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H: Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$
I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\i... |
H: Show that $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd
If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.
Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with
$$a^2\equiv b^2 \mod{8} \Rightarrow 4k^2+4k \equiv 4j^2+4j \mod{8}$$
$$\Ri... |
H: Drawing dynamic circles based on input value
Is there a formula that will allow me to calculate the radius of a circle based on an input value? The input value could be as small as zero or as large as $10^7$, or larger. The circle is restricted to a minimum radius of $10$ and a maximum radius of $100$.
Does anyon... |
H: Evaluation of a complex numbers partial sum
Let $w = e^{i\frac{2\pi}5}$. I would like to evaluate
$$w^0 + w^1 + w^2 + w^3 +...+ w^{49}$$
Can anyone please give me an idea how to evaluate the expression?
Thanks in advance
AI: This is the sum of the fifth roots of unity, ten times over. See here for a discussion o... |
H: Non-parallel vectors confusion
I've got a section in my textbook about non-parallel vectors, it says:
For two non-parallel vectors a and b, if $\lambda a + \mu b = \alpha a + \beta b$
then $\lambda = \alpha $ and $\mu = \beta $
Okay I get that you can equate coefficients and solve for mu and lambda, but how are ... |
H: how to dot product two vectors with different planes?
how to dot product two vectors with different planes?
I have vectors $A$,$B$ and $C$, vectors $A$ and $B$ is on $xy$ plane while vector $C$ is on $xz$ plane. I need to find the dot product of $A.C$ how should I do that? my book says that dot product of two vecto... |
H: How do you deduce X given a set of truth value?
given some truth value, how can we indeed deduce what the form is like? For example,
P Q R X
T T T T
T T F F
T F T T
T F F T
F T T T
F T F F
F F T F
F F F T
Here, given the truth value of X, can you deduce the form of X in term of P,Q,R?? I think if time are given on... |
H: How can i show this inequality?
Let $n>1$ and $a_1,...,a_n \in \mathbb{R}^+$ be such that $\sum a_i=1$. For evey $i$, define $b_i=\sum_{j=1,j\neq i}a_j$. Show that
$\sum_{k=1}^n \dfrac{a_k}{1+b_k}\ge \dfrac{n}{2n-1}$
Thanks a lot for any suggestion.
AI: Note that $b_i=1-a_i$. Thus using AM-HM inequality gives
\be... |
H: Finite equivalence class same cardinality
For an equivalence relation $\sim$, if each partition has a finite number of elements, and $X$ is an infinite set, then is it true that $|X/\sim|=|X|$?
I can prove injectivity one way by defining the map $f:X/\sim\rightarrow X$ by picking an element from the equivalence cla... |
H: Schur's Lemma in Group Theory
The analogue of well celebrated Schur's Lemma in group theory will be
"If $G$ is a finite simple group, and $\phi$ is a non-identity homomorphism from $G$ to $G$, then $\phi$ is an isomorphism".
The proof follows exactly same lines as it is in representation theory.
I would like to a... |
H: Prove $\sin \alpha+\sin \beta+\sin \gamma \geq\sin 2\alpha+\sin 2\beta+\sin 2\gamma $
Prove that $\sin \alpha+\sin \beta+\sin \gamma \geq\sin 2\alpha+\sin 2\beta+\sin 2\gamma $ where $\alpha$ $,\beta$ $,\gamma$ are the angles of a triangle
AI: use
$$\sin{2A}+\sin{2B}=2\sin{C}\cos{(A-B)}\le 2\sin{C}$$
$$\sin{2B}+\si... |
H: determinants of 2 matrices with given property
I have two $3\times3$ integer matrices $A$ and $B$ such that $AB=A+B$. I need to find all possibe values of $\det(A-E)$, where $E$ denotes the identity matrix. Any help is appreciated.
AI: Hints:
$$\bullet\;\;\;\;\;AB=A+B\implies (A-I)(B-I)=I\;,\;\;I:=\text{the unit (i... |
H: Is the function identically zero?
Let $f(x, y)$ be a continuous, real-valued function on $\mathbb{R}^2$.
Suppose that, for every rectangular region $R$ of area 1, the double integral of $f(x, y)$
over $R$ equals 0. Must $f(x, y)$ be identically 0?
