Blog November 2024

8/19/24, 8:34 AM Blog, November 2024 Blog Rastko Vuković November 2024 HOMEPAGE INFORMATICS BLOG SITES: November 2024 (Original ≽) November 2024 » Truth and Lies October 2024 » Representations September 2024 » Duality August 2024 » Games George Boole » srb July 2024 » MiniMax Question: Does Boolean algebra have anything to do with information theory? June 2024 » Options Answer: Yes, and it is a long and interesting story. George Boole (1815– 1864) was the first mathematician to think of and then succeed in calculating accuracy (⊤) as well as inaccuracy (⊥). A "statement" is a mathematical sentence that can only be true or false. A ∨ B is true if at least one of the statements A or B is true, and this is called a "disjunction." A ∧ B is a "conjunction" that is true only when both statements A and B are true. Then we have the statement A ⇒ B, if A then B, is always true except when A is true and B is false, which we call "implication." The negation of true (¬⊤) is false, and the negation of false (¬⊥) is true. There are 24 = 16 binary logical operations, as many true-false combinations in 2×2 = 4 positions, but all of them can be written by disjunction, conjunction, and negation. For example, the implication of A ⇒ B is equal to the statement ¬A ∨ B, which is easy to check by immediately including all (four) possibilities A, B ∈ {⊤, ⊥}. Polyvalent logics, with the values true, false, and at least one maybe, as useful as they are for the simplicity of some statements, are not needed as much because each of their statements can be reduced to statements of binary, Boolean algebra logic. May 2024 » Not really April 2024 » Memory March 2024 » Irregularities February 2024 » Processes January 2024 » Markov chain December 2023 » Drowning November 2023 » Freedom October 2023 » Uniqueness September 2023 » Differences August 2023 » Intelligence July 2023 » Timing June 2023 » Concepts May 2023 » Metrics Almost a century after its discovery, Boolean algebra was overlooked or unimportant until the advent of computers. The accuracy and inaccuracy became "current" or "no current," usually represented by the binary numbers 1 and 0. There is also an as-yet-unnoticed big step in understanding "falsehood" as equivalent to "truth," which is very useful in this information theory. In addition, note that by simultaneous substitutions of "true" and "false," ⊤ → ⊥ at the same time as ⊥ → ⊤, true statements become false and vice versa, and all tautologies (always true) become contradictions (always false). April 2023 » Forces However, truth and falsehood, like the presence of action and the absence of action, are essentially different phenomena. The first lasts, and the second disappears. The April 2022 » https://rvukovic.rs/bloge/2411.html March 2023 » Memories February 2023 » Vitality January 2023 » Minimalism May 2022 » 1/15 8/19/24, 8:34 AM Blog, November 2024 first one is bound by the law of conservation, the second one does not have to be taken into account, and the second one is easily forgotten, possibly even at an exponential rate (Dissonance). However, the properties of the latter are irreplaceable to the former, from building vitality to the ability to understand mathematics. Consensus » srb March 2022 » February 2022 » January 2022 » December 2021 » November 2021 » Question: Can a lie be true? October 2021 » Answer: Metaphorically speaking, yes. Also, if there is a consensus about, say, the past, that may not have happened to us. Thirdly, when we willingly or unwillingly agree to the lies that guide us into a new reality. September 2021 » When a fiction writer, using fictional characters and events, utters forbidden or hard-to-understand truths, it is by way of implication ⊥ ⇒ ⊤, which is the correct statement. It also happens when we speak ambiguously and deliberately inaccurately, causing the interlocutor to read truths between the lines. Or, if we say that, of course, the stars hang in the sky like lamps on the ceiling. Not expecting anyone to believe what we are saying. But this is not, in the full sense, "a lie that is true". Fiction, even when it continues to tell lies, will, in its own way, become reality (Pinocchio). When stories reach their readers, they can find their place in our vitality, in our memories that persist in connections with physical reality. I remind you that what is real can be perceived, lasts, and is true (Relationship). This is how fiction itself, as a "pure" falsehood, can be true. August 2021 » July 2021 » June 2021 » THEME: These topics mostly revolve around algebraic logic in an informatics way. These are questions that I was asked a long time ago, when they didn't seem so interesting until they piled up. Of course, I will not write the history or the teaching of mathematics here either, but as an introduction to new parts of information theory. A group of people in a mutual agreement can live in their "reality" and last unusually long in their "truth." Such is the ability of living beings in general, thanks to the excess of information (the amount of uncertainty) they possess and, on the other hand, the connection of those options with reality. The first side is the vitality that gives them the ability to choose, greater than that allowed by the principle of least effect, and the second is the substance that gives them constancy, therefore truth. After separation, the first disappears and the second survives. Physical reality inexorably runs its course oblivious to fiction and sooner or later reaches a schism with half-truths that would last. Therefore, the question of survival turns to the ability to adapt and to the art of vitality (consciously or not) to always be in the right places at critical moments. The evolution of living beings on earth seems to have "determined" that vitality does not have the skill of duration, so it moved from the maintenance of individuals to the maintenance of biological species. It is possible to live a lie "as if it were true," but not as long as if it were true. Finally, it is possible to defy nature and change it. It is the essence of "excess options," of vitality, in competitions (Strategies), in the economy, in science and technology. We communicate with our physical presence, we change things, and by acting on our body with our excess will, we further adapt to the environment. This is the third allegation when