Inequality Definition and 1000 Threads

The Rushbrooke inequality: H=0, T\rightarrow T_c^- C_H \geq \frac{T\{(\frac{\partial M}{\partial T})_H\}^2}{\chi_T} \epsilon=\frac{T-T_c}{T_c} C_H \sim (-\epsilon)^{-\alpha'} \chi_T \sim (-\epsilon)^{-\gamma'} M \sim (-\epsilon)^{\beta} (\frac{\partial M}{\partial T})_H \sim...

Homework Statement Let p > 1 and x > y > 0 Use the MVT to prove the inequality py^(p-1)[x-y] =< x^p - y^p =< px^(p-1)[x-y] The Attempt at a Solution The only way i only how to use the MVT is where i already have the function. Do you have to define the function from the problem...

5. Suppose that x, y and z are positive real numbers such that xyz = 1. (a) Prove that 27 \leq(1 + x + y)^{2} + (1 + y + z)^{2} + (1 + z + x)^{2} with equality if and only if x = y = z = 1. (b) Prove that (1 + x + y)^{2} + (1 + y + z)^{2} + (1 + z + x)^{2} \leq 3(x + y + z)^{2} with...

Homework Statement Prove: If a>b and c>d, then a+c>b+d Hint: (a-b)+(c-d)=(a+c)-(b+d)>0 Homework Equations The Attempt at a Solution How to use the hint to prove the inequality? My method, not sure it's right. Given c>d, c-d>0 Given a>b => a+(c-d)>b Thus a+c>b+d

Good day, I have this problem that appeared in some practical problem that I'm working on. I basically want to find the boundaries of a,b,c,d for which the following inequality is satisfied, if a,b,c,d \in ℝ^+ and the inequality is: -2 \cdot d + c - a \cdot (c \cdot d)^2 + a \cdot c +...

Does anyone know a simple proof for holder's inequality? I would be more interested in seeing the case of |∫fg|≤ sqrt(∫f^2)*sqrt(∫g^2)

Homework Statement Suppose \int_{-\infty}^{\infty}t|f(t)|dt < K Using Cauchy-Schwartz Inequality, show that \int_{a}^{b} \leq K^{2}(log(b)-log(a)) Homework Equations Cauchy Schwartz: |(a,b)| \leq ||a|| \cdot ||b|| The Attempt at a Solution Taking CS on L^{2} gives us...

Homework Statement \frac{1^2*3^2*5^2...(2n-1)^2}{2^2*4^2*6^2...(2n)^2}<\frac{1}{2n+1} Edit: Must be proven without using induction. Homework Equations The Attempt at a Solution I understand the LHS is the same thing as \frac{(2n-1)!}{(2n)!} And (2n)! = k!2^k & (2n-1)! =...

Homework Statement If 0 <= A <= B, prove that: A(B-A) <= (B/2)^2 Homework Equations - The Attempt at a Solution I've been blindly rearranging the terms trying to see a way to prove this but due to my complete lack of experience in proofs, I'm hoping someone here can give a little...

Hi, Is the following inequality true for x>0: Pr[X1<x]<Pr[X2<x] for X1<X2?

I want to prove cos2(x)/(n2 + 1) ≤ 1/(n2 + 1) I know this is an obvious inequality but I want to know if my reasoning is correct. For the expression cos2(x)/(n2 + 1) to be as large as possible the numerator must → ∞ but cos2(x) is bounded above by 1. cos2(x) = 1 for x = 2∏k where k ≥1...

How can I solve this inequality? (a-x+1)(a-x+2) ≤ a where a is a constant with unknown value. Thanks in advance.

I just got Spivak's calculus today, and I'm already stuck on the prologue problems: 1. The problem Find all x for which (x-1)(x-3)>0 2. The attempt at a solution We know that if ab>0, then either a>0 and b>0, or a<0 and b<0. Thus, if a=(x-1) and b=(x-3), then either (x-1)>0 and...

Is this inequality true ?? 0\leq \sum_{k=0}^{n} \frac{1}{(k+1)^2 (n-k+1)} \leq \frac{1}{\sqrt{n+1}} for all natural numbers n Is it true ?? Thanks

Obviously a violation of the CHSH inequality means that local realistic theories are untenable. If we sent two entangled photons towards detectors (far enough away that for information to travel, you'd require it to go faster than light). One reaches a detector before the other, so...

Proving the "triangle inequality" property of the distance between sets Here's the problem and how far I've gotten on it: If you are unfamiliar with that notation, S(A, B) = (A \ B) U (B \ A), which is the symmetric difference. And D(A, B) = m^*(S(A, B)), which is the outer measure of...

34. Turner's syndrome is a rare chromosomal disorder in which girls have only one X chromosome. It affects about 1 in 2000 girls in the United States. About 1 in 10 girls with Turner's syndrome also suffer from an abnormal narrowing of the aorta. a. In a group of 4000 girls, what is the...

