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H: Partial integration
We want to integrate
$$ f(x) = 2x \cos(x)$$
We use partial integration where $2x = g$ and $\cos(x) = f'$
I end up with
$$d (2x \sin(x)) - \sin(x) \cdot d2x$$
What confuses me is the term $\sin(x) \cdot d2x$
How would I be able to evaluate this term?
AI: $$ f(x) = 2x \cos(x)$$
We use partial in... |
H: Big-O Notation and Algebra
This is my first question here. Trying to simplify the following.
$$f = O\left(\frac{5}{x}\right) + O\left(\frac{\ln(x^2)}{4x}\right)$$
I give it a try as follows.
$$\begin{align}
f &= O\left(\frac{5}{x}\right) + O\left(\frac{2\ln(x)}{4x}\right), \\
f &= O\left(\frac{5}{x}\right) + O\lef... |
H: Can this equation be factored down? $\frac{(2^{y}-2) - 2^{y-x}}{2^{y}-1} $
Can this equation be factored down so as to be smaller? Or is this as small as it will go?
$$
\frac{(2^{y}-2) - 2^{y-x}}{2^{y}-1}
$$
AI: Let $y=3, x=1$, then $2^y-1=7$ and your expression becomes $\cfrac 27$. Whenever the denominator is a pr... |
H: Possible dimensions for a sum of two vector spaces
Let $U, W \subseteq \mathbb{R}^6$ vector subspaces from dimension $4$. Show that $U \cap W$ holds at least $2$ linearly independent vectors and at most $4$ linearly independent vectors.
Using the dimensions theorem I showed by assuming the opposite that $0 \leq dim... |
H: Is it sufficient to claim that the limits of $\frac{x+2}{x^2+x+1}$ and $\frac1x$ must be the same?
I was trying to evaluate this limit Calculate: $\lim\limits_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$ . After doing long division, I got like everybody else limit as $x$ approaches infinity of $(1+(x+2)/... |
H: Endomorphim Ring of Abelian Groups
In the paper
"Über die Abelschen Gruppen mit nullteilerfreiem Endomorphismenring."
Szele considers the problem of describing all abelian groups with endomorphism ring contaning no zero-divisors. He proved that there is no such group among the mixed groups. While $C(p)$ and $C(p^... |
H: Quadratic equation with parameter
Stuck solving this equation.
Full text:
For what real values of the parameter do the common solutions of the equation became identical?
1. y = mx - 1
2. x^2 = 4y
ans. m = +/- 1
2a. x^2 - 4y = 0
So i started by substituting 1. into 2a.
x^2 - 4 * (mx - 1) = 0
x^2 -4mx + 4 = 0
Solu... |
H: "Total" degree of a polynomial?
What is the difference between the "degree" of a polynomial and its "total degree"?
AI: The total degree of a polynomial in more than one variable is the maximal of the sums of all the powers of the variables in one single monomial. For example
$$\deg(x_1^2x_2x_3^4-3x_2+4x_1x_4^5-x_1... |
H: Relationship between number of conjugacy classes and number of irreducible representations of a group
For a finite group G the number of irreducible representations over an algebraically closed field F is at most the number of conjugacy classes whose sizes are coprime to the characteristic of F.
What about over fie... |
H: How to calculate the integral of $x^x$ between $0$ and $1$ using series?
How to calculate $\int_0^1 x^x\,dx$ using series? I read from a book that
$$\int_0^1 x^x\,dx = 1-\frac{1}{2^2}+\frac{1}{3^3}+\dots+(-1)^n\frac{1}{(n+1)^{n+1}}+\cdots$$ but I can't prove it. Thanks in advance.
P.S: I found some useful material... |
H: Are function spaces Baire?
Let $X$ and $Y$ be manifolds and suppose that $X$ is a compact, complete metric space and $Y$ is a complete metric space.
So, both $X$ and $Y$ are Baire spaces.
Question: For what values of $k\geq 0$ is the space $C^k(X,Y)$ (with the $C^k$-topology) a Baire space?
For $k=0$, the space $... |
H: Calculating the determinant of this complicated matrix
I am calculating the characteristic polynomial for this matrix:
$$A = \begin{pmatrix} 1 & 2 & \cdots & n \\ 1 & 2 & \cdots & n \\ \vdots & \vdots & \cdots & \vdots \\ 1 & 2 &\cdots & n \end{pmatrix}$$
First I was asked to figure out that $0$ is an eigenvalue, a... |
H: Introduction to Pseudodifferential operators
I'm interested in elementary introduction to pseduodifferential operators and its application to hyperbolic PDE's. I know measure theory, Fourier analysis and some elementary(linear) hyperbolic PDE's but not functional analysis, distributions, Sobolev spaces,etc. Can you... |
H: Dropping the orientable condition from the Thom isomorphism theorem.
