Inequality Definition and 1000 Threads

I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality. Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result. The problem is...

Homework Statement The problem is stated as follows: "The result in Problem 1-7 has an important generalization: If ##a_1,...,a_n≥0##, then the "arithmetic mean" ##A_n=\frac {a_1+...+a_n} {n}## and "geometric mean" ##G_n=\sqrt[n] {a_1...a_n}## Satisfy ##G_n≤A_n## Suppose that ##a_1\lt A_n##...

Hey! :o Let $D=\left \{x=(x_1, x_2)\in \mathbb{R}^2: x_2>\frac{1}{x_1}, \ x_1>0\right \}$. We have the function $f: D\rightarrow \left (0,\frac{\pi}{2}\right )$ with $f(x)=\arctan \left (\frac{x_2}{x_1}\right )$. I want to show using the mean value theorem in $\mathbb{R}^2$ that for all...

EDIT: I realize now that I have fundamentally misunderstood a crucial aspect of deriving the Bell inequality for this case which is the existence of the third axis. The setup of the problem did state that the axes were chosen at random. Therefore I can't just look at the possibility of choosing...

Homework Statement Attached I understand the first bound but not the second. I am fine with the rest of the derivation that follows after these bounds, Homework Equations I have this as the triangle inequality with a '+' sign enabling me to bound from above: ##|x+y| \leq |x|+|y| ## (1)...

I was musing about why the HUP is an inequality. If you analyse a wave packet the spatial frequency spectral width is inversely proportional to the spatial width. So there should be an equality such as Heisenberg's equation 3 in this paper. Has anyone got a simple explanation of where the...

Homework Statement Suppose that ##N \in \mathbb{N}## and that ##(s_n)## is a nonnegative sequence. Prove that ##\displaystyle \frac{s_{N+1} + s_{N+2} + \cdots + s_n}{n} \le \sup \{s_n ~:~ n > N \}## Homework EquationsThe Attempt at a Solution I need help explaining why this is true. Supposedly...

I have a vector B of length N, I would like to prove that: ∑n=0 to N-1 (|Bn|x) ≥ Nαx where: x > 1; α = (1/N) * ∑n=0 to N-1 (|Bn|) (i.e., The mean of the absolute elements of B). and ∑n=0 to N-1 (||Bn|-α|) ≠ 0; (i.e., The absolute elements of B are not all identical). I believe the above to...

Rule: Suppose a>0, then |x|>a if and only if x>a OR x<-a So |x|>|x-1| becomes: x>x-1 which is false (edit: or more accurately doesn't give the whole picture, it implies true for all x) OR x<-x+1 2x<1 x<1/2 which is false

I am trying to see how to derive the following inequality on page 36 in the proof of Lemma 11.3: https://arxiv.org/pdf/math/0412040.pdf I.e, of: $$\| fg \|_{Lip} \le \bigg(1+\ell \sup_{t\in T} |g'(t)|\bigg)\sup_{t\in T}|f'(t)| , \ \ supp \ f(1-g)\subset S^c$$ My thoughts about how to show...

Homework Statement This is a homework problem for a numerical analysis class. Use the following theorem to find bounds for the number of iterations needed to achieve an approximation with accuracy 10^-5 to the solution of the equation given in part (a) lying in the intervals [-3,-2] and...

I want to prove the following inequality: $$\sum\limits_{k\in\mathbb{N}}\Big(\int \big|f(x)\big|\big|g(x-k)\big|dx\Big)^2 \leq \big\|f\big\|^2 \sum\limits_{k\in\mathbb{N}}\Big (\int\big|g(x-k)\big|dx\Big)^2$$ where $$\|f\|^2=\int |f(x)|^2dx.$$ My attempt: Just prove the following inequality...

Suppose, that $f(x)=ax^2+bx+c$, where $a$,$b$ and $c$ are positive real numbers. Show, that for all non-negative real numbers $x_1,x_2,…,x_{1024}$ \[\sqrt[1024]{f(x_1)\cdot f(x_2)\cdot \cdot \cdot f(x_{1024})} \geq f\left ( \sqrt[1024]{x_1\cdot x_2\cdot \cdot \cdot x_{1024}} \right )\]

Hey! :o We consider the exponential distribution. I want to show that $$\mathbb{P}\left (\left |X-\frac{1}{\lambda}\right |\leq \lambda \right )\geq \frac{\lambda^4-1}{\lambda^4}$$ I have shown so far that \begin{align*}\mathbb{P}\left (\left |X-\frac{1}{\lambda}\right |\leq \lambda \right...

Homework Statement I Type I Cost per pupil I I Full session I $50 I I Half session I $30 I The above table shows the cost of lessons per month to students attending a private class. The class operates under the following limitations...

