Recent Questions - Mathematics Stack Exchange |
- $y'''+p(x)y''+q(x)y'+r(x)y=0$ such that $p,q,r$ are continues functions in $[a,b]$. $y_1,y_2,y_3,y_4$ are different solutions
- How to check if any number of points are coplanar?
- Are differentials of symmetric mappings symmetric?
- Can I get assistance on how to go about this?
- Giving initial values to a set of functions
- Nilpotent elements and the Cartan decomposition
- The New(?) method for find x; x^2< sqrt(p) mod p
- What's going on in this notation for the projective limit in Serre?
- Manipulation of Infinite Series Example
- Primality test for numbers of the form $(10^p-1)/9$ (and maybe $((10 \cdot 2^n)^p-1)/(10 \cdot 2^n-1)$)
- Proving that the metric space $(\mathcal{H}(\mathbb{D}), d)$ is complete using Complex Analysis
- What does $\mathbb{R}(p)$ mean?
- Real world asset acquisition problem with two or more unknowns who are interdependent.
- $F \le E$ extension. every element $\alpha \in E - \overline{F}_E$ over $\overline{F}_E$ transcendental
- What exactly does it mean that a limit is indeterminate like in 0/0?
- Upper bound: Given $L$-smooth convex $f$; $( y- x)^T \left( \nabla f(z)-\nabla f(x)\right)\leq(L/2) ( \| x-z\|^2+\| x-y\|^2+\| z-y\|^2)$?
- When a regular $F_{\sigma}$ set open?
- Atiyah singer index theorem and grothendieck riemman roch theorem
- Why $\lfloor \sqrt{n} \rfloor$ changes by at most one when incrementing $n$?
- Loewner order and norms of images: Does $A \preccurlyeq B$ imply $\|Ax\| \leq \|Bx\|$?
- How to prove r = n * 2(l·n) - l in specular reflection?
- Find triangles to fill rectangle
- Does there exists any non trivial linear metric space in which every open ball is not convex?
- Is there a relationship between the retrieved spectrum width with the ratio between actual spectrum width and resolution?
- Showing that the distribution of record times $(\tau_k)_{k\geq 1}$ doesn't depend on the distribution, $F$, of the records $X_i$
- Intersection of circle and geodesic segment on sphere
- $f(x)\bmod b$: solve for $ x$
- $a+b+c = 2001$ and $a>b>c>0$ how many ordered triples are possible?
- Submitting for publication in American Mathematical Monthly
- Find the value of $\lim_{n \rightarrow \infty} \sqrt{1+\left(\frac1{2n}\right)^n}$
| Posted: 17 Dec 2021 05:59 AM PST $y'''+p(x)y''+q(x)y'+r(x)y=0$ such that $p,q,r$ are continues functions in $[a,b]$. $y_1,y_2,y_3,y_4$ are different solutions that have common tangent point $(x_0,y_0)$ such that $x_0\in [a,b].$ $1.$ Find the maximal number of linearly independent solutions? $(y_i , i\in [4])$ $2.$ Is it possible that each pair of solutions $((y_i,y_j) , i\neq j\in [4])$ are linear dependent ? My try: First of all, I don't get how it is possible that two solutions have a common point $(x_0,y_0)$ since by getting an initial condition there are two possible solutions. $1.$I think the correct answer is $3$ since the o.d.e is third-order o.d.e. $2.$ I think it is possible if $y_0=0$. Lets take $y_1,y_2=c_1y_1,y_3=c_2y_1,y_4=c_3y_1.$ All these solutions are tangent in $(x_0,0)$ and linear dependant. Thanks! |
| How to check if any number of points are coplanar? Posted: 17 Dec 2021 05:56 AM PST How to check if any number of tridimensional points are coplanar? I have found just how to check if 4 points are coplanar, but I need to know if a few tenths of points are coplanar. |
