text stringlengths 83 79.5k |
|---|
H: Proving that UB is transitive follows from B is transitive
I'm really stuck as to how to prove that if B is a transitive set, it must be the case that the union of B is transitive as well. I guess I'm just a bit shaky on the concepts of proofs overall, but even though I can intuit that it should be the case, I'm no... |
H: What does the symbol $\subset\subset$ mean?
In some texts (mainly complex analysis or harmonic analysis) I sometimes see the following double subset symbol $\subset\subset$ for inclusion relation of two regions, e.g., $\Omega$ and $\Omega'$ are two regions in $\mathbb{C}$ such that $\Omega \subset\subset \Omega'$. ... |
H: Logic/Quantifiers and Proofs/counterexample
How do I negate the following statement?
Also please help me with this exercise:
AI: First question :
$$\neg(c): (\exists x,y \in \Bbb R)(\exists z \in \Bbb Z)(\forall a \in \Bbb R) \quad a \geq \frac{x+y}{2} \vee a > z $$
Second question:
Let $x \in U$ where $U$ is the ... |
H: Harmonic Oscillator Homework, Require Verification
A particle of unit mass movies on a straight line under a force having potential energy $$V(x)=\frac{bx^3}{x^4 + a^4}$$ where a,b are positive constants. Find the period of small oscillations about the position of stable equilibrium.
So, I differentiated $V(x)$ ... |
H: How to write product of three sums
I know that by the binomial theorem,
$\displaystyle \left(\sum_{n=0}^\infty a_nx^n \right)\left(\sum_{n=0}^\infty b_nx^n \right)= \sum_{n=0}^\infty \left(\sum_{k=0}^n a_kb_{n-k} x^n\right)$. How do I write the product of three sums
$\displaystyle \left(\sum_{n=0}^\infty a_nx^n \ri... |
H: How to verify the set of two vectors is a basis for the plane?
Verify that B={(3,2,0),(0,2,3)} is a basis for the plane 2x[1]-3x[2]+2x[3]=0 in R3.
How to solve out this question?
And I have a question about this question. I thought dim(Rn)=n.So in this question the basis should have 3 vectors rather than two. So my... |
H: System $\dot x=Ax-|x|^2x$ has one limit ccle
I have a system in $\mathbb R^2$, namely $\dot x=Ax-|x|^2x$ where $A$ is constant real matrix with complex eigenvalues $p+-iq (q>0)$.
I want to show that there exists at least one limit cycle for $p>0$ and none for $q<0$ using Dulacs Theorem and Poincare-Bendixson teore... |
H: Is $G=\{A\in M_2(\mathbb{R}): A^2=I_2\}$ a group?
I am investigating whether $G=\{A\in M_2(\mathbb{R}): A^2=I_2\}$ is a group. It is clear that associativity and existence of identity and inverse are satisfied. So it only remains to determine wether $G$ is closed under matrix multiplication. Then, we consider $A,B\... |
H: Probability of packages
Suppose mail is delivered 6 days each week. Someone sent you ten packages, all of which are scheduled to arrive this week. But you don't know what day any of the packages will arrive. What is the expected number of packages that you receive on a particular weekday?
AI: Assuming the packages ... |
H: Continuous functions on Closed/Bounded sets
Ok so I know that if K is closed and bounded, f(K) is also closed and bounded, I reckon both statements are false but not entirely sure, I just cant seem to come up with counter examples. Thanks
AI: A counterexample for closed: consider the set $S:=(x,1/x)$ in $\mathbb R... |
H: Solving for $y$ in $y= 14x + 1000 = y= 16x + 800$
Given $y= 14x + 1000 = y= 16x + 800$, solve for $y$.
I think I have it:
$x= 100$. Subbing that in would leave me with:
$y = 14x + 1000$
$y = 14 \cdot 100 + 1000$
$y = 2400$
$y = 16x + 800$
$y = 16 \cdot 100 + 800$
$y = 2400$
the lines break even at $y= 2400$ and... |
H: Exponential distribution problem
Suppose that waiting times for M103 buses are independent and exponentially distributed with the same parameter. On average, there are 4 buses per hour.
(a) What is the probability that there are no buses for 30 minutes?
This is clearly an exponential random variable with $\lambda$ ... |
H: Examples of groups of mapping over a set which is not a subset of the symmetric group.
We can find examples of sets $X$ for which there exists a group $G$ (with |G| > 1) under function composition which is a subset of $X^X$ but not a subset of the group $Sym(X)$. The catch is that the identity of $G$ may not be the... |
H: Use partial fractions to find the integral.
