These kinds of issues can be incredibly disruptive and distressing for non tech-savy users. You update your OS and suddenly it looks like a lot of your data are corrupted, with no explanation of how to get it back.
Forcing saving to OneDrive causes this issue a lot too. I was stunned to find that saving changes to an existing document will often try to save a new file in OneDrive instead. So if you don't notice this and go back to your original file, it will look like your changes weren't saved.
> If the proof is correct, Aristotle has a good chance at translating it into Lean
How does this depend on the area of mathematics of the proof? I was under the impression that it was still difficult to formalize most research areas, even for a human. How close is Aristotle to this frontier?
What do you think about the market for custom apps? Like one app, one customer? You describe future businesses as having one app/service and using AI to add more features, but you did something very different for your wife with AI and it sounds like it added a lot of value.
So much of the conversation is around these models replacing software engineers. But the use cases described in the article sound like pretty compelling business opportunities; if the custom apps he built for his wife's business have been useful, probably there are lots of businesses that would pay for the service he just provided his wife. Small, custom apps can be made way more cheaply now, so Jeven's paradox says that demand should go up. I think it will.
I would love to hear from some freelance programmers how LLMs have changed their work in the last two years.
One problem with the idea of making businesses out of this kind of application is actually mentioned in passing in the article
"I decided to make up for my dereliction of duties by building her another app for her sign business that would make her life just a bit more delightful - and eliminate two other apps she is currently paying for"
OP used Opus to re-write existing applications that his wife was paying for. So now any time you make a commercial app and try to sell it, you're up against everyone with access to Opus or similar tooling who can replicate your application, exactly to their own specifications.
so everybody is making their own apps for their specific problem? Sounds as it will get a mess in the end. So maybe it will be more about ideas and concepts and not so much about know how to code.
Yep vast numbers of personalized apps seems like it would end up being pretty messy. I think the challenge of betting on ideas and concepts is that once you've published something, someone else can take the idea and replicate it easily and cheaply, so it'll be harder to monetize unless you can come up with something that's hard to replicate.
I think you're misunderstanding my point. If you can crank out a custom app this quickly, you don't make a commercial app and then try to sell it on an app store. Customers pay you to make apps for their specific usecase. One app, one customer. And if a week later they want some new features, they pay you (or another freelancer) to add it.
Put another way, we programmers have the luxury of being able to write custom scripts and apps for ourselves. Now that these things are getting way cheaper to build, there should be a growing market that makes them available to more people.
Why do they pay you though, why not just do it themselves? With improving models and surrounding tooling the barrier to creating apps is lowered, and it's easier for a user just to create their own app, no 3rd party person needed.
- I try to get outside and go for a short walk. Even a cloudy sky is actually much much brighter than the lights inside a building, so it has a stronger affect on maintaining your circadian rhythm than indoor lights. It's also a good time to zoom out and reevaluate priorities. Or just enjoy looking at the sky.
- Read and revise my todo list. I find I actually spend a huge amount of time thinking about my to do list, and I think it pays off. One better decision of what to work on can save days, lead to new ideas, or even completely change the course of what I work on.
- Sometimes I guess I manage to squash a bug or complete some other minor task, but probably I'm more productive if I just use that time to think rather than context switch twice and rush some minor task.
- The big exception is if I can get something started that will run on its own for a long time. Then 10 spare minutes can save hours.
Least squares is guaranteed to be convex [0]. At least for linear fit functions there is only one minimum and gradient descent is guaranteed to take you there (and you can solve it with a simple matrix inversion, which doesn't even require iteration).
Intuitively this is because a multidimensional parabola looks like a bowl, so it's easy to find the bottom. For higher powers the shape can be more complicated and have multiple minima.
But I guess these arguments are more about making the problem easy to solve. There could be applications where higher powers are worth the extra difficulty. You have to think about what you're trying to optimize.
Guessing the original comment hasn't taken complex analysis or has some other oriented view point into geometry that gives them satisfaction but these expressions are one of the most incredible and useful tools in all of mathematics (IMO). Hadn't seen another comment reinforcing this so thank you for dropping these.
Cauchy path integration feels like a cheat code once you fully imbibe it.
Got me through many problems that involves seemingly impossible to memorize identities and re-derivation of complex relations become essentially trivial
Complex exponentials and complex logarithms are useful in some symbolic computations, those involving formulae for derivatives or primitives, and this is indeed the only application where the use of e^x and natural logarithm is worthwhile.
