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H: Is $\int_{0}^{\infty} e^{-(a^2x^2+\frac{b^2}{x^2})}dx=\int_{0}^{\infty}e^{-(b^{2}{X}^2+\frac{a^2}{X^2})}dX$ for arbitrary $a,b$ and fixed range?
The problem says,
If $\int_{0}^{\infty} \mathbb{e^{-(a^2x^2+\frac{b^2}{x^2})dx}=\frac{\sqrt{\pi}}{2a}.e^{-2ab}} \longrightarrow(i)$, then prove that $\mathbb{\int_{0}^{\i... |
H: Find the general solution of the D.E. $ x^2 y'' -2xy'+2y=x\ln(x) $ with $x>1$
$$ x^2 y'' -2xy'+2y=x\ln(x) $$ with $x>1 $.
We can tell that it is a Euler's equation. I started by setting $u=\ln x$,
$$\boxed{u''-2u+2=\dotsb \ ?} $$
Having problems with the $u$ replacement. What are next steps?
AI: Hint: Plug $y=... |
H: I don't understand this exponent simplification
I've been doing the Khan Academy math courses to brush up on my math foundations before starting my CS/math degree in the fall semester.
I just don't fully understand negative exponents, I stumbled upon the following exponent simplification:
Equation screenshot
I unde... |
H: Computation of Definite Integral of Rational Function
I am dealing with a definite integral of a rational function that seems quite hard to get a nice closed form/explicit expression for. Let $ -1 < z < 1 $, then my aim is to determine an expression for the integral $ I $ in terms of $ z $:
$$ I(z) = \int_{0}^{\inf... |
H: Paley-Zygmund inequality.
I was trying to understand the proof of the Paley-Zygmund inequality, but encountered the following step, which is not clear for me, i.e.
$$\mathbb{E}(X\cdot \ \mathbb{1}_{X < \lambda\mathbb{E}(X)}) \leq \lambda\cdot \mathbb{E}(X)$$
Sorry for this "brilliant" question, but I would really a... |
H: n total antennas of which m are defective, confused by reasoning?
My problem follows from Sheldon Ross' A First Course in probability. And the question can be viewed in more detail here:
Confused by combinatorical reasoning (n functional antennas, m defective problem)
Basically there are a total of n antennas, of w... |
H: $\det \varphi$ is not a zero divisor $\implies$ Endomorphism $\varphi$ of a free $R$-Module injective
Let $R$ be a commutative Ring with $1$, $M$ a free $R$-Module of rank $2$ and $\varphi \in \operatorname{End}_R (M)$. Show that:
If $\det \varphi$ is not a zero divisor in $R$, then $\varphi$ is injective.
Ok, so a... |
H: Reducing this summation in a recurrence relation
I'm reviewing some of the solutions for the CLRS textbook by Cormen... For practice and I'm stuck on a recurrence embedded inside a summation.
\begin{aligned}
T(n) &=1+\sum_{j=0}^{n-1} 2^{j} \\
&=1+\left(2^{n}-1\right) \\
&=2^{n}
\end{aligned}
I'm not sure how they c... |
H: Upper bound on the $\|\cdot\|_2$ norm of a tridiagonal matrix
Let $T\in M_{n}(\mathbb{R})$ be a tridiagonal matrix. What can we say about operator norm $\|T\|_2$?
I'm asking this question because we know that if $T$ were only diagonal, then $\|T\|_2$ is the largest absolute value of any diagonal entry of $T$, as sh... |
H: How to efficiently perform matrix inversion more than once $\left(A^TA + \mu I \right)^{-1}$ if $\mu$ is changing but $A$ is fixed?
How to efficiently perform matrix inversion more than once $\left(A^TA + \mu I \right)^{-1}$ if $\mu \in \mathbb{R}$ is changing but $A \in \mathbb{R}^{n \times m}$ is fixed?
Or do ... |
H: Expected number of unique transient states visited in an absorbing markov chain.
If I have an absorbing Markov chain represented as a transition matrix P - same notation as Wikipedia article.
$
P = \begin{pmatrix}
Q & R \\
0 & I_r
\end{pmatrix}
$
How would I compute the expected number of unique transient states vi... |
H: $(L^\infty ,\left \| . \right \|_\infty )$ is complete
I only have a question about the proof of this theorem.
In the proof , we took any cauchy sequence $(f_n)$ in $L^\infty$ and we want to show that this sequence converges. We also considered the set $A_k$ of measure zero and its countable union A to arrive to $... |
H: how much would you be willing to pay for this card game?
suppose there are 12 cards with number 1 - 12 on each card. All cards are faced down and are listed one by one in front you.
You get to pick one card, if the card has 1 on it, you get $1.