My first question is: what are some examples of un-oriented n-plane bundles over $ \mathbb{Z}$ where it is easy to see that there is no Thom class?
I would like to know some examples because the real question I have is: Where in the proof (In Miln... |
H: Why dividing by zero still works
Today, I was at a class. There was a question:
If $x = 2 +i$, find the value of $x^3 - 3x^2 + 2x - 1$.
What my teacher did was this:
$x = 2 + i \;\Rightarrow \; x - 2 = i \; \Rightarrow \; (x - 2)^2 = -1 \; \Rightarrow \; x^2 - 4x + 4 = -1 \; \Rightarrow \; x^2 - 4x + 5 = 0 $. No... |
H: Where can I learn about the lattice of partitions?
A set $P \subseteq \mathcal{P}(X)$ is a partition of $X$ if and only if all of the following conditions hold:
$\emptyset \notin P$
For all $x,y \in P$, if $x \neq y$ then $x \cap y = \emptyset$.
$\bigcup P = X$
I have read many times that the partitions of a set ... |
H: Generators of a subgroup of $SL_2(\mathbb{Z}/24\mathbb{Z})$
So I have this subgroup of $SL_2(\mathbb{Z}/24\mathbb{Z})$ which has $256$ elements. Is there a way in sage to get the list of its generators ? The "only" information I have on the group is the list of its elements.
If it is not implemented in Sage does an... |
H: How do I calculate a double integral like $\iint_\mathbf{D}e^{\frac{x-y}{x+y}}dx\,dy$?
The Problem is to integrate the following double integral $\displaystyle\iint_\mathbf{D}\exp\left(\frac{x-y}{x+y}\right) dx\, dy$ using the technique of transformation of variables ($u$ and $v$).
There is a given $D$ with $D:= \{... |
H: Maxima of sum of exponential.
I want to find the maxima of $f=e^{-x}+e^{-2x}$, $x \ge 0$. I know the maximum happens at $x=0$, however when I differentiate and equate to zero I get : $\,e^{x}=-2$ which leads to $x=\ln(-2)$. Can any one point why the differentiation method doesn't lead to the correct answer?
AI: Not... |
H: How to (quickly) prove that $24p+17$ is not a square number
Computer says $24p+17$ is not square number for $p<10^7$ so I guess it's not. I know that squares of odd numbers are all $8p+1$ but $24p+17$ passes the test
And how to solve problems like this in general? Thx in advance
AI: The number is $2 \pmod 3$. No su... |
H: Absolute value of infinite sum smaller than infinite sum of absolute values
A question emerging from an exercise in Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press.
The exercise consists in showing that if $\sum_{i=1}^\infty x_i$ converges, then
$|\sum_{i=k}^\infty x_i| \leq \... |
H: Changing bounds of integrals
If I have:
$$\int_{L}^{\infty }e^{\dfrac{-(x-\sigma \sqrt{T})^2}{2}}\,dx$$
let $y = x - \sigma \sqrt{T}$
$$\int_{L - \sigma \sqrt{T}}^{\infty }e^{\dfrac{-y^2}{2}} \, dy$$
Why does the lower bound change in the new integral?
AI: The lower bound was from $x=L$, after the transformation, w... |
H: Why are continued fractions for irrational numbers always convergent?
Like in the title:
Why are continued fractions for irrational numbers (i.e. infinite fractions) always convergent?
AI: Normally this is proved in steps.
The even convergents (i.e. the second, fourth, etc.) are an increasing sequence.
The odd co... |
H: $A \in M_{n \times n} (\mathbb R)$, $n\geq 2$, rank($A) = 1$, trace($A) = 0$. Prove A is not diagonalizable
Given:
$A \in M_{n \times n} (\mathbb R)$, $n\geq 2$, rank($A) = 1$, trace($A) = 0$. Prove A is not diagonalizable and find $P_A(x)$.
So I said:
if $n \geq 2$ and rank($A)=1$ then $A$ is not invertible. Th... |
H: Reference request, self study of cardinals and cardinal arithmetic without AC
I'm looking for references (books/lecture notes) for :
Cardinality without choice, Scott's trick;
Cardinal arithmetic without choice.
Any suggestions? Thanks in advance.
