Recent Questions - Mathematics Stack Exchange |
- The payoff in binomial model is a martingale.
- Proof involving lim sup and lim inf
- $-\cos(z)$ is a biholomorphic
- How to solve this differential equation of first order and higher degree solvable for y
- Non-zero measured single point sets under finitely additive measure are countable
- On $\int e^{ix^ x}dx$
- Help in finding CDF when PDF is given.
- Least squares with complex conjugate
- Is My Proof Correct? (Prove that if $a_i|b_i$ for all $1 \leq i \leq n$, then $(a_1 a_2 \cdots a_n)|(b_1 b_2 \cdots b_n)$)
- Stationary increments definition
- Determine where a point is on a plane
- I feel like this "proof" is incorrect, but I don't know why?
- How to understand this statement (measure theory)
- Solving two equations in polar coordinates where one of the equations is $r = 1$
- Determine whether the series diverges or converges
- Integral Domain Properties
- Hungerford exercise V.2.8
- the imaginary function and its implications
- Let $k$ be a positive integer, $S$ is a set of $k$ nonzero complex numbers, sastifying $a,b\in S\Rightarrow ab\in S$
- Prove that there exists a natural number $n$ that has more than 2017 divisors $d$ satisfying $\sqrt{n} \le d < 1,01\sqrt{n}$
- More thoughts on if a entire function has at least one coefficient of the Taylor series being real
- Can a probability density function diverge at an endpoint?
- Deriving the equation for radial wave function
- Let $G$ be a nilpotent group and $|G| = p_1p_2p_3$ where $p_i$ are different prime numbers. Prove that $G$ is an abelian group.
- PDF of $\frac{\min(X,Y)}{\max(X,Y)}$
- Doubt about the proof of a property of compact subspaces of a union set with the union topology
- Pre-1930 outlooks on the decidability of syntactic consistency
- What does$ f(f(x))=0$ mean?
- Measure space with range $[0,1] \cup [3, \infty]$
- What is the difference between counting and measuring?
| The payoff in binomial model is a martingale. Posted: 12 Sep 2021 07:54 PM PDT Let $V_N$ the payoff of a security at time $N$, recurssvely define \begin{equation} V_n=\frac{1}{r+1}(\tilde{p}V_{n+1}(H)+\tilde{q}V_{n+1}(T)) \end{equation} where $\tilde{q},\tilde{p}$ are the risk free probabilities. I am trying to proff that $V_n$ is a martingale, what I was doing is given a secuense of results $(\omega_1,...,\omega_n)$ of $H,T$, then \begin{equation} \begin{aligned} V_n(\omega_1,...,\omega_n)&=\frac{1}{r+1}(\tilde{p}V_{n+1}(\omega_1,...,\omega_n,H)+\tilde{q}V_{n+1}(\omega_1,...,\omega_n,T))\\ &=\frac{1}{1+r}\tilde{\mathbb{E}}(V_{n+1}|\mathcal{F}_n) \end{aligned} \end{equation} where $\{\mathcal{F}_n\}$ is the information given by $(\omega_1,...,\omega_n)$, but this means that is not a martingale, so I don't know what I am wrong, or what is my mistake?. Thank you very much for the help.  |
| Proof involving lim sup and lim inf Posted: 12 Sep 2021 07:54 PM PDT I have the following analysis problem:
I know that this question has been asked about in other posts (such as here: Problem involving Characteristic Function and Sequence of sets.), but I am having trouble just understanding what the question is even saying because I am confused by the notation. For example, what do $\chi_{\lim \inf A_i}(x)$ and $\chi_{\lim \sup A_i}(x)$ mean? Any clarification would be appreciated.  |
