Inequality Definition and 1000 Threads

Guess what? I just got my new calculus book last week! ^^ The book opens with the definition of the real numbers by Dedekind and goes to prove properties of this numbering system such as The supremum axiom and others. At the end of the chapter are about 30 exercises without their solutions...

Hi everybody, For part of my research, I need to solve an elliptic PDE like: Δu - k * u = 0, subject to : 0≤ u(x,y) ≤ 1.0 where k is a positive constant. Can anyone tell me how I can solve this problem? Thanks in advance for your help.

Homework Statement \sqrt[4]{2x + 1} - 0.1 < \frac{1}{2}x + 1 < \sqrt[4]{2x + 1} + 0.1 Homework Equations The Attempt at a Solution I'm having trouble getting just an 'x' by itself in the middle because of the 4th root. How should I solve this inequality? I tried everything but...

Homework Statement Show that the angles a, b, c of each triangle satisfy this inequality. \tan \frac{a}{2}\tan \frac{b}{2} \tan \frac{c}{2} (\tan \frac{a}{2} + \tan \frac{b}{2} + \tan \frac{c}{2}) < \frac{1}{2} Homework Equations The Attempt at a Solution I used the half angle...

Homework Statement Prove the following inequality for any triangle that has sides a, b, and c. -1<\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-\frac{b}{a}-\frac{a}{c}-\frac{c}{b}<1 Homework Equations The Attempt at a Solution I think we have to use sine or cosine at a certain point because...

Homework Statement Prove the following inequality for any triangle that has sides a, b, and c. -1<\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-\frac{b}{a}-\frac{a}{c}-\frac{c}{b}<1 Homework Equations The Attempt at a Solution I think we have to use sine or cosine at a certain point because...

Hey check out the attached picture. I think I solved the issue now but just to confirm perhaps... If I were to continue here using the comparison test, is the only problem that b-subn limit equals 0, so the comparison test is inconclusive? Otherwise, since our summation does not included...

Homework Statement From the inequality |a.b| <= |a||b| prove the triangle inequality: |a+b| <= |a| + |b| Homework Equations a.b = |a|b| cos theta The Attempt at a Solution Making a triangle where side c = a+b. Don't know how to approach the question. Thanks.

Homework Statement Prove llxl-lyll≤lx-yl (The triangle inequality: la+bl≤lal+lbl) The Attempt at a Solution For the first part, I assumed lxl≥lyl: lxl=l(x-y)+yl Then, by Triangle Inequality l(x+y)+yl≤l(x-y)l+lyl So, lxl≤l(x-y)l+lyl Subtract lyl from both sides to...

Lately I was studying the Bell and CHSH inequalities on Wikipedia (it has proven to be a good source to get an quick idea about everything). The articles are detailed and even provide the core of the proof in a mathematical derivation that is easy to understand. But it leaves me still with a...

I have been tasked with solving the following inequality: \frac{1}{x} < 4 Attached to this thread is my attempted solution. As you can see I begin with simply solving the inequality for x, and I obtain the result x > \frac{1}{4} Next, I convert the equation into what I thought was the...

Prove that $\displaystyle \sin x+2x \geq \frac{3x(x+1)}{\pi}\forall x\in \left[0,\frac{\pi}{2}\right]$

Homework Statement So I'm following along with my physics book and I get to the point where Mg * abs(sin(θ) - cos(θ)) <= μMg * (cos(θ) + sin(θ) Next they say: If tan(θ) >= 1 then sin(θ) - cos(θ) <= μ(cos(θ) + sin(θ)) => tan(θ) <= (1+μ) / (1-μ) Homework Equations The Attempt at a...

Homework Statement \frac{3x + 1}{2x - 6} < 3 Homework Equations The Attempt at a Solution \frac{3x + 1}{2(x -3} < 3 \frac{3x +1}{x - 3} < 6 Assume x < 3 3x + 1 > 6(x - 3) 3x + 1 > 6x - 18 3x + 1 - 6x + 18 > 0 19 > 3x x < 19/3 No Contradiction. Assume x > 3...

Homework Statement Prove that √x+y ≤ √x + √y for all x,y ≥ 0 Homework Equations The Attempt at a Solution square both sides: x + y ≤ x + 2√x√y + y subtracting x and y: 0 ≤ 2√x√y dividing by 2: 0 ≤ √x√y 0 ≤ √x√y is true for all x,y since the square root of a...

Homework Statement To prove the inequality (attached) Homework Equations The Attempt at a Solution I tried factoring out a 2 from each of the even terms in the denominator. This allowed me to cancel out all the terms (odd) on the numerator up to 1005. Leaves me with...

Hi everyone I have an inequality 2x - 4 < 1 I had to double check it to ensure I wrote it down correctly. 2x < 1 + 4 x < 2.5 2(2.5) - 4 < 1 1 < 1 Is this me or am I missing something? 2x - 4 < 1 reads to me as 2x - 4 should be less than < 1 and not equal to it?