| Are differentials of symmetric mappings symmetric? Posted: 17 Dec 2021 05:55 AM PST Call a mapping $f:R^n \rightarrow R^m$ symmetric (or anonymous) if $f(\sigma^{-1} x) = \sigma f(x)$ for all $x \in R^n$ and $\sigma \in S(n)$, where of course $S(n)$ is the set of all permutations on $n$ variables (I guess you could also call these intertwining functions). I've classified the set of all linear symmetric functions, in the context of game theory, but what I'm investigating is the case when $f$ is not necessarily linear. In particular, I'm trying to prove/disprove the existence of symmetric non-linear functions possessing symmetric differentials at each point $x \in R^n$ (or at least some points). If $m = 1$ I think I know where to start looking (e.g. ring of invariant polynomials) but when $m > 1$ the only approach I've thought of is considering the manifold $M$ of graphs of symmetric functions under an action of the symmetric group $\sigma^{-1} (x,f(x)) = (\sigma(x),\sigma(f(x)))$, which will map $M$ into itself, but am stuck at that point. Stipulating to the fact that I could be making some basic mathematical errors here, can anyone suggest an area to start exploring (assuming of course that the answer is not already well-known)? Thank You. |
| Can I get assistance on how to go about this? Posted: 17 Dec 2021 05:59 AM PST Consider the sequence $(x_n) _{n\in N}$ defined by $x_o=\dfrac{1}{2}$, and $x_{n+1}=x_n^2+\dfrac{\alpha^2}{16}$, $\alpha>0$. For what values of $\alpha$ does this sequence converge? diverge? Find $\lim_{n\rightarrow \infty}x_n$ for $\alpha^2=3$ Any form of directive will help |
| Giving initial values to a set of functions Posted: 17 Dec 2021 05:51 AM PST Lets assume we have a set of functions Thanks for any ideas! |
| Nilpotent elements and the Cartan decomposition Posted: 17 Dec 2021 05:49 AM PST Is it true that any nilpotent element of $\mathfrak{s}\mathfrak{l}_n(\mathbb{C})$ is contained in any one of the root spaces given by the Cartan decomposition of the semisimple Lie algebra $\mathfrak{s}\mathfrak{l}_n(\mathbb{C})$? In other words, could it be that a nilpotent element occurs only as a sum of elements coming from different root spaces? I am trying to prove the existence of an embedding of $\mathfrak{s}\mathfrak{l}_2(\mathbb{C})$ into $\mathfrak{s}\mathfrak{l}_n(\mathbb{C})$ for which any particular nilpotent element of the latter is contained in the image. It seems like a positive answer to my question above would help greatly. Thank you |
| The New(?) method for find x; x^2< sqrt(p) mod p Posted: 17 Dec 2021 05:41 AM PST Let $p-x>x>\sqrt{p}$ где $p$- composite, $x\in Z$ Lets find such $x$, for $ x^2<\sqrt{p} \; \pmod p$ There my new (?) method. $(A+x)^2 \equiv A^2+2Ax+x^2 \; \pmod p \;(1)$ Let $A^2 \equiv C \; \pmod p$ substitute to (1) and solve for x $C+2Ax+x^2 \equiv 0 \; \pmod p$ Ceiling the result $x=A \pm \lceil(\sqrt{A^2-C})\rceil$ Againe, substitute $x$ at (1), $( \pm \lceil\sqrt{A^2-C}\rceil)^2 \equiv Y\; \pmod p$. Check $Y <\sqrt{p}$ if this not true - Attention, most strange part Let $A \equiv Y^2 \;\pmod p$ and do the above again Strange. but this cycle frequently converges to sub $\sqrt{p}$ ; Pari/GP code \p300 {p=1234577*3001; c=ceil(sqrt(p)); for(u=c,p, b=lift(Mod(u^2,p)); c2=lift(Mod(b^2,p)); for(y=1,200, t=ceil(sqrt(b*b-c2)); b=lift(Mod(t^2,p)); c2=lift(Mod(b^2,p)); if(b<c, break()); \break at sub sqrt residual ); localprec(7); z=b/c/1.; if(z<1,print(z," ",t," ",issquare(b))); );} |
| What's going on in this notation for the projective limit in Serre? Posted: 17 Dec 2021 05:28 AM PST $\newcommand{\Z}{\mathbf Z}\newcommand{\Q}{\mathbf Q}$I am currently reading Serre's A Course in Arithmetic, and in Chapter 2 where he introduces the $p$-adics, he mentions the projective limit. My question is not so much about the projective limit itself, but about the notation he uses. First, for context, he constructs the $p$-adic integers $\Z_p$ as a certain projective limit, then the full $p$-adic numbers $\Q_p$ as the field of fractions of $\Z_p$. Here is the relevant excerpt from the text.
