Inequality Definition and 1000 Threads

Homework Statement Prove that for all numbers a, b, c, d: if 0 \leq a < b and 0 \leq c < d then ac < bd. This is problem 5 from chapter 1 of Michael Spivak's "Calculus", 4th Edition. It is the text for my real analysis course. I should also mention that this is not a homework problem...

Hi, Does the following inequality hold regarding the product of 2 matrices A and B: p(AB) <= p(A)p(B), where p denotes the spectral radius. Thanks!

While this is not technically an assignment for any particular class (that I'm aware of, at least), I think the nature of this problem makes it suitable for this forum. Please, inform me if I should direct my question elsewhere. Find x>3 such that ln(x)<x^0.1 (hint: The number is "huge")...

Hi, I am working on a calculus of variations problem and have a general question. Specifically, I was wondering about what kind of constraint functions are possible. I have a constraint of the form: f(x)x - \int_{x_0}^x f(z) dz \leq K If I had a constraint that just depends on x or...

Let be k \leq n poitive integers. How to show that \left (1+\frac1 n \right)^k \leq 1 + \frac{ke}{n} . It seems to me that it has something to do with Bernoulli's inequality. Thank you in advance!

Homework Statement I am attempting to show that -x \leq sin(x) \leq x for x>0 and thus \int^1_0 nxsin(\frac{1}{nx})dx converges to 1. Homework Equations I know that I need to use the fact that I have shown that the limit as T tends to infinity of \int^T_1 \frac{cos(x)}{\sqrt{x}}dx...

Homework Statement Homework Equations The Attempt at a Solution

I want to find value for m for which: 4m2 - 12m > 0 Say I do this algebraically: 4m(m-3) > 0 so m > 0 or m > 3 The answer however is 0 < m and m > 3, I know this as a fact as I have looked graphically. So, my question is, when done algebraically, how do I get 0 < m instead of m...

Homework Statement Show from definition that if f is measurable on [a,b], with m<=f(x)<=M for all x then its lebesgue integral, I, satisfies m(b-a)<=I<=M(b-a) Homework Equations The Attempt at a Solution I know that the definition is that f:[a,b]->R is measurable if for each t...

Homework Statement Prove the Triangle Inequality Theorum using the coordinate system. Homework Equations The corners of the triangles will be at (x1,y1), (x2, y2), (x3,y3) The Attempt at a Solution The proof that I know is proving that |x+y|<=|x|+|y|: -|x|<x<|x|, and...

Hi I'm in the process of trying to understand the proof to bessel's equality and inequality and I am stuck, I have got to the line http://img141.imageshack.us/img141/396/besselsequality.jpg Uploaded with ImageShack.us and I'm not entirely sure how it equates to the next line but according to...

My mathematical methods for theoretical physics course recently began looking at linear vector spaces. We defined the Banach and Hilbert Spaces and proved the Cauchy-Shwarz Inequality. There's one step in this proof that I can't really follow (in red): consider: w=x+uy (i'll drop the...

Homework Statement Hi Guys, I try to find the range for parameters phi1 and phi2 were the autoregressive process below is stationary. We have the process X(t)+phi1*X(t-1)+phi2X(t-2)=Epsilon(t) (1) Homework Equations We get the characteristic polynomial F(z)=z^2+phi1*z+phi2 (2) The...

I have the following question on metric spaces Let (X,d) be a metric space and x1,x2,...,xn ∈ X. Show that d(x1, xn) ≤ d(x1, x2) + · · · + d(xn−1, xn2 ), and d(x1, x3) ≥ |d(x1, x2) − d(x2, x3)|. So the first part is simply a statement of the triangle inequality. However, the metric...

Homework Statement Prove the following inequality: \frac{1}{6}\leq\int_{R}\frac{1}{y^{2}+x+1}\chi_{B}(x,y)dxdy\leq\frac{1}{2} where B={(x,y)|0\leq (x)\leq (y)\leq1} and R=[0,1]x[0,1] EDIT: The B region should be 0 less than or equal to x less than or equal to y less than or equal to 1...

I am having a bit of a problem with Chebychev's inequality which is: P(|X-\mu |\geq \alpha )\leq \frac{\mathrm{Var}(X)}{\alpha ^2} For a positive \alpha. Here X denotes a stochastic variable with mean \mu and finite variance. I am asked to give a direct proof of this result, using the...

Homework Statement Show that a2+b2 =>2ab, and hence, if x+y+z=c, show that x2+y2+z2 => 1/3 c2 Homework Equations The Attempt at a Solution How to prove this when we only have unknowns? The only thing i can think of for the first one is a (a+b)2= a2+b2 +2ab, but how to prove that a2+b2 =>2ab...

