Recent Questions - Mathematics Stack Exchange |
- Orthonormal basis on Hilbert space and strong operator topology
- We can solve the triangle with two data or always we need third information about triangle? [closed]
- colouring properties of functions
- Simplification of a term
- Laplace probability estimate
- Let $A^\star:\operatorname{dom}A^\star\subseteq F^\star\to E^\star$ be the adjoint of $A$. Then $\operatorname{ker}A=(\operatorname{im}A^\star)^\perp$
- Is this map from a direct limit of topological spaces inductively irreducible?
- Perturbing a partial isometry to part of a unitary
- If $\ z_k\ $ are complex numbers, sort of uniformly spread out on the unit circle, then what is $\ \sup \{\ \vert z_0 + \ldots + z_{n-1} \vert\ \}\ ?$
- Find minimum value of $[4+(x+5)^2]^{1/2}+ [9+(x-7)^2]^{1/2}$
- Given the vectors a (-5, 2); b (6, -2) and c (-2,2) solve the equation: 4a-2 [5 (b-2c) + 3a] + 6x = 4c + x
- Rudin exercise 1.18: understanding the use of "without loss of generality"
- PDE question ALj [closed]
- Union-Closed Families form a subcategory of a functor-structured category: can we describe it through categorical closure operators?
- Add a deterministic function to a stochastic process in order to get a martingale
- Given $\epsilon>0$, calculate $m_\epsilon \in \mathbb{N}$ such that for all $n \ge m_\epsilon$
- Dependence of perimeter of a triangle on the movement of one of its vertices
- Find $\lim_{n\to\infty}\frac{1}{n}\left(n+\frac{n-1}{2}+\frac{n-2}{3}+...+\frac{2}{n-1}+\frac{1}{n}-\log(n!)\right)$
- How do I prove that two maps are homotopic iff their corresponding paths are homotopic?
- Any finite abelian group of Möbius transformations is either isomorphic to $C_2 \times C_2$ or is cyclic.
- (Request for) simple constructive proof of existence of nonstandard model of PA
- Regarding an integer being a sum of three primes...
- Largest $T$ (where $0 < t < T)$ such that $u(x, t)=\frac{1}{2}(f(x-a t)+f(x+a t))$ satisfies the wave equation
- Question about universal derivation $\Omega_{A/k}$
- Decomposition of a stochastic process into a continuous and pure jump part
- Critic Loss in PPO
- About the definition of Determinant in Paul R. Halmos' Finite-Dimensional Vector Spaces
- Probability and Combinatorial Group Theory.
- integration of gamma with exp
- Number of hands with identical cards out of two decks
| Orthonormal basis on Hilbert space and strong operator topology Posted: 14 Jan 2022 09:54 AM PST
In the problem attached above, I need to prove that $A_n \to 0$ in the strong operator topology. So, I need to show that $||A_n(x)|| \to 0$ for all $x \in H$ $\implies ||(\phi_n,x)\phi_1|| \to 0$ $\implies ||(C_n)\phi_1|| \to 0$ ( as $x$ can be written as $\sum_{i}Ci\phi_i$) No, idea how to proceed. Also please give hints for 2nd part where I need to prove thay $A_n^*$ has no limit in strong operator topology. |
| We can solve the triangle with two data or always we need third information about triangle? [closed] Posted: 14 Jan 2022 09:49 AM PST We can solve a triangle with two data or always we need third information about triangle? And how is it understand? Throught experience in this problem? |
| colouring properties of functions Posted: 14 Jan 2022 09:46 AM PST on page 5 (857) here in the definition $AR_1^+$ isn't there missing $\max$ right before $x+1$ ? It should read $$\max C(x)\leq \max x+1$$ because on the left hand side there is "x" bounded but on r.h.s. it is not. With this $\max$ it is more akin to page 2 "limited". |
| Posted: 14 Jan 2022 09:46 AM PST Stuck on a problem about simplifying the term $(x^3-4)^2$. Applying the binomial theorem I'm left with $x^6-2x^3(-4)+16=x^6+8x^3+16.$ Howerver, every online calculator gives $x^6-8x^3+16$ as the correct answer and I did get to that answer by multiplying everything out manually. What did I do wrong? |
| Posted: 14 Jan 2022 09:44 AM PST I am trying to calculate the Laplace probability estimate according to this formula. The correct answer is 0,733 but I just don't know how to find the number of examples that belong to the Class and the number of all examples in the given set. I tried multiplying 4*12 as the number of all examples in the set, but doesn't seem to quite work. |
| Posted: 14 Jan 2022 09:42 AM PST I'm trying to do exercise 2.18 in Brezis' book of Functional Analysis. Could you have a check on my attempt?
