More on Science

Adam Frank
3 years ago
Humanity is not even a Type 1 civilization. What might a Type 3 be capable of?
The Kardashev scale grades civilizations from Type 1 to Type 3 based on energy harvesting.
How do technologically proficient civilizations emerge across timescales measuring in the tens of thousands or even millions of years? This is a question that worries me as a researcher in the search for “technosignatures” from other civilizations on other worlds. Since it is already established that longer-lived civilizations are the ones we are most likely to detect, knowing something about their prospective evolutionary trajectories could be translated into improved search tactics. But even more than knowing what to seek for, what I really want to know is what happens to a society after so long time. What are they capable of? What do they become?
This was the question Russian SETI pioneer Nikolai Kardashev asked himself back in 1964. His answer was the now-famous “Kardashev Scale.” Kardashev was the first, although not the last, scientist to try and define the processes (or stages) of the evolution of civilizations. Today, I want to launch a series on this question. It is crucial to technosignature studies (of which our NASA team is hard at work), and it is also important for comprehending what might lay ahead for mankind if we manage to get through the bottlenecks we have now.
The Kardashev scale
Kardashev’s question can be expressed another way. What milestones in a civilization’s advancement up the ladder of technical complexity will be universal? The main notion here is that all (or at least most) civilizations will pass through some kind of definable stages as they progress, and some of these steps might be mirrored in how we could identify them. But, while Kardashev’s major focus was identifying signals from exo-civilizations, his scale gave us a clear way to think about their evolution.
The classification scheme Kardashev employed was not based on social systems of ethics because they are something that we can probably never predict about alien cultures. Instead, it was built on energy, which is something near and dear to the heart of everybody trained in physics. Energy use might offer the basis for universal stages of civilisation progression because you cannot do the work of establishing a civilization without consuming energy. So, Kardashev looked at what energy sources were accessible to civilizations as they evolved technologically and used those to build his scale.
From Kardashev’s perspective, there are three primary levels or “types” of advancement in terms of harvesting energy through which a civilization should progress.
Type 1: Civilizations that can capture all the energy resources of their native planet constitute the first stage. This would imply capturing all the light energy that falls on a world from its host star. This makes it reasonable, given solar energy will be the largest source available on most planets where life could form. For example, Earth absorbs hundreds of atomic bombs’ worth of energy from the Sun every second. That is a rather formidable energy source, and a Type 1 race would have all this power at their disposal for civilization construction.
Type 2: These civilizations can extract the whole energy resources of their home star. Nobel Prize-winning scientist Freeman Dyson famously anticipated Kardashev’s thinking on this when he imagined an advanced civilization erecting a large sphere around its star. This “Dyson Sphere” would be a machine the size of the complete solar system for gathering stellar photons and their energy.
Type 3: These super-civilizations could use all the energy produced by all the stars in their home galaxy. A normal galaxy has a few hundred billion stars, so that is a whole lot of energy. One way this may be done is if the civilization covered every star in their galaxy with Dyson spheres, but there could also be more inventive approaches.
Implications of the Kardashev scale
Climbing from Type 1 upward, we travel from the imaginable to the god-like. For example, it is not hard to envisage utilizing lots of big satellites in space to gather solar energy and then beaming that energy down to Earth via microwaves. That would get us to a Type 1 civilization. But creating a Dyson sphere would require chewing up whole planets. How long until we obtain that level of power? How would we have to change to get there? And once we get to Type 3 civilizations, we are virtually thinking about gods with the potential to engineer the entire cosmos.
For me, this is part of the point of the Kardashev scale. Its application for thinking about identifying technosignatures is crucial, but even more strong is its capacity to help us shape our imaginations. The mind might become blank staring across hundreds or thousands of millennia, and so we need tools and guides to focus our attention. That may be the only way to see what life might become — what we might become — once it arises to start out beyond the boundaries of space and time and potential.
This is a summary. Read the full article here.

