Recent Questions - Mathematics Stack Exchange |
- Log and change of sign
- A simple way to explain Abel's impossibility theorem
- How to learn Strichartz estimates for wave equations?
- Actuarial Mathematics
- Why is the subscheme fixed if I fix the Hilbert Polynomial?
- Accelerated Collatz map is actually a commutative monoid homomorphism at each of its iterations $f^i, i \geq 0$.
- Meromorphic functions on smooth plane curves
- How to prove that $\mu = \sqrt{\frac{1+ \delta^2}{4}}$
- Is it wrong to think of area as a measure of the total number of points?
- Linear preconditioning on a symmetric and non-definite matrix
- Can you tell which option is correct.
- Proof of convergence for the Monte-Carlo method in the Linear Transport Equation
- Poincare constant for Gaussian distribution
- Kernel of Bounded Operator on Hilbert Space
- Let A be a square matrix so that AB = BA for every square matrix B of the same order. Prove that A is a diagonal matrix. [duplicate]
- One problem with conditional probabilities
- What's the difference between $\mathbb{R}^3$ and $\mathbb{R}_3$?
- associated prime ideal and localization of Noetherian ring
- Prove that if $\lVert T(v)\rVert <\lVert T^*(v)\rVert$ then there exists $u$ such that $\lVert T(u)\rVert >\lVert T^*(u)\rVert$
- $(3)$ is inert in $\mathbb{Q}[\sqrt{2}]$
- Prove $\sum_{cyc}{\sqrt{\frac{x+1}{x^2+16x+1}}}\geqslant 1$ and $ \sum_{cyc}{\sqrt{\frac{x+1}{4x^2+10x+4}}}\leqslant 1$ for $x,y,z>0,xyz=1$
- Would $ \lim_{\sigma\rightarrow0^+}\int_{-\sigma}^{\sigma}\frac{1}{\sigma}f(x)dx$ be equal to zero?
- For which values of n does insertion sort beat merge sort?
- models of coins and uniform distribution
- Expected run in a run test.
- Prove that $\lim_{n\to\infty} (\sqrt{n^2+n}-n) = \frac{1}{2}$
- Proving that $n\mid(nCr)$ for all $r$ ($1 \leq r \leq n-1$), only if $n$ is prime
- Calculating a tangent arc between two points on two circles
| Posted: 12 Feb 2022 04:06 AM PST I have $0.01 \geq 0.2^n \iff ln(0.01) \geq nln(0.2)$ Not changing the sign, is this correct? What if we had only one side negative and plugged in $ln$ ? |
| A simple way to explain Abel's impossibility theorem Posted: 12 Feb 2022 04:07 AM PST For philosophy students, I want to explain very briefly the theory of equations showing the very great innovations in thinking made by Abel (the study of them as structure), and Galois (the resolvability of their ambiguity as a dynamic reciprocal determination) (Deleuze), for which I need to explain first some rudiments of both theories, and it occurred to me that, in summary, it could be as follows:
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| How to learn Strichartz estimates for wave equations? Posted: 12 Feb 2022 04:03 AM PST I am a student planning to learn some knowledge about Strichartz estimates for wave equations. My goal is to understand the Strichartz estimates or a priori estiamtes of weak solutions for the linear wave equation as follows \begin{eqnarray} \partial_t^2u-\operatorname{div}(A(x)\nabla u)=0, \end{eqnarray} where $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ is a symmetric real matrix such that $ a_{ij}(x)\xi_i\xi_j\geq\lambda|\xi|^2 $ for all $ x\in\mathbb{R}^d $ and $ \xi\in\mathbb{R}^d $. So what should I start? I know that Sogge wrote some book about it and there may be some other books or papers abou the topic. What I want to ask is that what order should I follow? Now I have some basic knowledge about linear wave equations(mainly from "Evans pde") and harmonic analysis(mainly from "Stein harmonic analysis"). |
