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How to prove the following theorem?
Theorem:
If the binomial numbers $C(n,p)$ and $C(n,q)$ are equal, with $p$ different from $q$, then $p+q = n$.
($n$, $p$ and $q$ are natural numbers)
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Here's the probability (I think) that a particle in Brownian motion
(w/ standard deviation $\sqrt{t}$) will exceed $m$ between times $t_1$ and $t_2$:
$$\frac1{2\sqrt{2\pi}}\int_{-\infty }^m \frac1{\sqrt{t_1}}e^{-\frac{x^2}{2t_1}}\left(1+\mathrm{erf}\left(\frac{m-x}{2\sqrt{t_2-t_1}}\right)\right)\mathrm{d}x$$
or, in M... |
I need to re-arrange the following equation to make $d$ the subject [i.e., solve for $d$]:
$$a = \sqrt{\frac{d}{be}}.$$
The square root part is really confusing me, any help would be much appreciated.
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For a university project, I am implementing an automated regression analysis tool.
However, I have very little background in statistics.
So what books / articles / material would you suggest I could use to brush up on this topic, based on your experiences?
Thanks
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Question:
Let $a$ be a quadratic residue to a prime modulus $p$. Prove $a^{(p-1)/2} \equiv 1 \pmod{p}$.
My attempt at a solution:
\begin{align*}
&a\text{ is a quadratic residue}\\\
&\Longrightarrow a\text{ is a residue class of $p$ which has even index $c$ relative to a primitive root $g$}\\\
&\Longrightarrow a ... |
My algebra is so rusty, this should be an easy one!
The area of a rectangle is
$$A = W \times H$$
If you're given $A$ (say $150$) and a ratio of $W:H$ (say $3:2$) how can you calculate $W$ and $H$?
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I'm writing a program to solve this proof, but I don't know how to go about solving it.
If anyone has some insight it would be great help. Thanks
For every odd integer $n$, $3 \leq n \leq 199$, there exists an integer $m \geq 0$, and a prime number $p$, such that $n = 2^{m} + p$.
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I learned of the plate trick via Wikipedia, which states that this is a demonstration of the fact that SU(2) double-covers SO(3). It also offers a link to an animation of the "belt trick" which is apparently equivalent to the plate trick. Since I've thought most about the belt version, I'll phrase my question in terms ... |
Let a be a natural number.
Suppose
$a^4 \not\equiv 1 \pmod {29}$
$a^7 \not\equiv 1 \pmod {29}$
$a^{14} \not\equiv 1 \pmod {29}$
Calculate the order of a modulo 29.
Using Euler's Criterion, I have $a^{14} = a^{(29-1)/2} \not\equiv 1 \pmod {29}$.
How would I go about solving this problem, I have not seen this kind o... |
I'm not sure if this is the proper place to ask this, but I am looking for an English translation of Voronoy's paper on the generalized Pell's equation. It might be his doctoral dissertation; I'm not sure. I believe the title is "On a generalization of the Algorithm of Continued Fractions".
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This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and the derivative of the quotient of two functions is the quotient of their derivatives.
This might be an ill formed ques... |
So, we are considering the subset
$$ S = \{(x, y) \in \mathbb{R^2} | (x \text{ and } y \in \mathbb{Q}) \text{ or } (x \text{ and } y \notin \mathbb{Q})\} $$
And consider its complement $$ T = \mathbb{R^2} \backslash S $$
The set T is disconnected, actually I am fairly certain it is totally disconnected. I am just ha... |
Suppose you want to find $E(X)$ for a non-negative random variable $X$. So you can use $\int_{0}^{\infty} [1-F(x)] \ dx$. To get $1-F(x)$, all you have to do is integrate as follows: $\int_{x}^{\infty} f(t) \ dt$? In other words, instead of integrating from $t=0$ to $t= x$, we can just integrate from $t = x$ to $t = \i... |
I heard about it sometime somewhere and want to read about it now, but I can't recall what the name is:
Start with $a_1 = \ldots =a_n=1$. Choose a number between 1 and $n$ with probability $a_i/(a_1+ \ldots + a_n)$ to choose $i$. If $i_0$ is the number chosen, increase $a_i$ by 1 and now choose another number and so on... |
Is there a simple algorithm for efficient decomposition on this special kind of graph?
