Is it possible to reduce the study of the set of solutions of a polynomial equation to the study of simpler objects? Ways to remove a tattoo at home Benefits of tattoo removal with a Q-Switched Nd:YAG laser

The central concept that predetermines the structure of the vast majority of research in algebraic geometry is the concept of invariant. The idea of ​​invariants is easy to understand. Let's assume that there are two objects (in this case, two sets of solutions to certain equations), and we need to find out whether they are equal. It is very difficult, if not impossible, to do this - how to compare? But it is possible to establish some properties of objects, and if these properties turn out to be not identical, then the original objects are obviously not equal. For example, you can check whether two texts match by comparing their volume. If the size of the texts is different, you don’t have to look at them. In algebraic geometry, one of the simplest invariants is the dimension or connectedness of the desired set. The converse, of course, is not true: from the equality of two invariants one cannot conclude anything about the equality of the original objects. But even such partial knowledge is already good. And complete happiness will come if it is possible to prove the opposite statement (in other words, if the selected set of invariants uniquely defines the original object). The Hodge hypothesis is just one such tempting statement. If it turns out to be true, the study of a large and complex class of algebraic varieties (the so-called sets made up of pieces, each of which is a set of solutions to some polynomial equations) will actually be reduced to the study of much simpler objects.

4. Yang-Mills theory.

Preface. The equations of quantum physics describe the world of elementary particles. Physicists Young and Mills, having discovered the connection between geometry and particle physics, wrote their equations. Thus, they found a way to unify the theories of electromagnetic, weak and strong interactions. The Yang-Mills equations implied the existence of particles that were actually observed in laboratories all over the world, so the Yang-Mills theory is accepted by most physicists despite the fact that within the framework of this theory it is still not possible to predict the masses of elementary particles.

Problems with quantum Yang-Mills theory are like a ball that has hit the mathematical field. Physics requires from mathematics a theory that would describe the ideas and relationships accumulated by physicists, and mathematics cannot yet provide a suitable apparatus.

Interactions between any natural objects (bodies, particles, waves) are divided into four types: gravitational, electromagnetic, strong and weak. In physics, attempts have not stopped to create a theory that would explain all these interactions, the so-called general field theory. Yang-Mills theory is a mathematical language that has allowed physicists to describe three of the four fundamental forces of nature (gravity is not yet amenable, so it is too early to talk about a general field theory). Chen Ning Yang and Robert Mills published a short paper in 1954 that still serves as the basis for quantum field theory. What distinguishes quantum theories from classical ones? In the classics, the main object of study is a particle or body. Bodies interact with each other. Interaction (as has been generally accepted since the time of Newton) is carried out through fields that are created by particles and affect other particles. For example, a charged particle creates an electromagnetic field, a particle with non-zero mass creates a gravitational field. Let us also note one of the key ideas of physics, both classical and quantum: a particle is equivalent to a set of fields that it creates, because any interaction with other particles is carried out through these fields; from the point of view of physics, considering the fields generated by a particle is the same as considering the particle itself.

In the quantum approach, the same particle can be described in two different ways: as a particle with a certain mass and as a wave with a certain length. A single particle-wave is described not by its position in space, but by a wave function (usually denoted as y), and its location is probabilistic in nature - the probability of finding a particle at a given point x at a given time t is |Y(x,t)| 2.

How to describe the movement of particles? What laws predict the evolution of the wave function over time? In classical mechanics, motion is carried out according to the principle of least action. For a given mechanical system, it is possible to construct a function (called the Lagrangian), minimizing the integral of which predicts the behavior of the system - the trajectory of moving bodies. In quantum mechanics, the concept of “trajectory” loses its meaning, but the concept of the Lagrangian is retained, and with its help it is possible to predict the behavior of the wave functions of interacting particles.

