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H: Find an equation on the cone which is tangent plane is perpendicular to a given plane
Let $z=\sqrt{x^2+y^2}$ be cone. Find an equation of each plane tangent to the cone which şs perpendicular to the plane $x+z=5$
I have learnt the solution of such question for parallel at previous question I asked today.
But now... |
H: Simpson's Rule, find an $N$ for $\;\int_1^5 \ln x \, dx,\;\; $ error $\leq 10^{-6}$
$$\int_1^5 \ln x \, dx\;\qquad\text{ error } \leq 10^{-6}$$
I know that my $K_4 = 24$ since the fourth derivative is $24x^{-5}$
$$\frac{24 \cdot 4^5}{180 N^4} \leq 10^{-6}$$
$$\frac{24 \cdot 4^5}{10^{-6}} \leq 180 N^4$$
$$\frac{24 \... |
H: Distance of a function from a subspace
Let $f \in L^2([-a,a])$. Trying to find $\mathrm{dist}(f,S)$ in $L^2([-a,a])$ (where S is the subspace of real polynomials of max degree $2$, like $a+bx+cx^2$) and knowing that $\langle f,a\rangle=0$ and $\langle f,bx\rangle=0$, can we proceed by considering just
$$
\left(\in... |
H: Prove inequality by induction
Once again, I'm stuck in a demonstration by induction, this time, it's really proving that an inequality is valid. So, here is the inequality:
Prove that $\binom{2n}{n} \geq (n+5)^2 \ \forall n \geq 5, n \in \mathbb{N} $
Then, what I wanted to prove is that:
$\binom{2n+2}{n+1} \geq ... |
H: Solving for $x$ in $3^{2x+1} = 3^x + 24$
I'm having trouble solving this equation step by step:
$$3^{2x+1} = 3^x + 24$$
I've tried to take the log of both sides but then I am stuck with the right hand side being $\log(3^x + 24)$. I've found the answer to '$x$' by trial and error but cannot arrive at the answer othe... |
H: Find a point on a parabola that's closest to another point.
Find the point on the parabola $3x^2+4x-8$ that is closest to the point $(-2,-3)$.
My plan for this problem was to use the distance formula and then that the derivative to get my answer. I'm having a little trouble along the way.
$$ d = \sqrt{(x_1-x_2)^2+(... |
H: Show that $f \in c_0^*$ and $||f||=\sum_{j=1}^{\infty} \frac{1}{j!}$
Let $$\begin{eqnarray}
f: c_0 & \to & \mathbb{R}\\
(x_i)_1^{\infty} & \to & \displaystyle \sum_{j=1}^{\infty} \frac{x_j}{j!}\\
\end{eqnarray}$$
Show that $f \in c_0^*$ and $||f||=\sum_{j=1}^{\infty} \frac{1}{j!}$.
I can show that $f \in c_0^*$ b... |
H: Write an expression in powers of $(x+1)$ and $(y-1)$ for $x^2+xy+y^2$
Write an expression in powers of $(x+1)$ and $(y-1)$ for $x^2+xy+y^2$
I calculated
$f_x=2x+y $
$f_{xx}=2 $
$f_y=x+2y$
$f_{yy}=2$
And then what I need to do?
What is the formula to solve the question ?
AI: Let $u=x+1$, $v=y-1$, then
$$
x^2+xy+... |
H: Force for electron movement $F= \frac{k}{d^2}$
An electron is fixed at $ x = 0$. Electrons repel each other with formula $F= \frac{k}{d^2}$ where k is proportionality constant. Find the work done in moving a second electron along the x axis from x = 10 to x = 1.
I don't know what this means or where to start, there... |
H: Proving a statmenet about convergence of complex sequence
Let $x_k \in \mathbb C$ for $k \in \mathbb N \cup {0}$ and let $y_k = \frac{(x_0 + x_1 + ... + x_k)}{k+1}$. We want to prove that if $x_k$ converges to $x$ ($x \in \mathbb C$) as $k \rightarrow \infty$ then $y_k$ also converges to $x$ as $k \rightarrow \inft... |
H: $p$-adic Ring extensions vs. "ordinary" Ring extensions
I read about inverse limits in this post, and found the example by Arturo Magidin quite interesting (his "approximate" solution of $x^2 = -1$ in $\mathbb Z$).
By his construction we get a Ring which is an extension of $\mathbb Z$, i.e. the ring of $p$-adic (he... |
H: Why it is important to find largest prime numbers?
It always takes a lot of effort and money to find the next largest prime number. Why is it so important to do this work and what is the application those numbers?
