Approximating the Radial Distribution Function of the Electron in a Hydrogen Atom by a Normal Distribution Suggests That Magnetic Confinement Fusion Would Be Less Energy Efficient than Inertial Confinement Fusion

Since the position of the electron in a hydrogen atom cannot be determined, the region in which it resides is said to be determined stochastically and forms an electron cloud. The probability density function of the single electron in 1s orbit is expressed as φ2, a function of distance from the nucleus. However, the probability of existence of the electron is expressed as a radial distribution function at an arbitrary distance from the nucleus, so it is estimated as the probability of the entire spherical shape of that radius. In this study, it has been found that the electron existence probability approximates the radial distribution function by assuming that the probability of existence of the electron being in the vicinity of the nucleus follows a normal distribution for arbitrary x-, y-, and z-axis directions. This implies that the probability of existence of the electron, which has been known only from the distance information, would follow a normal distribution independently in the three directions. When the electrons’ motion is extremely restricted in a certain direction by the magnetic field of both tokamak and helical fusion reactors, the probability of existence of the electron increases with proximity to the nucleus, and as a result, it is less likely to be liberated from the nucleus. Therefore, more and more energy is required to free the nucleus from the electron in order to generate plasma.

Probability Theory Predicts That Winning Streak Is a Shortcut for the Underdog Team to Win the World Series

It is common for two teams or two players to play a game in which the first one to win a majority of the initially determined number of matches wins the championship. We will explore the probabilistic conditions under which a team (or player) that is considered weak may win the championship over a team (or player) that is considered strong, or a game may go all the way to the end, creating excitement among fans. It is unlikely to occur if the initially estimated probability remains constant when the weaker one wins each game against the stronger one. The purpose of this study is to identify probabilistically what conditions are necessary to increase the probability of such an outcome. We examine probabilistically by quantifying momentum gains to see if momentum gains by a weaker team (or player) winning a series of games would increase the likelihood of such an outcome occurring. If the weaker one gains momentum by winning a series of games and the probability of winning the next game is greater than the initial probability, we can see that such a result will occur in this study. Especially when the number of games is limited to seven, the initial probability that a weaker one will beat a stronger one in each game must be 0.35 or higher in order to win the championship and excite the fans by having the game go all the way to the end.

Modified Kuramoto Phase Model for Simulating Cardiac Pacemaker Cell Synchronization

It has been suggested that sick sinus syndrome, which is due to the dysfunction of the sinus node, may result from the sparser gap junctions and/or lower intrinsic frequencies of pacemaker cells that occur with aging. Hence, in this paper, the synchronization mechanism of pacemaker cells that lie in the sinus node of the heart is examined using a modified Kuramoto phase model. Although each element always interacts with all the others in the Kuramoto phase model, in the proposed model, each element interacts only with the neighbors over a certain time (called the interaction time) during Phase 4 of the action potential. The pacemaker cell elements are arranged on a square lattice, and each element connects with the elements surrounding it. The results indicate that the diversity of intrinsic frequencies of pacemaker cells may be necessary for synchronization. Moreover, increasing the proportion of invalid connections causes the elements to take more time to synchronize until eventually they do not synchronize at all, and decreasing the intrinsic frequencies of the elements prevents them from synchronizing. Probably these elucidate the cause of sick sinus syndrome.

Probability Theory Predicts That Chunking into Groups of Three or Four Items Increases the Short-Term Memory Capacity

Short-term memory allows individuals to recall stimuli, such as numbers or words, for several seconds to several minutes without rehearsal. Although the capacity of short-term memory is considered to be 7 &#177?2 items, this can be increased through a process called chunking. For example, in Japan, 11-digit cellular phone numbers and 10-digit toll free numbers are chunked into three groups of three or four digits: 090-XXXX-XXXX and 0120-XXX-XXX, respectively. We use probability theory to predict that the most effective chunking involves groups of three or four items, such as in phone numbers. However, a 16-digit credit card number exceeds the capacity of short-term memory, even when chunked into groups of four digits, such as XXXX-XXXX-XXXX-XXXX. Based on these data, 16-digit credit card numbers should be sufficient for security purposes.

