What is Fixed point: Definition and 101 Discussions
In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f if f(c) = c. This means f(f(...f(c)...)) = f n(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.
For example, if f is defined on the real numbers by
f
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x
)
=
x
2
−
3
x
+
4
,
{\displaystyle f(x)=x^{2}-3x+4,}
then 2 is a fixed point of f, because f(2) = 2.
Not all functions have fixed points: for example, f(x) = x + 1, has no fixed points, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point x means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.
Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point.In Galois theory, the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.
I did not use the hint for this problem. Here is my attempt at a proof:
Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##.
Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...
The first image is the question and the second is the answer.
My thinking is let's say North is positive, and South is negative. Fixed point O is the starting point. Then the question becomes +(2a-b)-(3a+2b). The answer should be -a-3b. I cannot fathom why the book gives the answer as a+b. Any...
“Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes…”
The reason this statement sounds like one of those theora is that I recall reading a Time-Life book on Mathematics, and there...
Today I read a book in mechanics and encountered a funny proposition about rigid body with fixed point. Perhaps somebody will be interested to propose it to students as a task. This proposition is almost correct:)
Consider a rigid body with a fixed point ##O##. Let ##Oxyz## be a coordinate...
Question: For the following functions, does the fixed point iteration for finding the fixed point in $\left [ 0,1 \right ]$ converge for all first points $ p_{0} \in \left [ 0,1 \right ]$?
Justify your answer.
a.$ g(x) = e^{\frac{-x}{2}}$
b.$ g(x) = 3x - 1$
Let me attempt for part a first...
Hey! :giggle:
Question 1 :
Let $g(x)-=x-x^3$. The point $x=0$ is a fixed point for $g$. Show that if $x^{\star}$ is a fixed point of $g$, $g(x^{\star})=x^{\star}$, then $x^{\star}=0$. If $(x_k)$ the sequence $x_{k+1}=g(x_k)$, $k=0,1,2,\ldots$ show that if $0>x_0>-1$ then $(x_k)$ is...
Show that if ##f## is a shrinking map ##d(f(x),f(y)) < d(x,y)## and ##X## is compact, then ##f## has a unique fixed point.
Hint. Let ##A_n=f^n(X)## and ##A=\cap A_n##. Given ##x\in A##, choose ##x_n## so that ##x=f^{n+1}(x_n)##. If ##a## is the limit of some subsequence of the sequence...
If I understand correctly (dubious), given a consistent theory C (collection of sentences), Kripke proposes to add a predicate T so that, if K = the collection of all sentences T("S") for every sentence S in C, ("." being some appropriate coding) then the closure of K∪C forms a new theory C*...
Homework Statement
Charged sphere with a mass of 15 mg and charge 2 nC moves with a speed of 15 cm/s towards a fixed point charge of 3 nC. How close will sphere approach charge?
Homework Equations
K=(1/2)*mv2
U=k*(Q1Q2/r)
The Attempt at a Solution
So I am not sure I approached correctly but...
Hey! :o
We have the function $f(x)=x^5-\frac{5}{16}$.
I have approximated the root of that function using three steps of Newton's method with initla value $x_0=\frac{1}{2}$ :
\begin{align*}x_1&=x_0-\frac{f(x_0)}{f'(x_0)}\approx \frac{7}{5} \\ x_2&=x_1-\frac{f(x_1)}{f'(x_1)} \approx...
Homework Statement
I have a system of coupled differential equations representing chemical reactions and given certain initial conditions for the equations I can observe oscillation behaviour when I solved the equations numerically using Euler's Method. However, then it asks to investigate the...
Homework Statement
Let ##(X,d)## be some metric space, and let ##f : X \to X## be such that ##d(f(x),f(y)) \le a d(x,y)## for every ##x,y \in X## for some ##a \in (0,1)## (such a map is a called a contraction map)
If ##f## is a contraction and ##X## is compact, show that ##f## has a unique...
This isn't a real homework problem (i.e. I made this problem up myself for my own purposes), but I figured this is the correct forum to post.
1. Homework Statement
In the following figure we have two rods connected to each other, and the bottom rod is connected to the blue structure (G)...
Let f be a continuous function of a metric space, M, to itself with a dense orbit and a fixed point.
I.e. there exists z such that the set {f(n)(z)} for all n ∊ N (where f(n) is the nth iterate of f) is dense in M, and there exists p such that f(p) = p.
Does this imply that f spreads?
I.e. does...
A smooth vector field on the phase plane is known to have exactly three closed orbit. Two of the cycles, C1 and C2 lie inside the third cycle C3. However C1 does not lie inside C2, nor vice-versa.
