I think I was still in German mode, it's called "electric punch" (Stromschlag) if translated literally, my brain went the easy route and tried to find the closest match.
> Adding one edge after the exact number of edges required to create a Hamiltonian cycle (number of edges equal to number of vertices) would appear to break the property.
A Hamiltonian cycle is a simple cycle (no repeated vertices or edges) that contains every vertex of the graph. If a graph G has a Hamiltonian cycle, then adding more edges to G will not make that cycle go away; it will still be there. So the property of "has a Hamiltonian cycle" is not broken by adding more edges.
As a simple example: consider the graph which is the cycle on 5 vertices. That is, the graph has 5 vertices, and is just one big cycle with 5 edges. This graph has a Hamiltonian cycle (the entire graph itself is one such cycle). If we add an extra edge to this graph, say between vertices 1 and 3, the original Hamiltonian cycle does not go away.
> How can there be a lower bound other than zero? However small the possibility, surely given an infinite number of cases, there are infinite possibilities of a particular structure being created.
The lower bound can be other than zero because they are looking at the threshold (probability) at which the probability of the object existing goes from "very low" to "extremely high". This is alluded to in the following quote:
"When edges are added to a random graph of N vertices with a probability of less than log(N)/N, for instance, the graph is unlikely to contain a Hamiltonian cycle. But when that probability is adjusted to be just a hair greater than log(N)/N, a Hamiltonian cycle becomes extremely likely."
I don't know the precise probabilities, but this would be something like: "When the probability of an edge being present is less than log(N)/N then the probability of there being a Hamiltonian cycle is 1/(N^2). When the probability of an edge being present is slightly more than log(N)/N then the probability of there being a Hamiltonian cycle becomes (1 - 1/(N^2))."
(Note that I plucked the above probabilities out of thin air just for the sake of illustration, just to give you an idea of the form that these statements take. For the precise probabilities, please ask Google.)
> This seems to suggest multiple Hamiltonian cycles in a graph, contradicting the earlier definition that every vertex must be connected.
This is no contradiction. There can be multiple Hamiltonian cycles in a graph. Consider the complete graph on n vertices; there are roughly n-factorial-many Hamiltonian cycles. Any permutation of the vertices corresponds to one such cycle. Different permutations can correspond to the same cycle, so the number is not exactly n-factorial. But you get the idea.
> Consider the complete graph on n vertices; there are roughly n-factorial-many Hamiltonian cycles. Any permutation of the vertices corresponds to one such cycle. Different permutations can correspond to the same cycle, so the number is not exactly n-factorial.
Isn't it just (n-1)!/2 as each cycle can be cyclically permuted (n possibilities) and (for n>=3) also be reversed? (Cases n=2 and n=1 have only one cycle.)
(That wasn't the point you were trying to get to, I know. I'm just thinking about the precise number.)
Thanks for the answers to some of my questions. I think the conclusion is that the article is above my comprehension, which is absolutely fine given who I imagine Quanta's readership is.
I configured Fastmail to send emails that I compose from my work address, via the work SMTP server. Not any other type of email. And indeed, they don't send emails that I compose from my non-work addresses via my work SMTP server: they use their own server to send these.
The support person who is dealing with this also turned up the fact that this is a change that they made some time in 2019. Earlier to that, all emails from Fastmail Calendar were sent from the "support@fastmail.com" address. They changed this in late 2019 so that Calendar now sends these mails from my work address (following some peculiar logic and email rewrite rules).
So yes, it is surprising that they send emails from my Calendar via my work SMTP server.
> You can also publish papers as an undergrad with your adviser's support in the US, is that not true in India?
It is just as true in India as it is in the US.
And: while it definitely helps if your college/department/advisor is there to lend you support, such support is not a prerequisite for getting published.
Yes as an undergrad I have 3 publications till now (in 3 years) in India. Still other institutes in India won't care how many publications I have when I am applying.
1. Identify research scientists (e.g: in Google/IBM/Microsoft/... research labs) and professors (in institutes/universities) that you would like to work with.
2. Write to them directly, attaching your publications and requesting an internship.
Depending on how your publications are perceived, there is a good chance that at least a few will choose to engage with you.
Indian institutes funded by taxpayer money are required to follow a prescribed set of rules for selecting PhD candidates, in the interest of ensuring fairness, preventing nepotism, etc. . The competitive exams that you seem to loathe are part of this process, and professors can do nothing about this.
They have much more leeway in choosing interns, though. The rest follows as per kubrickslair's recipe.
I can speak of theoretical computer science: if you have work that is worth publishing (as evaluated by your peers), then yes, it is possible to publish without any affiliation. I would be very surprised if a good result is rejected just because the author is not affiliated.
The operative word in the first sentence above is "if". In my opinion, unless you are in the Ramanujan or Erdős class, then it is highly unlikely that you come up with a publishable manuscript on your own. Briefly put: you need a lot of "context" to be able to do publishable work, and you are unlikely to get this context unless you interact with the research community.
The corollary to this: most correct work is not worth publishing. And most people bitter over their work not being accepted were not rejected because their work was incorrect, but because it was not notable. So at the end of the day, it's all a popularity contest, and that's what you're really engaging in.
A new proof of the Four Colour Theorem, by Ashay Dharwadker.
Abstract
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We present a new proof of the famous four colour theorem using algebraic and topological methods. Recent research in physics shows that this proof directly implies the Grand Unification of the Standard Model with Quantum Gravity in its physical interpretation and conversely the existence of the standard model of particle physics shows that nature applies this proof of the four colour theorem at the most fundamental level, giving us a grand unified theory. In particular, we have shown how to use this theory to predict the Higgs Boson Mass [arXiv:0912.5189] with precision. Thus, nature itself demonstrates the logical completeness and consistency of the proof. This proof was first announced by the Canadian Mathematical Society in 2000. The proof appears as the twelfth chapter of the text book Graph Theory published by Orient Longman and Universities Press of India in 2008. This proof has also been published in the Euroacademy Series Baltic Horizons No. 14 (111) dedicated to Fundamental Research in Mathematics in 2010. Finally, the proof features in an exquisitely illustrated edition of The Four Colour Theorem published by Amazon in 2011. The Endowed Chair of the Institute of Mathematics in recognition of this achievement was bestowed in 2012.
Title: Grand Unification of the Standard Model with Quantum Gravity
Author: Ashay Dharwadker
Abstract:
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We show that the mathematical proof of the four colour theorem [1] directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem, at the most fundamental level. We preserve all the established working theories of physics: Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein's law of gravity. Quantum gravity is a direct and unavoidable consequence of the theory. The main construction of the Steiner system in the proof of the four colour theorem already defines the gravitational fields of all the particles of the standard model. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with 't Hooft's table [8]. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.
India has been successfully using EVMs for elections since the turn of the century [1].
And an Indian General Election is no menial task. In the latest election of 2014, 814.5 million people were eligible to vote, making it the largest-ever election in the world. A total of 8,251 candidates contested for 543 seats in the Indian parliament. The average election turnout was around 66.38% [2].
You get shocked easily when you use the escalator.You wouldn't be electrocuted more than once.