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H: Nested intervals theorem - a special case on open intervals
I learned the nested intervals theorem in the class: If $I_n\ (n\in\Bbb N)$ is a sequence of bounded closed intervals, i.e. $[a_n,b_n]$, then $\bigcap I_n\ (n\in\Bbb N)\neq\varnothing$. In the proof of the theorem, we used another theorem: monotone non-dec... |
H: Prove that $f$ is continuous only at $x=0$
I cannot understand the solution to this problem given in my book.
Problem: Consider function $f$ defined for all $x$ by $f(x)=x$ if $x$ is irrational and $f(x)=0$ if $x$ is rational. Prove that $f(x)$ is continuous only at $x=0$.
Solution given in book: Recall that, arbit... |
H: $\mu(X \bigcap [0, r))>0,\mu(X \bigcap [-r, 0))>0$ for all $r >0 $ implies there some non zero $x$ , $x \in X,-x \in X$
Given some Lebesgue measurable set $X$ and Lebesgue measure $\mu$
$\mu(X \bigcap [0, r))>0,\mu(X \bigcap [-r, 0))>0$ for all $r >0 $ implies there some non zero $x$ , $x \in X,-x \in X$
AI: Not ... |
H: Why do mathematicians approach axiomatic proofs like this?
When doing proofs, I keep a tab open on 'Advice for students for learning proofs', this guidelines helps me take the right first steps when looking at statements.
But, with axiomatic proofs, I am on a shaky foundation. See this proof of 2.1.2 (a) below, the... |
H: total number of list of delegates in this question
The question is :
A conference is attended by 200 delegates and is held in a hall. The hall has $7$ doors labelled A,B,..G. At each door, an entry book is kept and the delegates entering through that door sign in the order in which they enter. If each delegate is ... |
H: The components of basis vectors in linear algebra under a change of basis
If I choose a non-orthonormal basis for $\Bbb{R}^2$, I have to label the basis vectors by their components. But doing this requires me to specify the components of one of the basis vectors relative to that of the other one. For instance if I ... |
H: The principal branch of $\sqrt{z}$ maps $\mathbb{C}-(-\infty,0]$ onto the right half plane - $\{Re(z)>0\}$
I have heard my professor say a couple of times that:
The principal branch of $\sqrt{z}$ maps $\mathbb{C}-(-\infty,0]$ onto the right half plane - $\{Re(z)>0\}$
but never realized why is that true.
Can anyon... |
H: Derivation of natural log inequality
While looking at a proof of this inequality
$(1+\frac{1}{x})^x < e < (1+\frac{1}{x})^{(x+1)}$
The authors take the natural log of on both sides and get
$xln(1+\frac{1}{x}) < 1 < (x+1)ln(1+\frac{1}{x})$
But then they write this last inequality as follows (which I don't understand... |
H: Problem on an isosceles right triangle, involving similarity and congruence
Given that $ABC$ is an isosceles right angled triangle with angle $\widehat{ACB}=90$ degrees. $D$ is the midpoint of $BC$, $CE$ is perpendicular to $AD$, intersecting $AB$ and $AD$ at $E$ and $F$ respectively. Prove that angle $\widehat{CDF... |
H: 5 is primitive root on modulo 47 then solve the $x^{20} \equiv21(mod47)$
I have tried to look powers of 5 in modulo 47 but It seems not working.
AI: As $5^3\equiv31\equiv-16,5^6\equiv(-16)^2\equiv21\pmod{47}$
Take discrete logarithm,
$20\cdot$ind$_5x\equiv$ind$_5{21}\pmod{46}\equiv6\equiv6-46$
ind$_5x\equiv-2\pmod{... |
H: Find the probability using Central Limit Theorem
Approximate the probability that $100$ elements, each of which works for time $T_{i}$, will provide $100$ hours of work in total. It is known that $E[T_{i}]=1$ and $D^2[T_{i}]=1$.
I have introduced a new variable
$$Z=\sum_{i=1}^{100}T_i$$
so that
$$E[Z]=E[nT_i]=n$$
$... |
H: Proof of $_()∩=1$,$$ is a holomorph, $$ automorphism group
I was told that this was a simple proof, by definition of the centraliser and by definition of the intersection with the automorphism group, but I just don't see it.
$A = \mathrm{Aut}(G)$
$H = G \rtimes A$
$C_H(G) = \{h \in H| hg = gh, \forall g \in G\}$
... |
H: Find the Probability Density Function
Given $ X\sim U(-1,1) $ and $ Y=e^{2X} $, how can I find the probability density function of $Y$ ?