Homework Statement Prove the Schwarz inequality by first proving that (x_{1}^{2} + x_{2}^{2})(y_{1}^{2} + y_{2}^{2}) = (x_{1} y_{1} + x_{2} y_{2})^{2} + (x_{1} y_{2} - x_{2} y_{1})^{2}. Homework Equations x_{1} y_{1} + x_{2} y_{2} \leq \sqrt{x_{1}^{2} + x_{2}^{2}} \sqrt{y_{1}^{2} +...

Homework Statement Solve |3x-7|-|x-8|>4 The Attempt at a Solution so i made columns... and using the columns i made a number line.. 7/3 on the left as a point, with a column on its left, and 8 with a column on its right and sharing a coumn in the middle with 7/3 so i have...

Homework Statement Prove the following inequality holds: ||x|^\alpha - |y|^\alpha | \leq |x-y|^\alpha \qquad (\forall x,y\in \mathbb{R}, \alpha \in (0,1]) Homework Equations The Attempt at a Solution I tried squaring both sides, getting: x^{2 \alpha} - 2 (|x||y|)^\alpha + y^{2 \alpha} \leq...

The problem Given the Schwarz inequality, x_{1}y_{1} + x_{2}y_{2} \leq \sqrt{x_{1}^{2} + x_{2}^{2}} \sqrt{y_{1}^{2} + y_{2}^{2}}, prove that if x_{1} = \lambda y_{1} and x_{2} = \lambda y_{2} for some number \lambda \geq 0, then equality holds. Prove the same thing if y_{1} = y_{2} = 0. Now...

Hi, Given $ A+B+C=\pi$, I need to prove $ cosA+cosB+cosC\leq \frac{3}{2}$. I wish to ask if my following reasoning is correct. First, I think of the case where A and B are acute angles, then I can use the Jensen's Inequality to show that the following is true. $ cos\frac{A+B}{2}\geq...

I am having trouble proving these. I cannot figure out how to get to the conclusion. Here is my attempt. The stuff in red is just side work and is not part of the proof. I always get stuck on these types of problems, can someone offer some tips on how to approach these kind of problems in...

Homework Statement Show that if |z| = 10 then 497 ≤ |z^{3} + 5iz^{2} − 3| ≤ 1503. The Attempt at a Solution I'm not an entirely sure how to begin this one, or if what I'm doing is correct. If I sub in |z| = 10 into the equation; |1000 + 500i - 3| = 997 +500i Then the modulus of...

Homework Statement By expanding (x-y)^2, prove that x^2 +y^2 ≥ 2xy for all real numbers x & y. Homework Equations The Attempt at a Solution expanding (x-y)^2 x^2 - 2xy + y^2= 0 Hence, x^2 + y^2 = 2xy But where does the ≥ come into it? and why? when you put values in...

inequality... how does this make sense?? Homework Statement Solve (x-1)(x-2)<0 Homework Equations The Attempt at a Solution Given this is a parabola graphical solution cuts the x-axis at 1 & 2 therefore sltn... 1<x<2 However, in my textbook the answer says...

Hi just a quick question I was curious about. Im not sure if the results from CERN about the faster than light neutrino have been verified, but given that this is true... as I understand it bell's inequality assumes 1. the reality of the external world, independent of us "observers". 2...

I am trying to show $|(n+z)^2|\leq (n -|z|)^2$ where is complex $|(n+z)^2| = |n^2 + 2nz + z^2| \leq n^2 + 2n|z| + |z|^2$ But I can't figure out the connection for the final piece.

Here's a nice problem. Suppose $a_1,a_2,...,a_n$ are postive real numbers satisfying \(a_1\cdot a_2\cdots a_n=1\). Show that $(a_1+1)(a_2+1)\cdots(a_n+1)\geq2^n$.

Homework Statement Show that if z_1,z_2 \in \mathbb{C} then |z_1+z_2| \leq |z_1| + |z_2| Homework Equations Above. The Attempt at a Solution I tried by explicit calculation, with obvious notation for a,b and c: my frist claim is not that the triangle inequality holds, just that...

As I can see from the formula of cauchy inequality: (a1^2+a2^2+...+an^2)^1/2 . (b1^2+b2^2+...+bn)^1/2 >= a1b1+a2b2 + ... + anbn Can I conclude from the above formula that: (a1+a2+...+an)^1/2 . (b1+b2+...+bn)^1/2 >= (a1b1)^1/2 + (a2b2)^1/2 +...+ (anbn)^1/2 by setting a1,...,an =...

