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H: Lottery Competition
Suppose I want to hold lottery competition. I just want two people to win the lottery and there are 100 people buying my lottery. What should the maximum probability (of each player wining the lottery) be to ensure that exactly two people win the lottery?
I obviously think it's $2/100=1/50$. But... |
H: On the proof of Schur's lemma in Fulton & Harris
I'm reading the book on representation theory by Fulton and Harris. I'm stuck with the proof of Schur's Lemma 1.7:
Schur's lemma 1.7 If $V$ and $W$ are irreducible representation of $G$ and $\phi:V\rightarrow W$ is a $G$-module homomorphism, then
1. Either $\phi$ ... |
H: On the definition of divisors in Riemann Surfaces
The sum notation for a Divisor $D$ in a Riemann Surface $X$ (as in Miranda's "Algebraic Curves and Riemann Surfaces") is
$$
D=\sum_{p\in X} D(p)\cdot p
$$
That is, $D$ assumes the value $D(p)$ at $p$. For example, a principal divisor of $f$ is the divisor
$$
div(f)=... |
H: Good source for Triebel-Lizorkin spaces?
I'm trying to look into different types of function spaces. At the moment, at least for function spaces involving integration, I only have $L^p$ and $W^{k,p}$. The next function spaces I thought I'd read about are Besov spaces and Triebel-Lizorkin spaces. Admittedly I usu... |
H: For two disjoint compact subsets $A$ and $B$ of a metric space $(X,d)$ show that $d(A,B)>0.$
I was thinking about the following problem: For two disjoint compact subsets $A$ and $B$ of a metric space $(X,d)$ show that $d(A,B)>0.$ I'm having doubt with my attemp. Please have a look and do comment:
Let $d(A,B)=\inf... |
H: Non trivial Automorphism
Prove that every finite group having more than two elements has a nontrivial Automorphism.
It is from Topics in Algebra by Herstein. I am not able to solve.
AI: Suppose that $G$ is abelian with exponent greater than 2 (i.e. not every element has order 2). Then mapping $g\in G$ to $g^{-1}$ i... |
H: Density of the image and closedness of the inverse of a bounded linear operator
Let $A \colon X \to X$ be a bounded linear operator, where $X$ is a Banach space.
$(Q1)$ Is it true that if $A$ is injective then the image of $A$ is dense in $X$?
$(Q2)$ Is it true that $A^{-1} \colon \text{rgA} \to X$ is closed?
My a... |
H: If finer topology is second-countable then coarses is second-countable
Let $(X,\tau_1)$ and $(X,\tau_2)$ be topological spaces and $\tau_2$ finer than $\tau_1$. Prove if $\tau_2$ is second-countable then $\tau_1$ is also second-countable.
My try:
Let $B=\{B_n,n\in \mathbb{N}\}$ be countable basis for $\tau_2$. Let... |
H: Why do we use open intervals in most proofs and definitions?
In my class we usually use intervals and balls in many proofs and definitions, but we almost never use closed intervals (for example, in Stokes Theorem, etc). On the other hand, many books use closed intervals.
Why is this preference? What would happen i... |
H: Is there a difference between 'inconsistent', 'contrary', and 'contradictory'
Is there a difference between 'inconsistent' 'contrary' and 'contradictory'? As far as I understand, two statements are inconsistent when they can not both be true; two statements are contradictory when they can not both bear the same tru... |
H: Finding the indefinite integral $\int \frac{3x+2}{(6x^2+8x)^7}\,\mathrm dx$
I'm not too familiar with how to solve this. Could anyone present a step by step guide on how to get the answer?
$$\int \dfrac{3x+2}{(6x^2+8x)^7}\,\mathrm dx$$
AI: Let $u = 6x^{2} + 8x$. Then $du = (12x + 8) dx \rightarrow du = 4(3x + 2) d... |
H: Show that $\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$
The question asks to prove the identity:
$$\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$$
where $n\in\mathbb{Z}$
I ha... |
H: Find the limit as n approaches infinite
We have the following function:
$$U_n = \sin \dfrac{1}{3} n \pi$$
What is the limit of this function as n approaches infinity?