AI: Just to add to the previous answers: Usually, part of the discovery of these mathematical curiosit... |
H: An Equivalence Relation: Introspection into a Particular Well-Defined Quotient
DATA:
Let $f:\mathbb{Z}\setminus \{0\}\rightarrow \mathbb{N}$ be a function defined by
$$f(n) = \{k~:~n=2^km,~m\in \cal{O}\},$$
where $\cal{O}$ is the set of odd integers.
Let $v:\mathbb{Q}\setminus \{0\}\rightarrow \mathbb{Z}$ be ... |
H: If every open subset of R is a disjoint union of open intervals, the number of the intervals is at most countable.
Q:
Assume that every open subset of R is a disjoint union of open intervals.
Show that the number of the intervals is at most countable.
Could you give me some help to solve this problem?
Since R is un... |
H: Dirac Orthonormality Proof - Can't Make Sense of Complex Integral
I'm having trouble rationalizing a particular statement that is, surely, present in many quantum mechanics textbooks. The following statement comes from the orthnormalization condition for eigenfunctions of the wavefunction, $\Psi (x, t) $, subject... |
H: What's $P$ and what's $Q$ in this classic proof of the irrationality of $\sqrt 2$?
In this proof extracted from the Wikipedia
A classic proof by contradiction from mathematics is the proof that
the square root of 2 is irrational. If it were rational, it could
be expressed as a fraction $a/b$ in lowest terms, w... |
H: how to show that $\int_0^1 \frac{t^{s-1}}{\sqrt{1-t^2}} d t = \frac{1}{2} B\left(\frac12, \frac{s}{2}\right) $
how to show that $$\int_0^1 \frac{t^{s-1}}{\sqrt{1-t^2}} d t = \frac{1}{2} B\left(\frac12, \frac{s}{2}\right) = \dfrac{\sqrt{\pi}\, \Gamma\left(\frac{s}{2}\right)}{2 \Gamma\left(\frac{s+1}{2}\right)}$$
AI:... |
H: How close could two local maxima be?
How close could two local maxima be?
Def
Let $f$ be a real function defined on a metric space $X$. We say that $f$ has a local maximum at a point $p \in X$ if there exists $\delta > 0$ such that $f (q) < f(p)$ for all $q \in X$ with $d(p, q) < \delta$.
Local minima are defin... |
H: Solve for x, when $ \log_3 (2 - 3x) = \log_9 (6x^2 - 19x + 2)$
How do you deal with the different bases when solving the equation:
$$\log_3 (2 - 3x) = \log_9 (6x^2 - 19x + 2)$$
I'm going round in circles trying to reconcile the bases.
AI: Raise both sides to the $9$th power to get:
$$ 9^{\log_3(2 - 3x)} = 9^{\log_9... |
H: The connection between mathematical induction and implication
What is the connection between mathematical induction and implication?
I always see that mathematical induction is about
$$P(k)\implies P(k+1).$$
From what I know, mathematical induction works by finding a way to transform $P(k)$ into $P(k+1)$ and if you... |
H: Revisted$_2$: Are doubling and squaring well-defined on $\mathbb{R}/\mathbb{Z}$?
Define a relation $\sim$ on $\mathbb{R}$ as follows: for any $a,b \in \mathbb{R}$,
$$a\sim b \iff a-b\in \mathbb{Z}.$$
Let $S=\mathbb{R}/{\sim}$. That is, $S$ is the set of equivalence classes of elements of $\mathbb{R}$ under the equi... |
H: Examples of a monad in a monoid (i.e. category with one object)?
I've been trying to figure out what having a monad in a monoid (i.e. a category with one object) would mean.
As far as I can tell it would be a homomorphism (functor) $T : M → M$, with two elements (natural transformation components) $\eta, \mu : M$, ... |
H: Solving a problem using Cauchy's residue theorem, is there more to it?
Let $z_1,...z_n$ be distinct complex numbers. Let $C$ be a circle around $z_1$ such that no other $z_j$ is in $C$ for $j>1$. Let $$f(z) = (z-z_1)(z-z_2)...(z-z_n)$$ Find $\oint_{C}{\dfrac{\mathrm{d}z}{f(z)}}$.
Attempt: Using Cauchy's Residue Th... |
H: I couldn't find the fault in $B_X(a,\epsilon)\times B_Y(b,\epsilon)=B_{X\times Y}((a,b),\epsilon)$
I know that the product of two balls of equal radius in metric spaces is not necessarily a ball in the product space.
But I couldn't identify the fault in the proof where I showed
$B_X(a,\epsilon)\times B_Y(b,\epsi... |
H: Why is the sample correlation coefficient not $1?$
A reasonable value for the sample correlation coeffcient
$\rho$
between daily maximum tem-
peratures and daily ice cream sales would be
$A) 0$
$B) 1$
$C) 0.7$
$D) -0.7$
I am taking intro stats for the first time and this was a question that confounded me.
