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H: Schwarz Inequality for Riemann-integrable
I want to show the Schwarz inequality,
$$
\left(\int_Qfg\right)^2\le\int_Qf^2\int_Qg^2,
$$
for Riemann-integrable functions $f:Q\subset\mathbb{R}^n\to\mathbb{R}$, where $Q$ is a rectangle.
But in the case where $\int_Qf^2 = 0$ I'm not seeing how to get $\int_Qfg=0$ from it.... |
H: Congruence mod p involving a product
Let $p$ be a prime, $p\equiv 3$ mod $4$. Numerically it appears that
$$
\prod_{n=1}^{p-1}\left(1+n^2\right)\equiv 4\mod p.
$$
How can one prove this? For $p\equiv 1$ mod $4$, the product is $0$ mod $p$ because $-1$ is a quadratic residue.
AI: Let's work over the finite field $\B... |
H: How are accumulation points related to topologies?
Is it possible to have two topologies, one strictly finer than the other, yet all of the accumulation points are the same?
AI: Since the closure of a set is the union of the set with the set of its accumulation points, if all accumulation points of all sets are the... |
H: Can any cyclic polynomial in $a, b, c$ be expressed in terms of $a^2b+b^2c+c^2a$, $a+b+c$, $ab+bc+ca$ and $abc$?
Problem. Let $f(a,b,c)$ be a cyclic polynomial in $a, b, c$.
Can $f$ always be expressed as $g(a+b+c, ab+bc+ca, abc, a^2b+b^2c + c^2a)$ for some polynomial $g(p, q, r, Q)$?
Motivation: If we want to pro... |
H: Prove that $|AUC| = |A|$, where $A$ is an uncountable set and $C$ is a countable set.
Let $A$ be an uncountable set and let $C$ be a countable set with $A \cap C = \{\}$. Show that $|A\cup C| = |A|$. I'm quite lost as to how to approach solving this problem. I know that CSB theorem is involved to conclude there is ... |
H: What would a graph where $A\propto Y^2$ look like?
Consider a relation between $X$ and $Y$ such that the area under the curve of this was proportional to the square of $Y$, what would such a graph look like.
I was inspired of this question when I was going through my Physics textbook which mentioned such a thing wh... |
H: Is there a closed formula for the sum of a geometric progression with binomial coefficients?
The title asks it all.
$$\sum_{i=0}^n{n\choose i}x^{i+1}=?$$
AI: The binomial theorem gives
$$(1+x)^n = \sum_{i=0}^n \binom n i x^i.$$
So your sum in question is simply $x(1+x)^n$. |
H: Find Lebesgue Integral: $\lim_{n\rightarrow\infty}\int_0^2f_n(x)dx$
Consider a sequence of functions $f_n:[0,2]\rightarrow\mathbb{R}$ such that $f(0)=0$ and $f(x)=\frac{\sin(x^n)}{x^n}$ for all other $x$. Find $\lim_{n\rightarrow\infty}\int_0^2f_n(x)dx$.
My attempt:
$\sup_{x\in(0,2]}|f_n(x)-f(x)|=\sup_{x\in(0,2]... |
H: An equivalent form of the definition of conditional probability
Problem: Suppose that the $\sigma$ algebra $\mathscr{G}$ is generated by a $\pi$ system $\mathscr{C}$. Please prove that $f\in\mathscr{G}$ is the conditional probability of the event $A$ with respect to $\mathscr{G}$ if and only if
$$\int_B f d\mathbb... |
H: Find $\int_0^{2\pi} \frac{x \cos x}{2 - \cos^2 x} dx$.
I have to find the integral:
$$\int_0^{2\pi} \frac{x \cos x}{2 - \cos^ 2 x} dx$$
I rewrote it as:
$$\int_0^{2\pi} \frac{x \cos x}{1 + \sin^ 2 x} dx$$
But nothing further. I plugged it in a calculator and the result was $0$. I can see that the following relation... |
H: Suppose that $S\sim T$, $P\sim Q$, $S⋂P=\varnothing $ and $T⋂Q=\varnothing$. Prove that $(S∪P) \sim (T∪Q)$.
By definition of equivalent sets, a set $S$ is equivalent to set $T$ if and only if the function $f:S\to T$ is one-to-one and onto. A set $P$ is equivalent to set $Q$ if and only if the function $f:P\to Q$ is... |
H: Switching limit and infinite product
Dominated convergence theorem for an infinite product states that:
$$\lim_{n \to ∞} \prod_{k=1}^{∞}(a_{kn}+1)=\prod_{k=1}^{∞}\lim_{n \to ∞} (a_{kn}+1)$$
If
There exists a convergent sum $$\sum_{k=1}^{\infty}b_{k}$$ such that (for all k)$$b_{k}{\ge}|a_{nk}|$$
To prove this base... |
H: Is this vector proof question wrong?