| Posted: 12 Sep 2021 07:49 PM PDT Problem statement: Show that $f(z) = -\cos(z)$ is a biholomorphism between the semi infinite strip $\{(x,y): 0<x<\pi, y>0\}$ to the upper half plane. To show holomorphic, we just notice that the imaginary part magnitude of $e^{iz} = e^{-y}e^{ix}$ is strictly smaller than $e^{-iz} = e^{y}e^{-ix}$. But is there an easy way to show biholomorphism without using complicated injective/surjective argument? Technique from Biholomorphic mapping of $\tan(z)$ does not seem to work in this case.  |
| How to solve this differential equation of first order and higher degree solvable for y Posted: 12 Sep 2021 07:54 PM PDT The question is $$x-yp=p^2$$. I arrived at the value of x which is $$x=(-p^2/2)+(1/2)(p/√(p^2-1)log|p+√(p^2-1)|$$ But the answer is given as $$x=p/√(p^2-1)(c+ sin^-1p)$$ Where $p=dy/dx$  |
| Non-zero measured single point sets under finitely additive measure are countable Posted: 12 Sep 2021 07:34 PM PDT Problem:
And there is an extended problem:
In fact I just have a little idea about the first question. I want to use the 'finitely additive' to say $D_\mu$ is countable(Construct an increase set list). But it seems impossible because we don't know whether$D_\mu$ is countable.  |
| Posted: 12 Sep 2021 07:32 PM PDT We know about the Fresnel Integrals: $$\mathrm{C(x)=\int \cos\ x^2 dx, S(x)=\int \sin\ x^2 dx}$$ which can also be written as: $$\mathrm{\int e^{ix^2}dx=C(x)+i\,S(x)}$$ To make a more interesting and tetration based integral with a rapidly oscillating part of $\mathrm{Re(x)\ge0}$ integral converges. To make a more general result possible, letks consider the goal integral: $$\mathrm{\int e^{ix^ x}dx=\int cos\ x^x dx+i\,\int sin\ x^x dx}$$ Then the exponential Taylor Series comes to mind. Is there a good way to integrate this function? I will add more. Please correct me and give me feedback!  |
| Help in finding CDF when PDF is given. Posted: 12 Sep 2021 07:53 PM PDT
I need to find the CDF of the given function but here the intervals are not continuous so I am confused.  |
| Least squares with complex conjugate Posted: 12 Sep 2021 07:26 PM PDT Using the least squares equations: $$Ax=b$$ $$ A^HAx = A^Hb$$ $$ x = (A^HA)^{-1} A^Hb$$ What is the impact of changing the conjugation such that: $$ x = (AA^H)^{-1} Ab^H $$ I acknowledge that this is an incorrect derivation of least squares. However, I am working on a software implementation and realized the conjugation could happen in the incorrect way and that results in a conjugated x vector solution. What is the impact/side effect of having a conjugated weight vector?  |
| Posted: 12 Sep 2021 07:23 PM PDT This is from Grimaldi, Exercise 6 from Section 4.3. I'm posting this question because the book's approach to the proof used mathematical induction, but my approach didn't, and I don't see why we need to use mathematical induction. Here is the question: Let $n\ in Z^+$ where $ n \geq 2$. Prove that if $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n \in Z^+$ and $a_i|b_i$ for all $1 \leq i \leq n$, then $(a_1 a_2 \cdots a_n)|(b_1 b_2 \cdots b_n)$. (In case the notation isn't universal, the book defines $b|a$ as meaning "b divides a" and equal to $a = bn$, with $n$ some integer.) Here is my proof. Let's choose some arbitrary $n \geq 2$. That means that $a_i|b_i$ for all $1 \leq i \leq n$, which means that $b_i = a_i x_i$, where $x_i$ is the relevant integer. Thus, $(b_1 b_2 \cdots b_n) = (a_1 x_1 \cdot a_2 x_2 \cdots a_n x_n ) = (a_1a_2 \cdots a_n)(x_1 x_2 \cdots x_n) $. Since $(a_1a_2 \cdots a_n)|(a_1a_2 \cdots a_n)(x_1 x_2 \cdots x_n)$, that means that $(a_1a_2 \cdots a_n)|(b_1 b_2 \cdots b_n)$. Also, since we chose an arbitrary $n\geq2$ at the beginning, it should be true for all $n$ by the rule of universal generalization, and we're done. Is that a valid proof? Or am I missing something and we do need to do mathematical induction here? I apologize if this is rather simple, I just want to make sure I'm not missing an obvious reason for why mathematical induction is required here.  |