| Posted: 12 Feb 2022 04:02 AM PST 6.10 A deferred life annuity on (60) provides for annuity benefits payable for 3 years, beginning at age 63, provided that (60) is alive. The first annuity payment is 1000, the second 2000, and the third 3000. Premiums are payable for 3 years, beginning at age 60. The second and third premiums are equal in amount and each double the amount of the initial premium. If (60) dies before age 63, there will be, at the end of the year of death, a return of all premiums paid prior to death, without interest. You are given that q60 = 0.1, q61 = 0.2, q62 = 0.25, q63 = 0.3, q64 = 0.4. The interest rates are 25 % per year for the first 4 years, and 50 % for the fifth year. Find (a) the initial premium, (b) 1V, (c) 4V |
| Why is the subscheme fixed if I fix the Hilbert Polynomial? Posted: 12 Feb 2022 04:03 AM PST Let $Y$ be a closed subscheme of $X=\mathbb P{^n}$ with Hilbert Polynomial given by $h(m) =\binom{m+r}{r}$
Can someone help me solve this? I know for $\mathbb P^r$ that is indeed the Hilbert Polynomial. |
| Posted: 12 Feb 2022 04:13 AM PST The following are my results related to the Collatz conjecture so far. As far as I know, no research has addressed the fact that iterates $f^i$ of the accelerated Collatz map seem to form monoid homomorphisms $f^i : (M, *) \to (M, \cdot)$ for each $i$, where $*_1$ for $i=1$ seems to be $x*_1 y := 3xy + x + y$ and $*_2$ seems to be: $$ x*_2 y :=(3x + \dfrac{2^{\nu_2(3x + 1)} - 1}{3} +1)*_1(3y + \dfrac{2^{\nu_2(3y + 1)} -1}{3} + 1) $$ The following only goes into how to derive $*_1$, but a similar procedure will give us $*_2$, and I expect that pattern to hold as we increase $i$. Let the and $N$ be the monoid $2\Bbb{N} + 1$ with the usual multiplication. Define $M = \{ x \in N : \nu_2(3x + 1) \in 2\Bbb{N}\cup 0\}$ which is closed under $\star_1$ you will later. Define $f: M \to N$ to be the accelerated Collatz function, i.e. $$ f(x) = \dfrac{3x + 1}{2^{\nu_2(3x + 1)}} $$ where $\nu_2(y) =$ the maximum power $k$ such that $2^k \mid y$. Proposition 1. The set $N$ is also closed under the operation $x*y := 3xy + x + y$, and this operator gives $N$ the structure of another monoid. Proof. The identity for this binary map is $0$ since $x*0 = 3x\cdot 0 + x + 0 = x$. And $U$ is closed under this operator: if $x,y$ are odd numbers then $x + y$ is an even number which, when added to the odd number $3xy$, yields an odd number. Finally and most difficult is associativity: $$ a*(b*c) = a*(3bc + b + c) = 3a(3bc + b + c) + a + 3bc + b + c = \\ 9abc + 3ab + 3ac + 3bc + a + b + c \\ (a*b)*c = (3ab + a + b)*c = 3(3ab + a + b)c + 3ab + a + b + c = \\ 9abc + 3ac + 3bc + 3ab + a + b + c $$ $\blacksquare$ Proposition 2. The accelerated Collatz function $f$ is itself a monoid homomorphism $(M, *_1) \xrightarrow{f} (N, \cdot)$, or iow $f(x*y) = f(x)f(y)$ for all $x,y \in M$. Proof. By definition $f(x*y) = \dfrac{3(x*y) + 1}{2^{a}}$ where $a = \nu_2(3(x*y) + 1)$. Let $b = \nu_2(3x + 1)$ and $c = \nu_2(3y + 1)$. We then have: $$ f(x)f(y) = \dfrac{3x + 1}{2^b} \dfrac{3y + 1}{2^c} = \dfrac{9xy + 3x + 3y + 1}{2^{b + c}} = \dfrac{3(3xy + x + y) + 1}{2^{b + c}} \\ =\dfrac{3(x*y) + 1}{2^{b+c}} $$ but $\nu_2$ satisfies $\nu_2(st) = \nu_2(s) + \nu_2(t)$ i.e. it is a completely additive arithmetic function. $\blacksquare$ Background Info on Commutative Monoids(This material was not easy to find or derive, so please appreciate it.) Define the kernel of $f$ to be $K = \ker f = \{ x \in M: f(x) = 1\}$. Already, this seems like a good object to be working with in the context of the Collatz conjecture, which is to prove that continued iterations of $f$ on odd input $x$ will eventually stabilize at $1$. If $x$ goes to $1$ after only one iteration, then $x \in K$. Lemma 1. $K$ is a submonoid of $(M, *)$. Proof. $f(x) = 1 = f(y) \implies f(x*y) = f(x)f(y)= 1\cdot 1 = 1$, where we've used proposition 2. Thus $x*y \in K$ as well. What remains to be shown is that $0 \in K$. Well, $f(0) = \dfrac{3(0) + 1}{2^{\nu_2(1)}} = 1$ and we're done. Lemma 2. Any time you have a submonoid $K$ of a commutative monoid $M$, there is a natural monoid congruence relation by which you can form the quotient monoid $M/K$ much like as in group theory. Proof. Define an equivalence relation on $M$ via $x \sim y \iff x*K = y*K$. Then $\sim$ respects $*$, ie. is a monoid congruence relation. This is true by commutativity: $x \sim y$ and $x' \sim y' \implies x*x'*K = x*y'*K = y'*x*K = y'*y*K = y*y'*K$. Therefore, the set of cosets $x*K, x \in M$ form the quotient monoid $M/K$ which has as monoid law $(x*K)*(y*K) := x*y*K$. Lemma 3. There is a first isomorphism theorem for commutative monoids. Namely, if $f: M \twoheadrightarrow N$ is a surjective monoid hom, then $M/\ker f \simeq N$. Proof. Let $K = \ker f$ and define $\bar{f}: M/K \to N$ by $\bar{f}(x*K) := f(x)$. We have that $\bar{f}$ is well-defined: if $x*K = y*K$, then in particular $\{x\} = x*\{1\} \subset y*K$ since $1 \in K$. This means that $x = y*k$ for some $k\in K$, which implies $f(x) = f(y*k) = f(y)f(k) = f(y)\cdot 1$ since $k \in K = \ker f$. Thus $\bar{f}$ is a well-defined map from $M/K$ to $N$, which is clearly surjective since $f$ is and we are free to choose any pre-image $x \in M$ to form $y = f(x) \in N.$ It is also a monoid homomorphism: $\bar{f}((x*K)*(y*K)) = \bar{f}(x*K*K*y) = \bar{f}(x*K*y) = \bar{f}((x*y)*K) = f(x*y) = f(x)f(y)$. And we're done. Question. Since we have a family of short exact sequences $0 \to \ker f^i \xrightarrow{\iota} (M, *) \xrightarrow{f^i} \text{im f} \to 1$ in the category of commutative monoids, and these sequences seem to be connected as in the diagram below:
where the horizontal maps are all monoid homs and the vertical maps are all set maps, and all squares commute, is there anything we can say about the inverse and direct limits along the vertical sequences of inclusions or $f$ mappings? By that I mean can we say that they are isomorphic with $M$ as monoids or even the trivial monoid? I do know that the inverse limit along the vertical identity maps must be the set $M$. |
| Meromorphic functions on smooth plane curves Posted: 12 Feb 2022 03:50 AM PST I was trying to solve an exercise about smooth plane curves but got stuck. Suppose $C$ is a degree d smooth curve in $\mathbb{P^2}$ and let $L_0$ and $L_1$ be two lines on $\mathbb{P}^2$. I was trying to solve an exercise which asked to show that $\frac{L_0}{L_1}$ is a well defined meromorphic function over $C$ and to geometrically interpret the zeros and poles as well as its orders. It seems like the zeros will be given the $d$ intersections of $C$ with the line $L_0$ with the orders dictated by the multiplicity and likewise for the poles. However I am not sure how to show that if both lines intersect at the same point in $C$ the function I was trying to work with, $[X,Y,Z] \to [L_0(X,Y,Z), L_1(X,Y,Z)]$, is even well-defined. Any help would be appreciated. |
| How to prove that $\mu = \sqrt{\frac{1+ \delta^2}{4}}$ Posted: 12 Feb 2022 03:51 AM PST Prove that $\mu = \sqrt{\dfrac{1+ \delta^2}{4}}$. This was a question asked in my exam. It is out of syllabus and not taught. How do I solve this? |
| Is it wrong to think of area as a measure of the total number of points? Posted: 12 Feb 2022 03:54 AM PST Consider the following mapping $f: M \to N$ between the points on a hemisphere and the points on a circle of radius r, illustrated by the figure below
