Inequality Definition and 1000 Threads

Hey everybody, I'm a little embarrassed to be asking this question as I feel it is extremely easy, but I will do so anyway. I'm self studying some algorithms, and the book that I'm using claims that a n^2 + b n + c \in \Omega(n^2) Of course, this is equivalent to saying that there...

Homework Statement What is the maximum value of 'n' such that the modulus of pi-22/7 < 10-n? Homework Equations The Attempt at a Solution I have worked out that pi > -10-n + 22/7 , and pi < 10n +22/7. I also know that 10-n is equivalent to 1/10n. I do not know where to go...

Homework Statement Firstly, I'd just like to point out that this is not actually a course related question. I have been trying to teach myself mathematics, and have been grappling with this for a couple of days. The book has no answer at the back for this particular question. Variables...

Homework Statement Regarding problem 1-6 in Spivak's Calculus on Manifolds: Let f and g be integrable on [a,b]. Prove that |\int_a^b fg| ≤ (\int_a^b f^2)^\frac{1}{2}(\int_a^b g^2)^\frac{1}{2}. Hint: Consider seperately the cases 0=\int_a^b (f-λg)^2 for some λ\inℝ and 0 < \int_a^b (f-λg)^2 for...

Homework Statement Prove that for every two distinct real numbers a and b, either (a+b)/2>a or (a+b)/2>bHomework Equations The Attempt at a Solution Proof: if two distinct numbers a and b then (a+b)/2>a Since a≠b and a,bεR, (a+b)/2>a=a+b>2a=b>a. Therefore (a+b)/2>a if b>a. and if two...

Homework Statement Givens: \forall x\ge 0:\quad f^{ \prime \prime }\left( x \right) \ge 0;\quad f\left( 0 \right) =0 Prove: \forall a,b\ge 0:\quad f\left( a+b \right) \ge f\left( a \right) +f\left( b \right) Homework Equations By definition, f is convex iff \forall x,y\in \Re \quad \wedge...

Hi everyone, I have a question on Rudin's proof of Theorem 1.33 part e. Here he prove the following statement: The absolute value of z+w is equal or smaller than the absolute value of z plus the absolute value of w -Yes, is the triangle inequality, where z and w are both complex numbers-...

Hi everyone, I am trying to solve an optimization problem using fmincon in Matlab with a nonlinear inequality restriction. Part of the objective function is undefined if this nonlinear inequality is violated. I also set up lower and upper bounds for fmincon. I use the "interior-point"...

Dear friends, I am interesting to find some functions g satisfying the following convolution inequality (g\astv)(t)\leqv(t) for any positive function v\inL^{1}[0,T] and * denotes the convolution between g and v.

Hello gurus, I've been trying to prove the following inequality for several days: \int_1^\infty \frac{\exp\left(-\frac{(x-1)^2}{2a^2}\right)}{x}dx > \ln(1+a)\quad \forall a>0. I've demonstrated by simulations that this inequality holds. I‘ve also proved that this inequality holds for large...

. . Let \ \ p_n \ \ = \ \ the \ \ nth \ \ prime \ \ number.Examples:p_1 \ = \ 2 p_2 \ = \ 3 p_3 \ = \ 5 p_4 \ = \ 7- - - - - - - - - - - - - - - - - - - - - - - - - - - - Let \ \ n \ \ belong \ \ to \ \ the \ \ set \ \ of \ \ positive \ \ integers. Prove (or disprove) the following:p_n \ +...

1. Homework Statement f(x)=(2x^3+2x^2-5x-2) / 2(x^(2)-1) f''(x)=(-12x^5-24x^3+36x)/(4x^8-16x^6+24x^4-16x^2+4) Find the intervals where f is concave up. 2. The attempt at a solution (I am having trouble interpreting the results at the end or if I've made a mistake somewhere): Attempt at...

Homework Statement I am asked to prove: 2n < (n+1)! , where n≥2 The Attempt at a Solution Base step: set n=2, then test 22 < (2+1)! 22 = 4 (2+1)!= 3! = 3(2)(1) = 6 so 4 < 6 , which is true. Induction hypothesis is 2k < (k+1)! Using this, prove 2(k+1) < [(k+1)+1]! = (k+2)! Attempt to...

Homework Statement Let a and b be real numbers a. The condition “a + b = 0” is ...for the condition “a = 0 and b = 0” b. The condition “a + b > 0” is ...for the condition “a > 0 and b > 0” c. The condition ab = 0 is .... for the condition a = b = 0 d. The proposition “ a + b > 2 and ab >...

Hello everybody, I've been trying to understand the CHSH proof as it is listed on Wikipedia: http://en.wikipedia.org/wiki/CHSH_inequality I got to this without any problem: E(a, b) - E(a, b^\prime) = \int \underline {A}(a, \lambda)\underline {B}(b, \lambda)[1 \pm \underline {A}(a^\prime...

Copied from the OWS May Day protest thread... This is a very common argument on here and I've seen it in other contexts as well. Usually it is in the context of a larger discussion about income inequality, but it is treated as a self-evident, throw-away claim that doesn't ever get...

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

Homework Statement {(x+y+z)^3-2(x+y+z)(x^2+y^2+z^2)}/xyz ≤ 9 Homework Equations AM-GM inequality x+y+z ≥ 3(√xyz)(cube root) and xy+yz+zx ≥ 3√(xyz)^2(cube root) The Attempt at a Solution This is my attempt but I don't know if I am using the AM-GM inequality correctly...

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...

EDIT: Turns out, the solution to my question is related to the determinant of a positive definite quadratic form. This is more or less straight from Landau's Statistical Physics Part 1 (3rd edition), Chapter 21. I don't understand how the inequality/condition (the last equation in this post)...

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?