I first tried to use my calculator as help, for n I chose some arbitrary large numbers, such as 100 and 1000. Then I I just took $n = 10^{50}$ and i... |
H: Integral of $\int \frac{x^4+2x+4}{x^4-1}dx$
I am trying to solve this integral and I need your suggestions.
$$\int \frac{x^4+2x+4}{x^4-1}dx$$
Thanks
AI: welcome to math.stackexchange this question were answered already.
Here is the link
use polynomial division, we get $$\int \frac{x^4+2x+4}{x^4-1} dx = \int 1 + \fr... |
H: Eigenvalues of $\sum_{i=1}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}$
Consider the cuadratic form
$$
\mathbf{x}^{\intercal}Q\mathbf{x} = \frac{x_1^2}{\lambda_1} + \sum_{i=2}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}\ .
$$
Is it true that the eigenvalues of $Q$ are $\lambda_i^{-1}$ ?
AI: The answer to your question is no. ... |
H: Perimeter or Calculus Word problem
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single strand of electric fence.
With 1400m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
Is there a way to solve this... |
H: Prove that there isn't a polynomial with $\text {f(x)}^{13} = {(x-1)}^{143}+(x+1)^{2002}$
Prove that there isn't a polynomial with $\text {f(x)}^{13} = {(x-1)}^{143}+(x+1)^{2002}$
We can easily find out that $\text {deg}(f) = 154$
Then?
AI: Hint: Consider the coefficients of $x^0,x^1$ on both sides of the equati... |
H: What is $\lim_{n \rightarrow \infty}\sum\limits_{k = 1}^n\frac{k}{n^2}$?
We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$
What is the limit of this function as $n \rightarrow \infty$?
My idea:
$$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + \dfrac{n}{n^2} = 0$$
Is this correct?
AI: HINT:
Su... |
H: A question regarding the Poisson Process
I have the following question: Buses arrive at a city according to a Poisson process with a rate of 5 per hour. What is the probability that the fifth bus of the day arrives after midday given they start arriving at 9 a.m.
What I think is the correct way to go about this que... |
H: cardinality of a set with repeating elements?
What is the cardinality of a set which has repeating elements ?
For example $S = \{1,1,1,2,2\}$
Is each individual element counted?
Please quote a reference text if possible.
AI: The set $A = \{1,1,1,2,2\}$ is identical to the set $B = \{1, 2\}$ because one can show $... |
H: When is an Integer a Rational Number, and are All Ratios Rational, Even $\frac{\sqrt{7}}{2}$?
$$\Bbb{Q} = \left\{\frac ab \mid \text{$a$ and $b$ are integers and $b \ne 0$} \right\}$$
In other words, a rational number is a number that can be written as one integer over another.
For an integer, the denominator is $... |
H: If $|f(z)| \leq |z|^2+\frac{1}{\sqrt{|z|}}$, show f is quadratic polynomial.
Suppose the function f is analytic in the punctured plane $z!=0$ (it means we excluded the zero) and satisfies the above condition, $|f(z)| \leq |z|^2+\frac{1}{\sqrt{|z|}}$, then show f is quadratic polynomial.
I think that if we multiply ... |
H: How to integrate $\int_0^\infty e^{-ty^2} \sin t dt$
My book suggests that I do some sort of limiting
$\lim_{A \to \infty} \int_0^A e^{-ty^2} \sin t d t$
But I'm not getting anywhere.
AI: Starting with the initial integral.
You can integrate it by parts.
$$\int_0^A e^{-ty^2} \sin t d t=-\int_0^A e^{-ty^2} d \cos(... |
H: Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& \int {{{\sin }^3}x} dx \cr
& = \int {\sin x({{\sin }^2}x} )dx \c... |
H: Can someone please explain $e$ in layman's term?
I never really understood what $e$ means and I'm always terrified when I see it in equations. What is it? Can somebody dumb it down for me? I know it's a constant. Is it as simple as that?
AI: The simplest way to understand it is, consider the following equation:
$$f... |
H: Dagger category generated by $\mathsf{Set}$ viewed as a subcategory of $\mathsf{Rel}$.
Whenever a category $\mathcal{C}$ is being viewed a subcategory of a dagger category $\mathcal{D}$, define that the dagger category generated by $\mathcal{C}$ is the least subcategory of $\mathcal{D}$ that is closed under the dag... |
H: Path-connectedness and compactifications
Is the compactification of a path-connected space path-connected? Why or why not?