AI: Hint: Consider $g(y)=\int_0^1 f(x,y)dx$.
Since the intgeral of $... |
H: Integration with change of variables (multivariable).
The following are the problems that I have been working on. It involves change in variables with 2,3 variables respectively.
(1)Let $R$ be the trapezoid with vertices at $(0,1),(1,0),(0,2)$ and $(2,0)$. Using the substitutions $u = y-x$ and $v = y +x$, evaluate... |
H: Calculating the probabilities of different lengths of repetitions of numbers of length 4
I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't total to 1
e.g.
Here is my rea... |
H: Searching for unbounded, non-negative function $f(x)$ with roots $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$
If a function $y = f(x)$ is unbounded and non-negative for all real $x$,
then is it possible that it can have roots $x_n$ such that $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$.
AI: The funct... |
H: Some questions on Proof of Structure Theorem
I understand the general idea of the proof of Structure Theorem for finitely generated modules over a principal ideal domain but I found it quite difficult to follow some lines of reasoning in the proof. I have spent hours tried to think about it and look for other sourc... |
H: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
Is the following true?
Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a
further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
I ... |
H: Find $a,b,c \in \mathbb {Q}$
Find $a,b,c \in \mathbb {Q}$ such that:
$\left\{\begin{array}{rl} x^3&\in \mathbb Q \\ x&\notin \mathbb{Q}\\ ax^2+bx+c &=0\end{array}\right.$
I tried Vieta's formulas, but seem like it didn't help.
I think $a=b=c=0$ is only solution.
AI: $$a^2x^3=-ax(bx+c)=-b(ax^2)-cax=b(bx+c)-cax=x(b^2... |
H: Solving Modular Equations With Identities
$4+2x≡7 \pmod 8$
Find all possible solutions and note any identities.
Identify how you found the solutions.
Explain what identities are.
AI: $\implies 2x\equiv3\pmod 8=3+8a$ for some integer $a$
So, as $3+8a$ is odd and $2x$ is even for integer $x,$ there is no solution
Mo... |
H: sum of polynoms of given property
I have $P(x)$ a polynomial with degree $n$ ,$P(x) \ge 0$ for all $x \in$ real.
I have to prove that:
$f(x)=P(x)+P'(x)+P"(x)+......+P^{n}(x) \ge 0$ for all $x$.
I tried different methods to solve it but I got stuck.Any suggestion or advice is welcomed.
AI: you can let
$$G(x)=f(x)e... |
H: Relation between a sum of a series and the limit of a sequence
I'm stuck on this question
Let $\{a_{n}\}$ a sequence of real numbers
I need to show the series $\sum_{n=1}^{\infty}(a_{n} - a_{n-1})$ and the sequence $\{a_{n}\}$ are the same nature (convergent or divergent). Additional, I need to give a relation bet... |
H: is a relation R total/linear/well-order
Let $\mathcal{R}$ be a relation on $\mathbb{N}\times \mathbb{N}$ i.e $\mathcal{R}\subseteq(\mathbb{N}\times \mathbb{N})\times (\mathbb{N}\times \mathbb{N})$ s.t $(x,y)\mathcal{R}(z,w)$ iff $x<z$ or $x=z \wedge y\le w$.
a)is $\mathcal{R}$ total order/linear order?
b) does the ... |
H: Find the smallest possible integer that satisfies both modular equations
Find the smallest positive integer that satisfies both. x ≡ 4 (mod 9) and x ≡ 7 (mod 8) Explain how you calculated this answer.
I am taking a math for teachers course in university, so I'm worried about guessing too much in fear that I will te... |
H: probability of getting a double six ($2$ dice) rolling them $24$ times
This is what I got. $\dfrac{1}{6} \cdot \dfrac{1}{6} = 2.78\% \cdot 24 = 66.72\%$
I believe that since it is a six sided dice, since you roll both of them simultaneously it would be $\dfrac{1}{6} \cdot \dfrac{1}{6}$.