... |
H: On upper central series
Let $G$ be a group and Z(G) be the center of $G$. This is the upper central series $$1=Z_{0}(G)\leq Z_{1}(G)\leq...,$$ defined by $\frac{Z_{n+1}(G)}{Z_{n}(G)}=Z\left(\frac{G}{Z_{n}(G)}\right)$.
Now prove that $Z_{i}\left(\frac{G}{Z_{j}(G)}\right)=\frac{Z_{i+j}(G)}{Z_{J}(G)}$.
Attempt: ... |
H: Why does the series $\sum_{n=1}^∞ \ln ({n \over n+1})$ diverges? And general tips about series and the logarithm
Why does the series $\sum_{n=1}^∞ \ln ({n \over n+1})$ diverges? I'm looking for an answer using the comparison test, I'm just not sure what I can compare it to.
And can I have some tips on what to look... |
H: Choices for course selection
I read this question on brilliant.org:
Winston must choose 4 courses for his final semester of school. He must take at least 1 science class and at least 1 arts class. If his school offers 4 science classes, 3 arts classes and 3 other classes, how many different choices for classes doe... |
H: Analog of modus ponens for semantics
To pose my question, I first must first quickly define a language, a model, semantics for such models, and a logical system called S4O.
Consider a language $L$ with a set $PV$ of propositional variables, Boolean connectives $\neg$ and $\vee$(with the other boolean connectives a... |
H: $a^2+b^3=c^5$Are there infinitely many solutions?
I am having troubles figuring whether there are infinitely many integer solutions to the following equation: $$a^2+b^3=c^5$$
This is just a problem I thought of on my own, so sorry in advance if this is already an open problem.
The way I tried to solve it is this:... |
H: $\mathbb{R}/{\sim}$: A Question about the Formal Definition of a Quotient
For an equivalence relation $\sim$ what is $\mathbb{R}/{\sim}$? I mean explicitly and formally...
AI: Define on $\mathbb R$ an equivalence relation $\sim$ that's reflexive, symmetric and transitive and for all $x\in \mathbb R$ let
$$[x]=\{y\i... |
H: If $f$ is any function and $X_1 ... X_n$ are IID, are $f(X_1), f(X_2), ..., f(X_n)$ IID?
Suppose that $f : \mathbb{R} \rightarrow \mathbb{R}$ be any function and let $X_1, X_2, ..., X_n$ be IID real-valued random variables drawn from any arbitrary distribution. Is it guaranteed that $f(X_1), f(X_2), ..., f(X_n)$ a... |
H: What makes a Maclaurin Series special or important compared to the general Taylor Series?
I realize that the Maclaurin Series is a special form of the Taylor Series where the series is centered at $x=0$, but I have to wonder what's special about it such that it deserves its own special designation? On that point, h... |
H: Use a known Maclaurin series to derive a Maclaurin series for the indicated function.
$$f(x)=x\cos(x)$$
I'm not quite sure how to do this. I did two others, which I presume is the right way to do it, as follows:
\begin{align}
e^x&=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots\\
e^{-x/4}&=1-\frac{x}{4}+\fr... |
H: $h:\mathbb{R}_{/\sim}\rightarrow \mathbb{R}^2$: A Bijection from a Quotient Space to the Unit Circle (Geometrically Considered)
NOTE: This is not a duplicate.
Define a relation $\sim$ on $\mathbb{R}$ as follows: for any $a,b \in \mathbb{R}$,
$$a\sim b \iff a-b\in \mathbb{Z}.$$
Let $S=\mathbb{R}_{/\sim}$. That is,... |
H: combination related question
suppose that dinner cooker has 500 mint,500 orange and 500 strawberry,and he wished to do packets containing 10 mint,5 orange and 5 strawberry,question is what is a maximum number of packets he can make by this way?
so as i think,it is a combination related problem,which means... |
H: Faster mental arithmetic with powers of 10
Please excuse me if this question is too vanilla. What's a faster way to do mental arithmetic involving powers of ten? I've always had to do this and I do it using scientific notation which I'm equivocal about, but am finding myself roaringly slow.
Here's what I do. Supp... |
H: Eigenvalues of a self-adjoint operator necessarily distinct?
Let's say we have a self-adjoint operator acting on an inner product space (real or complex), represented, of course, by a self-adjoint matrix.
