DNA computers get a new boost.

“Researchers have developed a DNA computer that performs mathematical calculations using molecular interactions rather than silicon chips. Credit: Shutterstock” (SciTechDaily, Scientists Build a World-First DNA Computer That Runs Without Continuous Power)

From. The point of view of nanotechnology. Molecular computers are one of the most important types of computers. Those systems could control extremely small machines and turn cells into biorobots.  The idea behind those computers is similar to a spring. The spring can also transmit data. As well as other things. Like fluids and air. In those cases, we can talk about acoustic computers. 

When we send an impulse through the medium. The wave. It can be 1, and the area between waves can be zero. The same way springs and spring-shaped molecules can transmit data. 

DNA-based molecular computers are a reality. Those systems are slower than silicon-based computers. But they are smaller. And they can operate without an external energy source.  Those systems. They don’t need as much energy as silicon-based systems. Those systems are slower than silicon. 

“To build the computer, the team combined a small drop of water and salt with short pieces of DNA and a longer DNA scaffold in a test tube. The mixture needs only a little heat to get started and is then cooled, allowing the computation to proceed.” (ScitechDaily, Scientists Build a World-First DNA Computer That Runs Without Continuous Power)

“Prof. Woods explained the motivation for exploring other forms of computing. “Silicon-based computers use so much energy – 23% of Ireland’s electricity goes into computing and data storage. We’ve been blinkered by only seeing one type of computer, but there are other examples around us, including our brain.” (ScitechDaily, Scientists Build a World-First DNA Computer That Runs Without Continuous Power)

“The researchers ran ten programs, including 100-bit computations. In one experiment, the computer added numbers. In. The range was about 11 million to 34 million. It took up to 14 hours to reach a result. The system also demonstrated robustness and reusability, performing up to 25 different calculations in a row.”  (ScitechDaily, Scientists Build a World-First DNA Computer That Runs Without Continuous Power)

They can control things like nanomachines and biorobots. In visions, they can operate inside cells. Or they can turn a cell into a robot. This system. It can be the next step in medicine. It could revolutionize things like surgery. And those systems can control intelligent tissues. 

Those intelligent tissues, like intelligent skin. They can operate as molecular microchips. That intelligent skin can use hybrid processors. Their microchips transmit data to them. Using neurons. The “intelligent skin”. It can be the plant cell. That means the molecular computer. It can be an intelligent plant or an intelligent leaf. The system can use the cells from electric eels.  That produce electricity as the power source. The electric eel’s cells. They can act as a power source. For regular microchips. 

Unlike electronic computers. DNA-based computers use molecular interactions. For. Data processing. Unlike a regular computer. The DNA computer. It doesn’t require a constant power supply. The DNA computer can use the liquids in the cell to operate. 

The DNA computer uses interacting strands in water. The system could also use interactions between water and DNA. That interaction can be vertical pressure waves. In. In that case. The pressure waves travel in the same line as the DNA. Or. Between two DNA bits. These systems. They can also be used. For programming cells. This ability to transport information into neurons. It will be the key to repairing neural damage. 

And a gate to one version of immortality. The DNA nanomachines can copy the internal brainwaves with extremely high accuracy. If. We think. About. The ability to clone humans. Those inner brainwaves play a critical role in mind upload. And copy those brainwaves into the descendants' brains. That thing requires more accurate systems. Than just using cortical brain waves. 

https://scitechdaily.com/scientists-build-a-world-first-dna-computer-that-runs-without-continuous-power/

Data center warfare is part of high-tech warfare.

“Ukrainian Drones Knock Out Supercomputers at Two Yandex Data Centers. Yandex confirmed fires and full shutdowns at its Sasovo and Kaluga sites on back-to-back days, October 8 and 9, after drones hit supercomputers tied to training its YandexGPT model and disrupted services for Russian Railways and MegaFon“.(Startupfortune, Ukrainian Drones Knock Out Supercomputers at Two Yandex Data Centers)

The third closed data center is in Vladimir. That data center provides data services to the Moscow region. And agencies like the GRU, FSB, and the Russian Aviation Institute can use that data. 

