Recent Questions - Mathematics Stack Exchange |
- Can Ricci's flow be used to prove Poincaré’s conjecture for $n=2$?
- spline interpolation cost function
- How do you combine probabilities of two events in a time series that have the same "prediction"?
- All entries of representation matrix of $A$(character $0$ ring) -linear map is integer?
- Congruence mod application
- Calculate the smallest integer n such that the function h(x) is differentiable at x=5
- Is the Laplacian operator and divergence operator commutable on the 2D riemannian manifold?
- Outside Roots of a Polynomial
- Linear transformation $\mathbb{P}^2 \rightarrow \mathbb{P}^2$
- Understanding the concept of $k$-algebra
- Show $4^n \frac{2}{\pi}\int_0^1\frac{x^{2n}\ln x}{\sqrt{1-x^2}}dx = -\binom{2n}{n}\int_0^1\frac{x^{2n}}{1+x}dx$
- Abuse of using conventional and nonconventional $\text{arcsec}(x)$ range
- Second cofactor for $4 \times 4$ matrix
- A fake proof for a function taking on all reals in any interval
- Regular surface that is homeomorphic to a plane
- Understanding Why $\lvert{x}\rvert < \min\{1,\delta\}$ in Spivak's Solutions.
- Prove $f'(0) = 0$ if $f$ is even
- Probability that $n$ randomly chosen spherical belt points are linearly independent
- Isn't Math basically a matter of combinatorics?
- Conditional expectation probability mass function
- How did they come up with the units of flow supplied by $b, c$, and $d$?
- Is it possible for a graph that is not a cycle to have both a Euler circuit and a Hamiltonian circuit? [closed]
- Probability of Sampling without Replacement
- Question about $y^\prime(x) = A y(x) + g(x) e^{ax}$
- Problem related to matching ( Tutte's and Hall condition)
- what is the actual curvature of earth, either per kilometer (km of sea-level distance, round, not straight line), either calculated
- For a recursion based on $x+y+z+2\sqrt{xy+yz+zx},$ does what happens in $\mathbb Z\left[\sqrt n\right]$ stay in $\mathbb Z\left[\sqrt n\right]$?
- Help finding n-order Maclaurin polynomial
- HINT for summing digits of a large power
- Fubini's theorem for Riemann integrals?
| Can Ricci's flow be used to prove Poincaré’s conjecture for $n=2$? Posted: 02 May 2021 07:59 PM PDT This question is concerning the conjecture described in this Wikipedia article. The conjecture has been proved for dimension two ($n = 2$). For $n=3$, conjecture is proved using Ricci flow. My question is, can Ricci flow be used to solve the conjecture for any $n$ or for $n=2$? Can Ricci flow defined for any $n$?  |
| spline interpolation cost function Posted: 02 May 2021 07:53 PM PDT What is the cost (objective) function, while performing spline interpolation optimization? Could you please write it to me how does the cubic spline interpolation optimization cost function looks like?  |
| How do you combine probabilities of two events in a time series that have the same "prediction"? Posted: 02 May 2021 07:46 PM PDT Suppose you're given a conintuous stream of symbols $A, B, C, A, B, A, C, A, B, \dots$, that you receive as time progresses. The symbols are not necessarily evenly-spaced in time. But assume for simplicity now that they are, so for instance the first symbol, $A$, occurs at time $t = 0$ and the next symbol, $B$, occurs at time $t = 1$, and so on. $t$ increments by $1$ each symbol in this ideal example. And you measure the probability that for instance $A \xrightarrow{t} B$ read "$A$ predicts $B$ in time $t$". For example in the first line, if we have read $9$ symbols so far from when our system starts up, then $A \xrightarrow{1} B$ has a probability (so far, measured over $9$ symbols) of $3/4$ since $A$ transitions to $B$ in one "tick" 3 times and transitions to $C$ 1 time making a total of 4 transitions and a probability of $3/4$. Now, similarly, $A \xrightarrow{2} C$ has probability of $1/3$ measured over the $3$ events that occur and $A \xrightarrow{2} A$ takes on the