Find the integral using partial factions.
$$\int\frac{(2x^2+5x+3)}{(x-1)(x^2+4)}\,dx$$
So do I do...$$\frac{2x^2+5x+3}{(x-1)(x^2+4)}=\frac{A}{x-1} + \frac{Bx+C}{x^2+4},$$
then get
\begin{align*}
2x^2+5x+3 &= A(x^2+4)+(Bx+C)(x-1) \\
2x^2+5x+3 &= Ax^2+4A+Bx^2-Bx+Cx-C?
\end... |
H: Reversing a conditional probability
If I'm given a P(X|Y) table and P(Y), how can I find P(Y|X)?
I understand that $P(Y|X)=\frac{P(X|Y)P(Y)}{P(X)}$ but how do i find P(X)?
Furthermore, If i'm told that random variable X is given a value, does this affect P(Y)?
AI: Assume that you are given a decomposition $\mathscr... |
H: Frobenius Norm Triangle Inequality
How can I go about proving the triangle inequality holds for the Frobenius norm?
I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.
AI: You can use the fact that it's a norm from inner product :
$$\langle A,B \rangle = \text{Trace}(A^TB)$$
The... |
H: From 1-1000, choosing at random, what is the probability that number is prime or composite with a prime factor p $\leq$ 29?
An integer $k \in \{1,2, \dots, 999, 1000\}$ is selected at random. What is the probability that $k$ is a prime number or a composite number with a prime factor $p\leq29$?
AI: It is much easie... |
H: A subgroup $H$ is normal in $G$ if and only if...
Prove that $H$ is a normal subgroup of $G$ if and only if $\{g^{-1}h^{-1}gh|g \in G, h\in H\}$ is a subgroup of $H$.
Attempt:
For the necessary condition, assume $H$ is normal in $G$. Then $g^{-1}h^{-1}g \in H$ for all $h \in H$. Then elements of the form $g^{-1}h^{... |
H: Exactly $\frac{p-1}{2} - \phi(p-1)$ incongruent quadratic nonresidues mod p that are not primitive roots mod p
Prove that there are exactly $$\frac{p-1}{2} - \phi(p-1)$$ incongruent quadratic nonresidues modulo $p$ that are not primitive roots modulo $p$.
I have been looking at this problem for quite some time, but... |
H: Prove that the limit is zero given integral is zero.
Let $f(t)$ be a real-valued function that is continuous, positive, and increasing on the real interval $(0,T)$. If
$$\int_0^T {\dfrac{f(s)}{s} ds} < \infty, $$
prove that $\lim_{t \to 0} f(t)= 0.$
I only have an intuitive idea of why this is true: if the conclusi... |
H: Help in a calculus question
I'm trying to solve this question:
In item (a)
I know $P'(200)=1/50$ and $P'(250)=8/625$, but how can I know the average rate of $P$ from $200\ in^3$ to $250\ in^3$? Sum and divide by two?
In item (b)
I think it's simple, because $V'=800/P^2$
How can I finish item (a)? the item (b) is c... |
H: Norm of a principal ideal
I am trying to prove $N(\sigma(x)\mathcal O_k) = $$N(x\mathcal O_k) $ where $N(I)$ is the number $|\frac{\mathcal O_k}{I}|$ and $ x\mathcal O_k $ is the principal ideal generated by $x$. $\sigma \in Gal(K/Q)$
As $ \sigma(\mathcal O_k)= \mathcal O_k $.
Can I say $|\frac{\mathcal O_k}{x\math... |
H: Complex Integration poles real axis
In class my professor said that
$$
\int_{-\infty}^{\infty}\frac{e^{iax}}{x^2 - b^2}dx = -\frac{2\pi}{b}\sin(ab)
$$
where $a,b > 0$.
However, since the poles are on the real axis, isn't the integral equal to
$$
\pi i\sum_{\text{real axis}}\text{Res}(f(z); z_j)\mbox{?}
$$
If that i... |
H: proving of inequality $(n!)^2 \leq n^n\cdot n!\leq (2n!)$, where $n\in \mathbb{N}$
How can we prove the inequality $(n!)^2 \leq n^n\cdot n!\leq (2n!)$, where $n\in \mathbb{N}$
$\bf{My \; Try}$::
$1\leq n$
$2\leq n$
$3\leq n$
....
....
