}(n+2)$
Prove the below:
$$\binom{n}{0}\cdot\binom{n+1}{n} +\binom{n}{1}\cdot\binom{n}{n-1} +\binom{n}{2}\cdot\binom{n-1}{n-2} +\cdots +\binom{n}{n}\cdot\binom{1}{0} = 2^{n-1}\... |
H: Ext of the rationals Q is a vector space over the rationals
How to see that Ext$_Z^1$(Q, A) is a vector space over Q (where Q is the rationals) for any abelian group A? Any help would be appreciated!
AI: If $a\in\Bbb Q$ then $a$ induces a map $\mu_a$ "multiplication by $a$" from $\Bbb Q$ to $\Bbb Q$. As $\text{Ext}... |
H: Given a range of numbers, is it possible to skip multiples of $2$ and $3$ without division?
First of all, is there a name for the observation that starting at $5$, alternating between adding $2$ and $4$, will skip all multiples of $2$ and $3$? E.g. $5+2=7$, $7+4=11$, $11+2=13$, $13+4=17$ (notice how $9$ and $15$ ar... |
H: Integrating $\int \sqrt{a^2+x^2} \ \mathrm{d} x$ with trig. substitution
I am trying to come up with all the formulas I have myself and I stumbled upon a roadblock again.
Integrating $\int \sqrt{a^2+x^2} \ \mathrm{d} x$ with Trig Substitution.
So I imagined a triangle where the hypotenuse is $1$, $\sin(y) = x$ oppo... |
H: Generic construction of a probability measure on a random variable X
Let's consider the usual setting for a probability space, that is:
$X:(\Omega,\mathscr{B}) \longrightarrow (\mathbb{R},\mathscr{B}(\mathbb{R})) $ . Where X is a random variable; now consider a probability measure on X as:
$P \circ X^{-1}(-\infty,x... |
H: Let continuous $T:X\to Y$ with $(X,||.||)$ and $(Y,||.||')$ normed spaces.Is it always true that $T(Bx)=By$?
I am looking for a case where you have continuous $T:X\to Y$ where $T(Bx)$ is something else than a open (or closed) ball or even sphere ($Bx$ is the closed ball of $X$ with centre 0 and radius 1). If you ha... |
H: Prove that no points on a circle of radius $\sqrt{3}$ can have both $x$ and $y$ coordinates rational
The problem is stated in the title.
$$x^2+y^2=3$$
Assume one coordinate is rational, i.e. $y=\frac{n}{m}$. Then $x^2+\frac{n^2}{m^2}=3$ , which implies : $$x=\sqrt{3-\frac{n^2}{m^2}}$$ $$\ \ \ \ \ =\frac{\sqrt{3m^2-... |
H: Proving that the absolute value of a polynomial can have arbitrarily large values
I am stuck in an exercise that asks me to prove that if
$a_n=c_0+c_1n+\ldots+c_pn^p$
then $\lim_{n\to\infty}\lvert a_n\rvert=+\infty$.
I tried to approach it in this way:
$\lvert a_n \rvert \geq \lvert \lvert c_p \rvert n^p-\lvert c_0... |
H: I need help answering and understanding this problem about Cauchy Sequences.
Context: I done this problem awhile back and was looking through my notes on it and my answer seems incorrect.
Let $(x_n)_{n{\in}\mathbb{N}}$ be a sequence such that $|x_n-x_{n+1}|\;{\le}\;2^{-n}$ holds for every $n\,{\in}\,\mathbb{N}$. Sh... |
H: Show that the kernel ker(B) is a vector subspace of the domain.
I want to know how to show that $\ker(B)$ is a vector subspace of the domain.
AI: Theorem: the kernel of any linear map $f : V \to W$ is a vector subspace of the domain. Proof: we must show that $\ker(f) = \{x \in V : f(x) = 0\}$ is closed under zero, ... |
H: Does $\sum _{n=1}^{\infty } \frac13 (\frac{n+1}{n})^{n^2}$ converge?
I am studying the convergence of
$$\sum _{n=1}^{\infty } \frac13 \left(\frac{n+1}{n}\right)^{n^2} $$
Let $a_n = \left(\frac{n+1}{n}\right)^{n^2}$
Using the root test:
$$ \lim _{n\to \infty }\left(\sqrt[n]{\left(\frac{n+1}{n}\right)^{n^2}}\right) ... |
H: How many combinations possible to make $10 from a set amount of coins
I have 50 coins. They add up $10. They are comprised of 10 cent, 20 cent and 50 cent coins. How many combinations are possible?