If a, b, c and d are not equal or 0 and (a.b)c=(b.c)a`,
show that a and b are parallel.
Since the dot product is a scalar, I can see that c and a are scalar multiples of each other, but what if b is prependicular to a and c? Wont the given equation still be satisfied whilst a... |
H: KKT $\min x_1^2+2x_2^2+x_1$
Given
$\min x_1^2+2x_2^2+x_1$
s.t : $x_1+x_2\leq a$
Prove:
a)for every $a\in\mathbb{R}$ the problem has a unique solution.
b)find the optimal solution as a function of $a$.
c)let f(a)be optimal value find explict formula for f(a) and prove it is convex.
So for a) i said the the target... |
H: How do you find the limit of $8+(-1)^n 8$? And the intuitive way of thinking why it approaches the limit.
Given $\lim_{n\to \infty} 8+(-1)^n8$ does this converge towards some number or does it diverge? What is the intuitive way of thinking about the limit of this function?
AI: Notice that if $n$ is even, then $(-1)... |
H: Question About An Induction Problem: What Am I Supposed To Prove?
I've encountered this induction problem, and I'm not sure what I'm supposed to prove. I would like an explanation on a few things about the problem; I'm not looking for a hint/solution to the problem. The problem here is:
What I'm confused about is... |
H: how to calculate （1987^718） mod 60
My attempt:
$1987=60×33+7$，$\phi（60）=16$，so $7^{16}≡1 \bmod60$,
$1987^{718} ≡ 7^{718} ≡ 7^{(16×44+14)} ≡ 7^{14} \bmod60$,
then I have no idea how to solve it.
What do you think about it? Could you please show me?
Regards
AI: Once you've reduced it to $7^{14} \pmod {60}$, note tha... |
H: It is true that $1 \leq k \leq n$ then $k^2+n+1 \leq 2k(n+1)$?
Problem:
Suppose $k \in \mathbb{N}$ and $n \in \mathbb{N}$.
Suppose $1 \leq k \leq n$.
It is true that $k^2+n+1 \leq 2k(n+1)$?
How can I prove it?
My attempt:
I tried using induction on $n$ but actually I am not sure the statement is true.
AI: We want t... |
H: Let S, T and P be three nonempty set. Prove that (a)S~S (b)If S~T, then T~S
(a) S~S means it is reflexive
(b) If S~T, then T~S means it is symmetry
Using the definition of equivalent sets,
set S is equivalent to T if and only if there exists a function f:S->T which is one-to-one and onto.
set T is equivalent to T i... |
H: Which is the canonical base for the Space of Complex matrices $A\in \Bbb{C}^{n\times n}$?
Which is the canonical base for the Space of Complex matrices $A \in \Bbb{C}^{n\times n}$?
I know in real matrices $\in \Bbb{R}^n$, the canonical base is trivial, just $n$ vectors with a $1$ in the $i$-th component for the $i$... |
H: Derivative of parametric equations
Consider the parametric equations $x=x(t), y=y(t)$.
I'm told that the derivative can be expressed as $$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ provided $dx/dt\neq 0$.
However, in the derivation of this, it was assumed that we can express $y$ as a function of $x$, i.e.... |
H: $ |T z|= |z| \,\,\forall z\in \mathbb{C} \implies ab=0 $
If $T: \mathbb{C} \to \mathbb{C}$ is given by $ T(z)= a z + b \bar z$ where $a,b \in \mathbb{C}$.
Then is it true that, $ |T z|= |z| \,\,\forall z\in \mathbb{C} \implies ab=0 $ ?
If yes, how can I show this?
Any help would be appreciated. Thanks in advance.... |
H: Probability function question - Uniform distribution
if $X∼Uc(1,8)$ and $Y=X^2$. I need to calculate $P(Y≤7)$.
So Isn't it just the cumulative uniform distribution function but to calculate $x$ first? meaning $(x-a)/(b-a)$.
So I do $x-1/8-1$, but what's getting me confused is if I can do this -
x = $2.645$ (the sq... |
H: what's the meaning of ordinates of a Gaussian distribution?