| Stationary increments definition Posted: 12 Sep 2021 07:28 PM PDT Let $X_t$ be a stochastic process on the reals. I was under the impression the definition of stationary increments was that $X_{t+s}-X_{s}$ has the same distribution as $X_{t}-X_{0}$ for all $s,t$. I also thought stationary processes have stationary increments (is this correct)? However, the wikipedia page for Levy process (https://en.wikipedia.org/wiki/Lévy_process) says: 'Stationary increments: For any $s<t$, $X_t-X_s$ is equal in distribution to $X_{t-s}$. Suppose now $X_t$ is a Gaussian Process with mean $0$ and covariance function $K(t-s), K(0)=1$. By construction $X_t$ is stationary and therefore has stationary increments. However, $X_t-X_s$ has distribution $N(0,2-2K(t-s))$, whereas $X_{t-s}$ has distribution $N(0,1)$, and they are not equal in general? Which definition is the correct one? I'm guessing $X_t-X_s$ is equal in distribution to $X_{t-s}-X_{0}$ would be the correct definition but the wikipedia page omitted $X_{0}$ because it's assumed to be almost surely 0 for Levy processes?  |
| Determine where a point is on a plane Posted: 12 Sep 2021 07:26 PM PDT Given the points $A(1,0,1), B(2,3,0), C(-1,1,4)$, determine whether the point $D(0,3,2)$ lies on the plane $(ABC)$  |
| I feel like this "proof" is incorrect, but I don't know why? Posted: 12 Sep 2021 07:37 PM PDT I'm in an introductory differential equations class, and while studying, I worked out the following "proof", which I feel is almost certainly incorrect but I'm not sure why. If I had to guess, it would be some sort of error caused by dividing by 0 or with the boundary wherein the solution is valid, but I'm not sure. Let $y = f(x)$. Then, $\frac{dy}{dx} = f'(x)$. Solving as a separable differential equation, $$\frac{dx}{dy} = \frac{1}{f'(x)} \rightarrow \int dx = \int \frac{1}{f'(x)} dy$$ $$x = \frac{y}{f'(x)} + C \rightarrow y = xf'(x) + Cf'(x)$$ $$f(x) = xf'(x) + Cf'(x)$$ I can find many counterexamples which show that this is not true, for example: $$f(x) = x + \ln(x) + \sin(x)$$ Plugging into the above formula $f(x) = xf'(x) + Cf'(x)$ gives $$f(x) = x + \ln(x) + \sin(x) \neq x\bigl(1 + \frac{1}{x} + \cos(x)\bigr) + C\bigl(1 + \frac{1}{x} + \cos(x)\bigr)$$ for which I can think of no constants $C$ for which this is true.  |
| How to understand this statement (measure theory) Posted: 12 Sep 2021 07:13 PM PDT Am reading an introductory text about measure theory. I need help in understanding the following concept ... The text mentions two theorems. Here $m^*$ refers to the outer measure of a set:
Using these two theorems, the text comments afterwards:
I am trying to understand the last line of the above quotation, i.e. the last inequality goes in the wrong direction... I was cracking my head on what the line goes in the wrong direction really means ... So far, I found two ways to interpret it: (1) It is the case that $m^*(G) - m^*(A) > m^*(G \setminus A)$. However, this does not seem to make any sense since it contradicts Theorem 1 above and goes against the additivity property of outer measure ... Theorem 1 should hold for all cases right ? (2) $m^*(G) - m^*(A) \leq m^*(G \setminus A)$ is simply undefined. The only way I can see this if $A$ is a non-measurable set. Among the above, I think (2) makes more sense ... but this means that given $A$ is non-measurable, then $m^*(A)$ should be some nonzero number greater than $m^*(G)$ right ? since if $m^*(A)$ is zero then it should be measurable ....  |