If it's 2, you get $2, so on and so forth. Now you have the chance to p... |
H: Describe $\mathbb{Z} \times \mathbb{Z} /\mathbb{3Z}\times\mathbb{Z} $
I am trying to describe this quotient group $\mathbb{Z}\times\mathbb{Z}/\mathbb{3Z}\times\mathbb{Z}$ Let's denote with $A$ and $B$ respectively $\mathbb{Z} \times \mathbb{Z} $ and $\mathbb{3Z}\times\mathbb{Z}$
$A / B:= \{a+B : a \in A\}$ my probl... |
H: Let $X$ and $Y$ be points contained in the disk of radius $r$ around the point P. Explain why $d(X, Y) \leq 2r$.
Working on the book: Lang, Serge & Murrow, Gene. "Geometry - Second Edition" (p. 23)
Let $X$ and $Y$ be points contained in the disk of radius $r$ around the point P. Explain why $d(X, Y) \leq 2r$. Use... |
H: Condition is true if only 1 out of 3 variables is true or if the 3 are false: better logic?
I have this condition:
(A is true OR B is true OR C is true) OR (A is false AND B is false AND C is false)
(edit: It's been pointed out that this formula is wrong for what I want)
So as the title says, I want the condition t... |
H: Weak-* closed subspaces and Preduals, a la von Neumann Algebras
Let $X$ be a Banach Space. Suppose that $M \leq X^*$ is a weak-* closed subspace. Is it true that $M$ has a predual? According to my understanding, taking the pre-annihilator $M^\perp = \{x \in X \vert \forall a \in M, \, a(x)=0\}$ we should get that... |
H: limit of the sequence $x_{n}:= \sqrt[n]{n \sqrt[n]{n \sqrt[n]{n\ldots}}}$
I was thinking what happens with the sequence $\{x_n\}_{n\in \Bbb N}$ where:
$$x_{n}:= \sqrt[n]{n \sqrt[n]{n \sqrt[n]{n\ldots}}}$$
When you look some terms, for example $x_{1}=1$, $x_{2}=\sqrt[]{2 \sqrt[]{2 \sqrt[]{2 ...}}}$, $x_{3}=\sqrt[3]{... |
H: Evaluating the $ \lim_{n \to \infty} \prod_{1\leq k \leq n} (1+\frac{k}{n})^{1/k}$
I am really struggling to work out the limit of the following product:
$$ \lim_{n \to \infty} \prod_{1\leq k \leq n} \left (1+\frac{k}{n} \right)^{1/k}.$$
So far, I have spent most of my time looking at the log of the above expressio... |
H: Derivative of the delta function at some point
The derivative of the delta function can be treated similar to the actual delta function. Suppose I have an expression like
$$\frac{\mathrm{d}}{\mathrm{d}x}\delta\left(x-x_0\right)$$
what does this mean for the integral
$$\int\mathrm{d}x\ f\left(x\right)\frac{\mathrm{... |
H: If the $p$-core of a finite group $G$ is trivial, what can be deduced from that about $G$
I am looking for (textbook) references for the following situation:
If $G$ is a finite group and its $p$-core $\mathcal{O}_p(G)$ is trivial, what can be deduced from this about $G$?
(Do groups with this property have a speci... |
H: Given matrix $A^2$, how to find matrix $A$?
Let $$A^2 = \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$$ Knowing that $A$ has positive eigenvalues, what is $A$?
What I did was the following:
$$A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$$
so
$$A^2 = \begin{pmatrix}
a^2 + bc & ab+bd \\
ac+cd & bc+d^2
\end{pm... |
H: What does "measurable" mean intuitively?
So if we have an outer measure $\mu$ on a set $\Omega$, we defined:
A subset A $\subseteq$ $\Omega$ is called $\mu$-measurable, if for all B $\subseteq$ $\Omega$:
$\mu$(B) = $\mu$(B $\cap$ A) + $\mu$(B \ A).
And i understand the definition, but i always thought we can only m... |
H: A cyclic vector with respect to a von Neumann algebra is also cyclic with respect to the commutant.
Let $\mathcal{M}\subseteq B(\mathcal{H})$ be a von Neumann algebra with a given cyclic vector $\xi\in\mathcal{H}$, i.e. $[\mathcal{M}\xi]=\mathcal{H}$ where $[\mathcal{M}\xi]$ is the closed vector subspace generated ... |
H: Evaluate $\lim_{n \to \infty} \frac{2^n}{3^n}$
As stated in the question, I'm trying to find the limit $$\lim_{n \to \infty} \frac{2^n}{3^n}$$
This is my attempt:
$$ \lim_{n \to \infty} \frac{2^n}{3^n}
= \lim_{n \to \infty} 2^n \cdot \lim_{n \to \infty} \frac{1}{3^n}$$
The first limit pulls to $\infty$ whereas the ... |
H: Prove that: For all sets $A$ and $B$, $A\cap B = A \cup B\Leftrightarrow A = B$.