In a Gaussian distribution, what's the meaning of the height (ordinate) at $x$?
according to [1], the funtion is called the probability distribution function of a Gaussian distribution, according to [2], it calculates the height of a Gaussian distribution.... |
H: Find the value of $p$ for which $f^{'\:}\left(\frac{1}{2}\right)\:=\:g^{'\:}\left(\frac{1}{2}\right)$
Given that $f\left(x\right)\:=\:tan^{-1}\left(2x\right)$ and $g\left(x\right)\:=\:p\:tan^{-1}\left(x\right),$ where $p$ is a constant. Find the value of $p$ for which $f^{'\:}\left(\frac{1}{2}\right)\:=\:g^{'\:}\le... |
H: Notation question: linear map $P(x_1,x_2,x_3,x_4,x_5)$
Consider the vector space $\mathbb{C}^5$ and the map $P:\mathbb{C}^5\mapsto\mathbb{C}^5$,
$P(x_1,x_2,x_3,x_4,x_5)=(x_1,x_2,0,0,0)$.
Am I supposed to read $x_1,x_2,x_3,x_4,x_5$ as 5 vectors or as the 5 entries of a single vector?
AI: They're five entries of a v... |
H: Multivariable Calculus proof explanation help, Cauchy Sequences
Proof here. This is a proof from the Advanced Calculus book by Gerald.B Folland. I understand all the steps except where the author goes on to say that $|\textbf{x}_k|<|\textbf{x}_{K+1}|+1$ for all $k>K$. How did he get to this inequality?
AI: It's jus... |
H: Dividing polynomial, multiple divisors and remainders
Problem:
If we divide $P(x)$ by $x-1$, then the remainder is $1$.
If we divide $P(x)$ by $x^2+1$, then the remainder is $2+x$.
What is the remainder when we divide $P(x)$ by $(x-1)(x^2+1)$?
My attempt:
I know $P(1)=1$ from first condition, I tried to see if I ca... |
H: Length of a line in a regular hexagon
The picture belongs to a regular hexagon. I don't understand that if a line is drawn from $E$ to $C$, how its length equals to $|KL|$ and how it is parallel to $|KL|$?
Would you mind drawing its explanation?
AI: The two lines are not parallel or have the same length. We know th... |
H: What does it mean for a point to be on the segment $[1,i]$?
I came up on a problem where I have to prove that there does not exist a point $a\in\mathbb{C}$ on the segment $[1,i]$ such that $i+1=3a^2$.
I am struggling to understand what is meant by $a$ being on the segment $[1,i]$. Does it mean that $a$ is on the se... |
H: Let $a > 1$ and $x > 0$. Prove that $a^x > 1$
What i'm proving is in the title. Essentially, I know that it holds for rational numbers. I want to prove it for all real numbers. The following is the definition for $a^x$ that I'll be using.
Let $\{r_n\}$ be any sequence of rationals that converge to $x$. Then, the ex... |
H: Gebrane's Hanoi
I need some help to solve a modification of the Tower of Hanoi problem found on a french forum.
The classic problem is described here.
This modification is called Gebrane's Hanoi problem (from the name of its inventor), the $n$ disks are numbered from top to bottom from $1$ to $n$ and are also place... |
H: Prove the following: If $\lim(x_n) = x$ and if $x > 0$, then there exists a natural number $M$ such that $x_n > 0$ for all $n\ge M$.
I recently encountered the following question:
Prove that if $\lim(x_n) = x$ and if $x > 0$, then there exists a natural number $M$ such that $x_n > 0$ for all $n\geq M$.
My solutio... |
H: Glueing together Riemann surfaces
In Miranda's , he wrote such a proposition:
proposition 1.6 of chapter 1: Let $X,Y$ be 2 Riemann surfaces. $U\subset X$, $V\subset Y$ are their open subsets. Suppose $\phi: U\rightarrow V$ is an isomorphism of Riemann surfaces, then there's a unique complex structure on $Z:=X\amalg... |
H: Why is $\left( -a,a \right)$ equal to $|x| < a$
I have a problem understanding the equality of an open Interval as given $\left( -a,a \right) = \{x \in R | -a< x< a\}$ to say $|x| < a$..
Maybe someone can get me intuitive to understand that?
AI: Since the end points are not contained in the interval, we have $|x|<a... |
H: Relation between areas in a trapezoyd
A trapezoid of ABCD vertices is inscribed in a circle, with radius R, being AB = R and CD = 2R and BC and AD being non-parallel sides. The bisectors of the internal trapezoidal angles, so that the bisector of  intercepts that of Dˆ at point Q, that of B intersects that of C N ... |
H: For $f(x) = e^x + x^3 - x^2 + x$ find the limit $\lim\limits_{x\to \infty} \frac{f^{-1}(x)}{\ln x}$.
