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H: How to formally justify the existence of a limit with two variables?
Problem: Find the limit of the following functions
a) $\displaystyle \lim_{x \to \infty, \ y\to \infty}$ $\frac{x+y}{x^2 + y^2} $
b) $\displaystyle \lim_{x \to 0,\ y\to 2} \frac{\sin(xy)}{x} $
I know how to make cases when the limit does not exist... |
H: Do the paths intersects? If so where
There are two unidentified objects in the sky. The path of the first object is given by $r(t)=\langle t,-t,1-t\rangle $ and the second object's path is $s(t)= \langle t-3,2t,4t\rangle$
Do the paths intersect? If so where and do the objects collide?
AI: You have two objects with ... |
H: Prove a relation is asymmetric.
Prove that if a non-empty relation $R$ on $A$ is transitive and irreflexive, then it is asymmetric.
I assume that I need to prove this one by contradiction, but I'm having a hard time wrapping my head around it. If a relation is transitive and irreflexive wouldn't it also be symmetr... |
H: Lambda Calculus: Reduction to Normal Form
I'm working on some problems where I'm supposed to reduce lamda terms to normal form. I'm not sure if I'm doing it right so if someone could let me know, that would be awesome.
$$(\lambda x.\lambda y.x*2+y*3)\; 5 \;4 $$
$$\rightarrow(\lambda y.5*2+y*3) \; 4 $$
$$\rightarro... |
H: Inverse of a real matrix plus identity times i
How would you proof that given a real square matrix $A$
then the inverse of the matrix ( $A + i I $) exists?
AI: It won't necessarily exist. For instance, if
$$
A = \left(\begin{array}{cc}0 & 1 \\ -1 & 0 \end{array} \right)
$$
then
$$
A + iI = \left(\begin{array}{cc}i... |
H: Is gluing the boundary of $2$ discs removed from a surface also a surface?
Suppose we have a surface $S$. Also, suppose we remove $2$ discs from the surface $S$ and we glue the boundary circles of these two discs together. Is the result a surface?
My beliefis that it is a sphere. But I am not sure how to show it. C... |
H: How do we explain to students that division by a vector does not make sense?
I have to give a presentation on vector analysis. One of many important things I want to emphasize is that a division by a vector does not make sense.
How do you explain to your students, for example, that division by a vector does not mak... |
H: Determine convergence $\sum_{n=1}^\infty\left(\frac{2 n + 2}{2 n + 3}\right)^{n^2}$
Does $$\sum_{n=1}^\infty\left(\frac{2 n + 2}{2 n + 3}\right)^{n^2}$$ converge?
Hi, I was wondering if anyone knows how to solve this problem? I think I can't use root test... because the result is 1 and it is meaningless. Thank you... |
H: Irreducible subsets of a topological space
I found this definition on Hartshorne, Algebraic geometry, page 3...
Definition A nonempty subset $Y$ of a topological space $X$ is irreducible if it cannot be expressed as the union $Y=Y_1\cup Y_2$ of two proper subsets, each one of which is closed in $Y$. The empty set i... |
H: Is Minkowski space not a metric space?
I've just started reading a book on functional analysis, and first definition given there is for a metric and metric space:
Let $\mathfrak{M}$ be an arbitrary set. A function $\rho\colon \mathfrak M\times\mathfrak M\to[0,\infty)$ is called metric if it has the following prope... |
H: If a cyclic group has an element of infinite order, how many elements of finite order does it have?
If a cyclic group has an element of infinite order, how many elements of finite order does it have?
I know that the order of the entire group must be infinite, for an element of the group must have an order less than... |
H: Some problems about functions.
1- Let $X = \{1,2,3,7,12\} $ and $Y = \{1,15,7,4,20\} $. We use notation $(x,y)$ to denote that the element $x \in X$ is assigned to (or paired with) the element $y \in Y$. For the relations defined below answer the following questions:
Does the relation define a function from $X$ t... |
H: Definiteness of A'BA
Let $A$ be a $(k \times n)$ matrix and $B$ a $(k \times k)$ matrix. In that case, is there a general result for the definiteness of the $(n \times n)$ matrix $A'BA$? If not, what if $B$ is known to be positive definite. Can the definiteness of $A'BA$ then be determined?