Katherine Kornei
3 years ago
The InSight lander from NASA has recorded the greatest tremor ever felt on Mars.
The magnitude 5 earthquake was responsible for the discharge of energy that was 10 times greater than the previous record holder.
Any Martians who happen to be reading this should quickly learn how to duck and cover.
NASA's Jet Propulsion Laboratory in Pasadena, California, reported that on May 4, the planet Mars was shaken by an earthquake of around magnitude 5, making it the greatest Marsquake ever detected to this point. The shaking persisted for more than six hours and unleashed more than ten times as much energy as the earthquake that had previously held the record for strongest.
The event was captured on record by the InSight lander, which is operated by the United States Space Agency and has been researching the innards of Mars ever since it touched down on the planet in 2018 (SN: 11/26/18). The epicenter of the earthquake was probably located in the vicinity of Cerberus Fossae, which is located more than 1,000 kilometers away from the lander.
The surface of Cerberus Fossae is notorious for being broken up and experiencing periodic rockfalls. According to geophysicist Philippe Lognonné, who is the lead investigator of the Seismic Experiment for Interior Structure, the seismometer that is onboard the InSight lander, it is reasonable to assume that the ground is moving in that area. "This is an old crater from a volcanic eruption."
Marsquakes, which are similar to earthquakes in that they give information about the interior structure of our planet, can be utilized to investigate what lies beneath the surface of Mars (SN: 7/22/21). And according to Lognonné, who works at the Institut de Physique du Globe in Paris, there is a great deal that can be gleaned from analyzing this massive earthquake. Because the quality of the signal is so high, we will be able to focus on the specifics.

Will Lockett
3 years ago
Thanks to a recent development, solar energy may prove to be the best energy source.
Perovskite solar cells will revolutionize everything.
Humanity is in a climatic Armageddon. Our widespread ecological crimes of the previous century are catching up with us, and planet-scale karma threatens everyone. We must adjust to new technologies and lifestyles to avoid this fate. Even solar power, a renewable energy source, has climate problems. A recent discovery could boost solar power's eco-friendliness and affordability. Perovskite solar cells are amazing.
Perovskite is a silicon-like semiconductor. Semiconductors are used to make computer chips, LEDs, camera sensors, and solar cells. Silicon makes sturdy and long-lasting solar cells, thus it's used in most modern solar panels.
Perovskite solar cells are far better. First, they're easy to make at room temperature, unlike silicon cells, which require long, intricate baking processes. This makes perovskite cells cheaper to make and reduces their carbon footprint. Perovskite cells are efficient. Most silicon panel solar farms are 18% efficient, meaning 18% of solar radiation energy is transformed into electricity. Perovskite cells are 25% efficient, making them 38% more efficient than silicon.
However, perovskite cells are nowhere near as durable. A normal silicon panel will lose efficiency after 20 years. The first perovskite cells were ineffective since they lasted barely minutes.
Recent research from Princeton shows that perovskite cells can endure 30 years. The cells kept their efficiency, therefore no sacrifices were made.
No electrical or chemical engineer here, thus I can't explain how they did it. But strangely, the team said longevity isn't the big deal. In the next years, perovskite panels will become longer-lasting. How do you test a panel if you only have a month or two? This breakthrough technique needs a uniform method to estimate perovskite life expectancy fast. The study's key milestone was establishing a standard procedure.
Lab-based advanced aging tests are their solution. Perovskite cells decay faster at higher temperatures, so scientists can extrapolate from that. The test heated the panel to 110 degrees and waited for its output to reduce by 20%. Their panel lasted 2,100 hours (87.5 days) before a 20% decline.
They did some math to extrapolate this data and figure out how long the panel would have lasted in different climates, and were shocked to find it would last 30 years in Princeton. This made perovskite panels as durable as silicon panels. This panel could theoretically be sold today.
This technology will soon allow these brilliant panels to be released into the wild. This technology could be commercially viable in ten, maybe five years.
