Recent Questions - Mathematics Stack Exchange |
- Euclidean domain and principal ideal domain
- Complex Analysis Differentiability by Analyzing Glad of Complex Surface
- If $\|x-y\|\ge c$ for all $x,y\in A$, then $A$ is unbounded
- 5 fair coin flips probability
- Is there a term for when two elements are in the same orbit under a group action?
- Why don't directly obtain the solution from the equality constraint in this convex optimization problem?
- Problem 16.6 How many parameters are there in a variable component mixture consisting of 9 Burr distributions?
- CLT for correlated weighted standard normal random variables
- $X \times \{y\}$ is homeomorphic to $X$
- Shouldn't the chain rule be applied instead of the product rule in linear operators?
- Question about spectral sequece and hypercohomology.
- Roots producing isomorphic trees
- *- homomorphism and self adjointness
- Purely geometric proof of convex polygon facts
- Question about formula for geometric sums
- What is the meaning of "w describes a halting TM" mean?
- Sequence lemma for general topological spaces
- Proof for a theorem about relations beween convergents in continued fractions
- Continuous mapping theorem and convergence in $L_p$
- How to find the elapsed time for one candle to be consumed when it is thicker by three units with respect of another?
- How to find $\int_1^\infty (7 + x \cdot 2^x)/(\ln x + 1 + 3^x) \, dx$
- The equation ${4 \choose k}=6$
- How can Zaremba's Conjecture be true if it fails for $d=1$?
- Geometry : Ratio of area of a triangle to the rectangle containing it
- How do derivatives w/r to polar variables behave at the origin?
- Boundedness of a sequence of bounded, measurable functions on a closed subset of a finite-measure set
- Algebraic structure of the set of (Lebesgue) measurable functions over $\Bbb R^n$
- For any integer $n\ge 1$, which of the following is/are true?
- How to count the number of strings of length 5 in which at least one symbol occurs two or more times.
- Proof that the Sturm-Liouville operator is positive
| Euclidean domain and principal ideal domain Posted: 02 Apr 2021 08:33 PM PDT We know that every euclidean domain is a principal ideal domain. But how to prove that not every principal ideal domain is an euclidean domain. I saw some examples of PID that aren't euclidean domains but i want to know the proof of it please not examples, im a student  |
| Complex Analysis Differentiability by Analyzing Glad of Complex Surface Posted: 02 Apr 2021 08:29 PM PDT How could tell if a complex function plotted on the graph(used matematica)'s $f'(0)$ exists without me providing the equation I used to arrive at this graph? Furthermore, is it possible to tell if Cauchy Riemann Partial Equations are differentiable based on merely looking at the graph without any calculations? I could not find any information on this specific method of finding these information[$f'(0)$ & C.R. eq. existence].
Here's the derivative the $f'(z)$ graph if it is useful:
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| If $\|x-y\|\ge c$ for all $x,y\in A$, then $A$ is unbounded Posted: 02 Apr 2021 08:27 PM PDT Let $A\subset\mathbb{R}^n$ an infinite set and $c\in\mathbb{R}^+$. Prove that if $\|x-y\|\ge c$ for all $x,y\in A$, then $A$ is unbounded. I tried using inequalities, but I'm stuck. Any hint?  |
| Posted: 02 Apr 2021 08:33 PM PDT A coin is flipped five times. For each of the events described below, express the event as a set in roster notation. Each outcome is written as a string of length 5 from{H, T}, such as HHHTH. Assuming the coin is a fair coin, give the probability of each event. (c) The first flip comes up tails and there are at least two consecutive flips that come up heads. When I did this via brute force, I came up with the following event of size 8: {THHHH, THHHT, THHTH, THHTT, TTHHH, TTHHT, THTHH, TTTHH}. Since there are $2^5=32$, possibilities for the 5 coin flips, we have a probability of $\frac{8}{32} = \frac{1}{4}$. This makes sense to me, the issue is that I tried to sanity check myself by doing this via complement. The