Graph G is bipartite graph, I want to just count number of perfect matching. Is there any algorithm exist using combinatorial+graph theory, by which i cant count this. I tried by traversing the graph, but as there are so many possibilities and I just want number of perfect matching I asked here.
My actual question is to find the number of transversal given a collection of set ...  After a little bit of study it has come down to:  How can we count the number of matchings in a bipartite graph with parts of size $m$ and $n$ such that it covers all $m$ vertices of the first part, $m \le n$?  I already know that there is a way to count the number of perfect matchings in a bipartite graph with $m=n$ using the permanent of its (square) incidence matrix.
I have very old Computer  What's the Best Operating System for Old Computer? How to get from Gatwick airport to Paddington? What's the most straightforward way to get from Gatwick airport to Paddington station in London using public transport?   I'm mostly interested in travel time &amp; general smoothness of the trip (e.g. buying just one ticket for the whole trip would be nice). Price is a secondary concern. Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$ Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction. I want to make the teeth a separate child object from the mouth. I want to know the best way to do this.   I have tried going into edit mode with Tab and trying to select all the vertices but that doesn't move the teeth. Is there an easy way to select just that part and make it a separate child mesh to the rest of the mouth? Opt in "mentor" privilege at 7.5k I am suggesting a Mentor privilege at 7.5k. It would be on a per-user per-site opt in basis.  At some reputation level (perhaps 7.5k) there could be an opt in "mentor" privilege.  If someone opts in then when a new user signs up, they become their Mentor. The new user is the Mentee.  They will get notified of any posts or suggested edits they make, and can help and support them.  While this would by no means help everyone (I'm sure there are more new people signing up than there are 7.5k people willing to help them), it could help mitigate the "new user's bad posts" and such like.  By no means would being someone's mentor mean you are the only person who can edit and help them, and you aren't obliged to do so - but if you have opted in then you should help.  How would this work?   Any Mentor would have no more than 5 active Mentees at once. Once the Mentee gets to 125 rep, the downvote privilege (or any other level deemed sensible) they are no longer a Mentee. If they aren't active for a month, they are no longer a Mentee. If the Mentor has 5 Mentees (i.e. cannot mentor anyone else) then any Mentee not active for 2 weeks is no longer a Mentee. The Mentee can opt out at any time, the Mentor can opt out, but if they opt out a lot (say 5 people in 2 months) they are blocked from the privilege for 2 weeks. The Mentor gets no extra rep. They can still get rep from answering their Mentee's questions and similar. Once the new user gets 20 rep on the site, a chat room is created for them to talk. No user with the association bonus will become a Mentee. This would (maybe) be opt in for sites. For example, this might not be at all useful for this site, or for , but on  or  I imagine this could be quite good.    appears to be asking the same thing, based on the title, but that is asking about people who know each other offline mentoring each other.    Every single one of these suggestions (especially the numbers) is open to discussion. Maybe 125 is to low to stop being mentored, or maybe it's too high. What do you think? Not able to execute a .sh file: /bin/bash^M: bad interpreter I wanted to execute a shell script:  -rwxr-x--x 1 root root   17234 Jun  6 18:31 create_mgw_3shelf_6xIPNI1P.sh   I tried to do a standard procedure, but I got this error:  ./create_mgw_3shelf_6xIPNI1P.sh  localhost 389 -l /opt/fews/sessions/AMGWM19/log/2013-06-06-143637_CLA-0  DEBUG   cd/etc/opt/ldapfiles/ldif_in ; ./create_mgw_3shelf_6xIPNI1P.sh  localhost 389 -l /opt/fews/sessions/AMGWM19/log/2013-06-06-143637_CLA-0 **ERROR  sh: ./create_mgw_3shelf_6xIPNI1P.sh: /bin/bash^M: bad interpreter: No such file or directory**   What does it mean? I was doing this as the root user under the root group.  Does it mean that the file does not have the correct permission for the root user? I want to resolve merge conflicts in my Git repository. How to do that ? Make grub keep its default boot under kernel updates I dual-boot my machine and I want it to default to boot into windows so that whenever I restart the machine remotely from my home it will be able to get back into Windows (instead of Ubuntu).  