I wasn't able to prove this statement. I'd much appreciate if you could lend a helping hand.
AI: Prove the contrapositive. Assume $A\ne B$.
Then without loss of generality there is $x\in A$ such that $x\not\in B$.
(If not, switch $A$ ... |
H: Is there an efficient way to calculate the following power series?
I want to find coefficients of a power series $K_p$ given by the equation:
$$\frac{1-z^2}{1+z^2-2z\cos(\theta)} = \sum_{p=0}^{\infty}K_pz^p$$
where $\theta$ is a constant. I have checked that $K_0=1, K_1=2\cos(\theta), K_2 = 2\cos(2\theta), K_3 = 2\... |
H: What does "programming" mean in mathematics?
In the past few years, I have came across some topics in Math and CS that have the word "programming" in them. For example, there are linear programming, quadratic programming and dynamic programming. However, I find it hard to pin down what "programming" mean.
A standar... |
H: Optional Stopping Theorem
Let $X=(X_n)_{n\in\mathbb{N}}$ be a stochastic process. The optional stopping theorem (OST) requires $X$ to be a martingale. The OST assures that under certain conditions on the stopping time $\tau$, it holds that $\mathbb{E}[X_\tau]=\mathbb{E}[X_0]$.
But just by being a martingale it woul... |
H: How do I evaluate $\int_{-R}^{R} \sqrt{R^2-x^2}\,dx$ using the change of variables $x = R\sin(w)$?
I'm trying to answer this question from my calculus 1 exam I did last December.
The question asks us to compute a definite integral by using the following change of variables $$ x=R \sin(w)$$ in the following equation... |
H: $f^{-1}(D-C)=f^{-1}(D)-f^{-1}(C)$
Please can you give me feedback on this proof?
Result: Let $f:A \rightarrow B$ be a function. Let $C$, $D \subseteq B$. Then $f^{-1}(D-C)=f^{-1}(D)-f^{-1}(C)$.
Proof: To show that $f^{-1}(D-C)=f^{-1}(D)-f^{-1}(C)$, it is sufficient to show that the set in each side is a subset o... |
H: Understanding Compact Sets (In Complex Analysis)
The definition that I have for a compact set is that
The way that I interpret this is that I can pick any sequence and it should have a limit point (i.e. point of accumulation). However, I can't understand a couple of things about this definition.
Why is it not tru... |
H: Evaluate $\frac{2021!+2020!}{2019!+2018!}.$
Evaluate $\displaystyle{\frac{2021!+2020!}{2019!+2018!}}.$
I think we can write this as $\displaystyle{\frac{2018!\cdot2019\cdot2020\cdot2021+2018!\cdot2019\cdot2020}{2018!\cdot2019+2018!}},$ but I don't know if this is the right direction.
AI: $$\frac{2021!+2020!}{2019!+... |
H: How do I proceed in proving that $\cos 20^o\cos 40^o\cos 60^o\cos 80^o = \frac{1}{16}$
I have taken the value of $\cos 20^o$ to be $x$.
Now,
$$\cos 20^o\cos 40^o\cos 60^o\cos 80^o = \dfrac{\cos 20^o}{2}\cos(40^o)\cos(80^o)$$
$$=\dfrac{x}{2}\Bigg( \dfrac{\cos(40^o + 80^o) + \cos(40^o - 80^o)}{2} \Bigg)$$
$$=\dfrac{... |
H: Is a countable infinite union of $\Sigma_1$ sets is $\Sigma_1$?
I’m reading Kunen’s book Foundations of mathematics. My question is whether a countable union of $\Sigma_1$ sets in $HF$ is also $\Sigma_1$ or not. I wonder if we can think $\Sigma_1$ sets as open sets in HF.
AI: $\Sigma_1$ subsets of $\mathit{HF}$ are... |
H: A quiz question in real analysis on finding whether a function is bounded, continuous under certain conditions
This particular question was asked my senior's quiz and I was unable to solve it. So, I am asking for help here.
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(x+1)=f(x)$ for all $x\in \mathbb{... |
H: Need for left limits in Stochastic Calculus theorems
A lot of the theorems in stochastic analysis are stated for cadlag processes (i.e. right continuous processes with left limits), but I am having a hard time seeing why the "left limits" part is important. It seems like for the most part just right continuity is ... |
H: Is every sine and cosine orthogonal to every other?
I've been learning about Fourier series, and haven't found an explicit statement of this requirement for constructing any arbitrary function using just sines and cosines, so I'm asking here. Is it true that $\sin{ax},\sin{bx},\cos{cx},\cos{dx}$ are all orthogonal ... |
H: What is the meaning of division of a formal power series by $x$?