Best,
Esben
AI: I assume... |
H: Help me Verifying that the equation is integrable and finding its solution
How can I verify that the equation is integrable and that find its solution;
$$2y(a-x)dx+[z-y^2+(a-x)^2]dy-ydz=0$$
Honestly, I tried too much, but I got too strange results,thus I couldnt show my efforts here so sorry. Thank you for helping... |
H: Linear program dual
We are trying to find the dual of the following linear program.
$$ \max_x \ 2x_1 \ + x_2 \ \ \ \ -- (1) $$
such that,
$$ x_1 + x_2 \leq 2 \ \ \ \ -- (2)\\ -x_1 - x_2 \leq -4 \ \ \ \ -- (3)\\
x_1 \geq 0 \\ x_2 \geq 0$$
I get the following answer.
$$ \min_\theta \ 4\theta_1 + 2\... |
H: what is a multiplicative group in prime order p?
on pg. 378 section 2 (Overview) it says "We let G be a multiplicative group of prime order p , and g be a generator of G. We let e : G x G --> $G_T$" be a bilinear map.
If somebody could please break each piece of this into smaller parts I would really appreciate it... |
H: Prove d to be a metric
Goodday.
The problem is as follows:
Let $\mathbb{Z}^\mathbb{N}:=\{x:\mathbb{N}\rightarrow \mathbb{Z} \}$.
We define a function $\text{d}:\mathbb{Z}^\mathbb{N} \times
\mathbb{Z}^\mathbb{N} \rightarrow \mathbb{R}$ by the following relation: $-\text{log
d(x,y)} = \text{inf}\{\text{n}\in \math... |
H: Understanding syntax for defining a relation.
Let T = {1, 2, 3, 4} and A = T * T. We can define a relation R on A;
(a,b)R(c,d) <=> (a/b)=(c/d)
For example:
(1,2)R(2,4) since (1/2)=(2/4)
Does this mean that ((1,2),(2,4)) ∈ R
or
(1,2) ∈ R and (2,4) ∈ R
AI: It means that ((1,2),(2,4)) ∈ R |
H: Given a circle γ(t) = (cos t, sin t, 0);Is γ an asymptotic curve of the unit sphere with centre in (0, 0, 0)?
I know that the dot product of γ// and N gives the normal curvature andig its 0, then it's called an asymptotic curve.
The equation of the given sphere is x^2 +y^2 + z^2=1---as its an unit sphere with centr... |
H: Limit of a sequence true or false statement
Suppose $S(0)=2$, and $$S(n+1)=\dfrac{(S(n))²+2}{2S(n)}.$$ Then $\lim_{n\to\infty} S(n)= \sqrt2$.
So I worked some of the terms out and I get $S(1)=\frac{3}{2}$, $S(2)= \frac{17}{12}$,...
I did the limit of $(x^2+2)/(2x)=∞$ so the statement is false or what am I missing h... |
H: conversion of discrete to continuous
Given $N_{j+1}-N_j=kN_j$
How can I substitute some time variable in to make $delta(t)$ small? Meaning change in time.
I need to show $N_j=e^{(j\ln(1+k))}$
How can I rewrite the given in terms of delta t in order to take limit to find derivative ?
AI: Actually this dynamics is so... |
H: Ordinary Differential Equation how to solve this
Hi i need to learn in fast way how to solve ode i still have problem with this.
I need to find $y = y(x)$
having
$y′ = 0$,
$y′′ = 0$,
$y′′′ = 0$,
$y'''' = 0$
I am guessing that $y= e^{rx}$ Then we have some function with degree of 4 but how to solve this. I am not e... |
H: A question about extreme points
If the extreme points of the unit ball of $C[0, 1]$ are $\pm{1}$, where $C[0, 1]$ is the Banach space of all continuous real-valued functions on the unit interval, then what would the extreme points of the unit ball be if we considered all continuous complex-valued functions on the u... |
H: If the expectation exists and finite, is it always integrable?
If the expectation of a random variable exists and finite, i.e., $EX<\infty$, is it always true that $E |X|<\infty$?
This question arises when I try to prove some stochastic process is a martingale. In the definition of martingale, we need the process ... |
H: Special function
I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ with $f(\overline{A})\subset\overline{f(A)}$. $\subset$ means $f(\overline{A})\neq\overline{f(A)}$.
Does anyone has an idea?
AI: Hint: Choose some $y$ that you want to be in $\overline{f(A)}$ but not in $f(\overline{A})$. Now look for a func... |
H: Using Matlab's bsxfun to subtract matrices of different sizes
My professor gave me some matlab script in which I found the following line:
differences = bsxfun(@minus, A, B);
where A is of size [200,50] and B is of size [1,50]. The documentation and forums didn't give me the answer I'm looking for. My current unde... |
H: Uniqueness of Inverses in Groups Implies Associativity Holds?
I was checking multiplication tables for groups with 4 elements, to see which tables "passed" the group axioms of closure, associativity, identity and inverses. But then I had a question, so hopefully someone can help me with this basic group theory ques... |
H: Inverse Trig Derivative
Here's a simple problem that I'm not getting.
y = arcsec (1/x), where o < x < 1.