remaing probability of $2/3$. Now $B \xrightarrow{1} A$ has probability $1/2$. My question is, if we wanted to predict the $10$th symbol, then what is a way of doing it since the last $A$ predicts an $A$ with probability $2/3$ and the last $B$ predicts an $A$ with probability of $1/2$. Clearly, it's not just summation of probablities since $1/2 + 2/3 \gt 1$. Suppose we considered all $t \in [1, T]$ and all symbol pairs $X \xrightarrow{t} Y$, for $X,Y \in \{A, B, C\}$, then what is a formula for measuring the probability of each symbol $A, B, C$ coming next in the sequence, assuming that our system is predictable in this way roughly speaking?  |
| All entries of representation matrix of $A$(character $0$ ring) -linear map is integer? Posted: 02 May 2021 08:00 PM PDT Let $A$ be a ring of character $0$.And $M$ be $A$-module. Let representation matrix of $A$-linear map $φ:M→M$ be $N$. $N$'s matrix entries are in $A$. In other words, representation matrix of $A$-linear map is in $M(A)$. If we admit this, $M$ also can be seen as module over arbitrary subring of $B$.So all entries are in $\mathbb Z$... . (Every character $0$ ring contains $\mathbb Z$ as subring...) Where am I missing?  |
| Posted: 02 May 2021 07:43 PM PDT Question: In the country of Zzyzzx the clocks have eight-hour faces. Suppose it is now five o'clock in Zzyzzx. What time will it be in 4725 hours? My answer: It is currently 5 hours. In 3 hours, the clock will return to zero (eight). After 4722 hours (3 hours subtracted above), the clock reads 2. The reason: 4722 ≡ 2 (mod 8) Another approach: (5 + 4725) mod 8 = 2 Is my answer correct? If yes, which one do you prefer? If no, can you suggest another way to do it? Thank you so much!  |
| Calculate the smallest integer n such that the function h(x) is differentiable at x=5 Posted: 02 May 2021 07:37 PM PDT can someone help me with this question? I have to calculate the smallest integer n such that the function h(x) is differentiable at x=5. Thanks!  |
| Is the Laplacian operator and divergence operator commutable on the 2D riemannian manifold? Posted: 02 May 2021 07:36 PM PDT I am trying to derive the Green function of the poroelastic media on a 2D membrane. Here I encounter a general math problem. In 3D space, We know that the Laplacian operator and divergence operator are commutable, which means that: $$D^{\beta}D_{\beta}D^{\alpha}v_{\alpha}=D^{\alpha}D^{\beta}D_{\beta}v_{\alpha}\quad(1)$$ where $D_{\alpha}$ is covariant derivative and $v_{\alpha}$ is a vector in terms of its covariant. However, we know that the differential operator is not commutable on manifold given that Christoffel symbols do not always vanish. So I am not sure about the eq. (1) is still valid or not on an arbitrary 2D manifold(i.e. spherical surface)? If not, what is the correct relation between the left side and right side of the equation? In other words, what extra term should we put there to correct the equation on a 2D manifold? Thank you for your help.  |
| Posted: 02 May 2021 07:21 PM PDT Consider the following polynomial $$P(z)=\frac{(z-a)(1-az)}{bz}+Q(z),\quad \text{with}\;a\in(0,1)\; \text{and}\;b\in \mathbb R_+,$$ and with $$Q(z)=1-\sum_{i={-n}}^m c_i z^{i},\quad \text{with}\;c_i\in\mathbb R\;\text{for all }i.$$ Suppose that $Q(z)$ has $m$ roots satisfying $|z|>1$. Under which conditions can I guarantee that $P(z)$ has $m+1$ roots satisfying $|z|>1.$? My guess is that $Q(a^{-1})>0$ would be sufficient, but have not been able to prove it. A related result: If $Q(a^{-1})>0$ and $m=0$, I can show that $P(z)$ has one root satisfying $|z|>1$ with the following argument: First notice that $P(a^{-1})=Q(a^{-1})>0$. Next, notice that the coefficient of highest order of $P(z)$ is $-a/b$ so that for $z$ high enough $P(z)<0$. It follows that there is a root of $P(z)$ bigger than $a^{-1}>1$.  |