Clearly this is a bijection between the two sets, and also if I'm not mistaken, a homeomorphism. Every point on the circle corresponds to a point on the hemisphere and vice versa, and yet one has double the area of the other. How is this possible? How is area rigorously defined in higher mathematics, so as to make sense of this? |
| Linear preconditioning on a symmetric and non-definite matrix Posted: 12 Feb 2022 03:35 AM PST Suppose we have a linear equation $Ax=b$, such that $A = W+iT$ where $W, T $ are positive definite matrices, and I have an iterative method for solving the equation. Now I want to find a preconditioner for the method. I know that if the matrix has a form $A = M-N$, where $A$ and $M$ are positive definite, than $M$ can be used as a preconditioner, because of: $$\kappa(M^{-1}A) \leq \frac{1+\rho(C)}{1 -\rho(C)}$$ Where $C = I-M^{-1}A$ is the iteration matrix of the method, such that $\rho(C)<1$. However, with my proposed equation, $A$ can't be a positive definite. My question is: can you still make an argument that $M$ can be used as a preconditioner for the equation? Thanks to all who answer! |
| Can you tell which option is correct. Posted: 12 Feb 2022 03:35 AM PST Let S = (x1,...,xn) and T = {yl,....ym} be subsets of the vector space V. Then (a). If S and T are both linearly independent then m=n (b). If S is a basis for V and if T spans V then men (c). If S is a basis for V and if T is linearly independent then m<n (d). If S is linearly independent and if T spans V then m<n |
| Proof of convergence for the Monte-Carlo method in the Linear Transport Equation Posted: 12 Feb 2022 03:34 AM PST There exists a well known procedure for simulating the transport equation using the Monte Carlo (MC) approach. I want to understand the mathematical connection (not physical) of this approach to the equation. Specifically, I have reviewed the Feynman-Kac approach and the proof of a Brownian motion being the solution of a diffusion equation (under some conditions). Now, I want to know if their exists similar methods for proving that the MC method converges to the solution of the linear radiation transport equation. A quick overview of the approach (for more info, Google the topic or see the excellent presentation "Nuclear Fission in Monte Carlo Particle Transport Simulations" accessible at https://t2.lanl.gov/fiesta2014/school/Brown.pdf): For the neutrons population $n$ in a fissionable media, a suitable model is: $$\frac{\partial{n\left(\vec{r},\hat{\Omega},E,t\right)}}{\partial{t}}+v\left(E\right) \hat{\Omega}\cdot \nabla{n\left(\vec{r},\hat{\Omega},E,t\right)}+\Sigma_{\text{t}}\left(\vec{r},\hat{\Omega},E,t\right) n\left(\vec{r},\hat{\Omega},E,t\right) v\left(E\right) = \int\limits_{0}^{+\infty}{\int\limits_{4 \pi}{\frac{\chi\left(\vec{r},E,t\right)}{4 \pi} \nu\left(\vec{r},E',t\right) \Sigma_\text{f}\left(\vec{r},E',t\right) n\left(\vec{r},\hat{\Omega}',E',t\right) v\left(E'\right)d{\hat{\Omega}'}}d{E'}} + \int\limits_{0}^{+\infty}{\int\limits_{4 \pi}{\Sigma_\text{s}\left(\vec{r},\hat{\Omega}' \rightarrow \hat{\Omega},E'\rightarrow E,t\right) n\left(\vec{r},\hat{\Omega}',E',t\right) v\left(E'\right)d{\hat{\Omega}'}}d{E'}} + Q_{\text{ext}}\left(\vec{r},\hat{\Omega},E,t\right)$$ with the boundary condition: $$\hat{\Omega}\cdot \hat{S}<0:n\left(\vec{r},\hat{\Omega},E,t\right)=0$$ where $\hat{S}$ is the normal unit vector on the boundary surface. Also, the initial condition is assumed to be: $$n\left(\vec{r},\hat{\Omega},E,0\right)=0$$ where in this equation, $n$ is the dependent variable and $\vec{r}$, $\hat{\Omega}$, $E$, and t, are the independent variables. Also:
The standard MC approach is as following:
Any help, regarding proof of convergence using stochastic methods for solving this equation or other partial differential (besides the usual diffusion-drift) or integro-differential equations, would be greatly appreciated. |
| Poincare constant for Gaussian distribution Posted: 12 Feb 2022 04:12 AM PST The minimum of constant $C$ to satisfy the following condition is called Poincare constant for a probability measure $\mu$. For any smooth function $f$ on $\mathbb{R}$, the relation $$\int_{-\infty}^\infty (f(x)-E[f])^2 \mu(dx) \le C \int_{-\infty}^\infty f'(x)^2 \mu(dx) $$ holds, where $E[f]$ is the expectation of $f(X)$. It seems that Poincare constant for 1-dimensional Gaussian distribution is known. However, I cannot find its value and its suitable reference. I am seeking the answer for this question. |
| Kernel of Bounded Operator on Hilbert Space Posted: 12 Feb 2022 03:38 AM PST I encountered this question on an exam recently and was not able to solve it. Suppose you have a bounded operator $T$ on a Hilbert space $\mathcal{H}$ such that $I-T$ is compact, where $I$ is the identity operator. Then $ker(T)$ is finite-dimensional. I'm not really sure how to even start with this problem. We know that since $I-T$ is compact, the image of the unit ball is compact. I thought it was true that $(I-T)\mathcal{H} = I\mathcal{H} - T\mathcal{H} = \mathcal{H} - T\mathcal{H} = (T\mathcal{H})^\perp$. However, this is not true, although I do not quite understand why. I would be grateful for some tips on how to solve this, and maybe someone can tell me why the equation above does not hold. |
| Posted: 12 Feb 2022 03:38 AM PST Let A be a square matrix so that AB = BA for every square matrix B of the same order. Prove that A is a diagonal matrix. I have no clue as to how to even start. |
| One problem with conditional probabilities Posted: 12 Feb 2022 04:08 AM PST The problem goes this way. There are 20 watches, 2 of them are broken. And there is an expert who can say if the watch is broken. But his verdict is only correct with probability 0.8 - in both cases, if the watch is broken and if it is not. So, then I choose one watch, show it to the expert and he says: it's broken. I choose then another watch of these 20. What is the probability that this second watch is broken? Thanks in advance. PS. I dont really understand how this information can help, but let it be. My math background: PhD, but it is 20 years ago, so I'm not an active mathematician anymore. The source of the problem: a student who i know. What I tried: naturally, Bayes theorem. But I still cannot solve it. PPS. I found the solution. It goes without Bayes but with the law of total probability. Its rather technical. |
| What's the difference between $\mathbb{R}^3$ and $\mathbb{R}_3$? Posted: 12 Feb 2022 04:08 AM PST My course problem booklet (mathematics BSc, second year module in algebra, unpublished) uses them in a context which suggests $\mathbb{R}_3$ might be the same as M$_{1,3}(\mathbb{R})$. I'm used to the idea that $\mathbb{R}^3$ is the home of elements which in some sense contain 3 elements from $\mathbb{R}$, but I'm not sure exactly what distinction is being made by the subscript vs. superscript. The subscript expression occurs in part (f) of the question shown below. Screenshot of question: |
