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H: Asymptotics of the solution of $x^x=n$
I need to find asymptotics of the solution of the equation
$$
x^x=n
$$
while $n\to\infty$. The only thing I understand is this solution grows very slowly. I can't find $x$ explicitly, I think this is impossible. So what is the necessary trick?
AI: How about
$$
x =
\frac{\oper... |
H: Matrix raised to a matrix: $M^N$, is this possible? with $M,N\in M_n(\Bbb K).$
I was wondering if there is such a valid operation as raising a matrix to the power of a matrix, e.g. vaguely, if $M$ is a matrix, is
$$
M^N
$$
valid, or is there at least something similar? Would it be the components of the matrix rais... |
H: If $f_n\colon [0, 1] \to [0, 1]$ are nondecreasing and $\{f_n\}$ converges pointwise to a continuous $f$, then the convergence is uniform
Suppose that $\{f_n\}$ is a sequence of nondecreasing functions which map the unit interval into itself. Suppose that $$\lim_{n\rightarrow \infty} f_n(x)=f(x)$$ pointwise and tha... |
H: Composition of measurable functions
Consider a locally bounded function $f: X \times W \rightarrow X$, where $X \subseteq \mathbb{R}^n$, $W \subseteq \mathbb{R}^m$, such that
for all $x \in X$ the function $w \mapsto f(x,w)$ is (Borel) measurable;
Consider a locally bounded, (Borel) measurable, function $g: W \righ... |
H: Acceleration along a 2D plane
It's been a long time since I've done trig, and I never new it very well. I have a problem I don't know how to solve.
I have an object on a 2D plane that I want to move. I have this objects x and y coordinates, I also have it's velocity, and the angle of the velocity in radians.
How c... |
H: Difference between power law distribution and exponential decay
This is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks like the one in the Power law article, which is the same... |
H: How to disprove this fallacy that derivatives of $x^2$ and $x+x+x+\dots\quad(x\text{ times})$ are not same.
Possible Duplicate:
Where is the flaw in this argument of a proof that 1=2? (Derivative of repeated addition)
\begin{align*}
x^2 &= \underbrace{x + x + x + \dots + x}_{x \text{ times}}, \\
\therefore \frac... |
H: Is there a function where it actually converges on the real line?
I am trying to come up with a function such that
$\int_{-\infty}^{\infty} f(x) dx$ coverges or $\int_{-a}^{a} g(x) dx$ converges where $g(x)$ is not defined on $x = -a$ or $x = a$ (so both will be improper integrals)
The integrand cannot have comple... |
H: Coin Arrangement Puzzle
Disclaimer: I'm not sure how math related this puzzle is (it could potentially be), but I thought it was an interesting puzzle, and I also don't know how to solve it so I wanted to see if anyone had any ideas.
You have a board divided in quarters and a coin is in each spot. You
do not kn... |
H: What is the inverse of $f(n)=\frac{n^2+n}{2}$?
I'm building an algorithm to determine whether a value is inside a series. To speed it up, I need the inverse function of the following series:
$$1 + 2 + 3+\cdots +n$$
What is the inverse function of $f(n) = \frac{n(n + 1)}{2}$?
AI: I don't understand much of your ques... |
H: Some case when the central limit theorem fails
If I understand correctly, for various versions of the central limit theorems (CLT), when applying to a sequence of random variables, each random variable is required to have finite mean and finite variance, plus some other conditions depending on the version of the CL... |
H: Is it safe to assume that the altitude of a triangle always cuts the base in half
While solving different questions , I realized that whenever I constructed an altitude it always bisected the base in half. From what I deduced from Wikipedia is that this is only true if the triangle is either isosceles or a right tr... |
H: Proof that Gauss-Jordan elimination works
Gauss-Jordan elimination is a technique that can be used to calculate the inverse of matrices (if they are invertible). It can also be used to solve simultaneous linear equations.
However, after a few google searches, I have failed to find a proof that this algorithm works... |
H: How to approximate $\sum_{k=1}^n k!$ using Stirling's formula?
How to find summation of the first $n$ factorials,
$$1! + 2! + \cdots + n!$$
I know there's no direct formula, but how can it be estimated using Stirling's formula?
Another question :
Why can't we find the summation of n! ?
Why there's no direct formul... |
H: For how many integral values of $R$ is $R^4 - 20R^2+ 4$ a prime number?