Thanks in adv :)
AI: As X is uniform in $(-1;1)$ it is well known that $F_X(x)=\frac{x+1}{2}$
Now let's start with definition of $CDF_Y$ and try to express it as function of $CDF_... |
H: Yes/No :Is $(x,y)$ is inner product space?
let $x= ( x_1, x_2,......, x_n)$ an $y= ( y_1, y_2,...,y_n)$ be arbitary vectors in $V_n$. Determine whether $\langle x,y\rangle$ is an inner product for $V_n$ where $\langle x,y\rangle $ is given by
$\langle x,y\rangle=\sum_{i=1}^{n}(x_i +y_i)^2 - \sum_{i=1}^{n} x_i^2 -... |
H: Identifying the type of random variable in experiment
A manufacturing company uses an acceptance scheme on items from a production line before they are shipped. The plan is a two-stage one. Boxes of 20 items are readied for shipment, and a sample of 10 items is tested for defectives. If any defectives are found, th... |
H: Finding MLE estimator for given density $f(x, \alpha, \beta)$
I'm having trouble with the following example problem of MLE:
Let $X = (X_1, ..., X_n)$ be a trial from i.i.d r.v. with density:
$$
g(x) = \frac{\alpha}{x^2}\mathbb{1}_{[\beta, \infty)}(x)
$$
where $\beta> 0$.
Write $\alpha$ in terms of $\beta$ to obta... |
H: For $\alpha + \beta \;$ limit ordinal, $\alpha, \beta > 0\;$ are ordinals, show that $\beta$ is a limit ordinal
Since $\alpha + \beta $ is a limit ordinal, we have that forall $x\in \alpha + \beta \;$, $x+1\in \alpha + \beta \;$
Lets assume that $\beta\;$ is not a limit ordinal, then there exists some $x\in \beta \... |
H: if $X\mid Y$ follows Bernoulli with parameter $g(Y)$ then what is $E[X]$?
The context is not important for the question but nevertheless here it is: $A$ is the adjacency matrix of a random simple graph (A is symmetric with zero diagonal and with entries in $\{0,1\}$). The graph is generated on a fixed number of ve... |
H: Show $\sum_{c|n} \mu(c)f(c) = \{1-f(p_1)\}\{1-f(p_2)\} \dots \{1-f(p_r)\}$
$n=p^{k_1}_1p^{k_2}_3...p^{k_r}_r$ and f is multiplicative function.I have tried convolution but it seems not solving.
AI: $$\sum_{c|n}\mu(c)f(c)=\prod_{i=1}^r\left(\sum_{p_i|n, q=0}^{k_r}\mu(p_i^q)f(p_i^q)\right)$$
But $$\sum_{p_i|n, q=0}^{... |
H: Does complement of a set is closed imply the set is open?
I gave a test on topology. On seeing the checked paper, I saw that the professor has deducted my marks for writing this in an intermediate step:-
"$A^c$ is closed implies $A$ is open."
I even asked this since I feel it follows from the definition of a closed... |
H: What is the meaning of this number displayed in my R environment?
My question is really silly, I want to know what the meaning of this number displayed in my R environment, is it $2.2\times e^{-16}$?
AI: 2.2e-16 means $ 2.2 \times 10^{-16}$. It is a shorthand convention for scientific notation hanging over, I thin... |
H: Definition of compactification
$(X',\tau')$ is a compactification of $(X,\tau)$ if
$(X',\tau')$ is compact
$(X,\tau)$ is a topological subspace of $(X',\tau')$
$X$ is dense in $X'$.
I was wondering why it is necessary to have the last condition. In my notes, I have written down that the third point is there to ... |
H: solving $3(x_{n+1} - x_{n}) = \sqrt{16+x^2_n} +\sqrt{16+x^2_{n+1}}$ with $x_1=3$
I was solving a question which led to the following recurrence:
$$3(x_{n+1} - x_{n}) = \sqrt{16+x^2_n} +\sqrt{16+x^2_{n+1}}$$
which I could not solve
my approach :
I tried putting $x_n= tan(t_n)$ and other trigonometric substitutions b... |
H: How do you mathematically describe "removing" a digit?