So since they are rolling t... |
H: Quadratic Functions
Consider the strictly convex quadratic function $f(x) = \frac{1}{2}x^tPx - q^tx + r,$ where $P \in \mathbb{R}^{n \times n}$ is a positive definite matrix, $q \in \mathbb{R}^n$ and $r \in \mathbb{R}.$ Let $\mathcal{H} := \{H: H \text{ is a }k- \text{dimensional subspace in } \mathbb{R}^n\}.$ Clea... |
H: A question regarding the Power set
In the proofs that I have seen so far for showing that the power set $2^X$ of a set $X$ cannot be in bijection to $X$, the common idea is to assume that there exists a surjection $f \colon X \to 2^X$ and then consider the set
$$
B = \{ x \in X \mid x \notin f(x)\}
$$
Then the argu... |
H: How can you prove Euler's phase angle formula for differential equations?
How can you prove this formula:
$C_1 e^{(\alpha + i\beta) t} + C_2 e^{(\alpha - i\beta)t}=Ke^{\alpha t}\cos {(\beta t + \phi)}$
This gives $x(t)$ in the second-order differential equation for an underdamped system with the characteristic eq... |
H: Understanding the (Partial) Converse to Cauchy-Riemann
We have that for a function $f$ defined on some open subset $U \subset \mathbb{C}$ then the following if true:
Suppose $u=\mathrm{Re}(f), v=\mathrm{Im}(f)$ and that all partial derivatives $u_x,u_y,v_x,v_y$ exists and are continuous on $U$. Suppose further th... |
H: Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
AI: The assertion that the integral is $0$ doesn't really make sen... |
H: Irreducible representation of tensor product
Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$.
What are the irreducible representaions of $\varphi_2 \otimes \varphi_3$?
AI: A result commonly attributed to Clebsch and Gordan gives you the answer:
$$
\varphi_2\ot... |
H: Multiplication in $\mathcal D'(R)$.
I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ goes to infinity in $\mathcal D'(R)$. And not in the case of $\sin^2(nx)$ as $n$ ... |
H: Tutte's 1947 proof and paper on a family of cubical graphs
It is known that if a graph is connected, cubic, simple and $t$-transitive, then $t \le 5$. A proof is given in [Biggs, Algebraic Graph Theory, Chapter 18], and this result is due to [Tutte, ``A family of cubical graphs,'' Proc. Cambridge Philosophical Soci... |
H: The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$
Suppose $x$ is invertible in the unital Banach algebra $A$.
How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
AI: If $x$ is invertible, $x \cdot v = \lambda v$ is equivalent to $v= x^{-1}... |
H: Action of $S_7$ on the set of $3$-subsets of $\Omega$
Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem:
Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and rank $4$ with subdegrees $1,4,12,18$.
I wanted to probe the pro... |
H: Rolling dice until each has taken a specific value
I'm facing the following problem.
Let's say I have $N$ dice in hand. I need to calculate how much time I should roll my dice to make all of them equal to some selected (pre-defined) number. Each time I roll the selected number with some dice, I remove these dice fr... |
H: Open Cover / Real Analysis
I have the next question: Let $K \subset $ $R^1$ consist of $0$ and the numbers 1/$n$, for $n=1,2,3,\ldots$ Prove that $K$ is compact directly from the definition (without using Heine-Borel).
I'm trying to understand compact sets so I would be grateful if someone could give me some exampl... |
H: Finite ultraproduct
I stucked when trying to prove:
If $A_\xi$ are domains of models of first order language and $|A_\xi|\le n$ for $n \in \omega$ for all $\xi$ in index set $X$ and $\mathcal U$ is ultrafilter of $X$ then $|\prod_{\xi \in X} A_\xi / \mathcal U| \le n$.
My tries:
If $X$ is finite set then $\mathcal ... |
H: Convergence in $L^p$ and $L^q$ - multiplication
We have: $X_n \rightarrow X$ in $L^p$ and $Y_n \rightarrow Y$ in $L^q$. Moreover $p,q>1$ are such that $\frac{1}{p} + \frac{1}{q} =1$. Prove that $X_nY_n \rightarrow XY$ in $L^1$. Please, can you help?
AI: Just use Hölder. We have $\def\norm#1{\left\|#1\right\|}$
\be... |
H: Why doesn't integrating the area of the square give the volume of the cube?
I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):
$$V = \int\limits_a^b {A(x)dx}$$
But this doesn't seem to work with the square. Since the size of the area of the square i... |
H: How to calculate the norm of an ideal?
Would someone please help explain how to calculate the norm of an ideal? I can't find a source that explains this clearly.