idea being that p(consecutive heads)+p(non-consecutive heads)=1. Since the first flip must be tails, we're really looking at the ways to achieve consecutive and non-consecutive heads in 4 flips. With that, there is only 1 way to see 0 heads, TTTTT, so $p(0heads)=\frac{1}{16}$. In the case of 1 non-consecutive heads flip, there are 4 positions for it to be, thus $p(1heads) = \frac{4}{16}$. Lastly, for the case of 2 non-consecutive heads flips, there are only two possibilities HTHT and THTH. Thus $p(2heads) = \frac{2}{16}$. Using the sum rule we end up with $p(non-consecutive\, heads) = \frac{7}{16} \Rightarrow p(consecutive\, heads) = \frac{9}{16}$. Then using the product rule, since the first flip has to be heads, we end up with a probability of $\frac{1}{2} \cdotp \frac{9}{16} = \frac{9}{32}$. I can't find the error in my logic for either of these methods but clearly there is one somewhere... If you can find it please let me know and thank you in advance.  |
| Is there a term for when two elements are in the same orbit under a group action? Posted: 02 Apr 2021 08:32 PM PDT Suppose a group $G$ acts on a set $X$, then the relation on the set $X$ given by $a\sim b \iff ga=b$ for some $g\in G$ splits $X$ into equivalence classes called orbits. If $a$ and $b$ are in the same orbit we say $a$ and $b$ are co-orbital (except for the fact that we don't say that, is there a word we can exchange for "co-orbital" so that the statement is somewhat true?)  |
| Posted: 02 Apr 2021 08:12 PM PDT In paper Generative Adversarial Imitation Learning, the authors derived the following Lagrangian(Page 3) $$ \arg\max_{c\in\mathbb R^{\mathcal S\times\mathcal A}}\min_{\rho\in\mathcal D}\bar{L}(\rho,c)=-\bar {H}(\rho)-\psi(c)+\sum_{s,a}(\rho(s,a)-\rho_{E}(s,a))c(s,a) $$ where $s$ and $a$ denote state and action, $c$ is the cost function, $\rho$ and $\rho_E$ are the state-action visitation distribution associated to policy $\pi_\rho$ and $\pi_E$ respectively, $\psi$ is a regularization function(which we assume is a constant and omitted in the following discussion), $\bar H$ is the entropy. Then, on Page 4, they derive
I'm confused about the explanation in the second paragraph, which explain why $\rho(s,a)=\rho_E(s,a)$. But why can't we directly derive it from the constraint? After all, the dual has a solution when the constraint is satisfied and the convexity of $-\bar H$ ensures that $\rho(s,a)=\rho_E(s,a)$ is optimum for the primal.  |
| Posted: 02 Apr 2021 08:11 PM PDT Problem 16.6 How many parameters are there in a variable component mixture consisting of 9 Burr distributions?  |
| CLT for correlated weighted standard normal random variables Posted: 02 Apr 2021 08:10 PM PDT Assume $U_k$ are correlated standard normal random variables. Let $R_k := a_k U_k$, $a_k >0$ ($\sum_{k=1}^{\infty}a_k < \infty$). I'm looking for CLT of $S_p := \sum_{k=1}^{p}\frac{R_k}{\sqrt{p}}$. Since $U_k$ are correlated, I'm looking at CLT for weakly correlated variables, but here identical distribution is also assumed, so not sure what to do with weights $a_k >0$ ($\sum_{k=1}^{\infty}a_k < \infty$). On the other hand, there are variants of CLT for non iid variables, but often independency is assumed. In my case, the form of dependency is specific (and probably irrelevant to the question) Which CLT would work in my case? Are there any known results that would work under combined (weak) dependency and non-identical distribution?  |
| $X \times \{y\}$ is homeomorphic to $X$ Posted: 02 Apr 2021 08:25 PM PDT Can I get a proof verification? The only thing I am unsure about is the proof of continuity of $f,f^{-1}$. I know the proof of bijectiveness is trivial. Prove:$X \times \{y\}$ is homeomorphic to $X$. Attempt: Take the fact that we can prove continuity by showing that the inverse image of basis elements are open. Define the map $f:X \times \{y\} \rightarrow X$ by $f(x,y)=x$. Then $f$ is one to one since $f(x,y)=f(w,y)\implies x=w$ and so $(x,y)=(w,y)$. $f$ is surjective, since for any $x \in X, f(x,y)=x$. So $f$ is bijective. Let $U$ be an open set in $X$. Then $f^{-1}(U)=U \times \{y\}=(X \times \{y\}) \cap (U \times Y)$, which is open in the subspace topology, so $f$ is continuous.Let $W$ be any open set in $X \times \{y\}$. Then $W=(U \times V) \cap (X \times \{y\})$ where $U \times V$ is open in the product topology. Then $f(W)=U$ is open in $X$ and so $f^{-1}$ is continuous.  |