Let us say you have the number 1234. If we were to "remove" the number 3, we would have 124. I have seen this operation in many programming tutorials, but I do not know what mathematical operation, if any, this maps onto. What is the mathematical term for this ... |
H: Eigenvectors of function of linear transformation
Let $T$ be a linear transformation and $f$ be any polynomial. I already know that if $Tv=cv$ for some eigenvalue $c$ and eigenvector $v$, we must have $f(T)v=f(c)v$. Thus, every eigenvector of $T$ is an eigenvector of $f(T)$. Is the converse necessarily true? I know... |
H: Prove derivative by induction
$f:(0, \infty) \rightarrow \mathbb{R}$
$f(x) = \sqrt{x}$
a) Calculate the first four derivatives
$f'(x) = \frac{1}{2}\cdot \frac{1}{\sqrt{x}}$
$f''(x) = -\frac{1}{4}\cdot \frac{1}{\sqrt{x^3}}$
$f'''(x) = \frac{3}{8}\cdot \frac{1}{\sqrt{x^5}}$
$f''''(x) = -\frac{15}{16}\cdot \frac{1}{\s... |
H: Finding $\lim_{n\to\infty} \frac1{3^n}\left(a^{\frac{1}{n}}+b^{\frac{1}{n}}+c^{\frac{1}{n}} \right)^n$ where $a,b,c>0$
$$\lim_{n\to\infty} \bigg(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3} \bigg)^n, \quad \textrm{$a>0$, $b>0$ and $c>0$.}$$
I had an idea to present the terms and decompose them as $1+(a-1), 1+(b-1)$ and $1+(c-... |
H: Issue with basic double integral
I have $(X,Y)$ random variable in $\mathbb{R}^2$ with joint pdf defined as:
$$f_{X,Y}(x,y)=\dfrac{c}{1+x^2+y^2}\mathbb{1}_{D}(x,y)$$
Where $c \in \mathbb{R}$ and $D=\{(x,y) \in \mathbb{R}^2 : x^2+y^2 \le 1\}$, and I have to calculate $c$. In other words, I have to find $c$ such that... |
H: Show if the function $\frac{x+2}{x^2+2x+1}$ is injective
$$\frac{x+2}{x^2+2x+1}=\frac{y+2}{y^2+2y+1} $$
The domain of the function is: $\forall x \in \mathbb{R}\smallsetminus\{-1\}$
If $y=x$ the function is injective:
$$\frac{x+2}{(x+1)^2}=\frac{y+2}{(y+1)^2}$$
$$(x+2)(y+1)^2=(x+1)^2(y+2)$$
Here it stops. I would l... |
H: Number of ways to select the election dates
A general election is to be scheduled on 5 days in May such that it is not scheduled on two consecutive days. In how many ways can the 5 days be chosen to hold the election?
My approach
There are 31 days in MAY month
$5$ days are election days and 26 days are non election... |
H: Is the ring $S=\left\{\begin{pmatrix} 0 & 0\\ 0 & a\\ \end{pmatrix} \in M_2(R) \right\}$ a field?
I am trying to proove if $S=\left\{\begin{pmatrix} 0 & 0\\ 0 & a\\ \end{pmatrix} \in M_2(R) \right\}$ is a field. My thought is that due to the fact that $\det=0$ for every a different than $0$ that means that there ar... |
H: Residue field of an infinite extension of $\mathbb{Q}_p$
Let $\zeta_1 \in \overline{\mathbb{Q}_p}$ such that $\zeta_1^p=p$, now let $\zeta_2 \in \overline{\mathbb{Q}_p}$ such that $\zeta_2^p=\zeta_1$ and so on with $\zeta_i^p = \zeta_{i-1}$.
I have to show that the resiude field of $K=\mathbb{Q} [ \zeta_1 , \zeta_2... |
H: Find equation of circle
If it is given parabola:
$${y}^2 = 4x$$
How can I find a equation circle (center on x axis) that thouch parabola from inside? $$r=2(sqrt){5}$$
I have done next:
$$ y^2 = 2px $$
$$ y^2=2*2*x$$
$$ p=2$$
$$ r^2=(x-p)^2+(y-q)^2$$
$$ 20=x^2-4x+4+y^2$$
What have I done wrong?
AI: Let us assume the... |
H: Calculate $\int_{\mathbb{R}^2}e^{-(|3x+4y|+|4y-3x|)}dxdy$
Calculate $\int_{\mathbb{R}^2}e^{-(|3x+4y|+|4y-3x|)}dxdy$
I set $3x+4y=u,4y-3x=v$ and then $|J|=\frac{1}{25}$.
Taking exhaustion of $\mathbb{R}^2$ to be $A_n=\{(u,v)\in\mathbb{R}^2:-n\leq u\leq n,-n \leq v \leq n\}$
$\int_{-n}^{n}\int_{-n}^{n}e^{-|u|+|v|}dud... |
H: Evaluate $\int_{-1}^{3} [ x+ \frac{1}{2}] dx$
Evaluate
$\int_{-1}^{3} [ x+ \frac{1}{2}] dx$ where $[.]$ denotes the greatest integer less than or equal to $x$.
My attempt : $\int_{-1}^{3} [ x+ \frac{1}{2}] dx= \int_{-1}^{0} [x +1/2]dx + \int_{0}^{1} [x +1/2]dx + \int_{1}^{2} [x +1/2]dx + \int_{2}^{3} [x +1/2]dx = ... |
H: Let $G$ be a group. Let $x,y,z \in G$ such that $[x,y]=y$, $[y,z]=z$, $[z,x]=x$. Prove that $x=y=z=e$.
Let $G$ be a group. Let $x,y,z \in G$ such that $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ (the commutators; $[x,y]=xyx^{-1}y^{-1}$). Prove that $x=y=z=e$.