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I'm looking for an approximate algorithm to solve this problem:
Given set of n points on 2D euclidean space, create a net of rectilinear edges, so that:
Every two points are connected with shortest edge, and
sum of all edges is minimal (minimized).
That's clearly a NP-hard problem and it's similar to Steiner tree pro... |
Is it known whether every finite abelian group is isomorphic to the ideal class group of the ring of integers in some number field? If so, is it still true if we consider only imaginary quadratic fields?
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In Chapter 1 of Polynomials by Victor Prasolov, Springer, 2001, the following theorem is proved. (p.3)
Theorem 1.1.4 (Ostrovsky). Let
$f(x)=x^{n}-b_{1}x^{n-1}-\cdots -b_{n}$,
where all the numbers $b_{i}$ are non-negative and at least one of them
is nonzero. If the greatest common
divisor of the indices of the positiv... |
I want to motivate the theory of $C_0$-semigroups to someone, and the following question was asked:
What is an example of a non-separable linear PDE?
Preferably a simple homogeneous one.
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Suppose $0.6$ of students submit assigned homework on time. In a typical class of $40$ students a professor has assigned homework. If a student submits homework on time, he gets $5$ points. Otherwise he gets $0$ points. Find the variance of the total number of points by all students.
So this is a binomial distribution... |
I have an interval $[2,t]$ containing some number of primes. I now want to divide this interval into two intervals $a=[2,m]$ and $b=]m,t]$ such that the number of primes in $a$ and $b$ is almost the same.
I am aware of the approximation $\pi(t)=\frac{t}{2\ln(t)}$, but can not seem to get any good and usable result out ... |
i have a question to special cases of Sasaki-manifolds. Let $(M, g, \xi, \eta, \Phi)$ a Sasaki-manifold.
In special case maybe $M=S^{2m+1} \cong \mathbb{C}^{m+1}$. This is a Sasaki manifold. But what is $\Phi$?
Analogous: Let $M=\mathbb{R}^{2m+1}$, it is also Sasaki, but what is $\xi$ or $\Phi$?
Thanks and best regards... |
I recently read an article in which the author describes how to find some functions that obey to certain recursion relationships. If we want to find a function that satisfies, for example,
$f(x^a) = x \cdot f(x)$, then the author explains we can proceed as follows:
Consider $ x > a > 1$ . Then $f(x) = f((x^{(1/a)})^a)... |
$M$, $N$ are sets, $f: M \rightarrow N$ is a function, $R \subseteq M\times M$ is an equivalence relation on $M$, $\pi: M \rightarrow M/R$ is the canonical projection (i.e. $\pi(x)=[x]_R$), and $\ker f \subseteq M\times M$ is defined as $\{(x,y) \in M \times M: f(x)=f(y)\}$.
Show that $R \subseteq \ker f \Leftrightarro... |
I was wondering if there are any general results to this problem. Explicitly, given an $n$-cube and $n$ distinct colors, in how many ways can we paint the faces of the cube. Here we consider two colorings the same if they can be obtained from one another by rotations.
For $n=3$ this number is $57$ by Burnside's lemma.... |
two integrals that got my attention because I really don't know how to solve them. They are a solution to the CDW equation below critical temperature of a 1D strongly correlated electron-phonon system. The second one is used in the theory of superconductivity, while the first is a more complex variation in lower dimen... |
Let $A$ be a commutative ring with identity. If $A$ has finite number of prime ideals $p_1,...p_n$ and moreover $\prod_{i=1}^n p_i^{k_i} = 0$ for some $k_i$. Are the prime ideals necessarily maximal?
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The rank of a finitely-generated abelian group is the size of the maximal $\mathbb{Z}$-linearly-independent subset; loosely this means the number of distinct copies of $\mathbb{Z}$ contained as direct summands. I have come across a situation where a notion somewhat like rank may be useful for finite abelian groups (whi... |
Can I draw a net for a Steinmetz solid from two cylinders with a compass?
That is, can we flatten the net? I often make a model using paper and a compass-- it looks about right... is it really a valid method of construction?