Background: a formal power series is defined as an expression of the form $\sum_{n\geq 0} a_n x^n$. If $f =\sum_{n=0}^\infty a_n x^n$ then, we write $\{a_n\}_{n\geq 0} \leftrightarrow f$. Two formal power series are equal if each of the components mat... |
H: Intersection of maximal ideals of $\mathbb{Q}[x]/(f(x))$ are nilpotent elements
I'm studying for an algebra qualifying exam (here is the practice exam to prove its not a HW problem) and I was trying to do problem 5(c). Here it is again:
Let $R=\mathbb{Q}[x]/(f(x))$ where $f\in \mathbb{Q}[x]$ is a non-constant polyn... |
H: For the elementary row operation of exchanging two rows, is it required that the rows be different?
For example, on this page, $i \neq j$ is specified for the elementary row operation of row addition, but not for row switching. Of course, I understand that if $i = j$ and you switch rows $i$ and $j$, then you are ju... |
H: Automorphism of the unit disk that fixes a point
I came across the following question:
For each $b \in \mathbb{D}$, construct an automorphism $\phi$ of the unit disk that is not the identity map such that $\phi(b)=b$.
I know that all automorphisms of the disk must have the form $e^{i\theta}\left(\frac{a-z}{1-\overl... |
H: Proof of continuous force of interest with infinitely compounded interest rate. For actuaries, delta and i upper infinity
Specifically, we know the following: $$(1+\frac{i^{(2)}}{2})^2 =(1+\frac{i^{(4)}}{4})^4=(1+\frac{i^{(12)}}{12})^{12}= 1+i, $$ Where $i^{(2)}$, $i^{(4)}$, and $i^{(12)}$ are the interest rates co... |
H: Examples of singular non locally constant functions
What are (as easy as possible) examples of functions $f$ with the following properties?
singular, i.e. continuous, non-constant, and differentiable almost everywhere with derivative zero,
non locally constant, i.e. $\exists x$ with $f'(x)=0$ but $\forall U $neigh... |
H: PDF of $Y = WX$
There are two independent random variables, $X \sim \mathcal{N}(0, 1)$ and $W$ whose PMF is given by
$$
P(W = w) = \begin{cases} \frac{1}{2} \hspace{3mm} \text{if} \hspace{3mm} w = \pm1 \\
0 \hspace{3mm} \text{otherwise}.
\end{cases}
$$
A third random variable is defined as $Y = WX$. I want to find ... |
H: Let $G$ be a group of order $2016 = 2^5 \cdot 3^2 \cdot 7$ in which all elements of order $7$ are conjugate.
Let $G$ be a group of order $2016 = 2^5 \cdot 3^2 \cdot 7$ in which all elements of order $7$ are conjugate. Prove that $G$ has a normal subgroup of index $2$
I know any subgroup of index $2$ must be normal,... |
H: Gradient of norms - general advice
I have something of the following sort:
$$ F(x): \mathbb{R}^n \to \mathbb{R} $$
Where $F(x)$ is a function mapping from one value to another. For example, I may have functions of the form
$$ F(x) = \|x - x_0\|_2^2 $$
or
$$ F(x) = \|Ax - b\|_2^2 $$
Now, I would like to know how to ... |
H: Database of solutions to this generalised Pell equation.
Does there exist a database of primary solutions to generalised Pell's equations of the form:
$$x^2 - 2w^2 = -N$$
for every constant $N \in \mathbb{Z}$?
AI: The tree diagrams below, if extended out enough layers, show all your "primary" solutions for $x^2 - 2... |
H: Biholomorphism between Riemann Surfaces.
Let's consider the homeomorphism $\psi : \mathbb C \rightarrow B_{1}(0), z \rightarrow \frac {z}{|z|+1}$.
Let $Z$ be be the Riemann Surface with topological space $\mathbb C$ induced by the chart $( \mathbb C,\psi)$.
I have to show that $Z$ is biholomorphic $B_{1}(0)$ .
I th... |
H: Kernel of a linear functional (definite integral)
I need to find the Kernel of a linear functional, in this case the linear functional is a definite integral (I have already proved that a definite integral is linear).