I tried to show it by proving that $zx^mz^{-1}=x^{2m}$ with inductio... |
H: Find the limit of the following expression $\lim_{x\to1}(3 \sqrt[3]{x}-2\sqrt{x})^{\frac{1}{\ln x}}$
$\displaystyle\lim_{x\to1}(3 \sqrt[3]{x}-2\sqrt{x})^{\frac{1}{\ln x}}$
How is this limit taken?
I was able to convert this expression to the following form
$\displaystyle\lim_{x\to1} \exp(\log_x(3 \sqrt[3]{x}-2\sqrt... |
H: what is a "powerset" with base larger than 2?
A powerset $P(S) $of some set $S$ can be treated as all different possible ways of partition $S$ into 2 ordered pair of disjoint subsets.
And I'm curious what is the equivalence of partition $S$ into more than 2 subsets?
For example, partition the set S into 3 ordered p... |
H: If $U \leq \mathbb{R}^4$, $\dim(U) =3$ and $\langle(0,0,0,1)\rangle \cap U = \{0\}$ then $U = \langle(1,0,0,0), (0,1,0,0), (0,0,1,0)\rangle$
Suppose $U \leq \mathbb{R}^4$, $\operatorname{dim}(U)=3$ and $\langle(0,0,0,1)\rangle \cap U = \{0\}$. Is it then true that $U = \langle(1,0,0,0), (0,1,0,0), (0,0,1,0)\rangl... |
H: Showing that $|\sin(a+x)-\sin(a)-\cos(a)x|\leq x^2$
I am trying to work through a small problem (finding a Fréchet Derivative), and I arrive at a function that is "obviously" less than $x^2$ for all $x$, which would be very nice to prove. I say "obviously" because by looking at the plots I visually notice that it i... |
H: Fatou lemma for $\{f_n+g_n \}$
Let $(E,\mathcal {A },\mu) $ be a finite measure space.
Take $\{f_n\}$ and $\{g_n\} $ two integrables sequences such that $\{f_n\}$ is positive.
Can we say that
$$
\int_E \liminf_n\big ( f_n+g_n\big )d\mu\leq \liminf_n\int_E \big (f_n+g_n\big )d\mu
$$
AI: Hint,
Take $E=[0,2]$, $\math... |
H: If $\mathbb{E}[X^2] < \infty$ and $ g : \mathbb{R} \to \mathbb{R}$ minimizes $ \mathbb{E}[(X -g(Y))^2]$, then $g(Y) = \mathbb{E}[X\mid Y]$.
Consider random variables $X$ and $Y$ with $\mathbb{E}[X^2] < \infty$. Show that if $g : \mathbb{R} \to \mathbb{R}$ is the function that minimizes $ \mathbb{E}[(X -g(Y))^2]$, ... |
H: 7th Degree Differential Homogeneous Operator
Question asks:
$1, 1 - i, i, i$ are the roots of $L ( r ) = 0$ (characteristic equation)
where $L(D)$ is a $7^{\text {th}}$ ( seventh) order linear, homogeneous differential operator with constant coefficients.
Find the differential equation $L (D) y = 0$ and its gen... |
H: Understand the definition of set $C = \{x : \exists~A (A \in F \rightarrow x \in A )\}$ where $F = \{(1,2,3), (2), (1,2)\}$.
Given set $F = \{(1,2,3), (2), (1,2)\}$.
Let $C = \{x : \exists~A (A \in F \rightarrow x \in A )\}$
I am not able to understand what $C$ means ?
AI: The implication $A\in F\to x\in A$ is eq... |
H: $a:A$ in $\Gamma$
I am taking an introductory course on type theory. I find the following sentence in my handout:
''$a:A$ in $\Gamma$'' or ''$\Gamma\vdash a:A$'' is equivalent to the following judgment
''$a(x_1,...x_n):A(x_1,...,A_n)<x_1:A,...x_n:A_n(x_1,...,x_{n-1})>$'', where $\Gamma=<x_1:A,...x_n:A_n(x_1,...,x_... |
H: How do I interpret this set builder notation into English terms?
How would I interpret this set builder notation into English terms?
If $A_\alpha$ is a set for every $\alpha$ in some index set $I\ne\emptyset$,$$\begin{align}\bigcup_{\alpha\in I}A_\alpha&=\{x\::\:x\in A_\alpha\text{ for at least one set}A_\alpha\te... |
H: If $\mathrm{M,N}$ are $3\times 2, 2 \times 3$ matrices such that $\mathrm{MN}=$ is given. Then $\mathrm{det(NM)}$ is?
If $\mathrm{M,N}$ are $3\times 2, 2 \times 3$ matrices such that $\mathrm{MN}=\pmatrix{8& 2 & -2\\2& 5& 4\\-2& 4&5}$, then $\mathrm{det(NM)}$ is?
($\mathrm{NM}$ is invertible.)