What is the derivative?
y' = 1 / { x^-1 * (x^(-2) - 1)^(1/2) } * -x^-2
y' = -x / { x^2 * (x^(-2) - 1)^(1/2) }
y' = -1 / { x * (x^(-2) - 1)^(1/2) }
That's what I'm getting.
But the answer should be y' = -1 / {1-x... |
H: Nested Cell Property: I'm missing something about this proof
This is a paraphrased version of the beginning of a proof given in The Elements of Real Analysis: Second Edition By Robert G. Bartle, on page 47.
Nested Cell Property. If $n\in\mathbb{\mathbb{N}}$
let $J_{n}$
be a non void closed cell in $\mathbb{R}$
... |
H: Why does $(3\sqrt3)^2 = 27$?
How does $(3\sqrt3)^2 = 27$?
I've tried to solve this using binomial expansion and using the FOIL method from which I obtain $9 + 3\sqrt3 +3\sqrt3 + 3$.
it has been a while since I've done this kind of thing so it may be something obvious that I can't see.
AI: Recall that $$(ab)^2 = (a)... |
H: Prove that $\mathbb{Z}$ is isomorphic to $\mathbb{Z}_x$
Let $\mathbb{Z}_x$ denote the ring of integers with the operations $\odot, \oplus $ defined as $a \odot b$ = $ a+b -a b$ and $a \oplus b$ = $a+b -1$. Prove that $\mathbb{Z}$ is isomorphic to $\mathbb{Z}_x$.
Not sure how to start.
AI: So we want to find a bijec... |
H: Recursively enumerable language of Turing machines
If you have the language $L_{h}=\{M_{i} | (\exists z \in \sum ^{*}) M_{i}\text{ halts on some input } z\}$
where $M_{i}$ are Turing machines, is $L_{h}$ recursively enumerable?
I'm fairly certain it is, but I'm having issues proving that to be the case.
The way ... |
H: sequence of sets with $\limsup A_n = \mathbb N$
Find a sequence of one-point-sets $A_n = \{\ell_n\}$ with $\ell_n\in\mathbb N$ for all $n\in\mathbb N$, such that
$$\limsup_{n\to\infty} A_n=\mathbb N$$
I know the definition of the $\limsup$ of a sequence of sets,
$$\limsup_{n\to\infty} A_n = \bigcap_{n=1}^\infty \b... |
H: Science Fair Project - Square roots
For a school science fair project, I need four or five square root algorithms to use. Googling gives sites like http://en.wikipedia.org/wiki/Methods_of_computing_square_roots with so many methods that I don't know how to narrow it down to four or five.
I desire the most standard ... |
H: Is the converse of Stolz Caesaro Lemma true?
Let $(a_n)_n, (b_n)_n$ be two real sequences s.t. $(b_n)_n$ is strictly increasing and unbounded. Prove that $$\lim_{n\rightarrow \infty} \frac {a_n}{b_n}= \lim_{n\rightarrow \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}$$ if limit on the right exists.
I could prove this. But... |
H: Equivalence classes for a Relation on a product set.
How do you determine the the equivalence classes for a relation on a product set?
Background:
Let $S=\left\{1,2,3,4\right\}$ and $A=S\times S$. The relation $R$ on $A$ can be defined by
$$\left(a,b\right)R\left(c,d\right) \iff a/b =c/d$$
For example:
$$\left(1,2... |
H: Subspace of $C^1 [0,1]$
Consider the inner product space of continuously differentiable functions, $C^1 [0,1]$ with inner product:$$\left<f,g\right> =\int_{0}^1f(x)\overline{g(x)}\,dx + \int_{0}^1f'(x)\overline{g'(x)}\,dx $$
Show $\left<f,\cosh\right> = f(1)\sinh(1)$ for any $f$ $\in$ $C^1 [0,1]$
Use part 1. to sho... |
H: Number Theory about least common multiple
Let a and b be positive integers and let [a,b] denote the least common multiple of a and b. Show that there exist integers x and y such that
$$ \left(\frac xa\right) + \left(\frac yb\right) = \left(\frac 1 {[a,b]}\right). $$
I need a nudge in the right direction - not sure ... |
H: How to find the determinant of this matrix
I have the following matrix:
$
\begin{bmatrix}
a & 1 & 1 & 1 \\
1 & a & 1 & 1 \\
1 & 1 & a & 1 \\
1 & 1 & 1 & a \\
\end{bmatrix}
$
My approach is to rref the matrix so that i can find the determinant by multiplying along the diagonals.