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Let $$ \theta(x) = \sum\limits_{p \leq x} \log{p} \quad \ ; \ \psi(x)=\sum\limits_{n=1}^{\infty} \theta(x^{1/n})$$ then how does one prove $$e^{\psi(x)}= \text{lcm}[ 1,2,\cdots , \lfloor{x\rfloor}]$$
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When I was trying to find closed-form representations for odd zeta-values, I used
$$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$
and rearranged it to
$$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1}... |
Alright, I'm trying to figure out how to calculate a critical value using t-distribution in Microsoft Excel... ex. a one-tailed area of 0.05 with 39 degrees of freedom: t=1.685
I know the answer, but how do I get this? I've tried TDIST() TINV() and TTEST() but they all give me different answers. This web calculator: h... |
Suppose $x \geq 0$, $y \geq 0$ and $0<p<1$. Why is the following inequality true?
$|x^{p}-y^{p}| \leq |x-y|^p$
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A vector $v \in \mathbb{Z}^n$ is primitive if there does not exist some vector $v' \in \mathbb{Z}^n$ and some $k \in \mathbb{Z}$ such that $v = k v'$ and $k \geq 2$.
For a paper I'm writing right now, I'd like to know that a "random" vector in $\mathbb{Z}^n$ is primitive. Let me make this precise.
Let $\|\cdot\|_{1}$ ... |
In 'An Introduction to Probability Theory and Applications' by W. Feller I encountered this apparently innocuous problem.
A throw of twelve dice can result in
$6^{12}$ different outcomes, to all of
which we attribute equal
probabilities. The event that each
face appears twice can occur in as
many ways as twe... |
Can we determine if a binomial coefficient $C(n,p)$ is even or odd, without calculating its value?
($p\lt n$, $p$ and $n$ are positive integers)
|
Which of the following mappings are linear?
(a)$\varphi : \mathbb{R} \rightarrow \mathbb{R}^{2}, \varphi(x) = (1 - x, (1 - x)^{2}),$
(b)$\varphi : \mathcal{F}(\mathbb{R},\mathbb{R}) \rightarrow \mathbb{R}, \varphi(f) = f(1),$
(c)$\varphi : \mathbb{R}[t] \rightarrow \mathbb{R}[t], \varphi(f) = t \cdot f', $
(d)Let $V$ b... |
Assume $M$ is a (finitely generated) $A$-module such that $\wedge^n M$ is free of rank $1$ for some $n \geq 1$. Does it follow that $M$ is free of rank $n$? Or at least locally free of rank $n$?
In general, is there a way of characterizing local free modules via exterior powers and tensor products?
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I know it is unknown if there are infinitely many primes of the form $n^2+1$. Is it known if there is a positive integer $k$ such that $|\{n\in\mathbb{Z}:n^2+1 \text{ has at most k prime factors}\}|=\infty$?
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Let ${f_{n}}$ be a sequence in $L^{2}(X,\mu)$ such that $||f_{n}||_{2} \rightarrow 0$ as $n \rightarrow \infty$. How to show that:
$\displaystyle \lim_{n \to \infty} \int_{X} |f_n(x)| \log(1+|f_{n}(x)|) d\mu = 0$
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If $f \in L(0,1)$ show that $x^{n} f(x) \in L(0,1)$ for $n \in \mathbb{N}$ and that $\displaystyle \int_{0}^{1} x^{n} f(x) dx \rightarrow 0$ as $n \to \infty$.
Is the following attempt correct?
Since $0<x<1$ then $0<x^{n}<1$ also $x^{n}$ is measurable (being cts) and $f$ is measurable because it is in $L(0,1)$ so $x^{n... |
It is a well-known fact that
$f\colon a \mapsto \left(\begin{array}{cc}
1 & 1\\
0 & 1\\
\end{array}\right), ~ b \mapsto \left(\begin{array}{cc}
1 & 0\\
1 & 1\\
\end{array}\right)$
is a monomorphism from the free monoid generated by $a$ and $b$ to the matrix monoid $\mathbb{Z}^{2\times 2}$.
Is there an efficient... |
Stuck with this problem from Zgymund's book.
Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $\|f_{n}\|_{p} \leq M < \infty$. Prove that:
$\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}$ such that $\dfrac{1}{p} + \dfrac... |
For a 2D Normal distribution $N(0, \left[ \begin{array}{cc} 1 & -1/4 \\ -1/4 & 1 \end{array} \right])$, I am now trying to maximize its density function over $\{ x\geq 10, y \geq 10 \}$.
My intuition is that the optimal solution is $(10,10)$. My problem is that I am not sure which way to find the optimal solution.