$\mathrm{det(MN)}$ ... |
H: $\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$
I'm having trouble proving that for any $x,y,z>0$ such that $x+y+z=1$ the following inequality is true:
$\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$
It seems to me that Jensen's inequality could do the trick, but I'm having... |
H: Understanding Rudin's proof of: Every bounded sequence in $R^k$ contains a convergent sequence
I am trying to understand the proof for:
Theorem 3.6b:
Every bounded sequence in $R^k$ contains a convergent subsequence
which is as follows in Baby Rudin:
This follows from (a), since Theorem 2.41 implies that every b... |
H: Big $\Theta$ arithmetic
I'm trying to understand this formula from this wikipedia article about amortized analysis.
In general if we consider an arbitrary number of pushes n + 1 to an array of size n, we notice that push operations take constant time except for the last one which takes $\Theta (n)$ time to perform... |
H: Prove that $d^2=a^2+b^2+c^2-2[ab\cos B+bc\cos C+ac\cos(α)]$
The angle between the AB and CD sides of an ABCD convex quadrilateral is equal to $\alpha$. Considering that AB = a, BC = b, CD = c, DA = d, prove that:
$$d^2=a^2+b^2+c^2-2[ab\cos B+bc\cos C+ac \cos(\alpha)]$$
I tried to prove this by Cosines Law, but I co... |
H: Convergence in probability vs almost surely using Borel-Cantelli Lemma
Looking at a sequence of independent rvs
$$
Z_n = \Bigg\{
\begin{array}{lr}
1 & w.p. \frac{1}{n}\\
0 & w.p. 1-\frac{1}{n}
\end{array}
$$
It is easy to see that
$$
P(|Z_n - 0|>\varepsilon) = P(Z_n = 1) = \frac{1}{n} \to_n 0
$$
so $Z_n \to_n 0$ in... |
H: Sum of Fourier series members with odd indices
I have the following Fourier series for the function $f(x) = e^x$ on $[-\pi, \pi]$
$$
\frac{e^{\pi} - e^{-\pi}}{2\pi} + \sum_{n = 1}^{\infty}\left[\frac{(-1)^n(e^{\pi} - e^{-\pi})}{\pi(n^2 + 1)}\cos nx + \frac{n(-1)^n(e^{-\pi} - e^{\pi})}{\pi(n^2 + 1)} \sin nx\right]... |
H: proving some interesting properties of these matrices
let X and Y be two matrices different from I, such that $XY=YX$ and $X^n-Y^n$ is invertible for some natural number n .If
$$X^n-Y^n = X^{n+1}-Y^{n+1} = X^{n+2}-Y^{n+2}$$,
then prove that $I-X,I-Y $ are singular and $X+Y=XY+I$
my approach:
I tried to pre multiply... |
H: frac{a}{b}+\frac{b}{a} \notin \mathbb{Z}$
I know that $a \neq b$, then $\frac{a}{b}+\frac{b}{a}$ would not equal to $m$ (an integer) so I set them into one that wasn't a fraction by squaring. So I got $\frac{a^2+b^2}{ab}$. How can I show that $a^2+b^2$ is not divisible by $ab$?
AI: I assume $a,b\in \Bbb Q-\{0\}$ un... |
H: a and b both divide c and are coprime; does ab then also divide c?
I believe that I intuitively understand that if $a$ divides $c$ and $b$ divides $c$ and if $a$ and $b$ are coprime, then their product $ab$ must also divide $c$.
What would be a convincing proof of that using elementary number theory?
AI: Here's a p... |
H: $\prod_{i=1}^\infty\left(\frac{i+x}{i+1}\right)^{1/i}\stackrel{?}{=}x$
$$\prod_{i=1}^\infty\left(\frac{i+x}{i+1}\right)^{1/i}\stackrel{?}{=}x$$
I do not have the knowledge needed to prove this (assuming it is true).
quick equivalent forms:
$$\sum_{i=1}^{\infty}\frac{\log(i+x)-\log(i+1)}{i}\stackrel{?}{=}\log(x) $$
... |
H: Class of Successor ordinals.
I understand fairly well why the class of all of the ordianls is not a set, because if we assume by contradiction that it is a set, then as an ordinal - we get that it contains itself - which it is a contradiction.
However, why is the class of all the successor ordinals isnt a set?
AI: ... |
H: When is $(x \textrm{ mod } a) \textrm{ mod } b = (x \textrm{ mod } b) \textrm{ mod } a$?
I don't know when the equation $(x \textrm{ mod } a) \textrm{ mod } b = (x \textrm{ mod } b) \textrm{ mod } a$ holds.
I am looking for non-trivial necessary of sufficient conditions on $a,b$, and $x$. Is there any special condi... |
H: What is meant by a power of a Markov kernel?
My lecture notes use powers of transition kernels, but I am not sure what is meant by these powers.
Let $E$ be a Polish space equipped with the Borel $\sigma$-algebra $\mathcal{E}$, and with a partial ordening $\preceq$ on $E$.