I attempted to do an rref and end... |
H: Proving that $\{f \in End(A): \forall a \in A:|a|<\infty \implies f(a)=0\}$ is an ideal
Let $A$ be an abelian group. I need to prove that
$I = \{f \in End(A): f(a)= 0 \ \text{for all $a$ of finite order}\}$,
is an ideal of $\text{End}(A)$. It isn't hard to prove that $I$ is a subgroup of $End(A)$, but it is quite ... |
H: Even function integration problem
Let $f$ be an even function. Show
$$\int_{-a}^af(x)dx=2\int_0^af(x)dx$$
So I thought of breaking it up into two integtrals with one from $-a$ to $0$ and $0$ to $a$. Then I have on the left side
$$\int_{-a}^0f(x)dx$$
Then I thought
$$\int_{-a}^0f(x)dx=-\int_0^{-a}f(x)dx=-\int_0^{... |
H: Linear Diophantine equation - Find all integer solutions
Using the linear Diphantine equation
121x + 561y = 13200
(a) Find all integer solutions to the equation.
(b) Find all positive integer solutions to the equation.
edit: The answer I have for (a) is an equation:
$x=16800 + 51n$
$y=-3600-11n$
where $n... |
H: example of a topological space such that there exists a sequence that escapes to infinity but has convergent subsequence
Find an example of a topological space such that there exists a sequence that escapes to infinity but has a convergent subsequence
This actually is from exercise 2.15 of introduction to smooth ma... |
H: Is my answer correct? (And what's the name of the rule?)$\lim_{n \to \infty} \frac{\left(n+3\right)!-n!}{n\left(n+2\right)!}$
Want to know if I solved this problem correctly:
$$\lim_{n \to \infty} \frac{\left(n+3\right)!-n!}{n\left(n+2\right)!} =\lim_{n \to \infty} \frac{1 \cdot 2 \ldots(n-1)n(n+1)(n+2)(n+3) - 1 \c... |
H: What are non-orthogonal eigenvectors?
Given a symmetric matrix $A$, the maximum of the trace, $Tr(Z^TAZ)$ under the assumption that $Z^TZ=I$ occurs when $Z$ has the eigenvectors of $A$, as $Tr(U^TAU)= \lambda_1 +\lambda_2+...\lambda_ d$ where $Z\in\mathbb{R}^{n\times d}$. I know that the eigen vectors being the sol... |
H: Limits of a recursively defined sequence
Let $x_1=a$ and define a sequence $\left(x_n\right)$ recursively by:
$x_{n+1} = \dfrac{x_n}{1 + \frac{x_n}{2}}$
For what values of $a$ is it true that $x_n$ approaches $0$?
AI: Whenever the sequence converges (to $L$, say) we must have $L=\frac L{1+\frac L2}$, i.e. $L=0$. So... |
H: $p\mapsto\Vert f\Vert_{L^p}$ is continuous.
Could someone help me prove the following:
Let $(\Omega,\mathscr{A},\mu)$ be a probability space. Let $f:\Omega\rightarrow\mathbb{R}$ be a non-negative measurable function.
How do I prove that the funtion $N_f:p\in [1,\infty)\mapsto \Vert f\Vert_p\in[0,\infty]$ is continu... |
H: Compute the determinant
a)
| -1 -2 -3 ..... -k | | -1 -2 -3 ..... -k |
| 1 0 3 ..... k | | 0 -2 0 ..... 0 |
| 1 2 0 ..... k | = | 0 0 -3 ..... 0 | =
| ............. k | | ................ |
| 1 2 3 ..... 0 | | 0 0 0 ..... -k |
= (-1)(-2)(-3)...(-k) =
= (-1)^k*1*2*...*k =
= (-1)^k... |
H: Prove $\limsup\limits_{n \rightarrow \infty} b_n \leq \limsup\limits_{n \rightarrow \infty} a_n$, given $b_n = \frac{a_1+ \cdots +a_n}{n}$.
I am working on this question for quite a long time. I think I will prove by contradiction, but I can not proceed. Anyone could give a hint? Thanks!
AI: Fix $m$ and pick $n\ge... |
H: Derivative of the $\sin(x)$ when $x$ is measured in degrees
So a classic thing to derive in calculus textbooks is something like a statement as follows
Is $\frac{d}{dx}\sin(u)$ the same as the derivative of $\frac{d}{dx}\sin(x)$ where $u$ is an angle measured in degrees and $x$ is measured in radians? and of course... |
H: Prove the following determinant identities without expanding the determinants
a)
$$\begin{vmatrix}
\sin^2 x & \cos^2 x & \cos 2x \\
\sin^2 y & \cos^2 y & \cos 2y \\
\sin^2 z & \cos^2 z & \cos 2z \\
\end{vmatrix} = 0;$$
$$\begin{vmatrix}
\sin^2 x & \cos^2 x & \cos^2x-\sin^2x \\
\sin^2 y & \cos^2 y & \cos^2... |
H: Uncountable product of separable spaces is separable?