I tried making equations and etc.
x + 2y + 5z = 100 cents
x + y + z = 50
y + 4z = 50 cents
x = 3z
However I don’t know... |
H: Do orthogonal projections play a role in diagonalizability?
I'm studying Linear Algebra by myself, and the textbook I use is the fourth edition written by Friedberg, Insel, and Spence. For now, I'm trying to get through Section 6.6 that concerns orthogonal projections and the spectral theorem. The following claim e... |
H: If $f(x)=x \sin (\frac{\pi}{x})$, is continuous everywhere, then find $f(0)$
$$\lim_{x\to 0} x\sin \left (\frac{\pi}{x} \right )$$
$$=\lim_{x\to 0} x \frac{\sin \left (\frac{\pi}{x} \right )}{\frac{\pi}{x}} \frac{\pi}{x}$$
$$=\pi$$
So the answer should be $\pi$, but it is actually $0$
Why is the value of limit $0$ ... |
H: Finding a Mistake for a Particular Form of Inequality
My book depicts that the following problem uses ${x^3\over (1+y)(1+z)}+{(1+y)\over 8}+{(1+z)\over 8} \ge {3x\over 4} $.
Let $x, y, z$ be positive real numbers such that $xyz = 1$. Prove that $$ {x^3\over (1+y)(1+z)}+{y^3\over (1+z)(1+x)}+{z^3\over (1+x)(1+y)}\g... |
H: What would be a good self study textbook for 1-variable calculus with a pinch of multivariable?
About 10 years ago I took calculus 1 and 2.
I know a few basics.
Like integrals are good for areas of curves. And derivatives are good for finding relative slopes.
I have decent grasp of simple derivation/integration lik... |
H: Rod cut at two points, expected length of largest part
This question is similar to this but I cant find what goes wrong in my answer. The full statement of problem is this:
If a 1 meter rope is cut at two uniformly randomly chosen points (to give three pieces), what is the average length of the largest piece?
Wha... |
H: Recursive sequence defined using $\cos x$
For any $x\in \mathbb{R}$ , the sequence $\{a_n\}$ , where $a_1=x$ and $a_{n+1}=\cos (a_n)$ for all $n$ is convergent . True/False
I think this is true. Well , my plan is to show $\cos x$ is a contractive mapping on the real line (complete metric space) and then the above... |
H: Let A be a non-zero vector in $R^n$ and let c be a fixed number. Show that the set of all elements X in $R^n$ such that $A*X \geq c$ is convex.
I have this problem. I solved it, but I think there is a problem in my solution. Maybe I've misunderstood something. I'd be glad if someone told me whether there are any pr... |
H: subgroups complementing terms of the lower exponent-p central series (Huppert-Blackburn Th. VIII.1.7b))
I am trying to understand the proof of Theorem VIII.1.7b) in Huppert-Blackburn "Finite Groups. II". It deals with what is elsewhere called the lower exponent-$p$ series of a group $G$, defined as:
$$\lambda_1(G)=... |
H: Why is: $\lim_{x\to 0+} x\ln x = \lim_{y\to \infty} -\frac{\ln y}{y}$ true?
In class the following equivalence was used: $$\lim_{x\to 0+} x\ln x = \lim_{y\to \infty} -\frac{\ln y}{y}$$
Why is does this hold and what is the general rule here (when is switching the values to which $x$ converges allowed)?
AI: By a cha... |
H: If $y = \frac{2}{5}+\frac{1\cdot3}{2!} \left(\frac{2}{5}\right)^2+\frac{1\cdot3\cdot5}{3!} \left(\frac{2}{5}\right)^3+\cdots$, find $y^2+2y$
If $$y = \frac{2}{5}+\frac{1\cdot3}{2!} \left(\frac{2}{5}\right)^2+\frac{1\cdot3\cdot5}{3!} \left(\frac{2}{5}\right)^3+\cdots$$ what is $y^2+2y$?
Attempt:
We know that for n... |
H: Solving $\sup_{x,y}\{a + bx + cy + d(x + y)^2\} $
My question is, how does one solve
$\displaystyle\sup_{\substack{x,y\in[0,1]^2 \\ x + y\leq 1}}\{a + bx + cy + d(x + y)^2\} $ when $a, b,c,d$ are constants ?
To be honest, I'm not exactly sure how to proceed (haven't had optimization classes for a long time). My vag... |
H: Does congruent triangles apply to this question?