| Solving two equations in polar coordinates where one of the equations is $r = 1$ Posted: 12 Sep 2021 07:55 PM PDT Problem: Looking at the plot we see that there are $8$ points where the two curves intersect. We also see that:
The above facts suggests that this system of equations will have $8$ solutions. However, it has only $4$ solutions because some of the points of intersection occur when $r = -1$. Now we need to find the possible values for $\theta$. \begin{align*} 2 \sin 2\theta &= 1 \\ \sin 2\theta &= \dfrac{1}{2} \\ \text{Let }u = 2\theta \\ \sin u &= \dfrac{1}{2} \\ \end{align*} Now we have $u = \dfrac{\pi}{6} \pm 2\pi$ or $u = \dfrac{5\pi}{6} \pm 2\pi$. The values for $\theta$ are $\dfrac{\pi}{12} \pm \pi$ or $\dfrac{5\pi}{12} \pm \pi$. Hence, the four points are: $$\left( 1, \dfrac{\pi}{12} \right), \left( 1, \dfrac{5 \pi}{12} \right), \left( 1, \dfrac{13 \pi}{12} \right), \left( 1, \dfrac{17 \pi}{12} \right) $$ The book's answer is: $$\left( 1, \pm \dfrac{\pi}{12} \right), \left( 1, \pm \dfrac{5 \pi}{12} \right), \left( 1, \pm\dfrac{13 \pi}{12} \right), \left( 1, \pm\dfrac{17 \pi}{12} \right) $$ According to the book, $\left( 1, -\dfrac{\pi}{12} \right)$ is a point of intersection. However, $$ r = 2 \sin ( 2 \theta ) = 2 \sin\left( -2\left( \dfrac{\pi}{12}\right) \right) = -1$$ Therefore it seems to me that $\left( 1, -\dfrac{\pi}{12} \right)$ does not satisfy both equations and it is not a point of intersection. What am I missing?  |
| Determine whether the series diverges or converges Posted: 12 Sep 2021 07:13 PM PDT I would appreciate help in determining if the following series diverges or converges: $\sum_{n=1}^{\infty} \left(\ln (n+1)-\ln n\right)^{\ln n}$. I know of one approach, but that gets very complicated, and I thought that someone in here might know of a slightly smoother way to do this? My approach: I have started by rewriting the expression to $e^{ln(n)ln(ln(n+1)-ln(n))}$. Then I applied the Limit Chain rule where $g_1(x)=ln(n)ln(ln(n+1)-ln(n))$ and $f_1(u)=e^{u}$ and started with solving $$(1)\lim_{x\rightarrow\infty}ln(n)ln(ln(n+1)-ln(n))=\lim_{x\rightarrow\infty}ln(n)\lim_{x\rightarrow\infty}ln(ln(n+1)-ln(n))=\infty\cdot\lim_{x\rightarrow\infty}ln(ln(n+1)-ln(n))$$ And to solve that I applied the Limit Chain rule again where $g_2(x)=ln(n+1)-ln(n)$ and $f_2(u)=ln(u)$ and started solving $$(2)\lim_{x\rightarrow\infty}ln(n+1)-ln(n)=\lim_{x\rightarrow\infty}ln(1+\frac{1}{n})$$ And here I need to apply the Limit Chain rule again where $g_3(x)=1+\frac{1}{n}$ and $f_3(u)=ln(u)$ and started solving $$(3)\lim_{x\rightarrow\infty}1+\frac{1}{n}=1.$$ And only from here I can start computing all $f(u)$, so for $f_3(u)$ we have that $\lim_{u\rightarrow 1}ln(u)=0$. For $f_2(u)$ we have that $\lim_{u\rightarrow 0}ln(u)=-\infty$ that we now can put in (1) and get $\infty(-\infty)=-\infty$. Now we can finally determine $f_1(u)=\lim_{u\rightarrow-\infty}e^u=0$ and say that the series is convergent.  |
| Posted: 12 Sep 2021 07:29 PM PDT I've been self-teaching some abstract algebra and am wondering if it is possible to prove that an integral domain's property that the product of any two nonzero elements is nonzero follows logically from an integral domain being nonzero. In other words, does 1 != 0 imply the product of any two nonzero elements is nonzero? If so, what is the proof, and if not, why?  |