$n\leq n$
So $1\cdot 2 \cdot 3 ..n \leq n^n$
So $n!\leq n^n\Rightarrow (n!)^2 \le... |
H: How find this inequality minimum $\sum\limits_{cyc}\sqrt{a^2+b^2+ab-2a-b+1}$
Let $0<a,b,c<1$, find this follow minimum
$$\sqrt{(a+b)^2-(a+1)(b+2)+3}+\sqrt{(b+c)^2-(b+1)(c+2)+3}+\sqrt{(c+a)^2-(c+1)(a+2)+3}$$
My try:
since
$$(a+b)^2-(a+1)(b+2)+3=a^2+b^2+2ab-ab-2a-b-2+3=a^2+b^2+ab-2a-b+1$$
so we only find this fol... |
H: Problems with axioms
I am currently exploring the idea of axiomatic truths. As of now, I have looked into axioms dealing with euclidean geometry and they are said to be self-evident truths. Each axiom in euclidean geometry seems very intuitive and easy to apply to geometry in many respects.
Do these axioms exist i... |
H: Transcendence of $\sqrt{\pi}$
So it is known that $\pi$ is transcendental. With a little thought I was able to prove that $k\pi$ and $\pi^{k}$ for all $k\in\mathbb{Z}$ was transcendental. After that I thought about $\pi^{b}$ for any rational number $b$ thinking this result wouldn't be to difficult but I got stumped... |
H: Can $\mathbb{R}$ be written as an ascending union of proper additive subgroups?
Can the group $\mathbb{R}$ be written as countable ascending union of proper subgroups? (i.e. does there exists a series of proper subgroups $H_1\leq H_2\leq \cdots $ such that $\cup {H_i}=\mathbb{R}$?)
AI: The real numbers $\mathbb{R}$... |
H: How do I know if my answer satisfies Rolle's theorem?
Given a function on $[0,2]$,
$$f(x) = x^3 - x ^2 -2x +2$$
I know the answer has to be between $[0,2]$, but for some reason, my answer isn't being accepted. I derived the function and got the following:
$$f'(x) = 3x^2 - 2x -2 \ .$$
I then set it equal to zero and... |
H: Center of $D_6$ is $\mathbb{Z}_2$
The center of $D_6$ is isomorphic to $\mathbb{Z}_2$.
I have that
$$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$
$$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$
My method for trying to do this has been just checking elements that could be candidates. ... |
H: Is $\sum_{n=1}^\infty \frac{e^{\delta n^d}}{\sqrt{n}}\exp\left[-\frac{e^{\delta n^d}}{\sqrt{n}}\right]<\infty$?
Let $\delta>0$ and $d>0$. I am pretty sure that
$$\sum_{n=1}^\infty \frac{e^{\delta n^d}}{\sqrt{n}}\exp\left[-\frac{e^{\delta n^d}}{\sqrt{n}}\right]<\infty$$
as it seems that the series in the form $\sum_... |
H: Find all numbers $p$ such that both $p$ and $p^2+14$ are primes.
Find all numbers $p$ such that both $p, \ p^2+14$ are primes.
I believe only $ p= 3 $ works.
How do I prove this using the complete set of residues Modulo 3. Would something like this work.
Assume $\mathbb F = \{p, \ p+1, \ p+2 \} $ be the set of c... |
H: The number of $j$-dimensional subspaces which contain a given $i$-dimensional subspace.
Let $V=F_q^n$ be a vector space of dimension $n$ over the $q$-element field $F_q$ ($q$ is a prime number). Given a $i$-dimensional subspace of $V$. What is the number of the $j$-dimensional subspaces of $V$ which contain the giv... |
H: Find a bijection between the two sets
I know the function to find the number of bijections is $f(n) = f(n-1) * n$ where $f(n)$ is the number of bijections of $n$, and $n = |A|$, i.e the number of elements in the set (in this case, matching parentheses). But I have no idea where to go from here, I know it's not wha... |
H: Modular quadratic equation (solve for 3-digit natural numbers)
$n^2 + 6n - 88$ is divisible by 97. Solve for all n if n is a 3-digit natural number.
Here's my progress so far
$$n^2 + 6n - 88 \equiv 0\pmod {97}$$
$$n^2 + 6n - 88 + 97 \equiv 0\pmod {97}$$
$$n^2 + 6n + 9 \equiv 0\pmod {97}$$
$$(n+3)^2 \equiv 0\pmod {9... |
H: Defining sets using pairs, check if definition satisfies the pair correctness property - Kuratowski ordered pair
I know that (a,b) = (c,d) if a = c and b = d, but I have no idea what to do here. I assume I'm supposed to show that {{a},{a,b}} = {{c},{c,d}} if a = c and b = d, but how do I verify that using the sets... |
H: Committee selection with no two consecutive people.
Assume that $10$ people are sitting around a round table. Determine the number of ways to choose a committee, where the committee is made up of two people who are NOT sitting next to each other.