“Yandex shut down its data center in Vladimir, east of Moscow, after a drone strike damaged its infrastructure on October 11, the company said in a Telegram alert. The outage disrupted Yandex services as the company assessed the damage and worked to restore operations; the Associated Press reported it was the third strike on Yandex facilities in four days.” (LetsDataScience, Yandex Shuts Vladimir Data Center After Drone Attack)

Things. like automated and semi-automated systems require complex codes. To operate.  Data centers provide a platform to run and train AI. To operate in that area. Denying drone communication. Using jammers is more complicated than people thought. Jammer bases are in the radio transmitters. Those transmitters send radio signals that should deny data communication between the operator and the drone. 

The problem is in radio transmission. That makes jammer systems vulnerable to anti-radiation missiles. The aircraft or drone. It can shoot anti-radiation missiles against those radio transmitters. And the radio jammer works only If. it can operate on the same frequency as the drone. This means that the opponent operator knows those frequencies. When. The drone is jammed. 

The operator can shoot anti-radiation missiles from ground vehicles. Or they can shoot those missiles from airplanes. Anti-radiation missile. It can be effective against drones. But only if the missile and its operators know the drone’s command frequency. The anti-radiation missiles are shot from rocket launchers. They can be destructive tools. If. The operators find the right radio frequencies. And these kinds of systems require lots of data. 

Data centers play a central role in modern warfare. Those centers serve as information tools. And disinformation tools. They serve as data processing systems. Those systems. They. Can be used to program drones. They can also carry out cyberattacks. Those attacks are important. If. Those operators want to know. 

Where. Somebody is moving. Things. Like. Hacked traffic surveillance cameras can tell where some interesting things are moving. Hacked telephone centers can uncover enemy plans. And the big problem is how to make software. That those attacks require. 

We must remember. That without cyberattacks, things like the elimination of the Iranian leaders are impossible. Reconnaissance aircraft and satellites cannot. recognize people. Who enter cars in a garage. The system sees cars. But it cannot recognize people. Who sit in a car. Or the person’s precise location. Who is in the house. 

Cyberattacks allow attackers to see things. Like operation plans. Hacked cameras and access controls also give more interesting data. Like. Who the next commander of enemy forces is. Or who has private conversations with enemy leaders. 

Updated firewalls and antivirus software can make old programs useless. The operators must generate new code. That the antivirus cannot detect. This requires AI-based systems. Attackers must generate lots of code.

 In a short period. These kinds of systems require data centers. 

Data centers are vulnerable and high-value targets.  Trolls are operating using those data centers. And destruction makes it hard to run AI agents. And bots on platforms. The remarkable detail is this. Some far-right and Pro-Trump campaigns. On the net turn silent. When the Ukrainian drones attacked the Yandex data centers.

Yandex attack doesn’t end Russian campaigns. But it decreases them by about 50%. Another thing is that Yandex is part of the “Russian Google” infrastructure. This means: a major Part of the Russian data travels through those data centers. Governmental organizations like Roskomnadzor use those data centers to control the internet. 

Yandex closed a Vladimir data center. Near Moscow. That data center could serve the FSB for data processing. At least it provided very important data capacity for the Moscow region. The FSB used that data. For. Targeting their strikes against opposition. Knocking out those supercomputers makes it harder to control the system. 