| Posted: 12 Sep 2021 07:19 PM PDT Assume $\operatorname{char}K = 0$ and let $G$ be the subgroup of $\operatorname{Aut}_K K(x)$ that is generated by the automorphism induced by $x\mapsto x + 1_K.$ Then $G$ is an infinite cyclic group. Determine the fixed field $E$ of $G.$ What is $[K(x) : E]$? I think that this exercise begin by using first part of Artin's lemma and get $K(x)$ is Galois over $E.$ I.e., $G < \operatorname{Aut}(K(x)/E)$, we get $E=G'=G'''=E''$ lemma V.2.6-(iv) of Hungerford. But I don't know how to proceed. Can someone help me?  |
| the imaginary function and its implications Posted: 12 Sep 2021 07:51 PM PDT $i = \sqrt{z^n-x^n-y^n}$ For n = 0 So with this coordinate description isn't the derivative taken with respect to r a valid algebraic expression and the resulting topology possibly valid in a representation of a black hole? Which then we can map a path along this surface and possible predict where a particle ends up at.  |
| Posted: 12 Sep 2021 07:53 PM PDT Let $k$ be a positive integer, $S$ is a set of $k$ nonzero complex numbers, sastifying $a,b\in S\Rightarrow ab\in S$. Show $\forall\ a\in S, a^k=1$, and find the sum of elements of $S$. Clearly, $a,a^2,\dots,a^{k+1}\in S$, and hence for some $i<j, a^i=a^j, a^{j-i}=1$. Then what to do?  |
| Posted: 12 Sep 2021 07:51 PM PDT
My reasoning was that $\tag 2 1.01\sqrt n-\sqrt n\ge2019$ must occur. Otherwise, there won't be enough space to fit in enough numbers in $(1)$. $(2)$ gives us that $n\geq 201900^2$. We also have that $201900^2<201900!$ so we can pick any factorial greater than or equal to $201900!$ and that satisfies the desired condition. Is this solution correct?  |
| More thoughts on if a entire function has at least one coefficient of the Taylor series being real Posted: 12 Sep 2021 07:15 PM PDT I have asked a question few days ago: The question is:
Thanks for @MartinR and @Conrad for their excellent explanation. Today, I was reviewing the proofs. I recall that the Liouville's Theorem stated that is $f$ is a bounded entire function, then $f$ is a constant. And because being a polynomial implies that there is some $N$ such that for all $k>N$, the derivatives are $f^{(k)}(z)=0$. I am wondering if there is a way to prove the question with using the Liouville's Theorem. Does anyone can help me figure it out?  |
| Can a probability density function diverge at an endpoint? Posted: 12 Sep 2021 07:04 PM PDT The essence of my question is in the title. Suppose we want to draw numbers from the unit interval $[0,1]$; can $p(x)$ be, say, of the form $$ p_1(x; \varepsilon) = \frac{\varepsilon}{x^{1-\varepsilon}}, $$ despite the fact that $p_1(x)$ is unbounded in the $x \to 0$ limit? Is there any 'legal' way to do away with the regulator $\varepsilon$ and take $\varepsilon \to 0$? (maybe this limit is just the Dirac $\delta(x)$, but it's certainly not a standard delta-sequence example) Clearly if we put a 'cut' on the possible values of $x$ and restricted ourselves to $[\varepsilon, 1]$ we could also do something like $$ p_2 (x; \varepsilon) = \frac{1}{-\log \varepsilon} \frac{1}{x}; $$ I would expect this to have the same limit, though as we've regulated it in a different way it has different properties. Any insight or references on this would be appreciated.  |