For example, I know that the norm $N_\mathbb{Q(\sqrt10)}(\langle2,\sqrt10\rangle)=2$, but not clear on how to get it myself.
Many thanks
AI: In general, g... |
H: Any way to simplify this gcd totient function
I have the following expression
$$\frac{gcd(a,b)}{\varphi(gcd(a,b))}$$
$a,b$ are known positive integers. Is there any way to rephrase this or simplify it?
AI: In general, $$\frac{n}{\phi(n)} = \prod_{p|n} \frac{p}{p-1}$$
where the product is over all primes that divide... |
H: Any arbitrary closed smooth curve bounds a orientable surface?
I've got a question that, given an arbitrary closed smooth curve $C:[0,1]\rightarrow\mathbb{R}^3$, can you always find a orientable surface $\Omega$ which satisfy $\partial\Omega=C[0,1]$ ?
I have no idea on this question, and I suppose that the surface ... |
H: Definition of Lebesgue-Stieltjes measure on $\mathbb R$
Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure
$$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$
$$ m(a,b]=F(b)-F(a+), \quad m[a,b)=F(b)-F(a) $$
I am wonde... |
H: $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
Let $n$ and $a$ be positive integers with $a > 1$. I need to show that $n$ divides $\phi(a^n -1)$.
Here, $\phi$ denotes the Euler totient function.
Could any one give me a hint?
AI: A group theoretic solution can be given, (though this solution require... |
H: What is the domain of $x^x$ when $ x<0$
I know that $x^x$ for all $x>0$
but what is negative values for that function which give a real number
for example $$f(-1)=(-1)^{-1}=-1\in R$$
I try to put sequence for that but i faild
is there any help
thanks for all
AI: $x^x$ is well defined as a real function for $$(0... |
H: $\lim_{x\to \pi/2} \;\frac 1{\sec x+ \tan x}$
how to solve it answer is $0$, but $\frac 1{\infty + \infty}$ is indeterminate form
$$\lim_{x \to \pi/2} \frac 1{\sec x + \tan x}$$
AI: Clarification:
$$\lim_{x \to \left(\frac{\pi}{2}\right)^+} \frac{1}{\sec x + \tan x} \to \frac{1}{\infty} = 0$$
$$\lim_{x \to \left(\f... |
H: The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-\left(\frac{-ab}{p}\right)$
What I need to show is that
For $\gcd(ab,p)=1$ and p is a prime, the number of solutions of the equation $ax^2+by^2\equiv 1\pmod{p}$ is exactly $$p-\left(\frac{-ab}{p}\right)\,.$$
I got a hint that I have to use Legendre sym... |
H: Find x for which for every "a" the equation has solution
$$a^{31x} \equiv a \mod 271$$
I need to find x variable, for which the equation has solution with any a. How can I do this?
Generaly, modular equations have solutions when $GCD(a^{31x}, 271) = 1$, or $GCD(a^{31x}, 271) = d > 1; d|a$
It also looks like I could... |
H: Proof that the interior of any union of closed sets with empty interior in a compact Hausdorff space is empty
The question is pretty much in the title, I need to show that given $X$ is a compact Hausdorff space and $\left\{ A_n\right\}_{n=1}^\infty$ is a collection of closed subsets of $X$ each with empty interior,... |
H: Can anyone help me finding recurrence relation in combinatoric?
Guys, I am having trouble finding recurrent relation.
A codeword is made up of the digits $0,1,2,3$ (order is important!). A codeword is defined as legitimate if and only if it has an even number of $0$’s. Let $a_n$ be the number of legitimate codewords... |
H: Proving that for $f\geq0$ on $X$, $\int_X f d\mu = 0$ iff $f = 0$ a.e.
Okay, so the question is the following:
Suppose $f \geq 0$ is a measurable function on the measure space $(X,\Sigma,\mu)$. Prove that
\begin{align} \int_X f d\mu = 0 \text{ if and only if } f = 0 \text{ almost everywhere.} \end{align}
I've sort ... |
H: How to calculate $\lim\limits_{x\to1^+}\frac{1}{(x-1)^2} \int\limits_{1}^{x} \sqrt{1+\cos(\pi t)}\,\mathrm dt$
Can anyone help me by calculating this limit?
