Recent Questions - Mathematics Stack Exchange |
- What is/are the main purpose(s) of topology?
- proof :K is positive integer ,if K-2 to be divisible by 2k,then k-2 to be divisible by 4
- Prove that the limit $\lim_{(x,y)\to(0,0)} \frac{y^4}{\sqrt{3x^2+y^2}}\cos(\frac{1}{x})$ exists using $\epsilon \text{ and } \delta$ definition
- Doubt in a numerical from Ratio and Proportion.
- Is it possible for a logical proof system to be both unsound and incomplete?
- Evans PDE 8.4 Theorem 2
- General method for solving 2nd degree homogenous recurrence relation with variable coefficients
- Why can't Pascal's Simplex with n terms be used to describe close-packing of equal n-spheres?
- two-dimensional number system without negative numbers
- equipartition question
- Finding expected result from a betting line
- Differential geometry exercice from Kristopher Tapp's book
- Show that $\mathcal{C}$ generates $\mathcal{E}_I$.
- Why is $\sqrt{-1}$ the only imaginary number we define?
- What data cleaning method is best to be adopted for price=$0?
- If $\lim\limits_{x \rightarrow 0} h(x)=0$ then, $\lim\limits_{x \rightarrow 0} (7h(x) +1 )^{\sqrt{2}/h(x)}$?
- The Lady or the Tiger , Logical Labyrinth by Smullyan
- Probability of two passengers will be staying at same inn
- A dice puzzle and an "obvious" fact I cannot prove
- Find the area of the large triangle
- Unitary-transformation invariant measure on subset of the sphere
- Prove $(\Box(p\supset q)\land\Diamond(p\land r))\supset\Diamond(q\land r)$ in K
- About the unicity of the analytic continuations of $\zeta$ and the continuation used to have $\zeta'(0) = -(\ln(1) +\ln(2) +\ln(3) + \ldots )$
- $\mathcal{O}_{\mathbb{C}_p}/ p \mathcal{O}_{\mathbb{C}_p}$ infinite
- Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$
- Invertibility of quasitriangular Hopf algebra element using Sweedler notation
- $L_p$ norm of the Dirichlet Kernel
- Converting Dirichlet Boundary Conditions to Neumann Boundary Conditions for the Heat Equation
- Find a non-zero vector u with terminal point Q(3,0,-5) such that u is opp directed to v=(4,-2,-1)?
- A formula for a sequence which has three odds and then three evens, alternately
| What is/are the main purpose(s) of topology? Posted: 21 Aug 2021 08:35 PM PDT
I was reading this article and I was wondering about what might be the main purpose of topology or at least one of the main purposes. Could it be that one of the main purpose is to see how n-1 dimensional plane behave when the plane is projected onto a n dimensional geometric object without breaking the geometrical rules of the n-1 dimensional plane (by giving it a curvature or a n dimensional geometric property to outside observers while the inside observers sees the n-1 dimensional plane as being a n-1 dimensional plane from the inside? Could you explain so that a layman can understand?  |
| proof :K is positive integer ,if K-2 to be divisible by 2k,then k-2 to be divisible by 4 Posted: 21 Aug 2021 08:32 PM PDT where the question from: I consider this from a problem what i try k=3,4,6 satisfy the condition  |
| Posted: 21 Aug 2021 08:36 PM PDT Prove that the limit $\lim_{(x,y)\to(0,0)} \frac{y^4}{\sqrt{3x^2+y^2}}\cos(\frac{1}{x})$ exists using $\epsilon \text{ and } \delta$ definition. I have so far reduced it algebraically to the point $\frac{y^4}{\sqrt{x^2+y^2}}$ but I am not sure if I have to use polar coordinates or something to continue, kinda lost from here. I eliminated the cosine because $\cos(\frac{1}{x})\leq 1$ so $$\frac{y^4}{\sqrt{3x^2+y^2}}\cos(\frac{1}{x}) \leq \frac{y^4}{\sqrt{3x^2+y^2}}$$ And likewise I eliminated the three from $3x^2$. Do I have to use polar coordinates like $r=x^2 + y^2$??? I did some more working and got $\delta=\epsilon^{\frac{1}{3}}$...  |
| Doubt in a numerical from Ratio and Proportion. Posted: 21 Aug 2021 08:20 PM PDT " In an office, one-third of the workers are women, half of the women are married and one third of the married women have children. If three-fourth of the men are married and one-third of the married men have children, then what is the ratio of married women to married men?" Doubt : I tried equating married men who have children with married women who have children in order to arrive at the relation between the two only to find out that the answer which I arrived after the calculation was incorrect. A slight guidance about the same is most welcome.  |
| Is it possible for a logical proof system to be both unsound and incomplete? Posted: 21 Aug 2021 08:20 PM PDT I know that there are simple examples of proof systems that are sound but incomplete and unsound but complete, but is there such a proof system that is both unsound and incomplete?  |
| Posted: 21 Aug 2021 08:18 PM PDT The theorem states that
My question is about the last few lines of its proof where he deals the case $g(u)=0$ a.e.
