Definition of a convention. Definition is not subject to proof. Though it is impossible to prove, it is said that axioms are acceptable if all are correct. It is interpreted that it is unnatural to think that axioms do not hold as abstraction of various facts and experiences in the real world. Combining definitions and axioms (combining methods are axioms), what we derive is called theorem. If there are a lot of guidance, many important things are used as a theorem, and intermediate results and the by-products from the theore arriving at the theorem are sometimes used as a lemma. I do not know whether it is true or false, but what the truth is revealed by the proof is called a proposition. Especially if it is A, what you can express in the form of B is called a clause.
Combining definitions and axioms The theorem is completed, and another theorem is derived from the theorem. The theorem is absolutely right thing, creating a theorem is to discover the laws of the world of numbers. Repeating this makes it possible to clarify the laws in turn in the world of mathematics. To discover the theorem and to prove the theorem is the purpose of mathematics.
It is called deduction to derive new logical consequences from definitions, theorems, axioms, so as to lead "from B" to "being B if A" . It is to guide new facts from what I know. It can be said that it leads concrete individual facts from abstract truth. On the other hand, induction enumerates various concrete facts and seeks abstract truth from which specific facts can be derived by deduction. It can be said that generalization of facts.
The denial of "B holds for all A" is "B does not hold for certain A". The denial of "B holds for A" is "B does not hold for all A". Under the law, you create a negative form of the proposition you want to prove, leading to its contradiction.
In the proposition "if it is A, B is satisfied," A is sufficient for B in the proposition, and B is required for A. Semantically, B is a necessary condition because A is a sufficient condition, "A must first be established for A to be established" because "B will be established if only A is established". When A = B, it is called "necessary and sufficient condition when A is B" "necessary and sufficient condition when B is A".
If "A if B" against the proposition "B if A" is reversed. "It is an integer if it is a real number" is opposite to "for an integer if it is a real number." As you can see in the example, the converse is not necessarily true. For "A if B" proposition, "If it is not A it is not B" is called the back. The back is not necessarily true. For "A if B" against the proposition "If not B is not A" is called a kinematic pair, this is true.
It is a spy law as a representative of methods that can be used for certification. In order to do a certain proof, we consider a proposition that denied it, perform deductive reasoning in combination with preconditions, and derive some logical consequences. If anything contradictory to the preconditions is found in the logical consequences, the first proposition is denied and the proposition that is the opposite is proved. If A it proves that the kinematic pair holds in the proof of the clause B. In proof that a proposition holds, it is often difficult to prove that the proposition holds for all variables. On the other hand, it is easy to prove that it does not hold for all variables, because it only needs to indicate a counterexample. Also, to prove that there is a solution, in other words, a solution exists, find a solution.
The proof of existence of a solution can use the intermediate value theorem. If f (x) is a continuous function with f (x 1) <0, f (x 2)> 0 , then f (x) = 0 < / var> The solution x exists between x1 and x2 . Mathematical induction is a technique to prove that a proposition holds for all natural numbers. It is a proposition that includes a natural number n , it proves that (1) n = 1 , (2) proves that n = k By proof that using
A set is a basic concept of mathematics. Modern mathematics is often described using the idea of a set. A group of numbers is called a group. For example, it is a set of integers from 1 to 5, a set of negative integers, a set of real numbers, and the like. Depending on the number of elements, there are finite sets and infinite sets. Mathematics expresses the attributes of a number as a set. In the addition of natural numbers, it is expressed as adding one element of a set of natural numbers and one element of the same set of natural numbers.
If all the elements of set A are elements of set B in the relationship between sets, set A is a subset. If all elements of set A are not elements of set B, set A is a complement of set B. Elements belonging to both A and B are elements belonging to A or B, elements belonging to B but not belonging to A, elements belonging to B, elements belonging to A but not belonging to B, It can be classified into elements not belonging to A or B.
A mathematical expression shows the relationship between variables. Variables include real numbers, integers, negative integers, and natural numbers less than or equal to 5. We use a set to express what a variable is (attribute).
When one variable and its corresponding evaluation value (profit amount, cost, number representing desirability and degree, etc.) are in a functional relation, how the evaluation value moves when the variable moves, maximum evaluation value ) What is the value of the variable that will be noticed. Many methods have been developed to analyze the nature of functions and to derive optimal values.
The function is a correspondence relationship in which another value is determined when a certain value is determined. y = f (x) when y is determined by specifying x . In the case of one-to-one correspondence, we can think of the inverse function x = g (y) . In the case of many-to-one correspondence, there is no inverse function.
By analyzing the function by differential method / integration method, it is possible to find out the form of the function and find the maximum value (minimum value). The necessary condition of the extreme value (local maximum value, local minimum value) is that the derivative of the function becomes zero, and from this expression, the value of the variable that brings the extreme value can be obtained.
Continuous if differentiable. The converse is not true.
Continuous means that the extreme value of the function always coincides with the value of the function. When the x approaches a in the definition of the limit of the function f (x) , f (x) Is approaching the value of. Intuitively this is understood, but mathematically it becomes the following description. A positive number satisfying | f (a + h) - b | <ε, | h |
What is the best result? It is the best result. Whether it is good or bad is measured with a ruler. The evaluation function is determined so that the maximum value or the minimum value of the function is the most desirable result. For example, it is a function that uses the profit amount and the cost amount as function values. Optimization is to move the variable and find a value that makes the evaluation function the optimum value. A variable to be moved is called a decision variable.