Tha... |
I am trying to do the following stochastic integral
$$
\int_0^T \mu(B_s) dB_s - \frac{\int_0^T (\mu(B_s))^2 ds}{2}
$$
where $ \{ B_t \}$ is a standard Brownian motion, and $\mu(x) = \frac{\beta}{2x} (x^2 - \frac{4\gamma}{\sigma^2} - \frac{1}{ \beta})$.
I currently only know how to integrate when the integrands are c... |
For every odd integer $n$, $3\leq n\leq 199$, there exists an integer $m\geq 0$, and a prime number $p$, such that $n=2^m+p$.
I created a software program in Java to solve this problem. The statement is actually false. The first odd number it fails at is 127. I really appreciate everyone's input on this.
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What is the easiest way to prove (if possible, without using ordinals etc. as my current math understanding of set theory counts only cardinals, and countable & uncountable sets) that the number of cardinalities that exists is not countable (that is, can't be put into bijection with $\mathbb{N}$)?
What exactly does i... |
My PhD supervisor called the following result something along the lines: "The theorem in the theory of Latin squares with the most incorrect proofs" (then, to my embarrassment, proceeded to point out an error in my "proof" at the time). I'll pose proving it as a problem.
Definition: A $k \times n$ *Latin rectangle* i... |
How many committees of $5$ can be chosen from $10$ people if each committee includes Alice and
excludes Bob?
How many committees of $5$ can be chosen from $12$ people if each committee
(i) includes Alice;
(ii) excludes Bob;
(iii) includes Alice and excludes Bob;
(iv) includes at least one of either Alice or Bob
Please ... |
Let $\{f_n\}$ be a sequence of smooth functions which converges to a function $f$. If the convergence is not uniform at a point $a$ the $f$ is discontinuous at $a$.
Is there any different type of convergence where if it happens at $a$ then $f$ is continuous at $a$ but is not differentiable.
EDIT
Let $\{f_n\}$ be a seq... |
I was reading Martin Gardner's Mathematical Games column on the Ulam spiral which appeared in the March 1964 issue of Scientific American. (The spiral actually featured on the cover of that issue.) Gardner makes the following statement:
The grid on the cover suggests that throughout the entire number series express... |
How to find the domain of the function $\sqrt{ \log_{\frac{1}{2}} x}$ ?
|
I've got a Python script that grinds out Amicable Pairs by trying out each integer: $i, i+1, i+2,\ldots$ . I've now found all APs with $1 \leq n \leq 65,000,000$. At this point I'm finding that it takes $24$ minutes per each additional million. To get into much larger $n$ I'll need to find a way to filter out substanti... |
Just today I had a bet with my friend over the following problem:
How many winning configurations can you have in a nxn Tic-Tac-Toe game where players win if a they get n/2 in either a row or column, consecutively.
n is even.
For example, in a 4x4 game, players win if a they get 2 of their symbols in either a row or co... |
I've been having some rather morbid thoughts lately, so, naturally, I decided to share:
Suppose I have a deadly disease, which has a chance of killing me every day that it is left uncured. Also, suppose that there's this operation I can go through, but it is not without its risks: If it is successful, I won't have to w... |
Let $S^1$ the unit circle in $\mathbb{R}^2$ and
$$d: S^1\times S^1\to\mathbb{R}$$
$$d(\theta_1,\theta_2) = \left\{
\begin{array}{ll}
|\theta_1-\theta_2| & \mbox{if } |\theta_1-\theta_2|\le \pi \\
2\pi-|\theta_1-\theta_2| & \mbox{else}
\end{array}
\right.$$
I'm trying to prove that this f... |
It is natural to understand the need for scalars (numbers), but why did we invent vectors? Who invented it and for what?
EDIT: As George Lowther pointed out, the problem is too broad; I added the following questions as concrete supplements.
It's easy for humans to understand the law of addition of scalar numbers, but ... |
$\lim_{x\to\infty} \frac{e^x}{x^n}$
n is any natural number.
Using L'hopital doesn't make much sense to me. I did find this in the book:
"In a struggle between a power and an exp, the exp wins."
Can I refer that line as an answer? If the fraction would have been flipped, then the limit would be zero. But in this cas... |
How do I calculate the height of a regular tetrahedron having side length $1$ ?