So far so good, but then he writes
I do not understand why $G(0)=0$ would follow.  |
| General method for solving 2nd degree homogenous recurrence relation with variable coefficients Posted: 21 Aug 2021 08:16 PM PDT Like in this case where recurrence is given by relation $a_{n+1} =(2n/(n+1)) a_n - [(n-1)/(n+1)]a_{n-1}$ , i know method for first degree , but for second and more what should be the approach ?  |
| Why can't Pascal's Simplex with n terms be used to describe close-packing of equal n-spheres? Posted: 21 Aug 2021 07:59 PM PDT Pascal's simplex is a generalization of Pascal's triangle into n dimensions, just as multinomial theorem is a generalization of binomial theorem. In Pascal's triangle, binomial coefficients are arranged in the close-packing arrangement for circles. In Pascal's pyramid, trinomial coefficients are arranged in a close-packing arrangement for spheres (there are multiple in 3 dimensions). This is also true for 1-dimensional spheres (aka line segments). My intuition is that this pattern would continue infinitely, but I know that doesn't mean anything, especially as a layperson. I should also point out that each the close-packing arrangement for spheres has cross-sections that look like the close-packing arrangement for circles, which has cross-sections that look like line segments stacked together, which is the close-packing arrangement in 1 dimension. Is this apparent pattern just a coincidence? If I've gotten anything wrong, please let me know.  |
| two-dimensional number system without negative numbers Posted: 21 Aug 2021 07:56 PM PDT Is there any existing literature for the number system that looks like this? two-dimensional number system without negative numbers Like the complex number system, this system exists on a plane. But instead of i and -1, it has two numbers-- called q and d here-- that form two new families of numbers moving away from the origin at 120° and 240°, respectively, relative to the ray representing the natural numbers. Three rays (the natural numbers, the q numbers, and the d numbers) divide the plane equally into three regions (instead of four regions as in the complex plane). Instead of subtraction, this number system has two new operations: One to move 120° up by x units (by adding xq) and the other to move 120° down by x units (by adding xd). We can still technically do subtraction, though: x-y = x+yq+yd Just as x + -x = 0, we have x + xq + xd = 0. Multiplication works as follows: \begin{array}{r|rrr} & 1 & q & d & \\ \hline 1 & 1 & q & d & \\ q & q & d & 1 & \\ d & d & 1 & q & \\ \end{array} Of course, q and d exist in the complex plane as well: q=-0.5+i$\frac{\sqrt{3}}{2}$ d=-0.5-i$\frac{\sqrt{3}}{2}$ -- I am not a mathematician, so that's all I could come up with given my very limited knowledge, and I would really like to know more, like if there are any known applications for this. I know it basically just takes two points in the complex plane and assigns some significance to them just to "remove" -1, i, and -i, but I wonder if perhaps the simpler facade actually leads to a practical benefit. I am also interested in knowing how this can be generalized in higher dimensions. Notice that for any number x, the points x, xq, and xd form an equilateral triangle on the plane. I reckon you could generalize the system for any n-dimensional space by constructing a regular n-simplex centered at the origin and constructing n rays emanating from the origin and passing through all vertices, but I don't know how multiplication would look like there aside from the vague idea that you could translate it from an appropriate Cayley–Dickson algebra.  |