I have the function:
$$f : \mathbb{R} \rightarrow \mathbb{R} \hspace{2cm} f(x) = e^x + x^3 -x^2 + x$$
and I have to find the limit:
$$\lim\limits_{x \to \infty} \frac{f^{-1}(x)}{\ln x}$$
(In the first part of the pr... |
H: Showing that $\mathbb{C}^5/\ker(P)\simeq \mathbb{C}^2$
Consider the vector space $\mathbb{C}^5$ and the map
$P:\mathbb{C}^5\mapsto\mathbb{C}^5,\,P(x_1,x_2,x_3,x_4,x_5)=(x_1,x_2,0,0,0)$
According to my lecture notes, this implies: $\mathbb{C}^5/\ker(P)\simeq \mathbb{C}^2$. But what exactly does this mean? That $\mat... |
H: Khan Academy - Factoring sum of squares
I am working on Precalculus - Factoring sum of squares
The video starts with example:
$36a^8 + 2b^6 = 6a^{4^{2}} + \sqrt{2}b^{3^{2}}$
Can someone explain this to me? What's happened here?
Thanks.
AI: Actually, the video starts with
$$36a^8+2b^6=(6a^4)^2+(\sqrt{2}b^3)^2$$
whi... |
H: Prove that a planar graph with all vertices degree $3$ must has a face with at most $5$ edges.
I have some problems when I prove "For a planar graph $G$ and $\deg(v) = 3$ for any vertex $v$, there is a face with at most $5$ edges". I want to prove with contradiction.
Suppose that every face has more than $5$ edges.... |
H: What is the "unit axis"?
As the highlighted text in the screenshot, what is the unit axis? And what mathematical properties the three coordinates X, Y, Z have?
AI: It means a vector of length 1. |
H: How to calculate the integral of $ \int\frac{1}{\cosh^{2}x}dx $
Ofcourse one can notice that
$ \left(\frac{\sinh\left(x\right)}{\cosh\left(x\right)}\right)'=\frac{\cosh^{2}\left(x\right)-\sinh^{2}\left(x\right)}{\cosh^{2}\left(x\right)}=\frac{1}{\cosh^{2}\left(x\right)} $
But im looking for a straight way to prove... |
H: How to solve such equations: $z^4 = -4$, $z \in \mathbb{C}$?
I have the following task:
Compute in each case all $z\in\mathbb{C}$ such that $z^4 = -4$ and $z^3 = 5i$.
I do not know how to solve such eqautions fast. Do you have any idea of how to solve such equations?
And one further question:
The multiplication o... |
H: Minimising variance of the expected return for two correlated investments
For two investments with returns $X$ and $Y$, we can allocate the cash in proportions $\alpha$ to $X$ and $1-\alpha$ to $Y$, making the total variance of our investment $$\text{Var}(\alpha X+(1-\alpha)Y).$$ In a textbook I'm told that the val... |
H: Is this result of the derivative of the composition of two functions correct?
I have the following functions:
$f(x,y)=xy,\; g(t)=(e^t,\cos(t))$
And I want to calculate the derivative matrix of $f$ after $g.$
I calculate the jacobian matrix of $f$, which was a matrix, like
$$
D_f=\begin{bmatrix}
y & x
\e... |
H: Showing that if $\phi:\Bbb{Z}\oplus\Bbb{Z}\to\Bbb{Z}\oplus\Bbb{Z}$ is an epimorphism of abelian groups, then it is an isomorphism.
I am a mathematician working in analysis and my knowledge in algebra is rusty.
Is there a direct argument showing that if
$\phi:\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{Z}\oplus\mathbb{Z... |
H: How would I cut one washer into two equal-area washers?
I'm trying to mill an asymmetric graphite crucible in the shape of a hollow cone (imagine an ice cream cone with the end bitten off). I require identical "horizontal resistance" at the wide top, as well as the narrow bottom (and throughout) - hence - the wall... |
H: Help understanding step in the proof of Taylor's theorem with remainder
Let $k\in\mathbb{N}$, $x_{0},x\in A\subseteq\mathbb{R}$ and $f:A\to\mathbb{R}$ satisfy that $f^{(j)}$ exists and is continuous on the closed interval between $x_{0}$ and $x$ and differentiable on the open interval between $x_{0}$ and $x$ for al... |
H: How many functions f : [n] → [n] are there for which there exists exactly one i ∈ [n] satisfying f(i) = i?
As the title asks, I'm looking for the number of functions $f : [n] \rightarrow [n]$ for which there exists exactly one $i \in [n]$ satisfying $f(i) = i$.
I know the number of functions $f : [n] \rightarrow [... |
H: $f(2-x)=f(2+x)$ and $f'(1/2)=0=f'(1)$. Find minimum number of roots of $f''(x)=0$.
Let $f$ be a non constant twice differentiable function satisfying $f(2-x)=f(2+x)$ and $f'(1/2)=0=f'(1)$. Find the minimum number of roots of $f''(x)=0$ in the interval $(0,4)$.