we willingly or forcibly agree to the lies that will lead us to such a changed reality. You can also look at it as a weakness of individuals due to which others manage them. When dreams become reality, lies become truth. So a lie can be true. So much for the macro world. But, in the small world, this "converting" is much more common, almost a normal phenomenon. Quantum World » srb Question: What about truth and lies in the quantum world? https://rvukovic.rs/bloge/2411.html 2/15 8/19/24, 8:34 AM Blog, November 2024 Answer: The extremely small world of physics communicates in packets of the smallest actions and interactions by which it exchanges information. Each such exchange is a gigantic event for them, like the collisions of the planets in our world. However, it is also a much rarer occurrence. The particles of the quantum world "live" in great uncertainty, so much so that "the path of the electron is determined by the measurement," as Heisenberg once noted, thus challenging the previous, classical physical point of view. After submitting the information to the apparatus in the measurement process, the electron is left with less uncertainty and thus greater certainty, with which its path becomes better defined — the explanation of the "strange" statements of Heisenberg from the point of view of this theory of information. The assumed objective uncertainty of this theory means that any state of the electron found by measurement could have been or could not have been before the measurement. This "is" and "is not" speak of reality, that is, of truth and untruth, and one of them is that the "lie" before the measurement becomes the "truth" after. Interestingly, measured phenomena always have "truth" and the law of conservation, while everything else comes down to the lack of it (Quantum Nature Experiment). Continuing this idea, we note that the measurement changes the past of the electron. Consistent with the informational interpretation (Relationship), reality exists as duration (by the law of conservation) and the development of history. Real closed systems contain constant aggregate information about the present and the past, so the electron after the first measurement and communication can again fall into a state of greater uncertainty. Similar reasoning reveals the discontinuous existence of electrons. Not only does a "lie" turn into a "truth" for him, but the reverse also happens to him. There is no continuity in the existence of individual quantum particles. Their space and time are "cellular," punctuated, and given the smallest "packages" of information, analogous to quanta for physical actions. But those positions are their own (proper), and the movements are relative, so the extended point of view of space-time is more important. On average, discrete positions become continuous, so, in essence, the space-time potential is uninterrupted. The discreteness (separations) of general space-time disappears as the coincidences decrease by moving from the world of the small to the world of the multitude. We also see this effect of the law of large numbers from the theory of probability in the fact that "the moon does not disappear when I am not looking at it," which was once an "argument" against Heisenberg's "non-existence of the path of the electron before measurement." Because large bodies communicate with multitudes at once, a single interlocutor is irrelevant to their stability. Vacuum » srb Question: Where do you see infinities in physics? Answer: The uncertainty of, say, the path of the electron before the measurement contains what we can get from it as information. This cannot be contested, as well as the statement "white is white," and then let's notice what the Borel–Cantelli lemma says. When the probability series is convergent: ∞ ∑ Pr(A k ) < ∞, k=1 https://rvukovic.rs/bloge/2411.html 3/15 8/19/24, 8:34 AM Blog, November 2024 then the most finite of the events Ak have cumulative probability one. In other words, whether there are infinitely many random outcomes or not, there are only finitely many actual ones. In this way, it is not possible to deny infinities, at least not as much as it is not possible to accept them in physics. A continuation of the above lemma establishes how a divergent sequence with probability one realizes infinitely many events. Between those two points of view is a gap that does not emit random events or emits them very rarely, with probabilities close to zero. These are uncertainties from which information does not follow or appears so rarely that it is not difficult to believe that they do not exist. By the way, there are already experimentally seen "impossible" emissions of particles from the vacuum. How many scientists have actually seen Andromeda or some other distant galaxy (List of galaxies), and you would be very naive to claim that such do not exist. I mean, we have long since outgrown the "baby period," i.e., all immature beliefs that it makes sense to deal only with what we can see, touch, and eat. So it is with infinities. This is especially true because we have mathematical proofs for them. Evidence that has too often proven to be stronger than our beliefs, intuition, and senses. Modal matrix » srb Question: What is operator diagonalization? Answer: Events ωk ∈ Ω, where k = 1, 2, 3, ..., form a distribution if they are disjoint (when one happens, the other will not) and they happen one at a time. If necessary, we add the event ω0 to the distribution if no event has occurred. The described (Vacuum) distributions give an equivalence to the superposition of quantum physics. The linearity (almost another name) of superpositions and their changes makes them suitable for states and processes. Let's denote them with x and A as applications of vectors and operators of linear spaces. It is particularly suitable when the vector x and the operator A are such that they satisfy the characteristic equation Ax = λx, where λ is some number (real or complex). Thus, λ is the eigenvalue of operator A, and x is their corresponding eigenvector. Vectors and linear operators, especially vector spaces of finite n dimension, have their own matrix representations. Then x = (ξ1, ξ2,..., ξn) vector-row, and x† column-vector, conjugate transposed x. The vector x is covariant, and the vector x† is contravariant. The first and second can be interpretations of dual spaces (Variance). However, the matrix notations of the operators depend on the choice of the basis of the space, and only in standard vector bases (a sequence of zeros and 1s in the kth place) are those matrices diagonal. A square matrix A has a diagonal matrix D if there exists P, an auxiliary matrix (modal matrix), so D = P-1AP. Matrix P = P[x1†, x2†,..., xn†] exists when the eigenvectors xj of the equation Axj† = λjxj†, for j = 1, 2,..., n, are linearly independent when all eigenvalues of λj are also different. This matrix P has column eigenvectors. Inverse P-1 = Q[x1, x2, ..., xn] is P† matrix which multiplied by it gives the unit matrix, QP = I. Along the diagonal unit matrix has ones and all other zeros. Note that the eigenvectors (xj) correspond to states in which their processes (A) do not change direction, so changes in intensity (λ j) can be seen as observable. As AP = [λ1x1†, λ2x2†,..., λnxn†], will be a QAP = D matrix with eigenvalues λ1, λ 2, ..., λn on the diagonal and other zeros. This means that by diagonalizing the matrix, the vector base is standardized while the observables are optimized. These coordinate transformations do not significantly affect states and changes. This can be seen by applying the resulting diagonal matrix: https://rvukovic.rs/bloge/2411.html 4/15 8/19/24, 8:34 AM Blog, November 2024 ⎛ λ1 0 0 λ2 D= ... ⎝0 0 ... ... 0⎞ 0 . . . λn ⎠ to the standard vector space basis: ⎛1⎞ 0 e⊤ , 1 = ... ⎝0⎠ ⎛0⎞ 1 e⊤ , 2 = ... ⎝0⎠ ..., ⎛0⎞ 0 e⊤ n = ... ⎝1⎠ which is then a set of eigenvectors, because D ek† = λk ek† for the same eigenvalues λk and indices k = 1, 2,..., n. Namely: † † λ k = x k Ax k = x k P DP † x k = (x k P )D(x k P ) † = e k D ⊤ k = λk . By default, xkxk† = ∥xk†∥² = 1, that the eigenvectors are normalized. Eigenvectors interpreted in information theory, otherwise such that xkxk† = 1 and ⎜⎟ xkxj† = 0 when k ≠ j, are special samples of normed states that do not communicate with each other. They are exemplary event structures that tolerate each other. Lipschitz » srb Question: What is the relation of observing about the p-norms that are not dual? Answer: The question is from the discussion about the applications of the functional as well as the uniqueness of the participants in the universe (Conjugated). It is very far from what Lipschitz (1832–1903), a mathematician born in Königsberg, did, and yet it is quite good for answering the question if we take Lipschitz's criterion and equivalence. The norms ∥⋅∥a and ∥⋅∥b of the vector space V are Lipschitz equivalent if there are constants μ, ν > 0 that ∥x∥a < μ ∥x∥b and ∥x∥b < ν ∥x∥a for all x ∈ V. Hence, we find the Lipschitz Equivalence metric d = ∥x - y∥, when: da(x, y) < μ db(x, y) ∧ db(x, y) < ν da(x, y) ∀x, y ∈ V, for some constants μ, ν > 0. It can be proved that every norm on ℝn is equivalent to the Euclidean norm as follows: ∥x∥ 2 = √|ξ 1 | 2 + |ξ 2 | 2 +. . . +|ξ n | 2 , where x = (ξ1, ξ2,..., ξn). Hence, we also define the Euclidean metric. The point behind this story is the possibility of reducing p-norms to Euclidean 2norms, and it is also the answer to the question. To some extent, the two given norms are reduced to Euclidean norms, and the intensities of the objects of perception are found in the same relations. Shifts occur that can produce new dual pairs. Abstract spaces are not only for the use of physical ones. They are applied in extremely many different ways, some of which are difficult for all of us to understand, and I believe that it is possible to apply them in ways that are completely incomprehensible to us at the moment. And only one group of those applications refers to the theory of observation. https://rvukovic.rs/bloge/2411.html 5/15 8/19/24, 8:34 AM Blog, November 2024 The focus of subjects of norms that are not dual to the object of observation is blurred—the primary subject will say. But no—the secondary can answer him— because what I see, you do not see. This ranges from the different frequencies of light or sound that individuals can perceive to perceptions in general, including the understanding of proofs or the application of theorems. What theorems of truth establish are, I emphasize, phenomena above mere fiction. Horizon II » srb Question: Where else do you see the touch of the infinite and the concrete? Answer: This question comes from previous conversations (Vacuum), but I'll build on it with the very previous one (Lipschitz). Let's start from Skolem's theorem (Actual vs Ideal), that theories always appear infinitely copied. I will add that they are inseparable parts of a practice that is ultimately unique. Uniqueness is a property of the practice, that is, of the subjects of "perception information," established and widely discussed here. Thus, an interesting question arises: how is it possible to have unique concrete phenomena in a possibly finite number and at the same time infinite templates of deductive theories Skolem. We guess the answer, firstly, in the infinite complexity of those concrete phenomena or, secondly, in the infinity of the concrete phenomena themselves. Including at least one, but not excluding both possibilities together. However, the first assumption is shaky. First, because of the finite divisibility of physical action and information (Packages), and in addition, because then the infinity of the contents of all, even the "smallest" packages, was established. The second possibility offered above is quite consistent: to put the physically measurable, communicative, sustainable, and true in the same basket with the label "Real". On the other hand, there is a vast, constantly expanding universe around us. If there is no end to future time, which is another (potential) infinity from everyday life, then there remain theories about physics beyond the limits of the cosmos visible to us. The universe is 92 billion light-years (the length of the path of light in that many years) in diameter, but its age is only 13.8 billion years. Absurd, but the universe is thought to have expanded faster than light in its beginning (inflation). In addition, with the known cosmology, the folding and stretching of space is possible for its parts and speeds greater than light in a vacuum (c ≈ 300,000 km/s). Investigating the theory of information, however, we will notice that the transmission of messages has losses (Markov chain) due to channel noise and misinformation, so