| Deriving the equation for radial wave function Posted: 12 Sep 2021 07:41 PM PDT I'm trying to solve Schrodinger's equation of an exciton using the separation of variables method: $\psi = RY$. Here's the equation I've already derived: $$ \frac{2\mu r^2}{\hbar^2}(E+\frac{e^2}{\epsilon r})+\frac{r^2}{R}\frac{\partial^2 R}{\partial r^2}+\frac{r}{R}\frac{\partial R}{\partial r} = -\frac{1}{Y}\frac{\partial^2Y}{\partial\theta^2} $$ Where the radial wave function is $R = R(r)$ and angular part is $Y = Y(\theta)$. Since the two sides of the equation only depend on a single variable, they are both constants. I've found the angular part is $$ Y_m(\theta) = \frac{1}{\sqrt{2\pi}}e^{im\theta} $$ I'm having trouble solving the left part to obtain the radial equation. How can I do that? Thanks:)  |
| Posted: 12 Sep 2021 07:18 PM PDT
It has been a long time since I worked with algebra and I am at a loss here. I was thinking of creating the sequence: $$ 0 \rightarrow G_1 \rightarrow G_{1,2} \rightarrow G_{1,2,3} = G $$ where $|G_1| = p_1$ and $|G_{1,2}| = p_1p_2$ and $|G_{1,2,3}| = p_1p_2p_3$. Now I think I will have to work my way from here considering their quotient groups are cyclic and I remember of a theorem where the quotient group $G/G'$ (where $G'$ is the commutator) is cyclic then the group is abelian. Probably it can be adapted. But the details of a proof escape me.  |
| PDF of $\frac{\min(X,Y)}{\max(X,Y)}$ Posted: 12 Sep 2021 07:38 PM PDT I am given that $X$ and $Y$ are iid Exponential with parameter $\lambda$. Then I am to compute pdf of $\frac{\min(X,Y)}{\max(X,Y)}$ I have derived that the pdf of $Z=\min(X,Y)$ . is given by:- $f_{Z}(z)=2\lambda e^{-2\lambda z}\,\, , z\geq 0$. And $T=\max(X,Y)$ is given by $f_{T}(t)=2\lambda (1-e^{-\lambda t})e^{-\lambda t}\,\, t\geq 0$. Now to derive the PDF of $\frac{Z}{T}$ I am thinking of using the Jacobian method with a substitution of $U$ and $V$ such that $U=\frac{Z}{T}$ and $V=Z$ . and then integrating the joint pdf of $U,V$ over $v$ to get the pdf of $U$. But I cannot find the joint pdf of $Z,T$ if Z and T are not independent. Is my thinking correct?. Am I in the right direction?. What is the correct method to solve this ?.  |
| Doubt about the proof of a property of compact subspaces of a union set with the union topology Posted: 12 Sep 2021 07:15 PM PDT In p.153 of Algebraic Topology from a Homotopical Viewpoint we can find the following lemma:
Here is the proof:
what is the property of Hausdorff spaces that is not satisfied in the reasoning?  |
| Pre-1930 outlooks on the decidability of syntactic consistency Posted: 12 Sep 2021 07:28 PM PDT There is a passage in (Skolem 1928) that has generated multiple conflicting interpretations (see discussion in Van Heijenoort's intro to (Skolem, 1928) in [1967b, p. 508] as well as (Goldfarb, 1971). Skolem states what is now recognizable as a key lemma of the completeness theorem: either for some $n$ there is no solution of level $n$, or, for every $n$, there are solutions of level $n$. ("Solutions" are truth-value assignments to propositional instances approximating to a first-order formula in normal form - see https://mathoverflow.net/q/373381/116705 for background). Instead of proving completeness by showing that the right-hand disjunct implies satisfiability (a direct consequence of the Lowenheim-Skolem theorem), Skolem gives a syntactic argument to show that "if there are solutions for arbitrarily high n, then the formula is (syntactically) consistent". Conversely, if there is an $n$ for which there are no solutions, the formula "contains a contradiction". The controversy stems from the fact that Skolem's switch to syntax and apparent avoidance of the Lowenheim-Skolem theorem seems unmotivated and the argument he gives is inconclusive. Of the interpretations that have been put forward, none take into account Skolem's aim of proving decidability for all first-order formulas. This of course was later discovered