The function: that revolutionizes calculus.

"Plot of Weierstrass function over the interval [−2, 2]. Like some other fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the global plot." (Wikipedia, Weierstrass function)

In the late 19th century (1872) mathematician Karl Weierstrass introduced a formula that is one of the most revolutionary in the world of mathematics. "In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve" (Wikipedia, Weierstrass function) The term fractal curve. 

This means that when we zoom that curve or otherwise use more and more accurate numbers and shorter number distances between the integers we can find more and more roughness in that interesting function. When we make Weierstrass function calculations and curve for the answers we can get quite a smooth curve. But when we start to use decimal numbers that turns the curve rough. The fact is that there is a limitless number of decimal numbers between two characters. 

"Animation based on the increasing of the b value from 0.1 to 5. (Wikipedia, Weierstrass function)

We can continue all decimal numbers as much as we want. So there is an unlimited number of decimal numbers between for example 0,1 and 0,11. That means we can put unlimited numbers between those numbers that seem to be very close in the number line. But we can put 0,1001 or 0,1009 or any number series between 0,1 and 0,11 and the only thing that we must be sure of is that the number series that we give is smaller than 0,11. (0,11>x). 

So Weierstrass formula is the thing that can introduce the situation that the little difference might be a big thing if we look at it in the right scale. The Weiserstrass formula was revolutionized in the 19th. Century. And sometimes people say that this thing is an early fractal. When we research this function we can see that there is a repeating form that is similar to small and large areas or periods. 

"Euler diagram for P, NP, NP-complete, and NP-hard set of problems (excluding the empty language and its complement, which belong to P but are not NP-complete)". (Wikipedia, P versus NP problem). 

"Diagram of complexity classes provided that P ≠ NP. The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by Ladner's theorem. "(Wikipedia, P versus NP problem). 

Weierstrass formula and P=NP. 

Normally P=NP is introduced as the computational or virtual problem. That means the P=NP or P≠NP is used to introduce similarity between P and NP. The non-proven question is: is it always that P=NP or P≠NP? And is that universal? But the P=NP is much more interesting and large-size complexity than other things. 

The Weierstrass formula has a connection with an unproven mathematical millennium problem called P=NP. Can the P-level be the same as the NP level? And can the P level interact with the NP level? That thing is one of the key questions of computing. 

The P=NP means that if we have two identical systems whose size is different when we move something in a smaller system same thing moves in a larger scale system. The P=NP means that when something happens at the P level same thing happens at the NP level. But the major problem is this. In the real world, the P cannot interact directly with the NP. So there forms a medium between those layers or spaces. And the P and NP don't exist in pure forms. The world is full of layers and spaces. Those spaces disturb information that travels between P and NP. 

That thing can cause the idea about the P=NP. The P=NP means that if something changes in the P-level same thing happens in the NP  level. So, P=NP means that we move something in the microcosmos same thing moves in the microcosmos. So the P=NP should be universal. That formula should have an effect between virtual and "real" (material) systems. But the thing is that the energy level that the virtual system can give to the material system is so weak that the virtual system cannot affect the material or physical systems. 

The P=NP problem is the thing that originates in Buddha, Siddharta Gautama. The Buddha introduced the microcosmos is connected with the macrocosmos. So when something happens in the microcosmos that thing reflects immediately into the microcosmos. The fact is that the P and NP levels are not determined and we can say that abstract, virtual mind, or imagination is the P level. 

And physical world is the NP level. If that thing is real, that means the telekinesis is possible. But the fact is that there is needed much higher or stronger force to make the visible interaction between those virtual and physical worlds possible. The P=NP means that if we move something in the macro cosmos we move the same thing in the microcosmos. As I wrote before the P=NP means that all cosmoses or layers. Without depending on their size have a twin in the microcosmos. 

https://www.quantamagazine.org/the-jagged-monstrous-function-that-broke-calculus-20250123/

https://en.wikipedia.org/wiki/NP-intermediate

https://en.wikipedia.org/wiki/P_versus_NP_problem

Should we stay in our comfort zone, or should we, try new things?

We all live in a bubble. Those bubbles are our social environment. Our opinions and interests. The fact is that all people are at least a little bit woke. We don't want to read all things that we might see. We don't also have time to read all the news, even the important ones that we see. We must select the information that we see and we want to read. We have only a certain number of hours in a day. So that means we must do many other things than just read the news. 

