Well, wouldn't the program itself be an input on which a human is unable to determine the result (i.e., if the program halts)? I'm curious on your thoughts here, maybe there's something here I'm missing.
The function we are trying to compute is undecidable. Sure we as humans understand that there's a dichotomy here: if the program halts it won't halt; if it doesn't halt it will halt. But the function we are asked to compute must have one output on a given input. So a human, when given this program as input, is also unable to assign an output.
So humans also can't solve the halting problem, we are just able to recognize that the problem is undecidable.
it's a _necessary_ but _not sufficient_ condition for a recession. so it does provide you information. if the yield curve does not invert, it is unlikely there will be a recession.
lepton number does not need to be conserved. it is an approximate symmetry of nature. if lepton number were conserved, neutrinos could not oscillate.
quarks, the particles composing protons and neutrons, have fractional charge; these would be the particles that would interact with an electron or positron. the charges wouldn't work out (charge is conserved (as far as we known...)) so there wouldn't be a fundamental electromagnetic interaction between a single quark and a e+/e- (i.e., an annihilation). But there are fundamental weak interactions between quarks and e+/e-; these processes are known as inverse beta decay and are used for pet scans.
mesons are composed of two quarks and can have neutral or non-neutral charge. baryons are composed of three quarks and can have neutral or non-neutral charge. both mesons and baryons are classified as hadrons
charge, baryon number, isospin and strangeness are all related by the Gell-Mann-Nishijima formula[1].
the overarching principle underlying this is that in quantum chromodynamics, the theory of the strong interaction, quarks are representations of an underlying SU(3) gauge group containing color charge (a quantum number) and electric charge quantized as -1/3e and +2/3e. QCD is a gauge theory (like all other theories of nature) and the gauge boson, the gluon, also pops out of demanding the theory be invariant under local SU(3) gauge transformations.
that said, no one has any clue why all of our physical theories are gauge theories :-)
I'm not sure what courses they had in mind, but Victor Shoup (one of the authors of the OP) has a book on number theory and algebra that goes over probability. That would probably be most useful if your goal is to study the applied crypto book.
The prerequisites are (self-reported) minimum, just calculus and mathematical maturity should be sufficient. I would check it out (it's free) and see if it's at an appropriate level.
Unfortunately I've yet to come across an introductory text or course on probability that is actually good :-(
a huge part of it is trying to solve / formulate easier versions of the problem, or problems that are similar or related to the original problem. Or making some stronger assumptions to get rid of the clutter / all of the moving variables and distill it down to the smallest form a human brain can handle! For many of the "huge" unsolved problems, there tends to be a program of "dominos" or "ledges" you hope to work on in some order that will make the original problem fall.
that way you don't just meander idolly from day to day, but instead gain some intuition for the central problem (and of course have publishable work to appease the grant gods / the university).
trying to code up some of the work to experiment is also useful, but that can be a research problem of its own :-)
but then the heap would grow down in such a diagram...
btw my understanding is that the heap came first in the logical development of C and other systems programming. so it made sense to have .text and other program data at the lowest virtual memory addresses and then have the heap grow towards higher memory addresses.
then when the stack became a thing it had to grow down...
a pattern that i converged on --- at least in postgres --- is to aggregate your data into json objects and then go from there. you don't need to know how many attributes (columns) should be in the result of your pivot. you can also do this in reverse (pivot from wide to long) with the same technique.
so for example if you have the schema `(obj_id, key, value)` in a long-formatted table, where an `obj_id` will have data spanning multiple rows, then you can issue a query like
```
SELECT obj_id, jsonb_object_agg(key, value) FROM table GROUP BY obj_id;
```
up to actual syntax...it's been awhile since i've had to do a task requiring this, so details are fuzzy but pattern's there.
so each row in your query result would look like a json document: `(obj_id, `{"key1": "value", "key2": "value", ...})`
XOR is used a ton in the theoretical underpinnings of cryptography. It's used in the one time pad which is essentially the "smallest" cryptographic scheme that is perfectly secure (perfectly secure has a mathematical definition in this context, it's not saying there can never be any attacks).
In general the reason why is that if you have two random variables x and y, where x has any distribution (so for example x could even be "attack normandy on june 6" with certainty) and y is uniformly distributed across all n-bit strings (so it could be any string of n zeros and ones with equal probability), then you can show that x ^ y appears as if it is also uniformly distributed across all n-bit strings as well.
Because of this property it's used frequently in many higher order methods as well.
i second tristan needham's visual complex analysis.
some other good ones:
* the one we used in my undergrad course was fisher's complex variables which is great if you're learning for the purposes of applications. it's a cheap dover book.
* rudin's real and complex analysis (if theory is your thing. note that rudin's books, while great, do require a good background in math).
* as the article mentions, eli stein has a series of books on the four main branches of analysis. i believe the second book is on complex analysis.
you might be able to do something like "predict the next prime number" or "predict the next zero of the Riemann zeta function".
you could try something like this for statements in a formal axiomatic system, but know that you're running up against things like the halting problem / entscheidungsproblem / godel incompleteness. so it may be possible to train a neural net to decide the veracity of a statement and do so more quickly than a human might, but you would inevitably be running up against things that are truly undecidable in nature. which is not like go or chess where although they are difficult, they are decidable.
think more like you're on the surface of the balloon as it's getting inflated. the "fabric" or elastic material on which you're standing is the thing that's expanding while the balloon inflates. you're still constrained to the surface of the balloon, however.
Incidentally one of the biggest benefits I see of using a text editor like vim / emacs is that it really encourages good code management.
It's not to save the ~10 minutes per year in faster key strokes to manipulate your code. It's about the way it shapes your thinking about how you code.
not necessarily. it could. but if x and y are truly uncorrelated variables that are observed with unlimited precision, then you still would not reject the null that they are uncorelated
i think it's largely culture. academics are "graded" based on how many papers they publish. depending on your geography, quality matters too.
but in most cases they aren't measured by how much non-peer reviewed publications they have.
there is also a technical barrier that works both ways. most don't have the capability to have a regularly updated website with content that otherwise would have been put in the file drawer. and the other side of that coin is what audience will actually find / read it online.
twitter is becoming the defacto medium of dissemination, however, so that may bode well for promoting other types of publishing medium.
edit
ALSO note that there are some academics who publish high-quality blogs. i'm thinking of [murat demirbas](http://muratbuffalo.blogspot.com/). i'm not familiar with any that publish actual results, however. the threat would be someone else might "steal" the idea and publish it elsewhere, particularly in competitive fields like bio.
You are right that Raft and Paxos provide for distributed consensus. However, they don't do so in an open, byzantine environment where some fraction of the nodes in the network may not be honest.
edit: there are byzantine variants of paxos --- i'm not 100% familiar with these (and tbh i'm take it or leave it on the whole thing), just wanted to note that i think the parent was trying to get at something slightly different than consensus in an environment with nodes that fail, i.e. "traditional" consensus algorithms
The function we are trying to compute is undecidable. Sure we as humans understand that there's a dichotomy here: if the program halts it won't halt; if it doesn't halt it will halt. But the function we are asked to compute must have one output on a given input. So a human, when given this program as input, is also unable to assign an output.
So humans also can't solve the halting problem, we are just able to recognize that the problem is undecidable.