Solar power will be the best once it does. Solar power is cheap and low-carbon. Perovskite is the cheapest renewable energy source if we switch to it. Solar panel manufacturing's carbon footprint will also drop.
Perovskites' impact goes beyond cost and carbon. Silicon panels require harmful mining and contain toxic elements (cadmium). Perovskite panels don't require intense mining or horrible materials, making their production and expiration more eco-friendly.
Solar power destroys habitat. Massive solar farms could reduce biodiversity and disrupt local ecology by destroying vital habitats. Perovskite cells are more efficient, so they can shrink a solar farm while maintaining energy output. This reduces land requirements, making perovskite solar power cheaper, and could reduce solar's environmental impact.
Perovskite solar power is scalable and environmentally friendly. Princeton scientists will speed up the development and rollout of this energy.
Why bother with fusion, fast reactors, SMRs, or traditional nuclear power? We're close to developing a nearly perfect environmentally friendly power source, and we have the tools and systems to do so quickly. It's also affordable, so we can adopt it quickly and let the developing world use it to grow. Even I struggle to justify spending billions on fusion when a great, cheap technology outperforms it. Perovskite's eco-credentials and cost advantages could save the world and power humanity's future.
You might also like

Dmitrii Eliuseev
2 years ago
Creating Images on Your Local PC Using Stable Diffusion AI
Deep learning-based generative art is being researched. As usual, self-learning is better. Some models, like OpenAI's DALL-E 2, require registration and can only be used online, but others can be used locally, which is usually more enjoyable for curious users. I'll demonstrate the Stable Diffusion model's operation on a standard PC.
Let’s get started.
What It Does
Stable Diffusion uses numerous components:
A generative model trained to produce images is called a diffusion model. The model is incrementally improving the starting data, which is only random noise. The model has an image, and while it is being trained, the reversed process is being used to add noise to the image. Being able to reverse this procedure and create images from noise is where the true magic is (more details and samples can be found in the paper).
An internal compressed representation of a latent diffusion model, which may be altered to produce the desired images, is used (more details can be found in the paper). The capacity to fine-tune the generation process is essential because producing pictures at random is not very attractive (as we can see, for instance, in Generative Adversarial Networks).
A neural network model called CLIP (Contrastive Language-Image Pre-training) is used to translate natural language prompts into vector representations. This model, which was trained on 400,000,000 image-text pairs, enables the transformation of a text prompt into a latent space for the diffusion model in the scenario of stable diffusion (more details in that paper).
This figure shows all data flow:
The weights file size for Stable Diffusion model v1 is 4 GB and v2 is 5 GB, making the model quite huge. The v1 model was trained on 256x256 and 512x512 LAION-5B pictures on a 4,000 GPU cluster using over 150.000 NVIDIA A100 GPU hours. The open-source pre-trained model is helpful for us. And we will.
Install
Before utilizing the Python sources for Stable Diffusion v1 on GitHub, we must install Miniconda (assuming Git and Python are already installed):
wget https://repo.anaconda.com/miniconda/Miniconda3-py39_4.12.0-Linux-x86_64.sh
chmod +x Miniconda3-py39_4.12.0-Linux-x86_64.sh
./Miniconda3-py39_4.12.0-Linux-x86_64.sh
conda update -n base -c defaults condaInstall the source and prepare the environment:
git clone https://github.com/CompVis/stable-diffusion
cd stable-diffusion
conda env create -f environment.yaml
conda activate ldm
pip3 install transformers --upgradeDownload the pre-trained model weights next. HiggingFace has the newest checkpoint sd-v14.ckpt (a download is free but registration is required). Put the file in the project folder and have fun:
python3 scripts/txt2img.py --prompt "hello world" --plms --ckpt sd-v1-4.ckpt --skip_grid --n_samples 1Almost. The installation is complete for happy users of current GPUs with 12 GB or more VRAM. RuntimeError: CUDA out of memory will occur otherwise. Two solutions exist.