Social media can be an opportunity. And it can be the worst thing in the world. It has shown that even famous people are humans. They have their opinions. The opinion is not the same thing as researched knowledge. 

One of the bubbles around us is a limited number of skills. That we have. Nobody knows everything. And nobody can do everything in the world. We must learn new things throughout our life. Without the ability to learn new things. We face the situation that our skills turn old. When our skills turn old, that is a bad thing for people, who can do only one thing. If we have only one skill, that skill is easy to teach to other people. 

Sometimes people say that they are more productive when they rest in their own comfort zone. But if we think carefully about those people who work only in their own bubble are humans. They have more than just one skill. And if they jump out of their comfort zone, that they know very well. Those people might find new dimensions for themselves. 

When we think about human nature, we should not think that when we learn something at the age of 21. We cannot learn new things after that. If we just make things that we learn at a young age, that makes us effective. But that makes us unilateral. Unilateral people are easy to replace. If we just make the same thing let's say 30 years, that is a long time. That means we should advance our skills and our work. 

Without advances, we would be in trouble. Even if we are effective in some works, while we use only one system we must realize that someday those systems turn old. And in that case, our employer must choose to retrain us to use the replacement system. Or does our employer hire new workers?

Single cells can learn things.

In some visions, the DNA pair or the protein pair can used to create the photons or acoustic bubbles between them. Then quantum computer can use those bubbles to make the quantum entanglement. That means the DNA or protein can be used as tools for acoustic qubit systems. But when we see that single cells can learn things we are facing an interesting thing. 

If that learning process is based on pressure that kind of acoustic system can be used to program cells. In some visions, the acoustic systems can control cells. 

Things like slime mold are things that show that the brains and neurons are not required in the learning process. And now researchers found that even individual cells learn things. That allows for improved medical research. The system can teach cells to find and terminate non-wanted cells. 

That allows creating. The control systems for nanorobots. The ability to learn. Makes it possible to create cells that operate as miniature robots. Those systems can clean the blood vessels from waste. 

This is one of the biggest breakthroughs in the research that can revolutionize nanotechnology and many other things. A short time ago. Researchers found that the neurons are not only cells that can store memories. Other tissue types participate in that process. 

Researchers noticed that the nervous system in the kidneys and nerve tissue cells can learn things. That thing can be a breakthrough for restoring brain damage. But the thing. That the single cells can learn. Is the key. To open new dimensions to the learning of other species. 

The big question in evolution is this. When did the species start to learn? Do things like amoebas and vegetables learn? The slime mold can learn things. So, can amoeba or some bacteria learn things cognitively? Those new findings can also make it possible to find out how things like bugs learn how to avoid their predators. And that can make it possible to create new ways to destroy harmful insects. Or program things. Like mosquitos to avoid malaria. 

The cell can store its memories in the DNA double helix or the protein helix structures. That thing makes a big advance in nano- and biotechnology. If researchers can crack this process. They can teach cells to deny things like virus entrance to the cells. The viruses can otherwise reprogram cells and make them make new things. 

The ability to teach cells will give new ways to fight against cancer. If cells include cognitive skills. The system can program them by using signals that the cell uses to learn. DNA-based learning is the thing. That requires the ability to change the DNA from those cells. That reprograms cells. 

https://scitechdaily.com/mind-blowing-discovery-scientists-discover-that-memories-are-not-only-in-the-brain/

https://scitechdaily.com/single-cells-can-learn-a-revolutionary-discovery-in-biology/

How much can AI affect our genomes?

In this text, I will not write about genetic engineering. I will write about the non-purpose processing of our genetic and environmental backgrounds. 

AI and evolution are the things that cause interesting ideas. The big question is: can AI and the internet affect evolution? And the answer is that those things are already affecting evolution. When we think about things like dating services. People search for people who are like themselves. The problem is simple. Similar types of people search for dates on the net. 

We know that DNA has some kind of role in people's behavior. And things like lifestyle are made by society. So our lifestyle and behavior are made by two major groups of abilities and features. Some of those abilities and features are hereditary. And some of them are formed by the environment and society. 