that in the end, it becomes a "black box." After many links, which could be the 13.8 billion years from the Big Bang to today, the output message is always the same no matter what the input was. But that theoretical "phantom exit" is exactly what we see as the "beginning" of the universe (Big Bang). That theory of mine (I still don't have collaborators) predicts a decrease in the information density of the present for the sake of an increasingly long past, from which it is then supplemented. The increasingly old past is an increasingly weak filling of the present, so the oldest one is from the time of the Big Bang. In other words, we see what that "black box" of the long transmission chain shows us. The basic thing in this theory is that what we see does not have to be what it is, so we will consider that beginning as a shaky idea. Aside from the fact that this "illusion" is derived mathematically and cannot be overcome by lower-level proofs. The sparse information of the present means fewer events and a slowing down of the flow of time, which in turn means an even faster expansion of more and more distant galaxies because, due to the limited speed of light, we see them from the past. This is how we arrive at a universe that does not recreate itself (rearranges its older elements) but is constantly created little by little, never being the same. Such constant change is yet another announcement of infinity, in addition to the justopened possibility that the universe itself is an infinity in current expansion. https://rvukovic.rs/bloge/2411.html 6/15 8/19/24, 8:34 AM Blog, November 2024 Life II » srb Question: How does life arise and survive? Answer: This is for "Information of Perception" a matter of theory that is otherwise very abstract, and I will only treat it "superficially," considering the majority of potential readers of the blog. Most recently, in the January 2022 blog, are the basic parts of the answer to the same question (Life), which I am now just explaining for a new topic. A lie is a diluted truth, and truth itself is less informative than uncertainty. According to this theory (minimalism), both are attractive, although not to the same extent. In some combinations, the truth of the lie is not repulsive, the physical substance captures it, and life arises. How, don't ask me. It would be like asking Darwin (1809–1882) for a description of the human genome, whose idea was only hinted at by Schrödinger (1887–1961), but the discovery of the genome is attributed to Watson and Crick (1953). Physical reality is subject to the laws of conservation. Those quantities can be defined in such a way that they are timeless; timeless are also truths, and lies tie their past to them. Fictions and falsehoods do not last differently. That lie that joins itself, like vitality, acquires the appearance of duration and participates in the decline in the present for the sake of an increasingly long past. Thus, it is further thinned and diluted, and the initial benefits of association are lost. Reality and fiction are linked together by the previous set of circumstances and by the system's spontaneous tendency towards less information. If the lie had some independent continuity in the diluted mixture (a living being), it would be more attractive. But it disappears as soon as it is separated, making such mixtures even more deadly. On the other hand, if a lie could survive on its own, it would be the truth. Durability defines physical reality. At least it is as far as this theory is concerned. Existence » srb Question: How to define non-interaction? Answer: Algebra-like orthogonality helps. Interpret states as vectors, which are usually strings or functions. The scalar product is then the information of perception. Specifically, the scalar product of the functions f1(x) = cos(x) and f2(x) = cos(2x) on the interval x ∈ [-π, π] is: π S = ⟨f 1 , f 2 ⟩ = ∫ cos(x) cos(2x) dx = −π π π cos x(1 − 2 sin 2 x) dx = ∫ =∫ −π π cos x sin 2 x dx cos x dx − 2 ∫ −π −π π π = sin x sin 2 x d(sin x) = 0 − 2 ⋅ 0 = 0. − 2∫ −π −π However, the sum of these functions, shown in the previous figure on the left, is: cos(x) + cos(2x) = 2 cos https://rvukovic.rs/bloge/2411.html 3x x cos 2 2 7/15 8/19/24, 8:34 AM Blog, November 2024 which is zero only with abscissas x = ±π and x = ±π/3. Except in those points, this sum is not zero when we interpret it as possibilities of communication, i.e., the interaction of two states, f1 and f2. Although they exist at a given interval, they do not communicate, do not interact — because their scalar product is zero. A superposition (Modal matrix) is a group of interfering waves or a bundle of indeterminacy that can collapse into a quantum physics outcome. We usually represent these elements as orthogonal states into which the package can be decomposed. The possibility of unpacking such packages, even when there are uncertainties, suggests that their contents are permanent and physically real. Their elements are truths. Cancellation » srb Question: Can non-interaction occur when overriding actual states? Answer: Yes. With the help of the picture on the right, let's try to understand one such case. Recall the function: y(x) = a⋅sin(μx + φ) represents a sinusoid of amplitude a, wavelength λ = 1/μ, phase shift φ. The sinusoid graphs are: y1(x) = sin(x), y2(x) = sin(3π/4 + x), f(x) = y1(x) + y2(x), respectively, blue, red, and green. By comparison, we easily find the amplitude, reciprocal wavelength, and phase shift for the first a1 = 1, μ1 = 1 and φ1 = 0, and for the second a2 = 1, μ 2 = 1 and φ2 = 3π/4. If we shear red so that its phase shift becomes φ2 = π, green will approach the abscissa and become f = 0, straight line. Then it is: y1(x) = sin(x), y2(x) = sin(π + x), f(x) = 0. Destructive interference by waves causes their cancellation or overriding to occur. Destructive interference by waves causes their cancellation or overriding to occur. Unlike the previous case (Existence), these coupled elements then do not exist. That these boundary waves at the interval x ∈ [-π, π] will not interact, or will not communicate, we check using scalar multiplication: π ⟨y 1 , y 2 ⟩ = ∫ y 1 (x)y 2 (x) dx = −π π =∫ sin(x) sin(π + x) dx −π =− 1 π ∫ cos(x + π/2) cos(π/2) dx = 0, 2 −π because cos(π/2) = 0. These states are orthogonal, therefore, without interaction. But the elements of this superposition package (y1 and y2) are non-existent; they are false. The second question is how the aforementioned "shearing" can translate physical reality, the existing state, into a non-existent one. Let's just say it is exactly as much energy consumption required for "shearing" as the states had before the disappearance. That's a separate topic. Complex II » srb Question: Do these conclusions about orthogonality and interference also hold for complex functions? https://rvukovic.rs/bloge/2411.html 8/15 8/19/24, 8:34 AM Blog, November 2024 Answer: Complex scalars are implied. Such, Φ ∈ {ℝ, ℂ}, when they are complex numbers, then scalars from the set ℂ, have long been used in quantum mechanics. Then we use Euler's recently named "cis" (cosine and sine) function, and the previous example (Cancellation) goes with the formula: y(x) = aei(φ + μx) = a[cos(φ + μx) + i⋅sin(φ + μx)], where the imaginary unit is i² = -1. We have the same thing with the previous of the previous case (Existence). See more details about the possibilities of complex numbers in my contribution (Complex Numbers), then (February 20, 2016) written for better students of the Gimnazija. The attachment may be difficult, but it is certainly mathematically more comprehensive than it should be when moving from those real examples to complex ones. However, there is no vector theory part about orthogonality bases. You can see it additionally, for example, in the attached link. See (Modal matrix) how operator diagonalization can be understood intuitively, without routine computation, as an example of reduction to an orthogonal basis. For quantum mechanics, a "basis" is generally a set of mutually orthogonal, perpendicular vectors, one for each "dimension" of space. They are then representatives of observables, or quantum states in a superposition of options, one of which can emerge by measurement. They are usually vectors of length, intensity, or norm one. Non-normed these vectors, also operators (dual space), are not suitable in functional analysis and are something we want to avoid most of the time. If two states are orthogonal, this means that ⟨y1|y2⟩=0. Their scalar product is zero. Physically, this means that if the system is in the state |y1⟩ then there is no possibility of finding the system in the state |y2⟩, and vice versa. In short, the two states are mutually exclusive. This important feature means to operators that the measurement results are unambiguous. From the position of information theory, let's add: that states that do not communicate will not pull each other into collapse, into that unique outcome. More specifically, when the eigenstates of an operator are not orthogonal, that operator is not Hermitian and is not observable. Non-Hermitian operators "appear" all the time but do not act directly on the classical domain. It is an additional phenomenon, in the theory of information that we are discussing, that complex numbers are also some truths and therefore some realities. The way to understand this is with "bypass". Collapsing » srb Question: Explain to me the transition of uncertainty into information? Answer: If we toss a fair coin, both outcomes have the same probability: "tails" p = 1/2 and "heads" q = 1/2. Mean uncertainty: S2 = -p⋅logb p -q⋅logb q equals 2 bits, binary system with logarithm base b = 2. It is exactly how much the outcome information (Shannon's) is for any of those two. It is impossible for both "tails" and "heads" to fall because, with one of them, the total uncertainty is exhausted. The laws of conservation apply to information and prior uncertainty. Moreover, they behave as various forms of energy that change their https://rvukovic.rs/bloge/2411.html 9/15 8/19/24, 8:34 AM Blog, November 2024 states with a constant total amount, with an important difference: when it can, the system moves to a state of greater certainty. This legality also applies to tossing an unfair coin, then p ≠ q, p + q = 1 and p, q > 0, in an arbitrary system of units (b > 0 and b ≠ 1) when the amounts of uncertainty and information delivered by the outcome of the toss are equal. The calculation is repeated with multi-probability distributions p1, p2,..., p n, positive numbers totaling one. Uncertainty and information then amount to: Sn = -p1⋅logb p1 - p2⋅logb p2 - ... - pn⋅logb pn. The formula is Shannon's information, the mean of logarithms, or the expectation of Hartley's information. Each of the outcomes will carry equal information, but the one with a higher probability will have priority. A quantum system is a representation of a vector space; a quantum state is a vector; and a quantum process is an operator (see previous answer). The distribution equivalent is superposition, the interference of multiple outcomes, which we write: |ψ⟩ = c1|1⟩ + c2|2⟩ + ... + cn|n⟩. Here ck are some complex numbers, and |k⟩ are vectors, for k = 1, 2,..., n, where |ck|² probability distribution of observables (physically measurable). Those vectors are orthonormalized: ⟨j|k⟩ = δj,k. The formula is a linear combination of the given base vectors, and the vector |ψ⟩ is the state of the measured system. Collapsing the superposition into some basic probability state pk = |ck|² it's the same story with the previous one. Due to the law of conservation, the information fed into the measuring devices has consumed the uncertainty, and therefore there is no duplication of results. The process is a linear operator, which can be interpreted as a square matrix, and these can be understood as a composition of smaller ones, block matrices. This will support the pooling of the pk probabilities of finding the k-th observable by measuring it as an action of the package of possibilities. So every pk = q1k + q2k + q3k +..., sum of a smaller or larger number of other probabilities. To continue, we do not lose generality if we assume that the previous number of outcomes grows indefinitely, n → ∞. Then each of the sequences of probabilities is a very small number, pk → 0, to which the Shannon item tends. It follows from L'Hôpital's rule: 1 p (ln p) ′ −p ⋅ ln p = − =− = p, ′ ( p1 ) − p12 therefore, that the limit value of the sum (-p⋅ln p) is equal to the probability (p), when these probabilities are small numbers. This is one of the reasons why, long ago (Exponential II), in the information of perception, i.e., sum of products, I sometimes treated factors as probabilities. The