So far so good; as you can see, he uses $\varprojlim(A_n,\phi_n)$ for the projective limit, which is in agreement with what I've been able to find elsewhere and seems to be common notation. However, a few pages later, he starts to use some notation that I do not recognise and is seemingly inconsistent. For example, he writes $\Z_p=\varprojlim\!\mathbf{.} A_i$ where you can see the clear "dot" between the limit symbol and $A_i$. Is this supposed to mean that the maps $\phi_i$ are omitted from notation, or something else? If so, why the dot symbol? Similarly, for a sequence of $(u_n)\in\Q_p$, he says that "a sequence $u_n$ has a limit if and only if $\lim\!\mathbf{.}(u_{n+1}-u_n)=0$". Again there is the appearance of this conspicuous dot. For the rest of the text, as far as I see, this dot persists in all notation in this context.
I am probably missing something obvious here; many thanks in advance! |
| Manipulation of Infinite Series Example Posted: 17 Dec 2021 05:28 AM PST I'm struggling to understand the manipulation of an infinite series shown in the text below.
We begin with the series $\sum_{n=0}^{\infty} u^n$ which we know converges since $|u|<1$. We then express each term as an infinite series using the series for $u$ and the product rule for power series to obtain series for terms with higher exponents. But then I don't understand on the penultimate line the justification for "unpacking" each of these series and then collecting like terms. I know absolutely convergent series can be rearranged and I know that $\sum_{n=0}^{\infty} u^n$ is absolutely convergent, but this, to my understanding, would only allow you to rearrange the terms in the original series $1+u+u^2+...$ So I could understand repositioning the infinite sums giving each $u^n$. But I can't understand breaking up those infinite sums and then freely repositioning the individual terms. To put it in a different way, I wouldn't think you could just remove the brackets around for example $(z^2/3!-z^4/5!+...)$ on the third to last line and freely move the terms. I would think those brackets would have to remain and you could just freely move the infinite sum contained in the brackets. What rule/property of infinite series allows the manipulation shown in the penultimate line of the above text? |
| Posted: 17 Dec 2021 05:50 AM PST This question is successor of Primality test for numbers of the form (11^p−1)/10 Here is what I observed: For $(10^p-1)/9$ : Let $N$ = $(10^p-1)/9$ when $p$ is a prime number $p > 3$. Let the sequence $S_i=S_{i-1}^{10}-10 S_{i-1}^8+35 S_{i-1}^6-50 S_{i-1}^4+25 S_{i-1}^2-2$ with $S_0=123$. Then $N$ is prime if and only if $S_{p-1} \equiv S_{0}\pmod{N}$. I choose $123$ because this is the $10_{th}$ Lucas number $L_{10}$. For the sequence, I choose the Lucas' polynomial $L_{10}(x)$ and alternate $+$ and $-$ for each part as shown in the sequence. (I don't know if these polynomials have a name). For the test I use PARI/GP. For example with $p = 19$ I found with PARI/GP: And $1111111111111111111$ is indeed a prime number. For $((10 \cdot 2^n)^p-1)/(10 \cdot 2^n-1)$ : I tested some extensions with $(20^p-1)/19, (40^p-1)/39$ and $(80^p-1)/79$ and it seems the primality test works for example where $S_0=15127$, the $20_{th}$ Lucas number with the Lucas polynomial $L_{20}(x)$ still with $+$ and $-$ alternated. Globally $L_{10 \cdot 2^k}$ for Lucas number and $L_{10 \cdot 2^k}(x)$ for Lucas polynomials with $+$ and $-$ alternated. Is there a way to explain this? I try to prove it by myself but it's not easy. If you found a counterexample please tell me. |