AI: Jech, The Axiom of Choice.
Herrlich, The Axiom of Choice.
Halb... |
H: Derivative of $\left(x^x\right)^x$
I am asked to find the derivative of $\left(x^x\right)^x$. So I said let $$y=(x^x)^x \Rightarrow \ln y=x\ln x^x \Rightarrow \ln y = x^2 \ln x.$$Differentiating both sides, $$\frac{dy}{dx}=y(2x\ln x+x)=x^{x^2+1}(2\ln x+1).$$
Now I checked this answer with Wolfram Alpha and I get t... |
H: Element of order $2n$ in symmetric group $S_n$
I've been recently reading some articles about orders of elements in $S_n$ and I know that in order to find max order in $S_n$ we can use Landau function though I think that for small $n$ it is better to do it "manually".
My question is: For what $n$ can $S_n$ contain ... |
H: Proof of exactness at the first two non-zero objects in the ker-coker sequence (snake lemma).
I am reading MacLane's chapter on Abelian Categories and I am proving the fact, needed for the snake lemma, that the sequence $0\to \text{Ke}f\to \text{Ke}g\to\text{Ke}h$ is exact at $\text{Ke}f$ and $\text{Ke}g$, where $\... |
H: Simplifying a quotient of complex numbers
Given the equation I am supposed to simplify :
$$\frac{(7 - 4i)}{(5 + 3i)}$$
I conclude that I should first multiply both the numerator and denominator by $(5 - 3i)$ (note : or by $7 + 4i$ but either will do), which leads me to :
$$\frac{(35 - 41i + 12i^2)}{(25 - 9i^2)}$$... |
H: Find the functions whose length is proportional to the area below them
I'm trying to solve this problem:
"Find all the functions $f : \mathbb{R} \rightarrow \mathbb{R}^+$, $f \in C^1(\mathbb{R})$ such that the area below $f$ in $[a,b]$ is proportional to the length of the graphic of $f$ in that interval"
This is my... |
H: Difficulties with partial integration
I have asked several questions on the site regarding this topic already, but I can't seem to grasp this at all. Consider the following example:
$$ h(x) = e^{2x} \sin x$$
We have to find the integral. I rewrote this to the form:
$$e^{2x} \sin x \space dx = d( -\cos x e^{2x}) - ... |
H: What stops me from making this conclusion?
Suppose I want to find $\sin^6x+\cos^6x$. What stops me from saying that $\sin^2t=\sin^6x$, and $\cos^2t=\cos^6x$? Of course this is wrong because $\sin^2t+\cos^2t=1$ and $\sin^6x+\cos^6x$ does not equal 1. So what stops me from making this substitution? The domain $and$ r... |
H: Definition of $C^k$ boundary
Can someone give me a resonable definition of $C^k$ boundary, e.g., to define and after give a brief explain about the definition.
I need this 'cause I'm not understanding what the Evan's book said.
Thanks!
AI: In $\mathbb{R^n}$, the boundary of a subset is $C^k$ if it's locally the gra... |
H: If $\mid \lambda_i\mid=1$ and $\mu_i^2=\lambda_i$, then $\mid \mu_i\mid=1$?
If $|\lambda_i|=1$ and $\mu_i^2=\lambda_i$, then $|\mu_i|=1$?
$|\mu_i|=|\sqrt\lambda_i|=\sqrt |\lambda_i|=1$. Is that possible?
AI: Yes, that is correct. Or, either you could write $1=|\lambda_i|=|{\mu_i}^2|=|\mu_i|^2$, and use $|\mu_i|\ge ... |
H: Is there a great mathematical example for a 12-year-old?
I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets.
Why? Because the maths department at her school is outrageously good, and set her the task of researching a mathematician, and understanding... |
H: Prove that $U$ is a self adjoint unitary operator
Let $W$ be the finite dimensional subspace of an inner product space $V$ and $V=W\oplus W^\perp $.
Define $U:V \rightarrow V$ by $U(v_1+v_2)=v_1-v_2$ where $v_1\in W$ and $v_2 \in W^\perp$.
Prove that $U$ is a self adjoint unitary operator.
I know I have to show t... |
H: About a condition for a continuous mapping to be open.
The text (Foundations of General Topology, by Pervin, Second edition) says
a (continuous) mapping $f$ of $X$ into $X^*$ is open iff $f(i(E))\subseteq i^*(f(E))$ for every $E\subseteq X$.