Running the optimized version
Try optimizing first. After cloning the repository and enabling the environment (as previously), we can run the command:
python3 optimizedSD/optimized_txt2img.py --prompt "hello world" --ckpt sd-v1-4.ckpt --skip_grid --n_samples 1Stable Diffusion worked on my visual card with 8 GB RAM (alas, I did not behave well enough to get NVIDIA A100 for Christmas, so 8 GB GPU is the maximum I have;).
Running Stable Diffusion without GPU
If the GPU does not have enough RAM or is not CUDA-compatible, running the code on a CPU will be 20x slower but better than nothing. This unauthorized CPU-only branch from GitHub is easiest to obtain. We may easily edit the source code to use the latest version. It's strange that a pull request for that was made six months ago and still hasn't been approved, as the changes are simple. Readers can finish in 5 minutes:
Replace if attr.device!= torch.device(cuda) with if attr.device!= torch.device(cuda) and torch.cuda.is available at line 20 of ldm/models/diffusion/ddim.py ().
Replace if attr.device!= torch.device(cuda) with if attr.device!= torch.device(cuda) and torch.cuda.is available in line 20 of ldm/models/diffusion/plms.py ().
Replace device=cuda in lines 38, 55, 83, and 142 of ldm/modules/encoders/modules.py with device=cuda if torch.cuda.is available(), otherwise cpu.
Replace model.cuda() in scripts/txt2img.py line 28 and scripts/img2img.py line 43 with if torch.cuda.is available(): model.cuda ().
Run the script again.
Testing
Test the model. Text-to-image is the first choice. Test the command line example again:
python3 scripts/txt2img.py --prompt "hello world" --plms --ckpt sd-v1-4.ckpt --skip_grid --n_samples 1The slow generation takes 10 seconds on a GPU and 10 minutes on a CPU. Final image:
Hello world is dull and abstract. Try a brush-wielding hamster. Why? Because we can, and it's not as insane as Napoleon's cat. Another image:
Generating an image from a text prompt and another image is interesting. I made this picture in two minutes using the image editor (sorry, drawing wasn't my strong suit):
I can create an image from this drawing:
python3 scripts/img2img.py --prompt "A bird is sitting on a tree branch" --ckpt sd-v1-4.ckpt --init-img bird.png --strength 0.8It was far better than my initial drawing:
I hope readers understand and experiment.
Stable Diffusion UI
Developers love the command line, but regular users may struggle. Stable Diffusion UI projects simplify image generation and installation. Simple usage:
Unpack the ZIP after downloading it from https://github.com/cmdr2/stable-diffusion-ui/releases. Linux and Windows are compatible with Stable Diffusion UI (sorry for Mac users, but those machines are not well-suitable for heavy machine learning tasks anyway;).
Start the script.
Done. The web browser UI makes configuring various Stable Diffusion features (upscaling, filtering, etc.) easy:
V2.1 of Stable Diffusion
I noticed the notification about releasing version 2.1 while writing this essay, and it was intriguing to test it. First, compare version 2 to version 1:
alternative text encoding. The Contrastive LanguageImage Pre-training (CLIP) deep learning model, which was trained on a significant number of text-image pairs, is used in Stable Diffusion 1. The open-source CLIP implementation used in Stable Diffusion 2 is called OpenCLIP. It is difficult to determine whether there have been any technical advancements or if legal concerns were the main focus. However, because the training datasets for the two text encoders were different, the output results from V1 and V2 will differ for the identical text prompts.
a new depth model that may be used to the output of image-to-image generation.
a revolutionary upscaling technique that can quadruple the resolution of an image.
Generally higher resolution Stable Diffusion 2 has the ability to produce both 512x512 and 768x768 pictures.
The Hugging Face website offers a free online demo of Stable Diffusion 2.1 for code testing. The process is the same as for version 1.4. Download a fresh version and activate the environment:
conda deactivate
conda env remove -n ldm # Use this if version 1 was previously installed
git clone https://github.com/Stability-AI/stablediffusion
cd stablediffusion
conda env create -f environment.yaml
conda activate ldmHugging Face offers a new weights ckpt file.