For how many integral values of $R$ is $R^4 - 20R^2+ 4$ a prime number?
I tried factorizing but couldn't conclude anything concrete.
Factorizing it, gives $(R^2 - 10)^2 - 96$. What should be my approach now?
AI: HINT $$R^4 - 20R^2 + 4 = (R^2... |
H: Why is $\sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|$ true?
Sorry if this is very basic but here's a question.
Let $\mathbf{v}=(v_1,\ldots, v_n)\in k^n$ where $k=\bar{k}$.
Why do we have
$$
\sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|,
$$
where the left-hand side can be thought of as the $2$-n... |
H: Test for convergence $\sum_{n=1}^{\infty}\{(n^3+1)^{1/3} - n\}$
I want to expand and test this $\{(n^3+1)^{1/3} - n\}$ for convergence/divergence.
The edited version is: Test for convergence $\sum_{n=1}^{\infty}\{(n^3+1)^{1/3} - n\}$
AI: By direct inspection, for every pair of real numbers $A$ and $B$,
$$
A^3 - B^3... |
H: Largest modulus for Fermat-type polynomial
Motivated by this question, I wonder:
Given $k\in\mathbb N, k\ge2$, what is the largest $m\in\mathbb N$ such that
$n^k - n$ is divisible by $m$ for all $n\in\mathbb Z$ ?
AI: To find the highest power of a prime $p$ dividing $m$, we find the highest $r$ such that $n\map... |
H: How to obtain the number of digits in n!?
How to obtain the number of digits in $n!$ ?
My approach :
I Used Stirling's formula to find out the approximate value of $n!$
Let the approximate value be $S$
Thus, number of digits in $\ = \left \lfloor \log S \right \rfloor$ + 1
where $\left \lfloor . \right \rfloor$ i... |
H: product of the numbers in each subset is equal
Possible Duplicate:
product of six consecutive integers being a perfect square
Find all positive integers $n$ such that
the set {$n, n + 1, n + 2, n + 3, n + 4, n + 5$} can be partitioned into two subsets
so that the product of the numbers in each subset is e... |
H: Chain rule and gradient
Let $\Gamma \subset \mathbb{R}^2$ be a curve. Define for a smooth function $f$, $$\nabla_\Gamma f = \nabla f - (\nabla f \cdot N)N$$ where $N$ is the unit normal.
Let $X:S \to \Gamma$ be a smooth regular parameterisation with $|\partial_s X(s) | > 0$.
Let $\tilde{f}(s) = f(X(s))$. How do I s... |
H: Asymptotics for sums of the form $\sum \limits_{\substack{1\leq k\leq n \\ (n,k)=1}}f(k)$
How can we find an asymptotic formula for
$$\sum_{\substack{1\leq k\leq n \\ (n,k)=1}}f(k)?$$
Here $f$ is some function and $(n,k)$ is the gcd of $k$ and $n$. I am particularily interested in the case
$$\sum_{\substack{1\leq k... |
H: Show there exists a positive, strictly increasing measurable function
Let $f$ be a nonnegative measurable function on $[0,\infty)$ such that $\int_0^\infty f(x)dx < \infty$ is finite. Show that there is a positive, strictly increasing measurable function $a(x)$ on $[0,\infty)$ with $\lim_{x\to\infty} a(x)=\infty$ a... |
H: Find the indefinite integral of $1/(16x^2+20x+35)$
Here is my steps of finding the integral, the result is wrong but I don't know where I made a mistake or I may used wrong method.
$$
\begin{align*}
\int \frac{dx}{16x^2+20x+35}
&=\frac{1}{16}\int \frac{dx}{x^2+\frac{20}{16}x+\frac{35}{16}} \\
&=\frac{1}{16}\int \fr... |
H: the sum $\sum \limits_{n>1} f(n)/n$ over primes
Let
$$
f(n)=\begin{cases}-1&\text{if $n$ is a prime integer},\\
1&\text{otherwise}.
\end{cases}
$$
Then, does the series
$$
\sum_{n>1} f(n)/n
$$
converge or diverge?