What is the configuration of the orbits?
Show that there must be at least one fixed point bounded...
Hi all!
For a talk I want to compare the values of the critical exponents found by Wilson and Fisher in their \epsilon = 1 paper (10.1103/PhysRevLett.28.240), with the experimental values measured up-to-date.
Can anyone provide a source for these measured values (\gamma, \nu, \eta)?
Thanks...
Not sure whether this is an intro physics or intro calculus/related rates problem.
1. Homework Statement
Suppose a point P lies at (x,y)=(0,1) meters.
A car is traveling at 30 meters/second along the x-axis towards +∞.
Define r to be the distance between P and the car at any time t.
I...
Homework Statement
Let be ##f \in C^{1}(\mathbb{R}^{n}, \mathbb{R}^{n})## and ##a \in \mathbb{R}^{n}## with ##f(a) = a##. I'm looking for a suffisent and necessar condition on f that for all ##(x_{n})## define by ##f(x_{n}) = x_{n+1}##, then ##(x_{n})## converge.
Homework Equations
##f(a) =...
Homework Statement
Homework Equations
The Attempt at a Solution
I started looking at this problem and I think I am going to have to use the intermediate value theorem for this proof, but I am not quite sure. I started looking at possible examples of these functions, but I know this is not...
Homework Statement
A person is standing on tiptoe, with the total weight supported by the force on the toe. A mechanical model for the situation is shown, where T is the force in the Achilles tendon and R is the force on the foot due to the tibia. Find the value of T. Assume the total weight is...
What is an example of a rigid body rotating when one point is fixed and there are no net applied torques? And the fixed point is not the center of mass.
I considered a cone rolling without slipping on a flat plane is such an example; the apex is the fixed point, but is there a net applied...
first of all i simply don't want to give up learning numerical methods ...
i am trying to follow fixed point iteration method from this link ...
http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf
fixed point iteration can be used to solve...
so if $a\left(t\right)=p+qt$
then $v\left(t\right)=\int a\left(t\right) dt = pt+\frac{q {t}^{2}}{2}+C$
if $v=3.5$ when $t=2$ then $1.75=p+q$
if so, now what? the answers are $4, -3; 4m$
loop quantum gravity and loop quantum cosmology gravity becomes weaker then repulsive instead of stronger, towards the Planck regime, due to the onset of quantum effects. gravity near Planck energies and densities, LQG/LQC becomes repulsive.
at the inflection point where LQG gravity strength...
Homework Statement
I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point...
Let (X, d) be a complete metric space, and suppose T : X → X is a function such that T^2 is a contraction. [By T^2, we mean the function T^2 : X → X given by T^2(x) = T(T(x))]. Show that T has a unique fixed point in X.
So I have an answer, but I am not sure whether it is correct. It goes as...
İn some articles, I see something...
For example,
Let we define a sequence by ${x}_{n}=f{x}_{n}={f}^{n}{x}_{0}$$\left\{{x}_{n}\right\}$. To show that $\left\{{x}_{n}\right\}$ is Cauchy sequence, we suppose that $\left\{{x}_{n}\right\}$ is not a Cauchy sequence...For this reason, there exists a...
Let $T$ be a continuous mapping of a complete metric space $X$ into itself such that ${T}^{k}$ is a contraction mapping of $X$ for some positive integer $k$. Then $T$ has a unique fixed point in $X$.
Proof:
${T}^{k}$ has a unique fixed point $u$ in $X$ and...
I see that if a mapping is contraction then it is contractive then it is nonexpensive and then it is lipschtiz...so, which class of mapping is general ? lipschitz or contraction ? which one ? thank you for your attention :)
Here I do not perceive the a sequence generated by fixed-point iteration. First would you like to explain this. How can it be that if lim n->∞ pn=P, then lim n-> ∞ Pn+1 ?
Source: Numerical Methods Using Matlab by Kurtis D. Fink and John Matthews.
Homework Statement
For which of them will the corresponding fixed point iteration xk+1 = g(xk) be locally convergent to the solution xbar in [0, 1]? (The condition to check is whether |g'(xbar)| < 1.)
A) 1/x2 -1
B)...
C)...
compute xbar to within absolute error 10-4.
Homework Equations
3. The...
Hello everyone. I'm studying the fixed point of theory in the context of QFT. First of all, let me say what I think I understood about fixed points and then I'll state my question.
Suppose we have a theory with a certain running coupling ##\lambda(\mu)##. If we have, for example, an UV fixed...