If people want to make dates through the internet. They must fill out the form. And that is the point where the AI affects our genomes. Because people search for similar people as they are, that thing causes segregation and the accumulation of certain genomes. 

**********************************************

When we select our spouses from the group there people have similar behavior as we have, and we advance things like segregation. In the same way, we can say that our behavior is the sum of hereditary and environmental things. If we only meet people who are similar to us. 

That makes our environment homogenous. We don't hear things that we don't want to hear. And that advances the segregation. That causes. Also a situation. Where we hear only similar opinions that we have. That advance makes our visions thinner than ever before. 

That destroys real discussions and the possibility. To see other points of view. Things are also governed by. We see only things that we can create ourselves. And that means. We cannot even get other opinions to our discussions. 

**********************************************

Genetic material and environment affect people's behavior. That means we choose people who have the same economic background and similar genomes. That is the thing that can affect the entire human population. 

The net allows us to make dates all over the world. And the AI is an effective tool to make the match with people who use the net. That means those people are more effective in producing descendants. Than people. Who search for their spouse from the local bar. 

The AI can search for people who have similar hobbies, favorite food, and other similar hobbies and work. That means. If the system gets ordered the person wants a spouse who plays some musical instruments. And the thing is that the algorithm can use the form and other information that it gets to select the perfect mate. The system can search things like the musical AI-experts from the net, and then offer them dates. The form that a person can feed can be similar to that used in animal breeding. 

The same methods can used with humans. When we use things like normal test-tube fertilization. There is always the possibility to select gemmates using a similar method. That the network dating tool uses. When it selects people whose dates are successful. In those cases, the system will not use things like genetic engineering. It uses the method. That has been a long time used in horse breeding. AI can affect us and our descendant's genomes in purpose or without purpose. 

The fact is that. AI can increase segregation when people select similar spouses and friends. As they are. The internet is the tool that can make us more isolated than ever. That is the sad thing. That destroys real discussion and helps us to select information narrower than ever before. Because those algorithms select our mates from the net using variables that make sure that our dates please us that makes it possible for us to hear and see only things that we can accept. That might look comfortable. But that destroys our ability to hear and see other choices. 

When the most advanced tool becomes the enemy of advance.

Why...

Above this text is the image that is not made by using AI. I took that image yesterday evening. That thing required a little bit of trouble and time. Walking into that point, taking my cell phone, and taking that picture took time. When I looked at that picture I realized why people use AI in many things. AI offers easy things to get the job done. That easy tool allows people to make thousands of lines of code in minutes. 

That is possible if the person uses some code libraries. And that makes a person effective. But that thing is not good for advancement and innovation. Also, that way of making programs is not good for data security. If some hackers get those code libraries that allow them to break the systems those codes are used. 

We know that thing. But we ever ask why person makes that thing? Why does that person use libraries and copy-paste? Why that person uses code, that somebody gave to the hard disk? The answer is this something forces a person to make things like that. The nice, well-done code in the hard disk makes work effective. And we can say that generative AI, large LLM, or whatever we call that thing is only to those libraries. 

We can be effective and always use that method to make things. That can make an impression. But then we can say that this method doesn't allow us to advance our programming skills. And that means the AI kills productivity and skills. But is the AI guilty? 

Do we bark the wrong tree? Should we prosecute the need to be effective for that thing? People use AI and code libraries because they have no time to learn things. People are so hurry, that they have no time to learn things. And if they make mistakes they are fired. That forces people to use AI. 

The biggest problem with algorithms is they use existing information. They don't call any telescopes or particle accelerators to create experiments that they can produce new things. They just seek existing information and reorder it. If we start to use AI more and more to become effective we are facing situations in which we don't get new innovations. 

New innovations are things that make the world more advanced. But when we make innovation. We process thought into the product. Without thought is not innovation. Because our mind is to bind to memories we are in trouble with new things.  That means we should create a mind that is not dependent on memories. But that thing is not very easy to make. Almost everything that we make is bound to memories. 

Algorithms do not create things. They just connect existent data into new entirety. And the question is this: do we let ourselves turn into the AI's? Mechanical minds can search existing information but cannot create anything new. 

If we don't let people innovate that means that we will not create new things. Innovations require time and space. And that thing is not a thing that fits into the idea of productivity and effectiveness. 