| Posted: 21 Aug 2021 07:42 PM PDT Let n points be chosen on the real segment, [0,1]. The expected values of those choices will partition [0,1] into n+1 equal segments. What is an intuitive way to see that this would be the case?  |
| Finding expected result from a betting line Posted: 21 Aug 2021 07:39 PM PDT If I have have a betting line at a value of 800 receiving yards for a NFL player that has implied probabilities of 50% for the over and 50% for the under, the expected receiving yards is 800 for these bets. I am unsure on what the expected receiving yards for a line with odds that aren't 50/50 For example: Betting line of 1000 receiving yards for a player with a 45% over, and a 55% under. My gut says to multiply the receiving yards by .95 since its 5% away and the expected value of these bettors is 950 yards. Or to multiply it by .9 (.45 * 2). But I am completely void of trying to find any logic behind these.  |
| Differential geometry exercice from Kristopher Tapp's book Posted: 21 Aug 2021 07:35 PM PDT I have problem with this exercice from Kristopher Tapp's book "Differential Geometry of Curves and Surfaces" EXERCISE 1.42. (Page 32) Let $\gamma: I \rightarrow \mathbb{R}^{n}$ be a unit-speed curve. Let $t_{0} \in I$ and assume $\kappa\left(t_{0}\right) \neq 0$. For sufficiently small $h>0$, prove that the three points $\gamma\left(t_{0}-h\right), \gamma\left(t_{0}\right)$ and $\gamma\left(t_{0}+h\right)$ are not collinear. My attempt: I think the question means: prove that there exists $\epsilon >0$ such that for all $h >0$, if $h<\epsilon$ then the three points $\gamma\left(t_{0}-h\right), \gamma\left(t_{0}\right)$ and $\gamma\left(t_{0}+h\right)$ are not collinear. I tried proof by contradiction. Assume that there exists a sequence $(h_{n})_{n\geq1}\subset \mathbb{R}^{*}_{+}$ such that $h_{n} \rightarrow 0 $ , and $(\alpha_{n})_{n\geq1}\subset \mathbb{R}$ such that $ \gamma\left(t_{0}+h_{n}\right)-\gamma\left(t_{0}\right)=\alpha_{n}\left( \gamma\left(t_{0}-h_{n}\right)-\gamma\left(t_{0}\right) \right)$ (It's easy to show that $\alpha_{n} \rightarrow-1 $ when $n \rightarrow \infty $), using Taylor expansion we get $\left\{\begin{array}{l} \left.\gamma\left(t_{0}+h_{n}\right)-\gamma\left(t_{0}\right)=h_{n} \gamma^{\prime}\left(t_{0}\right)+\frac{h_{n}^{2}}{2} \gamma^{\prime \prime}\left(t_{0}\right)+o\left(h_{n}^2\right)\right) \\ \gamma\left(t_{0}-h_{n}\right)-\gamma\left(t_{0}\right)=-h_{n} \gamma^{\prime}\left(t_{0}\right)+\frac{h_{n}^{2}}{2} \gamma^{\prime \prime}\left(t_{0}\right)+o\left(h_{n}^2\right)\end{array}\right.$ from this and the above equation we get $\left(1+\alpha_{n}\right) h_{n} \gamma^{\prime}\left(t_{0}\right)+\left(1-\alpha_{n}\right) \frac{h_{n}^{2}}{2} \gamma^{\prime\prime}\left(t_{0}\right)=o\left(h_{n}^{2}\right).$ I can't find any contradiction and I can't find where I can use $\kappa\left(t_{0}\right) \neq 0$.  |