| Linear transformation $\mathbb{P}^2 \rightarrow \mathbb{P}^2$ Posted: 02 May 2021 07:28 PM PDT Hi I dont understand how to do the following transformation Consider a transformation from $\mathbb{P}^2$ to $\mathbb{P}^2$ defined by $T(p(x)) = p(x + 3) − p(x)$ a) Proof that it is a linear transformation b) Find the linear matrix with the basis $\{1, x, x^2\}$  |
| Understanding the concept of $k$-algebra Posted: 02 May 2021 07:20 PM PDT If $k$ is a field then 'a finite $k$-algebra' (i.e. finitely generated as a $k$-module) is equivalent to saying that '$k$-algebra which is also a finite dimensional vector space over $k$'?  |
| Posted: 02 May 2021 08:00 PM PDT $$4^n \dfrac{2}{\pi}\int_0^1\frac{x^{2n}\ln x}{\sqrt{1-x^2}}dx = -\dbinom{2n}{n}\int_0^1\frac{x^{2n}}{1+x}dx$$ I have spent quite some time working on it but unfortunately it didn't get me anywhere, in fact I was trying to extract the LHS from the RHS and maybe that's why it went wrong. I'm looking for a real analytic proof. Your insight or help would be appreciated. Thanks in advance.  |
| Abuse of using conventional and nonconventional $\text{arcsec}(x)$ range Posted: 02 May 2021 07:44 PM PDT $$S = \int \frac{1}{\sqrt{x^2-1}} dx$$ Clearly $x = \sec(t)$ substitution will be used. The domain of the integrand is $x \geq1 \dots$ (i) or $x\leq -1 \dots$ (ii). Here is how I did, but this is not how most do it: $x = sec(t)$ (taking $0 \leq t \leq \pi$, $t \neq \pi/2$ as $t$ domain/$\text{arcsec}(x)$ range, which is conventional) $dx = sec(t)\ tan(t)\\$ (true for (i) and (ii)) $\sqrt{x^2 -1}= tan (t)$ (true for (i), $0 \leq t < \pi/2$) $- \sqrt{x^2 - 1} = tan (t)$ (true for (ii), $\pi/2 < t \leq \pi$) i) $S_i = \int sec( t) \ dt = ln\ |sec (t) + tan( t) | + c = \ln \left\vert x + \sqrt{x^2 - 1} \right\vert + C \ (true)$ ii) $S_{ii} = \int - sec(t) \ dt = -\ ln\ | sec( t) + tan( t) | + c = ln\ | sec( t) - tan( t) | = ln\ | x - (- \sqrt{x^2 - 1})| + C = ln\ | x + \sqrt{x^2 - 1} | + C \ (true, note \enspace (sec(t) + tan(t))(sec(t) - tan(t)) = 1)$ $S_i = S_{ii}, \quad \text{therefore} \quad S = ln\ | x + \sqrt{x^2 - 1} | + C$ for all valid domain. But this is not how most people do for ANY trig integration. I have never seen it done like this. For example, $sin^2 (t) + cos^2 (t) = 1 $ certainly doesn't need this because both $sin (t)$ and $cos (t)$ are positive for $arccos (x)$ and $arcsin (x)$ range respectively. Therefore it's always positive square roots. But $1 + tan^2 (t) = sec^2 (t)$ is clearly different, $tan (t)$ CAN be negative for SOME $arcsec (x)$ range, although $sec (t)$ is always positive for all $arctan (x)$ range. I read that some people like to take $\pi < t < 3\pi/2$ as $arcsec (x)$ range from here, where $tan (t)$ is positive and therefore $tan (t)$ is also (always) positive for all $arcsec (x)$ range, meaning I don't need to study the two cases (i) and (ii). I want to hear some justification from people who take conventional $arcsec (x)$ range and also conveniently use $\sqrt{x^2 - 1} = tan (t)$ for all possible x values. Their answer is correct indeed, because the negative sign is cancelled in the middle, miraculously for them, but they never anticipated it. when $x \leq -1$ and the moment they wrote $\int sec (t)\ dt$ instead of $- sec (t)$ , it is false. Isn't this a kind of abuse, like abuse of notation? Please note that this problem does not necessarily apply on the case where hyperbolic trigonometric identities and substitutions are used. I'm only curious about $x = sec (t)$ substitution case. Is this correct? Thanks for reading my question! Desmos for visualisation: https://www.desmos.com/calculator/izsg2x89ch  |
| Second cofactor for $4 \times 4$ matrix Posted: 02 May 2021 07:23 PM PDT Determinant of \begin{bmatrix}2&0&3&2\\3&1&-2&2\\1&0&3&4\\2&-2&3&1\end{bmatrix} = determinant of \begin{bmatrix}2&3&2\\1&3&4\\2&3&1\end{bmatrix} + determinant of \begin{bmatrix}?&?&?\\?&?&?\\?&?&?\end{bmatrix} How do I solve for the ? matrix?  |
