Consciousness is the manifestation of the subconscious, which is a probabilistic behavior, but whether this is substantive or combined, I think, should be the possibility of connection between different individuals, and this network-type connection structure is bound to lead to a pattern, that is, what we call the rule: modularity and hierarchical coupling. Pen × fun × Pavilion www. ｂｉｑｕｇｅ。 info This can be seen as a set of quanta with a certain tendency, satisfying a series of physical laws, such as Boltzmann distributions and operator operations of wave functions.

Modularity seems to be an independent entity at a certain level, and we will have a sense of separation. But it is also a living and evolving individual, which makes the whole robust and resistant to certain changes, while maintaining the ability to quickly return to a new homeostasis after a drastic change. The nature of the network, from a human perspective, seems to be a bit ruthless, but is this similar to the way of heaven? It is not for the sake of survival, it is not for the sake of destruction, and the way of heaven is not benevolent, and all things are used as dogs.

The focus is on determining whether a known equation has a radical solution. If there is, it is not necessary to investigate what the root of the equation is, but only to prove that there is a root solution--- and to prove that the original problem cannot be solved under the premise given and the meaning under consideration

Start with the substitution of the root of the equation. When he systematically studied the permutation properties of the roots of equations, he proposed some definite criteria for determining whether the solution of a known equation can be found by the radicals, but these methods led him to consider a set of elements called a group

The concept of a permutation group was defined for the first time by referring to a set of permutations with closed properties as a group. He believed that understanding the permutation group was the solution to the equation theory

Crucially, an equation is a system whose symmetry can be described by the properties of groups. From then on, he began to transform the problems of equation theory into problems of group theory to solve, and directly studied group theory.

a0xn+a1xn-1+a2xn-2+?+an-1x+an=0, each root is a possible permutation, and this polynomial is also a reflection of the world

Suppose that each transformation of its n roots x1, x2,?, is called a permutation, and n roots have n! possible permutations, and their sets form a group with respect to the multiplication of permutations, which is the permutation group of the roots. The solvability of the equation can be reflected in some properties of the permutation group of the root, which transforms the problem of solvability of algebraic equations into an analytical problem of the properties of the related permutation group and its subgroups. The permutation group from which the equation relates (it expresses the symmetrical nature of the equation)

It is a group in the coefficient domain of some equation. The Galois group of an equation is the maximum permutation group that satisfies this requirement for every polynomial function about the root whose function value is a rational number, or it can be said that for any polynomial function about the root with a rational value, each permutation in the Galois group makes the value of the function constant

Purely mathematical relationships, network structures

Existence theorem for differential equations (Proving that every polynomial equation has at least one root establishes an existence theorem.)

Mathematical truth entrusts the rationality of its discourse to its own linguistic system, and the meaning and judgment of mathematical propositions are integrated into the linguistic relations, syntactic transformations, and interactivity of its structure

The meaning of mathematics is intrinsically related to the context, and the rational premise of the existence of mathematical objects depends on the context

Pure mathematics is all propositional classes of the form 'p entails q', where p and q contain propositions of the same number of variables or variables, and p and q contain no constants other than logical constants. Logical constants are concepts that can be defined by the following objects: entailments, the relation of an term to the class to which it belongs, such and such concepts, the concepts of relations, and other concepts as those involving the general propositional concepts of the above forms. In addition to this, mathematics uses a concept that is not an integral part of the proposition it considers, the concept of true or false.

Problems arising from rational numbers and even natural numbers. He believes that it should be based on logic,

For arithmetic the rule is analytical judgment and is therefore a priori. According to this, arithmetic is only a form of further development of logic, and each arithmetic theorem is a logical law. The application of arithmetic to the explanation of natural phenomena is only the logical processing of the observed facts, and calculation is reasoning.

For linear regression, the cost function is the sum of squares of all modeling errors

Get more training instances, reduce the number of features, try to get more features, try to add binomial features,. Try to reduce the degree of normalization λ, try to increase the degree of normalization λ - to solve the high bias

The probabilities are the same, but the probabilities of the combination are not necessarily the same, which is the overall view. Combinations are information and need to be understood in conjunction with Bayesian probability formulas. If the probability of positive and negative is greater than that of positive and negative, does this reveal the superiority of palindromic structure? The whole is the understanding of probability at all levels, from the statistical probability of a single primitive, to the combination probability of a sequence, to the energy of a three-dimensional structure, the probability of the least entropy, to the survival probability of four-dimensional natural selection

Excellent Essay Writing: Innovate based on the work of certain predecessors (like variation by natural selection)

Interest, high goals, low start, insist on making progress every day, learn mathematics well, develop good habits, insist on thinking and exploring, accumulate ideas, quantitative change leads to qualitative change, read a whole field of papers, find the essence that is, classic papers, read a lot of books to build a network, find a starting point, multi-level traversal, knowledge guarantee, start writing papers and revisions, research procedural, continuous and innovative, according to a certain module (title, abstract, method, introduction, conclusion) The layer-by-layer depth of the constellative solution at different levels of the network is the selective expression of probability, the expression of a certain pattern, the existence of milestones is the key node, and the research is the intrinsic path connection derived from our previous experience

