Inequality Definition and 1000 Threads

Homework Statement Two numbers x and y are selected from a closed interval [0,4]. To find the probability that the two numbers satisfies the condition that y^{2}\leq x. 2. The attempt at a solution Don't have any idea

Homework Statement Ax ≤ b, assuming A is nxn and solution exists Homework Equations The Attempt at a Solution I don't know of any concrete methods offhand. A grad student suggested rearranging it to: Ax - b ≤ 0, zero vector Then I don't know where to go from here. I was...

I'm trying to understand a process called order finding as I need to know it for Shor's algorithm in quantum computing. The process is like this: For two positive integers x and N, with no common factors and x < N, the order of x modulo N is defined to be the least positive integer, r...

MGF of X denote M(x)=E[exp(tx)] exists for every t>0 . For t>0 Show p(tX >s^2 +logM(t)) < e^-s^2 .

Homework Statement Using the generalized triangle inequality, prove |d(x,y) - d(z,w)| ≤ d(x,z) + d(y,w) Homework Equations d(x,y) is a metric triangle inequality: d(x,y) ≤ d(x,z) + d(z,y) The Attempt at a Solution I know that this needs to be proved with cases: a) d(x,y) - d(z,w)...

Homework Statement Let k and n be positive integers. In how many ways are there integers a1≤ a2≤ ... ≤ ak≤ n. Homework Equations The Attempt at a Solution I don't really know where to begin. Simply using permutations doesn't seem to work. I know that for a1, there are n integers...

I found many information showed Schwarz inequality and Cauchy–Schwarz inequality are same on books and internet, but my teacher's material shows that: Schwarz inequality: \left\|[x,y]\right\|\leq\left\|x\right\|+\left\|y\right\| Cauchy–Schwarz inequality...

Homework Statement Find all real values of x that satisfy the following inequality. Homework Equations |x-3| > |x + 1| The Attempt at a Solution Splitting up the inequality into cases I get: 1. |x-3| > x + 1 and 2. |x-3| < -x - 1 1. x-3 > x + 1 or x-3 < -x - 1...

Homework Statement Determine m : |x-10|<{1}/{m} if its final form is : |x^{2}+{4}x-140|<1 Homework Equations To remove the modulus, square them... The Attempt at a Solution I have tried to assume that if |x-10|{m}<{1} then, I can find |x-10|{m}=|x^{2}+{4}x-140|...

Homework Statement Find the solution set of \large \frac{|x-1|(x-3)(x-5)^{2010}}{(|x|-3)(|x|+1)} \geq 0Homework Equations I am required to solve this using Wavy-Curve method The Attempt at a Solution The critical points are 3 and 5. But I don't know what to do with expressions involving...

Hi Given is a triangle on points x,y,z in the plane. This triangle has two points a and b on opposite sides (see Figure). I would like to show that the following inequality has to hold: \max {d(b,x), d(b,y), d(b,z)} + \max {d(a,x), d(a,y), d(a,z)} - d(b,a) > \min {d(x,y), d(x,z)...

Homework Statement Let x and y be real numbers such that x<y. There exists z that is a real number such that x<z<y. Homework Equations The Attempt at a Solution I wrote the following: We are given that x<y. We know from previous proof that (1/2)<1. Consider y such that y=x+1...

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 OK I have to argument for the fact that this inequality is true, where x > 1. |R_n \ln{x}| \leq \frac{1}{n+1}(x-1)^{n+1} And I have found out that the residual is equal to this: R_n \ln{x} = \frac{1}{n!} \int^x_a{f^{n+1}(t)(x-t)^{n}dt} Homework Equations...

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...

Hello, I need to use the Gronwall inequality to discuss existence/uniqueness of the solution to the initial value problem: x'(t)=xsin(tx) + t with initial condition x(t0) = x0. I can convert this into integral form x(t) = x_0 + \int\limits_{t_0}^{t} xsin(sx) + s ds Which of course can...

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 Hello! I am solving a problem that has a final answer in the book. However, my answer does not end up like the one even though I don't see any mistakes in my calculation by using a formula. More specific, my solution lacks power (n-1) as seen in the official answer and...

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...

Zero field magnetisation like a function of temperature vanished in ##T=T_c## as ##(T_c-T)^{\beta}##. Let ##M_1## be a magnetisation for temperature ##T_1##. Since ##\forall M<M_1##, ##(\frac{\partial A}{\partial M})_T=H=0## it follows that A(T_1,M)=A(T_1,0) for ##M \leq M_1(T_1)## Why only for...

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?