| A fake proof for a function taking on all reals in any interval Posted: 02 May 2021 07:52 PM PDT It is established that there exist functions that, on any interval, take on all real values (see Function whose image of every open interval is $(-\infty,\infty)$ and Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$? ). Below is a facile proof, which I believe incorrect: Any interval in $\mathbb{R}$ has cardinality $C$, equal to the cardinality of $\mathbb{R}$ itself. By definition of equal cardinality, a function exists between the two sets creating a one-to-one correspondence. QED. I believe this proof is incorrect, because it shows $f$ exists for any particular interval, but doesn't show that the same $f$ can be defined over multiple overlapping intervals. However, I don't find this objection to be very robustly formulated. Can you strengthen my objection to the "proof"? Can you provide another objection?  |
| Regular surface that is homeomorphic to a plane Posted: 02 May 2021 07:44 PM PDT I'm trying to solve the following exercise.
I have managed to show that such a surface is indeed a regular surface, but am struggling to come up with a suitable homeomorphism, or see why it is a homeomorphism. The official solutions state that "it is clear that the surface is homeomorphic to a plane", but I cannot understand why this is. Can anyone help me understand this?  |
| Understanding Why $\lvert{x}\rvert < \min\{1,\delta\}$ in Spivak's Solutions. Posted: 02 May 2021 07:20 PM PDT I am working my way through Spivak. I am not incredibly confident in my solution to the problem, and the solutions in the back of the book gives a subtlety that I don't have in my own proof. I'm wondering why. Claim: $\lim_{x \to 0} f(x) = \lim_{x \to 0} f(x^3).$ My Proof: First, suppose $\lim_{x \to 0} f(x) = L$. Then, by definition, $\forall \epsilon, \exists \delta$ such that $\forall x, \lvert{x - 0}\rvert < \delta \implies \lvert{f(x) - L}\rvert < \epsilon$. Since we may choose any $\delta \in \mathbb{R}^+$, consider $\lvert{x}\rvert < \sqrt[3]\delta$. Then $\lvert{x^3}\rvert < \delta \implies \lvert{f(x^3) - L}\rvert < \epsilon$. Thus, $\lim_{x\to 0} f(x^3) = L$. Conversely, suppose $\lim_{x \to 0} f(x^3) = L$. Then, again by definition, $\forall \epsilon, \exists \delta$ such that $\forall x, \lvert{x - 0}\rvert < \delta \implies \lvert{f(x^3) - L}\rvert < \epsilon$. If $\lvert{x}\rvert < \delta^3$, then $\lvert{\sqrt[3]{x}}\rvert < \delta \implies \lvert{f(\sqrt[3]{x}^3}) - L\rvert < \epsilon$. Thus, $\lim_{x\to 0} f(x) = L$. $\blacksquare$ Since this is the chapter on limits, the proof is written very pedantically. However, the crux of the argument for each case is:
My Question Spivak's Solutions has the following as justification for the first case:
I don't understand why Spivak shows that $\delta < \min{(1, \delta)}$, or why knowing this we can deduce $0 < \lvert{x^3}\rvert < \delta$.  |
| Prove $f'(0) = 0$ if $f$ is even Posted: 02 May 2021 07:20 PM PDT Let $f$ be an even function on the reals. Prove $f'(0)$ is either $0$ or undefined. Intuitively, $f(0) + h\cdot f'(0) \approx f(h) = f(-h) \approx f(0) - h \cdot f'(0) $ for small $h$. If $h \neq 0$, $f'(0)$ must be $0$. However, I'm having trouble formalizing this proof. I believe my difficulty is that there are multiple pairs of $\epsilon, \delta$ needed to be tracked separately, as the derivative is defined as a limit (one epsilon-delta pair), we need to show $f(0) + h\cdot f'(0) \approx f(h)$ (another epsilon-delta pair), and we need to de these for negative $h$ as well (two more epsilon delta pairs). Can you help me formalize my intuitive argument into a proper, clear, well organized proof?  |
| Probability that $n$ randomly chosen spherical belt points are linearly independent Posted: 02 May 2021 07:55 PM PDT Let $S = \{x : x \in \mathbb{R}^d, \|x\| = 1, |x_1| \leq k\}$ be a spherical belt defined by some constant $k \in (0, 1)$ that bounds the size of the first coordinate. Let $y_1, \dots, y_n$ be $n$ points randomly sampled from the uniform distribution over $S$. What is the probability that $y_1, \dots, y_n$ are linearly independent?  |