I am asking to what extent this problem is true:
Uncountable product of separable spaces is separable.
I need these statements to be explained if possible:
How to construct a countable subset of uncountable product of separable spaces.
Counterexample for the ... |
H: Finding derivative using product rule
When finding the derivative of:
$z=(1+t^2)6^t$
I am working it out to:
$$\frac{dz}{dy} = (1+t^2)(6^t)'+(1)'(t^2)'(6^t)$$
$$=(1+t^2)(ln(6))(6^t)+2t(6^t)$$
$$=6^t((1+t^2)(ln(6)))+2t(6^t)$$
I believe this to be the answer, but correct me if I am wrong. My biggest concern is that ... |
H: An algebraically closed field with characteristic $p>0$
I want to know about an algebraically closed field that is not of characteristic $0$.
I really don't know about infinite fields with characteristic $p$ so I will appreciate your comments.
AI: Any field has an algebraic closure—so the short answer to this is: J... |
H: Simplifying the sum of a fraction and an integer under a radical sign
I'm trying to help my little bro, a bit rusty here... Wolfram Alpha is telling me that:
$$
x\sqrt{1+{\frac{x^2}{16-x^2}}}
$$
simplifies to:
$$
4x\sqrt{\frac1{16-x^2}}
$$
I can't for the life of me figure out why. I'm thinking there's a simple ru... |
H: Order of the product of two commuting elements with coprime orders in a group.
I want to show that if $g,h\in G$ are group elements with finite coprime orders $m$ and $n$ and $gh=hg$ then the order of $gh$ is $mn$.
We have that
$$
(gh)^{mn}=g^{mn}h^{mn}=1
$$
using $gh=hg$, so $|gh|\leq mn$. On the other hand, $m... |
H: What is meant by 'runs through'?
I'm independently studying abstract algebra for fun (not my forte...) and I'm reading Herstein. He has a question in the chapter on rings:
Let $p$ be an odd prime and let$$\sum_{k=1}^{p-1}\frac{1}{k}=\frac{a}{b}$$ where $a,b\in{\mathbb{Z}}$. Show $p|a$. (Hint: As $a$ runs through ... |
H: Derivative of $n/x$
My assignment asks me to calculate the derivative of:
$$y = \frac{9}{x} + 6 \sec x$$
My first step, it seems, should be to break the equation up and find the derivatives of $\frac{9}{x}$ and $6 \sec x$, which would be $1$ and $6(\sec x\tan x)$, respectively.
This would leave with $ 1 + 6(\sec x ... |
H: Show that, if $\tan^2(x) = 2\tan(x) +1,$ then $\tan(2x) = -1$
(i) Show that, if $\tan^2(x) = 2\tan(x) + 1,$ then $\tan (2x) = -1$
AI: It's always the case that $\tan2x=2\tan x/(1-\tan^2x)$. If $\tan^2x =2\tan x+1$, then $2\tan x=\tan^2x-1$ and hence
$$\tan2x={2\tan x\over 1-\tan^2x}={\tan^2x-1\over1-\tan^2x}=-1$$ |
H: Question about Big-O notation
I'm learning Big-O notation in school and my friend and I have a hard time understanding some parts of it and we don't agree on some answers in the exercises.
There are two cases on which we don't agree, and they are particularly easy ones.
The following example contains both cases:
W... |
H: Calculating future value and Present Value
I have been stuck on this one for hours ... not too great at math can someone help. Thanks.
Isaac borrowed $\$4000$ at $11.5\%$ compounded quarterly $5.5$ years ago.
One year ago he made a payment of $\$1500$. What amount will extinguish the loan today?
I've tried a bunch... |
H: Can we turn $\mathbb{R}^n$ into a field by changing the multiplication?
Of course $\mathbb{R}$ is a field with usual addition and multiplication. When we move up a dimension into $\mathbb{R}^2$, however, there is not a clear way to multiply two vectors together to get something useful. In fact, if we define multi... |
H: Finiding the integral [0,1] of an interesting function
I have been working in this problem for 4 days, and I just can't get anywhere, so here it is:
Be f the function with domain [0,1], defined in the following way:
for every x between 0 and 1, we have the decimal expansion, the infinite succession of their digits
... |
H: probability question show that $P(A)>P(B)$
This question comes up several times in past exam papers so i would really like to work it out! Here is it:
Assume that $$0<P(C)<1.$$
How would I show that if
$P(A|C)> P(B|C)$ and $P(A|C^c)> P(B|C^c) $
Then $P(A)>P(B)$
My solution:
I have so far that the conditional pr... |
H: Find a positive integer $x$ less than $105$ satisfying the following simultaneous congruence equations.
$$x=2 mod 3$$
$$x=3 mod 5$$
$$x=4 mod 7$$
I have only learnt modulo for 2 weeks so far... really basic theorems.