Model it with the evaluation function, the decision variable, and the constraint condition for finding the maximum value (minimum value). The constraint condition is to prevent the evaluation function value from becoming infinite, but it is natural to think that there is always some constraint condition in the real world, and it is natural to move the decision variable under the constraint condition and maximize the evaluation function Finding a decision variable is an optimization. By formulating (modeling), for example, it can be written as: max f (x) subject to g (x)> 0, max f (x, y) subject to g (x, y)> 0 It will be like.
The condition of the local maximal value (minimum value) is zero f '(x) = 0 by differentiating the necessary condition. It is a local minimum if the differentiation of the second floor f '' (x)> 0 , it is a local minimum if f '' (x) <0 . In the case of the two variable function f (x, y) , the required condition is partially differentiated to zero fx (x, y) = 0, fy (x, y) = 0 < var>. This is a necessary condition and not sufficient condition. In the case of a model with more than one variable, f (x, y, a, b) divides notable decision variables and parameters (coefficients) .
Maximal value only represents a local mountain, not necessarily the maximum value. When the argument x of f (x) is a finite range, it is enough to examine all ranges, but in infinite cases it is not easy except when it can be differentiated over the whole area. There are the following methods that approach the optimum value in the sample survey regardless of the total survey. Newton method: When a differential coefficient is obtained at a point A, the surroundings of the candidate of the optimum value are set as the next search destination from the inclination. Hill climbing method: Based on the slope of the derivative coefficient at point A, we take the direction of search toward the ascent hill. However, there is a possibility that it will be sucked into a local mountain.
The mathematical programming method is a method of calculating the value of the decision variable that maximizes (or minimizes) the evaluation function on the premise that the decision variable satisfies the constraint condition. Constraint conditions are often inequalities including decision variables. For example, it is assumed that the production volume is less than a certain amount. The evaluation function is profit, total cost, and the like. For example, there are two decision variables of x, y and 2 x + 3 y < 10 , 5 x + 2 y < 5 Maximize the profit 7 x + 2 y when putting in , x > 0 , y > 0 x and y .
In the optimization model, when both the evaluation function and the constraint condition are linear functions, calculation of the optimum value becomes easy. In the case of max ax + by subject to cx + dy < 0 x > 0 y > 0 , if x, y satisfying the constraint is in a certain range, It exists. y = - (a / b) x + (k / b) , where k = ax + b y Since the Y intercept is (k / b) , plot the value of (x, y) that satisfies the constraint on the xy Draw a few straight lines that are inclined - (a / b) . Among the straight lines having an intersection with (x, y) that satisfies the constraint condition, a straight line with the largest intercept indicates an optimal solution. The point of (x, y) that satisfies the constraint that intersects with this straight line is the optimal solution, and the value of the evaluation function at this time is the highest value. When changing constraint conditions, it is the sensitivity analysis to see how the maximum value of the evaluation function and the optimal solution change. In the case of max ax + by subject to cx + dy < 0, x> 0, y > 0 , if subject to cx + dy < 1 There is a possibility that the new optimum value and the maximum value of the evaluation function can be found by that amount. By relaxing the constraint condition, if the further increase in the evaluation function can be expected, the option is sufficiently considered. Computation becomes complicated in nonlinear functional formulas (nonlinear formulas). The Lagrange multiplier method can be used.
In the case of max f (x, y) subject to g (x, y) = 0 , let L = f (x, y) + ag (x, y) x and y satisfying Lx = 0, Ly = 0, La = 0 are optimal solutions. f (x, y) increment. Consider the problem of finding the optimal route when transitioning from one state to another state. It is to find the optimal solution overall, but superimposing the partial optimal solutions becomes global optimization. In other words, think about the whole thing and consider the optimal solution at that point locally each time. Think about the problem of finding the best route. The necessary condition which is the optimum route is called Euler's theorem, and it is only necessary to select the optimum one among the routes that satisfy this requirement. Although an infinite candidate is considered as the optimum route, it is easy to find the optimum route easily by calculation. The optimum path that can be easily calculated is a linear path and a quadratic function (parabolic) path from point A to point B. There are two types of quadratic function (parabolic) paths, which move greatly at the beginning, gradually move slowly, move slowly at first, and gradually move bigger. When making decisions on the problem, a method of dividing the original problem into partial problems, hierarchizing it, solving individual partial problems, and finding the whole solution according to the hierarchical structure. This method is effective for subjective evaluation. Subjective evaluations are carried out for each of the partial items, and the consistency of the whole is observed. If it is combined in the hierarchy it will be the overall evaluation. When returning from state B to state A after transitioning from state A to state B, the phenomenon in which the path is different between going and returning.
When going from x = a to x = b it takes the path of f (x) but x = b To x = a take the path of g (x) . In general, it can be modeled as follows. With the state variable X
Y [t + 1] = f (X [t], Y [t]), Z [t] = g (t) X [t]) A basic, widely applicable model is a model in which the change of the state variable X depends on the value of the state variable X.
X '= f (X) , the simplest being the model which is proportional to the value of X , X' = a X Yes, X is X = be ct . Models that increase and decrease at a constant rate can be represented in this form. For example, when economic phenomena change at a constant growth rate, it can be expressed by this model. Calculation of Optimal Value
Convex function and concave function
Mathematical programming method
Linear programming
Sensitivity analysis
Optimization of Nonlinear Expressions
Dynamic optimization
Optimal route problem
Hierarchical Decision Making Method
Hysteresis phenomenon
Dynamic modeling
Differential equation