distributions in uncertainty are infinite (Vacuum), and the chances of any of them happening are nil. Even if the outcomes did not come in packages, they would have to be realized. However, the outcomes are grouped into collected possibilities, so it seems to us that there are not many choices. That is why we have packets of quanta with energy, for example, photons, E = hν, with Planck's constant h = 6.626070×10−34 J Hz−1 (Joule-second) countless frequencies ν. Or, say, they produce uncertainties in the position and momentum of the particle-wave, Δx⋅Δp ≈ h, always of the order of magnitude of Planck's constant, but of countless spacetime directions and factor intensities. *** The wave function |ψ⟩ is a cornerstone of quantum mechanics, ever since its inception, and the collapse of that function, though essentially nebulous, is a central concept of that physics. Since its discovery (Schrödinger, 1935), it has stood as a still unexplained measurement problem. By introducing the idea of information, that question becomes trivial, as can be seen from the said attachment. https://rvukovic.rs/bloge/2411.html 10/15 8/19/24, 8:34 AM Blog, November 2024 Diversification » srb Question: How can fiction act on matter (a living being)? Answer: Physical reality more than often ignores half-truths, which can be understood to begin with by the photoelectric effect (Einstein, 1905), but that is not all. The untruths and associated "fictions" that you mention in the question, together with physically unrealistic phenomena, I classify as effects that do not follow the law of conservation. Let's say that these are the infinities that we define by the fact that they can be their own (proper) real parts. From such come countless possibilities of "attacking" reality, or, on the other hand, "emerging" from reality itself, invisible to this one. Implied "fictions" are untruths as opposed to, for example, perceived conjugated reality, so I won't underline that further. They are so inconsistent, we would bring them into contradictions by deduction, and they are also unstable, so serious work on them could be unfeasible. A lot of it simulates "null space" algebra. The link mentions matrices: 0 1 A=( ), 0 0 0 0 B=( ), 1 0 otherwise representations of linear operators. We interpret the processes with such operators that, in this case, would cancel themselves. Multiplying these matrices by themselves results in zero matrices A² = 0 and B² = 0, which means the instability of the processes they represent. These matrices of theirs reveal them as "phantom processes," which disappear when isolated but persist when combined and joined together: 1 0 AB = ( ), 0 0 0 0 BA = ( ). 0 1 Thus (AB)(AB) = (AB)² = AB, a process that does not shut down, and so does (BA)² = BA. Moreover, their sum is the unit operator, AB + BA = I, isometry, i.e., the operator that preserves the norms, i.e., the process for which the law of conservation applies. A seemingly very different way of understanding it is with Parrondo's Paradox. We find it in combinations of non-winning games that result in a winning game. See this link to my contribution, but also the link to the picture on the top left that talks about applications in economics. Here is another application. Let's say that with the associated state A, the given system S loses 1 point of some "action" with each subsequent step (time interval). It is irrelevant whether system S perceives the change at all. When state B is attached to it, it reacts to the "action" stock, so when it is an even number, 3 units of that are added, and if it is odd, 5 units are subtracted from it. With state C, it alternately flip-flops, gains 1, then loses 1. Continuing with only A or only B, the system loses 100 points after exactly 100 moments (steps). Sticking to C alone, after n = 100 steps, it will remain with the initial value because the probabilities (p = q = 1/2) of wins and losses are equal, with variance σ² = npq = 25, that is, by dispersion around the mean value σ = 5. It is a binomial distribution. However, if the system has alternating first and second states, in the sequence ABABAB..., and starts with 100 points, after the first step A, it has 99 points left, and in the first step B, it loses another five. It has 94 points left, again an even number, which is why in the next cycle AB loses another six points, and it is left with an even number of 88. After 16 cycles, AB loses 16 ⋅ 6 = 96 points, so if it continues, it goes into the red. https://rvukovic.rs/bloge/2411.html 11/15 8/19/24, 8:34 AM Blog, November 2024 On the contrary, when it alternates with the second and then the first state, stringing BABABA... after the first B, it has 103 points. With the first A, it loses one and remains with 102 points. It's also an even number, so in the next cycle, BA gets two more points. After each cycle, BA gets two points, so after k = 1, 2, 3,... cycles, BA will have K = n 0 + 2k points. After k = 50 cycles and initial n0 = 100 points, it reaches K = 200 points. This combination of two losing affiliations became a winning one. Let's combine this accumulation of points (potential action) with the mentioned photoelectric effect, and there is one of the answers to the question. However, I do not expect that it is still possible to find good answers just like that, without future physiology or perhaps the unknown science of the tissues of living things. Quiescence » srb Question: Does the present change by linking fiction to reality? Answer: According to this information theory, the phenomena of quantum entanglement and the relativity of the flow of the present go together. Let's repeat that first. Photon coupling is one of the first to be measured. With the "cascade" transition, this was done by Alain Aspect and colleagues in the early 1980s, but also by John Clauser and Anton Zeilinger. In those experiments, they put a bunch of calcium atoms at a highly excited energy level where the electron is forbidden to return to its ground state by emitting a single photon. It decays by emitting two photons each, passing through an intermediate state with a short lifetime. The emission of one photon is followed within a few nanoseconds by the emission of another, within the uncertainty relations of the product of energy and time. After one, you know the other should be somewhere nearby. While different photons are