Homework Statement Given http://www.mathhelpforum.com/math-help/attachments/f33/20928d1298610998-function-msp281219ebge8he857gc6900005ba9285dff0f5h79.gif , find the values of ''a'' for which the value of the function f(x) <= 25/2. The answer is a<= 1/2. Homework Equations The Attempt at a...

Hi there, I am trying to prove the following. For any random vectors X,Y,Z,W in \mathbb{R}^d and deterministic d\times d matrices A,B the covariance \operatorname{\mathbb{C}ov}\left(X^TAY;Z^TBW\right) can in some way be bounded by the covariance...

Homework Statement The problem is Excercise 5. in page 88 of Folland's "real analysis: modern techniques and their applications", 2nd edition, as the image below shows. Homework Equations As the hint indicates, we should use Excercise 4. The Attempt at a Solution From Excercise 4, if...

So I multiplied the heat equation by 2u, and put the substitution into the heat equation, and get 2uut-2uuxx=(u2)t=2(uux)x+2(ux)2. I`m not sure where to go from there, I can integrate with respect to t, then I would have a u2 under the integral on the left side, but them I`m not sure where to...

for which f: R \rightarrow R such that \forall x,y\in R does | f(y) - f(x) | \mid \leq (y-x)^2 hold

How are these two equal??(equation, inequality) I study discrete mathematics and we are doing combinations at the moment. There is this example in the book(Discrete Mathematics and Combinatorics p. 30 Ex. 1.43) where it states that the number of integer solutions for: x1+x2+x3+x4+x5+x6<10...

hi, while trying to study complex analysis, i have a few problems. i already know that in complex number system, it's impossible for any order relation to exist. but i was confused to this fact when i saw the proof of triangle inequality. ; Let z,w be complex numbers. Then, triangle...

Dear all, Consider the system given by : http://www.freeimagehosting.net/image.php?53f7eed9ce.jpg where we are trying to solve for s and gamma using Newton's method. It turns out to be a simple implementation. Now, what if we need to impose an inequality constraint on the solution s : one...

Homework Statement If x,y\in R and x+y=1.then find max. and Min. value of (x^3+1)(y^3+1) (Without using calculus) Homework Equations here x+y=1 and (x^3+1)(y^3+1) The Attempt at a Solution I have done using Calculus...

Homework Statement then what are the operations that maintain the Inequality and what are the operations that don't? Homework Equations The Attempt at a Solution clearly addition and subtraction maintains it ,and so does multiplication and division by any number other than...

I'm trying to figure out how to prove this inequality: I know it's true (by graphing), but what's an algebraic way to prove it? 1+\frac{1}{3x^2}< x \tan \frac{1}{x} < \frac{1}{\sqrt{1-\frac{2}{3x^2}}} Thanks

Homework Statement I am trying to prove that \frac{1}{512} \left[101-(1-\alpha ) \gamma ^n \left(2 (37+64 \alpha )+27 (1-\alpha ) \gamma ^n\right) (1-\delta )-\frac{128 n (1-\alpha ) \alpha (1-\gamma ) \gamma ^{-1+n} (1-\delta )}{\alpha +\delta -\alpha \delta }\right]\geq0 for...

Homework Statement Prove without computation that 2<Integral[0,2] (1+x^3)<6 The Attempt at a Solution I know there is a theorem which says that if a function is bounded by two constants, then the integral of the function is also bounded by the integrals of the two functions. However...

Homework Statement Graph the following inequality in the complex plane: |1 - z| < 1 2. The attempt at a solution In order to graph the inequality I need to get the left side in the form |z - ...| |1 - z| < 1 |(-1)z + 1| < 1 |-1(z - 1)| < 1 |-1||z - 1| < 1 (1)|z - 1| < 1 |z - 1| < 1...

Homework Statement Solve the Inequality: (3x-7)/(x+2)<1 Homework Equations The Attempt at a Solution Cross Multiply: x+2>3x-7 Simplify: 9>2x Simplify More: 9/2>x My Answer: (-∞, 9/2) I put this as my answer but the answer is really (-2, 9/2) Can someone explain to me why this is? I know you...

Homework Statement Let V be a vector space with inner product <x,y> and norm ||x|| = <x,x>^1/2. Prove the Cauchy-Schwarz inequality <x,y> <= ||x|| ||y||. Hint given in book: If x,y != 0, set c = 1/||x|| and d = 1/||y|| and use the fact that ||cx ± dy|| >= 0. Here...

Is inequality bad for society as a whole?

Homework Statement Prove that ||a|-|b||\leq |a-b| for all a,b in the reals Homework Equations I know we have to use the triangle inequality, which states: |a+b|\leq |a|+|b|. Also, we proved in another problem that |b|\leq a iff -a\leqb\leqa The Attempt at a Solution Using the...

Question: I need to prove this inequality: Where x,y,x are non-negative and x+z<=2: (x-2y+z)^2 >= 4xz -8y. My attempt: I thought maybe choosing x as 0 and z as 0 will and then solving for y... but that only yields y+2 >= 0, which isn't really a solution, since I can't choose numbers...