Answer: $4$
I managed to rewrite the given equation a... |
H: Negative arc length value. True or not?
I saw a post about this and someone said the arc length is an integral of a positive function, so it is positive.
But by solving this exercise I found the arclength as a negative value.
The arc length for $ y= ln (1-x^2)$ from x=0 to x=1/2.
The result I found is $L= -ln(2)-l... |
H: $\forall x \in X: (y < x \implies y \in E) \implies x \in E$
Let $(X, \leq)$ be well-ordered and non-empty.
If $E \subseteq X$ satisfies
(i) $\min E \in X$
(ii) $\forall x \in X: ((y < x \implies y \in E) \implies x \in E)$
Then $E=X$.
Proof: Assume $E \neq X$. Then we can put $x:= \min X \setminus E$. If $y < x... |
H: Calculate probability based on depth
Task: Calculate probability based on depth
$$f(x)=\begin{cases} \frac{1}{2} & x \in [1,3]\\\ 0 & x \not\in [1,3]\end{cases} $$
$$P([1,2])=?$$
My take on it:
$$ \int_3^1 f(x)dx= [ \frac{x}{2} ]^{3}_{1} = \frac{3}{2} - \frac{1}{2} = 1$$
Is this correct?
AI: That's not correc... |
H: Convert English statements to logical forms
Everyone has a roommate who dislikes everyone
$ R(x,y)=$ $x$ and $y$ are roommates
$L(x,y)=$ $x$ likes $y$
so it becomes
$$\forall x \exists y (R(x,y) \land \lnot L(y,x))$$
Is this correct ? Thank you
AI: No, what you have written is that everyone has a roommate who d... |
H: Floor function of a product
I'm reading a book about proofs and I'm currently stuck in this problem.
Prove that for all real numbers $x$ and $y$ we have that:
$$\lfloor x\rfloor \lfloor y\rfloor \leq \lfloor xy\rfloor \leq \lfloor x\rfloor \lfloor y \rfloor + \lfloor x \rfloor + \lfloor y \rfloor$$
I though I could... |
H: Does the complex field with dictionary order have the least-upper-bound property?
If any clarification is necessary, here is a little definition of what "dictionary" order encompasses. In Rudin's book it was stated in the following way.
Let $z=a+bi$ and $w=c+di$ where $z,w$ are arbitrary complex numbers and $a,b,c,... |
H: get 2 numbers and return the second number with the sign of the first number
I know it's more about programming, but I can use arithmetic operators only so I think it fits the math community.
AI: If $x$ is not zero, its sign is $|x|/x$. Thus you'd just need to return $|xy|/x$, where $x$ is the first number, and $y$... |
H: How to re-write this recurrence in linear form?
I have the following recurrence
$$T(n) = \frac{1 - T(n - 1)}{T(n - 2)}$$
and $T(0) = a, T(1) = b$. I re-wrote it as such
$$T(n)T(n - 2) = 1 - T(n - 1)$$
I am now trying to re-write this in linear form by making some sort of substitution. What could I do to make this r... |
H: What is meant by this indicator function?
In Hastie et al. (2009) p.509, it is written that $N_k=\sum_{i=1}^N I(C(i) = k)$. To my understanding, $I$ should be an indicator function.
However $C(i)$ is defined as "Each observation is assigned to one
and only one cluster. These assignments can be characterized by a m... |
H: Expected number of red balls in urn of two-coloured balls with color substitution.
There are $m$ black and $n$ red balls in an urn. One randomly picks a ball from the urn. If the picked ball is the black one, person changes it with the red ball and returns to the urn. If the picked ball is red, he does nothing and ... |
H: A question on Lebesgue integral
Let $f\in L^1(\mathbb{R}^n)$. I want to prove that when $m(B-B’)\to 0$
$$\left|\int_{B’}f(x)dx-\int_{B}f(x)dx\right|\to 0.$$
In Riemann integral, if $f$ is absolutely integrable on $[a,b]$, then $|f|$ in bounded, namely $|f(x)|\le M$, on $[a,b]$, and hence we have
$$\left|\int_{B’}f(... |
H: Show that $F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$
I've been working on a recent exercise question where I was asked to show that:
$$F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$$
Now I cansee that the infinite sum is a power series, which lead... |
H: Is the theorem 5.7 given in Apostol's Mathematical Analysis correct?
I came across this theorem in Apostol's Mathematical Analysis and the proof makes sense and seems to be correct.
https://drive.google.com/file/d/1-sRgtgGlQ5N8h9Vq5eosftjcEbOavOv0/view?usp=drivesdk
However, I also encountered this counterexample in... |
H: If $A$ is finite and $f: A \rightarrow A$, $f$ is injective iff $\text{ran}f = A$: How Simple is My Proof Allowed to Be?
