Inequality Definition and 1000 Threads

given $|y+3|\le 4$ we don't know if y is plus or negative so $y+3\le 4 \Rightarrow y\le 1$ and $-(y+3)\le 4$ reverse the inequality $ y+3 \ge -4$ then isolate y $y \ge -7$ the interval is $-7 \le y \le 1$

I browsed the net and found : https://arxiv.org/abs/quant-ph/0408127 It is said the value of Bell's operator depends on the speed, so how can it be Lorentz invariant ?

Z can be any point on the argand diagram so if z molous is less than 2 , is that somehow giving us the distance from origin? But how i assumed mod sign only makes things positive therefore its not sqrt( (x+yi)^2 ) = distance ??

Prove: $\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\geq\frac{27}{2(a+b+c)^2}$ where a,b,c are positives

Summary: Given three points on a positive definite quadratic line, I need to prove that the middle point is never higher than at least one of the other two. I am struggling to write a proof down for something. It's obvious when looking at it graphically, but I don't know how to write the...

Given A>0, find a B>0 such that : for all x,y,a,b : if 0<x<1,0<y<1,0<a<1,0<b<1,and |x-a|<B,|y-b|<B,then $$|x^2y-a^2b|<A$$

Given 0<a<π/2 , b>0 find a c>0 such that : for all ,x : if 0<x<π/2 and |x-a|<c ,then $$|(\sqrt sinx +1)^2-(\sqrt sin a +1)^2|<b$$

##7x+12≥3x## ##7x−3x≥−12## ##4x≥−12## → ##x≥−3## or ## 12/x≥-4## but by substituting say## x=−4## we see that it also satisfies the equation implying that my solution may not be correct.

I have to prove that, for a non-increasing function ##g(x)## the following inequality is true: $$k^2\int_k^\infty g(x) dx\leq\frac{4}{9}\int_0^{\infty}x^2g(x) dx$$ This exercise is from the book Mathematical methods of statistics by Harald Cramer, ex. 4 pg 256 Following the instructions of the...

Concerning p.198 of Bell's famous 1964 paper http://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf How does TI explain Bell's move from the first equation to the second equation? Under TI, what is the physical significance of this move? Thank you.

Ok a student sent this to me yesterday so want to answer without too many steps I think the first thing to do is multiply every term by 12 $2(x+1)<12x-3(3x-2)$ Expanding $2x+2<12x-9x+6$

Hello, I'm working on solving linear equalities (with equations) and can anyone help with the below question. I know the answer is -16, but I can't figure out the steps that gets it to this. 4x-12≤6x+20 Once I've evened out the x's on both sides and got this to 2x, I'm then left with -12...

Hi, I am studying a paper by Yann Bugeaud: http://irma.math.unistra.fr/~bugeaud/travaux/ConfMumbaidef.pdf on page 13 there is an inequality (16) as given below- which is obtained from - , on page 12. How the inequality (16) is derived? I couldn't figure it out. However one of my...

I know there are numerous threads on this and I have read quite a bit such as EPR and Bell's inequality. I hope I can ask this the right way: A particle has 0 spin and gives off two children particles with spins -1/2 and +1/2 (we don't know which is which yet, or they have to end up this way...

Reading The Theoretical Minimum by Susskind and Friedman. They state the following... $$\left|X\right|=\sqrt {\langle X|X \rangle}\\ \left|Y\right|=\sqrt {\langle Y|Y \rangle}\\ \left|X+Y\right|=\sqrt {\left({\left<X\right|+\left<Y\right|}\right)\left({\left|X\right>+\left|Y\right>}\right)}$$...

The error ##e_{n}(y)## for ##\frac{1}{1-y}## is given by ##\frac{1}{(1-c)^{n+2}}y^{n+1}##. It follows that ##\frac{1}{1+y^2}=t_n(-y^2)+e_n(-y^2)## where ##t_n(y)## is the Taylor polynomial of ##\frac{1}{1-y}##. Taking the definite integral from 0 to ##x## on both sides yields that...

I do not know how to start thinking about part a and then got even more confused when I saw the answer be: P<x^2-5x. I ask for your guidance please.