Two identical rods $BA$ and $CA$ are hinged at $A$. When $BC = 8\ \textrm{cm}$, $\angle BAC = 30^\circ$ and when $BC = 4\ \textrm{cm}$, $\angle BAC = \alpha$. Show that $$\cos\alpha = \frac{6+\sqrt 3}{8}$$
I drew two diagrams and tried finding the length of $AC$ (w... |
H: Given two circles externaly tangent to each other and the common tangent line. Draw a third tangent circle.
We are given $\Gamma_A$ centered at $A$ and $\Gamma_B$ centered at $B$ tangent to each other externally at $C$. Line $DE$ is one common tangent to both not through $C$.
Is there a nice way to draw the red ci... |
H: differential equations solution team
Find the general solution set:
$$(x^2-y^2+1)dx+(x^2-y^2-1)dy=0$$
AI: That's an example of a first-order nonlinear ODE. As you state in your question, you are not looking for a hint nor a step-by-step solution, but for a general solution.
General solution:
$ e^{\frac{1}{2}(y(x)+... |
H: Derivatives of $ \frac{1}{r} $ and Dirac delta function
I am trying to understand the formula
\begin{equation}
\nabla^2\left(\frac{1}{|{\bf r}-{\bf r}'|}\right) = - 4 \pi \delta(\bf{r}-\bf{r}'), \qquad\qquad {\rm (I)}
\end{equation}
where ${\bf r}=(x,y,z)$. This is something heavily used in electrostatics and the s... |
H: Sum of the diameters of the incircle and excircle is congruent to the sum of the segments of the altitudes from the orthocenter to the vertices.
The problem is from Kiselev's Geometry Exercise 587:
Prove that in a scalene triangle, the sum of the diameters of the
inscribed and circumscribed circle is congruent to ... |
H: How to find the position vector for the point of intersection of a line and the perpendicular line through a point C
How could I find the exact coordinates of the point N for example which is the point of intersection of the line $L=\begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix} +t\begin{pmatrix} -3 \\ -2 \\ -3 \end{p... |
H: What does $ X = A + u U + v V $ mean in definition of triangle?
I'm trying to understand what does the algorithm does and I'm stuck in both 2.1.1 and 3 sections where the algorithm is explained.
Author says:
The triangles $ \triangle ABC $ and $ \triangle PQR $ are
$ X = A + u U + v V $ with $ U = B - A $, $ V = C... |
H: Evaluate: $\sum_{n=1}^{10}n\cdot (\sum_{r=1}^{10}\frac{r^2}{r+n})$
Evaluate: $\sum_{n=1}^{10}n\cdot (\sum_{r=1}^{10}\frac{r^2}{r+n})$
using the property of sigma operator I put $n$ in second sigma
So it becomes:
$\sum_{n=1}^{10}\cdot (\sum_{r=1}^{10}\frac{r^2n}{r+n})$
Which property I need to use now?
AI: $S= \disp... |
H: Solution to a Riccati type equation
I have the following Riccati type equation,
$$y'(x)-\frac{1}{4x}y(x)^{2}+\frac{y(x)}{x}-a\frac{x}{(1-x)^{2}}=0,$$
where $a$ is just a constant. I am struggling to find a substitution that makes the equation linear. I also tried to find a particular solution and use that method to... |
H: A question in real analysis related to connectedness
I am trying quiz questions of previous years and I couldn't think on how to solve this problem so, I am asking it here.
Question is :
Rationals are dense in $\mathbb{R}$ and closure of $\mathbb{Q} $ × $\mathbb{Q} $ = $\mathbb{R^2} $ and so is closure of complem... |
H: Proof verification: Baby Rudin Chapter 4 Exercise 8
I am trying to prove:
$f$ is a real, uniformly continuous function on the bounded subset $E$ in $\mathbb{R}^1 \implies f$ is bounded on $E$.
My attempt:
When I wrote this proof, I thought that the proof was correct. However, a friend of mine pointed out that t... |
H: The set of all vector spaces
How can I prove that the set of all vector spaces doesn't exist? (In other words, if I gather all vector spaces, then it cannot be a set)
AI: Take some fixed vector space $V$. Then for any set $A$ the set $V\times \{A\}$ is also a vector space in an obvious manner. Since the class of al... |
H: Bernoulli-like trial to obtain a given number of successes with high probabilty
I have a set of independent Bernoulli trials $X_i$ (biased coins with probability $p$). How many coins do I need to throw to obtain at least $c$ successes with a-priori probability at least $\delta$?