The Out of memory error prevented me from running this version on my 8 GB GPU. Version 2.1 fails on CPUs with the slow conv2d cpu not implemented for Half error (according to this GitHub issue, the CPU support for this algorithm and data type will not be added). The model can be modified from half to full precision (float16 instead of float32), however it doesn't make sense since v1 runs up to 10 minutes on the CPU and v2.1 should be much slower. The online demo results are visible. The same hamster painting with a brush prompt yielded this result:
It looks different from v1, but it functions and has a higher resolution.
The superresolution.py script can run the 4x Stable Diffusion upscaler locally (the x4-upscaler-ema.ckpt weights file should be in the same folder):
python3 scripts/gradio/superresolution.py configs/stable-diffusion/x4-upscaling.yaml x4-upscaler-ema.ckptThis code allows the web browser UI to select the image to upscale:
The copy-paste strategy may explain why the upscaler needs a text prompt (and the Hugging Face code snippet does not have any text input as well). I got a GPU out of memory error again, although CUDA can be disabled like v1. However, processing an image for more than two hours is unlikely:
Stable Diffusion Limitations
When we use the model, it's fun to see what it can and can't do. Generative models produce abstract visuals but not photorealistic ones. This fundamentally limits The generative neural network was trained on text and image pairs, but humans have a lot of background knowledge about the world. The neural network model knows nothing. If someone asks me to draw a Chinese text, I can draw something that looks like Chinese but is actually gibberish because I never learnt it. Generative AI does too! Humans can learn new languages, but the Stable Diffusion AI model includes only language and image decoder brain components. For instance, the Stable Diffusion model will pull NO WAR banner-bearers like this:
V1:
V2.1:
The shot shows text, although the model never learned to read or write. The model's string tokenizer automatically converts letters to lowercase before generating the image, so typing NO WAR banner or no war banner is the same.
I can also ask the model to draw a gorgeous woman:
V1:
V2.1:
The first image is gorgeous but physically incorrect. A second one is better, although it has an Uncanny valley feel. BTW, v2 has a lifehack to add a negative prompt and define what we don't want on the image. Readers might try adding horrible anatomy to the gorgeous woman request.
If we ask for a cartoon attractive woman, the results are nice, but accuracy doesn't matter:
V1:
V2.1:
Another example: I ordered a model to sketch a mouse, which looks beautiful but has too many legs, ears, and fingers:
V1:
V2.1: improved but not perfect.
V1 produces a fun cartoon flying mouse if I want something more abstract:
I tried multiple times with V2.1 but only received this:
The image is OK, but the first version is closer to the request.
Stable Diffusion struggles to draw letters, fingers, etc. However, abstract images yield interesting outcomes. A rural landscape with a modern metropolis in the background turned out well:
V1:
V2.1:
Generative models help make paintings too (at least, abstract ones). I searched Google Image Search for modern art painting to see works by real artists, and this was the first image:
I typed "abstract oil painting of people dancing" and got this:
V1:
V2.1:
It's a different style, but I don't think the AI-generated graphics are worse than the human-drawn ones.
The AI model cannot think like humans. It thinks nothing. A stable diffusion model is a billion-parameter matrix trained on millions of text-image pairs. I input "robot is creating a picture with a pen" to create an image for this post. Humans understand requests immediately. I tried Stable Diffusion multiple times and got this:
This great artwork has a pen, robot, and sketch, however it was not asked. Maybe it was because the tokenizer deleted is and a words from a statement, but I tried other requests such robot painting picture with pen without success. It's harder to prompt a model than a person.