AI: Let $f(n)=-1$ if $n$ is prime, and $f(n)=1$ otherwise. We show that the sum
$$\sum_{n=6}^\infty... |
H: linear-algebra bases polynomials
the answer that was provided was great help thank you, but i had a similar type of question to this once and i used the same method but was marked incorrectly.. it didnt state to compute the inverse and was marked wrong. is there another way of doing it? thanks in advance
Let $B := ... |
H: $T^2=I$ implies that $T$ is a normal operator
I need to show that if $T$ is an operator in an inner product space over the complex field and if $T^2=I$, then $T$ has to be normal.
AI: This is false. Let
$$T = \left[ \begin{array}{cc} 1 & -2 \\\ 0 & -1 \end{array} \right].$$
$T$ has eigenvalues $1, -1$ with eigenve... |
H: Integral of $1/x$ times a decaying function
Let $f:[1,\infty)\to\mathbb{R}$ be a measurable function with $\lim_{t\to\infty} f(t)=0$. I want to show that the function $x\mapsto \int_1^x \frac{f(t)}{t} dt$ is asymptotically sublogarithmic, i.e.
$$\lim_{x\to\infty}\frac{1}{\log x} \int_1^x \frac{f(t)}{t} dt = 0.$$
Al... |
H: Limsup of continuous functions between metric spaces
Let me start with a simple example:
Let $f_n:[0,1]\to[-1,1],x\mapsto \sin 2\pi nx$. For each $x\in[0,1]$, consider the sequence $\lbrace f_n(x):n\ge1\rbrace$ and denote by $F(x)$ the set of points of accumulation of $\lbrace f_n(x):n\ge1\rbrace$. This induces a m... |
H: Find $\lim \limits_{y\rightarrow\infty}\left (\ln^2y\,-2\int_{0}^y\frac{\ln x}{\sqrt{x^2+1}}dx\right)$
I have difficulty with this limit. Where to start?
$$\lim_{y\rightarrow\infty}\left (\ln^2y\,-2\int_{0}^y\frac{\ln x}{\sqrt{x^2+1}}dx\right)$$
AI: By simple integration by parts, we have
$$ \int_{0}^{y} \frac{\log... |
H: Proof that $\frac{(2n)!}{2^n}$ is integer
I am trying to prove that $\dfrac{(2n)!}{2^n}$ is integer. So I have tried it by induction, I have took $n=1$, for which we would have $2/2=1$ is integer. So for $n=k$ it is true, so now comes time to proof it for $k+1$, $(2(n+1))!=(2n+2)!$, which is equal to $$1 \times 2... |
H: Is there a pattern for reducing exponentiation to sigma sums?
The other day I was trying to find a method for cubing numbers similar to one I found for squaring numbers. I found that to find the square of a positive integer n, just sum up the first n odd integers.
$\sum_{t=1}^n 2t-1 = n^2$
Similarly, I found a met... |
H: Steps to get Inverse of Pentagonal
I have solved http://projecteuler.net/problem=44 by getting the inverse equation from Wikipedia http://en.wikipedia.org/wiki/Pentagonal_number:
Pentagonal:
$f(n) = \frac{n(3n - 1)}{2}$
Inverse Pentagonal:
$n = \frac{\sqrt{24f(n) + 1}+1}{6}$
am interested in the steps from Pentagon... |
H: For prime $p>2: 1^23^25^2\cdot\cdot\cdot(p-2)^2 \equiv (-1)^{\frac{p+1}{2}} \pmod p$
Possible Duplicate:
Why is the square of all odds less than an odd prime $p$ congruent to $(-1)^{(p+1)/(2)}\pmod p$?
If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$... |
H: When is Laplace variable $s =j\omega$?
Having an exam next week!
I've searched a lot, couldn't find anything I could understand.
When is the Laplace variable $s$ equal to $j\omega$? Because I know that, by definition, $s = \sigma +j\omega$
Thank you!
AI: $s=\sigma+j\omega $ means that $s$ is a complex variable with... |
H: Order of integration
I am reading a book by L. D. Landau titled Mechanics and there is a "changing order of the integral" step on page 28 that I don't get:
$$\int_0^a\int_0^E \left[{dx_2\over dU}-{dx_1\over dU}\right]{dUdE\over \sqrt{(a-E)(E-U)}}\\=\int_0^a\left[{dx_2\over dU}-{dx_1\over dU}\right] dU \int_U^a{dE... |
H: Sumset using first moment method
This is problem 1.1.6 from the book "Additive Combinatorics" by Tao and Vu.