Lemma Let $\lambda, \mu : \mathcal{E} \to [... |
H: Find the left hand limit of $f(x)$ at $x = 4$ if $f(x) =\frac{|x-4|}{x-4}$ if $x \neq 4$ and $f(x)=0$ if $x=4$
I just began limits for an introductory knowledge of calculus for Physics and I encountered this question in my Mathematics textbook.
So, what I know about limits is that if
$$\lim_{x \rightarrow a^-}f(x) ... |
H: Symmetric difference of A and B
In the two sets
$A = {\{1,2,3,4,5,6}\}$
$B = {\{1,3,9}\}$
Find the symmetric subtraction of those 2 sets.
Answer : $A \triangle B = (A \cup B) - (A \cap B)$ = ${\{1,2,3,4,5,6,9}\}- {\{1,3,9}\} = {\{2,4,5,6}\}$
Is it correct?
AI: $9$ should not be in the intersection of $A$ and $B$ so... |
H: Proving $(A - B) \times (C - D) = (A \times C - B \times C) - A \times D$
I am trying to prove $(A - B) \times (C - D) = (A \times C - B \times C) - A \times D$ using biconditionals, but cannot quite get there. For any ordered tuple $(\alpha, \beta)$, we have:
\begin{align*}
(\alpha, \beta) \in (A - B) \times (C - ... |
H: Is $n+1$-th order logic always more expressive than $n$-th order logic?
Is $n+1$-th order logic always more expressive than $n$-th order logic? That is to say, it means that third-order logic is more expressive than second-order, $11$-th order logic is more expressive than $10$-th order logic, etc. Is this true? An... |
H: What is the mathematical symbol for "nothing"?
The question might sound weird. But I have a situation coming up while writing a research paper. I will try to put it simply.
I want to define a random variable $X$ which takes the values from the set $\{0,1\}$. The probability of $X = 0$ is 0.4 and the probability of ... |
H: How to describe the solutions of a Pell equation which contains a rational number
Let $N(x,y) = x^{2}-dy^{2} $ with d a strictly positive integer and not a square and m an non-zero integer.
$$ N(x,y) = m
$$
is the general form of Pell's equations which is mostly studied in the literature.
But what happens when m is... |
H: Why is $\int \big( \nabla{p_{1}} \cdot \nabla{p_{2}} \big) = \int s_{1}p_{2}$ when $-\nabla^{2} p_{1} = s_{1}$ and $-\nabla^{2} p_{2} =s_{2}$
This came up while reading something about the porous medium equation, (see equation 10 of the link if curious) I don't have much background in PDE's so I apologize for my ig... |
H: An upper bound whose value is known for the series $ \displaystyle{\sum_{p \text{ prime}}}\frac{1}{p^{3/2}}$
We know, series $ \displaystyle{\sum_{p \text{ prime}}}\frac{1}{p^{3/2}}$ is Convergent since it is bounded above by the Convergent series $ \displaystyle{\sum_{n=1}^{\infty}}\frac{1}{n^{3/2}}$.
But can we s... |
H: Query regarding joint distribution
I came across a question which I never seen before, I just need guidance to which specific topic in probability such questions belongs, so that I can have a look at them out of curiosity. Here's the question :
.
AI: I would assume geometric probability means a measure proportional... |
H: Let $ba=a^4b^3$, show that $\mathrm{ord}(a^4b)=\mathrm{ord}(a^2b^3)$
Let $ba=a^4b^3$ show that $\mathrm{ord}(a^4b)=\mathrm{ord}(a^2b^3)$ where $a,b\in G$ and $G$ is a group.
I have toiled on this problem for the past 2.5 hours. I tried substitution of the elements with their expressions. I tried taking advantage of... |
H: Playing with the Möbius strip... mathematically
On the internet you often see these animations, where a Möbius strip gets cut along the center line, or other fun stuff.
But I have never seen a mathematical description of what is happening there.
What is the technique used to "cut" a topological space in half, like ... |
H: Number of $3$-colourings of this chain of $3$ triangles and $2$ trapeziums
Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?
Ok, so for the left... |
H: Inductive set, maximal elements and upper bounds
In my algebra notes :
Definition of upper bound
$x \in X$ is an upper bound of $Y$ if $y \le x\ \ \forall y \in Y$.
Definition of maximal element
We say $m \in X $ is a maximal element if $m\leqslant x\ \forall \in X\ \implies x=m$
Definition of inductive set
Let $X$... |
H: Markov chain on state space $S$, $A \subset S$ , $\forall i \in A, \sum_{j \in A} P_{ij} = 1$. Show that the Markov chain is not irreducible.
A Markov chain on state space $S$, and suppose there is a strict subset $A \subset S$ such that for all $i \in A, \sum_{j \in A} P_{ij} = 1$. Show that the Markov chain is no... |
H: Doubts about this summation (Geometric series)
I don´t understand why does in the numerator is $e^{{t}/{2}}$
AI: $ \sum\limits_{n=1}^{\infty} ar^{n-1}=a\sum\limits_{n=0}^{\infty} r^{n}=a\frac 1 {1-r}$ for $|r|<1$. Now multiply both sides by $r$ to get $ \sum\limits_{n=1}^{\infty} ar^{n}=a\frac r {1-r}$. |
H: Is Lagrange multipliers and (multivariable) extreme value theorem related?