Let $a_1 = 1$, $a_2 = 1$ and $a_n = a_{n-1}+a_{n-2}$ for each $n > 2$. Find the smallest real number, $A$, satisfying \[\sum_{i = 1}^{k}\frac{1}{a_{i}a_{i+2}} \leq A\] for any natural number $k$.

Hello, Here's an interesting question inspired by a homework probem (not mine), we know that circular orbit (for scjwarzchild black hole) exist only if L ≥ sqrt3 c Rsch=Lisco . Where does this inequality come from? do you have a lecture which can help me to understand? Thanks

I’m only an interested layman with no background in physics and just basic math. But I find a lot of physics fascinating and read up when I can. One thing I’ve struggled with for ages is understanding just exactly what a violation of Bell’s theorem means when referring to the tests done on...

Hi, I found in Kreyszig that if for any ##x_1\ and\ x_2\ \in \mathscr{D}(T)## then an injective operator gives: ##x_1 \ne x_2 \rightarrow Tx_1 \ne Tx_2 ## and ##x_1 = x_2 \rightarrow Tx_1 = Tx_2 ##If one has an operator T, is there an inequality or equality one can deduce from this, in...

I'm trying to do some practice Putnam questions, and I'm stuck on the following: For ##a,b,c \geq 0##, prove that ##(a+b)(b+c)(c+a) \geq 8abc## (https://www.math.nyu.edu/~bellova/putnam/putnam09_6.pdf) I started off by expanding the brackets and doing some algebraic rearranging, but I don't...

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of the Cauchy-Schwarz Inequality ... Duistermaat and Kolk"s proof of the Cauchy-Schwarz Inequality...

Please consider the following premises and correct me if I'm wrong in anyone: Based on the results of the experimental investigation of Bell's theorem and violation of the Bell's inequality, locality in tandem with reality is not applicable to quantum systems (no theory of local realistic...

problem statement: need to show: ||w||_p^2+||u||_p^2-2(||u||_p^p)^{\frac{2}{p}-1}\Sigma_i(u(i)^{p-1}w(i)) can be bounded as a function of ||w-u||_p^2 where p\in[2,\infty) work done: the expressions are equal for p=2, and i suspect that...

Determine the positive values of $a,\,b,\,c$ and $d$ such that $\log_{d+3} (a + 3)>\log_{c+2} (d + 2)>\log_{b+1} (c + 1) >\log_a (b)$.

Let $f$ be a positive and continuous function on the real line which satisfies $f(x + 1) = f(x)$ for all numbers $x$. Prove \[\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})}dx \geq 1.\]

Suppose that f is analytic on the disc $\vert{z}\vert<1$ and satisfies $\vert{f(z)}\vert\le{M}$ if $\vert{z}\vert<1$. If $f(\alpha)=0$ for some $\alpha, \vert{\alpha}\vert<1$. Show that, $$\vert{f(z)}\vert\le{M\vert{\frac{z-\alpha}{1-\overline{\alpha}z}}\vert}$$ What I have: Let...

I am trying to find the max and min values of the function ##f(x,y) = 2\sin x \sin y + 3\sin x \cos y + 6 \cos x##. By the Cauchy-Schwarz inequality, we have that ##|f(x,y)|^2 \le (4+9+36) (\sin^2 x \sin^2y + \sin^2 x \cos^2 y + \cos^2 x) = 49##. Hence ##-7 \le f(x,y) \le 7##. My question has...

Prove the inequality: \[\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2} < \frac{7}{4}, \: \:\: \: n\in \mathbb{N}.\] - without using the well-known result: \[\lim_{n\rightarrow \infty }\sum_{k=1}^{n}\frac{1}{k^2} = \frac{\pi^2}{6}\]

Suppose we have a function $f(x)$ which is infinitely differentiable in the neighborhood of $x_0$, and that: $f^{(k)}(x) \ge 0$ for each $k=0,1,2,3,\ldots$ for all $x$ in this neighborhood. Let $R_n(x)=\frac{1}{n!}\int_a^x f^{(n+1)}(t)(x-t)^n dt$ where $x_0-\epsilon <a<x<b<x_0+\epsilon$; I...

Homework Statement Solving an exercise I found myself with this problem: the solution ##c## needs to verify both ##\sum_{k=1}^c n\lambda^k\frac{e^{n\lambda}}{k!}\leq \alpha## and ##1-\sum_{k=1}^{c+1} n\lambda^k\frac{e^{n\lambda}}{k!}\geq \alpha##. Can an equation like this be solved for c...

Dear Everyone, Directions: Decide whether the statement is a theorem. If it is a theorem, prove it. if not, give a counterexample. $$n^2>n$$ for each negative integer n Examples might work for this inequality $$n^2-n>0$$ Let n=-1. Then $$(-1)^2-(-1)>0$$ $$1+1>0$$ $$2>0$$ Let n=-2. Then...