I know that I need L'Hôpital but how?
$$
\lim_{x \to 1^+}\frac{1}{(x-1)^2} \int_{1}^{x} \sqrt{1+\cos(\pi t)} \,\mathrm dt
$$
Thank you very much!!
AI: Since my... |
H: Question on Contractions
Let $S = \{x \in \mathbb{R}^n ; \|x\| \le 1 \}$ and $f: S \to S$ be a contraction. Determine one can have $f(S) = S$.
I really need some help with this question. In advance I wanted to give all the necessary definitions needed for this question.
Definition:
Let $(X, P)$, $(Y, P')$ be me... |
H: Proof of the uniqueness of maximal ideal
Let $R$ be a commutative ring with $1$. Let $M$ be a maximal ideal of $R$ such that $M^2 = 0$. Prove that $M$ is the only maximal ideal of $R$.
AI: A maximal ideal is prime. Let $\mathfrak{m}$ be any maximal ideal and let $x\in M$; since $x^2=0\in\mathfrak{m}$, you get $x\in... |
H: A family of sets such that the each subfamily has a different union
If it helps anything, please assume that everything below is finite.
Let $\mathcal A$ be a family of subsets of a set $X$. I want to consider the following independence condition (C) on $\mathcal A$.
(C). The function $\bigcup: 2^\mathcal A\to2^X... |
H: Problem solving earthquake problem magnitude logarithims
I need help solving a simlar type equation to this..... this one was easy though...
An earthquake off the coast of Vancouver Island was measured at 8.9 on the Richter Scale
and an earthquake off the coast of Alaska was measured at 6.5. How many times more int... |
H: System of differential equations: Inverse matrix of a fundamental matrix
I'm trying to show:
Let $A:[0,\infty[\to \mathcal{M}(n,\mathbb{R})$ a function and suppose that all solutions of the system of differential equations:
$$\vec{x'}(t)=A(t)\vec{x}(t) \ \ \ (\star)$$
are bounded when $t\geq 0$. If $X(t)$ is a fu... |
H: Odd and even functions- a direct sum?
Question:
Let $V$ be the vector space of all functions $\Bbb R\to \Bbb R$.
Show that $V=U \oplus W$
for $$U=\{f\ | \ f(x)=f(-x)\ \ \forall x\}, \quad W=\{f \ |\ f(x)=-f(-x) \ \ \forall x\}$$
What I did:
I did prove that $U \cap W$={$0$}. But proving that any function from $\mat... |
H: Determine the inverse function of $f(x)=3^{x-1}-2$
Determine the inverse function of $$f(x)=3^{x-1}-2.$$ I'm confused when you solve for the inverse you solve for $x$ instead of $y$
so would it be $x=3^{y-1}-2$?
AI: $$
y=3^{x-1}-2
$$
$$
y+2=3^{x-1}/\cdot\log_3
$$
$$
\log_3(y+2)=\log_3 3^{(x-1)}
$$
$$
\log_3(y+2)=x... |
H: The graph of the function $y=\log_bx$ passes through the point $(729,6)$. Determine $b$.
The graph of the function $y=\log_bx$ passes through the point $(729,6)$. Determine $b$.
Could someone show me a solution that is similar to mine if it is correct?
$\log_b 729 = 6$
$729 = b^6$
$b = 729^{1/6} = 3$
AI: Your solu... |
H: Closed form for Exponential Conditional Expected Value & Variance
I am wondering if there is a closed form for finding the expected value or variance for a conditional exponential distribution.
For example:
$$ E(X|x > a) $$ where X is exponential with mean $\lambda$.
Same question for variance.
What about for a joi... |
H: calculate $\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$
How to calculate $$\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$$? I've tried to use L'Hospital, but then I'll get
$$\lim_{t\rightarrow1^+}\frac{\pi\cos(\pi t)}{\frac{-\pi\sin(\pi t)}{2\sqrt{1+\cos(\pi t)}}}=\lim_{t\rightar... |
H: Elementary theory of an algebraic structures
Could someone elaborate me what the sentence "The elementary theory of finite fields is decidable" means? I'm not sure that for example if I take $x\in \mathbb{F}_4$ and $y\in \mathbb{F}_5$ then can I form a first order predicate logic sentence that contains both $x$ and... |
H: "Translating" one value $- \infty$ to $+ \infty$ to another ($+ \infty$ to $ \gt 0$)
Well even if I need to use the following in a computer game this is a math question. I have a world map which I can scroll with a scroll velocity $(f(x))$ with my mouse. And I have a zoom factor $(x)$ which is $0$ in its initial st... |
H: What can I do this cos term to remove the divide by 0?
