Magic Theory: Property Space
The first article, Introduction to the theory of magic , gave an overview of the main topics of the theory of magic. We now turn to the consideration of the property space, its features and characteristics.
Let M be given. Denote by U the set of all subsets of M. We will call U a universe, and M a generator set.
Independent properties
Let the set S of operators s <U, R, R> = f be also given. The angle brackets indicate the sets of which the operator acts, and, as usual, R is the set of real numbers. That is, s (u, t, y) = f, where u belongs to U, t and y belong to R, f (t) is a function from R to R, and y = f (t).
For given s and u, the operator thus defines the family of functions F [s, u], which is the domain of values of the operator for the specified element of the universe.
We also require the following condition:
if f0 and f1 belong to F, then f0 (t0) = f1 (t0) => f0 (t) = f1 (t) for any t> t0.
The operator s is called an independent strictly deterministic property, and the function f is called a trajectory. If s is applied to the element of the universe u [i], that is, the range of s (u [i], t, y) is not empty for at least one point (t, y), then they say that u [i] has the property s . It should be noted that the property s is equivalent to a vector field, when the trajectories are everywhere continuous and differentiable.
Dependent properties
The value of an independent strictly deterministic property in a closed system can always be determined if the value of this property is known at some point and thus one of the trajectories of the family of values is selected. In contrast, to find the value of the dependent property d, we need to know the trajectories of other properties that affect d. More formally
d (u, t, y, v [0], ..., v [n]) = f,
where v [i] = s [i] (u) (t), that is, the value at time t of the current trajectory of the i-th property of the element u.
Interdependent properties
In the most general case, properties can be related to each other so that a change in the value of a property from a group of interrelated properties will change the values of all other properties. Then, knowing the values of all the properties of the group at some point in time, we could determine the values of these properties in the future.
Stochastic properties
Although often the future values of a property are quite predictable from the value of a group of properties at the current moment, we may also encounter properties with probabilistic behavior, the subsequent values of which are given by the distribution function. In many practical cases, the evolution of stochastic properties is described by a Markov chain with continuous time.
Derived properties
Speaking about dependent properties, we have implied until now the dependence on the merits, due to objective reasons, sometimes resulting from hidden parameters. This connection can be called objective or ontological. On the other hand, a property can be dependent on another simply by definition; For example, if we introduce the concept of a double mass equal to the product of a mass by two, then the meaning and dependence of the double mass on the mass are not related to the physical meaning of the latter. A dependency of this kind will be called syntactic, and syntactically dependent properties - derived properties.
We define I as an ordered pair <u, S (u)>. Here u is an element of the universe, S (u) = {s [0], ..., s [n]} is the set of all properties of u.
Let the observation operator be given is (I, t) = ( [0] (s [0]) (t), ..., E [n] (s [n]) (t)) = (v [0], ... , v [n]).
v [i] belongs to R and is called the observed value of the i-th property at time t.
We will call the pair <I, E> an object. Since there is a one-to-one correspondence between the set of objects and the set of elements of the universe, we will freely use the expressions “properties of the element” and “properties of the object” interchangeably.
An arbitrary set of properties is called a genus. Consider the genus G = {s [0], ..., s [n]}. Let an Obj object be given such that the set of its properties S (Obj) is a superset of G. Then they say that Obj is an object of the genus G. But the properties belonging to the difference S (Obj) \ G are called additional or incidental to the genus G.
In practice, a genus can be identified by applying multiple operations (most often, an intersection operation) on the sets of properties of the specified objects. On this path, however, it is easy to admit a minor minor property in the definition of gender. Therefore, as we will see in the future, when determining the true name, the operation of fuzzy belonging to the set is used.
The area of instantaneous values (V) of a property is the union of the ranges of all trajectories of this property. If V has no singularities, that is, any value in the domain is achievable equally, then the metric function can be defined as
[1] r (v0, v1) = k * | v0-v1 |,
where v0 and v1 are values from the domain V, and k is called the conservativeness of the property.
But in the case of boundary singularities, it is necessary to introduce a metric with their account. Thus, a typical case is the asymptotic restriction of V from above; an example is the maximum speed limit, and many characteristics of human states and abilities behave in the same way: happiness, charisma, wisdom, etc. Let's define the function r as follows:
[2] r (v0, v1) = k * | v0 + 1 / (max - v0) ^ n - v1 - 1 / (max - v1) ^ n) |.
Far from the edge, when 1 / (max - v) ^ n is much less than one, function [2] is extremely close to function [1], and this almost coincidence encompasses the longer the interval, the larger the value of the exponent n.