| Proving that the metric space $(\mathcal{H}(\mathbb{D}), d)$ is complete using Complex Analysis Posted: 17 Dec 2021 05:48 AM PST Call the vector space of holomorphic functions on the unit disk $\mathcal{H}(\mathbb{D})$. Define for $f, g \in \mathcal{H}(\mathbb{D})$: $$d(f,g) = \sum^{\infty}_{k=1}2^{-k}\min\left(1, \sup_{|z|\leq1-\frac{1}{k}}|f(z)-g(z)|\right).$$ I think it's quite straightforward to prove that this defines a metric on $\mathcal{H}(\mathbb{D})$ (only my proof for the triangle inequality might be not entirely correct, but that another question in itself). Now I want to prove that this metric space $(\mathcal{H}(\mathbb{D}), d)$ is complete. For this I first wanted to prove that a sequence $(f_n)_n$ in $\mathcal{H}(\mathbb{D})$ converges in $(\mathcal{H}(\mathbb{D}), d)$ if and only if it converges uniformly on compact subsets of $\mathbb{D}$. I tried proving the first implication by noticing that since $\forall \epsilon > 0$ there exists a $N$ such that for all $n \geq N$ it holds that $d(f_n,f) < \epsilon$. Therefore the sum in $d(f_n,f)$ must be small in each term, so for the first few terms it must hold that $$\min\left(1, \sup_{|z|\leq1-\frac{1}{k}}|f(z)-g(z)|\right) = \sup_{|z|\leq1-\frac{1}{k}}|f(z)-g(z)|.$$ But does this hold? And how I proceed with the rest of the proof? For the converse statement, it must hold in any compact subset $K$ of $\mathbb{D}$ that $\forall \epsilon > 0$ there exists a $N$ such that $\forall n \geq N$ $\textbf{and}$ $\forall z \in K$ it holds that $|f_n(z),f(z)| < \epsilon$. But I'm now confused on how to proceed with the proof because in $d(f,g)$ the supremum is taken over all $z \leq 1 - 1/k$, not just the ones in some compact set $K$. Any help is appreciated! |
| What does $\mathbb{R}(p)$ mean? Posted: 17 Dec 2021 05:20 AM PST I am reading a paper, and I am curious about the notation used. Let $p$ be the differential operator, $p:=\frac{d}{dt}$. Let $F(p)$ be a filter, e.g., the first-order transfer function, $F(p) = \frac{1}{p+1}$. What does mean $F(p) \in \mathbb{R}(p)$, where $\mathbb{R}$ is the set of reals? |
| Real world asset acquisition problem with two or more unknowns who are interdependent. Posted: 17 Dec 2021 05:17 AM PST I am trying to create a calculator that allows users to specify asset classes (e.g. 500 in fund1, 500 in fund2, etc.) and the desired fraction of the total portfolio per asset class (e.g. 70% in fund1, 30% in fund2). The equation should determine how much to buy or sell per asset class (instead of move). I have the following formula for when only 1 fund changes, where $X_1$ is before, $X_2$ is after, and $dX$ is their difference. The same applies to the second fund $Y$. $a$ is the desired fraction of $X$. $$ dX = \frac{(aX_1 - X_1 + aY_2}{1-a} $$ The problem arises when $Y_2$ becomes a variable, because now any calculation of $dX$, will change $dY$, which will change $dX$ again and so on. Can anyone point me in the right direction? My apologies for the possible lack of proper tagging or wording, it has been a while since I did maths... Thanks! |
| Posted: 17 Dec 2021 05:45 AM PST Let $F \le E$ be a field extension. Prove that every element $\alpha \in E \setminus \overline{F}_E$ over $\overline{F}_E$ transcendental. Note: $\overline{F}_E = \{x \in E : x \text{ is algebraic over } F \}$. Edit: If $\alpha$ algebraic over $\overline{F}_E$ then $\overline{F}_E (\alpha)$ is algebraic over $\overline{F}_E$ and so $\overline{F}_E$ is algebraic over $F$. Hence $\overline{F}_E(\alpha)$ is algebraic over F and $\alpha$ is algebraic over $F$. Can you help me? |