| Shouldn't the chain rule be applied instead of the product rule in linear operators? Posted: 02 Apr 2021 08:29 PM PDT If I have the linear operators $A(x)$ and $B(x)$, and i want to calculate the derivative of $A(B(x))$, shouldn't it be: $$A(B(x))'= A'(B(x))B'(x)$$ Instead of $(AB(x))'= A'(x)B(x) + A(x)B'(x)$ ? This may not make much sense, but if you think about this linear operators as matrices, when we have $AB\vec{x} $, should't we undestand it as "$A(B(x))$", instead of "$A(x)B(x)$"? After all A is acting on $x$ transformed by "B", not in $x$ itself. I am making this stupid question because I am trying to study QM through Functional analysis and operator theory, but when I came to this example I thought about something that I hadn't thought before and now I am in doubt, sorry for the silly question.  |
| Question about spectral sequece and hypercohomology. Posted: 02 Apr 2021 08:03 PM PDT https://en.wikipedia.org/wiki/Hyperhomology In this Wikipedia, I have a question about the hypercohomology spectral sequece. According to above link, there are two spectral sequence. How we get the two spectral sequence?  |
| Roots producing isomorphic trees Posted: 02 Apr 2021 08:03 PM PDT Is there existing terminology for the following property of a rooted tree? I'd like to know how many choices of root node can produce an isomorphic tree. This is different from the number of isomorphic trees. For example, consider this tree.
If the node labeled '2' is the root, there are 4 total choices of root that produce isomorphic trees (2, 10, 8, and 15). If 3 is the root, then there are 2 total choices that would produce isomorphic trees (3 or 12). If 9 is the root, there is only one choice (9). Is there a name, either for (a) this property of equivalence between roots, or (b) the number of nodes sharing this property?  |
| *- homomorphism and self adjointness Posted: 02 Apr 2021 08:02 PM PDT Let $A$ be a $C^*$-algebra. Let $\pi: A \rightarrow A/I$ be the canonical *- homomorphism, where $I$ is a closed ideal of A Show that if $k$ is self-adjoint in $A/I$, then there exists a self-adjoint element $a$ in $A$ such that $\pi(a)=k$. We have that $k= h + I$ for some $h \in A$. Now we just have to show that $h$ is self-adjoint. Since $k$ is self-adjoint, $(h+I)^*= h+I = h^* +I$. Hence, $h=h^*$. Could someone let me know if my proof is correct? I feel like I didn't really use the *- homomorphism here. Thank you!  |
| Purely geometric proof of convex polygon facts Posted: 02 Apr 2021 08:24 PM PDT Suppose I have a polygon with all angles less than 180 degrees. Intuitively, this is a convex polygon, but how do I prove it? (using the line-segment definition of convex polygon, which is that if you choose any two points inside the polygon, you can draw a line segment connecting them that lies within the polygon). Edit: Non-self-intersecting polygon  |
| Question about formula for geometric sums Posted: 02 Apr 2021 07:52 PM PDT A long time ago when I was a high school students I've been taught the formula to sum the first $n$ terms of following geometric series Here are my questions: First: Is there another reason to encourage memorize formula as $S_n=\dfrac{a_1(1-q^n)}{1-q} ?$ Second: Now it is past about $5$ years since I learned about the formula for the first time. and I solved a lot of geometric series problem since that time( almost used the former formula every time). So can I use $S_n=\dfrac{a_1(q^n-1)}{q-1}$ in my head? doesn't that hurts? ( I want to use this because it is easier to calculate for example, $5^1+5^2+5^3+\ldots+5^{10}$ with $S=\dfrac{5(5^{10}-1)}{5-1}$ rather than having this unnecessary minus sign and see this as $S=\dfrac{5(1-5^{10})}{1-5}$  |