(I came across this question in my notes while studying for finals and I have no idea.)
AI: Let $\Gamma = \{(x, \sin(1/x)) \mid x>0 \}$ and $X=\Gamma \cup \{0\} \times [-1,1]$. Then $\Gamma$ i... |
H: Integrate over the uniform distribution on the simplex
Let $p=(p_1,\ldots,p_n)$ correspond to points in a simplex that add up to one, i.e. $p$ is a discrete probability distribution. I would like to compute an integral of the form $\int dp_1\ldots\int dp_n\sum_{i=1}^np_if(p_i)$ with $p$ uniformly distributed on the... |
H: Proving the quadratic formula (for dummies)
I have looked at this question, and also at this one, but I don't understand how the quadratic formula can change from $ax^2+bx+c=0$ to $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$. I am not particularly good at maths, so can someone prove the quadratic formula in a simple way, w... |
H: Are (semi)-simple Lie algebras not solvable?
Let $L\ne \{0\}$ be a non-abelian simple Lie algebra, then the only ideals are $L$ and $\{0\}$. We know that $L'$ is the smallest ideal of $L$ such that $L/L'$ is abelian. If $L'=\{0\}$, then $L/L'=L$, but $L$ is non-abelian. So $L'=L$ and hence $L^{(k)}=L\ne \{0\}$ for ... |
H: Integral, set and parametric representation
I am to compute the following: $\displaystyle\iiint\limits_V 1\, dx\, dy\, dz$,
where $V= \{{(x,y,z) \in \mathbb R^3 : (x-z)^2 +4y^2 < (1-z)^2} \text{ and } 0<z<1\}.$
Does anyone have idea what parametric representation should I take? I think there will be something ellip... |
H: Write the expression $\log(\frac{x^3}{10y})$ in terms of $\log x$ and $\log y$
What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$
This is what I got out of the equation so far. the alternate form assuming $x$ and $y$ are positive $$3\log(x)-\log... |
H: Mathematical Symbol
In the following paper, what does the symbol $\Phi$ in equation $3.1$ (page $3$) represent? Does it represent the normal distribution?
AI: $\Phi(x)$ typically (and which is what is also means in the article you have linked to) represents a suitably normalized error function, equivalently the cum... |
H: Abelian categories with direct sums
Does any abelian category admits direct sums?
If not, categories admiting direct sums have a special name?
I'm asking this since I am writing a proof that requires direct sums but I only know that the category is abelian.
Thank you very much for your answer!
Edit. In the tag desc... |
H: Solving for a matrix from its quadratic form
I have a set of vectors that I am trying to predict from another set of vectors using a matrix $W$. To find this matrix, I decide I want to minimize the $\ell^2$ norm of the error, e.g.:
$$
\text{find} \min_W \|y - Wx\|_2 \\
x,y \in \mathbb{C}^N \quad W\in \mathbb{C}^{N... |
H: How to find the limit of this function
We have the function $$\dfrac{\sqrt{n^4 + 100}}{4n}$$
I think the best method is by dividing by $n$, but I have no idea what that yields, mainly because of the square root.
AI: $\text{Note that $\sqrt{n^4+100} > \sqrt{n^4}$. Hence, we have}$
$$\dfrac{\sqrt{n^4+100}}{4n} > \dfr... |
H: Definition of Induced Module - Typo in Corps Locaux?
This is from the beginning of the section on group cohomology in Corps Locaux (English Edition).
Serre states that $A$ is an induced $G$-module if
(1) $A\cong A\otimes_\mathbb{Z}X$ for an abelian group $X$,
or, equivalently,
(2) $A=\bigoplus_{s\in G}s\cdot X$.
Is... |
H: Bifurcation value and description
Find the bifurcation of $a$ and describe the bifurcation that take place at each value
$\displaystyle dy/dt=e^{-y^2}+a$
I let $\displaystyle e^{-y^2}+a=0$ then solve for y. I got $y^2=-\ln(a)$ What do I do next to find $a$?