Assume that $10$ people are sitting around a round table. Determine ... |
H: Convergence of $\frac{1}{n}\sum_{i=1}^{n}\left[\frac{\left(\log\left(1+i/n\right)\right)^2}{1+i/n}\right]$
Sequence of real numbers
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\left[\frac{\left(\log\left(1+\frac{i}{n}\right)\right)^2}{1+\frac{i}{n}}\right]$$
Does $\lim\limits_{n \to \infty} S_n$ exist? If so, compute the value.... |
H: How to restrict the output values of a continued fraction?
I understand that a continued fraction of the form: $g(n_1,n_2,n_3,n_4,n_5,\ldots)= n_1 + \cfrac{1}{n_2 + \cfrac{1}{n_3 + \cfrac{1}{n_4 + \cfrac{1}{n_5+\cdots} } } }$
gives a unique irrational number for every sequence of natural numbers $(n_1, n_2,n_3,n_4,... |
H: Sylow subgroup of a subgroup of a finite nilpotent group is normal
Let $G$ be a finite nilpotent group and $H \le G$. Let $P$ be a Sylow subgroup of $H$. I'd like to either prove or disprove that, under these conditions, $P$ is normal in $H$.
Any hints how to get started?
(This is part of a larger proof that any su... |
H: How find this $\sum_{i=1}^{5}\tan^4{\frac{i\pi}{11}}$
show that:$$\tan^4{\dfrac{\pi}{11}}+\tan^4{\dfrac{2\pi}{11}}+\tan^4{\dfrac{3\pi}{11}}+\tan^4{\dfrac{4\pi}{11}}+\tan^4{\dfrac{5\pi}{11}}=2365$$
my try:
I think first we can find this value：$$x=\tan{\dfrac{\pi}{11}}+\tan{\dfrac{2\pi}{11}}+\tan{\dfrac{3\pi}{11}}+\... |
H: Solve the following complex number: $\frac{1 + i\tan \theta}{1 - i\tan\theta}$
How do I solve the following complex number?
$$\frac{1 + i\tan \theta}{1 - i\tan\theta}$$
I know how to solve arithmetic problems with complex numbers, but this is the first time I have a function and variable inside.
Thanks in advance!
... |
H: Cubic Poynomial : In the equation $x^3 +3Hx +G=0$ if G and H are real and $G^2 +4H^3 >0$ then roots of the.........
Question:
In the equation $x^3 +3Hx +G=0$ if G and H are real and $G^2 +4H^3 >0$ then roots of the equation are
(a) all real and equal
(b) all real and distinct
(c) one real and two imaginary
(d)... |
H: How to prove the two angles are equal?
It is from Young Double slit experiment. But How to prove the the two $\theta$ are equal, I meant, how $\angle EAD= \angle PEC$? I see from the both triagle have $90^0$ but what about others.
If we think θ→0 we get valid results. we can see the image from any article whoch i... |
H: find minimum of a function: $X^2 + X + 1$?
How do you find the minimum value of $X^2 + X + 1$?
I know it's $3/4$ from intuition. How do you prove it?
AI: For real $x,$ $$x^2+x+1=\left(x+\frac12\right)^2+1-\frac14\ge \frac34,$$ the equality occurs when $x+\frac12=0$
Alternatively, we can use Second derivative test |
H: Laplace Transformation spring question
Here is the question:
https://i.stack.imgur.com/yJyCO.jpg
I can't seem to get the answer.
Are those values in the writing like 1N/m even relevant?
Can someone give me some direction? Thanks!
AI: From that image, it just seems to me that they want you to use Laplace Transforms... |
H: $\lim\limits_{(x,y)\to(0,0)} \frac{x^2 \sin^2 (y)}{x^2 + 2y^2}$ how to prove this limit doesn't exist
Question is in the title.
I tried approaching from $x$ axis, $y$ axis, $y=x$, $y=x^2$, $y=x^3$... they all go to 0 as $(x,y)\to 0$. But wolframalpha says it doesn't exist.. how??? http://www.wolframalpha.com/input/... |
H: $\displaystyle 3^x+4^x=5^x$
Show that the equation $\displaystyle 3^x+4^x=5^x$ has exactly one root
I proved it graphically but I am in need of an analytic solution
AI: $3^x+4^x=5^x\iff\left(\frac35\right)^x+\left(\frac45\right)^x=1$. Since the exponential function on the left hand side is strictly decreasing, it f... |
H: why in $ C^*$-algebra generated by $ x$ that denote by $ A[x]$, $ A[x]$ is commutative?
suppose $ A$ is a $ C^*$-algebra and $ x$ is a normal element in $ A $. $ C^*$-algebra generated by $ x$ denote by $ A[x]$. then
1) $ A[x]$ is commutative.