| What exactly does it mean that a limit is indeterminate like in 0/0? Posted: 17 Dec 2021 05:31 AM PST Non-mathematician here (obviously...) Given this limit question: $ \lim_{x\to2} \frac{x^2-3x+2}{x^2-4} $ This will have the form $\frac{0}{0}$, which is indeterminate. This is because both the numerator and the denominator approach 0, as x gets closer to 2. And that's why we need to rewrite this in an equivalent form where division by 0 does not happen. If I understand correctly, this is only necessary due to the fact that division by 0 is not defined in mathematics. So when calculating the limit of a function, we get an indeterminate form, it means, that the function will give use some expression that is not defined in mathematics. Is this correct? To say that it's just a formal problem? Or is there more to it, when a limit of a function gives us an indeterminate form? I have also read things like "the expression does not give enough information", which kind of sounds like there could be more to it. |
| Posted: 17 Dec 2021 05:30 AM PST Given $L$-smooth convex $f$, I would highly appreciate if you can confirm whether the following bound is correct or not. \begin{align} \left( y- x\right)^T \left( \nabla f(z)-\nabla f(x)\right) \leq \frac{L}{2} \left( \| x-z\|^2+\| x-y\|^2+\| z-y\|^2 \right) \tag{$\clubsuit$}. \end{align} Attempt: Since function $f$ is both $L$-smooth and convex, I particularly make use of the following two inequalities (e.g., can be found here or many other books such as in Yurii Nesterov's book).
To this end, we rewrite the above two respective inequalities 1. \begin{align} f(x) &\leq f(y) + \left( x - y \right)^T \nabla f(z) + \frac{L}{2} \| x - z \|^2 \\ \Longleftrightarrow \left( y - x \right)^T \nabla f(z) &\leq f(y) - f(x) + \frac{L}{2} \| x - z \|^2 \tag{1} \end{align} 2. \begin{align} f(y) - f(x) - \left( y - x\right)^T \nabla f(x) &\leq \frac{L}{2} \| x - y \|^2 \\ \Longleftrightarrow \left( y - x\right)^T \nabla f(x) &\geq f(y) - f(x) - \frac{L}{2} \| x - y \|^2 \tag{2} \end{align} Now, \begin{align} \left( y- x\right)^T \left( \nabla f(z)-\nabla f(x)\right) =& \underbrace{\left( y- x\right)^T \nabla f(z)}_{ \text{upper bound using} \ (1) } - \underbrace{\left( y- x\right)^T \nabla f(x)}_{ \text{lower bound using} \ (2) } \\ =& \underbrace{\left( y- x\right)^T \nabla f(z)}_{ \leq f(y) - f(x) + \frac{L}{2} \| x - z \|^2 } - \underbrace{\left( y- x\right)^T \nabla f(x) }_{ \geq f(y) - f(x) - \frac{L}{2} \| x - y \|^2 } \\ \leq & f(y) - f(x) + \frac{L}{2} \| x - z \|^2 - \left[ f(y) - f(x) - \frac{L}{2} \| x - y \|^2 \right] \\ =& \frac{L}{2} \left( \| x - z \|^2 + \| x - y \|^2 \right) \\ \leq& \frac{L}{2} \left( \| x - z \|^2 + \| x - y \|^2 + \underbrace{\| z - y \|^2}_{\geq 0} \right) \end{align} This completes the proof of $(\clubsuit)$. |
| When a regular $F_{\sigma}$ set open? Posted: 17 Dec 2021 05:24 AM PST
My question is when an open set is regular $F_{\sigma}$ set? AS clearly every regular $F_{\sigma}$ set open, does the converse hold in all metrizable spaces or some other particular type of topological spaces? Also is there any relation between $F_{\sigma}$ set and regular $F_{\sigma}$ set? |
| Atiyah singer index theorem and grothendieck riemman roch theorem Posted: 17 Dec 2021 05:59 AM PST In complex geometry, we have Hizenbruch signature theorem, in algebraic geometry we also have hizebruch riemann roch theorem which can imply the riemann roch theorem for algebraic curves. If given a manifold which is not necessarily algebraic manifold, what is the relationship between atiyah singer index theorem in differential geometry and grothendieck riemman roch theorem in algebraic geometry? |
| Why $\lfloor \sqrt{n} \rfloor$ changes by at most one when incrementing $n$? Posted: 17 Dec 2021 05:57 AM PST I was looking at the function $f(n)=\lfloor\sqrt{n}\rfloor$, and I noticed that consecutive numbers never differ by more than $1$. Why or why not? |