to be impossible, and I'm inclined to think that the obscurity of the syntactic argument can be attributed to Skolem's struggle with this destined-to-fail task. The only interpretation I can think of that would support this is as follows: Perhaps recognizing the difficulty of deciding the key lemma above for the case of formulas of any prefix, Skolem tries to show how it can be replaced by a syntactic counterpart with more hope of being decidable: either the formula is contradictory, or the formula is consistent. To do this he would need to show exactly what he purports to show: (a) if there is a level with no solutions then the formula is contradictory, (b) if for every $n$, there are solutions of level $n$ then the formula is consistent. If he could then show that there was a decidable method for determining the consistency of a basic theory (essentially, propositional logic plus rule of substitution plus the functional form of the formula added as an axiom), this would also decide the alternative given by Godel's lemma, and from there the satisfiability of the formula via the LST. Would logicians in 1928 have had any reason to consider syntactic consistency to be a more tractable candidate for a decidability proof? References: GOLDFARB, WARREN D [1971] Review of Skolem 1970. The Journal of Philosophy 68, pp. 520-530. Skolem, Thoralf. [1928] On Mathematical Logic. English translation in van Heijenoort (ed.) [1967], pp. 508– 524. VAN HEIJENOORT, JEAN (ED.) [1967b] From Frege to Gödel; a source book in mathematical logic, 1879-1931. Cambridge, Harvard University Press.  |
| Posted: 12 Sep 2021 07:33 PM PDT I came across a question: If $f(x)=x^3-x+1$ then find the number of real distinct values of $f(f(x))=0$. Here is what I interpreted the $f(f(x))$ as: I assumed $a, b$ and $c$ to be the roots of $f(x)$, now if we put $a, b$ or $c$ in the $f(f(x))$ then it becomes $f(0)$ which will be equal to $1$. I saw a solution where they differentiated the polynomial $f(x)$. They made the graph of $f(x)$ using first order derivative test. Then for $f(f(x))$, they put in $x=a, b, c$ (assumed roots of $f(x)$ ). Then we got three lines for $x=a$, a line below $-1$, for $b$ a line between $0$ and $1$ and for $c$, a line between $1$ and $3$. I am unable to understand why we put $a,b$ and $c$ as $x$ and then how did we get these ranges?
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| Measure space with range $[0,1] \cup [3, \infty]$ Posted: 12 Sep 2021 07:55 PM PDT Let the space be $\mathbb R$ and the $\sigma$-algebra be $\mathcal P(R)$; define a measure $\mu$ on $(\mathbb R,\mathcal P(R))$ in the following way: $\mu(\{x\}) = x$ if $x \ge 3$. $\mu(\{n\}) = 2^{n}$ if $n$ is a negative integer. $\mu(\{x\}) = 0$ for other values of $x$. I think this defines a measure with range $[0,1] \cup [3, \infty]$. I am using what's on page 42 of Sheldon Axler's Measure, Integration, and Real Analysis which basically says that you can define a measure by defining its value on singleton sets (it's not stated as a theorem, so I am not sure if I have misunderstood anything). Am I correct?  |
| What is the difference between counting and measuring? Posted: 12 Sep 2021 07:06 PM PDT Are counting and measurement the same thing? I think that they are different since in my mind the idea of counting pertains to discrete objects while the idea of measurement pertains to continuous objects. I also think of counting as something which can be done directly with numbers while measurement requires a unit.  |
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