2) $ A[x]$ is the clouser of polynomials of two variable $ x$ and $ x^*... |
H: $f$ not measurable, but $\lvert f\rvert$ measurable
Do you know an example of a function $f\colon\mathbb{R}\to\mathbb{R}$ which is not $\mathcal{B}$-measurable but $\lvert f\rvert$ is $\mathcal{B}$-measurable?
AI: Pick any non-measurable set $A$. Define:
$$
f(x) = \begin{cases}
1 &: x \in A \\
-1 &: x \not\in A
\en... |
H: Why is this is the derivative?
We are using the Euler equation to calculate the minimum:
Euler equation: $-\frac{d}{dt}\hat{L}_{x'}(t) + \hat{L}_x(t) = 0$
We have the following $L = 12tx + x'^2$ ($x$ is a function of $t$)
Now calculating these derivatives my book says it equals $-2x'' + 12t = 0$
Can anybody pleas... |
H: Proof of binomial coefficient formula.
How can we prove that the number of ways choosing $k$ elements among $n$ is $\frac{n!}{k!(n-k)!} = \binom{n}{k}$ with $k\leq n$?
This is an accepted fact in every book but i couldn't find a proof.
AI: Well, let's first make sequences (ordered lists of distinct elements)... |
H: why from $ \int |f_n -f| \to 0$ we can conclude that $ \int f_n \to \int f$?
Let $(Ω,A,μ)$ be a measure space and $(f_n)$ a sequence of μ-integrable functions, which converges uniformly to $f:Ω→{\mathbb{R}}$. why from $ \int |f_n -f| \to 0$ we can conclude that $ \int f_n \to \int f$? thanks
AI: Because $$|\int f_n... |
H: Principle of inclusion and exclusion problem
I've got 5 bottles of rum, 4 bottles of vodka and 3 bottles of whisky. How many ways are to arrange them into a line when all the bottles of the same kind can't stand next to each other(the bottles of one kind are indistinguishable).
My answer was $\frac{(5+4+3)!}{5!4!3!... |
H: Value if term equals previous term count plus value
How would I calculate the value of the term in a range like this:
1:0
2:1
3:3
4:6
5:10
...
As you can see the value of any given term is equal to the value of the previous term plus the identifier of the previous term, but I have no idea what the formula would be ... |
H: On principal non-maximal ideal
Let $R$ be an integral domain but $R$ is not a field. Prove that, in $R[x]$, $\langle x\rangle$ is maximal principal ideal (that is, maximal among principal ideals) but not a maximal ideal.
AI: Since $R$ is not a field, there is nonunit element $a\in R$. Consider the ideal
$$
\langle ... |
H: Evaluate $\lim_{x\rightarrow 0} \frac{\sin (6x)}{\sin(2x)}$ without L'Hopital
I am trying to evaluate the following limit without L'Hopital's:
$$\lim_{x\rightarrow 0} \frac{\sin (6x)}{\sin(2x)}$$
I know I have to use the fact that $\frac{\sin x}{x} = 1$ but I don't know how to get the limit from the above to $\fra... |
H: Mean and Variance of probability distributions
I know how to calculate mean and variance of some given numbers but I have trouble computing them for probability distributions especially when it is a continuous probability distribution.
For example, can you show me how to calculate mean and variance of Gaussian dist... |
H: Divisibility by 37 proof
$\overline {abc}$ is divisible by $37$. Prove that $\overline {bca}$ and $\overline {cab}$ are also divisible by $37$.
$$\overline {abc} = 100a + 10b + c$$
$$\overline {bca} = 100b + 10c + a$$
$$\overline {cab} = 100c + 10a + b$$
When you add them:
$$\overline {abc} + \overline {bca} + \ove... |
H: Question on continuous functions on intervals
Problem Statement: Let $I = [a,b]$ and let $f: I \rightarrow \mathbb{R}$ and $g: I \rightarrow \mathbb{R}$ be continuous functions on $I$. Show that the set $E = \{x \in I: f(x) = g(x)\}$ has the property that if $(x_n) \subseteq E$ and $x_n \rightarrow x_0$, then $x_0 ... |
H: Fourier Series (Even and Odd Functions)
Let $f \in E$ (where $E$ is a linear space of complex-valued piecewise continuous functions defined on the interval $[-\pi,\pi]$) and $$f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx}\right]$$
denote its Fourier series. Define $$g(x)=\frac{f(... |
H: Is $\tan\theta\cos\theta=\sin\theta$ an identity?
A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like "$f=g$" is regarded as an "identity" when one of their domains is a proper subset ... |
H: Do the non-units in a commutative ring form an ideal?