| associated prime ideal and localization of Noetherian ring Posted: 12 Feb 2022 03:52 AM PST In Matsumura's Commutative Ring Theory he states in Thm 6.2 that: $A$ is a Noetherian ring which contains a multiplicative subset $S$ then for an $A$-module $M$ we have: $$Ass(M_S)=Ass(M)\cap Spec(A_S),$$here $Ass(M)$ is the set of all associated prime ideals(an ideal is prime also an annihilator of some $x\in M$) of $M$ and $Spec(A_S)$ is seen to be a subspace of $Spec(A)$ since there's a bijection between prime ideals of $A$ contained in $A-S$ and those of $A_S$. In the text in order to prove "$\subseteq$" direction,choose $P\in Ass(M_S)$ and since $p=P\cap A$ is already a prime ideal only need to prove $p=ann_{A}(m),m\in M$.And any $s\in S\subset A_S$ is a unit so $P$ can be choosen to be the form $ann_{A_S}(x),x\in M$. Here's what I'm confused about in the proof: he states that $p$ is finitely generated by Noether property of $A$ then $\exists t\in S,p=ann_A(tx)$. My thought is: By definition of $p$,$p=ann_{A_S}(x)\cap A=ann_A(x)\subseteq ann_A(tx)$ for any $t\in S$. If $atx=0$,then $attx=0\Rightarrow at\in p$,i.e. with $t\in S,t\notin p$ we have $a\in ann_A(tx)\Rightarrow a\in p$ thus $ann_A(tx)\subseteq p,\forall t\in S$.Then it seems like we get $p=ann_A(tx)$ without finiteness of $p$.So where's the mistake and how does the finiteness of $p $ is needed? |
| Posted: 12 Feb 2022 04:08 AM PST I was given this question: I didn't really get anywhere with this question, I mean, I saw an inequality with the norm and I thought about the projection on $\text{Im}T^*$ and maybe on $\text{Im}T$ but didn't really get anywhere. |
| $(3)$ is inert in $\mathbb{Q}[\sqrt{2}]$ Posted: 12 Feb 2022 03:32 AM PST Let $K=\mathbb{Q}[\sqrt{2}]$. Consider its ring of integers $O_K=\mathbb{Z}[\sqrt{2}]$ and the ideal ideal $(3)$. I want to show that $(3)$ is inert in $O_K$. By Gauss' reciprocity $\left(\frac{2}{3}\right)=2^{(3-1)/2}=2=-1\bmod{3}$, so $(3)$ is non-splitting in $O_K$. We also have $2=[K/\mathbb{Q}]=ef$, where $e$ is the ramification index and $f=[(O_K/P):(\mathbb{Z}/(3))]$. So, if I prove that $f=2$, then $e=1$ and we are finished. But I am stuck here. Another way to do this would be to just prove that $O_K/(3)$ is a field. But I also can't seem to think of something. For $O_K/(2)\to\mathbb{F}_2$ there is the isomorphism $a+b\sqrt{2}\mapsto a$. But for this case, it feels like it is not similar to the latter. I appreciate any hint or suggestion, thanks! |
| Posted: 12 Feb 2022 04:11 AM PST Source: https://artofproblemsolving.com/community/c6t243f6h2745656_inequalities_with_3 Let $x,y,z>0,xyz=1$ then $$ \sum_{cyc}{\sqrt{\frac{x+1}{x^2+16x+1}}}\geqslant 1 \ \ \ \ (1) $$ and$$ \sum_{cyc}{\sqrt{\frac{x+1}{4x^2+10x+4}}}\leqslant 1\ \ \ (2) $$ For this type of inequalities, I usually use local inequalities$$ x,y,z>0,xyz=1,\sum_{cyc}{\frac{1}{x^2+x+1}}\geqslant 1, \sum_{cyc}{\frac{x^2}{x^2+x+1}}\geqslant 1, \sum_{cyc}{\frac{x+1}{x^2+x+1}}\leqslant 2 $$ For (1) it's smooth,because$$ \frac{x^4+1}{x^8+16 x^4+1}-\left(\frac{1}{x^2+x+1}\right)^2=\frac{(x-1)^2 x \left(2 x^4+7 x^3+14 x^2+7 x+2\right)}{\left(x^2+x+1\right)^2 \left(x^8+16 x^4+1\right)} \geqslant 0$$ Then it's easy to prove. But for (2) I get $$\frac{x^2+1}{4 x^4+10 x^2+4}-\left(\frac{x+1}{2 \left(x^2+x+1\right)}\right)^2=\frac{(x-1)^2 x^2}{4 \left(x^2+2\right) \left(x^2+x+1\right)^2 \left(2 x^2+1\right)} \geqslant 0$$ Then $$ \sum_{cyc}{\sqrt{\frac{x+1}{4x^2+10x+4}}}\geqslant \sum_{cyc}{\frac{1}{2}\frac{\sqrt{x}+1}{x+\sqrt{x}+1}}\leqslant 1 $$ Failed. How can I prove (2)? Any solution is welcome. |