I'm looking at the proof for spectral theorem in which you build up a basis out of eigenvectors relying on the ... |
H: number leaving different remainders with different divisors
number when divided by 17 leaves remainder 3 and when divided by 16 leaves remainder 10 and is divisible by 15
find the smallest number in the series
i tried the conventional method but it gave me wrong answer
so please help
AI: By the Chinese Remainder T... |
H: Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$
Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$.
Is there an easy way to prove this?
AI: From $\sin(p(0))=p(0)$, we get $p(0)=0$. Therefore $\sin(p(2k\pi)) = p(\sin(2k\pi)) = p(0) = 0$ for every integer $k$. In... |
H: If $f$ is compact is $f$ continuous?
If $f$ is a compact function (image of every compact set is compact) is $f$ continuous?
Attempt: I can't find a counterexample. I can't prove it. I only know how to prove the converse.
AI: Let
$$f:\Bbb R\to\Bbb R:x\mapsto\begin{cases}
0,&\text{if }x\in\Bbb Q\\
1,&\text{if }x\in... |
H: Rationalizing a numerator
I'm having trouble rationalizing a numerator with radicals. After multiplying the conjugate I get 0. Does anyone know where I went wrong?
\begin{align}
\frac{\sqrt{2+y} + \sqrt{2 - y}}{y} & = \left(\frac{\sqrt{2+y} + \sqrt{2 - y}}{y}\right) \left(\frac{\sqrt{2+y} - \sqrt{2 - y}}{\sqrt{2+y}... |
H: Factoring a given polynomial
I am trying to factor the polynomial
$$(a-1)x^2 + a^2xy+(a+1)y^2.$$
The problem previous to it in the book uses the method of factoring a polynomial of the form
$$ax^2 + bx +c$$
by inspection, and the problem following it uses a formula related to cubes (I thought it's best you know).... |
H: Number of primitive roots modulo p; asymptotic behavior
I know that number of primitive roots modulo p is $\varphi(p-1)$, where $\varphi$ is Euler totient function. I'm actually interested in asymptotic behavior of $\frac{\varphi(p-1)}{p-1}$ (percentage of primitive roots among elements of $\mathbb{Z}_p^*$ ).
It's ... |
H: Can a "nearly" harmonic series converge to an irrational number (say, $\pi$)?
Suppose you take the set $X=\{\sum_{k \in A} \frac{1}{k}: A \in \mathcal{P}(\mathbb{N} \setminus \{1\})\}$. Suppose that we agree to introduce the symbol $\infty$ to encompass the cases where the series $\sum_{k \in A} \frac{1}{k}$ diverg... |
H: find angle in triangle
Let us consider problem number 21 in the following link
http://www.naec.ge/images/doc/EXAMS/math_2013_ver_1_web.pdf
It is from georgian national exam, it is written (ამოცანა 21), where word "ამოცანა" means amocana or problem. We should find angle $\angle ADE$. I have calculated angle $B$, wh... |
H: Estimating the integral $\sqrt{n}\cdot \int\limits_0^\pi \left( \frac{1 + \cos t}{2} \right)^n dt$
Consider the sequence $\{a_n\}$ defined by
$$ a_n = \sqrt{n}\cdot \int_{0}^{\pi} \left( \frac{1 + \cos t}{2} \right)^n dt.$$
An exercise in Rudin, Real and Complex Analysis, requires showing that this sequence is
con... |
H: Finding a Jordan base
Let $\frac{\mathrm{d} }{\mathrm{d} x}: \mathbb{R}\underset{\leqslant 3}{[x]}\rightarrow \mathbb{R}\underset{\leqslant 3}{[x]}$ be the derivative operator. I am trying to find a base $B\subset V$ and a Jordan block matrix $J$ so that $\left [ \frac{\mathrm{d} }{\mathrm{d} x} \right ]_{B}=J$.
I ... |
H: Show that $span({v_1,...v_n})=span({v_1,...,v_{n-1},w})$
This is what's given:
$v_1,...v_n,w ∈ V$ and $v_1+...+v_n+w=0$
then I need to show that $span({v_1,...v_n})=span({v_1,...,v_{n-1},w})$
I could think of a way to show that this is true if I was sure that every $v_i$ and w were $0$, but they arent, right?
Would... |
H: A morphism which fixes one root of an irreducible polynomial must also fix the others.
Let $E/K$ be a field extension, let $p(x)$ be an irreducible polynomial in $K[x]$ which splits in $E$ with roots $\alpha_1$, $\alpha_2$, etc., and let $\sigma$ be an automorphism of $E$ which fixes $K$. Then $\sigma$ fixes $p(x)$... |
H: What is periodic solution to a PDE?
