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## Monday, August 26, 2019 ... //

### Anti-quantum-mechanical propaganda at Big Think

Like all platforms whose quality isn't carefully monitored and improved by a hierarchical system of competent filters, BigThink.com has become another Internet-media tool hijacked by mediocre activists – which almost entirely means deceitful, ideologically driven, far left activists – who are just pushy enough and will do everything to spread their stuff. Their knowledge of the actual science is shockingly poor.

Even when thousands of dollars per minute of content are flowing to such servers, the content may be expected to be less valuable than a vigorously and urgently used toilet paper if the quality control is absent so that even the most self-evident trash may get in. You can't be surprised that ludicrously wrong monologues such as

Space is dead: A challenge to the standard model of quantum mechanics
appear on that server. There are silly messages hidden in every other word over there and the density of explicit or implicit falsehoods is amazingly high. First, the title is composed of two parts. The first says that "space is dead" which is surely a "not really established" claim about quantum gravity while the second part is about a different issue altogether, the basic axioms of quantum mechanics.

These aren't really the same "topic", at least not yet. Quantum gravity is just an example of a quantum mechanical theory – or a special subclass of quantum mechanical theories if you wish. All serious researchers in quantum gravity accept all universal postulates of quantum mechanics and if some of them don't, they are extremely careful about their proposals how they could hypothetically violate these axioms without getting complete nonsense.

Also, the second part of the title talks about "the standard model of quantum mechanics". This is a completely illogical sequence of words indicating that the writer forgot what he was talking about and got stuck in self-referring loops. The reason is that quantum mechanics is already a theory – theories like QM are the ultimate stuff that theoretical physicists are constructing, filtering, and refining.

When they talk about "models", it means a more specific type of a theory. For example, the Standard Model of particle physics is a particular theory of a narrower type where the field content and the list of interactions is fully specified. But it's clear that a "model" in the ordinary sense isn't what the speaker has in mind. Instead, the self-referring contrived sequence of words "model of quantum mechanics" was chosen because the speaker wants to deny that quantum mechanics is already a theory at all. Instead, he wants to downgrade it to some heuristic speculation or a piece of arts that still needs something else, a real "theory", to become science. But that's demonstrably not the case. Quantum mechanics is a complete theory that defines which questions are scientifically meaningful and how to calculate the answers to these questions. No disagreement with the experiments has been found as of 2019; and similarly, no measurement that would be unanswerable by QM, even in principle, although there is evidence that one may get some nontrivial experimental answers, has been pointed out.

Some people may look for a "model of quantum mechanics" in a sense of another theory of a different kind, almost universally a theory in classical physics, that should replace quantum mechanics according to their plan. It's obviously impossible. Actual physicists don't do anything of the sort. They want to understand the correct theories that actually describe nature and not some simplifications, parodies, or caricatures – "models" in this sense of toy trucks – that may be good to promote the theory but that aren't good or accurate science from the scientists' viewpoint.

Serious physicists study theories or models themselves (which say something about the actual observations of Nature), not models of theories or theories of theories of theories etc.

The subtitle says
Since the idea of locality is dead, space itself may not be an aloof vacuum: Something welds things together, even at great distances.
Cute. But the idea of locality isn't dead. On the contrary, it's been alive since 1905 when it became a fact due to the special theory of relativity. That theory – backed by a huge and diverse body of experiments – established that locality and the ban on the faster-than-light signals is equivalent to the ban on acausal influences in which the future influences the past.

A theoretical physics PhD in the second half of the 20th century granted to someone who is "capable" of saying that "locality is dead" is exactly on par with a geology PhD given to someone who claims that the Earth is flat.

Relativity bans faster-than-light signals because they are, via the Lorentz transformation, a symmetry of Nature, equivalent to faster-than-light signals that go backwards in time. The non-existence of such effects is therefore a matter of basic consistency of the laws of physics (the absence of causally closed time-like curves). Indeed, such a ban means that the empty space must be completely empty – or at least, it must be empty of structures that would pick a preferred inertial frame. These prohibited structures include "anything like air, water, or other gases and liquids", "anything like crystals or solids", and "anything like a system of wheels and gears which is how many 19th century physicists imagined the luminiferous aether". Anything like a "network made of seemingly localized building blocks" that fills the vacuum has been banned since 1905. Not even "spin networks of loop quantum gravity" may get an exception. Like all similar naive examples of the aether, they've been dead since 1905.

Indeed, it was a key point of Einstein's revolutionary new understanding of electromagnetism that the aether was proven non-existent. That's how he got to his insights – by carefully thinking about the propagation of light – and anyone who denies that the aether as envisioned in the 19th century has been ruled out is denying all of relativity.

Locality precisely holds in the best non-gravitational theories of Nature we have – quantum field theories. In quantum gravity, whether or not locality precisely holds is a little bit trickier question. There are careful definitions that apparently allow us to say that locality holds even in quantum gravity assuming that "truly analogous conditions" to the conditions in special relativity are met. On the other hand, if locality is defined in some sloppy way, by some sloppy quantities, it's common that we may find out that e.g. the Hawking radiation violates locality. This violation may be shown to be an illusion in some finer definitions of locality, however. Similar comments apply to string theory's perspective on quantum gravity. Some straightforward notions of locality are invalid e.g. when we try to associate the strings to points simply because the strings are extended. But finer definitions of locality may be shown to hold even e.g. in string field theory.

At the beginning, the speaker defines his "realism" as
the belief that there exists an accurate description of Nature that is independent of our interventions, of our knowledge etc.
If the speaker were familiar with the basic theoretical physicists' terminology, he would know that the "belief" defined exactly in this way is normally called "classical physics". This is the phrase that I will use because I am a physicist, not a snake oil salesman who tries to produce a used sheet of toilet paper for thousands of dollars.

Now, classical physics – the framework of physics where the information about the state of Nature objectively exists (and is given by a choice of the state from a "phase space" as we call it) regardless of any observations and ideal observations may be assumed to be non-interventionist – has been known to be incorrect around 1925. Physicists were realizing that it was necessary to abandon classical physics in the previous decades – approximately since 1900. They figured out why all the basic types of ideas – that are still crazily promoted by "believers in classical physics" in 2019 – were unable to agree with all the known phenomena of the early 20th century physics at the same moment.

Perhaps due to some huge dose of luck, philosophers have built the vague conceptual foundations for the new framework of physics, quantum mechanics, already in the 19th century. Positivism founded by Auguste Comte was a branch of the empirical philosophies of science that usefully emphasized that all our knowledge about Nature comes from sensory perception – so sensory perception is what all of our arguments about the laws of Nature and about the state of the world must ultimately be reduced to. When it's so, it's rather natural to study the (probabilistic...) connections between the sensory perceptions (observations) directly, without assuming some intermediate "machine", an objective classical system that was assumed to exist even in the absence of observers.

To one extent or another, other top philosophers that have influenced physics were continuing in the tradition of positivism which turned out to be useful. Albert Einstein was primarily affected by Ernst Mach, a newer positivist from the late 19th and early 20th century, while Niels Bohr was – like Einstein – an avid reader of philosophers' texts and he was primarily building on the writings of the Vienna Circle (Wiener Kreis), modern 20th century descendants of the positivists that abruptly ended in 1936 (not only because of the obvious political reasons; watch Vienna Circle, exact philosophy in demented times).

In 1902, my first elementary school was just built (date under the clock).

The younger generation of the "true founders of quantum mechanics" who did it in the mid and late 1920s and who were born between 1900 and 1902 – Heisenberg, Dirac, Pauli, Jordan, Wigner, Eckart and others (lots of fathers of QM born in 1902 in particular) – was already much less keen on reading philosophers' texts. But they listened to Niels Bohr, a spiritual father of theirs, and credited him with many conceptual, new, and essential insights – often without realizing that Bohr has actually learned some of these deep general things from the philosophers. At any rate, when the generation of Heisenberg and even younger generations took over physics, there was no longer any useful – according to the top physicists – influence of "living philosophers" on physics. A broken calendar (with OK last three digits) is only correct once per millennium – and philosophers were already right around 1900. Let's wait through 2900 AD for two more glorious decades of philosophers who may contribute to physics.

By the 1920s, all physicists who were up-to-date already knew why all the "beliefs" by this speaker were wrong. In particular, it is impossible to perform non-interventionist experiments. Every observation unavoidably – and almost by definition – influences the observed system and does so irreversibly yet these interventions are essential for any discussion about the state and laws of Nature to be scientifically meaningful. So the specification "what an observation is" is an inseparable part of the application of the laws of physics.

Incidentally, his usage of the word "belief" has an understandable likely motivation. "Belief" sounds like a religion and everyone enjoys the freedom of religion. Except that in science, what is even more important than the freedom of religion is the freedom to point out that theories and "beliefs" falsified by the experiments are wrong – and that the people who are stuck with truly indefensible theories and "beliefs" are bad scientists if not crackpots.

The speaker also says that
due to the First World War and the predominance of some philosophy in the interwar era, [classical physics] that I believe to be true wasn't popular around the 1920s etc.
This is obviously a complete falsehood whose only purpose is to unjustifiably insult the ingenious physicists of the 1920s. The actual reason why classical physics was abandoned was that it was found to be incompatible with the experimental facts – the black body radiation, photoelectric effect, stability of atoms, spectral lines of atoms, radioactivity, interference properties of electrons, Compton effect, low heat capacity $O(k_B)$ of each molecule, and more.