The question arises: how to take into account the fields of a quantum system when constructing this very Lagrangian? The answer to this question is provided by the so-called quantum field theories. The plural is not accidental: the Lagrangian can be constructed in different ways, the only question is which one describes reality better. Let's return to wave functions. When measuring, the probability of finding a particle at a given point is equal to the square of the modulus of the wave function. This means that the function can be multiplied by any complex number with unit modulus (phase shifted), and nothing will change: the probability of finding a particle at each specific point will remain exactly the same. In fact, we will never know the specific form of the wave function, and it does not interest us; but it is very interesting what operations can be performed on the wave function so that the properties of the system do not change.

Similarly, the Lagrangian in general is best characterized by those transformations that it “withstands,” that is, under which the properties of the system do not change. For example, the phase shift is subject to the Lagrangian, which describes the behavior of a single electron.

A set of such transformations in mathematics is called a group. Groups play a fundamental role in various fields of knowledge - it is the language in which modern science the concept of symmetry is formulated. The group of transformations that appeared in the electron example is called the gauge group. In mathematics it is denoted by U(1), and it is very simple - an ordinary circle on a plane (the set of all rotations around the origin). Similar theories for the strong and weak interactions lead to more complex gauge groups SU(2) and SU(3) (the latter is equivalent to a three-dimensional sphere lying in four-dimensional space).

To get to Yang-Mills quantum theories, there are two important steps left to take. The first step is to supplement the requirements of global invariance with the requirements of local invariance. In the previous example, the entire function had to be multiplied by a number with a unit module at once. But nothing would change if this multiplication occurred not in the entire space, but in some part of it. In mathematics, this is called the transition from groups of global transformations to groups of local transformations.

The second fundamental point is that in Yang-Mills theories one has to use so-called non-Abelian transformation groups. Because of this, the principle of superposition is violated: if several fields act on a particle at once, their combined effect can no longer be decomposed into the action of each of them individually. This happens because in this theory not only particles of matter are attracted to each other, but also the field lines themselves! Because of this, the equations become nonlinear and the entire arsenal of mathematical techniques for solving linear equations cannot be applied to them. Finding solutions and even proving their existence becomes an incomparably more difficult task.

Young and Mills proposed a general form of Lagrangians that were supposed to lead to success. Based on the Yang-Mills theory, the electric and weak theories were first combined, and then Murray Gell-Mann built the theory of the strong interaction. In this theory, which brought Gell-Mann the Nobel Prize, quarks appeared to explain the observed effects - particles with a fractional electric charge that make up protons, neutrons and other not at all elementary particles. The theory of strong interaction is called quantum chromodynamics

In order for a theory to describe the strong interaction, it must have three properties that are completely unusual for classical theories:

mass gap (“gap in the mass spectrum”, lower limit on  “energy spectrum”)

quark confinement: quarks cannot “get out” beyond the limits of elementary particles;

certain symmetry violations.

Numerous experiments have shown that quantum chromodynamics has these properties. However, this has not been proven mathematically. The task is to construct for each compact simple gauge group a quantum Yang-Mills theory in the four-dimensional space-time continuum that has the mass gap property - in other words, a theory whose Hamiltonian spectrum H (in the quantum case, the analogue of the classical Lagrangian is called the Hamiltonian ) would be separated from zero.

5. Birch and Swinnerton-Dyer hypothesis.

The Birch and Swinnerton-Dyer conjecture is related to the description of the set of solutions to certain algebraic equations in several variables with integer coefficients. An example of such an equation is the expression x 2 + y 2 = z 2. Euclid gave full description solutions to this equation, but for more complex equations, finding solutions becomes extremely difficult.

Getting rid of a tattoo is a long and painful process. Usually they go to a salon for this, but is it possible to remove a tattoo at home? How to do this as quickly as possible?

Among the commonly used means for removing unwanted tattoos are:

· vinegar essence;

· milk;

· potassium permanganate.

Let's consider these methods separately, assessing their effectiveness and harmlessness.

How to remove a tattoo at home?

How to remove a tattoo with iodine

Soak a cotton swab or disk in a 5% iodine solution, depending on the size of the tattoo. Smudge the design, being careful not to touch the skin without the tattoo. Repeat three times a day for several months until results are achieved. Do not cover the greased area with anything, otherwise there will be a burn.