| Would $ \lim_{\sigma\rightarrow0^+}\int_{-\sigma}^{\sigma}\frac{1}{\sigma}f(x)dx$ be equal to zero? Posted: 12 Feb 2022 04:10 AM PST Suppose a limit of the form $$ \lim_{\sigma\rightarrow0^+}\int_{-\sigma}^{\sigma}\frac{1}{\sigma}f(x)dx$$ where $f(x)$ is an integrable function. I think that this limit would be zero, since we would be taking an integral at a single point, even if the argument was divergent. Is this correct? |
| For which values of n does insertion sort beat merge sort? Posted: 12 Feb 2022 03:33 AM PST I reading "Introduction to Algorithms", where I asked solve exercise (ex. below) "Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size $n$, insertion sort runs in $8n^2$ steps, while merge sort runs in $64n \lg {n}$ steps. For which values of $n$ does insertion sort beat merge sort?" So I peek solution: $8n^2<64n\log_2(n) \implies n^2<8n\log_2(n) \implies n\leq43$ Please somebody explain me how from $n^2<8n\log_2(n)$ get to $n\leq43$. |
| models of coins and uniform distribution Posted: 12 Feb 2022 04:05 AM PST Here is the sentence which I'm confused with
I'm confused with the part uniform(0,1). ============================================================
========================================================== |
| Posted: 12 Feb 2022 04:07 AM PST The run test is used to know whether the given data is random or not. How can we derive the formula for the expected run with $n_1$ ups and $n_2$ downs. |
| Prove that $\lim_{n\to\infty} (\sqrt{n^2+n}-n) = \frac{1}{2}$ Posted: 12 Feb 2022 03:54 AM PST Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$ Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it spits out an error). Can anyone spot the error for me? $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| < \epsilon$ $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| < \epsilon$ $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} < \epsilon$ $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} > \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$ $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} > \frac{1-2 \epsilon}{2}$ $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} > \frac{1-2 \epsilon}{2}$ $\Rightarrow \sqrt{n} > \frac{1-2 \epsilon}{2}$ $\Rightarrow n > \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$ |
| Proving that $n\mid(nCr)$ for all $r$ ($1 \leq r \leq n-1$), only if $n$ is prime Posted: 12 Feb 2022 03:54 AM PST I'm trying to prove that $n\mid(nCr)$ for all $r$ ($1 \leq r \leq n-1$) if and only if $n$ is prime. Now proving that if $n$ is prime then $n\mid(nCr)$ is pretty easy, but how would you go about proving that $n\mid(nCr)$ only if $n$ is prime? Could you show that if $n$ is not prime, then there exists an $r$ such that $n$ does not divide $nCr$? If so how would you go about doing that? I thought it would be easy but then I realized that for some $r, n\mid(nCr)$ even if $n$ isn't prime which made things a little bit more complicated. I've managed to show that $n\mid(n-1)!$ for non-primes greater than or equal to $6$, which means that despite $n$ always canceling out with a product of factors in $r!(n-r)!$ in some cases, there will be another pair of factors that cancel out with the $n$ in $(n-1)!$, and sometimes there won't be. Maybe there's some kind of pattern but I can't find it unfortunately. (I'd prefer a method that doesn't use undergraduate and above level maths) |
| Calculating a tangent arc between two points on two circles Posted: 12 Feb 2022 03:34 AM PST How can I calculate the arc between two circles? The arc must be tangent to the two points on the circles. Here is a picture illustrating it. I'm trying to code and calculate the orange arc and the blue arc.
The points B, F, D, and E can be changed by the user and dynamically update. |
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