Synopsis
This exercise seems so obvious that I don't know how to put it into words. A simple diagram shows this to be true. But I've tried to write down a proof anyways, and I was wondering if an... |
H: Tensoring with projective module preserves injectivity.
Let $A$ be a ring, $P$ a projective left $A$-module and $E,F$ two
right $A$-modules. If $u:E\rightarrow F$ is an injective homomorphism,
the homomorphism $$u\otimes 1_P:E\otimes_A P\rightarrow F\otimes_A P$$
is injective.
Attempt:
Since $P$ is projecti... |
H: I do not understand an inequliaty that is used to prove the ratio test
I do understand:
$\exists q\lt1:\forall k\in\Bbb N_{0}:\lvert\frac{a_{k+1}}{a_{k}}\rvert \leq q$ is equivalent to
$\lvert a_{k+1}\rvert \leq q \cdot \lvert a_{k}\rvert$
Now the statement is that $
\sum \limits_{k=0}^{\infty}a_{k}$ converges abso... |
H: Constructing a model of ZFC in ZFC+Con(ZFC).
In ZFC+Con(ZFC), one can prove ZFC is consistent, so has a model. But how can I construct a model of ZFC in ZFC+Con(ZFC) formally?
I think the Con(ZFC) only says "the ZFC is consistent", not a technical statement, so if we formally construct a model of ZFC in ZFC+Con(ZF... |
H: Prove that $ \lim_{n\to\infty}\frac{\left(n!\right)^{2}2^{2n}}{\left(2n\right)!\sqrt{n}}=\sqrt{\pi} $
Prove that $ \lim_{n\to\infty}\frac{\left(n!\right)^{2}2^{2n}}{\left(2n\right)!\sqrt{n}}=\sqrt{\pi} $
What i want to do is to reach the form $ \sqrt{2\left(\prod_{k=1}^{n}\left(\frac{2k}{2k-1}\cdot\frac{2k}{2k+1}\r... |
H: Combinatorics: Why/how does this solution for counting three digit numbers work?
My homework book lists three different solutions for the exercise below, I used the most obvious one, (counting all possible configurations with one 2 and two 2's), but the book also gives this other (much shorter) solution. Sadly I do... |
H: Is the inverse of an element in a group different than the element (except for $e$)? Also, are all the subgroup of a cyclic group cyclic?
Suppose the cyclic group $G=\{e,a,a^2,a^3,a^4,a^5,a^6,a^7,a^8,a^9,a^{10},a^{11}\}$ under some operation, say *.
$$G=\langle a\rangle$$
Now, each element must have an inverse, s.t... |
H: Why the open-ball topology functor from $\boldsymbol{Met_c}$ to $\boldsymbol{Top_m}$ is not an isomorphism? Are they isomorphic at all?
Let $\boldsymbol{Met_c}$ denote the category of metric spaces whose morphisms are all continous maps. And let $\boldsymbol{Top_m}$ denote the category of metrizable topological spa... |
H: The probability that a killed random walk on $[-N,N]$ escapes before dying
Let $X_t$ be a continuous time simple random walk in $\mathbb{Z}$ starting at $0$, let $\tau^*$ be an exponential r.v of parameter $1$. What is the probability
$$
\mathbb{P}(\tau ^* \ge \tau_{N})?
$$
Where $\tau_N = \inf \{t \in \mathbb{R... |
H: How to end showing that a statement is true by using induction?
I am practicing how to show true statements using induction.
Since there are eventually different ways to show that a statement is true, I wanted to ask the following: When can you say that a true statement has been shown?
Example:
Prove that for any... |
H: finding the values for $x_1+x_2+x_3=5$ by restriction
Let $x_1+x_2+x_3=5$ and $1 \leq x_1 \leq 4$ , $0 \leq x_2 \leq 4$ , $0 \leq x_3 \leq 4$ .
How many $x_1$, $x_2$, and $x_3$ are there?
Firstly i found the all cases such that $C (4+3-1,4)=15$ but i stuck in the rest.I could not make inclusion exclusion part. Can... |
H: Weierstrass' M-test in reverse
Weierstrass' M-test says that the series of functions on some set $X$:
$$\sum_{n=1}^\infty f_n(x)$$
if $\forall n \in \mathbb{N}, \exists M_n$, \forall x\in X where $M_n \geq |f_n(x)|$, so the majorant series $\sum_{n=1}^\infty M_n$ converges, then the original series converges unifo... |
H: When the quotient $\frac {13n^2+n}{2n+2}$ is an integer?
Find the nonzero values of integer $n$ for which the quotient $\frac {13n^2+n}{2n+2}$ is an integer?