AI: You started well to let $e^{-y^2} + a =0$ and then sol... |
H: singularity of analytic continuation of $f(z) = \sum_{n=1}^\infty \frac{z^n}{n^2}$
How to show that all possible collection of analytic continuations of $\displaystyle f(z) = \sum_{n=1}^\infty \frac{z^n}{n^2} $ has singular point at $z = 1$. I know that $f(z)$ converges for $|z| \le 1$. Also is there a theorem that... |
H: Terminology for an element of a partition?
Suppose I'm dividing some region $\Theta \in \mathbb{R}^n$ into subregions $\theta_i, i=1,2,3$ such that $\theta_i \cap \theta_j = \varnothing, i\ne j$ and $\bigcup_i \theta_i = \Theta$. I might say (perhaps loosely, even technically incorrectly) that I am partitioning th... |
H: Counting couples of numbers
I have no trouble believing that, if $|n| \leq J$, then $$\#\{ (j_1,j_2) \in \{ 1,...,J \} \, | \, j_1-j_2 = n \} = J-|n|,$$
but can anyone explain it a little more formally?
Thank you in advance for any help.
AI: We can restrict to the case $n \ge 0$, since if $n \le 0$ then
$$\{ (j_1,j... |
H: can we divide by any term when we have an differential homogeneous equation?
I am asking because i think we divided by x here for whatever reason since the other side is equal to 0 and it wont affect the equation in any meaningful way.
Letting $y=ux$ we have
$$\begin{align}
(x-ux) dx + x(udx + x du) &= 0 \\
dx+ ... |
H: Remainders problem
What will be the reminder if $23^{23}+ 15^{23}$ is divided by $19$?
Someone did this way:
$15/19 = -4$ remainder and $23/19 = 4$ remainder So $(-4^{23}) + (4^{23}) =0$ but i didn't understand it
AI: $$\forall\,\text{prime}\;p\;\wedge\;\forall\,a\in\Bbb Z\;,\;(a,p)=1\implies a^{p-1}=1\pmod p$$
... |
H: Definition of continuous map
Let $X,Y$ be topological spaces, let $f:X\longrightarrow Y$ a function. There are several (equivalent) ways to define continuity of $f$, one of these says: $f$ is continuous if
$$(1)\qquad \forall A\subseteq X,\ \forall x\in X, \Big(x\in \overline{A}\Rightarrow f(x)\in\overline{f(A)}\Bi... |
H: Function Spaces
What is exactly the difference between $L^2$ space and ${\ell}^1$ space? I believe that one of them is the space of square of square integrable functions.
Does it have to do with one is for series and other for integration?
Thank You.
AI: $L^2(\Bbb R)$ is the space of square-integrable real functio... |
H: Test for polynomial reducibility with binary coefficients
I'm learning about Galois Fields, in particular $GF(2^8)$, as they are applied to things like the AES algorithm and Reed-Solomon codes. Each of these rely on an irreducible 8th degree polynomial with binary coefficients to serve as a modulus for generating ... |
H: Why are Haar measures finite on compact sets?
I'm working through the answer by t.b. to another user's question here:
A net version of dominated convergence?
because I am trying to work through a related problem and I think it will be illuminating.
From step two, "Then $KK'$ is compact and thus has finite Haar meas... |
H: Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?
Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define $ \mathbf{x}'=(|x_1|,...,|x_n|) $ .
Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$ for every norm on $ \mathbb{R}^n $ ?
NB: The answer is trivially yes for $p$-norms.
AI... |
H: How to integrate $\int_0^\infty \frac{1}{1+y^4} dy$
I tried the trigonometric substitution $y^2 = \tan \theta, sec^2\theta = 1 + y^4$
But now I'm stuck with $\frac12 \int \frac{\sqrt{\sin \theta}}{(\cos\theta)^{\frac92} } d \theta$
I ran out of imagination as what to try now
AI: Note that
$$\int_1^{\infty} \dfrac{d... |
H: how many empty sets are there?
Would I be correct in saying that in the category of sets, the "class of sets that are isomorphic to the empty set is a proper class"?
In other words, there are LOTS of initial objects in the category of sets, but they're all related to each other via unique isomorphisms?
Similarly, f... |
H: The number of words that can be made by permuting the letters of _MATHEMATICS_ is
The number of words that can be made by permuting the letters of MATHEMATICS is
$1) 5040$
$2) 11!$
$3) 8!$
$4) 4989600$
First of all I do not understand the statement of the problem, I would like if some one tell me with an example.