Hi, can you please give me some hints to show that \frac{|a-b|}{1+|a|+|b|} \leq \frac{|a-c|}{1+|a|+|c|}+\frac{|c-b|}{1+|c|+|b|}, \forall a, b, c \in \mathbb{R}. I tried to get this from |a-b| \leq |a-c|+|c-b|, \forall a, b, c \in \mathbb{R}, but I couldn't succeed. Thank you.

u_{n} = \sum_{k=1}^{n}\frac{1}{n+\sqrt{k}} Proof that: \frac{n}{n+\sqrt{n}} \leq u_{n} \leq \frac{n}{n+1} Ok, I've been working on that problem for about two hours now and I still don't have a clue how to proof this inequality. I guess it should be done by induction, but I have problems...

Homework Statement Hello, I'm little bit confused about a particular inequality in a proof: | (D_j f_i) (y) - (D_j f_i) (x) | ≤ | [(f'(y) - f'(x)]e_j | ≤ ||f'(y) - f'(x)|| The last part of the inequality confuses me. Is the absolute value (norm on R) less than any other norm on R^n?

Homework Statement Solve Ix+3I>2 *I is used for absolute value notation The Attempt at a Solution Considering both a) Ix+3I > 0 then Ix+3I= x+3 b) Ix+3I < 0 then Ix+3I= -(x+3) when solved this would yield to; a) x>-3 and x>-1 b) x<-5 and x<-3 from my general reasoning i...

Homework Statement Let a; b; c \in (1,∞) and m; n \in (0,∞). Prove that \log_{b^mc^n} a + \log_{c^ma^n} b +\log_{a^mb^n} c \ge \frac 3 {m + n} Homework Equations The Attempt at a Solution I do not even know where to start. A coherent explanation and possible solutions would...

In W. Hoeffding's 1963 paper* he gives the well known inequality: P(\bar{x}-\mathrm{E}[x_i] \geq t) \leq \exp(-2t^2n) \ \ \ \ \ \ (1), where \bar{x} = \frac{1}{n}\sum_{i=1}^nx_i, x_i\in[0,1]. x_i's are independent. Following this theorem he gives a corollary for the difference of two...

I am familiar with the proof for the following variant of the triangle inequality: |x+y| ≤ |x|+|y| However, I do not understand the process of proving that there is an equivalent inequality for an arbitrary number of terms, in the following fashion: |x_1+x_2+...+x_n| ≤...

The triangle inequality states that, the sum of any two sides of a triangle must be greater than the third side of the triangle. But the triangle law of vector addition states that if we can represent two vectors as the two sides of a triangle in one order ,the third side of the triangle...

Imagine a light source, double-slit, and a curved screen in vacuum, shaped so that all parts of the interference pattern are created simultaneously. Define distance as proportional to the time light requires to reach a point. Detectors at each slit can be operating or not. Call the source S...

Homework Statement Suppose U and V are unitary matrix, A and B are positive definite, Does: UAU-1 < VBV-1 implies A < B and vice versa?

Homework Statement Show that |x_1 + x_2 + · · · + x_n | ≤ |x_1 | + |x_2 | + · · · + |x_n | for any numbers x_1 , x_2 , . . . , x_n Homework Equations |x_1 + x_2| ≤ |x_1| + |x_2| (Triangle inequality)The Attempt at a Solution I tried using the principle of induction here, but to no avail...

I'm reading "An Introduction to Mathematical Reasoning," by Peter Eccles. It has some interesting exercises, and right now I'm stuck on this one: "Prove that \[\frac1n\sum_{i=1}^nx_i \geq \left(\prod_{i=1}^nx_i\right)^{1/n}\] for positive integers \(n\) and positive real numbers \(x_i\)."...

Homework Statement Prove Homework Equations (π^3)/12≤∫_0^(π/2)▒〖(4x^2)/(2-sinx) dx≥(π^3)/6〗 Also look at atachment The Attempt at a Solution I can't get round this one, since when you substitute x by 0 is always 0 and I don't know how to get ∏^3/12

Homework Statement Prove that \frac{1}{n}\sum_{i=1}^n x_i\geq {(\prod_{i=1}^n x_i)}^{1/n} for positive integers n and positive real numbers x_i Homework Equations There is also a hint. It states that it does not seem to be possible to prove it directly so you should prove it for n=2^m...

I've been making up problems to practice with, and I came across something I couldn't tell on my own, and that is, how do you know that your problem is supposed to be an inequality or you did something wrong? Should I just be looking up other peoples problems instead to try and practice with?

Prove that $\sin \frac{A}{2}+\sin\frac{B}{2}+\sin \frac{C}{2}\leq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$

Let x be in R^n and Q in Mat(R,n) where Q is hermitian and negative definite. Let (.,.) be the usual euclidian inner product. I need to prove the following inequality: (x,Qx) <= a(x,x) where "a" is the maximum eigenvalue of Q. Any idea?

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...