This makes those data centers very high-value targets. In the same way. In Kyiv. Data centers provide their support. For. Intelligence analysis. And. The ability to predict air strikes. Attacks against data infrastructure affect the ability to operate. 

https://dailyboulder.com/russian-disinformation-bot-traffic-plunges-overnight-after-ukraine-drone-strikes-cripple-multiple-putins-data-centers-report/

https://www.kyivpost.com/post/86708

https://tech-insider.org/ukraine-strikes-yandex-third-data-center-4-days-2026/

https://arstechnica.com/gadgets/2026/10/russian-drones-strike-ukraines-data-centers-by-exploiting-air-defense-gaps/

https://letsdatascience.com/news/yandex-shuts-vladimir-data-center-after-drone-attack-54325517

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

New attack uses vulnerability in microchips.

“Modern processors are fast, in part, because they guess. Rather than waiting to find out which way a program will branch, a chip predicts the likely path. And races ahead. When the guess is right. “ (MIT)

“Time is saved. When it’s wrong, the work is discarded, but traces of it linger. Since the Spectre vulnerability was disclosed in 2018, attackers have known how to read those traces to pull secrets out of memory they should never see.” (MIT)

MIT Schwarzman college of computing uncovers a new type of attack. That opens Linux passwords to an attacker. The attacker uses the vulnerability in a modern microchip log file. The microprocessor itself creates this file. Called a “guess file”. Officially, that file is called prediction machinery. But. I use the term “Guess file”. That is the log file. The microchip uses it to predict how the program routes itself. 

“AMD’s defense, called saferet, cleans the prediction machinery immediately before each use, leaving a vulnerable window just two instructions wide, which typically execute within tens of nanoseconds. The researchers hit it anyway, by slowing down the processor at that exact spot to make the target easier to strike.” (MIT)

There is one small vulnerability. The system needs to stop before it starts a new task. That opens the door to the processor. A couple of milliseconds is enough to slow the processor. The attacker can strike. We know. That hackers can exploit even the smallest windows and backdoors. They can get access to the system. Those vulnerabilities are growing bigger in the time of AI. AI can launch attacks. Using precise and fast reactions. The AI waits for the processor to slow down. Before it begins a new task. When the processor stops, the AI tries to slip malicious code into the processor. 

So the computer makes a bad guess. And that opens a path to at least the password file. 

“The defenses work by wiping or isolating the processor’s prediction machinery, removing anything an attacker might have planted. The catch, as PhD student Daniël Trujillo and MIT Assistant Professor Mengjia Yan point out, is that the wipe and the moment the predictions get used can’t happen at the same instant. There is always a gap. — sometimes only a handful of instructions wide. Anything that runs in that gap can dirty the machinery all over again. The researchers call this class of attack “TONTOU.” (MIT)

If hackers can infect microchips, they can infect an entire computer. The microchip is a small computer. There is a cache and an operating system. The Kernel is the program that connects the hardware to software. Every command that a user gives. And a program that runs on a computer travels through the processor. If malware. Can be written in the microchip. The results can be destructive. The cache is the point. There, the malware could write itself. 

If. The cache is not protected. That can cause infection. There is a small possibility that the attackers can write their malware in that memory. Normally, a microchip’s internal control programs run on ROM. But that cache is vulnerable. Researchers in an MIT laboratory created a method called “interrupt injection”. 

“To show what this means in practice, the team built a working exploit on an AMD system running a current Linux kernel. They first stripped away a defense that scrambles where the operating system sits in memory, succeeding in all 10 tries in about nine minutes each. That helped them read protected memory at roughly five bytes per second — slow, but fast enough to locate and copy “/etc/shadow,” the file storing the system’s root password hash, in half their attempts.” (MIT)

This file tells the computer which program requires attention. Normally, the processor destroys that file immediately after it uses it.  But researchers could slow the processor at a critical moment. And they could use that file to open the password file. This means. They got access to the entire system. 

But the “guess-file”. It has another option. That option is that. It determines which program or file can use the microprocessor’s resources. There is a small possibility. Tha. If hackers can create that file. And slip it into the microchips. They can gain access to the entire system. And they can slip in any malware. In a computer, using the microprocessors as the platform. 

https://computing.mit.edu/news/new-type-of-attack-can-slip-past-the-defenses-in-your-computers-processor/

https://en.wikipedia.org/wiki/Spectre_(security_vulnerability)

The new drones and solutions are shaping the next trends in the military.