... |
H: Half order derivative of $ {1 \over 1-x }$
I'm new to this "fractional derivative" concept and try, using wikipedia, to solve a problem with the half-derivative of the zeta at zero, in this instance with the help of the zeta's Laurent-expansion.
Part of this fiddling is now to find the half-derivative $$ {d^... |
H: Should $\mathbb{N}$ contain $0$?
This is a classical question, that has led to many a heated argument:
Should the symbol $\mathbb{N}$ stand for $0,1,2,3,\dots$ or $1,2,3,\dots$?
It is immediately obvious that the question is not quite well posed. This convention, as many others, are not carved in stone, and there... |
H: Proof $||A \underline x|| > 0 \Leftrightarrow \underline x \neq \underline 0$
If $n \geq m, A \in M(n,m)$ and $rg(A)=m$
Proof $||A \underline x|| > 0 \Leftrightarrow \underline x \neq \underline 0$
a)If $m \neq 0 \leftrightarrow A \neq 0_M$
Suppose that $\underline x = \underline 0 \Leftrightarrow ||A \cdot \... |
H: Remainder when $20^{15} + 16^{18}$ is divided by 17
What is the reminder, when $20^{15} + 16^{18}$ is divided by 17.
I'm asking the similar question because I have little confusions in MOD.
If you use mod then please elaborate that for beginner.
Thanks in advance.
AI: The key thing to remember about the operation... |
H: How to find the number of positive devisors of $50,000$
How to find the number of positive devisors of $50,000$, I would like to know that what mathematical formulae I need to use here as it is a big number to calculate mentally, I am sorry to ask if this is too silly question here to ask. Thank you.
AI: Write the... |
H: Find the inverse for arbitrary k
I need to find a, b, c, d, e, f, g, h (all of which are not zero)
such that for all k is in Real number, show A is invertible or this can't happen
$$A = \left(\begin{array}{ccc}
a&b&c\\d&k&e\\f&g&h
\end{array}\;\begin{array}{c}\end{array}\right)$$
My answer was this can't happen, b... |
H: Quadrature formula
How can we find a quadrature formula $\int_{-1}^1 f(x) dx=c \displaystyle \sum_{i=0}^{2}f(x_i)$ that is exact for all quadratic polynomials?
Thanks for help.
AI: Write:
$$\int_{-1}^{1} f(x)\ dx =\int_{-1}^{1} ax^2+bx+c\ dx =\frac{2a}{3}+2c$$
Moreover you want to have three points all having the s... |
H: Advocating base 12 number system
I had a calculus professor who suggested we should be using base 12 number system. What are the advantages of using such a system?
AI: As I see it, there are two advantages. First, it's not too different from base 10, so it comes fairly naturally. Second, 12 has many divisors, so ... |
H: Field of fractions of a finite $\mathbb{Z}$-module is finite extension of $\mathbb{Q}$
Let $A$ be a ring which is also a finitely generated $\mathbb{Z}$-module. If $A$ is an integral domain and $K$ is its field of fractions and $K$ has characteristic zero, then why is $K$ a finite dimensional vector space over $\ma... |
H: How to work with random variables?
If $X$ and $Y$ are independent random variables described by standard normal distribution could you please explain how to formally evaluate probabilities of occurrences such as $X-Y>0$ (intuitively it's $0.5$ of course) or $X^2-Y^2>0$?
Ultimately I'd like to be able to tell how li... |
H: How do we know which component belongs to which part in a separable differential equation
Take for instance dP/dt = kP
We get after separating: dP/P = kdt, but why shouldn't it be
dP/kP = dt instead, mathematically it doesn't make sense to say that k must belong absolutely to the right hand side of the equation.
A... |
H: $f(g(x))=x$ implies $f(x)=g^{-1}(x)$
Is it possible to find a necessary and sufficient condition to conclude when
$$f(g(x))=x \implies f(x)=g^{-1}(x) \wedge f^{-1}(x)=g(x),$$
if both functions are well defined?
AI: If either $f$ is injective or $g$ is surjective, then $f\circ g={\rm id}$ implies $\exists f^{-1},g^{... |
H: Projecting a surface segment of a cone onto a 2D plane?