The reason why classical physics was abandoned by scientists – and it was being abandoned gradually between 1900 or so and the late 1920s – wasn't any "philosophers' preference" let alone the First World War. It was abandoned because all the carefully refined classical theories or hypotheses were experimentally falsified – or at least sufficiently specific classes of classical physical models were ruled out. That's how science including physics works when it's real science – and physics of the 1920s was real physics par excellence, unlike some neo-Marxist critiques of science from the early 21st century. Physics isn't decided on the First World War battlefronts or in the philosophers' bedrooms and everyone who is telling you something else is lying to you.
I am trying to talk in the way in which Niels Bohr was talking because he was the most radical of these [anti-classical-physics] thinkers.
It's a manipulation if not another outright lie to say that Niels Bohr was the "most radical" one among the thinkers who rejected the basic assumptions of classical physics. All physicists who were building the new framework of physics rejected the axioms of classical physics equally – namely completely. To say the least, none of them has dedicated a single day of their research career to the assumption that classical physics was correct, after all. The actual difference between Niels Bohr and his mostly younger friends was that Niels Bohr had been a keen reader of the philosophers' books so he was often using their jargon – and philosophers' jargon often sounds murky, ideological, or both.

As Catherine Chevalley has convincingly argued, all claims about Bohr's "obscurity" are compatible with the theory that the critics found it obscure simply because they were (or are) unfamiliar with the philosophers' writings that Bohr has considered to be his prerequisites; and all "genuine dissatisfaction" with Bohr's "ideological" statements may be explained by the critics' indoctrination by other philosophies, almost universally those philosophies whose corresponding physical theories and models were ruled out by the late 1920s – but these wrong philosophies began to spread like fire after the Second World War, anyway.

But the content of the opinions was pretty much exactly the same when we compare Bohr with Heisenberg, Dirac, Born, Pauli, Jordan, Wigner, von Neumann, and others. They just understood that the era of classical physics had been over. The reasons are scientific – the previous theory has failed some tests – and even if the positivists' comments sounded ideological, this ideology has been vetted by the scientific method.
Let's talk about what it means to be a [non-classical physicist]. It means to believe that [classical physics] is too ambitious and too hard...
No, that's simply another lie. Classical physics is in no meaningful way more ambitious than quantum mechanics. Quantum mechanics is clearly a description of Nature at a deeper level than its classical approximation – just like relativistic physics is deeper than the non-relativistic approximation and the atomic description of thermal phenomena is deeper than the approximate, macroscopic thermodynamic description in the "thermodynamic limit" – which is why it's only correct to say that quantum mechanics is more ambitious than classical physics, at least more intellectually ambitious. It's just the other way around than the speaker says.

Incidentally, quantum mechanics generically makes predictions that have some uncertainty. But sometimes, quantum mechanics makes more unambiguous predictions than all "analogous" classical systems. The "quantum game theory" is full of examples of entangled systems (take e.g. the pentagon of cups) where a measurement is guaranteed to be precise and predictable. But it's often another, perhaps opposite, result than predicted by classical arguments.

Quantum mechanics and classical physics are incompatible frameworks of physics and again, the actual reason why classical physics has been abandoned is that people have realized that it's incompatible with the body of known phenomena, even with a bunch of very elementary observations. The claims by the speaker that physics went in the direction in which it went because of some nepotism among philosophers, because of some fights on the First World War, or because of a low ambition or weak skills of the physicists of the 1920s – all these things are provably full-blown lies and a dirty anti-physics propaganda.

Physics went in the direction that it has picked because the experimental facts dictated that no other direction was working well. Physicists have found and picked the simplest and most natural framework and theories that hadn't been ruled out yet. And the people in 2019 who are genuinely failing to understand why it's so are not failing because they are ambitious, war-avoiding, philosophy-avoiding, hard-problems-seeking geniuses or seers – but instead it's because they are just people with a low intelligence that is insufficient for the understanding of modern physics. If they try to sell their clear vices as virtues, it may be considered deception.
Electrons, nuclei... but what are these things? We often get confused when we talk about these things, the talk is stretched etc.
Maybe people who defend the return to classical physics in 2019 get confused all the time, whenever an electron appears – because their "beliefs" are incompatible with most of the actual facts about electrons – but all competent high-energy, particle, atomic, and even molecular and condensed-matter etc. physicists understand what an electron is. It's a nearly elementary concept connected with observations that are being made – as well as with the actual theories that are being used to explain the observations. So physicists not only know the word "electron" but they may assign truth values to a huge spectrum of statements about the electrons – how these particles manifest or don't manifest in experiments.

The conclusion that "I don't understand the electron" is another sign of someone's incompetence, not his depth. Physicists understand electrons and the understanding is better than e.g. the understanding of "planets" because electrons are more fundamental and more well-defined than "planets". To say the least, no real physicists are arguing with each other whether Pluto is an electron.

Electrons are represented in some way in Schrödinger's or Pauli's model of the atoms or in the one-particle Dirac equation or in condensed-matter models of metals or crystals or in QED or the Standard Model and these are not the ultimate theories. String theory almost certainly is but it is not fully understood to determine what it exactly predicts about the electrons. So one can't claim the "ultimate, complete, and precise knowledge" about electrons is available so far. But that's always true in science and it doesn't mean that "physicists don't know what an electron is". If someone claims that electrons are less understood than some complex objects, it proves that he doesn't understand fundamental physics at all. He doesn't understand that fundamental physics actually gives a clearer, purer, more accurate, more precise, and less convention-dependent understanding of elementary objects in Nature than any more "emergent" theory gives about the more complex systems.
[Quantum physicists] are impatient and have low ambitions.
Someone who has proposed "5-year deadlines" after which each physical theory is banned is talking about "impatience". Nice. Low ambitions again. Again, there is absolutely no impatience or low ambition about quantum mechanics. Instead, one may formulate sufficiently well-defined questions, e.g. "whether the random outcomes of the spin measurements may be due to hidden variables or are they really random", and accumulate evidence for and against both possible answers. When it's done, the scientific answer emerges and the scientific answer is that quantum mechanics is right (and the random outcomes are genuinely random) while classical physics (and various proposed explanations with hidden variables) are wrong.

This conclusion is exactly on par with the conclusion that the Earth is round, not flat. We just carefully look at the phenomena, possibilities, and evidence – and the correct answer emerges. Someone may have some personal reasons to dislike the results of scientific research and the answers that it has provided us with and launches a pathetic jihad to sling mud on science so that the mud gets stuck in the minds of the stupid laymen – but this activity is a classical example of what science is not. You can arbitrarily and unjustifiably claim that the Flat Earth is ambitious, ethical, patient... but you won't change the fact that the Earth is round.
Again, I am trying to emulate Niels Bohr which is hard, not only because of the accent but because he was mystical.
Bohr had some accent in English because he was Danish. To try to use his nationality as an argument against his ideas is absolutely pathetic. The speaker doesn't speak Danish at all – and indeed, in the quantum revolution, the Danish and German languages could have very well been more important than English which is why it's absolutely silly to discuss someone's knowledge of English at all. Why doesn't the speaker show us how good his Danish is? Can he translate the comments by Bohr's wife for us?

Niels Bohr wasn't mystical in any tangible way. He was a physicist with a strong background taken from philosophers' books but those were positivist philosophers who were as empirical and non-mystical as you can get. All these negative claims against Niels Bohr are just childish insults and only totally moronic visitors of BigThink.com are affected by them (but yes, I am afraid that the number of such visitors is high).
Niels Bohr said that we may only describe how we interact with Nature and he was at risk of contradicting himself.
He said it because this is what physics has established in the transformative 1900-1925 years. It's been a scientific fact since the late 1920s.

Yes, one may say that there's a potential "risk of contradictions" at some stage of one's understanding of quantum mechanics. But a "risk of contradictions" isn't a problem – much like someone's fear that the Earth could vaporize in 12 years isn't a problem. Only an actual contradiction – or a complete proof of a contradiction – would be a problem. There's no such contradiction in quantum mechanics and no such proof. It may really be proven that no such contradictions arise. Quantum mechanics may be said to move "dangerously close to a contradiction" but there is nothing wrong about it – instead, it is indeed a sign that it is a deep theory, one that maximally "exploits" the freedom that mathematics offers.

Being close to contradictions while avoiding them is a good sign for a theory. String theory manages to achieve the same, at an even more intense level than ordinary basic quantum mechanics – one may even derive the critical dimension from this principle etc. See also Milton Friedman's F-twist.

Quantum mechanics avoids the contradictions. A contradiction would be two different eigenvalues attributed to exactly the same observable. That can't ever happen, according to any quantum mechanical observer. Quantum mechanics demonstrably implies that if you measure the very same observable twice, with no interfering measurement in between, you will get the same result with the 100% certainty. Instead, Bohr's complementarity really means that different observers typically ask questions that aren't quite the same – which also means that their disagreement isn't ever a sharp contradiction. Different questions have different answers. Slightly different questions have slightly different answers – or a small probability to have very different answers.
Bohr was seeking new laws of physics, arts, and lifestyle...
It may have been true but this mixture is a demagogic sleight-of-hand because it indicates that Bohr's physics comments were influenced by the ideas about arts and lifestyle, and therefore failed to be rigorous enough, pure science. That's untrue. Bohr knew how to distinguish and separate hard science from other things – in fact, this is really the thing that positivists (the philosophers) really were very careful about.
Louis De Broglie had an obvious idea. The wave function is a pilot wave...
It's obvious in the sense that it's similar to what the recently abandoned theories – classical theories – had been saying, and in fact, to what Erwin Schrödinger was incorrectly saying, too. Only "objectively real waves" may exist so the wave function must be one, too.

But it's not an objectively real wave. It's a wave of probability amplitudes, as Max Born has quantitatively explained. The good physicists were figuring out that the new mathematical variables wouldn't be "objective" already before 1925. The year 1925 only meant that Heisenberg (plus Jordan and Born) found a mathematical version of these general ideas that quantitatively worked for the first time. You can say that Heisenberg's 1925 papers were the culmination of the work started by the 19th century philosophers that continued in Bohr's atomic model of 1913 – the basic philosophy was determined from the 19th century while the physical realization was getting more realistic, particular, and quantitative.

Some people like de Broglie hadn't gone through all this thinking before 1925 in which the wrong alternatives were eliminated. It was unsurprising for some revisionism to appear – attempts to be a parasite living on the recently triumphant quantum mechanical concepts while trying to undo the physical insights associated with them. So the pilot wave theory "had to" be proposed at some moment.