Algorithms are tools that make many things faster than humans. That is the reason why we use them. The question is always this: do we someday make algorithms that rule people? When we make things by using algorithms we can find an easy way to make things. Algorithms make our lives easier. They search for things from databases. Then, AI can make many things better than we can. We are afraid. That AI helps people to make things. 

Like fake doctoral theses. We are always afraid of that. People lose their creativity when they use AI. We ever ask why we leave our work to the AI. We see that things happen, never asking "why"? 

The problem with the algorithm is that AIs don't make anything new.  They just collect information from the networks. AIs build new entireties using information. That already exists. And that is the problem with AI. The problem is that the system requires effective working. During working days people don't cheer programmers and other people to make new decisions. The requirement for effectiveness is the thing that destroys creativity and advancement. 

People are afraid of remote working because it destroys innovation. But how many hours in week do we use for innovations? Do we really make innovations in our offices? Or do we just read newspapers and reports and make things that have been made 1000 times before? 

Many people think. That innovation is the case, where one person talks and the other listens. The problem with innovation and creativity is that people are afraid of failure. Failure is somehow a non-accepted thing, and that means we make many times things that we made many times before. 

Because. People are afraid of things like grammar mistakes they think that they must let the AI make their texts. Programmers are afraid of mistakes.  Fixing mistakes steals their and other worker's working time. That is the reason why the working environment forces people to use AI. And that is the best way to destroy productive thinking and innovation. 

The AI makes people look like effective. But that system uses only existing information. That is the limit of the AI. Even if we want to make images using AI, we must give orders to it. The AI is like the drawing program that follows orders that we give. That allows us to create new worlds. But that thing doesn't advance our painting skills. 

The memory makes humans effective. Imagination is the thing that connects memory bites together new way. That allows us to create new things. Memories and the connections with existing memories are the things that limit the imagination. The requirement for innovation is an idea, that doesn't come from emptiness. 

Memories are stored in our memory cells. When we use imagination we just reorder those things that we can call memory pixels into the new entirety. Those things are like puzzles that we can reorder. 

The big advance would be the mind. That is not connected with memory. The memory-independent mind is something that we cannot imagine. We use memories to create new entities. And it's hard to even imagine a tool that doesn't use memories. That thing is the tool that can make the next step for innovations and society. 

Can the photon itself be the origin of dark energy?

"AI-generated representation of an accreting supermassive black hole, surrounded by gas spiraling toward it along the equatorial plane (the accretion disk) and emitting powerful winds of matter as it falls in. This representation is based on a NASA’s artist’s concept that illustrates a supermassive black hole with millions to billions of times the mass of our sun. Credit: Emanuela Tortosa, edited" (ScitechDaily, Supermassive Black Holes Defy Physics to Become Cosmic Titans)

Dark matter and dark energy mysteries are heavily entangled with each other. Maybe supernovas, black holes, and new images of photons can solve another of those two mysteries. It's possible. That the photons are the source of dark energy. The ring-shaped structure means that energy that hits the ring travels into the middle of it. And maybe that thing is the source of the mysterious wave movement that rips the universe into pieces. 

If we think that black holes can turn all photons around it into one direction that causes a situation where the photons send the small energy string away from the black hole. And the other energy pike travels to the black hole. We know that photons can store energy that expands their size. That means the black hole's energy may expand the photon so much, that it can close the black hole in it. 

That means the photon can form a ring around the black hole. The model is that the photon is like a skyrmion. And that means it can turn so large. That it can form a ring around supermassive objects. If a photon can form that ring it can pump lots of energy into the black hole. But that is not sure. 

"A new theory, that explains how light and matter interact at the quantum level has enabled researchers to define for the first time the precise shape of a single photon. Credit: Dr. Benjamin Yuen" (ScitechDaily, Quantum Leap: Scientists Reveal the Shape of a Single Photon for the First Time)

Black holes. And especially supermassive black holes defy physics. The thing is that the black hole pulls wave movement and particles into it. Even if we see the black hole as the stoned object those monsters oscillate all the time. 

That thing means that they could make the same thing to gravity fields as time crystals make to electromagnetic wave movement. The black hole pulls energy fields into it. And some of those energy fields start other orbit the event horizon. When strings that form in the photons hit those fields they cause oscillation that we see as gravitational waves. 