Suppose A is a subset of an additive group Z. We need to show that there exists a d-element subset of Z, denoted B = { v1, v2, ... vd } with d = O( log |Z|/|A| ) such that
A + FS(B) is of size at least |Z|/2... |
H: Poisson integral in $R^2$
This question arises out of confusion about some aspects of the Poisson integral in $R^2$, simplified here to a function f on the boundary of the unit circle $ \partial C$ and on C,
$$u(r,\phi) = \frac{1}{2\pi} \int_0^{2\pi} \frac{ f (\theta) (1-r^2) d\theta}{1-2r \cos(\theta-\phi)+r^2}$$... |
H: Quadratic minimizing a certain maximum ratio
In connection with a CompSciSE question about largest eigenvalue of PSD matrics, I'd like to know which (nonzero) quadratic polynomial $f(x)$ minimizes the ratio:
$$\frac{\max_{0 \le x \le 0.8} |f(x)|}{|f(1)|}$$
[The goal is making a robust improvement on the simple powe... |
H: Implicit differentiation for $y\cos x = 4x^2 + 3y^2$
I am stuck on doing this implicit differentiation problem below.
$$y \cos x = 4x^2 + 3y^2$$
I am now stuck at the following equality and I don't know how to proceed. Can someone help me?
$$y(−\sin x) + (\cos x)y' = 8x + 6yy' $$
AI: Group the $y'$ terms to one si... |
H: Implicit Differentiation $y''$
I'm trying to find $y''$ by implicit differentiation of this problem: $4x^2 + y^2 = 3$
So far, I was able to get $y'$ which is $\frac{-4x}{y}$
How do I go about getting $y''$? I am kind of lost on that part.
AI: You have $$y'=-\frac{4x}y\;.$$ Differentiate both sides with respect to $... |
H: Prove that $\sum_{n=1}^\infty\frac{\sin(nz)}{2^n}$ is analytic on $\{z\in\mathbb{C}:|\operatorname{Im}(z)|<\log(2)\}$
Prove that $f(z)=\sum_{n=1}^\infty\frac{\sin(nz)}{2^n}$ is analytic on $A=\{z\in\mathbb{C}:|\operatorname{Im}(z)|<\log(2)\}$
I tried expanding $\sin(nz)$ in terms of $e^{inz}$ but that did not he... |
H: Left and Right Vector bundles
I am reading a paper that starts talking about 'left vector bundles' and I'm having trouble figuring out what they mean. The specific setup is as follows:
A quarternionic line bundle $L$ over manifold $M$ is a real smooth rank 4 vector bundle with fibers 1-dimensional quaternionic righ... |
H: Weakly compact operators on $\ell_1$
Is the following assertion true/known?
Let $V$ be a Banach space and let $T\colon \ell_1\to V$ be a bounded linear operator. Is it true that $T$ is not weakly compact if and only if there is a complemented subspace $X$ of $\ell_1$ (thus, isomorphic to $\ell_1$) such that $T|_X\c... |
H: Derivative of square root
What would be the derivative of square roots? For example if I have $2 \sqrt{x}$ or $\sqrt{x}$.
I'm unsure how to find the derivative of these and include them especially in something like implicit.
AI: $\sqrt x=x^{1/2}$, so you just use the power rule: the derivative is $\frac12x^{-1/2}$. |
H: Explanation of passage in Atiyah-MacDonald
On page 52 they write "...By (4.3) we can achieve (i)..."
where (4.3) is the lemma on the previous page that states that if $q_i$ are all $p$-primary then $\bigcap_i q_i$ is $p$-primary and (i) is the property of a minimal primary decomposition that $r(q_i)$ are pairwise d... |
H: Can someone help me with this conic?
$$\frac{(x+1)^2}{16} + \frac{(y-2)^2}{9} = 1.$$
I just started conics, but I thought you would multiply both sides by $16$ and then $9$ and then expand, which would get you $x^2 +y^2+2x-4y+5$.
Both signs are the same, but the foci is supposed to be: $(−1 \pm \sqrt{7},2)$, which ... |
H: Implicit Differentiation of $2\sqrt{x} + \sqrt{y} = 3$
I am trying to implicitly differentiate this problem below but I am stumped because of the square-roots.
$$2\sqrt{x} + \sqrt{y} = 3$$
AI: Our original is:
$$2\sqrt{x} + \sqrt{y} = 3 \tag{1}$$
Taking the derivative with respect to $x$ and recalling that the der... |
H: Prove that $\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$ for positive $p$ and $q$
I'm trying to prove that for $p,q>0$, we have $$\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.$$
The hint given suggests that we express $\Gamma(p)\Gamma(q)$ as a double integral, t... |
H: Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
Suppose $f$ is a real function defined on $\mathbb{R}$ which satisfies
$$\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0.$$
Does this imply that $f$ is continuous?