Proof: By the construction of $A^\star$, we have the identity $$\langle A^\star f, x \rangle_{E^\star, E} = \langle f, Ax \rangle_{F^\star, F}, \quad \forall f\in \operatorname{dom} A^\star, x\in \operatorname{dom} A.$$ Fix $x \in \operatorname{ker} A$. Then $Ax=0$ and thus $\langle A^\star f, x \rangle_{E^\star, E} = 0$ for all $f\in \operatorname{dom} A^\star$. This in turn implies $\langle g, x \rangle_{E^\star, E} = 0$ for all $g\in \operatorname{im} A^\star$. As such, $x \in (\operatorname{im} A^\star)^\perp$. Now we are going to prove the converse. Fix $x \in (\operatorname{im} A^\star)^\perp$, i.e., $\langle A^\star f, x \rangle_{E^\star, E} = 0$ for all $f\in \operatorname{dom} A^\star$. This implies $\langle f, Ax \rangle_{F^\star, F} = 0$ for all $f\in \operatorname{dom} A^\star$. We claim that $Ax=0$. Assume the contrary that $(x, 0) \notin \operatorname{graph} A$. By Hahn-Banach theorem, there is $(g_1, g_2) \in E^\star \times F^\star \cong (E \times F)^\star$ such that $$\langle g_1, z \rangle + \langle g_2, A z \rangle < \alpha < \langle g_1, x \rangle + \langle g_2, 0 \rangle = \langle g_1, x \rangle, \quad \forall z \in \operatorname{dom} A.$$ Because $\operatorname{dom} A$ is a linear subspace of $E$, we get $\langle g_1, z \rangle + \langle g_2, A z \rangle =0$ for all $z \in \operatorname{dom} A$. It follows that $|\langle g_2, A z \rangle| = |\langle g_1, z \rangle| \le \|g_1\| |z|_E$ for all $z \in \operatorname{dom} A$. Hence $g_2 \in \operatorname{dom} A^\star$ and thus $\langle g_2, A x \rangle = 0$. It follows that $\langle g_1, x \rangle = - \langle g_2, A x \rangle = 0$, which is a contradiction. |
| Is this map from a direct limit of topological spaces inductively irreducible? Posted: 14 Jan 2022 09:37 AM PST Let $q:Y\to X$ be a continuous surjective map between topological spaces. Then $q$ is inductively irreducible (Gruenhage) if $Y$ has a closed subset $V$ such that $q|V$ maps $V$ surjectively onto $X$ and no proper subset of $V$ has this property (in other words, $q|V$ is irreducible). Let $Y_1=[0,1)$ and $Y_{n+1}=Y_n\times Y_1$ $(n\ge 1)$. Let $Y$ be the disjoint union of the spaces $Y_n$. Define an equivalence relation on $Y$ by identifying $y\in Y_n$ with $(y,0)\in Y_{n+1}$ at each stage. Let $q:Y\to X$ be the quotient map. Then I think that $X$ is the direct limit of the spaces $Y_n$ (not that it matters for this question). Question: Is $q:Y\to X$ inductively irreducible? (I am hoping that the answer is no). |
| Perturbing a partial isometry to part of a unitary Posted: 14 Jan 2022 09:34 AM PST In Narutaka Ozawa's solution over at Mathoverflow, the following result is implicitly used:
Here $\sim$ is Murray-von Neumann equivalence. It's easy to see that if $1-v^*v \sim 1-vv^*$ then $v$ can be extended to a unitary in $M$. Why is this claimed result true? If $M$ is finite, then as indicated in this answer we always have that if $p\sim q$ then $1-p\sim 1-q$ (see also, for example, Exercise 6.9.6 in Kadison and Ringrose, Volume 2). By definition $u^*u\sim uu^*$ so the claim follows with $v=u$. Indeed, this would also hold if $u^*u$ (equivalently, $uu^*$) were finite, even if $M$ were not. Ozawa's wording suggests to me that in general, we might seek a projection $p\in M$ with $p\leq u^*u$ and set $v=up$. Then $v^*v = pu^*up = p$, and we want $\|u(1-p)\xi\|<\epsilon$, which seems plausible to achieve, perhaps? I have no clue how to get $1-v^*v \sim 1-vv^*$? Maybe instead a type-decomposition argument could work, but again I don't see how to get started. |
| Posted: 14 Jan 2022 09:51 AM PST Let $\ n\in\mathbb{N}\ $ and suppose $\ z_k = x_k + iy_k,\ $ where $ \vert z_k \vert = 1\ $ and $\ \frac{2k\pi}{n} < \arg(z_k) < \frac{2(k+1)\pi}{n}\quad $ for all $\ k\in \{ 0,\ \ldots,\ n-1 \}.$ What is $\ s_n = \sup \{\ \vert z_0 + \ldots + z_{n-1} \vert\ \}\ $ in terms of $\ n\ ?$ Also, what is $\ \lim _{n\to\infty} s_n\ ?$ At first, I just thought we put all points as far right as possible. But I don't think this is correct for all $\ n.\ $ For example, for $\ n=16,\ $ if we move all points as far to the right as possible, then I think we can make $\ \vert z_0 + \ldots + z_{n-1} \vert\ $ larger by moving $\ z_8\ $ clockwise towards the negative real axis, although maybe I am wrong about this. |