My attempt using definitions of modulo
From Equation 1, $3a=x-2 \rightarrow 15a=5x-10$
Fro... |
H: Uniqueness Proof: Related to Division Algorithms
Regarding the statement: Let $a\in \mathbb {Z}$, $b\in \mathbb {Z}$. Then there exists integers q and r such that $a = qb+r$ where $0\le r \le b$.
Let $S$ $=$ $\lbrace a-qb: q\in \mathbb {Z}, a-qb \ge 0 \rbrace$
Why does S have to be non-empty?
AI: If $a>0$, let $q=0... |
H: Finding the vertical asymptote of a given function
Given any function, how can I find its vertical asymptote? I know that for rationals I can do this by letting the denominator equal to 0. But how about a function like:
$\ln(1-\ln(x))$? You can find the horizontal asymptote of any function by finding the limit of t... |
H: Prove that decomposition of second order tensors into symmetric and skew components is unique.
Title says it all.
$A=A_{sym}+A_{skew}$
$A_{sym}= \dfrac{1}{2}(A+A^T)$
$A_{skew}= \dfrac{1}{2}(A-A^T)$
Anybody can help?
AI: Assume $T=S_1+A_1=S_2+A_2$ are two symmetric skew-symmetric decompositions of rank two tensor. T... |
H: How many multiples of 3 are between 10 and 100? (SAT math question)
In the figure above, circular region A represents all integers from 10 to 100, inclusive; circular region B represents all integers that are multiples of 3; and circular region C represents all squares of integers. How many numbers are represented... |
H: Show convexity of the quadratic function
Given symmetric positive semidefinite matrix $A$, let
$$F(x) := x^TAx + b^Tx + c$$
Can someone show that $F$ is convex using the definition (without taking the gradient)?
AI: By definition of convex, for any $x,y\in\mathbb R$, we have
$$f(\frac{x+y}2)\leq\frac12(f(x)+f(y))$$... |
H: Counterexample of certain non-primary ideals
Let $A$ be a commutative ring with unit. Let $I,J$ be primary ideals of $A$ such that $J$ is not contained in $I$ and $r(J)\subset r(I)$, $r(J)\neq r(I)$. Then $I\cap J$ is not necessarily primary.
I did not find counter examples in $\mathbb{Z}$. I am confused about po... |
H: How do I prove a "double limit"?
Prove $$\lim_{b \to \infty} \lim_{h \to 0} \frac{b^h - 1}{h} = \infty$$
I have never worked with double limits before so I have no idea how to approach the problem. Please don't use "$e$" in your solutions, since the above limit is part of the derivation of "$e$", so for all purpo... |
H: A First Order Definition of the Mod Function
Is there a good FOL definition of a $\bmod$ predicate in the language of Peano arithmetic? I tried $M(x,n,r) \equiv Ey(x=ny+r)$ but I don't like it very much.
AI: We want to say that $r$ is the remainder when $x$ is divided by $n$. I will assume that variables range over... |
H: How to maximise functions of this shape $y=2\cdot3^{-x}$
How can I find the maximum of $2\cdot 3^{-x}$? I know its close to $1$ because I have seen its graph, but when I differentiate the function and set it equal to zero (to get a maximum) I get $-2\cdot 3^{-x}=0$. What does that mean? How can I solve for the $x$ ... |
H: Finding a polynomial $k$ respects $e^{-\epsilon k}\leq \delta$
I am given $e^{-\epsilon k}$ and my goal is to find a polynomial $k$ (in $\epsilon$ and $\delta$) such that $e^{-\epsilon k}\leq \delta $ where $\epsilon,\delta,k>0$. The exercise shows that $k\geq$ $\frac{1}{\epsilon}$$ln(\frac{1}{\delta})$ is the solu... |
H: What are the every possible sums of these numbers?
Let $S=\{\pm a,\pm(a+b)\}$. If we take the sum of arbitrary $2$ elements of $S$, including duplication, the every possible sums are $\{0,b,2a+b,2a,2(a+b),-b,-(2a+b),-2a,-2(a+b)\}$.