My Attempt
I assumed $(2n+2 )| (13n^2+n)$ implies existence of integer $k$ such that $$13n^2+n=k(2n+2)$$
$\implies\ 13n^2+(1-2k)n-2k=0$
$\implies\ n=\frac{(2k... |
H: What would the notation for this binary string look like?
Every block of 1's of length $\ge 4$ cannot be followed by a block of 0's of length $\ge 4$, and any block of 1s of length 1, 2 or 3 must be followed by a block of 0s whose length is congruent to 1 mod 4.
The answer I came up with is this:
$$\{0\}^*\{(1111)(... |
H: Dose convergence in integral and measure imply convergence in L
Suppose a sequence of function $f_n \mapsto f$ converges in measure and in integral. by integral I mean $\int f_n \,d\mu \mapsto \int f \,d\mu$. Does that mean they converge in L1. It seems not entire true to me e.g. consider $f$ with compact support w... |
H: Proving $A - B \subset A - (B - A)$
I believe I have been able to prove that for sets $A$ and $B$, $A - (B - A) \supset A - B$, but my proof is not particularly elegant. I was hoping someone knew of a more clever or straightforward way to show this. My proof is:
Let $x \in A - B$. Then $x \in A$ and $x \not \in B... |
H: If $\operatorname{ker} T \neq \{0\}$, there is $S: V \rightarrow V$, $S\neq 0$, with $T \circ S = 0$.
Let $V$ be a vector space and $T$ be linear transformation $T: V \rightarrow V$ . If $\operatorname{ker} T \neq \{0\}$, prove that there's a linear transformation $S: V \rightarrow V$ such that $S$ is not the ze... |
H: Why traditional equation solving method gives different answer for some problems?
This question arose when I tried solving one of the math problems below,
Question: The average monthly income of Rakesh and Suresh is Rs. 5050. The average monthly income of Suresh and Ramesh is Rs. 6250 and the average monthly income... |
H: If $u(x,y)$ is harmonic, how do I show that $u(z(x,y),w(x,y))$ is harmonic?
Suppose that $u(x,y)$ is an harmonic function. Consider the transformation:
\begin{align}
z &= c + x\cos\theta + y\sin\theta, \\
w &= d - x\sin\theta + y\cos\theta,
\end{align}
where $d$, $c$, and $\theta$ are constants
Show that $u(z,w)$ ... |
H: show that $T$ and $T^*$ have different eigenspaces
I have the matrix $[T]_{E}^{E}$= $\left(\begin{matrix}
1 & 1 \\
0 & 1
\end{matrix}\right)$
$V=\mathbb{R^2}$ with the standard inner product, I need to show that $T$ and $T^*$ have different eigenspaces. so for $T$ it was easy I found the eigenvalue $\lambda=1$ an... |
H: Find $\iint_{U} \frac{x}{y+x^{2}}$
Could you help me with the following please:
Evaluate
$$\iint_{U} \frac{x}{y+x^{2}}dxdy$$
where $U$ is limited by $x = 1$, $y=x^{2}$, $y=4-x^{2}$. Suggestion consider $x=\sqrt{v-u}$ and find $y$ as a function of $u$ and $v$ and apply change of variable.
I have plotted the region a... |
H: Permutations/Combinations with Categories
I have 3 categories. Each of these categories has three options.
Category 1 options: A, B, C
Category 2 options: D, E, F
Category 3 options: G, H, I
So a permutation of these may look like this: A E H
I am trying to find the number of permutations with these categories.
Thi... |
H: Is the pushforward of flow equal to its time derivate?
Let $X \in \Gamma(M)$ with $\Phi^{X}_{t}(p)$ as the one parameter group at a point $p \in M$. I believe that in general the equality $$T_p\Phi^{X}_{t_0}(X_p)=X_{\Phi^{X}_{t_0}(p)}=(\frac{\partial}{\partial t}\Phi^{X}_{t}(p))(t_0)$$ (i.e. $(\Phi^{X}_{t_0})_*(X_p... |
H: What is the derivative of the following integral?
Consider the vector $(a_M, a_S) \in R^2_{+}$. Vector components are drawn from a joint log-normal distribution $G(a_M, a_S)$ with mean $(\mu_M,\mu_S)$ and covariance matrix $\Sigma$.