A Probabilistic Method to Determine Whether the Speed of Light Is Constant

Although the formula of mass-energy equivalence was derived from the hypothesis that the speed of light in free space is constant, conversely, the purpose of this research is to show that a method of probabilistically determining whether the speed of light is constant is derived from this formula. By considering the formula of mass-energy equivalence to be a function of the energy of an object moving at speed V, the probability density function (PDF) of the energy can be obtained using the inverse function of this formula, if the speed of light obeys a probability distribution. The main result is that the PDF of the energy diverges to infinity at a certain energy value regardless of the PDF of the speed of light. Thus, when the speed calculated from this value enters a certain range of the speed of light as V increases stepwise from below 299,792,458 m/s, the PDF of the energy should increase abruptly. If not, then the speed of light is constant. This is the method of probabilistically determining whether the speed of light is constant. An experimental method is proposed to confirm this.

Probability Theory Predicts That Group Survival May Be Guaranteed for Groups with More Than 10 Elements

The purpose of this study is to find out the critical number of elements needed for group survival. Taking a probabilistic approach, how the lifetime of a group consisting of several elements depends on the number of elements and the probability distribution of their lifetimes is investigated. Four probability distributions are examined: an exponential distribution, a uniform distribution, a parabolic distribution, and a pointed distribution composed of two parabolas. The lifetime of the group is defined as the expected value of the maximum lifetime of the elements in that group. The probability distribution of this maximum shifts to the right as the number of elements increases, and the expected value of the lifetime of each element eventually becomes less than the lower limit of this distribution. The number of elements in this case is defined as the critical number of elements needed for group survival. Hence, if the number of elements is larger than the critical number needed for group survival, the lifetime of the group is guaranteed to be longer than the expected lifetime of one element. The findings are in the following. The critical number needed for group survival is inversely proportional to the expected lifetime of one element, regardless of the probability distribution used. It decreases as for an exponential distribution, and as in case of the others, where N is the number of elements of the group. Because a probability distribution defined over a finite range is assumed to be reasonable in practice, a group consisting of more than 10 elements should survive well.

A Modified Right Helicoid Can Simulate the Inner Structure of the Cochlea in the Hearing Organ of Mammals

The purpose of this study is to develop a mathematical model of the spiral basilar membrane in the center of the cochlea, which plays an important role in the mammalian auditory system. The basilar membrane transmits sound vibrations, which are converted into electrical potential changes by the inner hair cells. The basilar membrane is thought to lie on a locally undistorted curved surface because the inner hair cells, which are arranged in an orderly fashion on the basilar membrane, respond to their location-specific frequencies. In mammals, the number of rotations of this surface and the rate of change of its width with each rotation are different. It turns out that by modifying the right helicoid, we can obtain a mathematical model that satisfies these points. In conclusion, even though the three-dimensional structure of the basilar membrane varies among species, this model can reproduce this structure. This further suggests that there are common genetic determinants of cochlear development in mammals. From a practical standpoint, this may be useful for creating cochlear implants.

A Mathematical Model Reveals That Both Randomness and Periodicity Are Essential for Sustainable Fluctuations in Stock Prices

Is it true that there is an implicit understanding that Brownian motion or fractional Brownian motion is the driving force behind stock price fluctuations? An analysis of daily prices and volumes of a particular stock revealed the following findings: 1) the logarithms of the moving averages of stock prices and volumes have a strong positive correlation, even though price and volume appear to be fluctuating independently of each other, 2) price and volume fluctuations are messy, but these time series are not necessarily Brownian motion by replacing each daily value by 1 or –1 when it rises or falls compared to the previous day’s value, and 3) the difference between the volume on the previous day and that on the current day is periodic by the frequency analysis. Using these findings, we constructed differential equations for stock prices, the number of buy orders, and the number of sell orders. These equations include terms for both randomness and periodicity. It is apparent that both randomness and periodicity are essential for stock price fluctuations to be sustainable, and that stock prices show large hill-like or valley-like fluctuations stochastically without any increasing or decreasing trend, and repeat themselves over a certain range.

Probability of Matching All Types of Prizes for the First Time Is Maximized at a Surprisingly Early Number of Trials

Suppose that when money or coins are placed in a box containing an arbitrary number of prizes of several different types, one of each type of prize will appear alone each time. Is the probability that all types of prizes will be together for the first time maximized when the majority of prizes are removed from the box? Simulations based on probability theory show that this probability reaches its maximum value in a surprisingly small number of trials, contrary to expectations. This will help us understand not only mathematical phenomena, but also real-world phenomena. Phenomena that do not occur without several substances or conditions seem unlikely to occur, but the results of this study suggest that, contrary to expectations, they are surprisingly likely to occur probabilistically.