| Loewner order and norms of images: Does $A \preccurlyeq B$ imply $\|Ax\| \leq \|Bx\|$? Posted: 17 Dec 2021 05:43 AM PST Does $A \preccurlyeq B$ imply that $\|Ax\| \leq \|Bx\|$ for all $x$? I assume that $A,B$ are symmetric matrices and $A \preccurlyeq B$ denotes that $B-A$ is positive semi-definite. I can see that $A \preccurlyeq B$ implies several related properties, like
But does it also imply $\|Ax\| \leq \|Bx\|$ for all $x$? What if we assume both $A,B$ to be positive semidefinite? |
| How to prove r = n * 2(l·n) - l in specular reflection? Posted: 17 Dec 2021 05:53 AM PST I asked this question where I understand all basics concepts of specular reflection. From that question I read the reflection of the vector
My first question is what does mean of projection of any vector( in this case And my second question is how to prove |
| Find triangles to fill rectangle Posted: 17 Dec 2021 05:58 AM PST I'm looking to solve the following problem: Given a rectangle R and a shape S within that rectangle, find all shapes that, if combined with S, would fill R (without overlapping). The shapes are give by a set of points that make up the path around that shape. This is probably a fairly common problem in computer graphics but I couldn't find anything when googling this. If it makes things easier the shapes could all be triangles. |
| Does there exists any non trivial linear metric space in which every open ball is not convex? Posted: 17 Dec 2021 05:27 AM PST $\Bbb{R^\omega}=\{(x_n)_{n\in \mathbb{N}}: x_n \in \Bbb{R}\}$ $d(x, y) =\sum_{j\in\mathbb{N}}{(a_j)} \frac{|x_j -y_j|}{1+|x_j -y_j|}$ Then $(\Bbb{R^\omega}, d) $ is a metric space. I know in a normed space any ball is convex. And it is easy to prove. The space $(\Bbb{R^\omega}, d) $ is not a normed space, I mean no norm on $\Bbb{R^\omega} $ can induce the metric $d$. So, I guess in that space, It may be possible to find an open ball which is not convex. My question :1) Can I pick any open ball to test whether it is convex or not?
For the last question can I take $(X, \|•\|)$ be any normed space and then define a metric $d(x, y) =\sqrt\|x-y\|$. I think it works. Isn't it? Here $d$ is not scaling equivalent, hence not induced by any norm. $B_{d}(x_0, r) =\{x\in X : \|x-x_0\|<r^2 \}=B_{\|•\|} (x_0, r^2) $ Hence, every open ball is convex. |
| Posted: 17 Dec 2021 05:57 AM PST I have a time domain data which when converted to frequency domain gives a Gaussian spectrum. If I have limited number of samples in time domain, my frequency domain resolution is bad. In that case, if we try to estimate the second moment of the spectrum, the estimation would be biased if the actual spectrum width (in reality) is less than the frequency resolution. The way I estimate the second moment is given below, $$ \sigma = \sqrt{ \int_{-f_m}^{f_m} \frac{1}{P_T} [f - \mu]^2 |S[f]|^2 df } $$ Here, $f$ is the frequency axis, $S[f]$ is the $DFT$ of the original signal in time domain, $df$ is the frequency resolution, $f_m$ is the maximum frequency allowed and $P_T$ is the total power contained in the signal $$P_T = \int_{-f_m}^{f_m} |S[f]|^2 df $$ Is there a functional relationship I can write in the following form? $$ \sigma = f( N, \frac{df}{\sigma_{True}} ) $$ Where, $\sigma_{True}$ is the true spectrum with and $N$ is the number of frequency points. |
| Posted: 17 Dec 2021 05:19 AM PST I read that it's possible to show that the distribution of a record time sequence doesn't depend on the distribution of the record sequence itself, but how would one do this? So $(X_i)$ is an iid sequence with common continuous distribution $F$. Then $X_1$ is the first record, and $\tau_1$ the first record time. From here on, the $(k+1)$st record time is $\tau_{k+1}$, given by $\tau_{k+1}=\min\{i>\tau_k: X_i >M_{\tau_k}\}$, for $k\geq1$, with $X_{k+1}$ being the $(k+1)$st record, and $M_{\tau_k}$ denoting the maximum of $X_{i-1}$'s until time $\tau_k$. It is supposedly then possible to show that the sequence $(\tau_{k})_{k\geq1}$ doesn't depend on $F$, but I'm unsure how this can be done? I tried looking at the probability \begin{align*} P(\tau_{k+1} \leq t) & = P(\min\{i>\tau_k: X_i >M_{\tau_k}\}\leq t)\\ & = P(i\leq t, i-1\leq t, \ldots), \end{align*} for some $t$, since if $i\leq t$, then all 'previous' $i$'s must necessarily also be less than or equal to $t$, but I don't know if this is the way to go. |