"Ukraine Tests World's First Purpose-Built Robot Aircraft Carrier for Drones" (Kyiv Post, Ukraine Tests World's First Purpose-Built Robot Aircraft Carrier for Drones)

The next trend in the military is networking. All actors share information about their ammunition capacity. Recognized vehicles. Enemies and their movements. Knowledge of the enemy is a key element. 

The new systems are integrated as part of the whole. Entire battle units and their support systems form a network that shares information immediately. The winner is not the fastest. It's the sharpest. It's the thing that can detect, follow, and shoot the enemy with the highest possible accuracy. The only accepted accuracy is destroying the enemy. This is the element. 

In tactical-scale work. The strategic scale is more complicated than just battlefield operations. 

The strategic scale means the entire system. Its ability to replace losses settles the war. Settling the war is not the same as winning battles. It’s a little bit the same. But if the actor uses all its anti-tank ammunition in one battle. That can mean a loss in the next battle. All ammunition is a single-use tool. That means resupply is the most important thing in the war. The new drones can carry other drones. 

Sea- and land-based solutions. They carry anti-drone, killer drones, and observation drones. Surface and underwater systems can release other drones. That can affect inland objects. They can also use their anti-drone systems as traps. That can affect things. Like aircraft and helicopters. And other ships and tanks. The USVs (unmanned surface vehicles).And UUVs (Unmanned Underwater vehicles). And UGVs (Unmanned Ground Vehicles). They can carry small, lightweight anti-aircraft missiles. Stinger-type missiles can cause damage to large aircraft.

"A photograph of Typhoon uncrewed surface vessel equipped with an Airstrike Pack. Image: Splash Industries" (The Defensepost, Typhoon USV Gains Ability to Launch One-Way Drones With a Single Button)

And if. The system carries an internal warhead. That can improve its ability to cause damage. The large-sized robot can carry other robots.  This means the UUV or USV can be dropped from an airplane.  Or some unmanned sea vehicle can carry a UGV to the operational area. This makes those systems very hard to predict. A drone is completely different from a manned vehicle. 

The AVs. (Autonomous Vehicles) like AUVs (Autonomous Underwater Vehicles) are replacing remote-controlled vehicles. Those AI-powered systems can operate independently without a remote pilot. This makes them immune to jammers. 

The drone must not care. About. Its crew. And the only problem is its communication. When. We think about small-sized FPV drones. Bigger drones like the MQ-9 Reaper and its successors. They can transport those drones. And. They can act as relay stations. This means that the small FPV drone can also be controlled from another side of the world. That gives small drones new capabilities. Larger drones. They can carry those small drones to operational areas. The fact is that. 

Things like Starlink systems. They can offer drones higher capacities. Than we ever imagined. The modem that uses jumping frequencies is hard to jam. But if jamming is successful. That makes those drones ineffective. Except. If they have AI-based systems. That can make them follow and hit the target even if the controller loses connection. 

When we think about manned submarines, their limit is food. Another thing. That limits their operations. It is psychological effects. Things like toilets can malfunction. And relationships in a closed environment can turn poisonous. This means the next thing that we can see at sea. It could be nuclear-powered drones. Full-scale nuclear submarine. It’s easy to transform into a drone. 

Small nuclear reactors. They can give small vehicles the ability to use things like lasers and microwave weapons. Nuclear power can solve problems with those drones' operational range. Nuclear batteries (RTGs) can give a small drone an unlimited operational capacity. But portable nuclear reactors can give the same ability. For. Larger-scale drones. The nuclear-powered tank Chrysler TV-8. It could be based on the new nuclear submarines. Those submarines can have caterpillar tracks. That allows them to operate. 