The problem with dark energy and the dark gravity effect is that. There is something that causes gravity waves and quantum fields to move. The movement of those things is the thing that we see as gravity and dark energy waves. The problem with dark energy is that nobody knows its origin. 

"Detecting axions via gamma rays from a supernova could unlock the secrets of dark matter, with UC Berkeley researchers pushing for advanced telescopes to ensure we don’t miss this fleeting opportunity. Credit: SciTechDaily.com" (ScitechDaily, A Nearby Supernova Could Finally Solve the Dark Matter Puzzle)

In some models, dark energy is the thing that forms when a gravity wave collapses. The idea is that the gravity wave is the energy ditch or lower energy area that travels in the Higgs field. If a gravity wave faces some other energy wave that energy wave can fall into the energy ditch or gravity wave. And that thing makes the energy impulse through space and time. 

But then we can think of things like supernovas as the things that can confirm the weakly interacting massive particles, WIMP's existence. The major question is always can the WIMP exists, and the other thing is that. Can regular, or visible material turn into the WIMP? 

The model is that when the energy travels out from the material elementary particles can turn flat. And it's possible that those flat particles can rotate around its axle. That turns the WIMP look like a twisted rag. When energy fields hit those particles they form the energy vacuum or energy shadow around that particle. The quantum fields start to travel into that bubble when the distance of the object is far enough. The WIMP is a theoretical object. 

Things like supernova explosions that happen near Earth can uncover the mystery. But it's possible that we must wait for Betelgeuse to explode as a supernova before we can make those observations. 

https://scitechdaily.com/a-nearby-supernova-could-finally-solve-the-dark-matter-puzzle/

https://scitechdaily.com/quantum-leap-scientists-reveal-the-shape-of-a-single-photon-for-the-first-time/

https://scitechdaily.com/supermassive-black-holes-defy-physics-to-become-cosmic-titans/

The engineered memories are a multipurpose tool.

Have you seen the movie "Total Recall"? That movie involves a scene where the main character makes a virtual trip to Mars. Virtual memories make it possible to create things and experiments that we never imagined. Virtual reality is a tool that we can use for education. Virtual reality allows us to go to the second reality or alternative reality. We can make a sandbox into a solarium and use the IR and UV light to make synthetic sunlight.  

Then we can put the virtual reality glasses on and look at the film of a sunny beach at some place like Hawaii. The AI can create characters that are connected with the AI chatbots. And those characters can discuss with us. That kind of virtual reality-based second life or alternative reality can allow us to create things like virtual trips to the sunny south. That thing allows us to see what kind of place that travel destination is. 

In real life, researchers think of the possibility of engineering memories. This thing can make it possible to create a treatment for brain damage. This type of research can be a new tool for many other things. All our experiences are based on memories. In the same way, everything that we learn and can do depends on memories. This is why this type of research interests many authorities, from civil to military. 

Engineered memories can help Alzheimer's patients. But there are many other things there that are useful. And one of them is educational. 

The ability to engineer memories is a tool for ultimate learning. That allows people to pass things like university studies in a very short time. This kind of ability is one of the things that people who live in isolated places might need. They need skills to fix their house and give emergency aid to injured people. And engineered memories can help them to make things that they need. There is always a doctor on board in expeditions but what if the doctor is injured? 

That thing requires a system that can teach things to people very fast. Otherwise. Those people might use the remote-AI-controlled robots. That makes those missions. That can require fast satellite communication. The helmet-based system where the action camera sends data to AI can make it possible to turn the AI assistant to the next level. The AI can tell the user what equipment that person must use, and it can tell what those things look like. But that doesn't replace the experience. 

But that is only one thing that this kind of tool that engineers memories can do. The ability to create synthetic memories is a tool, that can have many purposes. The ability to train military personnel in a very short time, and give them the target data straight into their brains is the thing, that makes this system interesting. 

Another thing is the ability to engineer memories makes it possible to brainwash people. This ability might interest people who are working with very high-risk law enforcement deep covert operations where the risk that the operator changes sides is high. And traitors can cause very big danger. 

https://scitechdaily.com/could-we-engineer-memories-new-johns-hopkins-findings-could-help-treat-alzheimers/

https://scitechdaily.com/neurons-rewrite-the-clock-new-discovery-reveals-how-we-truly-learn/