I couldn't find a question answering this concept but they seem to be related.
Extreme Value Theorem (two variables)
If f is a continuous function defined on a closed and bounded set
$A⊂\mathbb{R}^2$, then f attains an absolute maximum and ... |
H: Where is my mistake in calculating $\sum_{n=-\infty}^\infty|2f_N(n)-f_N(n+1)-f_N(n-1)|^2$?
Consider the problem
We assume $N$ is an even number. I notice that terms corresponding to $n > N+1$ and $n < -(N+1)$ are all $0$. The values at the endpoints $N+1$ and $N$ are $1$ and $9$, respectively, and the same for the... |
H: Is it true that $\lim_{x \to \infty} (e^{x}-P(x)) = \infty$?
I have already proven that $\lim_{x \to \infty} \frac{e^{x}}{P(x)}= \infty$. This means that $e^{x}$ grows faster than any polynomial function. Knowing this, can it be said that $\lim_{x \to \infty} (e^{x}-P(x)) = \infty$?
I believe this to be true, howev... |
H: Why does $dx$ disappear in the right-hand side?
I am reading some text about even functions and found this snippet:
Let $f(x)$ be an integrable even function. Then,
$$\int_{-a}^0f(x)dx = \int_0^af(x)dx, \forall a \in \mathbb{R}$$
and therefore,
$$\int_{-a}^af(x)dx = 2\int_0^af(x), \forall a \in \mathbb{R}$$
Why d... |
H: Sum of $5$ dice: Number of solutions diophantine equation
Calculate the probability that, when we throw $5$ dice, their sum is $18$. To do this, I figured I need to know how many solutions does this diophantine equation have:
$$\left \{\begin{array}[c]
xx_1+x_2+x_3+x_4+x_5=18 \\ 1\leq x_i \leq 6
\end{array} \right ... |
H: Combinatorics with a condition of maximum sum
I have to say that English is not my first language, so excuse me for any mistakes.
I was learning combinatorics to prepare myself for probability and I thought of this problem and I'm not sure if I got the right answer.
Let's say I have $3$ variables ($X$, $Y$ and $Z$)... |
H: Exact sequences (homomorphism of short exact sequences)
Provided is a homomorphism of short exact sequences:
$$
\begin{array}
& 0 & \rightarrow & F_1 & \overset{\varphi_1}{\rightarrow} & E_1 & \overset{\psi_1}{\rightarrow} & G_1 & \rightarrow& 0 \\
& & \varrho\downarrow & & \sigma\downarrow & & \tau\downarrow &... |
H: Baire category theorem for locally compact and Hausdorff proof
I understand steps i) and ii) but am confused how iii) is supposed to work. The induction starts at $1$ but $N_0$ is undefined and why does iii) deal with both $U_k$ and $U_{k-1}$? Isn't $U_{k-1}$ already defined?
I also don't get how they got the chai... |
H: Exponent notation (Tetration)
What is the meaning of this notation, and how does it work?
$$\exp_{10}^2(1.09902),\,\exp_{10}^3(1.09902)$$
I knew this notation when i was reading about tetration on wikipedia. Here is the link:
Tetration
and the values above are equal to the following values:
$$^4 3=\exp_{10}^2(1.0... |
H: Solve the following constrained maximization problem
Question: let $T\geq$ 1 be some finite integer, solve the following maximization problem.
Maximize $\sum_{t=1}^T$($\frac{1}{2}$)$^t$$\sqrt{x_t}$ subject to $\sum_{t=1}^{T}$$x_t\leq1$, $x_t\geq0$, t=1,...,T
I have never had to maximize summations before and I do n... |
H: For what $p_1,\ldots,p_n$ is $P\left(\sum_{i=1}^n X_i = m\right) $ maximized?
Suppose that I have $X_1,\ldots, X_n$ such that $X_i \sim \text{Bernoulli}(p_i)$, here we have that $X_i$ are independent but not neccessarily identically distributed. However, we have the constraint that $p_i>0.5$, let $m < 0.5 n$.
What ... |
H: Is the sum of two primitive elements primitive?
Let $K$ be a field of characteristic $0$, $\alpha,\beta$ algebraic elements in an algebraic closure of $K$.
Let $K(\alpha,\beta)/K$ be the associated field extension. If both $K(\alpha)$ and $K(\beta)$ are both proper subfields of $K(\alpha,\beta)$, then do we have $K... |
H: Absolute value inequalities with integrals
I don't understand why the following makes sense. Here, $1\leq t \leq 2$:
$$
\begin{align}
\int_1^2\Big|\int^t_1 f(x)x^2 \ dx \Big|\ dt &\leq \int_1^2 \int_1^t \Big|f(x)x^2\Big| \ dxdt\\
&\leq \int_1^2 \int_1^2 \Big| f(x)x^2\Big| \ dxdt\\
&\leq \int_1^2 \int_1^2 M \ dxdt,... |
H: Is $U$ diagonalizable?