| Find minimum value of $[4+(x+5)^2]^{1/2}+ [9+(x-7)^2]^{1/2}$ Posted: 14 Jan 2022 09:35 AM PST Is it possible to find minimum value of this expression without using derivatives? This expression obtain its minimum equal to $13$ if $x=-1/5$, but how to show it with an easier way? Any hint would help a lot! Thanks! |
| Posted: 14 Jan 2022 09:39 AM PST 4a-2[5(b-2c)+3a]+6x=4c+x 4a-2[5b-10c+3a]+6x=4c+x 4a-10b-20c-6a+6x=4c+x -2a-10b-20c+5x=4c 5x=24c+10b+2a 5x=24(-5;2)+10(6;-2)+2(-5;2) 5x=(-48,48)(60;-20)(10;4) 5x=(2;32) x=(2/5;32/5) Use vector operations a, b, c and x are vectors Vector operations Product by a scalar |
| Rudin exercise 1.18: understanding the use of "without loss of generality" Posted: 14 Jan 2022 09:46 AM PST Exercise 1.18 in Rudin states:
I'm trying to understand a solution to the first half of this problem. The $x = 0$ case is immediate, as any $y \neq 0$. So we suppose $x \neq 0$. So $x = (x_1, \ldots, x_k)$ has at least one non-zero component, $x_i$. The proof then says, "without loss of generality, permuting indices if necessary, we can assume $x_1 \neq 0$." It then constructs a $y$ on that basis. Why does this not sacrifice generality? It only treats a certain subset of $x$ with non-zero first components. Is the idea that I could have done this construction for any index, so I might as well do it for the $i$th decimal position? It seems more naturally to me to just pick an arbitrary $i \in \{1, \ldots, n\}$. |
| Posted: 14 Jan 2022 09:34 AM PST $u_{tt} = c_2u_{xx}$ , $0<x<1$, $t>0$, $u(x,0)=x(1-x)$, $u_t(x,0)=0$, $0≤x≤1$, $u(0,t)=u(1,t)=0$, $t>0$. |
| Posted: 14 Jan 2022 09:37 AM PST Let us consider the functor-structured category $S(\mathcal{P})$ induced by the powerset functor. As it is well-known it is the category having the pairs $(X,\mathcal{F})$ (with $\mathcal{F}$ any set system on $X$) as objects and with structure-preserving maps as arrows. One full subcategory of $S(\mathcal{P})$ is given for example by the category of union-closed families. Observe now that the objects $(X,\mathcal{P}(X)$, for any set $X$, are always union-closed, and arbitrary intersections of union-closed families on the same ground set is again a union-closed family on such a ground set. Therefore, given $(X,\mathcal{F}) \in Obj(S(\mathcal{P}))$, we can always find the smallest union-closed family on $X$ generated by $\mathcal{F}$. In other terms, we're defining a closure operator on the lattice $\mathcal{P}(X)$. Recent researches on categorical closure operators define a closure operator on a concrete category $(\mathcal{C},\mathcal{X})$, with $\mathcal{X}$ finitely complete and with suitable $(E,M)$-factorizations ($E \subseteq Epi(\mathcal{C})$ and $M \subseteq Mono(\mathcal{C})$). Given an $\mathcal{X}$-object $C$, we will always mean a subobject of $U(C)$ (here $U: \mathcal{C} \longrightarrow \mathcal{X}$ stands for the forgetful functor). Recall that if $X \in Obj(\mathcal{X})$, each $M$-morphism with codomain $X$ is called a $M$-subobject of $X$. A closure operator on $\mathcal{C}$ is a family of maps $c=(c_C: $M$-SubC \longrightarrow $M$-SubC)_{C \in \mathcal{C}}$ satisfying some suitable conditions (it is a "pre-closure" operator, in general idempotence it is not satisfied). Well, my question is: it is possible to use categorical closure operators to describe the category of union-closed families? Similarly, if I consider the functor-structured category $S(Q_2)$ (with $Q_2(\Omega):=\Omega \times \Omega$) and the category of preorders, it is possible to do the same thing as above? |