emitted in all directions, those from opposite directions are quantum entangled. As such, the sum of momentum and spin is canceled. One such entangled photon is, say, Alice, and the other is Bob. It forms one world of simultaneity. However, the presence of Alice and Bob is perceived by different observers when they are present at different times. So, the "phantom action at a distance" occurs because the spins of Alice plus Bob are zero, even though they are individually random, according to the measurement. We cannot know in advance which of the two spins A = +1 or A = -1 has Alice, or Bob, B ∈ {+1, -1}, but due to the conservation of spin A + B = 0. By measuring the first A, the information of that present has exhausted the uncertainty, and the second is certain B = -A. Alice and Bob may be very far apart at that moment, yet Bob will react "phantomically" by matching Alice's measurement, faster than light traveling between them. Of course, such a description quantum entanglement is not known to physics today because I keep it private. Now let's get to the question. Reality is what the law of conservation applies to. Starting from such statements, this theory of information is unique and still unknown to physics, and I hope, because of them, it will remain uncontradictory. Reality is what can be perceived, at least indirectly, as well as what is true. Therefore, when fiction, which is not subject to the law of conservation, attaches itself to reality in ways that make it sustainable, perceptible, and true (Pinocchio), it introduces unrest or increases the established information (quantity of options) of the present. An actual gain is followed by a loss of something similar in the present or in the past. Roughly divided, there are three situations in the present. The first is described by Alice and Bob, which transmits information about Alice's spin to an external observer (measuring device), which also determines Bob's spin. Loss of uncertainty in the present. The second is "revival," when fiction is attached to reality by erasing some evidence of the past and preserving the amount of uncertainty in the present. The third form would be "revival" by simply increasing the information of the https://rvukovic.rs/bloge/2411.html 12/15 8/19/24, 8:34 AM Blog, November 2024 present without stifling the flow of information from the past. Complex combinations of these. This picture is further complicated by the multiplicities of the world. Each subject perceives some conjugate image of objects (Lipschitz), which is not the same for different observers. Bats have a special sense of hearing; birds of sight; and dogs, bears, or elephants have exceptional senses of smell. However, not all frequencies of sound reach everyone, colors of light, or chemicals. Cats are good at seeing colors, but especially blue and yellow, while red and green are shades of gray. Each of these has conjugated perceptions with their present, which, in their own way, are subject to the aforementioned divisions. Reality » srb Question: Is "real" everything that is logically correct? Answer: It is so in the concept of this theory. A unique "essence" in the classical philosophical sense does not exist; that is, there are many such. They are all equal when they are interpretations of correct theories. We already know intuitively that what we perceive is not what it is, but we also found through functional analysis that even the best perceptions of "reality" are themselves conjugated spaces of objects and subjects, countless and often mutually equal. There is no one reality that we all observe as moviegoers in some world cinema. However, we are far from the fact that each subject can notice different shades of projection in others, and even further from the ability of the audience to realize that there is no one-size-fits-all film. Each of them is thus truly unique and equally "the center of everything.". Parallel lines are lines that lie in the same plane and do not intersect. In geometry, there are three possibilities: that one, at least two, and no parallel lines can be drawn through a given point outside of a given line. The first determines the Euclidean geometry, where the sums of the angles in each triangle are 180° and the ratios of the circumference and diameter of all circles are π = 3.14...; the second is the hyperbolic geometry, where the sums of the angles in the triangles are always less than 180°, and the quotients of the circumference and diameter of the circles are always greater than π; the third is the spherical geometry with greater than 180° of the sums of the angles of the triangle and the ratios of the circumference and diameter of the circle are always smaller than π. The model of the first geometry is the Euclidean plane. But the straight lines are the shortest between the points, wherein in the second case the "plane" is a saddleshaped surface and in the third, it is a sphere. Because such surfaces exist in Euclidean space, we know that each of these three geometries is at least as accurate as Euclidean. Inverse models were also found, Euclidean in the non-Euclidean plane, so we have equality of accuracy for the three geometries. Moreover, we have an absolute geometry that will leave the choice of parallel requirements to the user. This leaving the choice to the user is actually already present in simple algebraic equations like a + b = 5. Namely, by choosing a = 1, we were also choosing b = 4 to stay within accuracy. Choosing a = 2, we inevitably choose b = 3, and the possibilities are endless. Choices are the "essence" of this information theory. I am so consistent in this that I transfer the same to the "reality" of everything that is logically correct. It is noticeable that, at the same time, algebraic exactitudes behave as the present of known physical reality. After all, after proof of additional dimensions time, should we "put the key in the lock" and say we are not going any further. Especially after perhaps physical confirmations of parallel realities (Pseudo-reality). To be honest, I don't expect them, nor do I think that they can prove or disprove this mathematical theory. Adhesive » srb https://rvukovic.rs/bloge/2411.html 13/15 8/19/24, 8:34 AM Blog, November 2024 Question: In this interpretation of the "reality" of exact theories, where is the matter? Answer: In an attempt to submit some