I was asked to help someone with this problem, and I don't really know the answer why. But I thought I'd still try.
$$\lim_{t \to 10} \frac{t^2 - 100}{t+1} \cos\left( \frac{1}{10-t} \right)+ 100$$
The problem lies with the cos term. What can I do with the cos t... |
H: Generating functions combinatorical problem
In how many ways can you choose $10$ balls, of a pile of balls containing $10$ identical blue balls, $5$ identical green balls and $5$ identical red balls?
My solution (not sure if correct, would like to have input):
Define generated function:
$$\begin{align}
A(x) & =(x^... |
H: what is exactly analytic continuation of the product log function
When I solve in wolfram equation like this $xe^x=z$
they give me the solution $x=W_n(z)$
I know about $x=W_0(z) $ and $x=W_{1}(z)$ but for $n$ I searched in the internet but I didn't find anything can give me an expression about $x=W_n(z)$
Please can... |
H: Absolute value of an element in a C*-algebra
Is absolute value of a partial isometry a partial isometry itself?
AI: We have, as square roots are self-adjoint $\def\abs#1{\left|#1\right|}$
\begin{align*}
\abs v^*\abs v &= [(v^*v)^{1/2}]^*(v^*v)^{1/2}\\
&= (v^*v)^{1/2}(v^*v)^{1/2}\\
&= ... |
H: If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?
$X$ and $Y$ denote Hilbert spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? Homeomorphism means continuous map with continuous inverse.
I think the answer is yes, the only thing I am ... |
H: Possible description of closed subset of a projective variety
Let $k$ be an algebraically closed field. Let $P\subset \mathbb{P}^n(k)$ be a projective variety, and $X\subset P$ be a subset.
Suppose that $X_i = X\cap U_i$ is an affine closed subset for every affine open $U_i$ of $\mathbb{P}^n$, i.e. that for every ... |
H: All closed balls are compact each isometry is bijective
Let $(X,d)$ be a metric space in which all closed balls are compact and such that for any two points $x,y \in X$ there exists a function $u \in Iso(X,d)$ such that $u(x)=y$.
Prove that then each isometry $u: X \rightarrow X$ is bijective.
How can I prove this?... |
H: How to define this pattern as $f(n)$
Given a binary table with n bits as follows:
$$\begin{array}{cccc|l}
2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ &0&1&0&3 \\ &0&1&1&4 \\ &1&0&0&5 \\ &1&0&1&6 \\ &1&1&0&7 \\ &1&1&1&8
\end{array}
$$
If I replace each $0$ with a $1$ and each $1$ with $g(n)$ as f... |
H: REVISTED$^1$: Circumstantial Proof: $P\implies Q \overset{?}{\implies} Q\implies P$
To prove that if a matrix $A\in M_{n\times n} ( F )$ has $n$ distinct eigenvalues, then $A$ is diagonalizable is enough to show that the opposite holds? That is, if $A$ is diagonalizable, then $A$ has distinct eigenvalues?
Please do... |
H: Complex number question
For any complex numbers $z_1, z_2$, is the quantity $S$: $$
S = 4 \left(| z_1|^6 + |z_2 |^6\right ) + 4 |z_1|^3 |z_2 |^3 + \left(2 |z_1|^2\times \overline{z_1}^2\times z_2^2\right) + \left(2 |z_2|^2\times \overline{z_2}^2\times z_1^2 \right)$$ always real and nonnegative?
Here overline mean... |
H: Polynomial Equations for the Rank of a Power of a Matrix
If I have some $n \times n$ matrix $X$ (in my case, I happen to know that X is nilpotent and in Jordan normal form), how can I write the condition that $\text{rank}(X^r)= k$ as a polynomial equation (to represent a set of matrices as a variety) in terms of th... |
H: showing that the Euler's number is irrational
Our teacher wants us to do the following:
Suppose that e is rational i. e $e=\frac{a}{b}$ where $a,b\in\mathbb{N}$. Choose $n\in\mathbb{N}$ such that $n>b$ and $n>3$. Use the following inequality
$0<e-(1+\frac{1}{1!}+\frac{1}{2!}+....+\frac{1}{n!})<\frac{3}{(n+1)!}$ and... |
H: Find generator of principal ideal
The ideal $(9, 2 + 2\sqrt{10})$ of $\mathbb{Z}[\sqrt{10}]$ is a principal ideal; it is generated by $1+\sqrt{10}$.
