Inequality Definition and 1000 Threads

∫dQ/T≤∫dQ(rev)/T * , where both integrals are evaluated between the same thermodynamic coordinates- A and B , say. - I am having trouble interpreting this inequality. -( I understand the derivation in my textbook via the Clausius diagram(considering a reversible and an ireversible process...

Homework Statement Let f(x)=1-x-x3. Find all the real values of x satisfying the inequality, 1-f(x)-f3(x)>f(1-5x).Homework Equations The Attempt at a Solution I honestly don't know how to start with this one. Substituting f(x) directly in the inequality doesn't look like a good idea. I need a...

Homework Statement Solve for k: k2 - 16k < 0 In the answer it has 0 < k < 16, I do not know how they get there from the original question.

Solve 2x - |x+1| < 4.

Question http://puu.sh/52zAa.png Attempt http://puu.sh/52AVq.png I've attempted to use Riemann sums and use the integral to prove the inequality, not sure if this was the right approach to start with as I am now stuck and don't see what to do next. For part (b), I know that if (2√n...

Homework Statement Prove the inequality double integral (dA / (4+x^2+y^2)) is less than or equal to pi, where the double integral has a sub D where D is the disk x^2 + y^2 less than or equal to four Homework Equations The Attempt at a Solution I really have no idea, anyone want to...

Homework Statement |x + y| ≥ |x| - |y| [Hint: write out x = x + y - y, and apply Theorem 3, together with the fact that |-y| = |y|] Homework Equations Theorem 3: |a + b| ≤ |a| + |b| x = x + y - y |-y| = |y| The Attempt at a Solution |x + y| ≥ |x| - |y| x = x + y - y (don't know where to...

(I wasn't sure how to title this, it's just that the statement resembles Chebychev's but with two RV's.) Homework Statement Let \sigma_1^1 = \sigma_2^2 = \sigma^2 be the common variance of X_1 and X_2 and let [roh] (can't find the encoding for roh) be the correlation coefficient of X_1 and X_2...

Could I get some help with this question? Please refer to the attachment. Thanks

I have been reading about the derivation of Clausius' Inequality and there are a few things I do not understand. I have attached an image of the cycles. B) shows one carnot engine performing work ##d W_i## per cycle and delivering heat ##d Q_i## per cycle. For ##T'## to remain unchanged, it...

Hi, I'm trying get a better understanding of Bell's inequality in the form $$\left|E\left(\bf{a},\bf{b}\right) -E\left(\bf{a},\bf{c}\right)\right|\leq 1+E\left(\bf{b},\bf{c}\right)\enspace.$$ I'm considering the Bell state $$\left|\psi\right\rangle=...

I'm trying to really solidify my maths knowledge so that I'm completely comfortable understanding why and how certain branches of mathematics are introduced in physics and inevitably that leads me to a study of proofs. I usually skip proofs as I found them annoying, unintuitive and just...

I am given a statement to prove: Show (without using the Binomial Theorem) that \((1+x)^n\geq{1+nx}\) for every real number \(x>-1\) and natural numbers \(n\geq{2}\). I am given a hint to fix \(x\) and apply induction on \(n\). I started by supposing \(x\) is a fixed, real number larger than -1...

The following inequality can easily be proved on ##ℝ## : ## ||x|-|y|| \leq |x-y| ## I was wondering if it extends to arbitrary normed linear spaces, since I can't seem to prove it using the axioms for linear spaces. (I can however, prove it using the definition of the norm on ##ℝ## by using...

Prove that for any two numbers x,y we have [(x^2 + y^2)/2] >= x + y - 1 Solution) For any number a we have have a^2 > 0. So, (x-1)^2 + (y-1)^2 >= 0 And if we solve this we get the solution.I don't get the red part.

Using only the axioms of arithmetic and order, show that: for all x,y satisfy 0≤x, 0≤y and x≤y, then x.x ≤ y.y I'm really lost on where to start, my attempt so far was this as 0 <= x and 0 <= y, we have 0 <= xy from axiom (for all x,y,z x<=y and 0<=z, then x.z <=y.z). then we use the...

## x - |x-|x|| > 2 ## how would I go about solving something like this? my initial thoughts was to consider if x >= 0 I get 2-x < 0 then x > 2 in that case then consider if x < 0 which I get -|x+x| > 2-x then 2x > 2-x then x > 2/3 but I'm having troubles deciding which one is correct, and if...

Hi guys, Can you help me I am stuck: By finding the real and imaginary parts of z prove that, $$|\sinh(y)|\le|\sin(z)|\le|\cosh(y)|$$ i have tried the following: Let $$z=x+iy$$, then $$\sin(z)=sin(x+iy)=\sin(x)\cosh(y)+i\sinh(y)\cos(x)$$ $$|\sin(z)|=\sqrt{(\sin(x)\cosh(y))^2+(\sinh(y)...

