It is well known that an integral is nothing but a continuous form of a sum. Is it possible to do the same thing with a product? The answer is yes and done for the first time in this publication. The new operator is c...It is well known that an integral is nothing but a continuous form of a sum. Is it possible to do the same thing with a product? The answer is yes and done for the first time in this publication. The new operator is called inteduct. As an integral is a proper tool to calculate the arithmetic mean of a function, the inteduct gives the geometric mean of a function. This defines a new branch of mathematics. Most applications may lay way ahead. Only some are discussed here. One is applying the inteduct to probability theory. There it is possible e.g., to determine a function for a life expectation rather than just a numerical value. Another application is to distinguish chaos from randomness within numerically given values. At least for the logistic map there exists a direct connection between Lyapunov exponent and inteduct. To distinguish between chaos and randomness is particularly important in finance. While randomness implies ergodicity, chaos is non-ergodic. And many fundamental financial theories from portfolio theory to market efficiency require ergodicity.展开更多
The motivation of this paper is to show how to use the information from given distributions and to fit distributions in order to confirm models. Our examples are especially for disciplines slightly away from mathemati...The motivation of this paper is to show how to use the information from given distributions and to fit distributions in order to confirm models. Our examples are especially for disciplines slightly away from mathematics. One minor result is that standard deviation and mean are at most a more or less good approximation to determine the best Gaussian fit. In our first example we scrutinize the distribution of the intelligence quotient (IQ). Because it is an almost perfect Gaussian distribution and correlated to the parents’ IQ, we conclude with mathematical arguments that IQ is inherited only which is assumed by mainstream psychologists. Our second example is income distributions. The number of rich people is much higher than any Gaussian distribution would allow. We present a new distribution consisting of a Gaussian plus a modified exponential distribution. It fits the fat tail perfectly. It is also suitable to explain the old problem of fat tails in stock returns.展开更多
Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium ...Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium in parameter space. However, in real games in business and economics, the input parameters are not given. They are typically observed in several similar occasions of the past. Therefore they have a distribution and the average is used. Even if the averages are in an area of Nash equilibrium, some values may be outside making the entire result meaningless. As the averages are sometimes just guessed, the distribution cannot be known. The main focus of this article is to show this effect in an example, and to explain the surprising result by topological explanations. We give an example of two players having three strategies each (e.g. player and keeper in penalty shooting) where we demonstrate the effect explicitly. As the transition of Nash equilibrium to no equilibrium is sharp, there may be a special form of chaos which we suggest to call topological chaos.展开更多
文摘It is well known that an integral is nothing but a continuous form of a sum. Is it possible to do the same thing with a product? The answer is yes and done for the first time in this publication. The new operator is called inteduct. As an integral is a proper tool to calculate the arithmetic mean of a function, the inteduct gives the geometric mean of a function. This defines a new branch of mathematics. Most applications may lay way ahead. Only some are discussed here. One is applying the inteduct to probability theory. There it is possible e.g., to determine a function for a life expectation rather than just a numerical value. Another application is to distinguish chaos from randomness within numerically given values. At least for the logistic map there exists a direct connection between Lyapunov exponent and inteduct. To distinguish between chaos and randomness is particularly important in finance. While randomness implies ergodicity, chaos is non-ergodic. And many fundamental financial theories from portfolio theory to market efficiency require ergodicity.
文摘The motivation of this paper is to show how to use the information from given distributions and to fit distributions in order to confirm models. Our examples are especially for disciplines slightly away from mathematics. One minor result is that standard deviation and mean are at most a more or less good approximation to determine the best Gaussian fit. In our first example we scrutinize the distribution of the intelligence quotient (IQ). Because it is an almost perfect Gaussian distribution and correlated to the parents’ IQ, we conclude with mathematical arguments that IQ is inherited only which is assumed by mainstream psychologists. Our second example is income distributions. The number of rich people is much higher than any Gaussian distribution would allow. We present a new distribution consisting of a Gaussian plus a modified exponential distribution. It fits the fat tail perfectly. It is also suitable to explain the old problem of fat tails in stock returns.
文摘Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium in parameter space. However, in real games in business and economics, the input parameters are not given. They are typically observed in several similar occasions of the past. Therefore they have a distribution and the average is used. Even if the averages are in an area of Nash equilibrium, some values may be outside making the entire result meaningless. As the averages are sometimes just guessed, the distribution cannot be known. The main focus of this article is to show this effect in an example, and to explain the surprising result by topological explanations. We give an example of two players having three strategies each (e.g. player and keeper in penalty shooting) where we demonstrate the effect explicitly. As the transition of Nash equilibrium to no equilibrium is sharp, there may be a special form of chaos which we suggest to call topological chaos.