| Intersection of circle and geodesic segment on sphere Posted: 17 Dec 2021 05:33 AM PST I am trying to find an efficient way of computing the intersection point(s) of a circle and line segment on a spherical surface. Say you have a sphere of radius R. On the surface of this sphere are
where $\theta$ is the colatitude in $[0,\pi]$, $\phi$ is the longitude in $[0,2\pi]$, and $r$ is measured by the geodesic distance on the sphere (not straight line distance in Euclidean space). How would you
We can assume there is nothing pathological going on. $r$ is not zero and is not so large that it's bigger than the sphere, the line's endpoints are not identical, etc. |
| Posted: 17 Dec 2021 05:38 AM PST Is there a way to solve this for $x$ ? $$y=(ax^2+a-x)\bmod 2a^2$$ How to handle this kind of equation? From what I read, even a simpler form seems to be far from simple. When $a$ is odd, $y$ is odd, and I am only interested in $a$ odd for now. For a fixed odd $a$, each (integer $\ge 0$) value of $x<a^2$ maps $1$ to $1$ to an odd $y<2a^2$. Since there is a direct map, I think it is possible to reverse it. If it is only feasible with a prime, I can do with tha.t |
| $a+b+c = 2001$ and $a>b>c>0$ how many ordered triples are possible? Posted: 17 Dec 2021 05:21 AM PST $a+b+c = 2001$ and $a>b>c>0$ how many ordered triples are possible? I tried substituting these equations $c+x=b$ $b+y=a$ thus, the equation would be $3c+2x+y = 2001$ but I do not know how to solve this, is there a more general stars and bars formula? |
| Submitting for publication in American Mathematical Monthly Posted: 17 Dec 2021 05:48 AM PST I am a young computer science researcher. One of my working papers had some mathematical content so I presented it in a math department seminar. After the talk, one of the professors told me that he liked the talk and suggested that I submit the paper to American Mathematical Monthly. I was flattered, since this journal published many of the important results in my field (as well as in other fields, of course). Then, another professor who overheard the conversation said to his friend "do not do this to him! He is still young!". I was perplexed and did not have time to ask: why is the American Mathematical Monthly not a good option for a young researcher?
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| Find the value of $\lim_{n \rightarrow \infty} \sqrt{1+\left(\frac1{2n}\right)^n}$ Posted: 17 Dec 2021 05:45 AM PST Find the limit of the sequence as it approches $\infty$ $$\sqrt{1+\left(\frac1{2n}\right)^n}$$ I made a table of the values of the sequence and the values approach 1, so why is the limit $e^{1/4}$? I know that if the answer is $e^{1/4}$ I must have to take the $\ln$ of the sequence but how and where do I do that with the square root? I did some work getting the sequence into an indeterminate form and trying to use L'Hospitals but I'm not sure if it's right and then where to go from there. Here is the work I've done $$\sqrt{1+\left(\frac1{2n}\right)^n} = \frac1{2n} \ln \left(1+\frac1{2n}\right) = \lim_{x\to\infty} \frac1{2n} \ln \left(1+\frac1{2n}\right) \\ = \frac 1{1+\frac1{2n}}\cdot-\frac1{2n^2}\div-\frac1{2n^2}$$ Thank you |
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