Hello! (Wave) Using induction, I have showed the Bernoulli inequality, i.e. that if $a \geq -1$ and $n \in \mathbb{N}$ then $1+na \leq (1+a)^n$. Now I want to show that if $a \geq -1$ and $n \in \mathbb{N}$ the $1+\frac{1}{n}a \geq (1+a)^{\frac{1}{n}}$. How could we show this? Could we use...

I'd like to start off by saying I'm just a 52 yo interested layman with no back ground in physics so apologize up front for my ignorance! I understand the basic principle behind Bell's Inequality and how it disproves that when measuring the different spin states of a particle, the particle...

Show, that $5x \le 8\sin x - \sin 2x \le 6x$ for $0 \le x \le \frac{\pi}{3}$.

Let x(t) a positive function satisfied the following differential inequality $\frac{x'(t)}{1+{x(t)}^{2}}+x(t)f(t)<2f(t)$ , (1) with $0\leq t\leq T$ , $\arctan(0)<\frac{\pi }{2}$ and $f(t)$ is a positive function. Is x(t) bounded for all $T\geq 0$?

Homework Statement If ##\forall \epsilon > 0 ## it follows that ##|a-b| < \epsilon##, then ##a=b##. Homework EquationsThe Attempt at a Solution Proof by contraposition. Suppose that ##a \neq b##. We need to show that ##\exists \epsilon > 0## such that ##|a-b| \ge \epsilon##. Well, let...

Hi there, I'm having trouble understanding this math problem: |x| + |x-2| = 2 The answer says its: 0<=x<=2 I understand you need different "cases" in order to solve this. For example, cases for when x is less than 0, when x-2 is less than 0, etc. Thanks, blueblast

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with the proof of Proposition 2.1.25 ... Proposition 2.1.25 reads as follows: In the above proof, Sohrab appears to be using...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with the proof of Proposition 2.1.25 ... Proposition 2.1.25 reads as follows: In the above proof, Sohrab appears to be using...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Proposition 2.1.23/Exercise 2.1.24 (Exercise 2.1.24 asks readers to prove Proposition 2.1.23) ... Proposition...

Homework Statement I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Proposition 2.1.23/Exercise 2.1.24 (Exercise 2.1.24 asks readers to prove Proposition 2.1.23)...

Homework Statement a0 = 0, and for n > 0, $$a_n = a_{\frac {n} {5}} + a_{\frac {3n} {5}} + n $$ For the above equation, besides an, the subscripts are floored Prove that an ≤ 20n Homework Equations See above. The Attempt at a Solution I know how to do the question, my problem is starting...

Prove, that for all natural numbers, $a$ and $b$, with $b > a$: \[\frac{ab}{(a,b)}+\frac{(a+1)(b+1)}{(a+1,b+1)}\geq \frac{2ab}{\sqrt{b-a}}\] where $(m,n)$ denotes the greatest common divisor of the natural numbers $m$ and $n$.

If $x_1,x_2,...,x_{10} \in [0;\frac{\pi}{2}]$ and $\sum_{i=1}^{10}\sin^2x_i = 1.$ - then prove, that: \[ \frac{\sum_{i=1}^{10}\cos x_i}{\sum_{j=1}^{10}\sin x_j} \ge 3\]

Suppose, the four real numbers $a,b,c$ and $d$ obey the inequality:$(a+b+c+d)^2 \ge K b c$, when $0 \le a \le b \le c \le d$.Find the largest possible value of $K$.

Prove, that for any triangle: \[\sum_{\alpha \in \left \{ A,B,C \right \}}\frac{1}{1+\sin \frac{\alpha }{2}}\geq 2\]

Given any real No l,then prove,that there exist a>0 such that ,for all natural Nos k there exist n\geq k such that: |2-(-1)^n-l|\geq a

The Bell Inequality tests are only valid for positive numbers, which is reasonable because counts and probabilities cannot be negative. CHSH generates a negative number, which means CHSH experiments are invalid. Bell's Inequality can be violated by having a negative value. For example...

Given $n$ positive real numbers: $x_1,x_2,...,x_n$. Show, that: \[\frac{x_1^2}{x_2}+\frac{x_2^2}{x_3}+ ... + \frac{x_{n-1}^2}{x_n}+\frac{x_n^2}{x_1} \geq x_1+x_2+...+x_n\]

How do we use the graph to solve a given inequality. For example, say the graph of y = x^4 - 4x^3 + 6x^2 - 4x + 2 is given. The graph of y crosses the y-axis at one point. It does not touch or cross the x-axis. In what way can the picture, the graph help us solve either of the following...