Just to be completely clear, by height I mean if you placed the shape on a table, how high up would the highest point be from the table?
|
Let $f:\mathbb{R}\to \mathbb{C}$ be a Lebesgue measurable function. For each $z\in \mathbb{C}$ let $\text{Arg} (z)$ be the principal argument of $z$ (define it to be $0$ if $z=0$). Define $g(x)= \sqrt{|f(x)|}\exp(\frac{1}{2} i \text{Arg} (f(x))$. Then $g(x)$ is measurable and $(g(x))^2=f(x)$.
I am sure the above statem... |
I greatly enjoyed the Lights Out game described here (I am sorry I had to link to an older page because some wikidiot keeps deleting most of the page).
Its mathematical analysis is here (it's just linear algebra)
Now I just discovered a hexagonal version:
http://cwynn.com/games/lightsoutmobile.htm
(and hopefully ... |
I have two random variables $X$ and $Y$, both receiving values between 0 and 1.
I know that $E[X - Y] \ge 0$.
Can I get any inequality of the form:
$P(X - Y \ge \delta) \le F(\delta,X,Y)$
where $F(\delta,X,Y)$ is a (reasonable) function of $\delta$ and $E[X-Y]$?
Markov inequality would be good here, for example, by se... |
I've read about many attempts to find patterns in prime numbers. First, is there a mathematical way to prove there is not a pattern to prime numbers? Since there are ways to check if a number is prime or not, can these methods be combined into some large function that produces prime numbers? And in case it isn't obv... |
Is $L^2(\mathbb{R})$ a Banach algebra, with convolution?
I am pretty sure the answer is no, because I think that
$f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I can't seem to find a counterexample, or a proof that this is not the case, using Fourier Transform for example.
Can someone... |
$g(x) =\frac{3-5x}{5-x}$
I know that I am retarded for not being able to do this on my own. Algebraically you just have to solve for x. Even so, I can't do it.
|
Suppose $R=k[[x_1,...,x_n]]$ is a formal power series ring over the field $k$, what can we say about the structure of $R/p$ if $p$ is a prime ideal of $R$ such that dim$(R/p)=1$. In particular, are these subrings of power series rings in one variable?
Motivation: Given a subring of a power series ring in one variable w... |
I have a probability space
$\omega = 2^{\{1,\ldots,n\}}$
$\sigma$-algebra $2^\omega$
and $P(\{s\})=(p^{|s|})*(1-p)^{(n-|s|)}$
I assume that $n=2k$,$k$ natural number
I need to find a random variable that will distribute like $\mathrm{Bin}(k,p^2)$
Can you help me with this please?
Thanks.
benny
|
(a)Is there a linear mapping $f : \mathbb{R}^{3} \rightarrow \mathbb{R}^3$ with
$ f(1,1,1)=(3,2,7), f(0,2,1)=(2,1,-1), f(2,0,1)=(1,0,0)$?
(b)Is there a linear mapping $f :\mathbb{R}^{3} \rightarrow \mathbb{R}^3$ with
$ f(0,2,-1)=(3,-1,3), f(1,0,2)=(0,-2,1), f(-2,2,-5)=(3,3,1)$?
Substantiate your statements and if it e... |
Let $a$ and $b$ be real numbers and $p>0$. What is the best way to prove that $|a+b|^p<2^p(|a|^p+|b|^p)$?
|
The task:
$\lim_{x\to\infty} \sqrt{x^2+1} -x $
I've multiplied with the conjugate expression ($\sqrt{x^2+1} +x$), then I get this
$\lim_{x\to\infty} \frac{1}{\sqrt{x^2+1} +x} $
Is this correct so far? And what would be the next step?
|
I have a question about solving a system of geometric equations. I really hope someone here can help me, it's been several months since I try to solve the problem but without success. As I am not an expert in maths, I count on your help. Thank you in advance.
Problem:
I have three circles $C_1, C_2, C_3$. For example (... |
Let $\ell^n_p$ denote $\mathbb{R}^n$ with the $p$-norm; assume $n>1$. Now it's well-known that for $p\ne q$, $\ell^n_p$ and $\ell^n_q$ are never isometric unless $n=2$ and $\{p,q\}=\{1,\infty\}$, but I realized recently I didn't actually know how to prove this.
I can prove most of it - the only case I still need to ru... |
A diagonal of a Latin square is a selection of n entries in which no two entries occur in the same row or column. For example: the entries marked with an asterisk below form a diagonal.
1 2* 3 4
2 3 4 1*
3 4 1* 2
4* 1 2 3
Theorem: Every Latin square contains a diagonal in which no symbol appears thrice (or ... |
Could someone provide me with a good explanation of why $0^0=1$?