The problem is that every time Ubuntu upgrades the kernel, I have to reset the default boot item of grub back to windows. This is because the grub menu loader uses positions i.e. 6 for default OS to boot. And when Ubuntu installs a new kernel it changes that order.  I am looking for a way to configure grub to remember its default boot item under kernel updates. How to go with proving this : Let $R$ be a ring and let $I$ be an Ideal of $R$.  Prove that the factor ring $R/I$ is commutative iff $rs-sr \in I$ $\forall r$ and $s$ in $R$  any hint how to go with this... Using Fingerprint reader in 16.04 I have a Dell laptop with fingerprint reader inbuilt. I just ran lsusb on terminal and got the following result:   Bus 002 Device 003: ID 138a:0011 Validity Sensors, Inc. VFS5011 Fingerprint Reader Bus 002 Device 002: ID 8087:0024 Intel Corp. Integrated Rate Matching Hub Bus 002 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 003 Device 002: ID 046d:c077 Logitech, Inc. M105 Optical Mouse Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 001 Device 004: ID 05ca:1820 Ricoh Co., Ltd Integrated Webcam Bus 001 Device 003: ID 0a5c:21bc Broadcom Corp. BCM2070 Bluetooth 2.1 + EDR Bus 001 Device 002: ID 8087:0024 Intel Corp. Integrated Rate Matching Hub Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub   Is it possible to use fingerprint scanning instead of traditional login with 16.04 and Unity ? Thanks. Any way to send a personal message to another user? There has been quite a few times that I wished I could send a message to another user on SO - not ask a question for everyone to see, but just a short message informing them of something or requesting them to do something.  Are there any plans to allow this to happen in the future?  Related: I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's equations. These equations are invariant under many transformations (Lorentz transformation, rotations, spatial and temporal translations, etc. By the way is there a way, maybe from group theory, to find all the possible transformations that leaves the equation(s) invariant?) which imply as many conserved quantities thanks to Noether's theorem. In wikipedia I can see an equation that seems to give all the conserved quantities () but it involves the Lagrangian and I'm not sure whether the formula is valid for all systems whose Lagrangian is possible to obtain. A theorem about Cesàro mean, related to Stolz-Cesàro theorem Original Title: Tauberian theorems and Cesàro sum  Theorem (Landau-Hardy, From Rudin's Principle of Mathematical Analysis Exercise 3.14)  $\newcommand\abs[1]{\left\lvert#1\right\rvert}$ If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\frac{s_0+s_1+\dotsb+s_n}{n+1}\qquad(n=0,1,2,\dotsc)$$ Put $a_n=s_n-s_{n-1}$ for $n\ge1$. Assume $M&lt;+\infty$ and $\abs{na_n}\le M$ for all $n$, and $\lim_{n\to\infty}\sigma_n=\sigma$, then $\lim_{n\to\infty}s_n=\sigma$.  The outline of the proof  If $m&lt;n$, then $$s_n-\sigma_n=\frac{m+1}{n-m}(\sigma_n-\sigma_m)+\frac1{n-m}\sum_{k=m+1}^n(s_n-s_k)\tag{*}$$ Notice that $\abs{s_n-s_k}\le(n-m-1)M\,/\,(m+2)$, fix $\epsilon&gt;0$ and associate with each $n$ the integer $m$ that satisfies $$m\le\frac{n-\epsilon}{1+\epsilon}&lt;m+1$$ Then $(m+1)\,/\,(n-m)\le1/\epsilon$ and $\abs{s_n-s_k}&lt;M\epsilon$. Hence $$\limsup_{n\to\infty}\,\abs{s_n-\sigma}\le M\epsilon$$  Questions and thoughts  It seems that the equation (*) comes out strangely. I wonder how to discover such kind of strange identities. So is there any observation, even deeper, to look through that equation?  Thanks! Restore main APFS container to full disk How to know the types of windowing system, window manager and desktop environment of a Unix-like OS mod_pagespeed : install earlier version compatible with libc6 2.13 (Debian Wheezy) "Baggage" versus "luggage" In a photoelectric effect we remove electrons from a metal using high energy photons, the work function is the minimal energy required for this effect.  My question is why doesnt the work fucntion depended on time? The more electrons we take out from the metal, the more ionized it becomes (if we remove a lot of negative charge, it will turn positive), and as a result we would expect to need more energy for this effect (the possitive charge of the metal would make it harder to remove the remaining electrons).  Why does the work function remains constant no matter how much electrons we take out? Accessing laptop's hard drive when OS won't boot Why can templates only be implemented in the header file?