Formally, I want $\mathbb{P}[\sum_{i... |
H: why $\left(\left( \left(-\frac{1}{4}\right)^{-2}\right)^\frac{1}{4}\right) \neq \left(\left(-\frac{1}{4}\right)^{-\frac{1}{2}}\right)$?
Why $\left(\left( \left(-\frac{1}{4}\right)^{-2}\right)^\frac{1}{4}\right) \neq \left(\left(-\frac{1}{4}\right)^{-\frac{1}{2}}\right)$?
well, I tried this question but as far my ca... |
H: Proving the existence of minimum distance between two curves
Consider the following subsets of the plane:$$C_1=\Big\{(x,y)~:~x>0~,~y=\frac1x\Big\} $$and$$C_2=\Big\{(x,y)~:~x<0~,~y=-1+\frac1x\Big\}$$Given any two points $P=(x,y)$ and $Q=(u,v)$ of the plane, their distance $d(P,Q)$ is defined by$$d(P,Q)=\sqrt{(x-u)^... |
H: Finding convergence of sequence of random variables
Let ${X_n}$ be a sequence of independent random variables which takes values between $[0,1/3]$ Then show that $Z_n=X_1X_2...X_n$ converges to $0$ in probability.
What I think is that the sequence $Z_n$ converges to $0$ almost surely and almost sure implies converg... |
H: Is this polynomial irreducible over rationals?
$f(x)=1+x+\frac{x^2}{2!}+\dots+\frac{x^n}{n!}$.
I know that if n is prime, time $n!$ on both sides, since $n$ is prime, Eisenstein criterion shows $f(x)$ is irreducible. Is it also true for arbitrary finite positive integer $n$?
AI: Yes...but I have not seen a short pr... |
H: What does a two by two union of disjoint sets mean in $A_1, A_2, A_3$, for every pair $i,j \in {1, 2, 3}$ and $i \neq j$?
I know the definition of a disjoint set, but it is the first time that I have heard about a two by two union in three sets $A_1, A_2, A_3$ when
$\cap_i^3 A_i$ $=\varnothing$ and $A_i \cup Aj \n... |
H: Understanding Sum Property of determinants
If $A=B+C$, where $B=\begin{bmatrix}a &b &c\\
d& e& f\\
g &h& i\end{bmatrix}$ and $C=\begin{bmatrix}k &k &k\\
0& 0& 0\\
0 &0& 0\end{bmatrix}$, then $|A|=|B|+|C|$. Since $|C|=0$, so, $|A|=|B|$.
But, if we consider the properties of determinants, then $\left| \begin{arra... |
H: Understanding when to use residue theorem and when Cauchy's formula to solve integrals
This integral made me wonder, what should be used:
$ \underset{|z-3 \pi|=4}{\int} \frac{1}{z \sin{z}} dz$
Here $0$ is not a relevant pol since it's not in the circle. so the 3 relevant pols are:
$z_0 = -4 \pi \qquad z_1 = -3 \pi ... |
H: Line Integrals given points
I am stuck on the following problem:
Evaluate $\int_c xdx + ydy +zdz$ where $C$ is the line segment from $(4,1,1)$ to $(7,-2,4).$
I found the line equations (I believe that's what they're called) for $x, y,$ and $z$, getting $$x(t)=3t+4$$ $$y(t)=-3t+1$$ $$z(t)=3t+1$$
Then the derivatives... |
H: Can we add a structure notion on set relations without invoking choice?
To $\sf ZF$, can we add a primitive one place function symbol $\mathcal S$ [read as the structure of set relation _] such that for any binary relations $Q,R$ that are sets, we have: $$1. \ \ \mathcal S (Q)= \mathcal S(R) \iff Q \text{ isomorphi... |
H: Boundary of a cube
I was reading the chapter "Integration on Chains" from Spivak, and I was trying to understand the intuition behind the definition of the boundary of a cube:
Initially we define
$I^n\colon [0,1]^n\to\mathbb{R}^n$ by $I^n(x)=x$ for $x\in[0,1]^n$,
and then, for each $i$ with $1\le i\le n$, we define... |
H: Can Non-Continuous functions still be vectors within a vector space?
In Axler's "Linear Algebra Done Right", there is section of examples of subspaces. There is a sentence that goes:
The set of continuous real-valued functions on the interval $[0,1]$ is a subspace of $\mathbf{R}^{[0,1]}$
I am having trouble trying... |
H: What's the difference between $\mathbb Z\to\mathbb Z^+$ and $\mathbb Z^+\to\mathbb Z$?