Now, if $S=\{\pm a,\pm(a+b),\pm(a+2b)\}$, and if we take the sum of arbitrary $m$ el... |
H: How to solve $ 2x-3-2x^{ -1/2 }= 0 $ for $x$?
Equation to solve
$$
2x-3-2x^{ -(1/2) }= 0
$$
The answer should be $2.1777$. However I'm not too sure how the steps in between are constructed. Anyone can guide me how do I solve x for this equation?
Progress
One way that I have tried solving this is
$$
2x-3-2x^{ -(1/2)... |
H: List the numbers in order
How would I list these numbers in order without using a calculator?
Thank you
List these numbers in increasing order: $2^{800}$, $3^{600}$, $5^{400}$, $6^{200}$
AI: Note that $$2^{800} = 16^{200}$$ $$3^{600} = 27^{200}$$ $$5^{400} = 25^{200}$$
so the order is: $6^{200}, 2^{800}, 5^{400},... |
H: If $A$ is null set, then $\int\limits_A f dm = 0 $
Define $ \int_E f dm = \sup Y(E, f) $
where $ Y(E,f) = \{ \int_E \phi : 0 \leq \phi \leq f \} $ $\phi$ is simple
Suppose $A$ is a null set. We show $Y(A, f) = \{ 0 \}$. Pick $x \in Y(A, f)$. So, we have $x = \int_A \phi dm $ for some simple $\phi \leq f$. Let $\phi... |
H: Show that the quotient space $X^*$ is locally Hausdorff, but not Hausdorff.
Let $X$ be two disjoint copies of $\mathbb{R}$, that is say $X = (\{a\} \times \mathbb{R}) \cup (\{b\} \times \mathbb{R})$ for real numbers $a<b$ and consider X as a subspace of $\mathbb{R} \times \mathbb{R}$. Define an equivalence relation... |
H: biconnected components in a graph
i just started going through biconnected components can someone explain me this
Show that if G is a connected undirected graph, then no edge of G can be in two different
biconnected components
AI: HINT: Suppose that $G$ is connected, and $e=uv$ is an edge lying in the biconnected c... |
H: Use residues to evaluate $\int_{0}^{\infty} \frac{dx}{x^2 + 1}$
Use residues to evaluate $\int_{0}^{\infty} \frac{dx}{x^2 + 1}$.
Okay so these are the integrals in complex analysis I am a little uncomfortable with. I purposely chose a simple problem out of a book so that I can save the slightly more difficult pro... |
H: Solving optimization problems using derivatives and critical points
I have a homework question which I have completed 2/3 of; however I am stuck on the last part of the question.
The question is:
A drug used to treat cancer is effective at low doses with an efficacy that increases with the quantity of the drug. H... |
H: How is this set ascending?
In Royden's analysis, the proof for Lemma 10 in chapter 3 states the following:
For each $k$, the function $|f - f_k |$ is properly defined, since $f$ is real-valued, and it is measurable, so that the set $\{ x \in E : |f(x) - f_k(x)| < \eta\}$ is measurable. The intersection of a countab... |
H: Martingale: how do they simplify betting summation
$$
\sum_{i=1}^{n} B \cdot 2^{i-1}=B\left(2^{n}-1\right)
$$
How can I go from the summation to $B(2^n-1)$?
AI: It is a general fact that
$$\sum\limits_{i = 1}^{n} r^{i - 1} = \sum\limits_{i = 0}^{n - 1} r^{i} = \frac{1 - r^n}{1 - r}$$
Now select $r = 2$.
Proving th... |
H: what does eventually mean in the following question ? thanks
Suppose $\phi < f $, $g_n \to f $ pointwise, $g_n = inf_{k \geq n} f_k $. MY books says that 'eventually' $g_n \geq \phi$. What do they mean by eventually? Also to show this, they do the following:
Let $A_k = \{ x : g_k(x) \geq \phi \} $.
I understand tha... |
H: problems on split exact sequence
Let $0 \to A \stackrel f\to B \stackrel g\to C \to 0$ be short exact sequence of modules ($f:A \to B$, $g:B \to C$). Suppose that there exists $\alpha: B \to A$ and $\beta: C \to B$ such that $g \beta = id_C$, $\alpha f = id_A$. How can I prove that $f \alpha + \beta g = id_B$?
AI: ... |
H: what is the probality of taking first blue and last red ball when picking 6 balls?
There is 36 ballls.
12 are red R
12 are blue B
12 are yellow
What is the probability of taking first blue and last red ball when picking 6 balls? (Not returning them back).
Lets say A is any ball.