I have F: R2[x] to R, defined by F(p) = $$\int_0^1 p(x) \,dx$$
I am suspecting that it has somethi... |
H: Bound of Angle Between Minimal Vectors of a Lattice
I am considering a sublattice $\Lambda \subset \mathbb{Z}^2 \subset \mathbb{R}^2$ of dimension/rank 2. From $\Lambda$ I am pulling out a vector $\mathbf{v}_1 $ of minimal length, and a vector $\mathbf{v}_2$ of minimal length subject to the condition that $\mathbf{... |
H: If $f$ is bijection in a dense subset then $f$ is bijection in all space
Let $X=(X,\mathcal{T}_X)$ and $Y=(Y,\mathcal{T}_Y)$ be topological Hausdorff spaces and $f: X \longrightarrow Y$ be a continuous function. If $f:D \subset X \longrightarrow Y$, with $D$ dense in $X$, is a bijection (one-to-one and onto) then $... |
H: How could I find a slant asymptote of a function like x*e^(1/x)
Is there a general way of finding this. Usually what I find on the internet is dividing the function by ax + b but I can't seem to make it work
AI: Hint: Use differential geometry!
The oblique asymptotes have the equation:
$$y=kx+b, \space \text{ with ... |
H: Show that if $\sum_{n=1}^{\infty}a_{n}$ converges, then so do $\sum_{n=1}^{\infty}a^{2}_{n}$ and $\sum_{n=1}^{\infty}a_{n}/(1-a_{n})$.
Suppose that $0 < a_{n} < 1$ for $n\in\textbf{N}$. Show that if $\sum_{n=1}^{\infty}a_{n}$ converges, then so do $\sum_{n=1}^{\infty}a^{2}_{n}$ and $\sum_{n=1}^{\infty}a_{n}/(1-a_{n... |
H: A question in proof of Section - Transpose of Linear Transformation in Hoffman Kunze Linear Algebra
While studying Linear Algebra from Hoffman and Kunze I have a question in a proof
It's image :
My question is in 2nd last line of proof. I think in summation limit of i should be 1 to m instead of 1 to n as in the ... |
H: Proving $Y = WX$ and $X$ are uncorrelated
From my previous question here, I am able to prove $Y \sim \mathcal{N}(0,1)$, where PMF of $W$ is
$$
P(W = w) = \begin{cases} \frac{1}{2} \hspace{3mm} \text{if} \hspace{3mm} w = \pm1 \\
0 \hspace{3mm} \text{otherwise}.
\end{cases}
$$
and $X\sim\mathcal{N}(0, 1)$, independen... |
H: Are neighborhood bases always a subset of a basis?
The definition of basis and neighborhood basis are:
Let $(X,\tau)$ be a topological space, a base of $\tau$ is a subset $\mathfrak{B}$ of $\tau$ such that each open set $A \in \tau$ is union of elements of $\mathfrak{B}$
If $p \in X$, a subset $\mathfrak{B}_p\subse... |
H: Find all $a$ such that $y=\log_\frac{1}{\sqrt3} (x-2a) = \log_3(x-2a^3-3a^2) $
Find all values of parameter $a\in \mathbb{Z}$ such that $$y= \log_\frac{1}{\sqrt3} (x-2a)$$ $$and$$ $$y = \log_3(x-2a^3-3a^2)$$ intersect at points with whole coordinates.
This is what I did:
$$\log_\frac{1}{\sqrt3}(x-2a) = \frac{log_... |
H: Let $C_0(Q)=\{f:\Bbb{N}→\Bbb{Q}:(\forall ε>0)(\exists N∈\Bbb{N})(\forall m>N)(|f(m)|<ε)\}$. If $\{a_n\},\{b_n\}∈C_0(Q)$ then $\{a_n+b_n\}∈C_0(Q)$.
Suppose that we have $C_{0}(Q)=\{f:\mathbb{N}\rightarrow\mathbb{Q}:(\forall\epsilon>0)(\exists N\in\mathbb{N})(\forall m>N)(|f(m)|<\epsilon)\}$ and we want to show that ... |
H: Show that constant function is integrable.
An exercise is asked in the book to
Show that the constant function is integrable and find its value of integration.
I tried in the following way to show the statement
Suppose $f:\mathbb [a,b]\to \mathbb R$ such that $ f(x)=\lambda$ where $\lambda$ is any constant. Let... |
H: Using Gaussian Kernel to Define Distance
I want to define a region of ball with radius $R$ such that close to center the value is 1 and at the boundary, the value is 0. The gaussian kernel comes to my mind but I would like to know how can I set the $\sigma^2$ such that it works with given radius?
$$f(x,x')=e^{-\tfr... |
H: Diagonal arguments for uncountable lists?
The diagonal argument is a general proof strategy that is used in many proofs in mathematics. I want to consider the following two examples:
There is no enumeration of the real numbers. Because if there were such an enumeration of all real numbers, one could define a real ... |
H: Every irreducible closed has a generic point
If $X$ is a scheme and $Z$ an irreducible closed subset, then $\exists\,\xi\in Z$ with $Z=\overline{\{\xi\}}$
Here is my attempt:
Let $X=\bigcup_i U_i$ be an open affine cover. Since $Z$ is irreducible, then $Z\cap U_i$ is a closed irreducible subset of the affine sche... |
H: Find maximum point of $f(x,y,z) = 8x^2 +4yz -16z +600$ with one restriction
I need to find the critical points of $$f(x,y,z) = 8x^2 +4yz -16z +600$$ restricted by $4x^2+y^2+4z^2=16$.