If I have a PDE $$u_t = Au + f$$ with conditions $$u(0,x) = u(T,x)$$
then if it has a solution, why is the solution called periodic? Isn't it only true that $u(0) = u(T)$? It does not follow that $u(0+\epsilon) = u(T+\epsilon)$, which I would have thought is what periodic should ... |
H: I need help with an assignment question please for numerical methods
Consider the function $f (x) = xe^x - 2,$ we want to study the properties of $f (x)$ so that we can apply numerical methods to solve the equation $f (x) = 0$.
Which option is false ?
the function, $f (x)$ is well defined and continuous for all $x... |
H: What makes irreducible representations nice?
Let $\mathcal{A}$ be a C*-algebra and $(H,\pi,\Omega)$ a cyclic representation.
What does it intuitively mean if the representation is irreducible? From what I've read, irreducible representations are nice and I can be happy if my algebra can be represented in such a way... |
H: Seeking an analytic proof of a vector identity
Show that for any vectors $\bf{u_1},\bf{u_2},\bf{v_1},\bf{v_2}\in\mathbb R^3$, we have $$(\bf{u_1}\times\bf{v_1})\cdot(\bf{u_2}\times\bf{v_2})=
\left|\begin{matrix}
\bf{u_1}\cdot\bf{u_2} & \bf{u_1}\cdot\bf{v_2}\\
\bf{v_1}\cdot\bf{u_2} & \bf{v_1}\cdot\bf{v_2}
\end{matr... |
H: Discrete topology on infinite sets
I want to prove the following: Let $X$ be an infinite set and $\tau$ a topology on $X$. If every infinite subset of $X$ is in $\tau$, then $\tau$ is the discrete topology on $X$.
Proof. Let $x\in X$. There exist two infinite subsets $A$ and $B$ of $X$ such that $\{x\}=A\cap B$. So... |
H: Elliptic Cylinder Coordinates Integral
Could somebody show me an example of an integral that becomes easy when you change to elliptic cylinder coordinates $x = a\cosh(\eta)\cos(\phi)$, $y = a\sinh(\eta)\sin(\phi)$, $z = z$, or even (&?) an integral where you would think to change your variables to these coordinates... |
H: Proving that for certain ring of algebraic integers $R$, $R/bR$ is finite
This is a part of proof I try to understand.
The situation is the following:
Suppose that $a,b,x,y$ are algebraic integers such that $b \neq 0$ and $ax+by=1$. Set $K:=\mathbb{Q}(a,b,x,y)$ and $R:=O_K,$ that is, a subring of all algebraic int... |
H: Formula to fit a straight line to data
Theorem (Best Linear Prediction of $Y$ outcomes): Let $(X,Y)$ have moments of at least the second order, and let $Y'=a+bX$. Then the choices of $a$ and $b$ that minimize $Ed^2(Y,Y')=E(Y-(a+bX))^2$ are given by $$a= \left(E(Y) - \dfrac{cov(x,y)}{var(x)}\right)E(X)$$ and $$b=\d... |
H: Abelian $2$-groups
Is every abelian group $A$ where every element has order two isomorphic to a direct product of cyclic groups of order two, $A\cong C_2\times C_2\times\ldots$?
I ask because I used this "fact" in one of my old answers here (which is relevant to some work I am doing), and have just realised that th... |
H: Why does $\gamma=\lim_{s\to1^+}\sum_{n=1}^{\infty}\left(\frac{1}{n^s}-\frac{1}{s^n}\right)=\lim_{s\to0}\frac{\zeta(1+s)+\zeta(1-s)}{2}$?
To be clear, I'm having trouble with proving both equalities, and would appreciate a hint. I'm also not sure why $1^+$ must be used as opposed to $1^-$. I'm not sure about the def... |
H: Explicit isomorphism $S_4/V_4$ and $S_3$
Let $S_4$ be a symmetric group on $4$ elements, $V_4$ - its subgroup, consisting of $e,(12)(34),(13)(24)$ and $(14)(23)$ (Klein four-group). $V_4$ is normal and $S_4/V_4$ if consisting of $24/4=6$ elements. Hence $S_4/V_4$ is cyclic group $C_6$ or a symmetric group $S_3$ (re... |
H: Proving statement - $(A \setminus B) \cup (A \setminus C) = B\Leftrightarrow A=B , C\cap B=\varnothing$
I`m trying to prove this claim and I need some advice how to continue,
$$(A \setminus B) \cup (A \setminus C) = B \Leftrightarrow A=B , C\cap B=\varnothing$$
what I did is:
$$(A \setminus B) \cup (A \setminus C) ... |
H: Prove that $9\mid (4^n+15n-1)$ for all $n\in\mathbb N$
First of all I would like to thank you for all the help you've given me so far.