On the bottom of the sea. The miniature sub. It can look like the Soviet miniature submarine concept. That was designed for Spetsnaz. The submarine had those caterpillar tracks. That. It can travel along the seabed.

Miniature nuclear technology. That was created. For nuclear-powered cars like the Ford Nucleon and tanks like the Chrysler TV-8 in the 1960s. Those reactors' modernized versions. They can make it possible to create nuclear-powered flying vehicles, ground vehicles, corvettes, frigates, and missile boats. Or the nuclear-powered version of the Orca-class UUV.  Miniature reactors. They can turn many visions true. The ability to create systems with unlimited operational range is interesting. 

https://www.hisutton.com/Russian_Tracked_Submarines_Baltic_1980s.html

https://www.kyivpost.com/post/83090?

https://thedefensepost.com/2026/08/27/typhoon-one-way-drones/

https://www.twz.com/land/bradley-fighting-vehicle-lugging-load-of-swarming-ground-drones-is-a-sign-of-whats-to-come

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

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

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

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

https://en.wikipedia.org/wiki/Chrysler_TV-8

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

Intelligent glasses and AI.

Intelligent glasses and AI. They don’t destroy our ability to think; we destroy our ability to think ourselves. We use AI without criticism. We don’t check the sources that AI uses. 

Intelligent glasses and AI are tools that can bring human-AI fusion and singularity closer. Intelligent glasses have miniature cameras. That makes it possible to transmit information to AI. That tool is incredible for both free time and business work. Intelligent glasses allow a person to work in public places. 

When a computer screen is connected to those systems. 

That keeps users’ privacy at a high level. When a programmer uses data glasses and looks at the screen. The AI reads that text and gives advice. But then that advance can turn the AI into a programmer. The idea is the same with a chess game; there, the person uses headphones, AI, and a camera to make perfect moves. The AI gives orders on where the player should make moves. In this case, the real winner is the AI. We say that we are afraid. That AI takes our jobs. But at the same time, working life requires more effectiveness. 

When we cry that programmers lose their jobs to AI. 

We forget that today companies have no time to train new workers. They want full-stack coders. And where can they find those full-stack coders if new programmers cannot get any practice? Without practice. There are no masters. This means that workers are forced to use AI. People must be effective. And there is no time to give advice. People fight for their workplaces. So, that means there is no time to think. People must not think; they must act. And that is a sad truth. Reality is that AI is here. And we must accept it.

When we think about the jobs that AI takes. Well. Those jobs are already outsourced to India and other places; there is time to train programmers. The only problem is this. There is no space to think. The only goal that measures effectiveness is money. 

Only goal. What we can put to ourselves is how to turn expertise into leadership. Leadership means that expertise and precision should turn into money-making. And that turns people to use AI. When we think that an untrained programmer who uses AI can produce results as fine as an engineer, the company selects an untrained person. 

The reason is simple: the engineer gets a higher salary. The untrained person who makes similar-looking things has a lower price. The customer cares about what the product looks like and how it works. They don’t care about how many dead lines it includes. This is the philosophy. 

The low-trained. Person is easy to replace. The problem is that. Everybody wants to be lawyers. Or. Medical doctors. Those things require high university training. And the problem is that. The entrance exams require perfect success. And before those exams. The matriculation examination requires complete success. Those things force some people to use intelligent glasses and AI to make their answers. 

This means people give up their own willingness. And their ability to think and search for information is given to AI. When the entire working life revolves around effectiveness, measured by how many successful work outputs a person can make in a certain time. That leads people to use AI in their work. They don’t check facts. They might not trust AI. But they use its information without even clicking the source link. The attitude is this. Some experts made the AI. And that means AI will not make mistakes. So, why should anybody click any sources? By thinking that way. Clicking source links is a waste of time. 

When we look at results. We. See only information. 

That we see from outside. 

We can call that information loud information. That information is collected from interactions with humans and machines. 