This is easy enough to check once it's been found, but can anyone tell me some way to arrive at this (or some other) generator? That could be done with pencil & paper?... |
H: Unsure of attempt to determine $\dim(W_1+W_2)$ and $\dim(W_1 \cap W_2)$
Let $V=\mathbb{R}^4$. $W_1$ is a subspace of $V$ spanned by vectors $a_1=(1, 2, 0, 1)$ and $a_2=(1,1,1,0)$. $W_2$ is a subspace of $V$ spanned by vectors $b_1=(1,0,1,0)$ and $b_2=(1,3,0,1)$. Determine $\dim(W_1+W_2)$ and $\dim(W_1 \cap W_2)$.
A... |
H: What is the relationship between "recursive" or "recursively enumerable" sets and the concept of recursion?
I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am curious why ... |
H: Solving the equation $\dfrac{(1+x)^{36} -1}{x} =20142.9/420$ for $x$.
How would one solve for x in the following equation:
$\dfrac{(1+x)^{36} -1}{x} =20142.9/420$
I tried factorising the top but that didnt really help much.
$((1+x)^{18} - 1)((1+x)^{18}+1)$
Any help is appreciated thanks.
AI: This is a standard exam... |
H: Compact metric space group $\operatorname{Iso}(X,d)$ is also compact
Could you tell me how to prove that if metric space $(X,d)$ is compact, then the group $\operatorname{Iso}(X,d)$ is also compact?
The group $\operatorname{Iso}(X,d)$ is considered with topology determined by a metric $\rho$ on $\operatorname{Iso}(... |
H: deciding if a chain is a composition series (sanity check)
A small sanity check related to Question 2 from here: proof of the Krull-Akizuki theorem (Matsumura)
Let $C$ be an $A$-module, with $A$ commutative ring and suppose that there exists a chain of submodules $C=C_0 \supset C_1 \supset \cdots \supset C_m=0$ suc... |
H: Proof of Egoroff's Theorem
Let $\{f_n \}$ be a sequence of measurable functions, $f_n \to f$ $\mu$-a.e. on a measurable set $E$, $\mu(E) < \infty$. Let $\epsilon>0$ be given. Then $\forall \space n \in \mathbb{N} \space \exists A_n \subset E$ with $\mu(A_n) <\frac{\epsilon}{2^n}$ and $\exists N_n$ such that $\foral... |
H: Let $A$, $B$ be positive operators in a Hilbert space and $\langle Ax,x \rangle=\langle Bx,x \rangle$ for all $x$, show that $A=B$
Let $A$ and $B$ be positive operators in a Hilbert space $H$, and suppose that $\langle Ax,x\rangle=\langle Bx,x\rangle$ for every $x$ in $H$. Show that $A=B$.
AI: If they are positive ... |
H: How to calculate large exponents by hand?
How to calculate large exponents by hand like they did in ancient times?
Is it something to do with Prosthaphaeresis? for example calculate $2^{15}$.
AI: Use logarithms perhaps?
$$\log_{10} 2^{15} = 15\log_{10} 2 \sim 15\cdot0.3 =4.5$$
So that:
$$2^{15} \sim 10^{4.5} = 10^4... |
H: What's wrong with this Kuhn-Tucker optimization?
The function $u(x,y,z) = xyz$ is to be maximized, under constraints: $ 0 \le x \le 1, y \ge 2, z \ge 0 $ and $ 4 - x - y - z \ge 0 $
Now I'm not quite sure how to translate the x-constraint into proper inequalaties, but my first shot would be $ h_1 = x \ge 0$ and $h_... |
H: Mapping of a Lens-shaped region by a Möbius Transformation
Consider the 'lens' described by $\{z:|z-i|<\sqrt{2}\ \text{and}\ |z+i|<\sqrt{2} \}$ . We want to map this to the upper right quadrant using a Möbius transformation.