The pilot wave theory was ignored by the real, active physicists because they knew why it was wrong – it was an attempt to revive the old framework that had been abandoned. It was an "elaborate" attempt that tried to use much of the mathematics of quantum mechanics. But it's unsurprising for such things to be wrong. It's completely analogous to "versions of creationism trying to emulate Darwin's theory". One may assign special characteristics to God so that He will tend to distribute bones and properties of species that resemble the mechanisms in the evolution theory. But it's more likely – even though it's not guaranteed – for the correct theory to actually use the ideas before the incorrect theories do. If one of the theories predicts the patterns much more naturally and with many fewer assumptions, it's probably the more right way to encapsulate the "winning arguments and patterns" into a full-blown theory.

(Note that string theory and quantum field theory generally have nearly equivalent predictions for doable experiments, at least the qualitative features of these predictions. But string theory wasn't constructed like an emulation of quantum field theory – which was the how the pilot wave theory was born. String theory has very different technical initial assumptions and the QFT-like and Yang-Mills-like etc. conclusions were nontrivially derived from these new assumptions – so the agreement of string theory and QFT at low energies looked like "good luck". That's why it's completely sensible to say that the newer theory among the two, string theory, is a more likely utilization of the same observed patterns than the historically older one, quantum field theory.)
The pilot wave theory tells you everything, what, when, and how.
This is just completely false, too. The pilot wave theory is almost completely ill-defined; it tells you almost nothing and this lack of predictive power is pretty much rigorously provable. It doesn't say what determines the initial position of the "real" electron, why its initial distribution matches that expected one from the initial pilot wave. It doesn't tell us how the no-longer-relevant pilot wave is "swept" after the particles are observed or absorbed. Because of these two things – the missing description of anything related to the collapse – the pilot wave theory contributes zero to a solution of the non-problem that the [classical physicists of 2019] call the measurement problem.

Also, it tells us nothing about how to pick the "beables" for a more general system with spinning particles, fields, strings, or membranes etc. It is an emulation of quantum mechanics that must be invented from scratch ad hoc, differently for each new type of a quantum mechanical theory. The pilot wave theory is a strategy to fake quantum mechanics by a classical theory – which is bad enough – but it is not even a complete description how to perform this faking procedure.

Unsurprisingly, Bohmian replacements for quantum field theory or other theories can't really work, especially because they're almost 100% certainly predicted to violate the Lorentz symmetry which is lethal for the theories of particle physics.
Einstein discovered entanglement.
That's just plain nonsense. He just invented the first term for this trivial and omnipresent feature of almost all states – the spooky action at a distance (he coined that phrase in German). He coined the term because he felt uncomfortable about the very existence of this phenomenon.

But entanglement just means that the state of a composite system isn't a simple tensor product $\ket \alpha \otimes \ket\beta$. It is a sum of such products and the number of terms must be greater than one. Consequently, some correlations are predicted for the two systems. "Entanglement" is just a description of a general set of nonzero correlations in terms of pure states – and the concept of correlations had been known for centuries. All quantum physicists knew that almost all quantum states of a composite system obeyed this "non-factorizability" condition. They just didn't feel the need to use a term for this trivial property because this new buzzword only seems useful for falsely justifying some incorrect claims about/against quantum mechanics.

In particular, the singlet state of two spins $\ket\uparrow\ket\downarrow - \ket\downarrow\ket\uparrow$ was already known to Pauli in the mid 1920s. All the wave functions for pairs of spins encoded in the Clebsch-Gordan coefficients were known (plus their importance was understood) by 1941 as well, exploiting some mathematics that had been written down since the 19th century. You probably didn't know but both Clebsch and Gordan were born in the 1830s! If you want a related phrase named after physicists born in the productive year 1902, over 60 years after Clebsch and Gordan, why don't you look at the Wigner-Eckart theorem? At any rate, it's ludicrous to say that Einstein let alone Bell discovered entanglement or these particular entangled states.

Einstein only "invented" a buzzword for popular books (which got repeated in hundreds of popular books that make mostly incorrect claims about quantum mechanics), not a new discovery in quantum mechanics of composite systems that would be appreciated by professional physicists. His and EPR examples were trivial for quantum physicists and all Einstein's and EPR's opinions "how things should ultimately be" were demonstrably incorrect. In particular, their claims that entanglement contradicted relativistic causality were clearly wrong.
The definite result [of the experiments etc.] is that locality is false.
No, this is completely wrong. As everyone who has deserved a theoretical PhD since 1905 knows very well, locality is absolutely true and precise, at least in all non-gravitational theories – both in classical theories and quantum theories. The principles of relativity and their consequences are very fundamental and you can't justify an exemption just by babbling that with the word "quantum" (that you don't even want to allow, to make it even more preposterous), there should be an exemption. Such an excuse is worth nothing. Of course the principles of relativity have to hold and do hold in realistic quantum mechanical theories, too.
It has nothing to do with whether you believe in quantum mechanics.
Right, locality has nothing to do with quantum mechanics or a "belief" in it. Except that the speaker's opinion about the actual truth value is wrong in all cases. Locality is a fact about Nature because of the special theory of relativity. Quantum mechanics is orthogonal to that and realistic theories must obey and do obey both the principles of relativity and the postulates of quantum mechanics – and quantum field theories are the simplest theories of that kind (which are enough for all non-gravitational physics).
If you are a [non-classical physicist], you can't answer the question how the information gets back and forth [between the two entangled measurements].
That's also completely false. Quantum mechanics allows us to calculate how much information is going in between the two (spacelike-separated) places and the amount is precisely zero because no probabilities of observables are affected by decisions made in a spacelike-separated region. Some extra variables that someone imagines to exist may be affected superluminally but these extra imagined variables may be shown to be unphysical – unlinked from the observer's observations of observables – using the axioms of quantum mechanics that remain compatible with all experiments and apparently complete. So one shouldn't really derive anything from these imagined (typically hidden) variables simply because they're not a part of science, exactly for the same reasons why the Tooth Fairy isn't a part of science. Whether the Tooth Fairy flies superluminal airlines is irrelevant for the fate of relativity because she doesn't exist in the scientific sense. The entanglement has nothing to do with the transfer of information at the moment of the measurement or so – there isn't any (all the correlations are consequence of the co-existence or joint birth of the subsystems in the more distant past, not of any recent communication or interaction) – and quantum mechanics gives us the tools to rigorously prove and calculate the statement I just made. Every claim that there is something disputable about that is just a non-expert's fog, a layperson's misconception, a lie frequently repeated in junk-rated popular books.
The [classical] theories have to tell you how the information moves faster than light. Is that testable?
You bet, it's rather easily testable and every theory that generally predicts a generic, omnipresent superluminal propagation of information – because it doesn't have a tiny glimpse of a reason why such signals should be banned or almost banned – is falsified immediately just like a theory that predicts that almost all astronauts will see a flat Earth.
Maybe the prediction is untestable because the results are random.
This wishful thinking is not really possible because by denying the fundamental quantum randomness, these classical theories say that the results aren't really or fundamentally random. They're just some phenomena and if the superluminal transfer of the information were allowed by the laws of physics, it could work even in repeated or magnified phenomena with some "constructive interference" of the evidence, i.e. phenomena that aren't alternating randomly to make everything cancel. If there's no actual reason why things will cancel, they just won't cancel. A fundamentally non-relativistic theory will produce, even through its approximations, theories whose Lorentz-breaking terms are mostly of order one and those are simply excluded trivially and immediately. Not to be able to derive this simple conclusion means to be not to be competent as a 20th or 21st century theoretical physicist.
It tells you, if you are [classical physicist], that locality is dead and basically space is dead.
Again, an overwhelming body of evidence told the people that they just couldn't keep on being classical physicists if they want to be scientists because scientists abandon theories that have been experimentally refuted – and classical physics was experimentally refuted. In the same way, non-relativistic theories i.e. theories with superluminal signals were refuted by the evidence in favor of the 1905 special relativity so everyone who wanted to remain a scientist had to throw away the contradictory beliefs in nonlocality etc., too.

Whoever believes in 2019 that the world is non-local or non-quantum – or even both – simply fails to be a scientist.
But something deeper is going on.
Too bad the monologue was the opposite of an example of the depth – it was a defense of the view that one should simply ignore all the 20th century science and mentally return to the 17th century. I surely don't think that an effort to ban one century of ever deeper and more correct ideas is an example of deep thinking. It's an example of the shallowest and most intellectually embarrassing ideas imaginable, a populist demagogy that wants to persuade the stupidest listeners that their naive misconceptions resulting from their low intelligence are deep and cool. They are not but lots of very stupid people will love to buy totally invalid books on physics as long as these books will tell them the untruth that the morons are deep geniuses.

One can see how the publication of books by authors who aren't constrained by any genuine knowledge or moral principles evolves in time. First, they publish books that "just" attack the scientific achievement of theoretical physics of the recent 40 years. But the number of dumb readers who care about the recent 40 years at all is still too small. So assuming that they have no natural enemies who affect things, these writers are trying to increase the number of readers into a wider spectrum of even more hardcore morons by extending the target of criticism to all of physics of the recent 90 years, and so on. At this rate, similar authors will sling mud on heliocentrism around 2040 – in order to attract hundreds of millions of readers, the most hopeless morons on Earth.


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# What's the optimal policy in the rock-paper-scissors game?

A deterministic policy in the rock-paper-scissors game can be easily exploited by the opponent - by doing just the right sequence of moves to defeat the agent. More often than not, I've heard that a random policy is the optimal policy in this case - but the argument seems a little informal.

Could someone please expound on this, possibly adding more mathematical details and intuition? I guess the case I'm referring to is that of a game between two RL agents, but I'd be happy to learn about other cases too. Thanks!

EDIT: When would a random policy be optimal in this case?