Do the non-units in a commutative ring form an ideal?
The following are my thoughts on this. Have I made any incorrect assumptions?
Let $R$ be a commutative ring. Let $a, b \in N$ with $N$ being the set of non-units in $R$. We must show the following to prove $N... |
H: Following positive semi-definitness from matrix norm
How can I follow the following?
$$||A||_2 \le \sigma > 0$$
$$\Leftrightarrow A -\sigma I \mbox{ is positive semi-definit}$$
I always get it the other way around, i.e. that $\sigma I - A$ is positive semi-definit.
AI: I think you got it right. Obvious counterexam... |
H: Proof by induction that fibonacci sequence are coprime
I have a bit difficulty to proofe that two consecutive numbers are coprime. I have the following
The property $P(n)$ is the equation $(F_{n+1},F_n)=1$ where F_i the sequence of fibonacci is and $n \in \mathbb {N}$.
Induction hypothesis: $gcd(F_{n+1},F_n)=1$
$P(... |
H: $T$ be the operator from $C[0,1]$ to $C[0,1]$ defined by $Tf = f'+f''$. Show that the operator $T$ is unbounded.
$f \in C[0,1]$, the space of all continuous, complex-valued functions on $[0,1]$ with supremum norm.
$\|f\|=\sup_{x\in[0,1]}|f(x)|$.
Let $D$ be the set of $f \in C[0,1]$ such that the first derivative an... |
H: Equivalence of condition for Lebesgue-measurability
We had a proof in lecture, that showed a bunch of equivalences for Lebesgue-measurability. I have a problem understanding the following implication, where the professor said the reasoning was "trivial".
Let $A\subseteq\mathbb{R^n}$. If there exists $B\in F_\sigma... |
H: Norm of an operator induced by $L^2$-kernel is bounded by $L^2$- norm of the kernel
I am currently studying Hilbert Schmidt operators on my own using the book "Functional Analysis" (Vol 1) by Reed and Simon.
There it is stated that a function $K \in L^2(M \times M, d\mu \otimes d\mu)$ induces an operator
$$
A_Kf(x)... |
H: ZF Natural Even Numbers
Regarding $\Bbb N$ as constructed using ZF ($0=\emptyset, n+1=n^+=n\cup\{n\}$), how is the property "divisible by 2" expressed (using sets and logic)?
AI: We can define addition and multiplication on the natural numbers as defined. $m+n$ is defined to be the unique $k$ such that there is a b... |
H: does $(\overline{E})^{'}= E^{'} \cup ( E^{'})^{'}$ holds?
My question is as follows:
Suppose $E$ is a set in metric space $X$, let $\overline{E}$ denote the closure of E, let $E^{'}$ be the set of all the limit points of $E$. We all know that $\overline{E}=E\cup E^{'} $ Then my question is: Does the following equa... |
H: Equivalence of positive semi-definite matrices
Why can I follow
$$\sigma I - RB^{-1}R^\top \mbox{ is pos. semi-definite} \Leftrightarrow B- \frac{1}{\sigma}R^\top R \mbox{ is pos. semi-definite} $$
having $ \sigma > 0$ and $R$ being an upper triangle matrix?
EDIT: $B$ is positive definit and symmetric. $R$ orig... |
H: uniform convergence of series $\sum_{k=0}^{\infty} \frac{1}{1+k^2x} $
I know that the series
$\sum_{k=0}^{\infty}\frac{1}{\vphantom{\Large A}1\ +\ k^{2}\,x}$
is converges uniformly on $\left(a,\infty\right)$ for $a > 0$, but how can I show that it does not converge uniformly on $\left(0,\infty\right)$?
AI: If the s... |
H: Proving that total variation is equal to $\int_{a}^{x}|g|$
Question:Suppose $g$ is continuous on $[a,b]$. Let f(x)=$\int_{a}^{x}g$ where $x∊[a,b]$. Show that $\int_{a}^{x}|g|$ gives the total variation of $f$ on $[a,x]$.
I managed to prove that $V_{f}(a,x)≤\int_{a}^{x}|g|$. But I still could not find a way to prov... |
H: Density with irrational number and trig function
We may assume the following theorem:
Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$.
Consider the points $$\gamma(t)=(a\cos t+b\sin t, a\sin t-b\cos t, c\cos(\lambda t)+d\sin(\lamb... |
H: A simple Inequality: $\frac{a+b}{\max\{a',b'\}}\leq\frac{a}{a'}+\frac{b}{b'}$?