I have an inequality and tried to solve it and reached the following: Original question: Prove (1/a - 1)(1/b - 1)(1/c - 1) >= 8 when a+b+c = 1 and a,b,c positive After expanding and some eliminations, I still need to prove 1/a + 1/b + 1/c -1 >= 8 Any suggestion how to solve it?

Homework Statement Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem. Find b, such that maximum of the magnitude of the frequency response function...

Hello to everyone. This is my first time here so I hope I will not cause any unwanted trouble. Straight to the problem. I have one inequality for which I would like to prove, but I do not know how. The inequality has the following form:- (1+a)q < q/(1-a), where a < 1 and q can be any positive...

Homework Statement The problem is: for all 0≤a≤1 so i need to find the domain Homework Equations N/A The Attempt at a Solution I tried it like this: yet my solution is wrong,i am not so sure why. wolfram gives me this;

Write as one inequality with an absolute value x<-5 or 8<x not sure how you introduce the absolute value in this to solve it. thanks ahead

Homework Statement Prove the triangle inequality for the following metric d d\big((x_1, x_2), (y_1, y_2)\big) = \begin{cases} |x_2| + |y_2| + |x_1 - y_1| & \text{if } x_1 \neq y_1 \\ |x_2 - y_2| & \text{if } x_1 = y_1 \end{cases}, where x_1, x_2, y_1, y_2 \in \mathbb{R}...

Homework Statement Find all x for which \frac{x-1}{x+1}>0 \qquad(1)Homework Equations (2) AB > 0 if A,B >0 OR A,B < 0 (3) 1/Z > 0 => Z > 0 The Attempt at a Solution Since (1) holds if: (x-1) > 0 \text{ and } (x+1) > 0 \qquad x\ne -1 then we must have x>1 AND x>-1 and since (1) also...

Homework Statement I am doing the HW in Spivak's calculus (problem 4 (ii) ) on inequalities. The problem statement is: find all x for which 5-x2 > 8The Attempt at a Solution I know this is a simple problem, but bear with me for a moment. I want someone who is familiar with Spivak to tell...

Homework Statement Let a, x, and y be real numbers and let E > 0. Suppose that |x-a|< E and |y-a|< E. Use the Triangle Inequality to find an estimate for the magnitude |x-y|. Homework Equations The Triangle Inequality states that |a+b| <= |a| + |b| is valid for all real numbers a and...

Hi everyone I don't know if I can find someone here to help me understand this issue, but I'll try the jensen inequality can be found here http://en.wikipedia.org/wiki/Jensen%27s_inequality I have the following discrete random variable X with the following pmf: x 0...

I may have posted this back in the Old Country, but: let the polynomial: \[P(x)=x^n+a_1X^{n-1}+ ... + a_{n-1}x+1 \] have non-negative coeficients and \(n\) real roots. Prove that \(P(2)\ge 3^n \) CB

Prove that for positive real numbers a,b (a+1/b+1)^(b+1) is greater than or equal to (a/b)^(b). The case in which a<b is easy to prove, but after trying to represent the inequality with an integral, I'm a bit stumped. Any ideas?

Find all numbers x for wich: x+3^x<4 Relevant equations (PI) (Associative law for addition) (P2) (Existence of an additive identity) (P3) (Existence of additive inverses) (P4) (Commutative law for addition) (P5) (Associative law for multiplication) (P6) (Existence of a multiplicative identity)...

I am puzzled about this simple case, Suppose we have (A+B)T(A+B) <= (A+B1)T(A+B1), Can we say something about the relation between BTB and B1TB1? For example, is it correct if I say BTB <= B1TB1?

I'm trying to absorb a perplexing proof of Young's inequality I've found. Young's inequality states that if A,B \geq 0 and 0 \leq \theta \leq 1, then A^\theta B^{1-\theta} \leq \theta A + (1-\theta)B. The first step they take is the following: We can assume B \neq 0. (I get that.) But then...

The problem and my question and thoughts are all in the picture.

I found this problem the other day, seems interesting but I am still not sure about the solution Anybody can help x, y, z are numbers with x+y+z=1 and 0<x,y,z<1 prove that sqrt(xy/(z+xy))+sqrt(yz/(x+yz))+sqrt(xz/(y+xz))<=3/2 ("<=" means less or equal)

Homework Statement Let 1\leq p,q that satisfy p+q=pq and x\in\ell_{p},\, y\in\ell_{q}. Then \begin{align} \sum_{k=1}^{\infty}\left\vert x_{k}y_{k}\right\vert\leq\left(\sum_{k=1}^{\infty}\left\vert x_{k}\right\vert^{p}\right)^{\frac{1}{p}}\left( \sum_{k=1}^{\infty}\left\vert...

I am confused by the role of photon polarisation in Bell inequality experiments. The original logic of EPR as I understand it is based on the HUP such that QM predicts that measurement of momentum on one particle should affect the measurement of position of the other particle. Yet across...