My train of thought:
$$x>0\\
0^x=0^{x-0}=\frac{0^x}{0^0}$$
so
$$0^0=\frac{0^x}{0^x}=\,?$$
Possible answers:
$0^0\cdot0^x=1\cdot0^0$, so $0^0=1$
$0^0=\frac{0^x}{0^x}=\frac00$, which is undefined
PS. I've read the explanation on mathforum.org, but it isn'... |
I'm familiar with the "standard" proof using Galois theory that there is no general formula for solving an equation of fifth (or higher) degree using radicals (i.e. arithmetic and root-taking). However, now I'm wondering if other proofs of different nature were found (in particular ones relying on analysis rather than ... |
Why does $x$ divided by zero not equal $x$? After all, $x$ is not being divided by anything.
|
A homework problem states:
An aquarium 5 ft long, 2ft wide, and 3ft deep is full of water. Part C: find the hydrostatic force on one end of the aquarium.
Questions (I've completed part a and b. B asked for the force on the bottom which was 1875 lbs.:
What exactly does one end of the aquarium mean?
Is there a standard ... |
Show that the set of the R-homomorphism $A\rightarrow B$ with factorization through an injective module is an subgroup of $Hom_{R}(A,B)$.
It's a question from my last test in abstract algebra, that I can't solve.
|
Let $F \to E \to B$ be a fibration with $B$ simply connected (more generally, such that $\pi_1(B)$ acts trivially on the homology of $F$). Then there is a Serre spectral sequence $H_p(B, H_q(F)) \to H_{p+q}(E)$. One can do the same for singular cohomology. However, for reasonable spaces (specifically, locally contract... |
I am looking to expand my knowledge on set theory (which is pretty poor right now -- basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to your mind that would be useful for an undergrad math student who hasn't taken a set theory course yet?
Thanks a lot for... |
Let ( L, ≤ ) be a lattice, x, y, z ∈ L. Prove that : $x ∧ y ≤ x$ and $x ∧ y ≤ y$
Here is how I proceeded to solve it:
Approach 1 : Proceeding by definition:
$x ∧ y ∈$ { x, y }
Therefore, x ∧ y either x or y
If $x ∧ y = x, x ≤ x$ is true by reflexive property and $x ≤ y$ as L is a poset
If $x ∧ y = y, y ≤ y$ is true by ... |
Added later: Proof now in progress.
Let $(L, \leq)$ be a lattice, $x, y, z \in L$. Prove that $x ≤ y$ and $x ≤ z \iff x ≤ y ∧ z$.
I proceed to prove as follows:
The statement can be split into two implications :
(i) $x ≤ y$ and $x ≤ z \implies x ≤ y ∧ z$
(ii) $x ≤ y ∧ z \implies x ≤ y$ and $x ≤ z$
(i) Approach 1 : Pro... |
I read that one can prove by AM-GM-inequality that for all $a,b,c\in\mathbb{R}_+$ we have that
$$11(a^6 + b^6 + c^6) + 40abc(ab^2 + bc^2 + ca^2) \ge 51abc(a^2b + b^2c + c^2a)$$
How this can be done? Is it possible to prove stronger inequalities like
$$10(a^6 + b^6 + c^6) + 41abc(ab^2 + bc^2 + ca^2) \ge 51abc(a^2b + b^... |
Let $r \in \mathbb{N}$. Observe the equivalence relation $\sim \subseteq \mathbb{Z} \times \mathbb{Z}$ defined as $x \sim y :\Leftrightarrow (\exists k \in \mathbb{Z})(y=x+kr)$. Show that an operation $'+'$ on $\mathbb{Z}/\sim$ given by $[x]_\sim + [y]_\sim := [x+y]_\sim$ is well-defined.