| What is the meaning of "w describes a halting TM" mean? Posted: 02 Apr 2021 07:51 PM PDT I was originally looking for languages that weren't context sensitive. In Wikipedia (https://en.wikipedia.org/wiki/Chomsky_hierarchy), I believe the language $\{w | w \text{ describes a terminating TM} \}$ isn't a CSG. I'm not sure how a TM is "described" by a word. As far as I know a TM is a 7 tuple?? Also, could you give another example of a language which isn't a CSG.  |
| Sequence lemma for general topological spaces Posted: 02 Apr 2021 07:51 PM PDT I am struggling to come up with a proof for this. Any ideas? Prove:If $X$ is first countable:If $x \in \bar A$, there is a sequence of points $x_n$ of $A$ converging to $x$. Proof:Let $x \in \bar A$. Since $X$ is first countable, there is a countable neighborhood basis $\{B_n\}$ for $x$ such that $B_{1}\supset B_{2}\supset \cdots$ and $B_{n} \cap A \neq \varnothing$. Define a sequence converging to $x$ by setting $x_{n} \in B_{n}\cap A$ for each $n \in \mathbb{N}$. Then since each open set $U$ containing $x$ contains some $B_{n}$, for any $U$ open containing $x$, there is a $N\in \mathbb{N}$ with $x \in B_{N} \subset U$. So if $n \geq N$, then $x_{n} \in U$ so $x_{n} \rightarrow x$. I realize that this proof is definitely not correct, but I would like to see the correct way to do this, so I took a stab at it. What would be a good way to approach this problem? Also wouldn't this proof be needed as a lemma to prove the sequential characterization of continuity for first countable spaces? Edit: I realize I should probably not use a sequence of nested basis elements to solve this problem, but nested open sets. But if I do this, how should I define the sequence converging to $x$?  |
| Proof for a theorem about relations beween convergents in continued fractions Posted: 02 Apr 2021 08:19 PM PDT Upon reading about some properties of numerators and denominators in continued fractions here (chapter 2.3), I was unable to understand the following transmutation of the expression (circled in red in the image link): Image link In addition to not seeing how the equality in the last line was achieved, I was also under the impression that convergents of continued fractions (the quotients in square bracket notation) must by definition always be integers. The marked transmutation makes the last quotient a fraction.  |
| Continuous mapping theorem and convergence in $L_p$ Posted: 02 Apr 2021 07:59 PM PDT My question kind of generalizes the question in Analogue of continuous mapping theorem for convergence in $L_2$ and it is related to the answer by Nate Eldredge. Suppose that \begin{equation}\tag{1}\label{eq1} E|X_n - X|^p = O(b_n), \end{equation} where $p \geq 1$ and $b_n \to 0$ as $n \to \infty$. I also have that $X_n$ and $X$ are bounded a.e. In this case, what can I say about $E|g(X_n) - g(X)|^p$, where $g$ is a continuous function? Is it possible to find the rate of convergence as above? Since $X_n$ and $X$ are bounded a.e., I think that I can assume that $g$ is bounded and continuous, as discussed by Nate, which ensures $L_p$ convergence (we have that $g(X_n) \to g(X)$ in measure by the continuous mapping theorem, and in $L_p$ by the dominated convergence theorem), but I could not find any rate in my problem. In a more specific case, is it possible to find the rate of convergence for $E|X_n^r - X^r|^p$, with $r > 0$ and $p \geq 1$, assuming \eqref{eq1} and that $X_n$ and $X$ are bounded a.e.? Thanks in advance.  |
| Posted: 02 Apr 2021 08:18 PM PDT The problem is as follows:
I'm not sure how to solve this problem. The thing is that I don't know how to relate the thickness of the candles to make them work in an equation. The only thing which it comes to my mind is that the one which is thicker will burn out at a lesser rate than the one which is less thicker. I think one of the candles will have a rate of one third of the other based on their thickness but I don't know how to connect these ideas together. Can someone help me with the steps here?. My book gives the hint of using the fact that after an hour both candles consume the same amount of volume, but I don't know how to prove this or if this was stated on purpose by my book. Will this help?. It would help me a lot if the steps could explain what sort of interpretation should be done here. After working out this problem a little bit further I arrived to this conclusion: It seems that you can get some relationship between the descend lengths when you relate the volume of the candles hence it is given the radius for each. Since it establishes that the diameter of the thicker candle is three times of the other this can be arranged as follows: Let for the thicker candle: $\begin{matrix} &\textrm{candle 1 less thicker}&\textrm{candle 2 thickest one}\\ \textrm{diameter}&d&3d\\ \textrm{diameter in terms of radius}&\textrm{2r}&\textrm{6r}\\ \textrm{radius}&r&3r \end{matrix}$ Thus to get the speeds for each candle it is only needed to get the volume consumed: Letting: Thinest one: $v_{1}$ Thickest one: $v_{2}$ Therefore: Assuming for both we wait for $t$ time units $v_{1}=\frac{\pi(r)^2\cdot a}{t}$ $v_{2}=\frac{\pi(3r)^2\cdot b}{t}$ Then relating these consumed heights $a$ for the thinnest and $b$ for the thickest then we get: $\frac{v_{1}}{v_{2}}=\frac{1a}{9b}$ I don't know how to prove this thing but it seems that the speed of the thinnest is a ninth of the thickest one but since in the latter equation it appears as two variables. This doesn't help me much to understand. Does it exist a way to make my logic to work out? Anyways: If we follow what it is indicated: Assuming the length of the candles is $l$ then: $\left(l-\frac{a}{9}\right)-\left(l-a\right)=16$ Solving this I'm getting $a=18$ centimeters. Thus for the thickest candle it would be: (For one hour elapsed) it has descended. $\frac{18}{9}=2\,cm$ Assuming the thickest candle is a ninth of the speed of the thinnest candle. While for the other candle (thinnest) it would have descended $18$ centimeters. Then as it indicates that the thickest candle will have a height which is three times the size of the thinnest candle after it has elapsed $1+\frac{1}{2}$ hour I'm getting: For that amount of time: The height of the thickest candle: $l-2\cdot\frac{3}{2}=3\left(l-18\cdot\frac{3}{2}\right)$ This all gets the size of the candles to be $39$ centimeters. As it requests that how long will it take for the thickest candle to burn out it will be: $39\textrm{cm}\cdot\frac{\textrm{1 hour}}{\textrm{2 cm}}=17\frac{1}{2}\,h$ Which is equivalent to say $19$ hours and $30$ minutes or choice $4$. Which corresponds to the answer according to my book. Again for all of this to work I assumed these two things: The speed of the thickest candle is a ninth of the speed of the thinnest candle. And when the thickest candle descends $b$ this $b=\frac{1a}{9}$ But I got tangled with the equations as I don't know exactly how to justify these Therefore It would really help me a lot if someone could help me with this part because this makes me confused.  |
| How to find $\int_1^\infty (7 + x \cdot 2^x)/(\ln x + 1 + 3^x) \, dx$ Posted: 02 Apr 2021 08:01 PM PDT How to find $$\int_1^\infty \frac{7 + x \cdot 2^x}{\ln x + 1 + 3^x} \, dx$$ I tried to assume $x=e^y$, but I didn't get a supplement in the solution. Please guide me to the solution.  |
| The equation ${4 \choose k}=6$ Posted: 02 Apr 2021 08:08 PM PDT Find the solution of the equation $${4 \choose k}=6.$$ So $k$ must be a natural number $(n\in \mathbb{N})$ and we can find that when $k=1 \rightarrow {4\choose 1}=4$ and when $k=2\rightarrow {4\choose2}=6$, so the solution of the equation is actually $k=2$. Can we solve the problem without calculating the exact value of ${4\choose k}$? I tried: $$\dfrac{V_4^k}{k!}=6$$ but don't see how to solve this. Thank you in advance!  |
| How can Zaremba's Conjecture be true if it fails for $d=1$? Posted: 02 Apr 2021 07:51 PM PDT In the following photo is the definition of some sets and the conjecture of Zaremba. Am I missing something obvious or does the conjecture not certainly fail as $d=1$ will never be in the set $\mathfrak{D}_{\{1,\ldots,A\}}$? The are no options $b$ as a numerator for $d=1$ that satisfy $0<b<1$.