Hello. I am reading an introduction to induction example, and I am having the hardest time trying to determine what exactly happened in the proof. Can somebody please help? How can ##3^{k-1}## + ##3^{k-2}## + ##3^{k-3}## all of a sudden become ##3^{k-1}##+##3^{k-1}##+##3^{k-1}## and how can be...

here is the inequality: ##(\sum\limits_{i=1}^n |x_i-y_i|)^2= \ge \sum\limits_{i=1}^n(x_i-y_i)^2+2\sum\limits_{i \neq j}^n |x_i-y_i|\cdot |x_j-y_j|## does it have a name/is the consequence of a theorem? Thank you :)

Suppose that ##F(u,v)=a|u+v|^{r+1}+2b|uv|^{\frac{r+1}{2}}##, where ##a>1, b>0## and ##r\geq3.## How we can show that there exists a positive constant c such that ## F(u,v)\geq c\Big( |u|^{r+1}+|v|^{r+1}\Big). ##

Homework Statement What are the possible values of |2x−3| when 0<|x−1|<2? Homework Equations The Attempt at a Solution We know \left|x-1\right| becomes x-1 if x-1≥0 and -(x-1) if x-1<0. Now consider two cases. Case 1: 0<x-1<2 \Rightarrow 1<x<3 \Rightarrow -1<2x-3<3. Case...

Log x ((x+3)/(x-1) > Log x x ?? I've managed to find 4 conditions for this inequality: 1. -1 > x > 3 2. x > -3 3. x > 0 4. x ≠ 1 but I'm not sure how to write the solution. Is it " 0 < x & 1 < 0 < 3 " ? Thanks.

Homework Statement Prove; n^2 > n+1 for n = 2,3,4 by Induction Homework Equations The Attempt at a Solution p(n)= P(2) 2^2> 2+1 --> 4>3 Induction step: P(n+1): (n+1)^2 > (n+1) +1 (n+1)^2> n+2 n^2 + 2n + 1 > n+2 | -n n^2 +n + 1> 2 | -1 n^2 +n > 1 Is this correct, and how...

Hello all, I am currently reading about the triangle inequality, from this article http://people.sju.edu/~pklingsb/cs.triang.pdf I am curious, how does the equality transform into an inequality? Does it take on this change because one takes the absolute value of 2uv? Because before the...

Prove $$x^x \ge \left( \frac{x+1}{2} \right)^{x+1}$$ for $x>0$.

This form of a Bell inequality: n[x-y+] + n[y-z-] ≥ n[x+z+] is derived from spin measurements at A and B when detector settings are aligned. If it is correct that when a particle is measured at detector A and is spin up in the y direction , then its entangled twin at B is in superposition...

Hey everyone. I'm taking Calculus at UofT and I got a question in a problem set that kind of got me thinking, and well, I'm not sure if I'm doing it correctly. This isn't the exact question, but, how would you go about solving this inequality: ||(x^2)-4|-3|< 1

Let [x] be the floor function i.e. it produces the integral part of x. So for example if x = 1.5 then [x] = 1. I recently saw the claim $$[x] \geq x - 1$$ The strict part of the inequality makes perfect sense, but when does equality occur? Does it even occur at all? I have not been able to...

Here's a fun problem proof I came across. Show that $$\left| \frac { z- w }{1 - \overline{z}w} \right| < 1$$ given $$|z|<1$$, $$|w|<1$$. I attempted writing z and w in rectangular coordinates (a+bi) but to no avail. Any suggestions, forum?

Prove $$2^{\frac{1}{3}}+2^{\frac{2}{3}}<3$$.

Homework Statement Find δ (which is the input tolerance): lim (9 - 4x2)/(3 + 2x2) = 6 [SIZE="1"]x→-1.5 Homework Equations |f(x) - L| < \epsilon |x - a| < \delta The Attempt at a Solution lim (9 - 4x2)/(3 + 2x2) = 6 [SIZE="1"]x→-1.5 I need to get to: |x-(-1.5)| < \delta...

Homework Statement Prove If ## 0 \leq a < b ## and ## 0 \leq c < d ## then ## ac < bd ## The Attempt at a Solution not sure how to even start on this, was thinking if a = 0 or c = 0, then ac = 0, but bd > 0 (which is given) so bd > ac however this seems like I'm cheating because they give...