The skin will itch and peel, but you should not peel it. The layer of skin should peel off on its own, after which there will be a wound underneath. Until it heals, you cannot continue using iodine. Then continue until the tattoo disappears along with the dying layers of skin. You should not sunbathe at this time under any circumstances. If the itching is unbearable, you can lubricate the skin with a moisturizing antibacterial cream at night.

If you are allergic to iodine, this method is not suitable.

How to remove a tattoo with salt

This method will require salt and a lot of patience. Algorithm of actions:

· prepare a salt mass from two tablespoons of salt and a spoon of water

Apply the paste with a cotton pad to clean, shaved skin with a tattoo.

Rub in circular movements for half an hour

Rinse with cool water, dry and apply a bandage.

repeat every day for one to four months

The pattern will fade after just a couple of treatments, but it will take quite a long time for it to completely disappear. In principle, there are no ways to quickly remove a tattoo at home. This is always a long and rather painful process.

Just literally 5 years ago it was believed that if you decided to get a tattoo, it would be forever. But with the development of modern technologies, removing an unwanted or simply boring drawing is not difficult. The most popular method is laser tattoo removal. There are other methods, which, however, are becoming less and less relevant every year due to their poor effectiveness (moreover, some can even lead to skin injuries). Therefore, today laser therapy is confidently replacing all other methods, and most specialists prefer laser as the most advanced tattoo removal technology.

Today, the most popular method is laser tattoo removal. The operating principle allows you to destroy dye granules without damaging the skin. Therefore, the method is suitable for absolutely any skin type and is the safest. Experienced tattoo artist Marina Orlova tells how the procedure goes, what determines the success of tattoo removal, how long it takes to wait for the hated design to completely disappear.

How does the procedure itself work?

The specialist passes the laser emitter over the entire area covered with the pattern. Under the influence of the laser, the dye particles split and disintegrate into small pieces, which will be easily removed over the next 30 days on our own body. As a result, the tattoo will fade and disintegrate before your eyes. It is very important that the surface of the skin is not damaged and infection is excluded.

Who is the easiest to remove a tattoo?

People with fair skin, who have unwanted tattoos on their arms, chest, buttocks and legs are the main lucky ones. Tattoos on dark-skinned people or tattoos on ankles or fingers, that is, on those areas of the body where the skin is quite thin, are more difficult to remove. After the first session, the master will be able to more accurately say how many sessions will be needed to completely remove the tattoo.

What defines success

Complete tattoo removal depends on several factors: the size, location, age and color of the tattoo, as well as the patient's health and, therefore, the body's ability to heal itself. Old tattoos (more than 10 years) are more difficult to remove than fresh ones, professional pigment is more difficult to work with than “homemade” ones. To completely remove a tattoo, a series of procedures are required at intervals of 3-4 weeks.

More developed than ever before. Tattoo parlors are open in almost all major cities. Any adult who is willing to pay for this procedure can acquire their own body art. Ideally, a tattoo should be an organic addition to the image and delight its owner until the end of his days. But in practice it is often found bad tattoos, which not only do not bring pleasant emotions, but can also seriously ruin the life of their owner.

What is "partak"?

It is worth noting that unsuccessful tattoos can be divided into two categories:

1) poorly executed;

2) ugly or simply boring.

In the first case, we are talking about “partaks” - this is what masters and everyone interested in the art of applying body images call unsuccessful tattoos in their jargon. This term is suitable to refer to all ugly tattoos when the intended picture looks distorted and sometimes downright unaesthetic. Where do “partaks” come from? There are several options: insufficient qualifications or inexperience of the specialist, the use of low-quality equipment and paints, improper skin care during the healing period. People who are tired of the existing image on their body often want to remove tattoos. It is for this reason that psychologists advise thinking three times (or better, within a few months): do you really want to wear the chosen image for the rest of your life?