| Isn't Math basically a matter of combinatorics? Posted: 02 May 2021 07:25 PM PDT As I discover the foundations of mathematics, I begin to understand that it is a matter of arbitrarily defining axioms and combining them - arriving at what we call theorems. Having said that, it looks to me the following: We could fix a finite number of symbols ($\lnot$, $\forall$,...) and state all the arbitrarily useful axioms with them (say the construction of natural numbers, the definition of the addition operation). Given that, let's say we define $n$ axioms, what would be left for us would be just to combine them in any possible way. For example, $n(n-1)$ statements would arrive by the combination of two (distinct and with order) of the original ones, $n(n-1)(n-2)$ by the combination of three. After that, for example, we could pick one of these $n(n-1)$ and combine with one of the original ones, with two of them. By construction, all provable (derivable by a combination of the axioms) statements would be achieved. Finally, we could state a statement with unknown truth value with the symbols and just combinatorially see if it is achievable. The previous text is certainly bad written and not rigorous at all; in top of this, to try to clarify this thought that chases me, an analogy also comes to my mind: Given the rules of chess (our axioms), if we want to know if a certain configuration of the pieces (unknown statement) is provable (derivable by the rules), why can't we just go on trying all the moves from the beginning: moves pawn a, moves pawn b, ..., moves pawn a and knight b... I know this is probably distant of making sense, but I can't see why this procedure would be so impracticable. Therefore I am humbly coming here in the seek for a glimpse of clarity to end this train of thought that chases my mind and that I simply can't develop in any sense.  |
| Conditional expectation probability mass function Posted: 02 May 2021 07:45 PM PDT In a probability example, there are these random variables: $X_i, N_t,$ and $Z_t=E(\sum\limits_{i=1}^{N_t} X_i | N_t)$, where $S(N_t)=\mathbb{N}_0$ and $X_i$ is a continuous random variable. I have proved that $E(\sum\limits_{i=1}^{N_t} X_i | N_t=n)=\frac{nt}{2}$ Furthuremore, we have $E(N_t)=λt$ Now, I want to calculate $E(Z_t)$. Firstly, what is the most accurate way to calculate it? Secondly, I have calculated $E(Z_t)$ like this: $E(Z_t)=\sum\nolimits_z zp(Z_t=z)=\sum\limits_{n=1}^{\infty} \frac{nt}{2}p(Z_t=\frac{nt}{2})=\frac{t}{2}\sum\limits_{n=1}^{\infty} np(Z_t=\frac{nt}{2})$ Here I use something that I'm not so sure. And that is $p(Z_t=\frac{nt}{2})=p(N_t=n)$. Is it true? If yes, how can I prove it? If it holds we have $\frac{t}{2}\sum\limits_{n=1}^{\infty} np(Z_t=\frac{nt}{2})=\frac{t}{2}\sum\limits_{n=1}^{\infty} np(N_t=n)=\frac{t}{2}E(N_t)=\frac{t}{2}(λt)$ Thirdly, I think every conditional expectation like $E(A|B)$, assumes the value corresponding to $B=n$ with the probability $p(B=n)$. Is this correct?  |
| How did they come up with the units of flow supplied by $b, c$, and $d$? Posted: 02 May 2021 07:30 PM PDT I am a bit confused on how the numbers $60, 40$, and $40$ came about as the flow $b, c$, and $d$ supply respectively. This example is under "Flow Networks with Supplies and Demands." I understand they added vertex a to show that multiple sources can be combined into one, but what made them choose those specific values? Example: "Consider the network of solid edges in Figure 4.4 with supplies and demands. Vertex $b$ can supply up to $60$ units of flow, and vertices $c$ and $d$ can each supply $40$ units. Vertices $h, i$, and $j$ have flow demands of $50, 40$, and $40$ units, respectively."