So the order would look like this:
B... |
H: Show that the vector field $X(x, y, z)=(xy-z^2, yz-x^2, x^2+z^2+xz-1)$ is tangent to the set $x^2 + y^2 + z^2 = 1$
I know I need to find functions $F(t)$, $G(t)$, and $H(t)$ such that $F(0)=x$, $G(0)=y$, and $H(0)=z$ and $F'(0)=xy-z^2$, $G'(0)=yz-x^2$, and $H'(0)=x^2+z^2+xz-1$. It's also necessary that $(F(t))^2 +... |
H: Definition of a group
What defines a group mathematically, please explain both in Mathematical language and in English if possible.
My current understanding:
Four things are required to define a group:
Closure - Any binary operation completed upon two elements of a group, must always equal a third element that is a... |
H: $ \lim_{x\to 0}\frac{\tan x-\sin x}{\sin(x^3)}$
$$ \lim_{x\to 0}\frac{\tan x-\sin x}{\sin(x^3)} =[1]\lim_{x\to 0}\frac{\sin x/\cos x-\sin x}{x^3}\\ =[2]\lim_{x\to 0}\frac{1-\cos x}{x^2}\\ =\frac{1}{2} $$
My question is how [1]=[2]?
$\tan x-\sin x=\tan x(1-\cos x)=x(1-\cos x)$?
AI: We have $\frac{\sin x}{\cos x}-\si... |
H: Changing the order of $\lim$ and $\inf$ for point-wise converging monotonic sequence of functions
Suppose $f_n: X \rightarrow \mathbb{R}$, where $X$ is some arbitrary subset of $\mathbb{R}^N$. Suppose that
$$ \forall n\geq0, \forall x \in X, \; f_n(x) \leq f_{n+1}(x) $$
Let $\{f_n\}$ be such that each $f_n(x)$ is c... |
H: Let $v$ and $w$ be eigen vectors of $T$ corresponding to two distinct eigen values $ \lambda _1$ and $ \lambda_ 2$ respectively
Problem:Let $v$ and $w$ be eigen vectors of $T$ corresponding to two distinct eigen values $ \lambda _1$ and $ \lambda_ 2$ respectively
Then which of the following is true ?
$1)$ For non... |
H: linearity of expectation in case of dependent events
I could understand the linearity of expectation in case of independent events, but why does it work in case of dependent events too. It seems counter - intuitive. In case of dependent events, each outcome influences subsequent outcomes, hence they cannot be just ... |
H: The limit of $\sin(n^\alpha)$
(1) It is easy to prove that $\lim\limits_{n\to\infty}{\sin(n)}$ does not exist.
(2) I want to ask how to prove that $\lim\limits_{n\to\infty}{\sin(n^2)}$ does not exist.
(3) Furthermore, $\lim\limits_{n\to\infty}{\sin(n^k)}$ does not exist. ($k$ is a positive integer.)
(4) In addition... |
H: Find dimension of even polynomials
Let $V$ be a the vector space over $\mathbb R$ of all polynomials with real coefficients. Let $W$ be the subset of all polynomials with only even powers in their expression.
So $p(X) \in W$ means $p(x)=\sum_{n=0}^k a_nx^{2n}$.
I showed that $W$ is a subspace of $V$, but I need to... |
H: One-relator Groups and Subgroups.
I am currently working on "The Group Algebra of a torsion-free one-relator Group can be embedded in a Field." (Tekla and Jacques Lewin) In this Paper we find a corollary (the last one), which I am about to apply on something else. This corollary can only be aplied, if the group $G$... |
H: Parametric surfaces - Parameterization of torus
A rotational surface area is created when a curve in the $xz$-plane, with parameterization $\def\i{\pmb{i}}\def\k{\pmb k}$ $r=x(t)\i + z(t)\k$ , $t \in [a,b]$, rotates around the $z$-axis. This surface is parameterized by;
$$t \mapsto \bigl( x(t)\cos\theta, x(t)\sin\... |
H: Symmetric difference of sets and convergence in integration.
Let $(X,\mathcal{M},m)$ be a space of measure and $f_n,f \in L^1(m)$ such as $||f_n - f||_1 \rightarrow 0.$ Suppose that we also have $A_n,A \in \mathcal{M}$ and $m(A_n \triangle A) \rightarrow 0.$
I want to prove that $\displaystyle \int_{A_n}f_n dm \ri... |
H: Proving that representation is differentiable
Let $V$ be a real or complex finite-dimensional vector space, and let $\pi$ be a continuous representation of $(\mathbb{R}, +)$ on V with:
$$\pi(t + s) = \pi(t) \pi(s), t, s \in \mathbb{R} \: \: (1)$$
$$\pi(0) = I \: \:(2)$$
a) Prove that $\pi: \mathbb{R} \rightarrow L(... |
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