Statistics don’t involve silent information. Silent information is our thoughts. Things that nobody sees from the outside. If. We don’t tell that information to outsiders. That means information stays silent. 

We must realize that records and statistics involve only information that we see from outside. And an attitude that “someone else does”. Or, “who cares if it is wrong or right?” “They don’t check it anyway. “ That is a dangerous path. If people don’t dare or care to tell that somebody makes bad work. That escalates false information and bad code through the entire system. 

It doesn’t matter what we think. What we do and what others see matters. The thing that people see. It is what we do. When AI gives a wrong answer. The answer is wrong to us. Only. If we know that it is wrong. But if we still use the wrong answer. We mark it right anyway. And that is dangerous. The AI thinks in terms of statistics. And this means that. The AI can turn wrong into something that seems right. This means that, in the worst case. The entire system operates on wrong information. 

A new quantum computer solves a math problem in 15 minutes. That binary computers can never solve.

“Scientists used a new error correction method to encode 70 logical qubits and solve a problem that is intractable for classical computers. The quantum computation took about 15 minutes, while leading classical methods would require an impractical amount of time. Credit: Shutterstock” (ScitechDaily, Quantum Computer Solves a Problem in 15 Minutes That Classical Methods Can’t Practically Compute)

“Researchers have demonstrated a quantum computation that appears to exceed the practical capabilities of leading classical simulation methods while also addressing a longstanding problem: how to verify the result.”(ScitechDaily, Quantum Computer Solves a Problem in 15 Minutes That Classical Methods Can’t Practically Compute)

A quantum computer completed a difficult calculation in about 15 minutes, while leading classical simulation methods would require prohibitive amounts of time. Just as importantly, the experiment included a way to establish confidence that the quantum result was accurate.”(ScitechDaily, Quantum Computer Solves a Problem in 15 Minutes That Classical Methods Can’t Practically Compute)

“The system completed 2,415 logical two-qubit operations and 468 logical “T gates,” two measures of quantum circuit complexity. Encoding the circuit reduced the effective logical error rate to one tenth of the physical error rate, allowing the computation to maintain high fidelity despite the large number of operations.”(ScitechDaily, Quantum Computer Solves a Problem in 15 Minutes That Classical Methods Can’t Practically Compute)

"Common quantum logic gates by name (including abbreviation), circuit form(s) and the corresponding unitary matrices". (Wikipedia, Quantum logic gate)

There are two major problems in quantum computers. First is that. Nothing that another quantum computer can check or inspect the result. The other problem is. How to make sure a quantum computer gets the right answer. This means. The quantum computer may make many errors in the process. From the beginning of the formula. To the complete answer. The danger is that the system can get the right answer. But using the wrong methods. The danger is that. The control formula is made using binary computers. It can give the right answer. Or an answer that seems right. 

Even if a quantum computer makes mistakes. And. If a quantum computer works incorrectly. That can cause errors in extremely long calculations. That take years. If. A computer solves a problem in one year. Inspection takes another year. So, making an answer and proving it takes two years. And this is one problem. 

“Random circuit sampling (RCS) has long served as a benchmark for testing whether quantum computers can outperform classical systems. In this task, a quantum computer produces patterns so complicated that classical computers cannot efficiently recreate them.” (ScitechDaily, Quantum Computer Solves a Problem in 15 Minutes That Classical Methods Can’t Practically Compute)

A quantum computer solved a mathematical problem. That was impossible for binary computers in 15 minutes. This means that there are limits to quantum and binary systems. The binary system is faster and better. When. Calculations are easy and small. But when complexity in calculations grows. That turns. Quantum computers. More suitable. The big problem is how to detect errors in quantum systems. The only thing that can inspect calculations that are too complex for binary systems is another quantum computer. 

When a quantum computer inspects and tries to detect errors. 