Calculate distance for derived property
We now consider some derived property s = s (a, b, ...), where a, b, ... are basic properties for s. The transformation s0 → s1 then consists in the corresponding transformations of the basic properties a0 → a1, b0 → b1, ... Moreover, these transformations occur in such a way as to minimize the use of mana, that is, along the minimum common path. Take, for example, momentum
p = m * v
in a non-relativistic area. Let the current value be p0 = m0 * v0. We want to increase the momentum to p1, which can be achieved by changing the speed or mass, or both properties simultaneously. The distance is calculated as
r = k1 * (m - m0) + k2 * (v - v0) = k1 * (p1 / v - m0) + k2 * (v - v0)
To find the minimum distance, we require the equality of the first derivative to zero:
dr / dv = - k1 * p1 / v ^ 2 + k2 = 0, =>
v = sqrt (p1 * k1 / k2),
m = p1 / v = sqrt (p1 * k2 / k1),
r = k1 * (sqrt (p1 * k2 / k1) - m0) + k2 * (sqrt (p1 * k1 / k2) - v0) =
= 2 * sqrt (p1 * k1 * k2) - k1 * m0 - k2 * v0.
Of course, for subluminal velocities we would have to take into account the growth of mass with speed, and the calculation of the target velocity and rest mass would become more complicated. Additionally, we should take into account the change in the resistance of the object depending on the state, but for simplicity and clarity, we have now neglected this.
Transformation rules
')
Imagine that we need to raise a pound weight to a height of 10 meters.
It’s pretty obvious how to calculate the distance between two property values. But how could we determine the distance between a property and its absence? To clarify this question, we note first of all that non-existence is not something completely separate from existence, something point; ghosts, people from the other world or from other worlds, sometimes appearing at dusk, indicate the possibility of not existing in different degrees, with different intensity, so to speak.
Accordingly, with respect to each property, it is customary to talk about the level of non-existence, expressing the value of a property by a complex number, the real part of which is equal to the value of the property, and the imaginary one determines the level of non-existence, that is, the degree of remoteness from our world. We assume - conditionally - that if the imaginary part is zero, then the object has a property, and vice versa. It would be more correct to say, of course, that an object always has all possible properties anyway; Some properties lie in our world, while others - in parallel worlds.
Let two states be given s0 = a0 + i * b0 and s1 = a1 + i * b1. The distance between the corresponding worlds is calculated as Kw * | b0 - b1 |, where Kw is another universal constant - the distance coefficient between parallel worlds.
Each parallel world has its own metric. It is known, for example, that there is a world with a discrete metric in which the distance between any two different states is one. Therefore, complex and energy-intensive transformations are often performed through suitable parallel spaces. In addition, extremely close spaces sometimes influence each other, mix with each other, and the laws of such interaction and their use are studied in the course of necromancy.
Properties
The necessary is given in intuition.
We know what home, family, food is,
what are the circles of the day, year and life.
But start looking for precise definitions,
and here you are in the middle of the unfamiliar world
where everything, including you, is in question.
Objection to thinking
Let M be given. Denote by U the set of all subsets of M. We will call U a universe, and M a generator set.
Independent properties
Let the set S of operators s <U, R, R> = f be also given. The angle brackets indicate the sets of which the operator acts, and, as usual, R is the set of real numbers. That is, s (u, t, y) = f, where u belongs to U, t and y belong to R, f (t) is a function from R to R, and y = f (t).
For given s and u, the operator thus defines the family of functions F [s, u], which is the domain of values of the operator for the specified element of the universe.
We also require the following condition:
if f0 and f1 belong to F, then f0 (t0) = f1 (t0) => f0 (t) = f1 (t) for any t> t0.
The operator s is called an independent strictly deterministic property, and the function f is called a trajectory. If s is applied to the element of the universe u [i], that is, the range of s (u [i], t, y) is not empty for at least one point (t, y), then they say that u [i] has the property s . It should be noted that the property s is equivalent to a vector field, when the trajectories are everywhere continuous and differentiable.
Dependent properties
The value of an independent strictly deterministic property in a closed system can always be determined if the value of this property is known at some point and thus one of the trajectories of the family of values is selected. In contrast, to find the value of the dependent property d, we need to know the trajectories of other properties that affect d. More formally
d (u, t, y, v [0], ..., v [n]) = f,
where v [i] = s [i] (u) (t), that is, the value at time t of the current trajectory of the i-th property of the element u.
Interdependent properties
In the most general case, properties can be related to each other so that a change in the value of a property from a group of interrelated properties will change the values of all other properties. Then, knowing the values of all the properties of the group at some point in time, we could determine the values of these properties in the future.
Stochastic properties
Although often the future values of a property are quite predictable from the value of a group of properties at the current moment, we may also encounter properties with probabilistic behavior, the subsequent values of which are given by the distribution function. In many practical cases, the evolution of stochastic properties is described by a Markov chain with continuous time.