$a,b,c,d,e,f,g \in N$ $a<b<c<d<e<f<g$ $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}+\dfrac{1}{e}+\dfrac{1}{f}+\dfrac{1}{g}=1$ please find one possible solution of a,b,c,d,e,f,g (you should find it using mathematical analysis,and show your logic,don't use any program)

prove |a+b| \leq |a| + |b| i've proved it considering all the 4 cases for a and b but the book went about it a different way: (|a+b|)^2 = (a+b)^2 = a^2 + 2ab + b^2 \leq a^2 + 2|a||b| + b^2 = |a|^2 + 2|a||b| + |b|^2 = (|a|+|b|)^2 it then goes on the conclude that |a+b| \leq |a| +...

Hello, if you guys would turn to page 117 in Spivak's Calculus, there is the proof for theorem 3. At the last line he stated that this last inequality ##|f(x)-f(a)|<f(a)## implies ##f(x)>0##. How can you check this fact? Can we assume first that ##f(x)-f(a)<0## to eliminate the absolute...

Homework Statement (-3/x) < 3 Homework Equations Dividing/multiplying an inequality causes the inequality sign to change. The Attempt at a Solution I keep getting the wrong solution. I tried two methods. I cannot get the textbook solution (x < -1) Method one: -3 < 3x...

Hi PF Somebody gave me this link could you help me to understand why (9) has to be inserted in (7) when the efficiency \eta < 1?

Homework Statement Given that -pi < x < pi, solve the following inequality in radians root(2) - 2sin(x-(pi/3)) < 0 The Attempt at a Solution root(2) - 2sin(x-(pi/3)) < 0 - 2sin(x-(pi/3)) < -root(2) sin(x-(pi/3)) > (root(2)) -pi<x<-pi -pi - (pi/3) < x - pi/3 < pi...

Homework Statement In a triangle ABC, prove by vector method cos2A+cos2B+cos2C≥ -3/2. Homework Equations The Attempt at a Solution I can change the LHS of the inequality to the form (cos2A i + cos2B j + cos2C k).(i+j+k)

Prove that $$ \frac{y^x-1}{xy^{x-1}(y-1)}<1$$ where x,y \in ℝ, x>1 and y>1. I was able to prove it using calculus, but am wondering if there was another way of doing so, like exploiting some inequality-theorems which involves real numbers. I'll be glad if anyone can show me a way and...

Homework Statement ##\frac{a}{4}>\frac{a}{2}+6## The Attempt at a Solution ##\frac{2a}{2}>\frac{4a+48}{2}## ##a>2a+24## So do I just plug random numbers in and see what I get? I realized right away that it has to be a negative number so I stuck in -30 and got ##-30>-60+24## Well that's...

Homework Statement Given: |x-y| < K x+y > K - 2 0 < K < 1 Prove: \frac{|1-K+x|}{|1+y|} < 1 The Attempt at a Solution I have tried using the fact that |x-y| < K \Rightarrow -K < x-y < K \Rightarrow y-K < x < y+K to write \frac{1-K+x}{1+y} < \frac{1+y}{1+y} = 1 But I can't figure out...

I am struggling with this question. I need a different perspective. Any recommendation is appreciated. Please click on the attached Thumbnail.

Homework Statement Let f be an analytic function on the disc |z|<1 and satisfies |f(z)|≤M if |z|<1. Show that |f(z)| \le M \left| \frac{z-a}{1-a'z} \right| when |z|<1 where a' is the complex conjugate of a Homework Equations This section uses maximum modulus principle, but I really don't...

Forum, do you have any idea how to solve the trigonometric inequality $$\cos (x) < \sin (x)$$ strictly algebraically? The conventional(?) approach is to first solve $$\cos(x) = \sin(x)$$ and then draw the graphs for each function in order to find the correct interval. However, I would love to...

Homework Statement Prove Bernoulli's Inequality: if ##h>-1## (1+h)^n \geq 1+hn Homework Equations Binomial Theorem (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k} The Attempt at a Solution If ##h=0## (1+0)^n=1 1=1 If ##h>0## This (1+h)^n \geq 1+hn Implies...

Show that $$\frac{1}{44}>\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right)\cdots\left( \frac{1997}{1998}\right)>\frac{1}{1999}$$

In the EPR scenario the correlation results are explained with the conservation laws of classical mechanics as applied to spin. The Bell type inequalities are derived on expected spin values. But the violations of these inequalities are then explained with QM: That simultaneous knowledge of...

Homework Statement If ##A+B+C=\pi##, prove that ##\cos A+\cos B+\cos C \leq 3/2##. Homework Equations The Attempt at a Solution I don't really know how to start. ##A+B=\pi-C##. Taking cos on both sides doesn't seem of much help. I need a few hints to start with.