The network of the whole discipline is the reverse solution of various intrinses

The intrinsic nature of the network serves as an observable indicator, just like the Turing test

The eigengenis is an approximation, and through the determination of the boundary, the probability of the Bayesian network can be solved to limit the explosive growth of the combination

The explosive growth of combinations requires certain eigens to constrain them, logic is a reliable method of linking existing knowledge, and distributions are higher-order existences, just as calculus is the study of variable forces

Big Background-Small Background-Problem Formulation (Mathematical Language Description, Existing Model Selection, Purpose Determination)-Method Selection-Effect Achievement: The Nature of Selective Expression, the Nature of the Network

Concise, informative

Each stage is done to the extreme, and many different stages and different depths of modification

Topological properties, high-dimensional spatial invariants

Number of vertices (zero dimension) - number of edges (1 dimension) + number of faces (2 dimensions) = 2 (topological invariant)

When it is 0, it means that there is a ring

Number of vertices (zero dimension) - number of edges (1 dimension) + number of faces (2 dimension) - number of volumes (3 dimension)

Code decipherment, structure emergence, pattern discovery

The Fourier analysis of the trend of the period is coupled for a specific period

The multivariate of differential equations is the multi-level coupling, which is an intrinsic of the network, and the expression of the network is diverse, we must grasp the most critical skeleton structure, which is coupled with the form of selective expression

Separable variables, using equivalence in the final form, can derive the high-dimensional equivalence, i.e., the integral. However, the existence of indefinite integrals makes the operation of ascending or decreasing dimension finite, i.e., convergent

The initial value is an anchor that enables the determination of indefinite integrals

What we call solving is a kind of dimensional processing such that both ends of the equation are variables that we can understand, rather than changes in variables i.e., differentiation, which is implicit. We can also reduce the complex relationship of variables, such as y=x to a bunch of operations

Linearity is intrinsic, and nonlinearity is expression

There are appropriate equations for inseparable variables, and partial derivatives for the order of the different variables of a network's coupled differential equations are equivalent. First reduce the dimension, then ascend, then reduce the dimension, and find the equivalent term

The multiplication of integral factors, like extraneous terms in logical algebra, can be simplified, but it is the key to maintaining stability, allowing for the construction of a proper equation

Second-order linear homogeneous differential equations, extracting eigenequations, according to their solutions, can be guessed, and the structure of the coupling

The eigensolution is a combination of exponential e and trigonometric functions sin and cos, which shows that periodicity is the essence. The linear solution of the function may be the form of selective expression, which is consistent with the physical meaning of the polymorphism of Schrödinger's cat of the wave function. are all expressions of probability

The introduction of imaginary roots, according to Euler's formula, e^ix=cosx+isinx, acts as an intermediary

Initial anchoring of differential equations of different orders requires a different number of initial values

The solution of a non-homogeneous differential equation is a general solution and a special solution, with multiple anchors, which is a linear property

Guessing is the direction to the solution, and then using the method of undetermined coefficients to find the solution

The linear nature combined with the infinite approximation of calculus is a high-dimensional operation, i.e., ergoria

Laplace transform, is a global operation, extracting the linear property (time domain --- frequency domain), lf(t)=f(s), relying on the nature of calculus and periodicity, which is the law of exponential operation, variable differentiation is divided into algebra, linear transformation makes the transformation from the whole to the individual respectively f(a+b)=f(a)+f(b) and then the non-stop order can be turned into the same order operation, and then the original function is obtained from the Laplace transform table

Consider the convergence speed of multiplying functions

Linear Relationship Between Dimensions Revealed by Laplace Transform: Superposition of Dimensions (Low Dimension = S * High Dimension - Initial Value)

Hypothesis: Integration before Differentiation = Differentiation before Integration,

The Dirac function is a simulation, sudden burst. It is also an intermediate of communication convolution

Convolution is the inverse Laplace transformation of the integral function, which is the superposition of the results of the system's processing of the signal (the superposition of the effects of the changes)

The chain rule, the algorithm of high-dimensional variable couplings, can be decomposed into the sum of different low-order coupling terms. It is an idealized way of processing, where one variable is treated as normal and the rest are treated as constants, and these steps are repeated

Partial integrals can be coupled with period-based operations to give the results of the coupling

The analytic solution of an unsolvable equation can be simulated by a computer, and in the case of humans, it is a quick way to find the eigenpath

The mathematical basis of genetic algorithms is the hierarchical game of networks: probability. It has natural selection, which is equivalent to the probability of the network's path collapse. It is a learning Markov chain that can use the Bayesian system. Networks are geometrically based operations that can be predicted

Finding extrema is a series of local optimal solutions

Any level is intelligent, because based on the connection, there will inevitably be a network structure formed, which will show a certain ordered nature. Human highly intelligent organisms are a collection of this hierarchical coupling, which is not only the evolution of the network (power-law distribution, decreasing dependency), but also resistance to change

Evolution is also a structure, like survival of the fittest, because it can change in a certain pattern in order to survive

Statistical decision-making-hierarchical game, finding patterns, probability distribution functions

Network changes are dynamic, and there may be general fluctuations. And there can be rapid mutations for specific eigens, which is a probability distribution function. We understand it in terms of Markov chains: the expression of different probability distributions is a network of reality

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