I need help with the proof of the Fixed Point Theorem for a metric space (S,d) (Apostol Theorem 4.48)
The Fixed Point Theorem and its proof read as follows:
In the above proof Apostol writes:
" ... ... Using the triangle inequality we find for m \gt n,
d(p_m, p_n) \le \sum_{k=n}^{m-1}...
I found an issue with the fixed point library, and Heiko Oberdiek found the offending code and submitted a correction to fix the problem in the fixed point library.
The details can be found here.
At present, you can issue the command below to over come the problem. Eventually someone will...
Investing function fc(x) = (6/x)+(x/2)-c where 0<= c <=3
a) Use alegbra to find the positive fixed point (in terms of c) and identify its exact interval of existence
b) Use algebra and calculus to find the exact interval of stability of the fixed point
c) Use algebra to find the points of the 2...
Hello! ;) I have a question.
Let $\varphi:[-1,1] \to [-1,1]$ with $L=0.8$ at $[-1,1]$, $\varphi$ has a unique fixed point $x^{*}$ and the sequence $(x_{n})$ with $x_{n+1}=\varphi(x_{n}) ,n=0,1,2,...$ is well defined and coverges to $x^{*}$ for any $x_{0} \in [-1,1]$.Then if the 9th approximation...
Homework Statement .
Let ##(M,d)## be a metric space and let ##f:M \to M## be a continuous function such that ##d(f(x),f(y))>d(x,y)## for every ##x, y \in M## with ##x≠y##. Prove that ##f## has a unique fixed point
The attempt at a solution.
The easy part is always to prove unicity...
Homework Statement .
Let ##X## be a complete metric space and let ##T:X \to X## such that there exists ##n \in \mathbb N##: ##T^n## is a contraction. Prove that there is a unique ##x \in X## such that ##T(x)=x##.
The attempt at a solution.
Sorry but I am completely lost with this exercise...
Homework Statement
If all the lines given by the equation (3\sin \theta + 5\cos \theta )x+(7\sin \theta - 3\cos \theta )y+11(\sin \theta - \cos \theta)=0 pass through a fixed point (a,b) forall theta in R then |a-b|=
Homework Equations
The Attempt at a Solution
Dividing both sides by...
Hi everyone, I has been learning numerical method recently, i am very wonder how fixed point iteration method and Newton raphson method works (a more insight explanation rather than mathematical proof ) thanks!
Homework Statement
Let g(x)= (2/3)*(x+1/(x^2)) and consider the sequence defined by pn= g(pn-1) where n≥1
a) Determine the values of p0 \in [1,2] for which the sequence {pn} from 0 to infinity converges.
b) For the cases where {pn} converges (if any), what is the rate of convergence...
W= z+2 /z-2 drawing mapping find image in w plane line Re(z)constant and im(z)=constant find fixed point from mapping
In my textbook have just W = z-1 / z+1 .
Thank a lot for your help.
Homework Statement
OK, I need to solve for the fixed point of the equation
2sinπx + x = 0
on the interval [1, 2]. I know the answer to be ~1.21... but I need to prove it.
The Attempt at a Solution
I really just need help solving for a proper equation of x. I tried x = -2sinπx...
Homework Statement
Apply fixed point iteration to find the solution of each equation to eight correct decimal places
x3=2x+2
The Attempt at a Solution
I have tried to rewrite the equation for in every possible way to solve for one x and pluggin in my guess( have tried...
If you have a surjective function from the unit square, [0,1] X [0,1] onto a bigger square, such as [0,3] X [0,3], will there always be a fixed point under any injection of the unit square into the big square (i.e. will there always be x s.t. f(x)=i(x), where i is an injection?)
It seems to...
Hello!
I have a neuroscience question about local field potenitals (LFPs).
I was reading a jounral article where the LFP in a given location was recorded overtime, and then the LFP at ecah time point was divided into different frequency groups: i.e. the LFP at a given point in space...
If we want to solve $$f(x)=0$$ we can re-write the equation as
$$g(x)=x$$ and use the fixed point method, i.e, $$x_{n+1}=g(x_n)$$ starting with a guess $$x_0.$$ I was wondering if something similar can be done with
$$\Lambda(x,y)=h(x,y).$$
Hello,
I am currently trying to figure out if its possible to go from a number in (unsigned) Q3.5 format -> 8 bits given.
And print it on a display in decimal.
I only have a 8 bit microcontroller to work with.
For example 011.11010 should have 3.8125
Getting to 3 is easy but I...