Derived properties
Speaking about dependent properties, we have implied until now the dependence on the merits, due to objective reasons, sometimes resulting from hidden parameters. This connection can be called objective or ontological. On the other hand, a property can be dependent on another simply by definition; For example, if we introduce the concept of a double mass equal to the product of a mass by two, then the meaning and dependence of the double mass on the mass are not related to the physical meaning of the latter. A dependency of this kind will be called syntactic, and syntactically dependent properties - derived properties.
Object and genus
Don Pedro comes from a noble family.
Rod Jose - from the lowest.
A skilled magician will easily change the first and second.
Arguments for equality
We define I as an ordered pair <u, S (u)>. Here u is an element of the universe, S (u) = {s [0], ..., s [n]} is the set of all properties of u.
Let the observation operator be given is (I, t) = ( [0] (s [0]) (t), ..., E [n] (s [n]) (t)) = (v [0], ... , v [n]).
v [i] belongs to R and is called the observed value of the i-th property at time t.
We will call the pair <I, E> an object. Since there is a one-to-one correspondence between the set of objects and the set of elements of the universe, we will freely use the expressions “properties of the element” and “properties of the object” interchangeably.
An arbitrary set of properties is called a genus. Consider the genus G = {s [0], ..., s [n]}. Let an Obj object be given such that the set of its properties S (Obj) is a superset of G. Then they say that Obj is an object of the genus G. But the properties belonging to the difference S (Obj) \ G are called additional or incidental to the genus G.
In practice, a genus can be identified by applying multiple operations (most often, an intersection operation) on the sets of properties of the specified objects. On this path, however, it is easy to admit a minor minor property in the definition of gender. Therefore, as we will see in the future, when determining the true name, the operation of fuzzy belonging to the set is used.
Metrics
Man feels no more happiness
and the rate of change.
Happiness is completely destitute
grows exponentially from the first crumbs,
then goes into the realm of linear happiness,
and then slows down and tends asymptotically
to absolute happiness.
Therefore, the wise ruler
subjects are always in the first interval,
led to the linear domain by sovereign
and thrown back by his enemies.
Management principles
The area of instantaneous values (V) of a property is the union of the ranges of all trajectories of this property. If V has no singularities, that is, any value in the domain is achievable equally, then the metric function can be defined as
[1] r (v0, v1) = k * | v0-v1 |,
where v0 and v1 are values from the domain V, and k is called the conservativeness of the property.
But in the case of boundary singularities, it is necessary to introduce a metric with their account. Thus, a typical case is the asymptotic restriction of V from above; an example is the maximum speed limit, and many characteristics of human states and abilities behave in the same way: happiness, charisma, wisdom, etc. Let's define the function r as follows:
[2] r (v0, v1) = k * | v0 + 1 / (max - v0) ^ n - v1 - 1 / (max - v1) ^ n) |.
Far from the edge, when 1 / (max - v) ^ n is much less than one, function [2] is extremely close to function [1], and this almost coincidence encompasses the longer the interval, the larger the value of the exponent n.
Calculate distance for derived property
We now consider some derived property s = s (a, b, ...), where a, b, ... are basic properties for s. The transformation s0 → s1 then consists in the corresponding transformations of the basic properties a0 → a1, b0 → b1, ... Moreover, these transformations occur in such a way as to minimize the use of mana, that is, along the minimum common path. Take, for example, momentum
p = m * v
in a non-relativistic area. Let the current value be p0 = m0 * v0. We want to increase the momentum to p1, which can be achieved by changing the speed or mass, or both properties simultaneously. The distance is calculated as
r = k1 * (m - m0) + k2 * (v - v0) = k1 * (p1 / v - m0) + k2 * (v - v0)
To find the minimum distance, we require the equality of the first derivative to zero:
dr / dv = - k1 * p1 / v ^ 2 + k2 = 0, =>
v = sqrt (p1 * k1 / k2),
m = p1 / v = sqrt (p1 * k2 / k1),
r = k1 * (sqrt (p1 * k2 / k1) - m0) + k2 * (sqrt (p1 * k1 / k2) - v0) =
= 2 * sqrt (p1 * k1 * k2) - k1 * m0 - k2 * v0.
Of course, for subluminal velocities we would have to take into account the growth of mass with speed, and the calculation of the target velocity and rest mass would become more complicated. Additionally, we should take into account the change in the resistance of the object depending on the state, but for simplicity and clarity, we have now neglected this.
Transformation rules
- The transformation time is zero. Although preparatory and related actions may take some time, the actual movement from the initial to the final state occurs instantly.