Source: W. Rudin, Principles of Mathematical Analysis, Chapter 4, exercise... |
H: If $K$ is an extension field of $\mathbb{Q}$ such that $[K:\mathbb{Q}]=2$, prove that $K=\mathbb{Q}(\sqrt{d})$ for some square free integer $d$
I think I have the later parts of this proof worked out pretty well but what's really stumping me is how to go from knowing $[K:\mathbb{Q}]=2$ to knowing that $K = \mathbb{... |
H: codimension of "jumping" of the dimension of fibers
Let $f:X\rightarrow Y$ be a dominant morphism of projective (and smooth if you like) varieties over an algebraically closed field $k$ such that $n=\dim(X)=\dim(Y)$. Then $f$ is proper, so by Chevalley's upper semi-continuity theorem, $\dim(X_y)$ is upper semi-cont... |
H: Finding the equation of the line tangent to $y^2(y^2-4)=x^2(x^2-5)$
I am looking to find the equation of the line tangent to
$$y^2(y^2-4)=x^2(x^2-5)$$
at the point $(0,-2)$.
I have a feeling I need to implicitly differentiate here?
Am I on the right track?
What do I do after finding $y'$ to actually find the sol... |
H: Issues computing a directional derivative
I am trying to do a problem which wants me to compute the directional derivatives at $(0, 0)$ of $$f(x, y) = \frac{xy}{\sqrt{x^2 + y^2}}, \quad f(0, 0) = 0.$$
There are two equations I know for computing the directional derivative, and them seem to be inconsistent for some ... |
H: KL divergence between Bernoulli Distribution with parameter $p$ and Gaussian Distribution
I am trying to find the Kullback–Leibler divergence between Bernoulli Distribution on two points $T, -T$ with parameter $p$ and Gaussian Distribution with mean $\mu$ and variance $\sigma^2$. My attempt is as follows:
Let
$$
b(... |
H: Is there a domain without unity in which every element is a product?
In this answer, Fortuon Paendrag provides an example of a ring without unity such that every element is a product of some two elements. The example has zero divisors. Can a ring without a unity and without non-zero zero divisors satisfy this condi... |
H: Explicitly write down $g\in GL(n,\mathbb{C})$ so that $gAg^{-1}$ is upper triangular, where $A\in M_n(\mathbb{C})$
This is an elementary question which is do-able by hand but I am actually looking for suggestions or book references since I am sure that someone did this somewhere:
suppose
$$
A = \left( \begin{arra... |
H: Some digit summation problems
What is the sum of the digits of all numbers from 1 to 1000000?
In general, what is the sum of all digits between 1 and N?
f(n) is a function counting all the ones that show up in 1, 2, 3, ..., n. so f(1)=1, f(10)=2, f(11)=4 etc. When is the first time
f(n)=n.
So for the first q... |
H: Show $x+\epsilon g(x)$ is 1-1 if $g'$ is bounded and $\epsilon$ is small enough.
Problem: Suppose $g$ is a real function on $\mathbb{R}$ with bounded derivative (say $|g'|<M$). Fix $\epsilon>0$, and define $f(x)=x+\epsilon g(x)$. Prove that $f$ is one-to-one if $\epsilon$ is small enough.
(A set of admissible valu... |
H: Derivative of $f(x)= (\sin x)^{\ln x}$
I am just wondering if i went ahead to solve this correctly?
I am trying to find the derivative of $f(x)= (\sin x)^{\ln x}$
Here is what i got below.
$$f(x)= (\sin x)^{\ln x}$$
$$f'(x)=\ln x(\sin x) \Rightarrow f'(x)=\frac{1}{x}\cdot\sin x + \ln x \cdot \cos x$$
Would that b... |
H: Explanation for why $1\neq 0$ is explicitly mentioned in Chapter 1 of Spivak's Calculus for properties of numbers.
During the first few pages of Spivak's Calculus (Third edition) in chapter 1 it mentions six properties about numbers.