What's the difference between $\mathbb Z\to\mathbb Z^+$ and $\mathbb Z^+\to\mathbb Z$? Are they same, and if not, why not?
I encountered this problem in following multiple choice question:
Which one of the following functions is... |
H: Fundamental solution of a first order distributional equation
Which is a solution of $u^{'}+\alpha u=\delta_0$?
What about $u^{'}+f(x)u=\delta_0$?
(Where $u$ is a distribution over an open set $\Omega$ with $\Omega\subseteq\mathbb{R}$, $\alpha$ a real constant and $f\in C^{\infty}(\Omega)$ and $\delta_0$ is the Dir... |
H: The bisector of the exterior angle at vertex C of triangle ABC intersects the circumscribed circle at point D. Prove that AD=BD
The bisector of the exterior angle at vertex $C$ of triangle $ABC$ intersects the circumscribed circle at point $D$. Prove that $AD=BD$.
So what I'm wondering is how to prove this? I've al... |
H: Prove quotient of graded ring with graded ideal is a graded ring
Let $S=\oplus S_i$ be some graded ring and let $I\subset S$ be a graded/homogeneous ideal of $S$. That is to say, $I=\oplus I_i$, where $I_i=S_i\cap I$ (this is equivalent to the property that $I$ has a set of homogeneous generators). Put $R=S/I$. I w... |
H: Prove relationships in momentum problem.
I'm working on a situation where an object $A$ in motion hits a stationary object $B$, and the two exchange momentum. Object $A$ is drawn as a box for simplicity but is meant as something like an arrow that can potentially pierce through object $B$, imparting some of its mom... |
H: Proved by defining the boundary (delta and epsilon)
I would be happy if anyone could give me a detailed answer. I was unable to express Delta using Epsilon.
My thought was (I will use keyboard keys) |x-a| = |x -(- 7)| = |x + 7| <delta | 3x +20 -1 | = |x + 7 + 2x + 12| < |x + 7| + 2| x + 6| I now have an expression ... |
H: Kan extension "commutes" with a certain left adjoint
Let $\mathcal{A},\mathcal{B}$ be small categories, $\mathcal{C}$ a cocomplete category and $\mathcal{D}$ an arbitrary category. Consider functors $F:\mathcal{A}\rightarrow\mathcal{B}$, $G:\mathcal{A}\rightarrow\mathcal{C}$, $R:\mathcal{D}\rightarrow\mathcal{C}$ a... |
H: Is $L^2(\mathbb R)$ isometrically isomorphic with $\ell^2(\mathbb Z)?$
Is $L^2(\mathbb R)$ isometrically isomorphic with $\ell^2(\mathbb Z)?$
My thoughts:
We can define an operator $\mathcal L:L^2(\mathbb R)\rightarrow \ell^2(\mathbb Z)$ : $\mathcal Lf=\{\hat f(ξ)\}_{ξ\in \mathbb Z}$
(obviously $\mathcal L$ is lin... |
H: Equivalence of contrapositive and contradiction proofs with quantifiers
I have read that contraposition proof is a special case of contradiction proof. For example, the conditional statement: $P \rightarrow Q$, both proofs suppose $\neg Q$. If we show the contradiction $P \wedge \neg P$, then both proofs are equiva... |
H: If $a$, $b$, $c$, $d$ are positive reals so $(a+c)(b+d) = 1$, prove the following inequality would be greater than or equal to $\frac {1}{3}$.
Let $a$, $b$, $c$, $d$ be real positive reals with $(a+c)(b+d) = 1$. Prove that $\frac {a^3}{b + c + d} + \frac {b^3}{a + c + d} + \frac {c^3}{a + b + d} + \frac {d^3}{a + ... |
H: Real analysis existence of continuous functions
Does there exist continuous onto function from $[0,1)$ to $(0,1)$.
I have made some conclusions. Such function can not be one in a neighborhood of zero.
Please help.
AI: Consider the function $f:[0,1)\rightarrow(0,1)$ defined as $f(x) = 1/2 + 1/2*x*\sin(\frac{1}{1-x})... |
H: How to find the vertices of a triangle formed from 3 vector lines
Can someone help me find the answer to this question?
Three lines which form a triangle have vector equations:
$$r=16i-4j-6k+\lambda(-12i+4j+3k)$$
$$r=16i+28j+15k+\mu(8i+8j+5k)$$
$$r=i+9j+3k+v(4i-12j-8k)$$
Find the position vector of each of the thr... |
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