EDIT: $i(E)$ is the interior of set $E$ in $X$. Similarly, $i^*$ is the... |
H: Subsequential Limits
I'm working through Rudin's PoMA at the moment, and I've been learning about subsequential limits. However, I'm somewhat confused and I have a question, which is more conceptual than an actual exercise.
I know that when a sequence converges the $\lim \space \sup$ and $\lim \space \inf$ are equa... |
H: Expansion of $(z-1)^2 / (z-2)(z^2+1)$ in $z = 2$
I have to expand
$$
f(z) := \frac { (z-1)^2 }{(z-2)(z^2+1)}
$$ around $z = 2$. I wanted to write $f(z) = \frac 1 {z-2} g(z)$ and then expand $g(z)$. Some hints would help a lot :)
AI: You've got a good notion, but I suggest a slightly different approach. First, set$... |
H: If $X=[x_{ij}]_{n \times n}$ then how prove $X^n=0$
Let $n\in \mathbb N$ and $A_1,A_2,..,A_n$ be arbitrary sets. Now define $X=[x_{ij}]_{n \times n}$ where
$$x_{ij}=
\begin{cases}
1 , & \text{$A_i$$\subsetneq$}A_j \\
0 , & \text{otherwise} \\
\end{cases}.$$
How do you prove $X^n=0$?
Thanks in advance.
AI: Hint: b... |
H: Structure of p-adic units
I am trying to understand the structure of the $p$-adic units. I know that we can write
$$\mathbb{Z}_p^\times \cong \mu_{p-1} \times 1 + p\mathbb{Z}_p,$$
where $\mu_n$ are the $n$th roots of unity in $\mathbb{Z}_p$. My question is: what more can we say about the structure of $1 + p\mathbb{... |
H: Why can't you find all antiderivatives by integrating a power series?
if
$f(x) = \sum\limits^{\infty}_{n=0}\frac{f^{(n)}(0)}{n!}x^n$
why can't you do the following to find a general solution
$F(x) \equiv \int f(x)dx$
$F(x) = \int (\sum\limits^{\infty}_{n=0}\frac{f^{(n)}(0)}{n!}x^n) dx = \sum\limits^{\infty}... |
H: Multiplying the long polynomials for $e^x$ and $e^y$ does not give me the long polynomial for $e^{x+y}$
As an alternative to normal rules for powers giving $e^xe^y=e^{(x+y)}$ I am multiplying the long polynomial of the taylor series of $e^x$ and $e^y$. I only take the first three terms:
$$ \left(1+x+\frac{x^2}{2!}+... |
H: function in $L^1\setminus L^2$
I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded).
Does anyone know such a function?
AI: Try $f(x) = \frac{1}{\sqrt{x}}$ on (0,1) |
H: Relationship between frequency and period of harmonic motion (free undamped motion)
From $\frac{1}{8}x''+16x=0$ we obtain $$x=c_1\cos8\sqrt{2}t+c_2\sin8\sqrt{2}t$$ so that the period of motion is $\frac{2\pi}{8\sqrt{2}}=\frac{\sqrt{2}\pi}{8}$ seconds.
From $20x''+kx=0$ we obtain $$x=c_1\cos\frac{1}{2}\sqrt{\frac{k... |
H: How do you solve $w^4=16(1-w)^4$?
Giving you answer in Cartesian form.
$\dfrac{w^4}{(1-w)^4}=16$ Are you supposed to let $w=x+yi$?
$w^4=x^4+4x^3yi-6x^2y^2-4xiy^3+y^4=16$
I then know that you get routes 2,-2,2i,-2i But I dont know how?
I have done $x^4+y^4-6x^2y^2=16$
and $4x^3yi-4xy^3i=0$ => $x^3y=xy^3$ (which ... |
H: Calculus Leibniz' notation
I'm currently doing integration by parts, and I'm finding that the notation is what makes it tough for me. So I looked it up and found that:
$$\int u(x)v'(x)dx= u(x)v(x) - \int u'(x)v(x)dx$$
But the wikipedia said that this above equality is the same as:
$$ \int u dv = uv - \int v du $$
I... |
H: Let $A,B$ be $n \times n$ matrices so that $AB = 0$ If $A,B \neq 0$ what do I know about $A$ and $B$?
Let $A,B$ be $n \times n$ matrices so that $AB = 0$ If $A,B \neq 0$ what do I know about $A$ and $B$?
I want to expand my knowledge about matrices arithmetics and so. Supposing the above what do I know about both $... |
H: Long differential equation question?