Once again, I'm having some issues with a typical exam problem about divisibility. The problem says that:
Prove that $\forall n \in \mathbb{N}, \ 9\mid4^n + 15n -1$
I've tried usi... |
H: Solving a simple quadratic equation
I have problems every time I face a quadratic equation. What can I do to learn how to solve them? Can anyone please show me how to solve the one below and explain the basic principle of solving quadratic equations.
$$x^2- xa - ab = 0$$
AI: There is a formula:
$$Ax^2 + Bx + C = 0 ... |
H: Find the value of $\cos^{12}\theta + 3\cos^{10}\theta + 3\cos^{8}\theta + \cos^6\theta + 2\cos^4\theta + 2\cos^2\theta - 2$
We are given that $\sin\theta + \sin^3\theta + \sin^2\theta = 1$
Find the value of $\cos^{12}\theta + 3\cos^{10}\theta + 3\cos^{8}\theta + \cos^6\theta + 2\cos^4\theta + 2\cos^2\theta - 2$
Now... |
H: form of groups of motions of tessellations
I have read from the book "Mathmatics and Its History" by John Stillwell. In Section 18.6 it is about complex interpretations of geometry. The book says:
The triangle and hexagon tessellations have similar group of motions, generated by
$z \mapsto z+1 ,z \mapsto z+\tau,z ... |
H: Stuck at proving convergence of the series that is dependent on a converging series
Suppose $\sum_{n=1}^{\infty}{a_n}$ converges, and $a_n > 0$. Does $$\sum_{n=1}^{\infty}{\dfrac{\sin(\sqrt{a_n})}{\sqrt{n}+na_n}}$$ converge or diverge?
Attempt: I was able to prove that it diverges, as shown below, but could not fin... |
H: Fix point of squaring numbers mod p
Take the set of integers $\{0, 1, .., p-1\}$, square each element, you get the (smaller) set of quadratic residues. Repeat until you get a fix point set. The size of this set is a function of $p$. Does this function have a name? How can I efficiently calculate it?
Some values:
$$... |
H: Finding diagonal and unitary matrices
Let $A=\begin{pmatrix}
1 & 1+i\\
1-i & 2
\end{pmatrix}$ I'm trying to find a diagonal matrix $D$ and a unitary matrix $U$ so that
$U^\star AU=D$.
(We define $U^*=\overline{U}^t$ ).
I found the eignvalues: $\lambda_1 = 3$, $\lambda_2 = 0$.
The eignvectors are: $V_1=\begin{p... |
H: Linear homeomorphisms mapping an orthonormal basis into another orthonormal basis
Consider $L^2(A)$ and $L^2(B)$. If $\{a_i\}$ is an o.n basis of $L^2(A)$, how many linear homeomorphisms $F:L^2(A) \to L^2(B)$ do there exist such that $Fa_i$ is an orthonormal basis of $L^2(B)$?
Is this a very restrictive assumption... |
H: find parameter for maximize area
suppose that we have Cartesian coordinate system.and suppose that we have three point which depend on parameter $t$,where t belongs to $(0,1)$;points are
$A(cos(3-t),sin(3-t))$
$B(cos(t),sin(t))$
$C(-cos(t),-sin(t))$
goal: find $t$ for which area of triangle $ABC$ is maximum
fi... |
H: Residues at poles
What is the residue of $$f(x)=\frac{1}{(x^2+1)^a}$$ at $x^2=\pm i$, where $0<a<1$ ? My intuition tells me that there must be a non-zero residue, but my attempts to compute tells me the residue is $0$. How can this be so when $x^2+1=0$ when $x=\pm i$ ?
AI: Although $(1+z^2)^a$ is not analytic in a ... |
H: A Well-Defined Bijection on An Equivalence Class
DATA:
Let $f:X\rightarrow Y$ be a surjective function. Define a relation $\sim$ on $X$ by
$$a\sim b~\iff~f(a)=f(b).$$
Let $S=X/{\sim}$, namely let $S$ be the set of equivalence classes of elements of $X$ under the equivalence relation $\sim$. Define a function $q:X\... |
H: How many numbers can I make with subseries of $\sum_{n=1}^{\infty} \frac{1}{2^n}$?
Given $\sum_{n=1}^{\infty} \frac{1}{2^n}$, what real numbers in $\left[ \frac{1}{2},1 \right]$ can I generate with subseries of this series?
Obviously we have every power of $\frac{1}{2^n}$ (by taking single terms), as well as 1 itse... |
H: Equation $(a-3)cb=a(c+b)$ for natural numbers.