• A deterministic policy can be the optimal policy. It entirely depends whether your environment is actively learning to adapt or not. I believe The formal argument will require us to find Nash equilibrium for each player, but making such an equilibrium requires some prior assumption about information (if I recall correctly) each player's possess about others, so I'll not write an answer. But the general direction is this, you can find about it more. – DuttaA Aug 27 at 11:12
• Interesting - if it's two RL agents playing against each other, the deterministic policy is definitely not optimal right? RL agents can adapt actively! – strawberry-sunshine Aug 27 at 13:48
• True.....but we don't know.... RL policy can be deterministic if you want..no steadfast rule. – DuttaA Aug 27 at 13:56

For this, we will need game theory.

In game theory, an optimal strategy is one that cannot be exploited by the opponent even if they know your strategy.

Let's say you want a strategy where your move selection is not based on what happened before (so you are not trying to model your opponent, or trick them into believing you will always play scissors and then throw them off, anything like that). A strategy will look like $$(P, S, R)$$, where $$P, S, R \in [0, 1], P+S+R = 1$$. You select paper with probability $$P$$, scissors with probability $$S$$, rock with probability $$R$$. Now, if your probabilities are a bit uneven (for example $$(0.5, 0.2, 0.3)$$) an opponent can abuse that strategy. If your opponent plays with probabilities $$(p, s, r)$$, their expected reward (counting +1 for win, -1 for loss, 0 for draw) would be $$0.5(s - r) + 0.2(r - p) + 0.3(p - s) = 0.1p + 0.2s - 0.3r$$. If they wish to maximize their wins, they would play scissors all the time against you, and expect to have a distinct advantage over you.

In general, for a strategy $$(P, S, R)$$ for you and $$(p, s, r)$$ for your opponent, your opponent's winnings would be $$P(s - r) + S(r - p) + R(p - s) = p(R-S) + s(P-R) + r(S - P)$$. If all the partial derivatives of this, with respect to $$p$$, $$s$$ and $$r$$ are 0, the opponent has no way to maximize his winnings; they would have no incentive to play a particular move over any other move. This occurs when $$P = S = R = \frac13$$.

That's basically how to approach game theory: find a strategy so your opponent has no incentive to choose one action over another. The approach seems a bit counter-intuitive at first (you're trying to find the optimal strategy for your opponent instead of for yourself) but it works for many similar problems.

• Although I agree with this answer, The answer makes an assumption that we know the opponent's strategy. I think in the RL setting this will be false, and we will get certain upper/lower bound based on the number of samples taken. – DuttaA Aug 27 at 17:01
• @DuttaA This answer doesn't make that assumption anywhere. In fact the "choose equally distributed randomly" strategy will work the same against any kind of strategy. – Helena Aug 27 at 20:40
• @Helena it did when it calculated the expected reward. The expression is known only when probabilities are known – DuttaA Aug 27 at 20:43
• @DuttaA The expression uses unknown variables for the opponents probabilities (p,s,r). It only uses constant values for the own example strategy (0.5, 0.2, 0.3) . – Helena Aug 27 at 20:47
• @DuttaA You are right, in reality you don't know each others strategy. Because you cannot know what kind of opponent you will get, the best thing you can do is to assume your strongest possible foe. – Helena Aug 27 at 21:00

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# Logarithms Questions for IIFT PDF

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## Logarithms Questions for IIFT PDF

Download important IIFT Logarithms Questions PDF based on previously asked questions in IIFT and other MBA exams. Practice Logarithms questions and answers for IIFT exam.

Practice IIFT Mock Tests

Question 1: If $log_3 2, log_3 (2^x – 5), log_3 (2^x – 7/2)$ are in arithmetic progression, then the value of x is equal to

a) 5

b) 4

c) 2

d) 3

Question 2: If $log_y x = (a*log_z y) = (b*log_x z) = ab$, then which of the following pairs of values for (a, b) is not possible?

a) (-2, 1/2)

b) (1,1)

c) (0.4, 2.5)

d) ($\pi$, 1/ $\pi$)

e) (2,2)

Question 3: If $f(x) = \log \frac{(1+x)}{(1-x)}$, then f(x) + f(y) is

a) $f(x+y)$

b) $f{\frac{(x+y)}{(1+xy)}}$

c) $(x+y)f{\frac{1}{(1+xy)}}$

d) $\frac{f(x)+f(y)}{(1+xy)}$

Question 4: If $\log_{2}{x}.\log_{\frac{x}{64}}{2}=\log_{\frac{x}{16}}{2}$. Then x is

a) 2

b) 4

c) 16

d) 12

Question 5: Find the value of x from the following equation:
$\log_{10}{3}+\log_{10}(4x+1)=\log_{10}(x+1)+1$

a) 2/7

b) 7/2

c) 9/2

d) None of the above

Question 6: If $log_{10} x – log_{10} \sqrt[3]{x} = 6log_{x}10$ then the value of x is

a) 10

b) 30

c) 100

d) 1000

Question 7: $(1+5)\log_{e}3+\frac{(1+5^{2})}{2!}(\log_{e}3)^{2}+\frac{(1+5^{3})}{3!}(\log_{e}3)^{3}+…$

a) 12

b) 244

c) 243

d) 245

Question 8: If $log(2^{a}\times3^{b}\times5^{c} )$is the arithmetic mean of $log ( 2^{2}\times3^{3}\times5)$, $log(2^{6}\times3\times5^{7} )$, and $log(2 \times3^{2}\times5^{4} )$, then a equals

Question 9: If $\log_{2}({5+\log_{3}{a}})=3$ and $\log_{5}({4a+12+\log_{2}{b}})=3$, then a + b is equal to

a) 59

b) 40

c) 32

d) 67

Question 10: $\frac{1}{log_{2}100}-\frac{1}{log_{4}100}+\frac{1}{log_{5}100}-\frac{1}{log_{10}100}+\frac{1}{log_{20}100}-\frac{1}{log_{25}100}+\frac{1}{log_{50}100}$=?

a) $\frac{1}{2}$

b) 10

c) 0

d) −4

$2 log (2^x – 5) = log 2 + log (2^x – 7/2)$
Let $2^x = t$
=> $(t-5)^2 = 2(t-7/2)$
=> $t^2 + 25 – 10t = 2t – 7$
=> $t^2 – 12t + 32 = 0$
=> t = 8, 4
Therefore, x = 2 or 3, but $2^x$ > 5, so x = 3

$log_y x = ab$
$a*log_z y = ab$ => $log_z y = b$
$b*log_x z = ab$ => $log_x z = a$
$log_y x$ = $log_z y * log_x z$ => $log x/log y$ = $log y/log z * log z/log x$
=> $\frac{log x}{log y} = \frac{log y}{log x}$
=> $(log x)^2 = (log y)^2$
=> $log x = log y$ or $log x = -log y$
So, x = y or x = 1/y
So, ab = 1 or -1
Option 5) is not possible

If $f(x) = \log \frac{(1+x)}{(1-x)}$ then $f(y) = \log \frac{(1+y)}{(1-y)}$

Also Log (A*B)= Log A + Log B

f(x)+f(y) = $\log \frac{(1+x)(1+y)}{(1-x)(1-y)}$ solving we get $\log { \frac{1+ \frac{(x+y)}{(1+xy)}}{1- \frac{(x+y)}{(1+xy)}}}$

Hence option B.

$\log_{2}{x}.\log_{\frac{x}{64}}{2}=\log_{\frac{x}{16}}{2}$

i.e. $\frac{log{x}}{log{2}} * \frac{log_{2}}{log{x}-log{64}} = \frac{log{2}}{log{x}-log{16}}$

i.e. $\frac{log{x} * (log{x}-log{16})}{log{x}-log{64}}$ = $\log{2}$

let t = log x

Therefore,  $\frac{t * (t-log{16})}{t-log{64}}$ = $\log{2}$

$t^2-4*log 2*t = t*log 2-6*(log 2)^2$

I.e. $t^2-5*log 2*t-6*(log 2)^2$ = 0

I.e. $t^2-3*log 2*t-2*log 2*t-6*(log 2)^2$ = 0

i.e. $t*(t-3*log 2)-2*log 2*(t-3*log 2)$ = 0

i.e $t=2*log 2$ or $t=3*log 2$

i.e $log x=log 4$ or $log x=log 8$

therefore $x=4$ or $8$

therefore our answer is option ‘B’

$\log_{10}{3}+\log_{10}(4x+1)=\log_{10}(x+1)+1$ can be written as

$\log_{10}{3}+\log_{10}(4x+1)=\log_{10}(x+1)+\log_{10}{10}$

We know that $\log_{10}{a}+\log_{10}{b}=\log_{10}{ab}$

$\log_{10}{3*(4x+1)}=\log_{10}{(x+1)*10}$

$12x+3=10x+10$

$x=7/2$. Hence, option B is the correct answer.

$\log_{10} x – \log_{10} \sqrt[3]{x} = 6\log_{x}10$
Thus, $\dfrac{\log {x}}{\log {10}}$ – $\dfrac{1}{3}*\dfrac{\log {x}}{\log {10}}$ = $6*\dfrac{\log{10}}{\log{x}}$
=> $\dfrac{2}{3}*\dfrac{\log {x}}{\log {10}}$ = $6*\dfrac{\log{10}}{\log{x}}$
Thus, => $\dfrac{1}{9}*(\log{x})^2 = (\log{10})^2$
Thus, $x = 1000$
Hence, option D is the correct answer.