However, whenever your symbolic computation produces a mathematical model that will be used for numeric computations, i.e. in a computer program, it is more efficient to replace all e^x exponentials and natural logarithms with 2^x exponentials and binary logarithms, instead of retaining the complex exponentials and logarithms and evaluating them directly.
At the same time, it is also preferable to replace the trigonometric functions of arguments measured in radians with trigonometric functions of arguments measured in cycles (i.e. functions of 2*Pi*x).
This replacement eliminates the computations needed for argument range reduction that otherwise have to be made at each function evaluation, wasting time and reducing the accuracy of the results.
Even when you use the exponential e^x and the hyperbolic logarithm a.k.a. natural logarithm (which are useful only in symbolic computations and are inferior for any numeric computation), you never need to know the value of "e". The value itself is not needed for anything. When evaluating e^x or the hyperbolic logarithm you need only ln 2 or its inverse, in order to reduce the argument of the functions to a range where a polynomial approximation can be used to compute the function.
Moreover, you can replace any use of e^x with the use of 2^x, which inserts ln(2) constants in various places, (but removes ln 2 from the evaluations of exponentials and logarithms, which results in a net gain).
If you use only 2^x, you must know that its derivative is ln(2) * 2^x, and knowing this is enough to get rid of "e" anywhere. Even in derivation formulae, in actual applications most of the multiplications with ln 2 can be absorbed in multiplications with other constants, as you normally do not have 2^x expressions that are derived, but 2^(a*x), where you do ln(2)*a at compile time.
You start with the formula for the exponential of an imaginary argument, but there the use of "e" is just a conventional notation. The transcendental number "e" is never used in the evaluation of that formula and also none of the numbers produced by computing an exponential or logarithm of real numbers are involved in that formula.
The meaning of that formula is that if you take the expansion series of the exponential function and you replace in it the argument with an imaginary argument you obtain the expansion series for the corresponding trigonometric functions. The number "e" is nowhere involved in this.
Moreover, I consider that it is far more useful to write that formula in a different way, without any "e":
1^x = cos(2Pi*x) + i * sin(2Pi*x)
This gives the relation between the trigonometric functions with arguments measured in cycles and the unary exponential, whose argument is a real number and whose value is a complex number of absolute value equal to 1, and which describes the unit circle in the complex plane, for increasing arguments.
This formula appears more complex only because of using the traditional notation. If you call cos1 and sin1 the functions of period 1, then the formula becomes:
1^x = cos1(x) + i * sin1(x)
The unary exponential may appear weirder, but only because people are habituated from school with the exponential of imaginary arguments instead of it. None of these 2 functions is weirder than the other and the use of the unary exponential is frequently simpler than of the exponential of imaginary arguments, while also being more accurate (no rounding errors from argument range reduction) and faster to compute.
I want to add that any formula that contains exponentials of real arguments, e^x, and/or exponentials of imaginary arguments, e^(i*x), can be rewritten by using only binary exponentials, 2^x, and/or unary exponentials, 1^x, both having only real arguments.
With this substitution, some formulae become simpler and others become more complicated, but, when also considering the cost of the function evaluations, an overall greater simplicity is achieved.
In comparison with the "e" based exponentials, the binary exponential and the unary exponential and their inverses have the advantage that there are no rounding errors caused by argument range reduction, so they are preferable especially when the exponents can be very big or very small, while the "e" based exponentials can work fine for exponents guaranteed to be close to 0.
The air quality issue alone is mind-boggling. The air quality index nominally tops out at 500, corresponding to 'hazardous.' Major Indian cities blow past this threshold on a regular basis in the winter months. In Delhi, poor air quality is responsible for one in seven deaths annually [0]. People born in Delhi now are estimated to lose 8-12 years in life expectancy, depending on the study [1]. This is the norm for now, but it's hard to imagine how much worse things can get.
I was in India for a wedding a few years back and spent a couple days in New Delhi. I remember stepping out into the 6AM brisk morning air and feeling like I was going to cough up a lung.
It tasted like what I imagine a finely aged glass of acid rain would taste like.
You know how when you open the weather app on your phone, in normal places it says things like: sunny, cloudy, rainy? The weather app just showed SMOKE (this was an actual weather report).
This is partially a result of agricultural burning in the surrounding states which is one of the fastest (and cheapest) ways to clear out the fields for the next crop.
Forcing saving to OneDrive causes this issue a lot too. I was stunned to find that saving changes to an existing document will often try to save a new file in OneDrive instead. So if you don't notice this and go back to your original file, it will look like your changes weren't saved.
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