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| Posted: 02 May 2021 07:39 PM PDT If it is possible, could you provide an example?  |
| Probability of Sampling without Replacement Posted: 02 May 2021 07:54 PM PDT Let $Y_1, Y_2, \dots , Y_{10}$ be a random unordered sample without replacement from the set $\{1, \dots, 100\}$. (This means that the numbers $Y_1, \dots, Y_{10}$ are not necessarily ordered.) (a) What is the probability that $|Y_1 − Y_2| = 1$? (b) Let $Z$ be the number of indices $1 ≤ k ≤ 9$ with $|Y_k − Y_{k+1}| = 1$. (E.g. if the sample is $3, 1, 4, 5, 11, 14, 13, 99, 43, 72$ then $Z = 2$.) Find $\mathbb E[Z]$. Hint: part (a) should help. So my rationale for this problem was that the numbers have to be right next to one another, so then I used the complement, $\Pr(\text{$Y_1$ and $Y_2$ are next to one another}) = 1 - \Pr(\text{$Y_1$ and $Y_2$ are not next to one another}) = 1 - (99/100 * 98/100) = .0298$ Is that the right way to do it? And then for part b I am not sure what formula to begin with.  |
| Question about $y^\prime(x) = A y(x) + g(x) e^{ax}$ Posted: 02 May 2021 07:28 PM PDT Considering the differential equation $$y^\prime(x) = A y(x) + g(x) e^{ax}$$ where $A \in \Bbb R^{2 \times 2}$ and $a \in \Bbb R$ is no eigenvalue of $A$, and $g(x)$ a polynomial vector, then there exists a particular solution $$y_p(x) = h(x) e^{ax}$$ where $h(x)$ is a polynomial vector. I tried some approaches, but I couldn't solve it. differentiating $y_p$ I got $y_p'(x)=h'(x)e^{ax}+h(x)ae^{ax}$, inserting it into the equation yields $$h'(x)e^{ax}+h(x)ae^{ax}= A h(x) e^{ax} + g(x) e^{ax}$$ $$e^{ax}(h'(x)+h(x)a)= e^{ax}(A h(x) + g(x))$$ Thats how far I got. My problem now is that a is no eigenvalue.  |
| Problem related to matching ( Tutte's and Hall condition) Posted: 02 May 2021 08:02 PM PDT Let G be a graph without a perfect matching, thus, Tutte's conditions fail for G which implies that there exists a set $S ⊂ V (G)$ such that the number of odd components in $G − S$ exceeds $|S|$. Now let us assume that G is bipartite with bipartition $V (G) = V' \sqcup V''$. Show that Hall's conditions fail for at least one of the sets $S ∩ V'$and $S ∩ V''$ I tried to draw a graph that fails Tutte's condition as shown below: But when I verified it for hall's theorem it did not fail: Since Hall's condition states that for a bipartite graph with parts $V'$ and $V''$ the condition $ \forall A \subseteq V', \lvert N(A) \rvert \geqslant \lvert A \rvert$ is necessary and sufficient for the existence of a matching that covers all vertices in $V'$ Am I doing something wrong? Could you please help me to solve this. Thanks.  |
| Posted: 02 May 2021 07:16 PM PDT First of all, the following question with it's answers are the main reason I put this question, because as I will show the question wording is bad. How to calculate curvature of Earth per surface kilometer Basic data: at a distance (ground distance, as in a ship on the sea) of aproximately 10.000 kilometers, the drop (h in that drawing) should be close to the radius of the earth (aproximately 6371 KM), for simplification we don't take into account the small differences resulting from the fact that Earth is not a perfect sphere! That is basic geometry (because it corresponds to 90 degrees from the sphere). However with the complicated math in that question, the conclusion states there is a 8 cm per kilometer drop, which at 10.000 kilometers equals 80.000 cm, equals 800 meters??? If after 2, 3, 10, 2000 km (random values) and so on, the drop per kilometer is another value, it means the linked question is bad-worded (per surface kilometer would mean a constant value, not a non-linear one)... I added this paragraph after @zabop's comment I also found this resource, but the formula there is for straight line distance: http://www.revimage.org/what-is-the-curvature-of-earth-per-km/ The excel formula in the second link is good to calculate the drop for any straight line distance between 0 and 6371 km (after a distance equivalent to more than 90 degrees the h would no longer make sense): Later edit Another case (a tall object that would still be visible beyond sea-level horizon line) here What would an excel formula for sea-level distance (round distance, not straight line) be?  |
| Posted: 02 May 2021 07:54 PM PDT This problem has a geometric origin which I'll outline below, but I believe the concepts and explanation are algebraic. Given a function on triples $$K((x,y,z))=x+y+z+2\sqrt{xy+yz+zx}$$ we build a general recursion as follows.