We have the equation
$$xy'-y=(x+y) \ln \left(\dfrac{x+y}{x}\right)$$
To solve this equation, I first thought about replacing variables, but my friend suggested that I solve this with Lagrange. How can this be solved this Lagrange, because it seems odd to me?
Thank you.
AI: Try... |
H: Does $\sum_1^\infty\bigr(-\frac{1}{3}\bigl)^n \bigl(\frac{(-2)^n+3^n}{n}\bigr)$ converge?
This a follow-up question about whether or the not the values on the circle of this
Q : Calculate the Radius of convergence of $\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$ converges
Mainly I need to check if this one converges:... |
H: Bayes rule with multiple conditions
I am wondering how I would apply Bayes rule to expand an expression with multiple variables on either side of the conditioning bar.
In another forum post, for example, I read that you could expand $P(a,z \mid b)$ using Bayes rule like this
(see Summing over conditional probabilit... |
H: How far is it true that statements dependent on Axiom of Choice are not constructive.
Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ultrafilter on $\mathbb{N}$.
It ... |
H: What is the solution to this parametric equation problem?
How can I find the parametric equations of the line passing through the point $(-5,7,-2)$ and perpendicular to both the vectors $(2,1,-3)$ and $(5,4,-1)$?
AI: If you know the parametric equation for a line when a point on it and its direction vector is given... |
H: I want to study $\sqrt[n]{n}$ and its behavior.
As I was studying some limit problems, I came across
$$\sqrt[n]{n}$$
and astoundingly found out that the graph of this has a maximum when $n = e$.
I thought there is no way that this is not a famous fact and I am very interested in it. I looked up some words such as... |
H: Solving $f_n=\exp(f_{n-1})$ : Where is my mistake?
I was trying to solve the recurrence $f_n=\exp(f_{n-1})$.
My logic was this : $f_n -f_{n-1}=\exp(f_{n-1})-f_{n-1}$.
The associated differential equation would then be $\dfrac{dg}{dn}=e^g-g$.
if $f(m)=g(m)>0$ for some real $m$ then for $n>m$ we would have $g(n)>f(... |
H: Indeterminate Limits
I have been studying independently through various online courses and I still have trouble understanding what to do with certain limits. I am hoping for some guidance on the following two problems to help me solve them (I do not need to answer as much as help understanding where I am going).
$$... |
H: Residue Theorem to Compute Integrals of Rational Functions
Any help would be very much appreciated. Thanks.
$$\int_{-\infty}^{\infty}\frac{x^2}{x^4-4x^2+5}dx$$
Integral for the above using Residue Theorem.
AI: Check the singularities of the function, in this case check the roots of the denominator. Consider only t... |
H: definition of divisor functions
I have a question about the definition of divisor functions when I was reading primes in tuples by Goldston, Pintz, and Yıldırım:
Let $\omega(q)$ denote the number of prime factors of a squarefree integer $q$. For any real number $m$, we define $d_m(q) = m^{\omega(q)}$. This agrees ... |
H: Input size measurement according to polynomial presenation
There's a paragraph in my book (Complexity and cryptography by Talbot and Welsh, chapter 4) that I don't fully understand:
Let $\mathbb Z[x_1,\dots,x_n]$ denote the set of polynomials in n
variables with integer coefficients. [...] We have to be careful ... |
H: Exercise 3.15 [Atiyah/Macdonald]
I have a question regarding a claim in Atiyah, Macdonald. A is a commutative ring with $1$, $F$ is the free $A$-module $A^n$. Assume that $A$ is local with residue field $k = A/\mathfrak m$, and assume we are given a surjective map $\phi: F\to F$ with kernel $N$. Then why is the fol... |
H: Does "=" have to be interpreted as equality?
To put it briefly: In model theory, we are allowed to interpret any relation symbol in any way we like. So why do people seem to require that "$=$" is interpreted as the actual equality?
Let me elaborate a little more. In model theory, as I imperfectly understand it, one... |
H: Let G be a group of order 24 that is not isomorphic to S4. Then one of its Sylow subgroups is normal.
Let G be a group of order 24 that is not isomorphic to S4. Then
one of its Sylow subgroups is normal.
This is the proof from my textbook.
Proof
Suppose that the 3-Sylow subgroups are not normal. The number of 3... |
H: Expand function into a Maclaurin's series
The function is given with:
$f(x)=\dfrac{x^{2012}}{(1-x^3)^2}$
I have no idea what to do here and honestly, I don't really understand what a Maclaurin's series is. I know the definition but I don't understand the concept enough to be able to solve problems like this.