information to physical actions, to the product of changed energy ΔE and spent time Δt, otherwise sized around Planck's constant, ΔE⋅Δt ≈ h, let's treat laws the same way we treat matter. A consistent step in that direction is to accept gravity as a consequence of the principle of least action. This is the finished chapter, which you can see for yourself in the book "Minimalism of Information" (2.5 Einstein’s general equations). This is followed by the less well-known principle of more frequent occurrence of more probable outcomes as a causal principle of the least action, and the more probable outcome is less informative. Therefore, the deep cause of attraction is the principled tendency of the system towards less informative states, more certain states, and, therefore, towards rules. One of the obstacles on that path is the misinterpretation of entropy, which has been running through physics since the time of Shannon (1948). In short, calming the oscillations of molecules by cooling gas is a process of decreasing information, not increasing it (which is the common understanding of entropy). Shards of glass from a cup that fell from the table and broke are a mess for a housewife, but they are orderly like a soldier on parade. Thermodynamic processes spontaneously increase entropy, thus leading to less information. As Einstein once observed, places with a slower flow of time are gravitationally attractive. When we have constant uncertainty products ΔE⋅Δt = const, that means those quanta with smaller energy changes in longer time intervals are more attractive, i.e., they are most attractive than ubiquitous (Δt → ∞) infinitesimal energies (ΔE → 0). When trying to subsume information under physical actions, these limit values will be the laws of nature. They will be attracted to the natural phenomena with which they interact. Not everything communicates with everyone, nor do those who interact do so with the same intensity, so the laws of nature are also selective. The presented method is a rounded reduction of some information to physical effects, from extreme uncertainties to extreme certainties. Phenomena and regularities are not dual by means of adjoining or conjugation in algebra or analysis but rather are various forms of the same. Let's say that the realities are glued in thin layers. Expanse » srb Question: Can we discuss the origin of these laws and correct theories in general? Answer: Here we are now in the field of infinity, and the fantastic surprise is that the answer to this question is yes. For a long time, philosophy and mathematics were confused, trying to penetrate the secrets of unlimited quantities. Newton and Leibniz finally succeeded in this with the discovery of infinitesimal calculus. Then Cantor deepened the topic with the discovery of different cardinals and infinite numbers of elements in sets. Sticking to colloquial thinking, the ancient Greek philosopher Zeno of Elea (490– 430) "proved" that there is no movement because we first go halfway, then half of the other half, then half of the rest, and there is no end to it. Logically, he belonged to the Eleatic school founded by Parmenides. An Infinitesimal is a non-zero quantity that is closer to 0 than any non-zero real number is. Archimedes of Syracuse (287–212) was the first to use such "numbers" and successfully calculated the area of a circle and the volume of a ball, although he did not believe in their existence. Much later, Newton (1643–1727) and Leibniz https://rvukovic.rs/bloge/2411.html 14/15 8/19/24, 8:34 AM Blog, November 2024 (1646–1716) again, each for their own needs, found the infinitesimal calculus while also discovering analysis. Newton for calculations of the movement of celestial bodies under the influence of gravity, and Leibniz for the search for a universal language. A step further in unlocking infinity was the achievement of Cantor (1845–1918). He declared as equivalent those sets among whose elements there is a bijection (mutually unique mapping), so he considered such to have an equal number of elements, i.e., equal "cardinals" when those numbers are infinite. He proved that countable infinities (labels ℵ0) have sets of natural numbers (ℕ), integers (ℤ), or rational numbers (ℚ), but the set of real numbers (ℝ) is uncountably infinite, a continuum. When we understand the proof (2. Proposition) that there are more real numbers from the interval (0, 1) than natural numbers, let's apply the result to the way of writing real numbers from that interval in the binary system 0.b1b2b3..., digits bk ∈ {0, 1}. At each position of that countably infinite sequence of digits bk, k ∈ ℕ, can be two values. Their combinations are 2⋅2⋅2⋅... → 2ℵ0, which is the cardinal of the continuum 𝔠 = 2ℵ0 > ℵ0. However, we see that we can thus form increasingly many cardinals 𝔠1 = 2𝔠, 𝔠2 = 2𝔠1, ..., i.e. that the enumeration of those cardinals is analogous to natural numbers. Equivalently, they can be ordered by size. Moreover, between any two monotonically ordered adjacent cardinals, it is optionally possible to insert a third, different from both. The analogy of cardinals with rational numbers will no longer surprise us, after Löwenheim–Skolem's theorem, that theories always appear infinitely copied. This long introduction to the "revelation" of infinity through mathematics will help, I hope, to consider it "real" as well. In terms of being perceptible to our (even if only intellectual) perceptions and accurate, the power of seeing infinity from a frog's perspective and being bound by finite physical interactions is extraordinary, but the nature of truth is not given. That's how I see it—that the truth can show itself. There is no stable, permanent, non-contradictory, or consistent system of lies as opposed to truth. Well, if we could see the infinities from a bird's-eye view, we could see the emergence of laws and correct theories. However, we are barely able to recognize the evolution of biological species on Earth, so there is no one to talk about the genesis of all-time phenomena. It would be a "fairy tale" about the sequence, but also the denser accumulation of actual infinite phenomena, only some of which are among the laws (of mathematics, science, philosophy, or similar) known to us. November 2024 (Original ≽) Copyright © 2002 - 2024 Rastko Vukovic https://rvukovic.rs/bloge/2411.html Template by OS Templates 15/15