How can the inequality ##cosx \ge(1-x^2/2)## be proved? Would you please explain how to prove this inequality? This is the only equation that I could think of. ##1\ge cosx \ge 0## but I cannot use it here. Source: Thomas's Calculus, this is from an integration question there. Thank you.

Let $a, b$ and $c$ be non-zero real numbers, and let $a\ge b \ge c$. Prove the inequality: $$\frac{a^3-c^3}{3} \ge abc\left(\frac{a-b}{c}+\frac{b-c}{a}\right)$$ When does equality hold? Source: Nordic Math. Contest

If $x^2+y^2+z^2+xyz=4$ and that $x,\,y,\,x\ge 0$, prove $3!\sqrt{x}+2!y+1!z^2\le 13$.

Homework Statement ##\left|\left(\frac{x}{2}\right)^2\right| < 1## Homework EquationsThe Attempt at a Solution The absolute value situation is throwing me off for some reason. Would it be correct to split this into two equations? ##-\left(\frac{x^2}{4}\right) > -1## and ##\frac{x^2}{4} < 1##...

Hello! $$\lim_{n\rightarrow \infty }\frac{1}{n}ln(a^{\frac{n}{1}}+a^{\frac{n}{2}}+...+a^{\frac{n}{n}} ), \ a>1$$ I solved the limit by using the following inequality: $$a^{n}\leq a^{\frac{n}{1}}+a^{\frac{n}{2}}+...+a^{\frac{n}{n}}\leq n\cdot a^{n}$$ After I applied a $ln$ and $1/n$ I got $lna$...

Hi all, I was wondering if there exist any theorems that allow one to relate any joint distribution to its marginals in the form of an inequality, whether or not ##X,Y## are independent. For example, is it possible to make a general statement like this? $$f_{XY}(x,y) \geq f_X (x) f_Y(y)$$...

I need to find the solutions of the following inequation: (1-sqrt(1-4x^2)/x < 3 I put the conditions x different from 0 and 1-4x^2>=0 and I got [-1/2,0)U(0,1/2] which is the right answer but I'm confuse because I usually subtract 3 to get (1-sqrt(1-4x^2)/x - 3 < 0 then, after I made some work...

Homework Statement Prove bernullis inequality: If h>-1 then (1+h)^n ≥ 1+ nhHomework EquationsThe Attempt at a Solution How can I prove something that is false for h =1 n<1 ?

Homework Statement (1/x) + (1/(1-x)) > 0 Homework EquationsThe Attempt at a Solution 1+x-x/(x-x^2) > 0 1/(x-x^2) > 0 x-x^2 > 0 x> x^2 only occurs when 0<x<1 but in the solutions Spivak tells me "x>1 or 0<x<1"

Homework Statement Prove: if 0<a<b then a^(1/n) < b^(1/n) Homework EquationsThe Attempt at a Solution I've already proved a^n < b^n in another problem. So I have Assume a^n < b^n => \sqrt[n^2] a^n < \sqrt[n^2] b^n => a^(1/n) < b^(1/n)

Hello. I am bewildered by so many different notions of probability distribution percentages, i.e. the proportion of values that lie within certain standard deviations from the mean. (1) There is a Chebyshev's inequality: - for any distribution with finite variance, the proportion of the...

I am working on a proof problem and I would love to know if my proof goes through: If $A, B$ are sets and if $A \subseteq B$, prove that $|A| \le |B|$. Proof: (a) By definition of subset or equal, if $x \in A$ then $x \in B$. However the converse statement if $x \in B$ then $x \in A$ is not...

I am reading Reinhold Remmert's book "Theory of Complex Functions" ... I am focused on Chapter 0: Complex Numbers and Continuous Functions ... and in particular on Section 1.3: Scalar Product and Absolute Value ... ... I need help in order to fully understand Remmert's derivation of the...

What is error in the picture?

Homework Statement Prove that ##\forall n \in \mathbb{N}## $$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$ Homework Equations Peano axioms and field axioms for real numbers. The Attempt at a Solution Okay so my first assumption was that this part...

Homework Statement I would like to know how the boundary of the inequality change when the origin of the coordinate system changes. Homework Equations The original inequality is[/B] $$ r_0 \le x^2+y^2+z^2 \le R^2$$ I would like to know the boundary of the following term, considering the...