“Cover up”: covering up boring tattoos

So, what to do if you have a bad tattoo and want to get rid of it? You should start with a visit to a good tattoo parlor. One of the most simple options corrections - “cover up” (“ new job over the old one"). It's about about covering an existing tattoo with a new one. If you just don’t like your unsuccessful tattoos a little, you can try to agree on making minor changes to the design. However, few craftsmen like to finish other people’s drawings, and it is quite possible that you will have to spend some time searching for a specialist who will agree to help.

Is it possible to remove a tattoo?

Tattoo removal is a procedure for which about 30 percent of tattoo owners sign up. It is indeed possible to completely remove pigment from the skin. Today there are two options: laser and surgical removal. In the first case, unsuccessful tattoos are literally burned out of the layers of skin using a special medical device with a laser. Surgical removal is an outdated method, which involves cutting out entire layers of skin along with the tattoo. It is worth noting that complications are possible with any procedure. Moreover, tattoo removal is not a cheap procedure, and after it is performed, scars will most likely remain.

An unsuccessful tattoo or, as it is popularly called, “partak” can become a real problem and obstacle in life, for example, at the time of hiring, joining organizations with conservative morals, or even in relationships with the opposite sex. A damaged area of ​​skin will not add elegance and aesthetics appearance, but, on the contrary, will create additional discomfort every day. Then the owners of an unsuccessful tattoo resort to removing the ink from the skin, since modern world it has become much easier and more effective, but some, at their own peril and risk, decide to carry out the procedure at home. Let's take a closer look at how to remove a tattoo yourself with visual examples.

Types of paint and classic mixing methods

The main classification of tattoo ink is based on color scheme, since the base of the solution remains unchanged - only pigment is added to create a certain shade. Initially, there was only one color for a tattoo - black, which limited the artist’s capabilities, but in the modern world there are already many different colors of paint to create the most vivid and colorful picture. It is worth noting that the appearance of colored inks significantly complicates the process of tattooing due to its composition.

There are several most popular ways to get rid of a tattoo:

• Coverup or overlap. If you have a completely democratic attitude towards tattoos, but you are simply not satisfied with the image on your skin, made, for example, in your youth out of stupidity, then you can simply cover up your “partak”. To do this, it is necessary, after consulting with the artist, to select a sketch you like, which will be larger and brighter than the current tattoo, and put a new one on top of the old one, fresh tattoo. This method allows you to completely hide the unsuccessful image and become the owner of a pre-thought-out beautiful tattoo.

  Find a tattoo artist with extensive experience in covering up old tattoos.

• Tattoo cream. There are many special creams, the effect of which is aimed at completely destroying the tattoo. Interacting with chemical composition paints, cream helps the skin actively reject the tattoo, and over time the design will disappear. The procedure lasts about a month, at the end of which the tattoo becomes covered with a crust and falls off, after which the wound heals. This is a relatively effective method: as a rule, two procedures are enough to achieve the desired effect.
• Mechanical peeling. The procedure involves exfoliating the top layer of skin, which contains the bulk of the paint, and sanding it using a sharp cutter or brush. Afterwards, the recovery process and healing of the wound begins. This method is suitable for those with small tattoos, since after the procedure, as a rule, scars and stains remain.
• Chemical peeling. The presented method, like the previous one, is aimed at exfoliating and removing paint particles, but with the help of special acids and chemical solutions. This is a rather painful and ineffective process that requires repeated sessions.
• Physical laser removal. This is the most effective and practical method, because a laser can remove tattoos of any size, complexity and color. There are certain types of lasers designed for specific shades. During the procedure, the pigment of the paint is split and destroyed, followed by its natural removal from the body through the lymph. The laser also fades the paint, making it less noticeable. After several procedures, you can achieve complete removal of the tattoo from the body.

  Among the proposed tattoo removal methods, laser removal is the most effective and safe.

