What is Inequality: Definition and 1000 Discussions
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that
z
≤
x
+
y
,
{\displaystyle z\leq x+y,}
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
‖
x
+
y
‖
≤
‖
x
‖
+
‖
y
‖
,
{\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|,}
where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute values.
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces.
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...
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...
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...
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 $n$.
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 $K$.
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...
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...
Solve the inequality.
x^2 + 4x - 32 < 0
Factor LHS.
(x - 4) (x + 8) < 0
x - 4 = 0
x = 4
x + 8 = 0
x = -8
Plot x = 4 and x = -8 on a number line.
<--------(-8)----------(4)----------->
Pick a number from each interval.
Let x = -10 for (-infinity, -8).
Let x = 0 for (-8, 4).
Let x = 6...
Homework Statement
Find the solution of the inequality ## \sqrt{5-2sin(x)}\geq6sin(x)-1 ##
Answer: ## [\frac{\pi(12n-7)}{6} ,\frac{\pi(12n+1)}{6}]~~; n \in Z##Homework Equations
None.
The Attempt at a Solution
There are two cases possible;
Case-1: ##6sin(x)-1\geq0##
or...