| Add a deterministic function to a stochastic process in order to get a martingale Posted: 14 Jan 2022 09:55 AM PST I was wondering if there were a way to transform the process $$ X(t) = w^4(t) $$ with $w(t)$ which is a standard Brownian motion, into a martingale, by adding a deterministic function. That is: $$ Y(t) = w^4(t) + g(t) $$ with g(t) that is a deterministic function. I tried to apply Ito's Lemma on $Y(t)$ in order to derive a condition on the term associated with $dt$ (which is equal to 0 for a martingale) but this leads to this condition: $$ \frac{dg}{dt} = -12w^2(t) $$ which does not seem to help much. What I also know is that $E[w^4(t)] = 3t^2$ but removing this quantity from $X(t)$ does not give a martingale either. |
| Given $\epsilon>0$, calculate $m_\epsilon \in \mathbb{N}$ such that for all $n \ge m_\epsilon$ Posted: 14 Jan 2022 09:46 AM PST I plan to solve the following exercise given $\epsilon>0$, calculate $m_\epsilon \in \mathbb{N}$ such that for all $n \ge m_\epsilon$ it is verified that $|x_n-x|<\epsilon$. In this particular case we have that $x_n= n^{2}a^{n}$ and $x=0$, also $|a|<1$ It is clear in a way that $$0 \le x_n=|x_n-0|=|n^{2}a^{n}|< \epsilon$$ Therefore, we can consider that $$|a^{n}|<\frac{\epsilon}{n^{2}}$$ I am not sure if the above is entirely true, and I have not been able to find the value of m_e requested. Any help on this? |
| Dependence of perimeter of a triangle on the movement of one of its vertices Posted: 14 Jan 2022 09:35 AM PST Can someone provide a rigorous or at least reasonable argument for why the perimeter of the triangle $\Delta PAB$ in the below image varies as the point $P$ moves horizontally? This is part (b) of 2000 AMC 10 Problems/Problem 5. |
| Posted: 14 Jan 2022 09:49 AM PST I want to determine the limit of the following sequence $$x_n=\frac{1}{n}\left(n+\frac{n-1}{2}+\frac{n-2}{3}+...+\frac{2}{n-1}+\frac{1}{n}-\log(n!)\right)$$ From the foregoing, consider $$n+\frac{n-1}{2}+\frac{n-2}{3}+...+\frac{2}{n-1}+\frac{1}{n}=\sum_{k=1}^n\frac{n+1-k}{k}$$ Also try to consider Stirling's approximation, so you would have to find the limit of $$x_n=\frac{1}{n}\left(\sum_{k=1}^n\frac{n+1-k}{k}-\log(\sqrt{2πn}\left(\frac{n}{e}\right)^n\right)$$ I don't know if my previous statement is completely true, besides, from this expression it is difficult for me to find the requested limit. Any help please? |
| How do I prove that two maps are homotopic iff their corresponding paths are homotopic? Posted: 14 Jan 2022 09:35 AM PST I have the following problem:
My idea was the following: Let us first define $$p:[0,1]\rightarrow \mathbb{S}^1;\,\,\,s\mapsto (\cos(2\pi s),\sin(2\pi s))$$We have just shown that $p$ is a homeomorphism, furhtermore $\gamma_f(t)=f(p(t)),\,\, \gamma_g(t)=g(p(t))$. Now let us assume that $\gamma_f, \gamma_g$ are homotopic through loops, i.e. we have $$H':[0,1]\times [0,1]\rightarrow U$$such that
Now since $p$ is a homeomorphsim, we can consider $p^{-1}:\mathbb{S}^1\rightarrow [0,1]$. Now let us define $$H:\mathbb{S}^1\times [0,1]\rightarrow U;\,\,\,((x,y),t)\mapsto H'(p^{-1}(x,y),t)$$ I claim that $H$ is a homotopy of maps. Indeed
Now since this are all equivalences we are done in my opinion. Does it works like this? Thanks a lot for your help. |
| Posted: 14 Jan 2022 09:42 AM PST I want to solve the following question Show that if a non-trivial element of $\mathcal{M}$ has finite order, then it fixes precisely two points in $\mathbb{C}_{\infty}$. Hence show that any finite abelian subgroup of $\mathcal{M}$ is either cyclic or isomorphic to $C_{2} \times C_{2}$. My Proof: I managed to do the first part by using the fact that there are three conjugacy classes, meaning that either all,1 or 2 elements are fixed. Since we are dealing with an non identity element, it follows that either 1 or 2 elements are fixed. Then writing out the mobius transforms in matrix forms in some convenient basis and abusing the fact that the order is finite, we get that the only possibility is that we have only two fixed points. The second part of the question is where I am having issues. I was able to prove that if $f,g \in H$, where $H$ is the subgroup in question, then $f$ and $g$ fix the same two elements, or interchanges them (just by looking at the fact that it is abelian). Using the fact that it has finite order I was also able to prove that each element has order $2$. But here is where I am stuck. I know that I have to show that for any other $h \in H$, we have that $$ h \in \langle f,g \rangle $$ but I can't seem to prove that. I tried