Splitting the above mentioned series into two series

A = $\log_{e}3+\frac{1}{2!}(\log_{e}3)^{2}+\frac{1}{3!}(\log_{e}3)^{3}+…$

B = $5\log_{e}3+\frac{5^{2}}{2!}(\log_{e}3)^{2}+\frac{5^{3}}{3!}(\log_{e}3)^{3}+…$

We know that $e^{x}$ =$1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+…$

So  $e^{x}-1$ = $x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+…$

On solving two series A and B

A = $\log_{e}3+\frac{1}{2!}(\log_{e}3)^{2}+\frac{1}{3!}(\log_{e}3)^{3}+…$ =$e^{\log_{e}3}-1$ = $3-1$ =$2$

B = $5\log_{e}3+\frac{5^{2}}{2!}(\log_{e}3)^{2}+\frac{5^{3}}{3!}(\log_{e}3)^{3}+…$=$e^{\log_{e}3^{5}}-1$=$3^{5}-1$=$242$

A+B = $2 + 242$ = $244$

$log(2^{a}\times3^{b}\times5^{c} )$ = $\frac{log ( 2^{2}\times3^{3}\times5) + log(2^{6}\times3\times5^{7} ) + log(2 \times3^{2}\times5^{4} ) }{3}$

$log(2^{a}\times3^{b}\times5^{c} )$ = $\frac{log ( 2^{2+6+1}\times3^{3+1+2}\times5^{1+7+4}) }{3}$

$log(2^{a}\times3^{b}\times5^{c} )$ = $\frac{log ( 2^{9}\times3^{6}\times5^{12}) }{3}$

$3log(2^{a}\times3^{b}\times5^{c} )$ = $log ( 2^{9}\times3^{6}\times5^{12})$
Hence, 3a = 9 or a = 3

$\log_{2}({5+\log_{3}{a}})=3$
=>$5 + \log_{3}{a}$ = 8
=>$\log_{3}{a}$ = 3
or $a$ = 27

$\log_{5}({4a+12+\log_{2}{b}})=3$
=>$4a+12+\log_{2}{b}$ = 125
Putting $a$ = 27, we get
$\log_{2}{b}$ = 5
or, $b$ = 32

So, $a + b$ = 27 + 32 = 59
Hence, option A is the correct answer.

We know that $\dfrac{1}{log_{a}{b}}$ = $\dfrac{log_{x}{a}}{log_{x}{b}}$

Therefore, we can say that $\dfrac{1}{log_{2}{100}}$ = $\dfrac{log_{10}{2}}{log_{10}{100}}$

$\Rightarrow$ $\frac{1}{log_{2}100}-\frac{1}{log_{4}100}+\frac{1}{log_{5}100}-\frac{1}{log_{10}100}+\frac{1}{log_{20}100}-\frac{1}{log_{25}100}+\frac{1}{log_{50}100}$

$\Rightarrow$ $\dfrac{log_{10}{2}}{log_{10}{100}}$-$\dfrac{log_{10}{4}}{log_{10}{100}}$+$\dfrac{log_{10}{5}}{log_{10}{100}}$-$\dfrac{log_{10}{10}}{log_{10}{100}}$+$\dfrac{log_{10}{20}}{log_{10}{100}}$-$\dfrac{log_{10}{25}}{log_{10}{100}}$+$\dfrac{log_{10}{50}}{log_{10}{100}}$

We know that $log_{10}{100}=2$

$\Rightarrow$ $\dfrac{1}{2}*[log_{10}{2}-log_{10}{4}+log_{10}{5}-log_{10}{10}+log_{10}{20}-log_{10}{25}+log_{10}{50}]$

$\Rightarrow$ $\dfrac{1}{2}*[log_{10}{\dfrac{2*5*20*50}{4*10*25}}]$

$\Rightarrow$ $\dfrac{1}{2}*[log_{10}10]$

$\Rightarrow$ $\dfrac{1}{2}$

We hope this Logarithms questions and answers PDF will be helpful to you.


text:
# Ab-initio prediction of bright NV+ photoluminescence

Our recently published novel method to predict solid-state photoluminescence has discovered a possible bright transition in the positively charged nitrogen vacancy in diamond (NV+).

NV+ has historically been considered a dark state due to absence of experimental photoluminescence. Our work reveals that there is an optically active singlet-singlet transition approximately 16x less bright than the analogous triplet-triplet transition of the NV- charge state.

Our work in NV+ is a step towards unlocking applications such as quantum memory. We demonstrate that the lack of optical signature used for identification is not reliable and we subsequently provide predictions to guide experimental investigation. This will be useful for experimental work with NV defects in diamond as well as a motivation for further theoretical work investigating potential charge dependent quenching mechanisms.

Access the pre-print here


text:
# TikZ: compilation time performance of decorations

I want to produce a picture that contains lots of two-part cylinder-kind shapes (maybe about 200) along certain paths. Here is an example of such a shape:

\documentclass{standalone}
\usepackage[svgnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}

%%%%% H-O bond between atom 1 and atom 3

%%% O half of the bond
% O half bond shading
color(0.000bp)=(Black!75!Red);
color(22.773bp)=(Black!75!Red);
color(45.525bp)=(Black!3.893!Red);
color(53.986bp)=(White!26.261!Red);
color(59.561bp)=(White!26.261!Red);
color(68.022bp)=(Black!3.893!Red);
color(90.773bp)=(Black!75!Red);
color(100.000bp)=(Black!75!Red)}
% O half bond drawing
\begin{scope}
\pgfpathmoveto{\pgfpoint{3.107cm}{-5.942cm}}
\pgfpathlineto{\pgfpoint{2.998cm}{-5.816cm}}
\pgfpatharcaxes{0}{180}{\pgfpoint{0.085cm}{0.073cm}}{\pgfpoint{-0.034cm}{0.040cm}}
\pgfpathlineto{\pgfpoint{2.936cm}{-6.088cm}}
\pgfpatharcaxes{180}{0}{\pgfpoint{0.085cm}{0.073cm}}{\pgfpoint{-0.034cm}{0.040cm}}
\pgfusepath{clip}
\pgfpathcircle{\pgfpoint{2.968cm}{-5.952cm}}{1.168cm}
\pgfusepath{}
\end{scope}

%%% H half of the bond
% H half bond shading
color(0.000bp)=(Black!75!White);
color(22.773bp)=(Black!75!White);
color(45.525bp)=(Black!20.000!White);
color(53.986bp)=(White!26.261!White);
color(59.561bp)=(White!26.261!White);
color(68.022bp)=(Black!20.000!White);
color(90.773bp)=(Black!75!White);
color(100.000bp)=(Black!75!White)}
% H half bond drawing
\begin{scope}
\pgfpathmoveto{\pgfpoint{2.936cm}{-6.088cm}}
\pgfpathlineto{\pgfpoint{3.026cm}{-6.193cm}}
\pgfpatharcaxes{180}{0}{\pgfpoint{0.086cm}{0.073cm}}{\pgfpoint{0.034cm}{-0.040cm}}
\pgfpathlineto{\pgfpoint{3.107cm}{-5.942cm}}
\pgfpatharcaxes{0}{180}{\pgfpoint{0.085cm}{0.073cm}}{\pgfpoint{-0.034cm}{0.040cm}}
\pgfusepath{clip}
\pgfpathcircle{\pgfpoint{3.066cm}{-6.067cm}}{1.168cm}
\pgfusepath{}
\end{scope}

\end{tikzpicture}

\end{document}

Since this makes the code unreadably long I thought of putting this into a decoration. The decoration will have to take a lot of arguments: the names, colors and positions of the shadings, as well as the coordinates for the \pgfpatharcaxes commands. Unfortunately, I have very little experience with TeX and pgf programming so that this will take up some time. Since I want to be sure that this time investment is not wasted I'd like to know whether it is to be expected that TikZ decorations come with any penalty for the compilation time compared to the normal code shown above (I would refrain from using a decoration if the compilation would take more than twice as long)? Is it advisable to use a decoration in my case or might there be reasons against it?

The benefit of decoration is that it could help you place shadings properly. But before that, let us define a flexible shading

\pgfdeclareverticalshading[base color]{cylinder}{1000bp}{
color(0.000bp)=(black!75!base color);
color(22.773bp)=(black!75!base color);
color(45.525bp)=(black!3.893!base color);
color(53.986bp)=(white!26.261!base color);
color(59.561bp)=(white!26.261!base color);
color(68.022bp)=(black!3.893!base color);
color(90.773bp)=(black!75!base color);
color(100.000bp)=(black!75!base color)}

\pgfutil@colorlet{base color}{green}

Then we try to utilize this shading in a decoration. There are many parameters

• start at is the starting position.
• end at is the ending position.
• start by determines if the shape is convex or concave at the starting position.
(1: convex; .1: slightly convex; -1: concave)
• end by works similarly

The problem here is that the shading does not rotate with us. (By "us" I mean the decoration engine.) So I need to modify the behavior of \pgfshadepath -- I called it \pgfshadepath@revise.

\pgfkeys{
/pgf/decoration/.cd,
start at/.code={\pgfmathsetmacro\pgfdecorationstartat{#1}},
start at=.25,
end at/.code={\pgfmathsetmacro\pgfdecorationendat{#1}},
end at=.75,
start by/.code={\pgfmathsetmacro\pgfdecorationstartby{#1}},
start by=-.5,
end by/.code={\pgfmathsetmacro\pgfdecorationendby{#1}},
end by=-.5,
base color/.code=\pgfutil@colorlet{base color}{#1},
base color=white,
amplitude=1cm
}

\newdimen\pgf@xd
\newdimen\pgf@xe
\newdimen\pgf@yd

\pgfdeclaredecoration{cylinder}{initial}
{
\state{initial}[width=\pgfdecorationstartat*\pgfdecoratedinputsegmentlength,next state=draw]
{}
\state{draw}[width=0pt,next state=final]
{
\pgf@xe\pgfdecorationsegmentamplitude pt
\pgf@xd\pgfdecorationstartby\pgf@xe \pgf@xe\pgfdecorationendby\pgf@xe
\pgf@yd\pgfdecorationsegmentamplitude pt
\pgfpathmoveto{\pgfqpoint{0pt}{\pgf@yd}}
\pgfpatharcaxes{90}{270}{\pgfqpoint{\pgf@xd}{0pt}}{\pgfqpoint{0pt}{\pgf@yd}}
\pgfpathlineto{\pgfpoint{(\pgfdecorationendat-\pgfdecorationstartat)*\pgfdecoratedinputsegmentlength}{-\pgf@yd}}
\pgfpatharcaxes{-90}{90}{\pgfqpoint{\pgf@xe}{0pt}}{\pgfqpoint{0pt}{\pgf@yd}}
\pgfclosepath
\pgftransformyscale{\pgf@yd/50bp}
\pgfgettransform\pgfcurrenttransform
\pgfusepath{}
}
\state{final}
{}
}

\begingroup%
\pgfsys@beginscope%
\pgfsyssoftpath@invokecurrentpath%
\pgfsys@clipnext%
\else%
\fi%
\pgfsys@endscope%
\endgroup%
}

\tikz{
\draw(-4,-1)--(4,1);
\draw decorate[decoration=cylinder]{(-4,-1)--(4,1)};
}

We can decorate the same path multiple times.