This system calculates all calculations backward. That means that getting the answer takes 15 minutes. And confirming that answer takes another 15 minutes. This looks impressive. But there are many problems. The biggest problem is that the quantum computer. It follows similar laws as other computers.

Increasing calculation capacity. It provides resources for more complex simulations and modelling. So, researchers bring more complicated formulas to computers. Increasing computing power. That increases the complexity of formulas. 

When researchers want to make a simulation. To simulate gas flow. They search for what that gas flow does. And then they try to make a fractal that follows those routes. The same way. It is possible to create material simulations at the quantum level. This thing. It makes it possible. To create very high-accuracy simulations. Those simulations are needed to control those materials with extreme accuracy. 

https://postquantum.com/quantum-computing/rcs-benchmark/

https://scitechdaily.com/quantum-computer-solves-a-problem-in-15-minutes-that-classical-methods-cant-practically-compute/

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

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

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

The Jacobian conjecture is partially solved. (Jacobian conjecture is disproved in three or more dimensions)

 

Jacobian conjecture is disproved in three or more dimensions. But in the two-dimensional model. It remains open. 

“A remarkably simple three-dimensional function has exposed an unexpected limit to a century-old mathematical conjecture. Credit: Shutterstock. An AI-assisted counterexample disproves the Jacobian conjecture above two dimensions while leaving its original two-dimensional form open.” (ScitechDaily, AI Helps Crack an 87-Year-Old Math Conjecture With One Tiny Formula)

AI is the basis for new mathematics. The ability to share problems and connect them again makes that tool impressive. In cases like the Riemann conjecture, AI is the best tool in the business. The system can share the number line across different servers. And then that makes this system an impressive code-breaking tool. In this case, the AI tries to make calculations. That encryption system is made backwards. And that opens the original bits to an attacker. The Riemann conjecture generates binary numbers. That connects to some other formula. 

The last impressive thing that we see is the solution to the Jacobian conjecture. That thing is important in linear algebra. This makes it a tool that is suitable for economics. In that case, the Jacobian matrix is used. For resource optimization. The next examples are things. There, the Jacobian matrix is used. 

Mathematics: An important part of methods for solving differential equations.  

Engineering: Helps model and optimize complex systems.  

Economics: Analyzes economic models and optimizes resources.  

Computer Science: Used in neural networks and machine learning algorithms to aid optimization.  

So, the Jacobian matrix is a versatile tool that provides deep insight into both mathematical and practical problems.

AI  shows that the Jacobian Conjecture is wrong. If. There are three or more dimensions. This means that all polynomial functions cannot be introduced backwards. And that means all calculations. They cannot be checked simply by calculating all calculations backward.

But. The Jacobian conjecture is wrong only in dimensions ≥3. The two-dimensional problem is open. So, dimensions 2≥ are still open. This means. That.

Encryption algorithms that involve the Jacobian Conjecture. It should have three or more dimensions. 

“In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse.” (Wikipedia, Jacobian conjecture). The conjecture can benefit the Jacobian matrix in its models. 

“In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi (1804-1851).”(Wikipedia, Jacobian matrix and determinant) 

“If m = n, then f is a function from Rn to itself and the Jacobian matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant. “ (Wikipedia, Jacobian matrix and determinant) 

In texts. The Jacobian determinant sometimes is referred to as "the Jacobian".(Wikipedia, Jacobian matrix and determinant) 

“The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ Rn if the Jacobian determinant at p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it is negative, f reverses orientation. The absolute value of the Jacobian determinant at p gives us the factor by which the function f expands or shrinks volumes near p; this is why it occurs in the general substitution rule.” (Wikipedia, Jacobian matrix and determinant)

https://scitechdaily.com/ai-helps-crack-an-87-year-old-math-conjecture-with-one-tiny-formula/

https://esimerkkeja.com/jacobin-matriisi-esimerkki-ja-sen-sovellukset-matematiikassa/

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

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

https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Jacobian_determinant