Firstly, I'd like to apologise - I do not know the correct terms for what I am asking.
Assume that the top/bottom of the highlighted portion there is actually aligned with the base.
To help explain:
I need to wrap that section of the cone using a piece of pap... |
H: relationship of polar unit vectors to rectangular
I'm looking at page 16 of Fleisch's Student's Guide to Vectors and Tensors. The author is talking about the relationship between the unit vector in 2D rectangular vs polar coordinate systems. They give these equations:
\begin{align}\hat{r} &= \cos(\theta)\hat{i} +... |
H: Show that if some nontrivial linear combination of vectors $\vec{u}$ and $\vec{v}$ is $\vec{0}$, then $\vec{u}$ and $\vec{v}$ are parallel.
I've never been that great at writing proofs, but I'm getting a bit better. I think I have the answer correct, but I don't know if I'm missing anything. My logic seems right bu... |
H: Preimages of a function: Is the following proposition true or false?
Let $g: ℤ \times ℤ → ℤ \times ℤ$ be defined by $g(m,n) = (2m, m – n)$.
Is the following proposition true or false? Justify your conclusion.
For each $(s, t) ∈ ℤ \times ℤ$, there exists an $(m, n) ∈ ℤ \times ℤ$ such that $g(m,n) = (s, t)$.
I unders... |
H: Special numbers in patterns and the reasons they are special
I know there are several big list questions out there (e.g. Patterns that break down at certain numbers) that touch on classifications of mathematical structures where certain numbers don't fit in, but I'd like to find more explanations on why these numbe... |
H: Finding the x value after a matrix multiplication?
I have the following solution of a problem, and I was wondering about a hopefully quite simple thing in it:
I was wondering how do they get from [5,10,5] to 5x? I am pretty sure there is a simple explaination for that.
Thanks in advance..
AI: Because $5\left[\beg... |
H: Differentiate $\log_{10}x$
My attempt:
$\eqalign{
& \log_{10}x = {{\ln x} \over {\ln 10}} \cr
& u = \ln x \cr
& v = \ln 10 \cr
& {{du} \over {dx}} = {1 \over x} \cr
& {{dv} \over {dx}} = 0 \cr
& {v^2} = {(\ln10)^2} \cr
& {{dy} \over {dx}} = {{\left( {{{\ln 10} \over x}} \right)} \over {2\... |
H: Inverse Laplace transform of the function: $F(s)=e^{-a\sqrt{s(s+r)}}$
I would like to find inverse Laplace transform of the function: $$F(s)=e^{-a\sqrt{s(s+b)}}$$
which $a$ and $b$ are positive real numbers and $s$ is a complex variable. It would be appreciated if someone can help me.
AI: See my solution to this pr... |
H: Relationship between 2 Dimensional Quadratic systems and roots
Given four points
$(x_1, y_1)
(x_2, y_2)
(x_3, y_3)
(x_4, y_4)$
How does one construct a system of two equations:
$a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$
$b_1x + b_2x^2 + b_3y + b_4y^2 + b_5xy = c_2$
such that the set of solutions of this system i... |
H: Better than Runge-Kutta-Fehlberg 4(5) at high order?
I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using $128$-bit floating-point number (quad arithmetics).
So, what would b... |
H: The set of complex numbers of modulus $1$ is a group under multiplication
Show that $C=\{z\in \mathbb{C} \mid |z|=1\}$ is a group under complex multiplication.
I'm a little confused because isn't the identity the only element with order $1$? What is this set?
AI: Hint: prove that if you multiply two unitary compl... |
H: Necessary and sufficient condition for an Euler trail between two vertices
Graph theory question one of those obvious math proofs so its going to be a pain to prove.
Show that G=(V,E) has an Euler trail between (different) vertices u and v if and only if G is connect and all vertices except u and v are of even degr... |
H: Spivak problem on Schwarz inequality
I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality:
$$
x_1y_1 + x_2y_2 \leq \sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2} \ .
$$
It says to prove that if $x_1=\lambda y_1$ and $x_2 = \lambda y_2... |
H: Why is $\{n=4r+1,r = {n-1\over 2}\}\subset \mathbb{P}$ true under these conditions?
Let $p=p_k$, $q=p_{k+1}$ and $r=p_{k+2}$, where $p_m$ denotes the $m$th prime.