| Show that $\mathcal{C}$ generates $\mathcal{E}_I$. Posted: 21 Aug 2021 07:30 PM PDT I can see that $\cup_{i \in I} \mathcal{E}_i \subset \mathcal{C}$. But, if we want to say that $\mathcal{C}$ generates $\mathcal{E}_I$, we have to show the equality $\cup_{i \in I} \mathcal{E}_i = \mathcal{C}$, right? But, I don't know how to show this. Suppose $A \in \mathcal{C}$. Then, there exists $\cap_{i \in J} A_i = A$. I don't think $\cap_{i \in J} A_i$ is necessarily an element of $\cup_{i \in I} \mathcal{E}_i $ because not every subset of $A_i \in \mathcal{E}_i$ is an element of $\mathcal{E}_i$. I think I am missing something...  |
| Why is $\sqrt{-1}$ the only imaginary number we define? Posted: 21 Aug 2021 08:17 PM PDT So I was wondering why the only imaginary number we define is $\sqrt{-1}$ why do we not consider stuff like $\log{0}$ or $\frac{1} {0} $ or even $-1!$ as "imaginary"? I don't understand why we picked $\sqrt{-1} $, is there something special about it?  |
| What data cleaning method is best to be adopted for price=$0? Posted: 21 Aug 2021 07:22 PM PDT Suppose I have an AirBnb dataset. In EDA, i noticed that some room price are 0 (as shown in the example below). Is removing them a best approach since its not possible that a room price is 0? Or should I replace with the mean values? AirBnB name Host Host_is_superhost Price Min_nights Max_nights Availability_365 Long Lat Ritz Stay Rina Y 0 3 300 300 3.4 -1.2 Cartel Josh Y 0 4 30 10 1.2 -1.23  |
| Posted: 21 Aug 2021 08:15 PM PDT This question is about limits. Please help how to find the solution.Written Question  |
| The Lady or the Tiger , Logical Labyrinth by Smullyan Posted: 21 Aug 2021 07:17 PM PDT I won't repeat the question (#12 - the fourth day) but for the last of the lady or tiger set in chapter 2 I believe either the solution is problematic or how parts of the puzzle is worded are problematic. I believe this is mainly due to that from sign 3 and sign 5 he uses the word "either" as in either A or B when it's pretty clear from the solution he means Logical OR, but where I and others would be only be able to understand it as Logical XOR. If you assume logical XOR the Lady can be in either room 1 or room 7. See these two Solutions where (b) is the solution given and (a) is also valid given "either" means XOR. If the above reasoning is correct then it would be better if the puzzle just dropped the word "either."  |
| Probability of two passengers will be staying at same inn Posted: 21 Aug 2021 07:33 PM PDT There are three passengers on an airport shuttle bus that makes stops at five different inns. Find the probability that two passengers will be staying at same inn I think there are two events: Just 2 passengers at same inn or just 3 passengers at same inn. So $ \binom{3}{2} $ is the number of ways in which you can choose the 2 passengers out of 3 passengers that are in same inn, $ \binom{5}{1} $ is the number of ways in which you can choose 1 inn out of 5 inns that the 2 or 3 passengers are , $ \binom{3}{3} $ is the number of ways in which you can choose 3 passagers out of 3 passengers that are in same inn Result: $ \frac{\binom{3}{2}\binom{5}{1}+\binom{3}{3}\binom{5}{1} }{5^3}$ Is this correct?  |
| A dice puzzle and an "obvious" fact I cannot prove Posted: 21 Aug 2021 07:55 PM PDT BackgroundThis question is inspired by this 538 "Riddler Classic" puzzle, and the following puzzle explanation below is copied from there:
QuestionWe are interested only in the general $n$ case, and we are interested in a question tangential to the puzzle's solution. Assuming perfect play, consider the expected value (EV) of your sum when starting with $n$ dice, which we'll denote $E_n$. The question is: Is it possible that, for some $n$, the following holds? $$E_{n+1} > E_n + 6$$ Alternatively, should you ever choose to re-roll a six when playing optimally? Intuitively, it seems like the answer to both of these equivalent questions should be "no". But I cannot think of a rigorous argument to prove it.  |
| Find the area of the large triangle Posted: 21 Aug 2021 07:08 PM PDT
The triangles are placed like you see in the image below:
Question: What is the area of the large triangle?Note: I have a hard time believing this is a hard problem but I've tried to find the base of the blue triangle but with no success. Would appreciate if someone could give me some insight on how to solve it!  |
| Unitary-transformation invariant measure on subset of the sphere Posted: 21 Aug 2021 08:00 PM PDT It is stated as a fact in this correction paper that
Could anyone point me to a resource elaborating on this? This is mainly a graph theory paper which I am trying to digest, but I haven't taken a course in measure theory. I would like to know how to construct this measure/what it looks like.  |
| Prove $(\Box(p\supset q)\land\Diamond(p\land r))\supset\Diamond(q\land r)$ in K Posted: 21 Aug 2021 08:31 PM PDT $(\Box(p\supset q)\land\Diamond(p\land r))\supset\Diamond(q\land r)$ Here's what I have so far:
But now I'm stuck. This is problem 2.1c from Hughes and Cresswell. I think some instance of what they call K7 could be used: $\Diamond(p\supset q)\equiv(\Box p\supset \Diamond q)$ Something like this: $\Diamond(((p\supset q)\land(p\land r))\supset(q\land r))\equiv(\Box((p\supset q)\land(p\land r))\supset\Diamond(q\land r))$ But I don't know how to prove $\Diamond(((p\supset q)\land(p\land r))\supset(q\land r))$. Thanks so much for any and all help.  |
| Posted: 21 Aug 2021 07:39 PM PDT The analytic continuation of an analytic function is unique. I've come across different representations of those, for example: $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s), s\neq1,$ AKA functional eq. $\zeta(s)=\frac1{s-1}+\sum_{n=1}^\infty \frac{(-1)^n\gamma_n}{n!}(s-1)^n, s\neq1,$ from here, etc. Now, on one hand if one takes the derivative of the functional equation and evaluates at $0,$ one gets $-\ln\sqrt{2\pi},$ e.g. proof here. On the other hand, in this proof, is mentioned:
That is, one ultimately gets $ - \lim_{n \to \infty} \ln(n!)$ In that same proof, mentions that $\zeta'(0)=-\ln\sqrt{2\pi}$ Ultimately, one would get to "$\infty!=\sqrt{2\pi}$" Why is that? So far, what I understand is that if $f$ is unique, then regardless of how it's presented, say $f'$, then $f(x)=a$ should be equal to $f'(x)=a.$ But in this example looks like $f(x)=a$ and $f'(x)=b.$ I don't know much about this. Excuse my ignorance in advance :)  |
| $\mathcal{O}_{\mathbb{C}_p}/ p \mathcal{O}_{\mathbb{C}_p}$ infinite Posted: 21 Aug 2021 07:20 PM PDT Let $\mathcal{O}_{\mathbb{C}_p}$ the ring of integers of complex $p$-adic numbers $\mathbb{C}_p$ which are defined as completion of alg closure $\overline{\mathbb{Q}_p}$ with respect $\vert \cdot \vert_p$ (extended from $\mathbb{Q}_p$ to $\overline{\mathbb{Q}_p}$. It is known that $\mathcal{O}_{\mathbb{C}_p}$ is not a Noetherian ring (e.g. A. M. Robert, A Course in p-adic Analysis, page 135). How this fact can be explicitely used to show that $\mathcal{O}_{\mathbb{C}_p}/ p \mathcal{O}_{\mathbb{C}_p}$ is infinite?  |
| Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ Posted: 21 Aug 2021 07:11 PM PDT We all know that: $$\sum_{x=0}^\infty \frac{1}{x!}=e$$ But what if we replaced $x!$ with $!x$ also called the subfactorial function also called the $x$th derangement number? This interestingly is just a multiple of $e$ and an Incomplete Gamma function based sum. This will get us a new number as the $n=0$ and 1 terms diverge as a result of the reciprocal. I also use the Generalized Exponential Integral function and the Round function. The OEIS entry for the constant is A281682: \begin{align*} S &=\sum_{x=2}^\infty \frac{1}{!x} =\sum_{n=2}^\infty \frac{1}{\operatorname{Round}\bigl(\frac{x!}{e}\bigr)} =e\sum_{x=2}^\infty\frac{1}{Γ(x+1,-1)}= \\ &= e\sum_{n=3}^\infty \frac 1{Γ(x,-1)}=\sum_{X=0}^\infty \sum_{x=2}^\infty\frac{1}{Γ(X+1)Γ(x+1,-1)} =-e\sum_{x=2}^\infty\frac{(-1)^x}{E_{-x}(-1)}= -e\sum_{x=-\infty}^{-2}\frac{(-1)^x}{E_x(-1)} =1.63822707… \end{align*} We can also use the Abel-Plana formula, and the alternate series version, to find an integral representation of the sum. You can also use other representations of the summand, but this integral is probably hard to work with. Note the Abel-Plana formula may not work with the constant: \begin{align*} S &=\sum_{x=0}^\infty\frac{1}{!(x+2)} =\frac{1}{2} + \int_0^\infty \frac{dx}{!(x+2)} + i\int_0^\infty\frac{\frac{1}{!(2+ix)}-\frac{1}{!(2-ix)}}{e^{2\pi x}-1} \, dx \\ \implies -\frac{S}{e} &=\sum_{x=0}^\infty\frac{(-1)^x}{E_{-x-2}(-1)}=-\frac{1}{2e} + \frac i2\int_0^\infty \left[\frac{1}{E_{-ix-2}(-1)}-\frac{1}{E_{ix-2}(-1)}\right] \operatorname{csch}(\pi x) \, dx \end{align*} See this nice closed form result of I do not think this simple looking problem has been posted so far. The sum does not need to be in closed form. You also can rewrite it in terms of a better sum. I am more looking for an evaluation or manipulation of the sum. Please correct any mistakes and give me feedback! Possibility of a closed form: Because $$\sum_{x=2}^\infty \frac{1}{!x} =e\sum_{x= 2}^\infty\frac{1}{Γ(x+1,-1)}= e\sum_{n=3}^\infty\frac 1{Γ(x,-1)} $$ one may notice the relation to the Mittag-Leffler function: $$\text E_{a,b}(x)=\sum_{n=0}^\infty \frac{x^n}{Γ(ax+b)}$$ The only problem is if there existed a function for the incomplete gamma function analogue of the Mittag-Leffler function. Maybe one can find this function or use the already known one? Please do not make up any new function.  |
| Invertibility of quasitriangular Hopf algebra element using Sweedler notation Posted: 21 Aug 2021 08:24 PM PDT The question concerns part of a theorem in the book Foundations of Quantum Group Theory, Shahn Majid (Cambridge University Press, 1995). More specifically, Theorem 2.3.4 (p.55-57) which I'll rewrite below in terms of what's impeding my progress through the book. Let $(H, \mathcal{R})$ be a quasitriangular Hopf algebra and let $\chi$ be a counital 2-cocycle. Define: $$ U=\chi^{(1)}(S\chi^{(2)}), $$ where $S$ denotes the antipode. Show that $UU^{-1}=1$. To start with, we define $U^{-1}=S(\chi^{-(1)})\chi^{-(2)}$. The proof as is given in the book goes as follows, $$ UU^{-1}= \chi^{(1)}(S\chi^{(2)})(S\chi^{-(1)})\chi^{-(2)} $$ $$ \qquad\qquad\qquad\qquad\quad\,\,= \chi'^{-(1)}\chi^{(1)}(S(\chi'^{-(2)}_{(1)}\chi^{-(1)}\chi^{(2)}))\chi'^{-(2)}_{(2)}\chi^{-(2)}$$ $$ \,\,= \chi^{-(1)}_{(1)}(S\chi^{-(1)}_{(2)})\chi^{-(2)}$$ $$ =1. \quad\qquad\qquad\quad\,\,\,\,$$ The author then proceeds to say the following:
My problem is that no matter what I try, I can do nothing to arrive at the precise form given in the second row of the proof, and I definitely have no clue how the last line equates to $1$, even though most of the major steps have been written. I tried inserting several identity relations (several "ones") in super specific ways, but that brought me nowhere. I am also unsure which specific antipode properties are being used here. Any help with filling in the gaps will be greatly appreciated.  |
| $L_p$ norm of the Dirichlet Kernel Posted: 21 Aug 2021 08:18 PM PDT I am trying to show that $$ \left(\int_{-1/2}^{1/2} \left|\dfrac{\sin(\pi(2N+1)x)}{\sin(\pi x)}\right|^p dx \right)^{1/p} \approx N^{\frac{p-1}{p}}$$ for all $1<p\leq \infty$ by using the fact that $|\sin(x)| \approx |x|$ if $x \in [-1/2, 1/2)$. I have tried using the change $y=Nx$ and I end having the result $2N+1$. I do not know what I am doing wrong. I would appreciate any help.  |
| Converting Dirichlet Boundary Conditions to Neumann Boundary Conditions for the Heat Equation Posted: 21 Aug 2021 08:06 PM PDT I'm solving the heat equation on a two dimensional square domain. The problem is defined as: $$ u_{xx}+u_{yy} = 0 \hskip{0.5cm}\text{for} \hskip{0.5cm} 0 \leq x \leq 1, 0 \leq y \leq 1 $$ with the following boundary conditions: $$ u(0,y)= 0, \hskip{0.5cm} u(1,y)= 0, \hskip{0.5cm} u(x,1)= 0, \hskip{0.5cm} u(x,0)= sin(\pi x) $$ I'm able to solve this eqations both numerically and analytically, however I'm having trouble with the following: I would like to convert the boundary condition on $u(x,0)$ to an equivalent Neumann boundary condition. I've tried the following: $$ u_x(x,0) = \pi cos(\pi x) $$ However this is not resulting in a satisfactory result. Is my thinking correct? edit: The exact solution I get is: $$ u(x,y)= sin(\pi y) \cdot \left[ cosh(\pi x) - coth(\pi) \cdot sinh(\pi x) \right] $$ Edit: I've got it working now using $u_y(x,y) = \pi cosh(\pi) / sinh(\pi) sin(\pi x)$ as the neumann boundary condition. Thank you for your help, Jan Willem  |
| Find a non-zero vector u with terminal point Q(3,0,-5) such that u is opp directed to v=(4,-2,-1)? Posted: 21 Aug 2021 07:08 PM PDT So for this question, this is how i approach Let's treat initial point as (a1,a2,a3) and we got the terminal point, so vector of u is (3-a) (0-b) and (-5-c) I am kind of stuck here as I as if the question was same direction, we just need to find the scalar multiple but opposite direction, how do we approach? Thanks for the help in advance :)  |
| A formula for a sequence which has three odds and then three evens, alternately Posted: 21 Aug 2021 07:44 PM PDT We know that triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36... where we have alternate two odd and two even numbers. This sequence has a simple formula $a_n=n(n+1)/2$. What would be an example of a sequence, described by a similar algebraic formula, which has three odds and then three evens, alternately? Ideally, it would be described by a polynomial of low degree.  |
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