My attempt at a proof:
Let $x... |
What is the :
most useful
prettiest
way to visualise extra dimensions in shapes and charts?
|
$$T(n)=1+2\sum_{i=1}^{n-1}T(i) , \quad n > 1$$
$$T(1)=1$$
any hint or how to solve?
|
While experimenting with certain sums, I came to the following sum:
$$S_n = \sum_{k=0}^n \frac{n-k+1}{(n+1) (n+2) (n-k+2)}.$$
After rewriting the summand as
$$\frac{n-k+1}{(n+1) (n+2) (n-k+2)} = \frac{n^{\underline{k}}}{(n+2)^{\underline{k}}(n-k+2)^2}$$
and then feeding the sum to Mathematica, i obtained
$$S_n = \fra... |
I should prove or give a counterexample for:
$\partial (\bigcup_{i \in \mathbb{N}} A_i) \subset \bigcup_{i \in \mathbb{N}} (\partial A_i)$
Where $\partial$ is the boundary and (X,d) is a metric space with $A_{i} \subset X $ for each $i \in \mathbb{N}$
I cannot make up a counterexample for finite sets $A_{i}$ but I th... |
Let $\Omega \subset R^n$ be open, and let us equip $\mathcal E = C^\infty$ with the locally convex topology induced by the following family of semi-norms:
For $K \subset \Omega$ and $\alpha, \beta \in \mathbb N _ 0 ^n$, let $|f|_{K,\alpha,\beta} = sup_{x \in K} | \partial_x^{\alpha} f(x) |$.
It is easy to see that comp... |
How to find center of ellipse from two points (these are just points on the ellipse, not related to foci), and two radii ($r_x$ and $r_y$, from standard definition of the ellipse $\frac{x^2}{r_x^2} + \frac{y^2}{r_y^2}=1$) of that ellipse? (there will be two centers, I assume)
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Investigating this sequence led me to the following conclusions:
if $a_{1}\geq 2$ then $a_{n}\geq 2$ and $a_{n}$ is decreasing, thus it converges and is bounded.
if $-1\lt a_{1} \lt 2$ then $a_{n}$ is increasing, thus it converges and is bounded.
if $a_{1}=-1$ or $a_{1}=2$ the sequence is constant and is bounded.
if $... |
I am a student taking engineering course and wish to learn more about Fourier Transforms. It seems very useful. Would highly appreciate it if anyone could advise me where to start.
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My consideration might be total nonsense (as a high school student, I lack the mathematical knowledge to really check my idea), but I was just wondering whether one could find a continuous generalization of the Taylor series comparable to the continuous Fourier transform.
Just like the Fourier transform decomposes a fu... |
I am trying to simplify this expression by as usual the expansion way,
$$\biggl(\frac{ 1+x^2}{1-x^2}\biggr)^2 = \frac{1}{1-y^2}$$
After some steps I am getting:
$$4x^2 - y^2 - 2x^2y^2 - x^4y^2 = 0$$
The answer suggested in my module is $x^2y = 2x - y$
For the answer to be correct I think what I should get is
$$4x^2... |
How to prove :
$$ \ln \biggl(\frac{x+1}{x-1}\biggr) = 2\biggl[\frac{1}{x} + \frac{1}{3x^3} + \frac{1}{5x^5} + \cdots \biggr]$$ where $|x| \gt 1$
I am not able to get how to proceed on this one ?
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Let $f_n,f \in L^p$ and $f_n \to f$ uniformly. Further let $g \in L^q$. Now, clearly $\int (f_n - f) g$ is defined. How do I show that
$$\int |f_n - f||g|$$
can be made arbitrary small if I choose $n$ large enough? If we just naively apply Hölder we will not be able to make it small but just finite. Further, we canno... |
There are 1000 light bulbs and 1000 tutors. All light bulbs are off. Tutor 1 goes flipping light bulb 1,2,3,4... tutor 2 then flips 2,4,6,8... tutor 3 then 3,6,9...etc until all 1000 tutors have done this. What is the status of light bulb 25? 93? 576? Is there a general solution to find out the status of any light ... |
Let $\mathbb{Z}_p$ be te ring of p-adic integers and let $T$,$T'$ be two free $\mathbb{Z}_p$-module with a continuous action of $G_{\mathbb{Q}_p}$ (the absolute Galois group of $\mathbb{Q}_p$).
It is not hard to see that if $T$ and $T'$ are of rank one, then there is a non zero $G_{\mathbb{Q}_p}$-equivariant map $T \to... |
Let's say you have a single-lane road (single in this case meaning single lane in each direction, or what you could also call two-lane). Let's say you have a random number of vehicles on a given 10-mile stretch of road. The speed limit is 45 mph, but some people go faster and some go slower.
Is there any math that desc... |
Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be bijective map with following properties:
1) $f|_{\mathbb{Q}^2}=id$;
2) Image of any line under map $f$ is again a line.
Is it right that $f=id$?
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