I conjecture that whenever $n$ is prime, where $n$ is defined as follows:
$$n = 1+\left(\left\lfloor{p\over q}\right\rfloor+r\right)
\left\lfloor{(p+r)(q... |
H: How to place a limit that it's inside the integral, outside.
I did this:
$$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx $$ just to have a way to approximate $\ln t$. $$\ln{t}=\lim_{h\rightarrow 0}\frac{x^{h}-1}{h}$$
The second expression may be correct, but I ... |
H: How do I solve for $dy/dx$ if $y=\ln (\sin x+\ln x)$?
Solve for $\frac{dy}{dx}$ if $y=\ln(\sin x+\ln x)$.
I know how to solve for integrals involving $du$ and $u$, but how do I do this type of problem (I think it's the opposite of the integral problem)?
AI: Use the "good old" chain rule: remember that?
$$y = \ln(... |
H: To prove that $2^{3n}+2^n +1$ is not a perfect square.
Question: Prove that $2^{3n} + 2^n + 1$ cannot be a perfect square for any natural $n$.
I attempted this question and failed in two different ways.
1) I considered a polynomial $p(x) = x^3+ x + 1 - m^2$ (for some natural $m$) and factorized the polynomial assu... |
H: What is $(\operatorname{monad}(0), \leq)$ isomorphic to?
Suppose, given $\epsilon \in \operatorname{monad}(0)$ and $\epsilon \neq 0$, is it true for each $x \in \operatorname{monad}(0)$, $$x = \sum_{r_i \in s \subset \Bbb R, s \text{ is finite}}a_i \epsilon^{r_i}$$
$a_i \in \Bbb{R}$ is a constant coefficient.
Let $... |
H: Demonstration using the Pigonhole principle
I was thinking about the following problem:
Let $n\in\mathbb N$ be odd. If I have a symmetric matrix in $M_n(\mathbb{N})$, i.e. a square symmetric matrix of size $n$, for which each column and each row consists of all numbers between $1$ and $n$, then the diagonal consis... |
H: Integrating $\int{\frac{1}{1+e^{x}}}dx$, Partial Fractions(?)
I need help with this integral:
$$H(x) = \int{\frac{1}{1+e^{x}}}dx$$
It should be easy, but I'm stuck. I thought about using a u-substitution but I didn't get any further. Am I meant to use partial fractions? I'm not yet very comfortable with partial fra... |
H: Why does $7^{2\ln x}\cdot \ln(7) \cdot (2/x)$ equal to $7^{2\ln x}\cdot \ln(49) /x$?
While reviewing, I came upon this problem which has the derivative
$7^{2\ln x}\cdot \ln(7) \cdot (2/x)$
simplified to
$7^{2\ln x}\cdot \ln(49) /x$
How/why is it simplified like that?
AI: The simplification relied on the following f... |
H: Multiples of one number in base-$10$
How can I prove that all the natural numbers has one multiple in base-$10$ such that this numbers is written just with zeros and ones?
For example, let $n=3$ then, exists al least one number, the $111$, such that is in base-10 and is written just using zeros and ones, in this ca... |
H: Counterexample to upper continuity
Let $M$ be a $\sigma$-algebra of subsets of a set $X$ and let $\mu:M\rightarrow[0,\infty)$ be a finitely additive set function. I'm trying to decide if it's automatically true that for all ascending chains $\{A_k\}$ in $M$:
$$\mu\big(\bigcup_{k=1}^{\infty}A_k\big)=\lim_{k\rightar... |
H: What is the relation between graded modules and finitely generated modules
The reason I ask this question is I found two different statements about Hilbert's syzygy theorem from Jacobson's Basic Algebras 2nd and Wikipedia. Please have a look at the following pictures. The first one is from the former and the second... |
H: If $\lim_{t\to\infty}\gamma(t)=p$, then $p$ is a singularity of $\gamma$.