Any he... |
H: Why should we take intersection in this order statistics problem
Let $X_1,\dots,X_n$ be a sample of i.i.d RVs, $X_j\sim F$. Denote by $X_{(1)}\le X_{(2)}\le\dots\le X_{(n)}$ the order statistics for the sample. Find the DF of $X_{(1)}$ and $X_{(n)}$.
My take:
$\mathbb{P}(X_{(1)}\le t)=1-\mathbb{P}(X_{(1)}>t)=1-(1-F... |
H: Is there a name for this type of relation?
Let $S$ be a set. Let $\sim$ be a binary relation on $S$. Suppose $\sim$ follows these three rules.
$x\sim x$ for all $x\in S$ (reflexivity).
If $x\sim y$, then $y\sim x$ for all $x, y \in S$ (symmetry).
If $x_1\sim x_2 $, $x_2\sim x_3 $, $x_3\sim \cdots$, then there exi... |
H: Proving divisibility in elementary number theory problem
Find all positive integers n such that $(n+1)\mid(n^2+1)$.
What I have done so far.
I noticed that $ n^2 + 1 = (n + 1 - 1)^2 + 1 = (n + 1)^2 -2(n + 1) + 2$.
Hence, for the relation to be true, we must have that $(n+1)\mid 2$, that is $n=1$.
How would I prove ... |
H: Linear combination (vectors in space)
First of all, sorry if my question is too easy for you guys, and sorry for my por English.. I have serious trouble with vectors haha Can someone please help me?
Given the vectors $$\vec{u} = 4\vec{i}+\vec{j}-3\vec{k}$$ $$\vec{v} = 3\vec{j}+\vec{k}$$ $$\vec{w} = 2\vec{j}+3\vec{k... |
H: Is partition function increasing function?
I have some exercises which require knowing the number of partitions of particular numbers, so I used some python code which I found on internet to compute the values of the partition function for the values I want.
But I noticed that $p(10)=p(12)=57$ and $p(11)=51$ where ... |
H: Some problems about the proof of a theorem
There's a theorem in my book (Complexity and cryptography by Talbot and Welsh, chapter 4) where I don't understand some parts of its proof:
THEOREM: Suppose $f \in \mathbb Z[x_1,..., x_n]$ has degree at most $k$ and is
not identically zero. If $a_1,...,a_n$ are chosen i... |
H: De Morgan's laws in natural deduction?
We are asked to use natural deduction to prove some stuff. Problem is, without De Morgan's law, which I think belongs in transformational proof, lots of things seem difficult to prove. Would using de Morgan's laws be a violation of "In your natural deduction proofs, use only n... |
H: The group $\mathbb{Z}^\mathbb{N}/\mathbb{Z}^{(\mathbb{N})}$ can't be embedded in a product $\mathbb{Z}^A$ for any $A$
How the tittle says I need to prove that:
There isn't a group monomorphism $\psi: \mathbb{Z}^\mathbb{N}/\mathbb{Z}^{(\mathbb{N})} \to \mathbb{Z}^A$ for any $A$
and, of course, this is equivalent t... |
H: Proving a subset
I need to prove that A ⊂ B if and only if, A ∩ B = A
This seems straightforward to understand and then explain in words, as the two statements are equivalent, but I don't understand how I would correctly explain this in a formal sense?
AI: Hint: to show that $X=Y$ you have to show both inclusions: ... |
H: What is the greatest integer function, and how do you integrate it?
$[x]$ denotes the greatest integer $\leq x$. Let $f(x)=[x]$ and let $g(x)=[2x].$
I am having hard time understanding this. What is meant by "greatest integer?" Can anyone refer me to any visual/graphical explanation for $[x]?$
For example, how w... |
H: What is the solution for $a^x + bx = c$?
What is the solution for $a^x + bx = c$? Also, can anyone refer me to a good article/book/etc that covers general solution methods for exponential functions?
Thanks,
AI: This can be solved in terms of the Lambert W function, which is defined as the solution for $w$ to $z = w... |
H: Transformation to spherical coordinate system
If I have a sphere $T: x^{2}+y^{2}+z^{2}\leqslant 10z$ by transformation to the spherical coordinate system by the:
$
x=r\cos\theta\sin\varphi\\
y=r\sin\theta\sin\varphi\\
z=r\cos\varphi
$
What is values for $\varphi$ I will get?
$0\leqslant\varphi\leqslant\pi$ or $0... |
H: $X$ is an odd number, $Y$ is a natural number more than 36. If $\frac{1}{X}+\frac{2}{Y}=\frac{1}{18}$, find the set $(X,Y)$?