Find in at least two different ways the smallest $\alpha$, such that \[\sqrt[3]{x}+\sqrt[3]{y} \leq \alpha \sqrt[3]{x+y}\] - for all $x,y \in \mathbb{R}_+$

Dear all, I am trying to solve this inequality: \[\frac{2}{x^{2}-1}\leq \frac{1}{x+1}\] I've tried several things, from multiplying both sides by \[(x^{2}-1)^{2}\] finding the common denominator, but didn't get the correct answer, which is: \[2<x<3\] or \[x<-1\]How to you solve this one ?

Find the smallest natural number $n$ for which $\sin \left(\dfrac{1}{n+1934}\right)<\dfrac{1}{1994}$.

Hi. I have looked through an example of working out a trig integral using the residue theorem. The integral is converted into an integral over the unit circle centred at the origin. The singularities are found. One of them is z1 = (-1+(1-a2)1/2)/a It then states that for |a| < 1 , z1 lies...

I am not sure what examples to give, need help on this. Have attached the theorem as well.

For any metric ##(X, \rho)## and points therein, prove that ##|\rho(x, z) - \rho(y, u)| \leq \rho(x, y) + \rho(z, u)##. I know that this will involve iterated applications of the triangle inequality...but I still need another hint on how to proceed.

Hello! (Wave) Let $A$ be a $n \times n$ complex unitary matrix. I want to show that the eigenvalues $\lambda$ of the matrix $A+A^{\star}$ are real numbers that satisfy the relation $-2 \leq \lambda \leq 2$. I have looked up the definitions and I read that a unitary matrix is a square matrix...

Dear Everyone, I am currently in an Introduction to Complex Analysis; I have a question: Use established properties of moduli to show that when $\left|{z_3}\right|\ne\left|{z_4}\right|$: $\frac{\Re{({z}_{1}+{z}_{2})}}{\left|{z}_{3}+{z}_{4}\right|}\le\frac{\left| {z}_{1} \right|+\left|...

I am working a bunch of problems for my Real Analysis course.. so I am sure there are more to come. I feel like I may have made this proof too complicated. Is it correct? And if so, is there a simpler method? Problem: Show that $liminfa_n \leq limsupa_n$. Proof: Consider a sequence of real...

Homework Statement Go through question number 4 Homework Equations The Attempt at a Solution See basically the question is asking us to find the range of the given function x/(x^2+x+1). So,I began solving it this way... I am stuck at this step. I asked my friend for a hint and he told me to...

Let $f$ be a positive and continuous function on the real line which satisfies $f(x + 1) = f(x)$ for all numbers $x$. Prove the inequality: $$\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})} \,dx \geq 1.$$

Homework Statement For x,y,z ## \in \mathbb {R^+} ##, prove that ## \sqrt {x (3 x +y) } + \sqrt {y (3y +z) } + \sqrt {z(3z +x)} \leq ~ 2(x +y+ z) ##Homework Equations The Attempt at a Solution I don't know which inequality among the above two has to be applied. I am trying to solve it by...

This doesn't really fit in any other place. I'm interested in alternatives to the Gini coefficient for studying inequality, such as were discussed in this thread and this one. What do I see as the shortcomings of Gini? By example: in Lower Slobovia, everyone has nothing. It's a perfectly...

I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality. Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result. The problem is...

Homework Statement The problem is stated as follows: "The result in Problem 1-7 has an important generalization: If ##a_1,...,a_n≥0##, then the "arithmetic mean" ##A_n=\frac {a_1+...+a_n} {n}## and "geometric mean" ##G_n=\sqrt[n] {a_1...a_n}## Satisfy ##G_n≤A_n## Suppose that ##a_1\lt A_n##...

Hey! :o Let $D=\left \{x=(x_1, x_2)\in \mathbb{R}^2: x_2>\frac{1}{x_1}, \ x_1>0\right \}$. We have the function $f: D\rightarrow \left (0,\frac{\pi}{2}\right )$ with $f(x)=\arctan \left (\frac{x_2}{x_1}\right )$. I want to show using the mean value theorem in $\mathbb{R}^2$ that for all...

EDIT: I realize now that I have fundamentally misunderstood a crucial aspect of deriving the Bell inequality for this case which is the existence of the third axis. The setup of the problem did state that the axes were chosen at random. Therefore I can't just look at the possibility of choosing...