Let $a$, $b$, and $c$ be positive integers. Suppose that $c \leq b \leq a$ and that they satisfy the relation
$$
(a-3)cb=a(c+b).
$$
What can be said about the solutions?
AI: This equation can be rewritten as
$$\frac{3}{a}+\frac{1}{b}+\frac{1}{c}=1.$$
Now
If $c>5$, t... |
H: Proving the length of a circle's arc is proportional to the size of the angle
How can I prove that:
The length of the arc is proportional to the size of the angle.
Every book use this fact in explaining radians and the fundamental arc length equation $s = r\theta$. However no book proofs this fact.
Is this fact s... |
H: An elementary problem in Group Theory: the unique noncyclic group of order 4
Following the advice given in this question, I have started to study Group Theory from the very basics. My reference text is Abstract Algebra by Dummit and Foote. While going through the exercises (page 24) I found one problem which requir... |
H: Prove that in an obtuse triangle the orthocentre is the excenter of the orthic triangle
Consider an obtuse angled $\Delta ABC$ with altitudes $AD, BE, CF$ concurrent at $H$. Consider the orthic triangle $\Delta FED$. Extend $ED$ to $D'$ and $EF$ to $F'$. Prove that $\angle FDH = \angle HDD'$ and $\angle DFH = \angl... |
H: law of large number modified statement
The weak law of large number states that, given $Y_n = \sum_{k=1}^{n} X_k$, where $X_k$ are random variables independent and identically distributed with finite expectation $\mu$,
$$
\forall \delta>0, \forall \epsilon>0 \, \, \exists N>0\, \,\, s.t.\, \, \, P ( |Y_n/n - \mu| ... |
H: numerical linear algebra 101
since I'm a programmer and I need linear algebra, I'm starting considering how to teach myself a little of numerical linear algebra, not really optimize things right from the start, but I would like to get how to decompose and "linearize" a matrix or a vector in a form that can be writt... |
H: What's the symbol m in this sum?
I'm supposed to write some code to calculate the inertia moments of a shape, but I am afraid I have been given too little information.
The matrix that I must obtain is this one:
$$
\begin{vmatrix}
J_{xx} = \sum \limits_i m_i y_i² &
J_{xy} = -\sum \limits_i m_i x_i y_i\\
J_{xy} = -... |
H: Verify that the six matrices form the group
I solved the problem myself and I want to check if my solution is legitimate.
My solution usually has partial errors or is not solid enough.
Thank you!
Verify that the six matrices $$\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0&0&1\end{bmatrix},\begin{bmatrix}0&1&0\\0&0&1\\1&0... |
H: Is there an injective operator with a dense nonclosed range?
Let $H$ be an infinite dimensional separable Hilbert space.
Is there an operators $A \in B(H)$ such that $Im(A) \subsetneq \overline{Im(A)} = H$ and $Ker(A) = \{0\}$ ?
Bonus : We can build such operators by using some compact or shift operators (see... |
H: Resolving a counterexample to the most fundamental probability concept; mutually exclusive and independence
Suppose you roll a die $(1-6)$ and toss a coin each once. Let $A$ be the event that I get either heads or tails (let's say tails) and $B$ be the event that I roll a number $(1 - 6)$, (let's say $2$).
So the s... |
H: Is there a specific reason to choose $3$ and $1$ for evaluating these limits?
I'm reading Gemignami's Calculus and Statistics.
There are two examples of limits that left me confused:
Example 5. Let $f$ be defined by $f(x)=x^2$. We now evaluate
$$\lim_{x\rightarrow 0}\frac{f(\color{red}{3}+h)-f(\color{red}{3})}{h}$... |
H: Proof that the topology of uniform convergence on $C(\mathbb R, X)$ is finer than the topology of pointwise convergence.
The proof is from R. Engelking, General Topology and it relies on the following fact
$$
f \textrm{ is continuous} \Leftrightarrow f(\overline{A}) \subseteq \overline{f(A)}. \qquad (1)
$$
Propos... |
H: Why is it so difficult to prove that the discrete Fourier transform (DFT) cannot be calculated in faster time than $N \log N$?
As the title says, why is it so difficult to prove that the discrete Fourier transform (DFT) cannot be calculated in faster time than $O(N \log N)$?
This is a famous open problem in mathema... |
H: Partitions and Orbit Sizes
If $U,V \subset S_n$ are subgroups with $S_n//U = \{id,g_2,...,g_e\}$ and $\alpha_j$ is $\frac{1}{j}$ times the number of $i\in [e]$ s.t $[V:V \cap g_i U g_i^{-1}]=j$ then $(\alpha_1,...,\alpha_e)$ is a partition of $e$.