$X$ is an odd number, $Y$ is a natural number more than 36. If $\frac{1}{X}+\frac{2}{Y}=\frac{1}{18}$, find the set $(X,Y)$ ?
Re arranging the given equation, we have,
$\frac{2}{Y}=\frac{X-18... |
H: Find all positive integers $x$ such that $13 \mid (x^2 + 1)$
I was able to solve this by hand to get $x = 5$ and $x =8$. I didn't know if there were more solutions, so I just verified it by WolframAlpha. I set up the congruence relation $x^2 \equiv -1 \mod13$ and just literally just multiplied out. This lead me to ... |
H: A simple adjoint operator question
I'm trying to solve this problem:
Let $\Omega$ a bounded open of $\mathbb{R}$, consider the Hilbert real spaces $X = L^2(\Omega)$ and $Y = \mathbb{R}^{2\times 2}$, with the inner products:
$$\langle u,v\rangle_X\ =\ \int_{\Omega}uv\quad\forall\ u,v\in X, \quad\quad \langle A,B\... |
H: expected number of repeats in random strings from different sized alphabets
The question is just for fun, and I feel like I'm missing a clever way of thinking about it.
suppose that you are on an alien planet, and you are trying to learn their language. You break into one of the aliens houses and get on his compute... |
H: \exists quantifier and being explicit about quantity
I have a question regarding the use of the $ \exists $ quantifier when trying to signify explicit quantities.
Consider
$F:$ the set of all fruits
$A(x):$ x is an apple
$R(x):$ x is rotten
If I wanted to express 'One apple is rotten' would this be logically sound?... |
H: function lifting on $S^1 \times S^1$
Let $f:S^1 \times S^1 \to S^1 \times S^1$ a continuous function and $p:\mathbb{R}^2 \to S^1 \times S^1: (t,s) \mapsto (e^{2\pi i t},e^{2\pi i s})$ a covering map. if $F: \mathbb R ^2 \to \mathbb R ^2 $ is a lifting of $f \circ p$
prove that there exists $(d_1,d_2),(e_1,e_2) \in ... |
H: Limits and continuous functions
I have always been told that if $f(x)$ is a continuous function at $a$ so that $f(a) = L$, then $\lim_{x\to a}f(x) = L$. Please, could someone explain in detail why this is true?
AI: It is true because of the way we define "continuous" and because of the meaning of $\lim_{x\to a}f(x)... |
H: If $\int^{\pi}_0 x f (\sin x) dx = k \int^{\pi/2}_0 f (\sin x) dx$, find $k$.
Problem : If $\int^{\pi}_0 x f (\sin x) dx = k \int^{\pi/2}_0 f (\sin x) dx$, find $k$.
Solution : Period of sine function is $2 \pi$
I don't know whether we can use the period of this function to solve this problem.
AI: Let $ I := \int... |
H: When integrating $A/x$, why use the logarithm instead of $x$ raised to a power?
$\int\frac{15}{x}dx$ would be $$15\int\frac{1}{x}dx = 5\ln|x|+c$$
This seems like a silly question but I'm feeling exceptionally dense today. Why would you apply the logarithm rule, why wouldn't raising $x$ to a $-1$ exponent work?
AI:... |
H: Is there a simple group of any (infinite) size?
I'm trying to show that for any infinite cardinal $\kappa$ there is a simple group $G$ of size $\kappa$, I tried to use the compactness theorem and then ascending Löwenheim-Skolem, but this is impossible as the following argument shows:
Suppose $F$ is a set of senten... |
H: Find the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}\leq \frac{1}{100}$
Find the smallest positive integer $n$ such that $\sqrt n-\sqrt{n-1}\le \frac{1}{100}$.
First I multiplied by the conjugate and got
$$\frac 1{\sqrt n + \sqrt{n-1}} \le \frac{1}{100},$$ or $\sqrt n+ \sqrt{n-1} \ge 100$. Now I... |
H: Clarification on some mathematics formula
In the book I am reading (complexity and cryptography by Talbot and Welsh, chapter 4), there is a theorem on $\textbf{BPP}$ where I don't understand a few steps of its proof, it's totally independent of the concept I am reading, I don't understand the mathematical steps (t... |
H: Calculating algorithm sizes
I am reviewing BigO notation from this document: http://www.scs.ryerson.ca/~mth110/Handouts/PD/bigO.pdf
This document contains this information and the accompanying table:
In the following table we suppose that a linear algorithm can do a problem up to size 1000 in 1 second. We then use... |
H: How find $\int(x^7/8+x^5/4+x^3/2+x)\big((1-x^2/2)^2-x^2\big)^{-\frac{3}{2}}dx$
How can I compute the following integral:
$$\int \dfrac{\frac{x^7}{8}+\frac{x^5}{4}+\frac{x^3}{2}+x}{\left(\left(1-\frac{x^2}{2}\right)^2-x^2\right)^{\frac{3}{2}}}dx$$
According to Wolfram Alpha, the answer is
$$\frac{x^4 - 32x^2 + 20}{2... |
H: Does the analog of homological algebra studying maps where, say, $d \circ d \circ d = 0$ have a name?