(P1) If $a,b,c$ are any numbers, then $a+(b+c)=(a+b)+c$
(P2) If $a$ is any number... |
H: Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
Problem: Let $f$ be defined for all real $x$, and suppose that
$$|f(x)-f(y)|\le (x-y)^2$$
for all real $x$ and $y$. Prove $f$ is constant.
Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 1.
AI: Here's a proof more elementary.
Let $c=f(0)... |
H: Prove existence of a real root.
Problem: If
$$C_0+\frac{C_1}{2}+\cdots + \frac{C_{n-1}}{n}+\frac{C_n}{n+1} =0,$$
where $C_0,...,C_n$ are real constants, prove that the equation
$$C_0+C_1x+\cdots +C_{n-1}x^{n-1}+C_nx^n=0$$
has at least one real root between $0$ and $1$.
Source: W. Rudin, Principles of Mathematical ... |
H: Derivative goes to $0$, then function goes to $0$.
Problem: Suppose $f$ is defined and differentiable for $x>0$, and $f'(x)\rightarrow 0$ as $x\rightarrow +\infty$. Put $g(x)=f(x+1)-f(x)$. Prove that $g(x)\rightarrow 0$ as $x\rightarrow +\infty$.
Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exe... |
H: How to prove that the space $ \omega_1\times R $ has countable extent?
How to proof that the space $ \omega_1\times R $ has countable extent? The topological space $\omega_1$ is the first uncountable ordinal with order topology.
A space $X$ has countable extent if every uncountable subset of $X$ has a limit point... |
H: How to prove an integer is (not) a power of some other integer?
I assume that is nigh-impossible to prove when the conditions on the integers are very general. However, my algebra professor told me that the following is true:
If $n$ is a composite positive integer, $(n - 1)! + 1$ is not a power of $n$.
I assume t... |
H: How to match a discrete distribution to a continuous distribution in information theoretic sense?
Let
$$
S \sim N(\mu, \sigma^2)
$$
be a normally distributed random variable with known $\mu$ and $\sigma^2$. Suppose, we observe
$$
X = \begin{cases} T & \text{if $S \ge 0$}, \\ -T & \text{if $S<0$},\end{cases}
$$
whe... |
H: What is the meaning of "countable spread"?
I encountered an example that said:
A Tychonoff 2-starcompact space of countable spread which is not $1\frac{1}{2}$-starcompact.
My question is this: What's the meaning of "countable spread" ?
AI: The question was answered in comments. Since we don't like leaving questio... |
H: If $f(0)=0$ and $f'$ is increasing, then $\frac{f(x)}{x}$ is increasing.
Problem: Suppose $f$ is continuous for $x\ge 0$, differentiable for $x>0$, $f(0)=0$, and $f'$ is monotonically increasing.
Define $g(x)=\frac{f(x)}{x}$ for $x>0$. Prove that $g$ is monotonically increasing.
Source: W. Rudin, Principles of Math... |
H: reference in Montgomery/ Zippin
In a paper, the authors use the reference [M–Z, Theorem in 4.13] where [M-Z] denote the book
D. Montgomery and L. Zippin, Topological Transformation Groups. Interscience
Publishers, New York–London, 1955.
Unfortunately, I have no access to the edition of 1955 of this book. I would b... |
H: How to find the eccentricity of this conic?
How to find the eccentricity of this conic?
$$4(2y-x-3)^2-9(2x+y-1)^2=80$$
My approach :
I rearranged the terms and by comparing it with general equation of 2nd degree, I found that its a hyperbola. Since this hyperbola is not in standard form $x^2/a^2-y^2/b^2= 1$, I don'... |
H: Multivariable Calculus - Gradient And Laplacian
I'm currently reading a couple of papers, which uses the following identity, which I can't figure out how to prove or see:
$$\int F(x,t) \Delta_x F(x,t) dx = -1 \int \left| \nabla _x F(x,t) \right| ^2 dx $$
Can someone help me figure out why this equality is true?
Th... |
H: Existence of Consecutive Quadratic residues
For any prime $p\gt 5$,prove that there are consecutive quadratic residues of $p$ and consecutive non-residues as well(excluding $0$).I know that there are equal number of quadratic residues and non-residues(if we exclude $0$), so if there are two consecutive quadratic re... |
H: $|2^x-3^y|=1$ has only three natural pairs as solutions
Consider the equation $$|2^x-3^y|=1$$ in the unknowns $x \in \mathbb{N}$ and $y \in \mathbb{N}$. Is it possible to prove that the only solutions are $(1,1)$, $(2,1)$ and $(3,2)$?