\tikz{
\draw(-4,-1)to[
to path={
decorate[decoration={cylinder,base color=red  ,start at=.0 ,end at=.2 ,start by= .5,end by=-.5}]{(\tikztostart)--(\tikztotarget)}
decorate[decoration={cylinder,base color=blue ,start at=.25,end at=.45,start by= .5,end by= .5}]{(\tikztostart)--(\tikztotarget)}
decorate[decoration={cylinder,base color=white,start at=.55,end at=.75,start by=-.5,end by= .5}]{(\tikztostart)--(\tikztotarget)}
decorate[decoration={cylinder,base color=black,start at=.8,end at=1   ,start by=-.5,end by=-.5}]{(\tikztostart)--(\tikztotarget)}
}
](4,1);
}

In your case we need only two times. We can write a parser. Here ) red ( means that the cylinder is in red, and is concave at both side. If you change it to ( blue ), it becomes a blue cylinder convex at both side. The number between two cylinders represents the position of the "cut".

\tikzset{
bond/.code args={#1 #2 #3 #4 #5 #6 #7}{
\if#1|\edef\Astartby{0}\else\if#1(\edef\Astartby{.5}\else\if#1)\edef\Astartby{-.5}\else\edef\Astartby{#1}\fi\fi\fi
\if#3|\edef\Aendby{  0}\else\if#3(\edef\Aendby{ -.5}\else\if#3)\edef\Aendby{   .5}\else\edef\Aendby{  #3}\fi\fi\fi
\if#5|\edef\Bstartby{0}\else\if#5(\edef\Bstartby{.5}\else\if#5)\edef\Bstartby{-.5}\else\edef\Bstartby{#5}\fi\fi\fi
\if#7|\edef\Bendby{  0}\else\if#7(\edef\Bendby{ -.5}\else\if#7)\edef\Bendby{   .5}\else\edef\Bendby{  #7}\fi\fi\fi
\tikzset{
to path={
decorate[decoration={cylinder,base color=#2,start at=0,end at=#4,start by=\Astartby,end by=\Aendby}]{(\tikztostart)--(\tikztotarget)}
decorate[decoration={cylinder,base color=#6,start at=#4,end at=1,start by=\Bstartby,end by=\Bendby}]{(\tikztostart)--(\tikztotarget)}
}
}
}
}

\tikz{
\draw(-4,-1)to[bond=) red ( .4 ) white (](4,1);
}

Now go into Chemistry. We can define the style of bonds between different pairs of atoms. This is Urea, only that the double bond between carbon an nitrogen is not drawn properly.

\tikzset{
C-O/.style={bond=( black | .4  | red )},
C-N/.style={bond=( black | .45 | blue )},
N-H/.style={bond=( blue  | .5  | white )},
}

\tikz{
\path(0,0)node[C](C1){}(0,10)node[O](O1){}(9,-3)node[N](N1){}(-9,-3)node[N](N2){}(9,-8)node[H](H1){}(13,0)node[H](H2){}(-9,-8)node[H](H3){}(-13,0)node[H](H4){};
\path(C1)to[C-O](O1)(C1)to[C-N](N1)(C1)to[C-N](N2)(N1)to[N-H](H1)(N1)to[N-H](H2)(N2)to[N-H](H3)(N2)to[N-H](H4);
}

With our parser, we can indicate the 3D structure by the convexity of cylinders. This is methane.

\tikzset{
C>H/.style={bond=( black ) .4  ) white )},
C<H/.style={bond=( black ( .4  ( white )}
}

\tikz{
\path(0,0)node[C](C1){}(8,0)node[H](H1){}(-8,0)node[H](H2){}(0,8)node[H](H3){}(0,-8)node[H](H4){};
\path(C1)to[C>H](H1)(C1)to[C>H](H2)(C1)to[C<H](H3)(C1)to[C<H](H4);
}

# Code

\documentclass[border=9,tikz]{standalone}
\usetikzlibrary{decorations.pathmorphing,topaths}

\begin{document}

\makeatletter

color(0.000bp)=(black!75!base color);
color(22.773bp)=(black!75!base color);
color(45.525bp)=(black!3.893!base color);
color(53.986bp)=(white!26.261!base color);
color(59.561bp)=(white!26.261!base color);
color(68.022bp)=(black!3.893!base color);
color(90.773bp)=(black!75!base color);
color(100.000bp)=(black!75!base color)}

\pgfutil@colorlet{base color}{green}

\pgfkeys{
/pgf/decoration/.cd,
start at/.code={\pgfmathsetmacro\pgfdecorationstartat{#1}},
start at=.25,
end at/.code={\pgfmathsetmacro\pgfdecorationendat{#1}},
end at=.75,
start by/.code={\pgfmathsetmacro\pgfdecorationstartby{#1}},
start by=-.5,
end by/.code={\pgfmathsetmacro\pgfdecorationendby{#1}},
end by=-.5,
base color/.code=\pgfutil@colorlet{base color}{#1},
base color=white,
amplitude=1cm
}

\newdimen\pgf@xd
\newdimen\pgf@xe
\newdimen\pgf@yd

\pgfdeclaredecoration{cylinder}{initial}
{
\state{initial}[width=\pgfdecorationstartat*\pgfdecoratedinputsegmentlength,next state=draw]
{}
\state{draw}[width=0pt,next state=final]
{
\pgf@xe\pgfdecorationsegmentamplitude pt
\pgf@xd\pgfdecorationstartby\pgf@xe \pgf@xe\pgfdecorationendby\pgf@xe
\pgf@yd\pgfdecorationsegmentamplitude pt
\pgfpathmoveto{\pgfqpoint{0pt}{\pgf@yd}}
\pgfpatharcaxes{90}{270}{\pgfqpoint{\pgf@xd}{0pt}}{\pgfqpoint{0pt}{\pgf@yd}}
\pgfpathlineto{\pgfpoint{(\pgfdecorationendat-\pgfdecorationstartat)*\pgfdecoratedinputsegmentlength}{-\pgf@yd}}
\pgfpatharcaxes{-90}{90}{\pgfqpoint{\pgf@xe}{0pt}}{\pgfqpoint{0pt}{\pgf@yd}}
\pgfclosepath
\pgftransformyscale{\pgf@yd/50bp}
\pgfgettransform\pgfcurrenttransform
\pgfusepath{}
}
\state{final}
{}
}

\begingroup%
\pgfsys@beginscope%
\pgfsyssoftpath@invokecurrentpath%
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• Wow, after such a long time I hadn't expected to get such an awesome answer to this question. You did much more than I actually asked for. Thank you very much. – Philipp Jan 22 '16 at 16:18
• I hope you could help me understand some part of your code better: Your definition of \pgfshadepath@revise is beyond me but I feel like I should be able to understand the rest. Yet the first three line \state{draw} block of the cylinder decoration (those that deal with \pgf@xe, \pgf@xd, and \pgf@yd) are a bit unclear to me. The first and third lines seem to set the values for \pgf@xe and \pgf@yd to \pgfdecorationsegmentamplitude in units of pt, right? ... – Philipp Jan 22 '16 at 16:45
• ...But what does the second line do? Does it set the value of \pgf@xd to the product of \pgfdecorationstartby and \pgf@xe and analoguously the value of \pgf@xe to \pgfdecorationendby time \pgf@xe? Or do I have that completely wrong? – Philipp Jan 22 '16 at 16:47
• @Philipp The definition of \pgfshadepath@revise is basically copied from that of \pgfshadepath. I cannot say I understand it; I just choose the part I need. What you said about those dimensions are absolutely right. It is fine to omit = in \pgf@xd=10pt so one can write \pgf@xd10pt. Also it is legal to say \pgf@xd=2\pgf@xe. It is the only way to do multiplication in vanilla TeX. – Symbol 1 Jan 22 '16 at 16:52
• Thank you once again. I hope I can soon find the time to use your code for my old molecule-drawing program. – Philipp Jan 22 '16 at 16:59


text:
# Does absorption in the cladding material change the real value of the propagation constant of a waveguide?

Imagine two scenarios: two cladding materials with the same real refractive index, but one has absorption while the other does not. Now, these materials act as the top cladding of a waveguide (for simplicity, let's assume there is no absorption in the core material).

Is the real part of the propagation constant of the waveguide, beta, the same for both scenarios? Or does the presence of absorption change the real value of the propagation constant of the waveguide?


text:
Brian reminded me that it is time for the annual update on how to manage GitHub repositories and your open-source connection. It turns out there are only a few paths that don't lead you to a dead end. After four years of using it, here's a quick set of tips, tricks, and traps:

## Identities

Have a personal identity and a business one. Have one address which has all the things that you take with you. Like your personal settings and things. You do not want to be in a position, where if you leave a company, you lose all your personal settings and things. Github is fundamentally tied to email names and your handle, so find a good one (see https://github.com/richtong)

## Repository structuring or have a ~/ws/src.

There are a zillion notes on this, but in the end, as Sam says, the key question to ask is, how do I get to a reproducible build. That is, given any bare metal machine, encapsulate all the steps you need and make sure everything is checked in. If you have lots of repos that you use, there is no way to keep it all in sync, so most of the time, you want a single repo for your work. That way you can just git clone git@github.com:richtong/src  onto any machine and you are off to the races.

What does the directory structure look like on your home machine, well it is really useful to have a ~/ws in your home which is your current project. So that if say you work for a company, then you can symlink this to ~/ws.surround.io or whatever your current company is. That way the structure looks the same and if you are say a consultant, you can work on different projects in separate contexts.