I don't have an application in mind or anything; I'm just curious.
We can think about homological algebra as the study of endomorphisms $d$ such that $d \circ d = 0$. Most of homological algebra seems to follow fr... |
H: Find all $f(x)$ if $f(1-x)=f(x)+1-2x$?
To find one solution I assumed that $f$ is even and rewrote this as $f(x-1)-f(x)+2x=1.$ By just thinking about a solution, I was able to conclude that $f(x)=x^2$ is a solution. However, I am sure that there are more solutions but I don't know how to find them.
AI: HINT:
As $f(... |
H: Error in Neukirch's "Algebraic Number Theory"?
I found what I believe is an error in Neukirch's book, in Chapter 1 Section 3 (Ideals). Exercise 5 states
The quotient ring $\mathcal{O}/\mathfrak{a}$ of a dedekind domain by an ideal $\mathfrak{a} \neq 0$ is a principal ideal domain.
I believe I can prove every ide... |
H: Problem understanding a proof
In the book I am reading (complexity and cryptography by Talbot and Welsh, chapter 4), there's this example:
Choosing an integer $a \in_R \{0,\dots,n\}$ using random bits.
We assume that we are given an infinite sequence of independent random
bits. To choose a random integer $a \i... |
H: If $f(a)$ is divisible by either $101$ or $107$ for each $a\in\Bbb{Z}$, then $f(a)$ is divisible by at least one of them for all $a$
I've been struggling with this problem for a while, I really don't know where to start:
Let $f(x) \in \mathbb{Z}[X]$ be a polynomial such that for every value of $a \in \mathbb{Z}$, ... |
H: Question involving Legendre symbols
Let r,p,q be distinct odd primes. Let 4r divide p-q. Show that
(r/p) = (r/q)
Where (a/b) is the Legendre symbol.
I'm sure we are suppose to use the law of quadratic reciprocity. I don't think this question is suppose to be difficult, but I cannot figure it out!
AI: Hint:
$$4r\mid... |
H: Closest point in $y = \sqrt{x}$ to the origin is at $x=-1/2$?
When I solve for the point in $y = \sqrt{x}$ closest to the origin using calculus, I get $x = -1/2$. And this is the case for ALL functions $y = \sqrt{x + c}$ using the distance formula $d^2 = x^2 + y^2$.
Why is it that the extreme point is somewhere the... |
H: How can $\frac{x}{\pi}-\frac{n-x}{1-\pi}$ be the correct derivative of $x\log\pi+(n-x)\log(1-\pi)$?
I am learning statistics and come across this calculation for Maximum-Likelihood estimator for the Binomial distribution. I don't understand the step from second to third row where they took the derivative,
my att... |
H: How to show this equality of probability on the unit disk
I came up with this problem which I think is so intuitive but fails to give more rigorous and convincing argument.
Let $(X,Y)$ be uniformly distributed in the disk $D:=\{(x,y):x^2+y^2\le 1\}$
For $x\in(-1,1)$ and small $\Delta>0$, $\mathbb{P}(X\in(x,x+\Delt... |
H: Find the tangent line of $\frac{x^2}{y+1}+xy^2=4$ at $y=1$ and where $y
I want to find the tangent line for the function $\frac{x^2}{y+1}+xy^2=4$ at the point $y=1$ and where $y<x$.
First step: finding the point so I inserted y=1 and get : $$x^2+2x-8 \rightarrow x_1=2 ,x_2=-4$$
$x_1 = 2$ satisfies the conditions.
S... |
H: Is there a formula in permutations and combinations if we are to find the sum of number of 1's in binary expansion of a number from 1 to n
We are given $N$. Suppose $f(x) =$ number of $1$'s in the binary expansion of $x$.
We have to calculate $f(1) +f(2) +f(3)+ \dots +f(N)$.
So is there a formula for this sum direc... |
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