AI: Yes. Levi ben Gerson (1288-1344), also known as Gersonides, proved this. The ... |
H: combinatoric problem related to drug
i want to choose optimal decision from following problem
Imagine having been bitten by an exotic, poisonous snake. Suppose the ER
physician estimates that the probability you will die is $1/3$ unless you receive
effective treatment immediately. At the moment, she can offer you a... |
H: Example of non-decomposable ideal
An ideal $I$ of a commutative unital ring $R$ is called decomposable if it has a primary decomposition.
Can you give an example of an ideal that is not decomposable?
All the examples I can think of are decomposable. Thanks.
AI: Zero ideal in $C[0,1]$ is not decomposable.
More ge... |
H: Cech cohomology of $\mathbb A^2_k\setminus\{0\}$
I'm trying to prove, via the Cech cohomology, that $S=\mathbb A^2_k\setminus\{0\}$ with the induced Zariski topology is not an affine variety. Consider the structure sheaf $\mathcal O_{\mathbb A^2_k}\big|_S:=\mathcal O_S$ (which is quasi coherent), i must show that $... |
H: Is locally free sheaf of finite rank coherent?
Let $\mathcal{F}$ be a locally free sheaf of finite rank of scheme $X$, is $\mathcal{F}$ coherent?
By the definition of locally free sheaf, there exists an open cover {$U_i$} of $X$ such that $\mathcal{F}|_{U_i}$ is isomorphic to the sheaf $\widetilde{\mathcal{O}(U_i)^... |
H: Generalized Laplacian operator?
Suppose a surface $S$ is endowed with a metric given by the matrix $$M=\begin{pmatrix} E&F\\F&G\end{pmatrix}$$
And $f,g$ are scalar functions defined on the surface. What then is the (geometric) significance of the scalar function given by ${1\over \sqrt{\det(M)}}{\partial \over \par... |
H: Introductory (online) texts on Bayesian Network.
I would like to ask for some recommendation of introductory online texts on Bayesian Network.
What I am searching for is some accessible and instructive text not necessarily covering the subject in great depth, but explaining the main ideas. Simply an accessible intr... |
H: cardinality: The cardinality of the set of all relations over the natural numbers.
I have to find the cardinality of the set of all relations over the natural numbers, without any limitations.
It seems to be א, but I can't find a function/other way to prove it.
help anyone?
thanxs.
AI: Recall that $R$ is a relation... |
H: All Bipartite Graphs on n number of vertices
I need to find a list of all connected bipartite graphs on 15 vertices.
http://mapleta.maths.uwa.edu.au/~gordon/remote/graphs/index.html#bips lists all graphs on 14 or fewer number of vertices.
http://oeis.org/A005142 says there are 575 252 112 such graphs.
AI: Try
gen... |
H: How to derive the equation for a bézier curve
So, I remember a while back there was a maths competition and we were given a curve that we needed to write an equation for. I just skipped the question since I didn't even know where to begin. I remember it was one among the last few questions of the paper and it was w... |
H: Are there rngs whose rngs of matrices are commutative?
If $R$ is a unital ring and $M_{2\times 2}(R)$ is a commutative ring, then $R$ is a trivial ring because if $$\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}=\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}\begin{pmatrix}0 & 1 \\ 0 & 1 \end{pmatrix}=\begin{pmatrix}0 & 1 \... |
H: Logical function plotting software in Linux
I want to plot function like this.
if $x > 6000$ plot function $y = 6000+ \frac{x}{15}$ otherwise plot y = $6000$
Suggest me any open source plotting software that has linux version. I have tried KmPlot, KAlgebra, kst they does not seem to has such option.
KAlgebra su... |
H: Calculating Distance - Issue
I am having difficulty with the following problem.
A man covers a distance on a scooter. Had he moved 3Kmph faster he would have taken 40 min less. If he had moved 2kmph slower he would have taken 40 min more. What is the distance (Ans=40)?
Here is what i came up with
$s = (v+3)(t ... |
H: Does this weird sequence have a limit?