I constructed the lagrangian function $$L(x, y, z, \lambda ) = 8x^2 +4yz -16z +600 - \lambda (4x^2+y^2+4z^2-16) $$
but I'm very confu... |
H: If matrix $X$ & $Y$ anti-commute then show that the two matrices are linearly independent
Show that if matrix $X_1$ & $X_2$ anti-commute then show that the two matrices are linearly
independent and $X_i ^{\,2}\ne0$
I know $X_1X_2=-X_2X_1$ from the definition then I tried the following:
$$X_1^{-1}X_1X_2=-X_1^{-1}X_2... |
H: Whay does an operator commuting with a finite rank operator have eigenvalues?
Let $A,B\in B(H)$ be such that $AB=BA$ where $B\neq 0$ is a finite rank operator. Does it follow that $A$ has eigenvalues? If yes, why please?
Thanks a lot.
AI: Hint: $A$ maps the range of $B$ (which is a finite-dimensional space) to itse... |
H: Evaluating $\int _0^1\frac{\ln \left(x^2+x+1\right)}{x\left(x+1\right)}\:dx$
How can i evaluate this integral, maybe differentiation under the integral sign? i started expressing the integral as the following,
$$\int _0^1\frac{\ln \left(x^2+x+1\right)}{x\left(x+1\right)}\:dx=\int _0^1\frac{\ln \left(x^2+x+1\right)}... |
H: Formally Introducing the Intersection Symbol into ZFC Set Theory
I am currently reading Lectures in Logic and Set Theory: Volume 2, Set Theory by Tourlakis. In the book, he formally introduces the power set notation, $\mathcal{P}(A)$, as well as union, $\bigcup A$, into the formal, first-order theory of sets as una... |
H: $\nabla f(x)^T(y-x) \geq 0$ if $x$ is optimal for a convex $f(x)$.
In the Convex Optimization book by Boyd and Vandenberghe, it states that if $x \in X$ and $X$ denotes the feasible set, and $f(x)$ is a convex objective function, then $x$ is optimal IFF
$$\nabla f(x)^T(y-x) \geq 0$$
if $x$ is optimal for a convex $... |
H: if $\{a_{n}\}\in C_{0}(Q)$ and $\{b_{n}\}\in C(Q)$ then $\{a_{n}\cdot b_{n}\}\in C_{0}(Q)$.
Suppose that we have $C_{0}(Q)=\{f:\mathbb{N}\rightarrow\mathbb{Q}:(\forall\epsilon>0)(\exists N\in\mathbb{N})(\forall m>N)(|f(m)|<\epsilon)\}$ and we have $C(Q)=\{f:\mathbb{N}\rightarrow\mathbb{Q}:(\forall\epsilon>0)(\exist... |
H: Existence of ordered bases $\beta$ and $\gamma$ for $V$ and $W$, such that $[T]_{\beta}^{\gamma}$ is a diagonal - Questions about proof
Let $V$ and $W$ be vector spaces such that $\text{dim}(V) = \text{dim}(W)$, and let $T:V \to W$ be linear. Show that there exist ordered bases $\beta$ and $\gamma$ for $V$ and $W$,... |
H: Division of two polynomial expressions
Is $$1/(X^n - 1), n \in N$$ a polynomial?
Intuitively I would say yes, because 1 is a polynomial($ X^0$) and so is $X^n - 1$. But Sage (The CAS) appears to disagree, when I type the expression in and call the function is_polynomial() I get False. Can somebody explain why this ... |
H: Consider the operator on $C[0,1]$ $T(f)=f(\sin(x))$
Consider the operator on $C[0,1]$ $T(f)=f(\sin(x))$
Show that this operator is not compact. I honestly do not know how to show this. One idea is to find a weakly convergent sequence, whose image does not converge strongly. But I do not know a nice sequence that co... |
H: Definition of Mapping Cylinder and Cone
I think I understand the idea of the mapping cylinder and cone, except that I am confused about what is, and is not, included.
Specifically suppose we have a CW complex X and a CW subcomplex A; in other words, a CW pair (X,A). Let Y denote X without A joined, and consider th... |
H: What does it mean for a category to have a semi-automorphism?
A while back, somebody asked me about why automorphisms are always isomorphisms. I bobbled the question a bit. Invertability is always one of those nice things that I take for granted. But he got me wondering. If I have a morphism whose source and ta... |
H: If $f$ us periodic and even, what I can conclude about of $\int f \;dx$?
Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a periodic, even and differentiable function. If $L>0$ is the minimal period of $f$, what can I conclude about $$I :=\int_{0}^{L} f(x)\; dx?$$
By the hypotheses we have
$$f(0)=f(L) \quad \text{... |
H: When does a monic arrow in $\mathcal{C}^\rightarrow$ implies the corresponding arrows are monic in $\mathcal{C}$?
Let $\mathcal{C}$ be a category, and let $(\varphi, \psi)$ be an arrow in the arrow category $\mathcal{C}^\rightarrow$, where $\varphi : a \rightarrow a'$, $\psi : b \rightarrow b'$, and there exist $f ... |
H: Limit of $\frac{x^a-a^x}{a^x-a^a}$
Limit of $$\lim_{x\to a} \frac{x^a-a^x}{a^x-a^a},$$ where $$a\in (0,\infty), a\neq1.$$
I know to use L'Hopital, but i am confused with the requirments of $a$.
AI: The function that you are computing the limit of,
$$f(x)=\frac{x^a-a^x}{a^x-a^a}\,,$$
is not defined for either $a=1$ ... |
H: What is the number of modes in prime product series?
Let $S$ be the number of possible solutions for the formula below
$$a * b \equiv c \pmod{x}$$
Where $a,b,c,x \in \mathbb{Z}$ and $0 < a \leq b < x$
I would like to find $c$ and $x$ for which $S/x$ will be the smallest. I tested all the $c$ and $x$ values from $5<... |
H: Why is a stochastic matrix a $l^2$ contraction
If $P$ is a doubly stochastic matrix i.e. $P=(p_{ij})_{1\leq i,j \leq n}$ is s.t. the row sums $\sum_j p_{ij}=1$ for all $i$ and $\sum_i p_{ij}=1$ for all $j$, then may I know why
$$||Px||\leq ||x||$$
for all $x\in \mathbb{R}^n$ where $||\cdot||$ is the $l^2$ norm?
AI:... |
H: Extension of a Continuous function on a dense subset to its closure.
A continuous function on $\mathbb Q\cap [0 \, 1]$ can be extended to a continuous function on [0 1] -Prove.
We have the result that a uniformly continuous function on a set A can be extended continuously to $\overline A$. I am unable to apply the... |
H: Steepest-descent optimization procedure with step size given by harmonic sequence
Here is a minimization procedure I've "dreamed up." I'm hoping to gain a better understanding of its mathematical properties and practical efficiency.
Given a (locally) convex function $f(x):{\mathbb{R}}^n \to \mathbb{R}$, initial $x_... |
H: Split $n \ge 45$ into 30 positive integers and prove there exists some consecutive numbers whose sum is 14
Literally, there is an integer sequence $A = \{a_1, a_2, ..., a_{30}\}$. Given that $30\le\sum_ia_i\le45$ and $a_i > 0$, prove that $\exists c\le d(\sum_{i=c}^da_i=14)$.
AI: Hint: Let $A_i = \sum_{j=1}^i a_i$.... |
H: Why does compactness of a subset in a Euclidean space imply that it is closed and bounded?
I'm just getting started on topology, and having trouble reconciling with the Heine-Borel Theorem. That is:
For a subset S of Euclidean space $\mathbb{R}^n$, the following are equivalent:
S is closed and bounded
S is compact... |
H: Deriving a function based on its properties
Suppose I have a function $\Lambda(t)$ for any $t>0$. This function has the following three properties:
$\Lambda(t)$ is differentiable.
$\Lambda(t)$ is strictly increasing.
$\Lambda(T) = \Lambda(T+S) - \Lambda(S)$ for any $T,S>0$.
It is stated that the function has the ... |
H: About the smallest value of B
I am trying to solve this problem:
We know that there's a inequality: $$(3n-1)(n+B)\geq A(4n-1)n$$
When $A=\frac{3}{4}$, what is the smallest possible value of B.
So, what I did is that:
$$B\geq \frac{\frac{3}{4}n(4n-1)-n(3n-1)}{3n-1}$$
We can deduce that:
$$B\geq \frac{3n(4n-1)-4n(3n-... |
H: What is the difference between $P(S = s, N = n )$ vs $P(S = s | N =n)$
What is the difference between $P(S = s, N = n )$ vs $P(S = s | N =n)$?
I know the former is a joint probability and the latter is a conditional probability, and that $P(S=s, N=n) = P(S=s | N=n)P(N=n)$; however, I can't seem to distinguish betwe... |
H: Is there a surjective function such as $f:2^\Bbb N \rightarrow \Bbb N$
This seems like a pretty simple question but nothing helpful came to my mind.