- As a result of the transformation, a transition occurs to a different trajectory of the changed property; the subsequent natural evolution of the property value follows this path.
- If the result of a magical operation is a completely unnatural value, then the transition to the nearest trajectory occurs according to a linear law.
s = s0 + BNV * k * t.
Coefficient k - conservative property,
| BNV | | - a fundamental constant describing the rate of return to the natural trajectory. - After transformation, the object tends to preserve the new state. This happens by cyclically repeating the auto-spelling of the property value adjustment, and the object is absorbed by the mana from the magic background
m = M - (M - m0) / exp (ka * t),
where m0 is the initial amount of accumulated mana,
M - the maximum amount of mana that an object can accumulate,
ka is the coefficient of mana absorption rate, depending on the density of the magic background.
Why this equation has this form will be explained later, for now, note that it is similar to the kinetic equation of a first order chemical reaction.
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Example: Levitation
I think now about missed opportunities.
The fact that it should have diligently learned many things
but first of all - levitation.
Last word sentenced to hang.
Imagine that we need to raise a pound weight to a height of 10 meters.
- We will change the height from 0 to 10 m. The required mana is m ~ k * 10, where k is the conservatism of the height.
However, there is one difficulty. The area of space where the body moves is usually occupied by at least air. The problem becomes even more obvious if we try to reduce the height (z coordinate) of the body standing on the ground. In such cases, the law of the occupied state is valid: if the state in which the Obj1 object is moved is occupied by the Obj2 object, then Obj1 and Obj2 exchange states. This law applies not only to teleportation. Suppose we are going to make a king out of a peasant, but there can be only one king (in our world). Then, as a result of the success of our magic, the king and peasant will change statuses.
Now note that Obj2 is generally larger than Obj1, respectively, to compensate for the Obj2 transformation, it will be necessary to transform objects adjacent to Obj1. Another question arises whether to consider something as a separate object or part of a group. This is the case, for example, with clothes or a closed padlock on a chest. This, however, is the subject of a separate discussion; at the moment it is important that the magician must provide mana for the whole operation, which, as we have seen, consists of several parts.
In the case of simple levitation, an exchange of the body and an equal volume of air occurs, that is, the mana is proportional to k * r, as we already said. - Immediately after the magic operation, the height of the weight begins to change along the natural trajectory of this property h = h0 - v0 * t - g * t ^ 2. Since v0 = 0, then h = h0 - g * t ^ 2.
- After the Taq time interval (this is one of the universal constants, called the auto-spell period), the weight tries to restore the height h0 using the aq auto-spell. Manu to move it accumulates from the background, as indicated above. If mana is not enough, then it is all spent to move to a height of h1 <h0.
- The weight again falls, but the initial speed is now greater than zero and increases with each repetition of auto-spells, and, consequently, the path passed before the height correction increases.
- With each repetition of aq, fatigue (a function of the spell) accumulates, in the intervals between repetitions it falls. When the critical value is reached, the auto-spell cycle is terminated, and the object goes into a state of prolonged relaxation. By the way, the magician fatigue accumulates similarly.
Levels of non-existence
Not completely dead and a little alive -
Here is the main material observations and experiments
non-existence specialists — necromancers.
Tales of nothingness
It’s pretty obvious how to calculate the distance between two property values. But how could we determine the distance between a property and its absence? To clarify this question, we note first of all that non-existence is not something completely separate from existence, something point; ghosts, people from the other world or from other worlds, sometimes appearing at dusk, indicate the possibility of not existing in different degrees, with different intensity, so to speak.
Accordingly, with respect to each property, it is customary to talk about the level of non-existence, expressing the value of a property by a complex number, the real part of which is equal to the value of the property, and the imaginary one determines the level of non-existence, that is, the degree of remoteness from our world. We assume - conditionally - that if the imaginary part is zero, then the object has a property, and vice versa. It would be more correct to say, of course, that an object always has all possible properties anyway; Some properties lie in our world, while others - in parallel worlds.
Let two states be given s0 = a0 + i * b0 and s1 = a1 + i * b1. The distance between the corresponding worlds is calculated as Kw * | b0 - b1 |, where Kw is another universal constant - the distance coefficient between parallel worlds.
Each parallel world has its own metric. It is known, for example, that there is a world with a discrete metric in which the distance between any two different states is one. Therefore, complex and energy-intensive transformations are often performed through suitable parallel spaces. In addition, extremely close spaces sometimes influence each other, mix with each other, and the laws of such interaction and their use are studied in the course of necromancy.
The mentioned issues require further disclosure.
- Observation operator
- Mana accumulation
- Fatigue
- The law of the busy state
- Parallel Worlds
Source: https://habr.com/ru/post/190936/
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