Any ideas as how to proceed? Edit + Message to the moderators: My question is not the same as the other one as the techniques and method used in the other question have not been covered in my course. This is a question from a first year course on group theory from the University of Cambridge. |
| (Request for) simple constructive proof of existence of nonstandard model of PA Posted: 14 Jan 2022 09:41 AM PST I know of two straightforward nonconstructive proofs of the existence of nonstandard models of arithmetic.
The compactness proof seems to be very common. For example, it is used here and here. (Tangentially, the fact that Wikipedia does not include a Löwenheim–Skolem argument makes me wonder whether "non-standard models of arithmetic" are generally assumed to be countable or if there's something else I'm missing something. The argument seems very simple.) I'm wondering if there's a simple constructive proof of the existence of nonstandard models of arithmetic. I'm especially curious if there's a way of extending the compactness argument (2) to pick a particular model satisfying the new axioms regarding $k$. |
| Regarding an integer being a sum of three primes... Posted: 14 Jan 2022 09:36 AM PST If I knew that every even number up to $4 \times 10^{14} $ is the sum of two primes, roughly how many primes would I need to find in order to show that every odd number up to $10^{22}$ is the sum of three primes? Why does this make efficient primality testing important? Well... if we know that every even number up to $4 \times 10^{14} $ is the sum of two primes, then if we add $3$ to each even number then we get every odd number up to $4 \times 10^{14}+1 $ as a sum of three primes. We can add primes bigger than $3$, say $5$ or $7$ to the even numbers verified to be the sum of two primes, but eventually we would get gaps. I am not sure how we can systematically extend the first assertion about twin primes to the second assertion about prime triplets, and roughly how many primes would I actually need? I am also not really sure about the last question about primality testing. |
| Posted: 14 Jan 2022 09:53 AM PST Suppose we have the wave equation $u_{tt} = a^2 u_{xx}$ for $t > 0$. How can I show that for some sufficiently small $T$ (where $0 < t < T$) $u(x, t)=\frac{1}{2}(f(x-a t)+f(x+a t))$ satisfies the equation? How do I find the largest such T? I know how to show that u(x,t) satisfies the equation, but I don't know how to find the largest such $T$. |
| Question about universal derivation $\Omega_{A/k}$ Posted: 14 Jan 2022 09:33 AM PST Let $k$ be a ring, let $A$ be a $k$-algebra. The universal derivation $\Omega_{A/k}$ is the (unique) $k$-module representing the functor of the $k$-derivations of $A$; suppose that $\Omega_{A/k}=0$. If $k$ is an algebraically closed field, we can deduce that, for every maximal ideal $m$, holds $m/m^2=0$. In what settings this is this true? I would say yes for $k$ a generic field, but I don't understand if the arguments we used in the course to get this result are valid when $k$ is just a ring. Thanks in advance for any clarify; also, any reference that could help is welcome |
| Decomposition of a stochastic process into a continuous and pure jump part Posted: 14 Jan 2022 09:41 AM PST We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,T]},P)$. Let $X$ be an adapted làdlàg stochastic process (i.e. the left and right limits exist). For $0<t<T$ denote by $\Delta_+ X_t := X_{t+}-X_t$ its right and by $\Delta X_t := X_t -X_{t-}$ its left jumps. In this paper on page 1894 it is claimed that the continuous part of $X$ is given by \begin{equation} X_t^c:=X_t -\sum_{s<t}\Delta_+ X_s -\sum_{s\leq t}\Delta X_s. \end{equation} However, since $\sum_{s<\cdot}\Delta_+ X_s$ is right-continuous, shouldn't we take the left limit of this sum, i.e. \begin{equation} X_t^c:=X_t -\left(\sum_{s<t}\Delta_+ X_s\right)_- -\sum_{s\leq t}\Delta X_s \end{equation} so that $X^c$ is continuous? |