How to remove a tattoo with a laser (video from experts)

Removal at home with iodine and other means

Although the tattoo removal procedure in professional studios and salons is quite expensive, it is better to overpay the extra money than to jeopardize your own health and life by resorting to the following method of tattoo removal:

1. Iodine. After the procedure, the treated skin receives a severe unsafe burn, becomes crusty and falls off over time along with the paint. After some time, it is necessary to repeat the session again, respectively, to burn the skin to the tattoo again. In this case, there is a risk that in the future you will have to start treating the skin for a severe burn.
2. Potassium permanganate. When resorting to this method, you need to take potassium permanganate powder, pour it only onto the required area, then moisten it with water, for example, using a spray bottle, wrap it with gauze and leave for three hours. The session must be repeated daily. That is, every day your skin will be subject to strong physical and chemical exposure, irritated and peeled for three hours.

   In cosmetology, there is a procedure for glycolic peeling of facial skin. Home experiments with exposure time and concentration of the substance cannot be called successful

3. Glycolic acid. The principle is also based on burning and getting rid of the top layer of skin. This is a rather unsafe and painful method.
4. Milk. Some riskier tattoo owners inject milk under the skin with a syringe to break down and remove the ink. But in this case, immediate rotting and scarring of the skin should be expected.
5. Sand and stone. This truly barbaric method involves simply erasing the tattoo using sand mixed with cream and applied to the desired area. It is necessary to apply pressure and move a stone along the skin with sand and thereby erase the layers of the epidermis. During the procedure, the skin will begin to burn and it will definitely turn red. There is also a risk of infection, since the sand contains a sufficient amount of unsafe bacteria and viruses. By rubbing it into the skin with a stone, you simultaneously tear off the skin “live” and provide free access for infection to your body.
6. Vinegar. In this case, vinegar is applied to the surface of the tattoo for five minutes, after which it is neutralized with peroxide and washed off with water. Exposing the skin to acetic acid for even five minutes is unsafe and can result in severe chemical burns.
7. Salt. This is old folk way, the most secure of all of the above. Its essence lies in the fact that you need to rub dissolved salt into the tattoo area with massaging movements for half an hour. The first result will be noticeable only after four months, but for greater effect, you can add lemon juice to the salt. This method is natural and relatively safe, but not very effective, so you should resort to it if you need to remove a tattoo that has faded over time or paint of a light, unsaturated color, such as white.

Important! It should be noted that independent attempts to remove a tattoo at home can lead to suppuration of the area, the appearance of wounds, scars, painful sensations and the need for professional treatment of damaged skin.

Consequences of building a house yourself (photo)

Are Some Tattoos Really Unremovable?

There are quite a lot of rumors that some tattoos made in an unconventional way cannot be removed. But let's look at each individual case in more detail.

Professional tattoo machines provide a very dense arrangement of pigment in the skin, and special needles allow you to introduce additional amounts of pigment into the skin. Therefore, the removal of such tattoos is much slower

• Gel pen. The use of gel pens as tattoo ink is popular in the military or prisons. The gel pen can be removed from the skin very easily using a laser. Such a dye does not present any difficulties in removing, especially if it is black. Colored gels are less common, but are also amenable to the procedure, even to remove blue and green More sessions will be scheduled.
• Mascara. The use of ink for tattooing is long gone, but there are still owners of such designs. Mascara is usually black in color and is easy to remove with a laser.
• Zhzhenka. Burnt rubber is the most popular in prisons. This is a rather dangerous method that causes suppuration and scars on the skin. Nevertheless, such a dye is easier to remove with a laser than others, due to the large particles and their loose arrangement.
• Special paint for tattoos. Professional dyes are much more difficult to remove because they are permanent, but still, after several sessions of laser treatment, you can completely get rid of the tattoo.

As you can see, any tattoo of any complexity and pigmentation can be removed, but this should only be done in professional studios using methods that are safe for health.

Removing a tattoo is a labor-intensive and difficult process that requires special safety precautions, because we are talking about your health. Removing tattoos at home is a less expensive and even cheaper method that does not require special equipment and professional knowledge, but do not forget that these are risky techniques that may not only not help, but also harm your health and the body as a whole. To ensure the safest tattoo removal procedure, you must contact a specialized studio or salon.