We know $a \geq a' \geq 0$ and $b\geq b' \geq 0$ . How we can prove:
$\frac{a+b}{\max\{a',b'\}}\leq\frac{a}{a'}+\frac{b}{b'}$
AI: Hint: Assuming that $a',b'>0$, use the fact that $\max\{a',b'\} \ge a',\ \max\{a',b'\}\ge b'$. |
H: What is the probability space for Poisson process?
I'm studying discrete stochastic processes using notes and lectures of prof. Gallager [1], and I was wondering if somebody could intuitively explain what the probability space (usually $\Omega$) for Poisson processes is. Or how to understand that?
Particularly, in ... |
H: Expectation of $\frac{1}{1+X}$ for Gamma
I am trying to evaluate the following integral:
$$ \int_{0}^{\infty} \frac{1}{c+x} \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} dx $$
I have tried simple transformation and by-parts and nothing worked. Then I found this simple solution on internet at http... |
H: Explicit computation $\operatorname{Tor}(M,N)$
Let $R=\mathbb{C}[t]/t^2$ the ring of dual numbers. Using the homomorphism $\phi:R \to \mathbb{C}=R/(t)$ we have that $\mathbb{C}$ is a $R$-module, infact we have
$$\psi: \mathbb{C} \times \mathbb{C}[t]/t^2 \to \mathbb{C} $$
taking $\psi(a,b)=a\phi(b)$. So we have that... |
H: Is entailment biconditional or conditional?
When we say a KB entails Q it means that it is never the case that KB is true and Q is false. Does this mean entailment is similar to the conditional statement KB -> Q? I'm confused because our textbook keeps using "if and only if".
AI: Entailment is closest to the materi... |
H: First steps in algebraic geometry
I'm reading some introductory material in algebraic geometry. I'm trying to closely follow
this http://people.fas.harvard.edu/~amathew/287y.pdf. I keep getting confused with some basic
notions. I'd appreciate if someone could help me clarifying some points.
Let $X$ be a projective ... |
H: How to prove that $\lim_{x \to 0} \sin(x) = 0$ using the epsilon-delta definition?
How do I prove that
$\lim_{x \to 0} \sin(x) = 0$
using the episilon-delta definition of a limit?
Do I have to divide the domain of $x$ into 4 cases for each quadrant?
Update:
Based on the input from @vadim123,
For any $\epsilon... |
H: Models of the full theory of a structure
I'm reading Model theory: an introduction, by David Marker. I'm at page 14, where it says:
...one way to get a theory is to take $\operatorname{Th}(\mathcal{M})$, the full theory of an $\mathcal{L}$-structure $\mathcal{M}$. In this case, the elementary class of models of $\o... |
H: Regularity of Lebesgue measure
Let $A\subseteq \mathbb{R}$ be a Lebesgue measurable set of measure $m(A)=p>0$. Then for all $0<q<p$, show that there is a subset $B\subseteq A$ with $m(B)=q$.
Which Theorem do i have to use here, regularity or density of Lebesgue measure or something else ?
AI: Consider $f(x) = \int... |
H: Same eigenvectors for A and A+rI: why?
I read [Lambiotte 2010, "Multi-scale modularity in complex networks"] that the eigenvectors of an (adjacency) matrix A and the matrix A + rI, with r a scalar and I the identity matrix, are the same. Why is this so? Thanks!
AI: This is a special case of a more general result. L... |
H: Is $y(x)=\frac{1}{2}M\left[1-\cos\left(\frac{\pi}{M}x\right)\right]$ an integer
Let
${\rm y}\left(x\right) = \frac{1}{2}M\left[1-\cos\left(\frac{\pi}{M}x\right)\right]$. Is ${\rm y}\left(x\right)$ an integer for each $x = 1,2,\ldots,M$ when
$M\to\infty$ ?.
AI: $\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{... |
H: Matrix with Functions as Entries
What do we call a matrix with functions as entries?
$$\textbf{f(x)}=\begin{bmatrix}
f_{11}(x) & f_{12}(x) \\
f_{21}(x) & f_{22}(x)
\end{bmatrix} $$
AI: Recall that you denote by $M_{2\times 2}(\mathbb{C})$ the set of matrices with entries in the complex numbers.
You can define mat... |
H: Question about sheaves on projective varieties.
I am new to algebraic geometry, and really can't get idea of this:
For any product $X_{1} \times X_{2}$ of a projective varieties, with projections $p:X_{1}\times X_{2} \rightarrow X_{1}, q:X_{1}\times X_{2}\rightarrow X_{2}$ and let $L_{1},L_{2}$ be sheaves on $X_{1}... |
H: Inequality of scalar-product and norm
Why does the following inequality hold, given $A$ is symmetric and $\lambda_{\min} (A)$ is the smallest Eigenvalue of $A$?
$$v^\top A v \ge \lambda_{\min} (A) \; ||v||^2$$
AI: Assuming that $A$ is real and symmetric, it can be orthogonally diagonalized, that is, for some orthog... |
H: Set intersection problem.