| Posted: 14 Jan 2022 09:38 AM PST TL,DR: How precisely is the critic loss in PPO defined? I am trying to understand the PPO algorithm so that I can implement it. Now I'm somewhat confused when it comes to the critic loss. According to the paper, in the objective that we want to maximize, there is a term $$ -c_1 (V_\theta(s_t) - V_t^{targ})^2 $$ which is the loss for the critic ($"-"$ in the beginning, since the objective is maximized). I didn't really know what $V_t^{targ}$ was supposed to be at first, after some research online it looks like that's what we're getting from GAE for $\hat{A}_t$, so it's our advantage estimate. On the other hand, $V_\theta(s_t)$ seems to be the output of our critic network, which estimates the value of the state. This would mean that we are trying to get the value of the state close to the advantage estimate, which doesn't make any sense, as the advantage of an action is defined as the action's value minus the value of the state (hence we would end up with "(value of the action - value of the state) - value of the state" in the critic loss). This can also be seen in the definition of the (generalized) advantage estimate, where the value of the state is deducted already. So I suppose I'm misunderstanding something about the critic loss? |
| About the definition of Determinant in Paul R. Halmos' Finite-Dimensional Vector Spaces Posted: 14 Jan 2022 09:52 AM PST In §53 he defines determinants: $\bar{A}$w($x_1$, $x_2$, ..., $x_n$) = w($Ax_1$, $Ax_2$, ..., $Ax_n$), where A is a linear transformation on an n-dimensional vector space V and w is an alternating n-linear form on V. "$\bar{A}w$ is an alternating n-linear form on V, and, in fact, $\bar{A}$ is a linear transformation on the space of such forms". Then he replaces $\bar{A}$ with detA and has (detA)w($x_1$, $x_2$, ..., $x_n$) = w($Ax_1$, $Ax_2$, ..., $Ax_n$). Does he imply that determinant is a linear transformation on the space of alternating n-linear forms on the n-dimensional vector space V? But I think determinant is an alternating multilinear map over the vector space. What's the meaning of employing the alternating n-linear form w here? |
| Probability and Combinatorial Group Theory. Posted: 14 Jan 2022 09:45 AM PST If this is too broad or is otherwise a poor question, I apologise. I learnt recently that the probability that two integers generate the additive group of integers is $\frac{6}{\pi^2}$.
I'm looking for any results of probability applied to group theory, preferably combinatorial group theory, in manner such as the one above. |
| Posted: 14 Jan 2022 09:52 AM PST I want to do the integration for $$\int_{0}^{\infty}\prod_{k=1}^{n}(kx-1)e^{-ax} \mathrm{d}x$$ where $a$ and $n$ are parameters, can I have a simple expansion of $$\prod_{k=1}^{n}(kx-1)$$ Or can I use other techniques to figure it out? I just want the result to be simple. |
| Number of hands with identical cards out of two decks Posted: 14 Jan 2022 09:47 AM PST Two identical complete decks of cards, each with 52 cards, have been mixed together. A hand of 5 cards is picked uniformly at random from amongst all subsets of exactly 5 cards (order of cards in the hand is not important).
I tried something, but I probably go wrong somewhere (I have the oposite probability in a textbook and that one summed up with solution of 1. and 2. is not 1).
My problem is that each of the two probabilities is bigger than 1 (which is definitely wrong) and that they are equal (which my intuition refuses to believe). |
| You are subscribed to email updates from Recent Questions - Mathematics Stack Exchange. To stop receiving these emails, you may unsubscribe now. | Email delivery powered by Google |
| Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States | |
No comments:
Post a Comment