I'm trying to solve this question:
Let $X$ be a vectorial field of class $C^1$ in an open set
$\Delta\subset \mathbb R^n$. Prove if $\gamma(t)$ is a trajectory of
$X$ defined in a maximal interval $(\omega_-,\omega_+)$ with
$\lim_{t\to... |
H: How prove this $\left(\sqrt{a^2+b^4}-a\right)\left(\sqrt{b^2+a^4}-b\right)\le a^2b^2$
let $a,b\in R$,and such that
$$\left(\sqrt{a^2+b^4}-a\right)\left(\sqrt{b^2+a^4}-b\right)\le a^2b^2$$
prove that $$a+b\ge 0$$
I think this is very beatifull problem, have you nice methods? Thank you,
I have see this problem
$$\lef... |
H: Using Spherical coordinates find the volume:
Inside the surfaces $z=x^2+y^2$ and $z=\sqrt{2-x^2-y^2}$
I integrated over the ranges:
$0 \leq \theta \leq 2\pi$
$ 0 \leq \phi \leq \frac{\pi}{2}$
$0 \leq r \leq \sqrt{2}$
I get $\frac{\pi}{2}(4\sqrt{2} -4).$
There answer is the same except a $-\frac{7}{2}$ instead of th... |
H: Relationships of Eigenvalues in Algebraic Closure
Suppose that $k$ is a field, and $A \in M_n(k)$ is a matrix that becomes diagonalizable over $\overline{k}$, the algebraic closure of $k$. Let $\lambda_1, \ldots, \lambda_n$ denote the (not necessarily distinct) eigenvalues of $A$ in $\overline{k}$.
Must each $\la... |
H: $H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true:
$$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$
The cohomological dimension of $I$ is defined to be the supremum of the set of i... |
H: Values of a parameter $x$ in an infinite series that makes it converge
I am required to find the values of $x$ in the following infinite series, which cause the series to converge.
$$\sum_{n=1}^\infty \frac{x^n}{\ln(n+1)}$$
I tried to use the ratio test, and found that the series converges when $x$ is in $(-1,1)$. ... |
H: Helpful to review certain calculus topics before first real analysis course?
This is my first time posting, so I apologize in advance if my question is inappropriate here. I wanted to know if it would be beneficial for me to review certain calculus topics before I take my first real analysis course. I have notice... |
H: Basic independent probability question
This question is a homework question.
The question states:
An airline can seat 100 people. Historically, the airline has noticed that each customer shows up independently and with probability $0.80$. The airline sells 101 tickets. What is the probability that there will be at ... |
H: $A^3 + A = 0$ then $rank (A) = 2$
Let $A$ be a $3\times 3$ non-zero real matrix and satisfies $A^3 + A = 0$. Then prove that $rank (A) = 2$.
As $A$ is satisfying $A^3 + A = 0$, so $0$ is an eigen value of $A$.So $\operatorname{rank} (A) < 3$. So $\operatorname{rank} (A) = 0,1,\text{or}\, 2$. Clearly $\operatorname{... |
H: What is the derivative of $\ln(4^x)$?
What is the derivative of $\ln(4^x)$ (which I believe is also equal to $x\ln4$)?
Is it $\dfrac{1}{x\ln4}$?
AI: Recall that for $a>0$, we have
$$\log(a^b) = b \log(a)$$
Also, note that
$$\dfrac{d}{dx}\left( cx\right) = c$$
I trust you can finish it from here. |
H: Finite Extensions and Roots of Unity
Two questions; the hint I've been provided is that they are, in fact, related.
Prove that a finite extension of $\mathbb{Q}$ contains finitely many roots of unity.
What is the largest (finite) order an element of $GL_{10}(\mathbb{Q})$ can have?
I am not sure how to approach t... |
H: A limit of integrals
Let $f:[0,T]\to\mathbb{R}$ be a Lebesgue integrable function. For each $h>0$ we define the piecewise function $f_h$ by
$$f_h(t)=f(h\left[\frac{t}{h}\right])\quad\mbox{for}\quad t\in[0,T].$$
Can we affirm that
$$\lim_{h\to 0}\int_0^Tf_h(t)dt=\int_0^Tf(t)dt ?$$
*N.B :*$\left[ x\right]$ is the en... |
H: Derivative of Linear Map
I'm reading Allan Pollack's Differential Topology and got stuck on this argument: In the second paragraph of page 9, section 1.2 he said
"Note that if $f:U\to \mathbf{R^m}$ is itself a linear map $L$, then $df_x=L$ for all $x\in U$. In particular, the derivative of the inclusion map of $U$... |
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