The two circles meet at $z=1,-1$ and so if we use the map $f:z \mapsto \frac{z-1}{z+1}$ we... |
H: Lipschitz condition normed vector space
Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition?
Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, as the last term here contains already the lipschitz condition.
AI: you are ri... |
H: Why do injective holomorphic functions have nonzero derivative
For some open sets $U$, $V$ in the complex plane, let $f:U\rightarrow V$ be an injective holomorphic function. Then $f'(z) \ne 0$ for $z \in U$.
Now I don't understand the proof, but here it is from my text. My comments are in italics.
Suppose $f(z_0... |
H: If $F$ and $R$ are subspaces of vector space $E$, then $F \cap R \neq \varnothing$
I need to prove the following:
let $F \cap R$ intersection of vector subspaces $F$ and $R$ of vector space $E$, then $F \cap R \neq \emptyset$
Thanks in advance!
AI: Hint: What vector must every vector space (or subspace, which is in... |
H: Show $60 \mid (a^4+59)$ if $\gcd(a,30)=1$
If $\gcd(a,30)=1$ then $60 \mid (a^4+59)$.
If $\gcd(a,30)=1$ then we would be trying to show $a^4\equiv 1 \mod{60}$ or $(a^2+1)(a+1)(a-1)\equiv 0 \mod{60}$. We know $a$ must be odd and so $(a+1)$ and $(a-1)$ are even so we at least have a factor of $4$ in $a^4-1$. Was thi... |
H: Composition of Linear Rotations and Reflections
Prove that if $T_{1}$ is a rotation of $R^{2}$ about O, and $T_{2}$ a reflection in a line through O, then $T_{1}$$\circ$$T_{2}$ and $T_{2}$$\circ$$T_{1}$ are both reflections in a line through O.
I'd prefer a hint than the answer, because I'm not sure how to begin. T... |
H: Can a knight visit every field on a chessboard?
I was doing excercises about graphs theory and I came across a quite interesting excercise (which probably has something to do with Hamiltonian Cycle):
"Is it possible to step on every field of a 4x4 or 5x5 chessboard just once and return to the starting point using a... |
H: orthogonal matrix and elementary matrix
Answer is False. But I can't think of the counter example... Could anybody have it?
Let A be an orthogonal 4 x 4 matrix such that $$ Ae_1 = e_2, Ae_2 = e_3, Ae_3 = e_1$$ Then $$Ae_4 = e_4 $$
AI: Let $$A = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 ... |
H: Dimension of vector space and symmetric matrix
Why the following statement is true?
I am so frustrated that I could not have any clue on this problem.
The dimension of the vector space of all symmetric 4 by 4 matrices is 10.
Please help me.
AI: Hints:
A symmetric $\,n\times n\,$ matrix is completely characterized b... |
H: Cocartesian squares in the category of abelian groups.
Recently, I've been doing a recap of (basic) category theory and found an old exercise I seem to be unable to solve. The question is as follows.
Let $A, B$ be abelian groups, $A'<A$ and $B'<B$ subroups and $\phi:A\to B$ such that $\phi(A')\subseteq B'$. Let $\p... |
H: power series quotient of polynomial functions
I have given $g(x)=\sum_{k=1}^\infty k^2x^k$. Why can you now write $g:(-1,1)\rightarrow\mathbb R$ as a quotient of two polynomial functions?
I just know the radius of convergence is $\lim\limits_{n\rightarrow\infty}\sup\sqrt[k]{k^2}=1$ and by using the fact for $|x|=1$... |
H: Composition of Reflection Is a Rotation?
Prove that if $T_{1},T_{2}$ are reflections in lines through O then $T_{1}\circ T_{2}$ is a rotation about O.
Once again, a hint would be preferable to an answer. I'm not familiar with these types of linear transformations, as I am not accustomed to thinking of them as funct... |
H: How do i prove that $\frac{1}{\pi} \arccos(1/3)$ is irrational?
How do i prove that $\frac{1}{\pi} \arccos(1/3)$
is irrational?
AI: Let $\theta = \arccos\dfrac13$ so that $\cos\theta=\dfrac13$.
If $\theta$ is a rational multiple of $\pi$, say $\theta=\dfrac mn \pi$, then $\cos(n\theta)=\pm1$. Now $\cos(n\theta)=T... |
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