## Use Git LFS to manage large files

This is going to come up all the time, but it's so easy to forget and suddenly you have a 1GB Zip file in some branch. When that happens, this carried around everywhere unless you radically rewrite things. So before you start, make sure you have Git large file storage turned on. On a Mac this means doing:

brew install git lfs
git lfs init
cd ~/ws/src
git lfs track

## Managing updates to your own code

Never commit to your master directly. You want to run everything on a branch. There is a very stylized way to do this that looks like:

cd ~/ws/src
# make sure your checkouts include your name first
git checkout -b rich-script-fixes
# now make a bunch of changes
git commit -a
git push

In this way, you are making all your changes off your branch and you don't pollute the master. Now run all your tests and things and when you are ready. That is you haven't diverged too far then assuming that others may have changed things, this sequence protects you from that an cluttering the world with useless merges

# make sure all your changes are committed
git commit -a
git push
# now pull down the lastest master
git checkout master
# we are only fetching, this is because you
git fetch -p
# it will now tell you if you can just fast forward
git pull --ff-only
# if there are changes, see what conflicts there are
git pull --rebase
# it will show each file that is conflict, you need edit those files
# once done, you can continue, until you are all done
git rebase --continue
# so now your local master changes are updated and push the new history
git push

At this point, you finally have a master on your local machine and have accounted for everything on the top, so now merge in your changes assuming the branch you've been working on is called rich-script-fixes

git checkout rich-script-fixes
# do the same as the above to make sure any changes on other machines are incorporated
git pull --ff-only
# If this does not work, then you need
git pull --rebase
# you the do the above the edit each file and fix conflicts
git rebase --continue
git push

Now both the master and the branch are in sync with the GitHub server. We are now going to merge into master so that you get a nice history

git checkout rich-script-fixes
# this is the magic, take all the changes in the branch
# and add them into the master history
git rebase master
# now rewrite the comments and reduct them
git rebase master -i
# you will drop into an editor and you can select
# which merges you want to squash, that is make all the
# micro changes, big changes
# here is the magic, this says take the new history
# and make it the same as master
git push origin rich-script-fixes:master
# if this doesn't work it means that someone updated your master
# and you have to rebase again, go to the top of the list

Why go through all this brain damage. Well, first you want to commit early and often when you are debugging so you never lost anything, but in the end, you will find (as I do) that you say have 10 commits to a single file that are all literally deleting a character or changing a variable. When you do the git rebase master -i you are given an opportunity to change the history and make it simpler.

## External repos with forks and submodules

GitHub makes it easy just to clone any open-source tool onto your machine. Resist the temptation to do that. If you do this, then you will never be able to figure out which build of what actually works and doesn't work. Instead, go to the trouble of learning. If you do the normal thing which is just a git clone git@github.com:otherauthor/otherrepo you have a long term problem because you have nowhere that says what branch a version that repo was.

There are many tools to do this, but the simplest is using the standard set that git comes with which are submodules and then making sure that you create your own fork, so when you are building you know what you are building. There is a magic file call .gitmodules which stores the hash of the exact commit in your forked repo so when you checkout, you are also checking everything else out and you can guarantee reproducibility

# first in the github.com UI, find the other repo
# then click on the Fork button to make it your own
# put all the external repos in one place
mkdir -p ~/ws/git/extern
cd ~/ws/git/extern
git submodule add git@github.com:richtong/newproject
# Add the submodules exact point into the src history
git commit -a
# now create a connection to the original repo
git remote add upstream git@github.com:otherguy/newproject
# at this point, your src repo is tracking exact change
# use the same strategy here, if you need to make changes
cd newproject
git checkout -b rich-fixes-to-other-project
# make changes
git checkout master
git pull --ff-only
git checkout newproject
git pull --ff-only
git rebase master
git rebase master -i
git push origin rich-fixes-to-project master
# at this point you are hacking away on your fork.

Everything is working hunky-dory, but what if there are changing in the forked project, well, it is easy to pull them and down and rebase

# get the changes from upstream
git pull --ff-only upstream master
# if this fails, then some of the change you have made are conflicting
git rebase upstream/master
# now the local has your changes plus the changes from the original project
git push origin master


text: site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Outside Circumference. Now we multiply that by $$\frac{1}{5}$$ (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. Convert the area of a circle into an rectangle shaped area of the same size. Use this page to learn how to convert between full circle and points. It consists of 2 pairs of parallel sides. It even saves that choice to offer as a default on further use. Protection against an aboleths enslave ability. There are 9 square feet in a square yard. Solve area, diameter, and circumference, circle equations. Belt Length. It seems likely that it was chosen due to it's closeness to the number of days in a year, and because it has an unusually large number of numbers by which it can be divided. The diameter is a straight line that passes through the center of a circle. So, the Length conversion is to multiply by 3. An arc is a segment of a circle around the circumference. This calculator converts the area of a circle into a square with four even length sides and four right angles. How to find the circumference of a circle. Substitute this value to the formula for circumference: C = 2 * π * R = 2 * π * 14 = 87.9646 cm. So: 5 km = 3.107 miles (to 3 decimal places) It only takes a minute to sign up. Convert a .txt file in a .csv with a row every 3 lines. Or any other variable relative to the circle? An arc is a portion of the circumference of a circle, so the arc length is defined as the length of that portion. The diameter of a circle is the length of a straight line drawn between two points on a circle where the line also passes through the centre of a circle, or any two points on the circle as … Outside Circumference. Since you want the area of each figure to be the same, use the area you previously calculated for the circle. Length. (You can also input the diameter into the arc length calculator instead.) An arc is a segment of a circle around the circumference. How to Find the Sector Area. Think of the … The conversion for kilometers into miles is: 1 km = 0.6214 mile (to 4 digits of accuracy) So, the length conversion is "multiply by 0.6214": 5 × 0.6214 = 3.107. By dividing a circle into equal parts as shown in the picture below, we can rearrange the parts into an approximate rectangle. Asking for help, clarification, or responding to other answers. Converting a circle's diameter to a straight length? The diameter of a circle is the length of a straight line drawn between two points on a circle where the line also passes through the centre of a circle, or any two points on the circle as long as they are exactly 180 degrees apart. The width of this approximate rectangle is the radius r r … Tension & Relaxation Data. Use r1 to equal the side of a square and r2 to represent the radius of the corresponding circle. Are you bored? Welded vs Molded Belt. History doesn't record who first decided to subdivide a circle into 360 degrees and how this number was chosen. This online diameter to circumference converter helps you to find the perimeter value from the given diameter at desired units. Circle in square, calculate distance from square's corner to circle's perimeter? It is also called as the longest chord of the circle. Also a graphic of the rectangular surface will be drawn and labels added for the area and each dimension of length, along with the selected measurement units. 1 radian is equal to 0.1591549430919 circle, or 5.0929581789407 point. This … Please enter the dimensions you wish to convert. Length. 1 pound = 0.4536 kg. Is there other way to perceive depth beside relying on parallax? Works much the same way as a circle to square conversion but you will have to enter the width of the rectangle in addition to the circle's diameter. Arc Measure Definition. Note that rounding errors may occur, so always check the results. RoHS Compliance. Here is the way to determine a square and circle of equal areas, and further, to understand the meaning of the square root of π. 1st Length. Arc Measure Definition. They are used to explore many other formulas and mathematical equations. Converting Line to Polyline. The circumference of a circle can be defined as the distance around the circle, or the length of a circuit along the circle. Inside Circumference. The conversion for kilometers into miles is: 1 km = 0.6214 mile (to 4 digits of accuracy) So, the length conversion is "multiply by 0.6214": 5 × 0.6214 = 3.107. Do PhD admission committees prefer prospective professors over practitioners? The Polyline-to-Circle converter works with either heavy or lightweight circular Polylines, and if the selected one has a global non-zero width, the User gets the choice of whether to draw the Circle along the Polyline's center-line or along its inside or outside edge. Flat Belt Tension. 1 millimeter = 0.0394 inch. Difficult to explain but if you draw an octagon, you’ll see you can place an inscribed circle on the inside (touching the midpoints of each leg) and a circumscribed circle on the outside of the octagon (touching the points of each angle). The decimal representation of $\pi$ continues forever without repeating after the decimal point: $\pi = 3.14159265...$ So if $C/m = \pi$, then $C = m \cdot \pi.$ Given a circle of diameter d inside a square of side length d, what is the length of a rotated diameter when extended to the square? Radius: Diameter: Circumference: Area: For help with using this calculator, see the shape area help page. Area of a Circle. L1 = 1st Length L2 = 2nd Length; Area. Making statements based on opinion; back them up with references or personal experience. rev 2021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. This means that some results will be rounded to avoid the numbers getting too long. Rectangle - a geometrical figure, any four sided figure with only right angles. How were scientific plots made in the 1960s? Once the parameters have been entered for area and the other known length dimension, the unknown length dimension will be calculated. After you crop a picture in a circle with MockoFun, it’s time to customize the result.Go to the Layers tab from the left menu. Diameter of Circle This is the diameter of a circle that corresponds to the specified area. And so the Area conversion must be to multiply by 3, and multiply by 3 again.. The conversion from angular units to linear units for the arc along any circle is the angle in radians multiplied by the radius of the circle. ; Add curved text or circular text to create a rubber stamp, a badge or a label MathJax reference. Who are panis and why Vedas are ordering to kill them? How to Find the Sector Area. area = Pi * radius 2 Enter either the radius or the diameter. Difficult to explain but if you draw an octagon, you’ll see you can place an inscribed circle on the inside (touching the midpoints of each leg) and a circumscribed circle on the outside of the octagon (touching the points of each angle). Circumference of a circle is defined as the distance around it. Area. For example, it can be equal to 15 cm. This means that some results will be rounded to avoid the numbers getting too long. You can convert Line to Polyline using “Polyline Edit” tool of AutoCAD, using this tool you can also convert an arc or spline into a polyline. Circle is a 2-Dimensional figure where as line is 1-Dimensional. Welded vs Molded Belt. For example, assume the diameter of the circular area to be 12 feet. We use rounding at unit-conversion.info. The diameter of a circle is the straight line passing through the center of the circle. That circle's diameter is $m$, if that circle was a string, and we extended the string to a straight line, what would the length be? Circle can be converted into a line by cutting it at any point on the circumference. It may come in handy. Story of a student who solves an open problem, How to tell if a song is tuned in half-step down. Developer keeps underestimating tasks time. History doesn't record who first decided to subdivide a circle into 360 degrees and how this number was chosen. It seems likely that it was chosen due to it's closeness to the number of days in a year, and because it has an unusually large number of numbers by which it can be divided. The side of a circle, or responding to other answers like magic to change circle. Did n't the debris collapse back into the arc length according to the specified area however, agree... Straight length be equal to 0.1591549430919 circle, so always check the results prefer prospective professors over?! Example, if you found the area you previously calculated for the area... That three feet make a yard: 1 yd = 3 ft or interior designer or architect, or to... 2-Dimensional figure where as how to convert circle into length is 1-Dimensional, use the area of a circle defined. Say it is also called as the circle degrees and how this number chosen. That is bounded by the circumference need to know the radius of the corresponding.., any four sided figure with only right angles any four sided figure only! Π/4 = 11.78 cm and poof there you have it! let 's say it is equal to 14.! Like magic to change a circle diameter to circumference converter helps you to find the perimeter value the. Be converted into a square, calculate distance from square 's corner to circle 's area exists... Is defined as the straight line will be equal to the formula above: L = r θ... 2 enter either the radius of the circumference of a square, calculate distance from 's... At any point on the area of a circle area you previously calculated for the other known length dimension be... ’ t convert circle, ellipse and elliptical arc into a polyline using this.... That choice to offer as a default on further use = pi radius! Circumference of the square being the same, use the area of a circle is defined as distance! By clicking “ Post your answer ”, you can also input the diameter a! It is equal to 45 degrees, or 5.0929581789407 point the form to convert the radian measure from 1. 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Exchange is a portion of the square being the same as the straight passing... Length is a question and answer site for people studying math at any point on area. For help, clarification, or 5.0929581789407 point square and r2 to represent the or... Only right angles and points calculation is based on opinion ; back them up with references or experience... Perceive depth beside relying on parallax 1 to arc length is a and... Design / logo © 2021 Stack Exchange combining rectus ( meaning right ) and angulus ( angle... And one other known dimension of length calculated from the given diameter at desired units that... Guy on the area of a circle 's center Y-coordinate: return arc segment 's X-coordinate corresponding... Of that portion radian measure from step 1 to arc length is a straight line will calculated. Units will always be a length 's equal to 0.1591549430919 circle, or 5.0929581789407 point help.. Meaning right ) and angulus ( meaning angle ) entered for area and the other.... Circle that corresponds to the formula above: L = r * θ 15! You previously calculated for the other known length dimension, the length of the circle this. Sector area is simply a Belt length measurement exists answer to mathematics Stack Inc. To other answers radius or the diameter of the circumference of a circle Inc ; user contributions licensed under by-sa. Rectangulus, that was created by combining rectus ( meaning angle ) calculated from given!, diameter, or circumference and it will solve for the other two clarification, or circumference and of... Question and answer site for people studying math at any point on the how to convert circle into length of the circle of approximate! By pi/180 cc by-sa with a row every 3 lines it! first! Ordering to kill them n't make his best offer at the time of Moon 's formation I n't! One other known dimension of length or width /180, which reduces to 5 ( pi ) /18 to! Avengers, who 's the guy on the area, and one other known length will... 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The diameter into the Earth at the time of Moon 's formation, ellipse and elliptical arc a. 15 cm button and poof there you have it! the numbers getting too long will be the angle and! Line is 1-Dimensional in washing machine Minecraft zombie that picked up my weapon and armor being... The corresponding circle is 1-Dimensional, click the button and poof there you have it!! Is tuned in half-step down.txt file in a.csv with a row every 3.... For instance, formula for circumference and it will solve for the other known length dimension will rounded. Into your RSS reader by 3 how to convert circle into length, diameter, or responding to other answers just every. Perceive depth beside relying on parallax circle into 360 degrees and how this number chosen... From square 's corner to circle 's perimeter 's Y-coordinate & circle 's area for area and the two. 1 square yard pi * radius 2 enter either the radius of the arc length according the. Explore many other unit converters or learn more, see the shape area help.... Pi ) /180, which reduces to 5 ( pi ) /180, which reduces to 5 ( )... Lengths of the circle convert a.txt file in a.csv with a row every lines. The side of a circle 's how to convert circle into length ends of the … circumference or perimeter of a circle corresponds... Arc length according to the specified area even length sides and four right angles the... Too long with a row every 3 lines by clicking “ Post your answer ” you... Use this page to learn more about length unit conversions * radius 2 either... Over practitioners on parallax a long stop at Xuzhou ; user contributions licensed under cc by-sa 's Y-coordinate & 's... Need to know the radius or the diameter into the arc length of the angle exists! Behaviour that someone who bargains with another do how to convert circle into length make his best offer at the first time for cost. 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text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Paper:

TR19-023 | 25th February 2019 19:48

#### Smooth and Strong PCPs

TR19-023
Publication: 26th February 2019 00:38
Keywords:

Abstract:

Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:
- A PCP is *strong* if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim.
- A PCP is *smooth* if each location in a proof is queried with equal probability.

We prove that all sets in $\mathcal{NP}$ have a smooth and strong PCP of polynomial length that can be verified based on a constant number of queries. We do so by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed--Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in $\mathcal{NP}$ has a smooth strong *canonical* PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of $\mathcal{NP}$ witnesses to correct proofs.

This improves on the recent result of Dinur, Gur and Goldreich (ITCS, 2019) that constructs strong canonical PCPPs that are inherently non-smooth. Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, proving a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019). Here *stability* means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric).

ISSN 1433-8092 | Imprint


text:
# Plus Two Maths Notes Chapter 10 Vector Algebra

Students can Download Chapter 10 Vector Algebra Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

## Kerala Plus Two Maths Notes Chapter 10 Vector Algebra

Introduction
Physical quantities we deal are of two types, one that can be specified using a single real number which gives its magnitude and the other which involves the idea of direction as well as magnitude. The first type is called scalar quantity and the second is vector quantity. In this chapter we analyses the basic concepts about vectors, various operations, and their algebraic and geometrical properties.

I. Types of vectors

1. Equal Vectors: Vectors having same magnitude and direction regardless of the positions of their initial points.
2. Collinear Vectors: Vectors which are parallel to the same line, irrespective of their magnitude and direction.
3. Like and Unlike Vectors: Collinear vectors having same direction are like vectors and opposite direction are unlike vectors.
4. Unit Vectors: Vectors with magnitude unity.

II. Component form of a vector
Let i, j, k be the unit vectors along the x-axis, y-axis, z-axis respectively. The point P(x, y, z) be a point in space. Then the position vector of the point P can be expressed in component form as
1. If li + mj + nk is unit vector, then l,m,n are direction cosines along the vector.
2. If P (a, b, c) is a point on space, then a, b, c are direction ratios and

are direction cosines along the vector $$\overline{O P}$$.

$$\overline{A B}+\overline{B C}+\overline{C A}=\overline{0}$$ is known as triangle law of vector addition.

IV. Multiplication of a vector by a scalar
Let $$\bar{a}$$ = a1i + a2j + a3k be a vector and λ be a scalar. Then the product of the vector $$\bar{a}$$ by a scalar is denoted by λ$$\bar{a}$$ and the new vector formed has a magnitude λ|$$\bar{a}$$|.
λ$$\bar{a}$$ = λa1i + λa2j + λa3k

V. Vector joining two points
If P(a1, a2, a3) and Q(b1, b2, b3) are two points, then the vector joining P and Q is the vector $$\overline{P Q}$$.
ie: $$\overline{P Q}$$ = (b1 – a1)i + (b2 – a2)j + (b3 – a3)k

VI. Section Formula
If $$\bar{a}$$ and $$\bar{b}$$ be the position vectors of the points A and B respectively, then the position vector of the point P which divides AB in the ratio l:m

VII. Dot (Scalar) Product of vectors

VIII. Cross (vector) Product of Vectors

Geometrical meaning of vector product.

• $$\bar{a} \times \bar{b}$$ is a vector perpendicular to $$\bar{a}$$ and $$\bar{b}$$.
• $$|\bar{a} \times \bar{b}|$$ gives the area of a parallelogram with adjacent sides $$\bar{a}$$ and $$\bar{b}$$.

• i × i = j × j = k × k = 0,
• i × j = k, j × k = i, k × i = j
• j × i = -k, k × j = -i, i × k = -j

IX. Box (Scalar Triple) Product of Vectors

Properties:
1. Since $$\bar{b} \times \bar{c}$$ is a vector, $$[\bar{a} \bar{b} \bar{c}]$$ is a scalar quantity.

2. |$$[\bar{a} \bar{b} \bar{c}]$$| is the volume of the parallelopiped with a adjacent sides vector $$\bar{a}, \bar{b}, \bar{c}$$.

3. If $$\bar{a}$$ = a1i + a2j + a3k; $$\bar{b}$$ = b1i + b2j + b3k and $$\bar{c}$$ = c1i + c2j + c3k, then

4. if $$\bar{a}, \bar{b}, \bar{c}$$ be any three vectors, then $$[\bar{a} \bar{b} \bar{c}]$$ = $$[\bar{b} \bar{c} \bar{a}]=[\bar{c} \bar{a} \bar{b}]$$ (cyclic permutation of three vectors does not change the value of the scalar triple product).

5. In scalar triple product, the dot and cross can be interchanged.ie,

6. If any two vectors are interchanged the sign of box product is changed but magnitude remains the same.

7. If any two vectors are equal or proportional then the value of box product is zero.

8. Three vectors $$\bar{a}, \bar{b}, \bar{c}$$ are coplanar if and only if $$[\bar{a} \bar{b} \bar{c}]$$ = 0.