Is $ A \cap B' = A - B$ where $A \cup B$ is the universal set?
I am an absolute beginner at Sets, So please dont vote down my question because it might be too easy for you. $ B'$ refers to the complement of set $B$.
AI: It doesn’t matter whether $A\cup B$ is the universal set or not. If ... |
H: Any finite set in $k^n$ is an algebraic set.
I'm trying to show that given a field $k$, and a finite set of points $\{a^i: i = 1\dots n\} \subset k^n$ is an algebraic set or equivalently is the set of common zeros of some set of polynomials $S \subset k[x_1, \dots, x_n]$.
For the case $n = 1$, we have $Z(x_1 - a_1... |
H: Group isomorphism: $\mathbb{R}/\mathbb{Z}\cong S^1$
Let $\mathbb R$, $\mathbb{Z}$ be the groups of real numbers and integers respectively under addition, and $S^1$ denote the group of complex with modulus $1$ under multiplication.
Then show that $\mathbb{R}/\mathbb Z\cong S^1$.
My idea is to build a homomorphism wi... |
H: Is there a name for a block-diagonal matrix with blocks of the form $\begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix}$?
Is there a name for a real square matrix of the form
$$\begin{pmatrix}
0 & a_1 \\ -a_1 & 0 \\
& & 0 & a_2 \\ & & -a_2 & 0 \\
& & & & \ddots \\
& & & & & 0 & a_k \\
& & & & & -a_k & 0
\end{pmatrix}$$
... |
H: prime number related proof
I want to prove if following is true for every integer a,b and c
$$a^2 - b^2 = cp $$
then p|(a+b) or p|(a-b) where p is a prime number. Any suggestion, help would be highly appreciated. Thanks in advance
AI: $$a^2-b^2=cp\implies a^2\equiv b^2\pmod p$$
If $p|a, b^2\equiv0\pmod p\impli... |
H: How to divide by 12 quickly?
Let $n\in\mathbb N$ be divisible by 12 and $n/12<100$. Is there a way of computing $n/12$ rather quickly using mental arithmetic (e.g. for 972/12, 1044/12, etc.)?
For example, the number 11 seems to have a nice property. When we consider $836/11=d$ then
770<836, but 880>836, so the firs... |
H: differentiability and local Holder continuity
My analysis is really rusty, so apologies if this is a stupid question.
If $f\in C^1$ in a compact set $\Omega$, does this mean $f$ is Holder continuous for any $\alpha$ in $\Omega$?
I have tried googling but I couldn't find this result,
I have tried to do this myself.... |
H: Frobenius Norm Unitary Operators
For something I'm working on, I have a matrix $A$ with other matrices $U$ and $V$ which are unitary ($U^*U = I$ and $V^*V = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\| =\|AV\| = \|UAV\|$. Now, I solve out the first portion, but everything else is giving... |
H: Prove $f_n(x)=(1-x/n)^n$ converges uniformly on non-negative reals
I need to prove $f_n(x)=(1-x/n)^n$ converges on non-negative reals. I can easily prove it converges to $f(x)=e^{-x}$, but it is unclear to me whether this convergence is pointwise or uniform.
I have attempted the proof by taking the Taylor expansion... |
H: Algebraic solution to: Do the functions $y=\frac{1}{x}$ and $y=x^3$ ever have the same slope?
The exercise doesn't specify how it must be answered, so I chose a graphical proof because I couldn't come up with an algebraic one. Sketching the graphs of $y=\frac{1}{x}$ and $y=x^3$, I noticed that $y=x^3$ always has a ... |
H: polynomial division, gcd, question
We are asked to show that there are polynomials $p,q \in Q[t]$ such that:
$p(t)*(t^4+2t^2+1)+q(t)*(t^4-3t^2-4) = t^2+1$
Is the answer the same for $t+5$ instead of $t^2+1$?
What I tried doing:
I don't really know why, but I thought maybe finding the gcd of $(t^4+2t^2+1)$ and $(t^4... |
H: How many automorphisms a countable field has?
Let $\mathbb{B}$ a countable algebraically closed field (car=0) of infinite transcendence degree. How many automorphisms $\mathbb{B}$ has? Are they $2^\omega$?
AI: Your field is isomorphic to the algebraic closure of the field $\mathbb{Q}(x_1,x_2,\dots,x_n,\dots)$, wher... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.