Now, I have the following expression
$$N_M(W_M)=\int^{\infty}_0 \int^{a_MW_M}_0 a_... |
H: LC differential equation
I want to solve the differential equation for the charge of an LC circuit:
$$d^{2} Q / d t^{2}+(1 / L C) Q=0$$
I create the characteristic equation:
$$\mu^{2}+\frac{1}{\mathrm{LC}}=0 \Rightarrow \mu^{2}=-\frac{1}{\mathrm{LC}} \Rightarrow \mu = \pm i \sqrt{\frac{1}{\mathrm{LC}}}= \pm i w_0\\... |
H: Checking Optimization function whether its convex or not
The optimization function is defined as
$\frac{1}{2}.x^T.A.x$ where $A=\begin{pmatrix}
1 & 0.5 \\
0.5 & 1
\end{pmatrix}$
How to check if this is a convex or not? I know about the second derivative test and it gives
$A$ which should be greater than $0$ in or... |
H: Discrepancy in calculating the volume formed by revolving the region above $y=x^3$ under $y=1$ and between $x=0$ and $x=1$ about the $x$-axis
Here's the problem:
Find the volume formed by revolving the region above $y=x^3$ under $y=1$ and between $x=0$ and $x=1$ about the $x$-axis
Now here's the integral I came ... |
H: how to solve this inhomogeneous second order differential equation
I want to solve this inhomogenenous differential equation of second order:
$$ x''+2x'+x= \sqrt{t+1}e^{-t} $$
with initial conditions $ x(0)= \pi $ and $ x'(0)= \sqrt{2} - \pi $
The solution is $ y= y_p+y_h $, so the particular solution added with ... |
H: Extension of a map holomorphic on the unit disk to map holomorphic on the complex plane
I want to prove a statement regarding the holomorphic extension of a function holomorphic on the unit disk.
Let $D \subset \mathbb{C}$ be the (open) unit disk and $f: \bar{D} \to
> \mathbb{C}$ be a continous map, such that $f$ ... |
H: Finite Abelian group and subgroup
Suppose $G$ is a finite abelian group and $H$ be a proper subgroup. Let $a$ be an element in $G$ not in $H$. Does there always exists an $m>0$ s.t. $a^m \in H$? If it is there what is the proof?
AI: This holds for any group $G$ and subgroup $H$ where $a \notin H$ has finite order: ... |
H: $2^X$ separable $\implies$ $X$ separable
Let $X$ be a $T_1$ space and let $2^X$ have the Vietoris topology. I know from an article that $2^X$ is separable only if $X$ is separable, but the article omits the proof as it is apparently obvious.
Given $D$ is a countable dense subset of $2^X$, how can I derive a countab... |
H: Evaluating volume of a sphere with triple integral in cylindrical coordinates
I need to evaluate $\iiint_V \frac{1}{x}\: dV$ where $V$ is the inside of a sphere given by $x^2+y^2+z^2=x$.
I write the equation as a sphere with centre in $\left(\frac{1}{2},0,0\right)$ and radius of $\frac{1}{2}$. Now I know that my sp... |
H: Question about product of generating functions in a proof that for positive integer $n$, $\sum_{k=0}^n(-1)^k\binom nk\binom{2n-k}n=1$.
I was recently looking at a copy of the Mathematics Magazine from 2004 and was reading Q944 (here). It asks this:
Show that for positive integer $n$, $$\sum_{k=0}^n(-1)^k\binom nk... |
H: Contour Integral of $\frac{1}{1+e^z}$
How do I calculate $\displaystyle\int_{C_1(1)}(1+e^z)^{-1}\text{d}z$?
I have tried parametrizing $C_1(1)$ by $z=1+e^{i\theta}$ with $\theta \in [0,2\pi]$, but this does not help much as I would have a double exponential.
Any ideas on how to progress are welcome.
AI: That integr... |
H: Series with limit
The problem is the following. Let $(a_{n})_{n\geq0}$ such that $\lim_{n\to\infty}a_{n}=\alpha\in\mathbb{C}$ and let $(b_{n})_{n\geq0}$ a succession of positive real numbers. We know that the series
$$
\sum_{n=0}^{\infty}b_{n}z^{n}
$$
converges for every $z\in\mathbb{C}$. Prove that, considering ... |
H: Use of Bernoulli equation $x^2y' +2xy=y^3$
I am not sure what am I doing wrong here, I have $$x^2y'+2xy=y^3$$
I should apply the Bernoulli equation, so I mark $$z=\frac{1}{y^2}$$ $$z' = -\frac{2y'}{y^3}$$
$$-\frac{z'}{2} = \frac{y'}{y^3}$$
Now I just substitute $z$ and I get $$-\frac{x^2z'}{2}+2xz=1$$
How I should ... |
H: quantiles of a monotonic function of two independent R.V.s
Consider two independent random Variables $X, Y$, and $X>=0, Y>=0$. $f(X, Y)$ is monotonic to $X, Y$ respectively.
Suppose that we know the 0%, 25%, 50%, 75%, 100% quantiles for both $X, Y$ and we have access to $f(\cdot, \cdot)$.
Is it possible to find so... |
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