Homework Statement Prove that if a,b > 1, then a+b < 1+ab The Attempt at a Solution Just want to know if this makes sense: first let a+b < 1+ab become 1<(1+ab)/(a+b) ==> 0<(1+ab-(a+b))/(a+b). Factoring the numerator: 0<(1-a+ab-b)/(a+b) ==> 0<(1-b)+a(b-1)/(a+b) So the next...

Hi, I was reading Heinz Pagels' description of the nail gun experiment in the chapter about "Bell's Inequality" from his book, The Cosmic Code: Quantum Physics as the Langauge of Nature, 1982, pp. 160-176. He describes the record of hits and misses after "turning polarizer A clockwise by...

Got some exciting news from a PF Mentor: And the actual paper: http://iopscience.iop.org/1367-2630/12/12/123007 Violation of Leggett inequalities in orbital angular momentum subspaces J Romero, J Leach, B Jack, S M Barnett, M J Padgett and S Franke-Arnold J Romero et al 2010 New J...

Homework Statement Let f(z) be an entire function such that |f(z)| less that or equal to R whenever R>0 and |z|=R. (a)Show that f''(0)=0=f'''(0)=f''''(0)=... (b)Show that f(0)=0. (c) Give two examples of such a function f. Homework Equations The Attempt at a Solution...

Homework Statement let n\inN To prove the following inequality na^{n-1}(b-a) < b^{n} - a^{n} < nb^{n-1}(b-a) 0<a<b Homework Equations The Attempt at a Solution Knowing that b^n - a^n = (b-a)(b^(n-1) + ab^(n-2) + ... + ba^(n-2) + a^(n-1) we can divide out (b-a) because b-a #...

Hello, i've met during problem solving with inequality \max\{A+B,C\}\le\max\{A,C\}+\max\{B,C\} where A,B and C are real numbers. I don't know whether it holds, but I need to prove that. Thanks for reply...

Homework Statement find n0,c1,c2 for which the following is true: c1 nb <=(n-a)b<=c2(n-a)b , for all n > n0Homework Equations http://en.wikipedia.org/wiki/Binomial_theorem" ?The Attempt at a Solution c1 nb <=(n-a)b<=c2(n-a)b c1 nb <=nb-nb-1a+nb-2a2-...-ab<=c2nb c1<=1-a/n + a2/n2-...

Homework Statement For f nonnegative and continuous on [0,1], prove. \left( \int f \right) ^2 < \int f^2 With the limits from 0 to 1. Homework Equations The Attempt at a Solution I was trying to use Upper sums, i.e. \inf \sum \Delta x_i M_i(f^2) = \inf \sum \Delta x_i...

Prove that in any triangle ABC with a sharp angle at the peak C apply inequality:(a^2+b^2)cos(α-β)<=2ab Determine when equality occurs. I tried to solve this problem... I proved that (a^2+b^2+c^2)^2/3 >= (4S(ABC))^2, S(ABC) - area but I don't know prove that (a^2+b^2)cos(α-β)<=2ab :(...

Is it true in general that: |\int f(x)dx| < \int |f(x)|dx Not sure if "Triangle Inequality" is the right word for that, but that seems to be what's involved.

Homework Statement Prove that 2 \leq 1+ \sum(m=1 to n) 1/m! \leq 1 + \sum (m=1 to n) (1/(2^(m-1))) < 3 The Attempt at a Solution I've proved by induction that 2m-1 \leq m!, so it just follows that 1 + (1/(2 ^ (m-1))) \geq 1 + (1/m!), and their sums are the same inequality...

suppose that g:[0,1] \rightarrow \re is continuous, g(0)=g(1)=0 and for every c \in (0,1), there is a k > 0 such that 0 < c-k < c < c+k < 1 and g(c)=\frac(1}{2} (g(c+k)+g(c-k)). Prove that g(x) = 0 for all x \in [0,1] Hint: Consider sup{x \in [0,1] | f(x)=M } where M is maximum of f on [0,1]...

1. Suppose that f: (a,b) --> R is convex. Prove Jensen's inequality: if x1,...,xn\in(a,b) and c1,...,cn >= 0 s.t. \sum(c_j)f(x_j) >= f(\sum((c_j)(x_j)) both summations from j = 1 to n 2: Convex: whenever x1, x2 \in(a,b) and 0 <= c <= 1, we have cf(x1) + (1 + c)f(x2) >= f(cx1 + (1-c)x2)...

This was already posted by someone else but an answer wasn't received so I thought I'd repost. Any help is appreciated. Homework Statement Use the Maclaurin series for cosx and the Alternating Series Estimation Theorem to show that \frac{1}{2} - \frac{x^2}{24} < \frac{1-cosx}{x^2} <...