Yesterday I was trying to come up with an example of the weirdest sequence I could think of, and I came up with this. I'm not even sure if this could be called a sequence, but here it goes:
We'll define the sequence $a_n$ in the following way. We'll keep a list of previously c... |
H: Open math problems which high school students can understand
I request people to list some moderately and/or very famous open problems which high school students,perhaps with enough contest math background, can understand, classified by categories as on arxiv.org. Please include statement of the theorems,if possibl... |
H: Is this kind of simplicial complex necessarily homotopy equivalent to a wedge of spheres?
Suppose $d \ge 2$ and $S$ is a finite simplicial complex of dimension $2d$, such that
(1) $S$ is simply connected, i.e. $\pi_1 ( S) = 0$, and
(2) all the homology of $S$ is in middle degree, i.e. $\widetilde{H}_i ( S, \mathbb{... |
H: Inspecting Direction field
This question is from Boyce and Diprima, page no 38, question 22.
Draw a direction field for the given differential equation. How do
solutions appear to behave as $t$ becomes large? Does the behavior
depend on the choice of the initial value a? Let $a_o$ be the value
of $a$ for which t... |
H: Find the probability of the given event.
Team $A$ is playing team B in a series of three games. Team $A$'s probability of winning any particular game is nonzero, independent of other games, and is $1.6$ times as large as its probability of winning the series. What is the probability of team $A$ winning the series?
... |
H: Recurrence relation, error in generating function - where did I go wrong?
This is the homework. If you are interested in precise formulation, it is as follows: write a recurrence relation and a generating function that would generate a sequence of trites (elements of a set $\{0, 1, 2\}$), where subsequences of 01 a... |
H: Taking stalk of a product of sheaves
Let $(\mathscr{F}_\alpha)_\alpha$ be a family of sheaves on $X$, and $\prod_\alpha\mathscr{F}_\alpha$ the product sheaf. If $x\in X$, is it true that
$$\left(\prod_\alpha\mathscr{F}_\alpha\right)_x\simeq\prod_\alpha(\mathscr{F}_\alpha)_x \ ?$$
I think $(\oplus_\alpha\mathscr{F}_... |
H: Is there any famous number theory conjecture proven impossible to be find out the truth or false?
Is there any famous number theory conjecture proven undecidable?
Is there any history about it?
i would like to know any number theory conjecture by the types of undecidable.
AI: Perhaps Hilbert's Tenth Problem and Ma... |
H: Is $3^n - 2^n$ composite for all integers $n \geq 6$?
I made a conjecture about the values of n for which $3^n - 2^n$ is not prime, but I didn't succeed in proving the conjecture. My conjecture is the following: "Suppose n is an integer greater than or equal to 6. Then $3^n - 2^n$ is not prime". I tried to prove th... |
H: Find the minimum number of tests
Sorry for the title, but I couldn't think of something else, its not actually homework, but rather a question from a maths question book I am currently stuck on, but still I've tagged it in homework.
The question is as follows:
An institute holds 32 mock tests, students have the ... |
H: Find max vertical distance
What is the maximum vertical distance between the line
$y = x + 20$
and the parabola
$y = x^2$ for $−4 ≤ x ≤ 5?$
What steps do I take to solve this? Do I have to use the distance formula and what do I do with the points it gave me?
If anyone could just bounce me in the right direction ... |
H: Sheaf of rings with vanishing stalk?
How common is that a sheaf of rings has a vanishing stalk? To define the rank of a locally free sheaf of $\mathscr{O}$-modules, for instance, $\mathscr{O}_x=0$ may cause some problem, since the rank of a free $A$-module is not well-defined if $A=0$. It would make life easier if ... |
H: Minimum value of $p^2x + q^2y + r^2z$ if $pqxyz = 54r$
What is the minimum value of $p^2x + q^2y + r^2z$ if $pqxyz = 54r$, where $p, q, r, x, y$ and $z$ are positive real numbers?
I tried applying Cauchy's here but it didn't yield any significant result. Please help.
AI: The problem might be incorrect since as such... |
H: Find $(x, a, b, c)$ if $x! = a! + b! + c!$
Find $(x, a, b, c)$ if $$x! = a! + b! + c!$$
I want to know if there are more solutions to this apart from $(x, a, b, c) = (3, 2, 2, 2)$.
AI: First note that $a,b,c < x$, since $1 \leq n!$. This means that $a,b,c \leq x-1$. This implies that $$x! = a! + b! + c! \leq 3 (x-1... |
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