Now let $U=V \in \{V_4,D_4\}$ and $n=4$ ($V_4$ is the group with onl... |
H: solve easy problem with group action
Here is a simple problem which can be found in every elementary group textbook:
$H,K$ are finite subgroups of group $G$, then
$$|HK|=\dfrac{|H|\cdot|K|}{|H\cap K|}$$
Could you help me to prove it with group action? Thank you in advance
AI: Show that the group $H^{\text{op}... |
H: do fibres of morphisms of Noetherian rings have finite Krull dimension?
Let $f:A \rightarrow B$ be a morphism of Noetherian rings. Let $p \in Spec(A)$ and let $C=B \otimes \kappa(p)$ be the fibre over $p$. Is it true that $\dim C < \infty$? How can we see that?
Remark: $B \otimes \kappa(p) \cong B_S/pB_S$ where $S$... |
H: Show $\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) = \cos^2x - \sin^2x$
Show $\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) = \cos^2x - \sin^2x$
I have got as far as showing that:
$\cos(x+y)\cos(x-y) = \cos^2x\cos^2y -\sin^2x\sin^2y$
and
$\sin(x+y)\sin(x-y) = \sin^2x\cos^2y - \cos^2x\sin^2y$
I get stuck at showing:
$\cos^2... |
H: A tough calculation involving hyperbolic contangents.
From here: http://en.wikipedia.org/wiki/Brillouin_function
Define $$B_j(x)=\frac{2j+1}{2j} \coth \left( \frac{2j+1}{2j} x \right) - \frac{1}{2j} \coth \left( \frac{1}{2j} x \right)$$
I want to do this calculation ($m,j$ are integers):
$$\langle m \rangle = \sum_... |
H: Calculating $\lim_{x\to 0}\left(\frac{1}{\sqrt x}-\frac{1}{\sqrt{\log(x+1)}}\right)$
Find the limit
$$\lim_{x\to 0}\left(\frac{1}{\sqrt x}-\frac{1}{\sqrt{\log(x+1)}}\right)$$
AI: Hint
$$\frac{\log(1+x)}{x}\to 1$$
$$\frac{\log(1+x)-x}{x^2}\to -\frac{1}{2}$$
Further Hint
$$\displaylines{
\frac{1}{{\sqrt x }} - \f... |
H: Prove $A \bigtriangleup B = B \bigtriangleup A$
I`m trying to prove the following statement:
$$A \bigtriangleup B = B \bigtriangleup A$$
I know that:
$$A \bigtriangleup B = (A \cup B )\setminus (A \cap B )= (A \setminus B) \cup (B \setminus A)$$
I can do that with truth table. but want to prove it by formal way.
An... |
H: Successive Differentiation and division
I am working on successive differentiation. I have ran into some confusion and would like some help with the process of differentiating when dealing with division. Here is the problem that sparked my intentions to post here: $$y = \frac{x^2 + a}{x + a}$$ I am reading Calculus... |
H: How to denote an 'atomic' morphism in category?
I want to distinguish between two disjoint classes of morphisms in a category: (1) those morphisms that are composed of other morphisms (other than identities) and could conceivably be factored into a sequence of other morphisms; and, (2) those morphisms that cannot b... |
H: Addition of ideals
Given a ring $R$ and ideals $A,C$
suppose we have $A + B' =A + B = C.$
I was wondering then what can we say about
relation between $B$ and $B'$.
Clearly, $B$ may not equal $B'$,
but can we say something?
Does it follow that $B= B' + D$
where $D$ is an ideal contained in $A$?
Thanks!
AI: In $\mat... |
H: bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$
I understand that both $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ are of the same cardinality by the Shroeder-Bernstein theorem, meaning there exists at least one bijection between them. But I can't figure out what such a bijection would be. The pap... |
H: How to find $k[x_1,\dotso,x_n]/I$ concretely?
I want to know if there is a way to find $k[x_1,\dotso ,x_n]/I$ in specific cases. For example how can we find concretely the ring $\mathbb{C}[X]/(X^2+1)$? How does one mod out $(X^2+1)$?
AI: Over $\;\Bbb C\;$ :
$$x^2+1=(x-i)(x+i)\implies $$
$$\Bbb C[x]/(x^2+1)=\Bbb C[x... |
H: How to evaluate the definite integral?
How to evaluate the definite integral?
$$\int \frac{7}{3x+1}dx$$
I am having difficulties to finish the question:
Below is what I did:
$$ =\left.\frac{7}{3}\ln|3x+1|\right|_0^4$$
$$=\frac{7}{3}\ln(\dots.$$
AI: Your integration is just fine:
All you have left to do is evaluate ... |
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