Assume that $V$ is an $n$-dimensional vector space, and $T, U \in \mathcal{L}(V)$ are such that $TU = UT$, and the characteristic polynomial of $T$ has $n$ distinct roots.
Is $U$ diagonalizable?
Justify your answer.
(image)
Well clearly $T$ and $U$ are commutative and since T has $n$ distin... |
H: Why is it that for three numbers $i, j, n \in \mathbb{N}$, if $i \equiv j \pmod{n}$ and $i, j \leq n$, then $i = j$?
For $i, j, n \in \mathbb{N}$, if $i \equiv j \pmod{n}$ and $i, j \leq n$, then $i = j$.
Might be a trivial question, but I don't see why this holds. Can someone explain to me why this is true?
AI: ... |
H: Elegant check of equalities in a 3Blue1Brown post on the Leibniz formula?
At this point on the presentation the Leibniz formula:
$$1- \frac 1 3 + \frac 1 5 - \frac 1 7+\cdots= \frac \pi 4$$
is expressed with "a few lines of calculus" as
$$\begin{align}
1- \frac 1 3 + \frac 1 5 - \frac 1 7+\cdots & =\int_0^1 \left(1... |
H: Solving Inverse trig problems using substitution?
I have this problem
$$\arccos\left(\frac{x+\sqrt{1-x^2}}{\sqrt{2}}\right)$$
The answer comes out be $\arcsin(x)-\frac{\pi}{4}$
I've realized that this problem can be solved by using something called substitution, but i really dont get the idea of how you can just su... |
H: A formula for Ramanujan's tau function
In his paper On certain Arithmetical Functions published in Transactions of the Cambridge Philosophical Society, XXII, No. 9, 1916, 159-184, Ramanujan makes some bold claims about the tau function defined as follows: $$\sum_{n=1}^{\infty} \tau(n) q^n=q\prod_{n=1}^{\infty} (1-q... |
H: For which $\alpha$ the integral converges $\iint_{\mathbb{R}^2}\frac{dx\,dy}{(x^2-xy+y^2+1)^\alpha}$?
For which $\alpha$ the integral converges $$\iint_{\mathbb{R}^2}\frac{dx\,dy}{(x^2-xy+y^2+1)^\alpha}\,?$$
I changed to polar coordinates but got stuck at after calculating the integral of $r$
$$\int_{0}^{2\pi}dt\in... |
H: Demonstration of Peano's Existence theorem by delayed argument
I am studying the proof of the Peano Theorem presented in the book A Short Course in Ordinary Differential Equations by Qingkai Kong.
The theorem is the Lemma 1.3.2 in the page 13 of the book. I have a specific doubt when "uniformly bounded" is demonstr... |
H: Why is $\frac{d}{dx}\sin^{-1}\left(2x\sqrt{1-x^2}\right)=\frac2{\sqrt{1-x^2}}$ wrong?
In our text book's(Higher Math 1st Paper-by S U Ahmed) differentiation chapter there is a section about replacing $x$(inside inverse trigonometric function) with trigonometric functions. A example problem was $\frac{d}{dx}\sin^{-1... |
H: Lambda Calculus: What does $\lambda x.(\lambda xy.xy)(xx)$ or $\lambda x.\lambda y(xx)y$ mean?
I'm trying to understand fixpoints in the Lambda Calculus with the example of computing the fixpoint of $\lambda xy.xy$. I might have a vague intuition what $\lambda x.(\lambda xy.xy)(xx)$ means/does but I cannot go any f... |
H: Find the unit digit of: $\left\lfloor{10^{20000}\over 100^{100}+3}\right\rfloor$
Find the unit digit of:
$$\left\lfloor{10^{20000}\over 100^{100}+3}\right\rfloor$$
I am completely clueless about how to deal with such huge powers.
I noticed that the numerator is $({100^{100}})^{100}$, which brings some sort of sim... |
H: Question about proof of Leibniz's integral rule.
Leibniz's integral rule :
Suppose $p(x)$ and $q(x)$ are differentiable on $\Bbb R$. Let $f:\Bbb R^2\to \Bbb R$ and $\frac{\partial f}{\partial y} $ are continuous on $\Bbb R$. Define $F(y)=\int _{p(y)} ^{q(y)} f(x,y) dx$ , $y\in \Bbb R$. Then $F$ is differentiable on... |
H: Books on advanced real analysis topics/metric spaces
There is a class in my uni called Real Analysis, which comes after 3 semesters of "analysis" and 1 semester of complex analysis. I am looking for good books that cover the material, but most of the books I can find are either more basic and cover mostly material ... |
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