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| Geometry : Ratio of area of a triangle to the rectangle containing it Posted: 02 Apr 2021 08:36 PM PDT $ABCD$ is a rectangle. $P,Q$ and $R$ are the midpoint of $BC$,$CD$ and $DA$. The point $S$ lies on the line $QR$ such that $SR:QS = 1:3$. The ratio of the triangle $APS$ and rectangle $ABCD$ is .... ? I am unable to think of the way to reach to the solution. The figure that comes out upon plotting all the lines and points does not show any way to come to the solution. I am unable to solve this. Can someone please help me on this? Thanks in advance!  |
| How do derivatives w/r to polar variables behave at the origin? Posted: 02 Apr 2021 08:38 PM PDT While argument is canonically undefined at the origin, since $u(r,\theta) = u(0)$ we could define $u(0, \theta):= u(0)$ [i.e., a constant function with respect to $\theta$]. So is $\frac{\partial^2 u}{\partial\theta^2}$ is undefined at the origin (because argument is undefined there) or defined to be zero (because we can define $u(0,\theta)$ as constant function in $\theta$? If the latter, then suppose that $u$ is harmonic near $z=0$. Does that mean that $$\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r}\mid_{r=0} = 0?$$ And what would that even mean, since $r = 0$ and the equation $x + \infty = 0$ has no solution in $\mathbb{R}\cup\{\infty\}$? Not to mention if $\frac{\delta u}{\delta r} = 0$. Yet there is no reason to believe that a harmonic function behaves differently at the origin than at any other point. What am I missing here?  |
| Posted: 02 Apr 2021 08:08 PM PDT $ \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\abs}[1]{\left \lvert #1 \right \rvert} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\ve}{\varepsilon} $The Problem: Suppose we have a sequence $\set{f_k}_{k \in \N}$ of functions such that, $\forall k \in \N$,
On this premise, we want to show that, $\forall \varepsilon > 0$, $\exists F \subseteq E$ a closed subset such that $\mu(E \setminus F) < \varepsilon$, and $\exists c \in \R$ such that $\abs{f_k(x)} \le c$ $\forall k \in \N$ and $\forall x \in F$. Thoughts & Attempts: This feels like a sort of "uniform boundedness" thing, in the sense that, even those there's a finite bound for all of the $f_k(x)$, there is one finite one that works for them all. Though obviously that's an intuitive notion, and searching brings up seemingly unrelated results. That said, I have some ideas floating around, but nothing feels quite concrete in terms of piecing them together, or finding the desired conclusion. For instance, some of the thoughts going around:
For reference's sake, here's Egorov's theorem, (mostly) as given in my text (Theorem $4.17$ in Measure & Integral: An Introduction to Real Analysis by Wheeden & Zygmund):
Can anyone give me some ideas as to where to go? Be that in terms of fleshing out my ideas and clarifying my problem spots, or putting me on the right path?  |
| Algebraic structure of the set of (Lebesgue) measurable functions over $\Bbb R^n$ Posted: 02 Apr 2021 08:24 PM PDT $ \newcommand{\MM}{\mathcal{M}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\set}[1]{\left\{ #1 \right\}} $In my analysis class, we have been discussing certain properties of the set of measurable functions; to introduce some notation, we'll define $$ \mathcal{M} := \MM(E) := \left\{ f : E \subseteq \R^n \to \R \; \middle| \; f \text{ is measurable} \right\} $$ under the usual definitions of measurability. We have observed the following properties in this class:
These in particular ensure that $\MM$ is a vector space. But we also have