Is there a surjective function such as:
$f:2^\Bbb N \rightarrow \Bbb N$
AI: Yes, simply for cardinality reasons there will be many such maps.
A concrete example: the f... |
H: prove that $\phi(a)=\frac{\int_{0}^{0.5} (\frac{u}{1-u})^{2a-1} du}{\int_{0.5}^{1} (\frac{u}{1-u})^{2a-1} du}>1 \Longleftrightarrow a<0.5$.
Let $a\in (0,1) $, the question is how to prove
$$\phi(a)=\frac{\int_{0}^{0.5} (\frac{u}{1-u})^{2a-1} du}{\int_{0.5}^{1} (\frac{u}{1-u})^{2a-1} du}>1 \Longleftrightarrow a<0.5... |
H: Verify if the sum of two subspaces are equal to $\mathbb R^3$
I'm having trouble in this exercise because i am not sure if I'm dealing with it in the right way. The exercise is to consider the following subspaces of $\mathbb R^3$:
$$U=\{(x,y,z) \in \mathbb {R}^3\mid x=y=z=0\}$$
$$V=\{ (x,y,z) \in \mathbb {R}^3\mid... |
H: why $x \in \mathbb{Q}^c ?$
i have some confusion in sorgenfrey line, my doubt is marked in red line and red box given below
My Doubts : we know that $\mathbb{Q}$ is countable and $\mathbb{Q}^c$ is uncountable
Here we have already assume that $( S,T)$ has a countable basis , then why $x \in \mathbb{Q}^c$ ?
From ... |
H: Expressing a presheaf as a colimit of representables
I don't understand how the highlighted isomorphism follows. And why is every object in $\mathbf {Set}\times\mathbf{Set}$ is a sum of copies of $(1,\emptyset)$ and $(\emptyset,1)$?
Next, right after the density theorem, there's this example:
By the theorem, the ... |
H: Find the generator of $(Z,*)$
let $Z$ be the set of integers
let $\ast\ $ be an operation defined on $Z$ by $a*b=a+b-1$ $\forall \ $ $a,b$ $\in \ $$Z$
Is $(Z,*)$ cyclic if so find the generator of $(Z,*)$.
I think it is not cyclic group since if it is $1$ its inverse same as $1$ therefore there is no generator but ... |
H: What is the concrete meaning of fixing an extension field through a subgroup of automorphisms in $x^3-2$?
The automorphisms corresponding to the extension fields of the splitting polynomials of $x^3-2$ are enumerated in this answer:
we know that the automorphism of $\mathbb{Q}(\sqrt[3]{2}, \omega_3)$, which fix $\... |
H: How do we calculate the rotation of 3D vectors?
Consider three vectors as 3D axis in a unit sphere:
$$A = (1,0,0)$$
$$B = (0,1,0)$$
$$C = (0,0,1)$$
If we rotate the sphere around y-axis by $\theta$ and then around the x-axis by $\phi$. How do we calculate the new vectors?
I came up with a solution of
$$A_x = \cos(\... |
H: Locus of the circumcenter of triangle formed by the axes and tangent to a given circle.
A circle centered at $(2,2)$ touches the coordinate axes and a straight variable line $AB$ in the first quadrant, such that $A$ lies the $Y-$ axis, $B$ lies of the $X-$ axis and the circle lies between the origin and the line $... |
H: Are these two statements about the probability tending to Normal distribution equivalent?
Are these two statements equivalent:
$P(X_n < \beta) \to z(\beta)$
For all fixed $\alpha \text{ < } \beta \in \mathbb{R} \quad P(\alpha < X_n < \beta) \to z(\beta) - z(\alpha)$
$z(\beta) = P(X < \beta)$ where $X$ ~ $N(0,1)... |
H: Halving number at regular intervals vs same number at doubling intervals.
The infinite series: $\sum_{n=0}^\infty 2^{-n}$ converges to 2.
If I was given an apple after one day, half an apple after another day, a quarter of an apple after another day, and so on, I would never quite get two apples.
In my (very limit... |
H: Why is the horizontal asymptote of $\lim_{x \to \infty} \frac{x}{x^2+1}$ $y=0$?
I figured, the limit of below is:
$$\lim_{x \to \infty} \frac{x}{x^2+1} = \frac{\infty}{\infty^2 + 1} \approx \frac{\infty}{\infty} = 1$$.
Should the horizontal asymptote be $y=1$ then?
AI: You have $$\lim_{x\rightarrow\infty}\frac{x}{x... |
H: How can I comment these explanation?
There are column vectors and there exists comments about it. it's given in the image. i don't understand inconsistent, consistent because the translation of these terms into my language is unmeaningfull.
could you comment on the image. really thanks
AI: The notation consistent a... |
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