In general, the members of $S$ will contain nested radicals. For example, if we start with the triple $(1,2,3)$ and use a restricted recursion $x_{n+1}=K((1,2,x_n)),x_1=3$ we get $$ S'=\left\{3,6+2 \sqrt{11},9+2 \sqrt{11}+2 \sqrt{2+3 \left(6+2 \sqrt{11}\right)},\\12+2 \sqrt{11}+2 \sqrt{2+3 \left(6+2 \sqrt{11}\right)}+2 \sqrt{2+3 \left(9+2 \sqrt{11}+2 \sqrt{2+3 \left(6+2 \sqrt{11}\right)}\right)},\dots \right\} $$ However, Mathematica is able to denest these to $$ S'=\left\{3,6+2 \sqrt{11},15+4 \sqrt{11},30+6 \sqrt{11},\dots\right\} $$ suggesting the conjectures that
I've done some hand and computer calculations to understand why the radicals denest in particular cases, but I'm guessing that if the conjecture is true it will be "obvious" to someone more versed in this area. Some background and observations:
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| Help finding n-order Maclaurin polynomial Posted: 02 May 2021 07:25 PM PDT EDIT AND PLEASE NOTE: I DON'T want solutions that are nicer or more elegant but presume knowledge of other infinite series and/or don't come from the nth-derivative because I'm precisely studying how Taylor and Maclaurin series relate to derivatives. My mathematical analysis course hasn't even made it to infinite series yet so there's really no point anyway. I can't find a pattern in the derivative functions of $(1-x^2)^{-1}$. All I know is that odd terms will probably be 0 and some terms in the derived functions will also be 0 as the derivative of $(1-x^2)^n$ will always have a $2x$ factor, but other than that I'm stumped. Any help will be appreciated.  |
| HINT for summing digits of a large power Posted: 02 May 2021 07:22 PM PDT I recently started working through the Project Euler challenges, but I've got stuck on #16 (http://projecteuler.net/problem=16)
(since I'm a big fan of generality, my interpretation is to find a solution to the sum of digits of $a^b$ in base $c$, and obviously I'm trying to solve it without resorting to "cheats" like arbitrary-precision numbers). I guess this is simpler than I'm making it, but I've got no interest in being told the answer so I haven't been able to do a lot of internet searching (too many places just give these things away). So I'd appreciate a hint in the right direction. I know that $2^{1000} = 2^{2*2*2*5*5*5} = (((((2^2)^2)^2)^5)^5)^5$, and that the repeated sum of digits of powers of 2 follows the pattern $2, 4, 8, 7, 5, 1$, and that the last digit can be determined by an efficient pow-mod algorithm (which I already have from an earlier challenge), but I haven't been able to get further than that… (and I'm not even sure that those are relevant).  |
| Fubini's theorem for Riemann integrals? Posted: 02 May 2021 07:32 PM PDT The integrals in Fubini's theorem are all Lebesgue integrals. I was wondering if there is a theorem with conclusions similar to Fubini's but only involving Riemann integrals? Thanks and regards!  |
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