This seems like a particularly rich structure, moreso than your average vector space -- we also have closure under multiplication (and it presumably has an inverse). We also have that suprema and infima of sequences, and likewise the limit suprema and infima, also are in the set. Of course, we can also introduce the $L^p$ norms and make this into whichever we need among a normed space, inner product space, and metric space. What is the name for such a structure (if one exists), one satisfying conditions such as these? (Perhaps not all of them necessarily, but more than just your usual vector space. Perhaps mentioning other facts or operations as necessary that also hold for $\MM$.) What would be some other examples of such structures? (Examples other than those tied to measurable functions and measurability, if possible.) Update: (April 2nd, 2021) One term that could apply, as noted in the comments, is the notion of an algebra over a field, and a commutative one in particular. We could even say it's unital ($f \equiv 1$ is the obvious identity). Once we equip the vector space $\MM$ with the usual pointwise multiplication, we have this structure, but there's a lot more to this, namely with the convergence properties. To some degree, as I've mulled over this, I wonder if they somehow could reflect a "closedness" or "sequential compactness" property. Of course, these are topological properties, and thus we would need to induce a topology. The continuous functions on a compact set have one induced by the supremum norm. Taking a compact domain, then, perhaps $\MM$ would admit these as a subalgebra, since continuous functions are measurable, and that would lead somewhere?  |
| For any integer $n\ge 1$, which of the following is/are true? Posted: 02 Apr 2021 08:15 PM PDT For any integer $n\ge 1$, let $d(n) =$ number of positive divisors of $n$ $v(n) =$ number of distinct prime divisors of $n$ $\omega(n)$= number of prime divisors of $n$ counted with multiplicities. for example $\omega(p^2)=2$, for a prime $p$. Then which of the following is/are true $1.~$ If $n\ge 1000$ and $\omega(n)\ge 2$, then $d(n)> \log~n$ My try: If $n=p_1^{n_1}\cdots p_k^{n_k}$ be the factorization of $n$, then $d(n)=(n_1+1)\times \cdots \times (n_k+1)$, $v(n)=k $ and $\omega(n)=n_1+\cdots n_k$, Using this, (4) is false, Counter example, $\omega(2^63^2)=\omega(5^47^4)=8$ but $d(2^63^2)=21 \ne d(5^47^4)=25$. I am not able to conclude other options please help.  |
| Posted: 02 Apr 2021 08:02 PM PDT Suppose that A is a set of 8 (distinct) symbols and consider strings (i.e. sequences) over A. How can I calculate the number of strings of length 5 which at least one symbol occurs two or more times. I started by calculating the total number of strings of length 5 by doing $8^5$ ( since we have 8 choices for each number) and then I subtracted the amount of strings of length 5 that do not have any repetition ($ 8\times 7\times 6\times 5 \times 4$) and I got the wrong answer. I think this is because my logic is wrong. Can someone help me?  |
| Proof that the Sturm-Liouville operator is positive Posted: 02 Apr 2021 07:59 PM PDT Define the Sturm-Liouville operator $L$ on $[a,b]$ such that $Lu = \frac{1}{\omega}[-(pu')'+qu]$, $p$ and $\omega$ being strictly positive real valued function, and $q$ a positive real valued function. The inner product is defined as such : $\int_a^b u(x)\overline{v(x)}\omega(x) \, \mathrm{d}x $ The question asks to prove that the operator is positive, i.e. $(Lu,u)\ge0$ (I believe they really mean positive semi-definite). I got so far: $(Lu,u)= \int_a^b q|u|^2 \mathrm{d}x + \int_a^b p|u'|^2 \mathrm{d}x - [pu'\bar{u}]^b_a $ Now I don't really understand how the boundary conditions come into play... any one knows?  |
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