I hope Stable Diffusion's general effects are evident. Despite its limitations, it can produce beautiful photographs in some settings. Readers who want to use Stable Diffusion results should be warned. Source code examination demonstrates that Stable Diffusion images feature a concealed watermark (text StableDiffusionV1 and SDV2) encoded using the invisible-watermark Python package. It's not a secret, because the official Stable Diffusion repository's test watermark.py file contains a decoding snippet. The put watermark line in the txt2img.py source code can be removed if desired. I didn't discover this watermark on photographs made by the online Hugging Face demo. Maybe I did something incorrectly (but maybe they are just not using the txt2img script on their backend at all).
Conclusion
The Stable Diffusion model was fascinating. As I mentioned before, trying something yourself is always better than taking someone else's word, so I encourage readers to do the same (including this article as well;).
Is Generative AI a game-changer? My humble experience tells me:
I think that place has a lot of potential. For designers and artists, generative AI can be a truly useful and innovative tool. Unfortunately, it can also pose a threat to some of them since if users can enter a text field to obtain a picture or a website logo in a matter of clicks, why would they pay more to a different party? Is it possible right now? unquestionably not yet. Images still have a very poor quality and are erroneous in minute details. And after viewing the image of the stunning woman above, models and fashion photographers may also unwind because it is highly unlikely that AI will replace them in the upcoming years.
Today, generative AI is still in its infancy. Even 768x768 images are considered to be of a high resolution when using neural networks, which are computationally highly expensive. There isn't an AI model that can generate high-resolution photographs natively without upscaling or other methods, at least not as of the time this article was written, but it will happen eventually.
It is still a challenge to accurately represent knowledge in neural networks (information like how many legs a cat has or the year Napoleon was born). Consequently, AI models struggle to create photorealistic photos, at least where little details are important (on the other side, when I searched Google for modern art paintings, the results are often even worse;).
When compared to the carefully chosen images from official web pages or YouTube reviews, the average output quality of a Stable Diffusion generation process is actually less attractive because to its high degree of randomness. When using the same technique on their own, consumers will theoretically only view those images as 1% of the results.
Anyway, it's exciting to witness this area's advancement, especially because the project is open source. Google's Imagen and DALL-E 2 can also produce remarkable findings. It will be interesting to see how they progress.

Vitalik
4 years ago
An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3

wordsmithwriter
3 years ago
2023 Will Be the Year of Evernote and Craft Notetaking Apps.
Note-taking is a vital skill. But it's mostly learned.
Recently, innovative note-taking apps have flooded the market.
In the next few years, Evernote and Craft will be important digital note-taking companies.
Evernote is a 2008 note-taking program. It can capture ideas, track tasks, and organize information on numerous platforms.
It's one of the only note-taking app that lets users input text, audio, photos, and videos. It's great for collecting research notes, brainstorming, and remaining organized.
Craft is a popular note-taking app.
Craft is a more concentrated note-taking application than Evernote. It organizes notes into subjects, tags, and relationships, making it ideal for technical or research notes.
Craft's search engine makes it easy to find what you need.
Both Evernote and Craft are likely to be the major players in digital note-taking in the years to come.
Their concentration on gathering and organizing information lets users generate notes quickly and simply. Multimedia elements and a strong search engine make them the note-taking apps of the future.
Evernote and Craft are great note-taking tools for staying organized and tracking ideas and projects.
With their focus on acquiring and organizing information, they'll dominate digital note-taking in 2023.
Pros
Concentrate on gathering and compiling information
special features including a strong search engine and multimedia components
Possibility of subject, tag, and relationship structuring
enables users to incorporate multimedia elements
Excellent tool for maintaining organization, arranging research notes, and brainstorming
Cons
Software may be difficult for folks who are not tech-savvy to utilize.
Limited assistance for hardware running an outdated operating system
Subscriptions could be pricey.
Data loss risk because of security issues
Evernote and Craft both have downsides.
The risk of data loss as a result of security flaws and software defects comes first.
Additionally, their subscription fees could be high, and they might restrict support for hardware that isn't running the newest operating systems.
Finally, folks who need to be tech-savvy may find the software difficult.
Evernote versus. Productivity Titans Evernote will make Notion more useful. medium.com