The closed forms of two kinds of generating functions for the Pascal-type triangle are derived.A certain of properties like the Pascal triangle are found.Explicit expression of any element and closed forms of several ...The closed forms of two kinds of generating functions for the Pascal-type triangle are derived.A certain of properties like the Pascal triangle are found.Explicit expression of any element and closed forms of several kinds of sums are given.A number of combinatorial identities associated with the triangle are constructed.展开更多
Thousands of landslide data being taken as the nation wide statistics of sampling and the two state variables of landslide being processed with two methods described in the references, the main types of lithologica...Thousands of landslide data being taken as the nation wide statistics of sampling and the two state variables of landslide being processed with two methods described in the references, the main types of lithological groups of landslides in China have been sieved and selected.On the other hand, through the displacement table of Pascal Yanghui triangle used in the information encoding theory, the mark weight of sampling can be calculated and the main lithological groups which have close relationship with landslide occurrence can be gained.In comparison with the both results, the characteristics of main sliding lithological groups are determinated, and the main distribution regions of landslides can be prognosticated.展开更多
Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal ...Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal triangle elements and pascal matrix formation, etc. This paper explains about its functions and properties of N-Summet-k. The result of variation between N and k is shown in tabulation.展开更多
Objective:We through the anatomy of cadavers to study the"Kambin’s triangle"in the safe working area of lumbar intervertebral foramen and to provide anatomical reference for clinical lumbar fusion through K...Objective:We through the anatomy of cadavers to study the"Kambin’s triangle"in the safe working area of lumbar intervertebral foramen and to provide anatomical reference for clinical lumbar fusion through Kambin’s triangle approach.Methods:five complete cadaveric specimens were taken,the soft tissue of the lumbar back was removed,the transverse process,upper and lower articular processes and part of the vertebral lamina were bitten,the Kambin’s triangle area of the lumbar spine was completely exposed,the bottom edge and height of the Kambin’s triangle were measured,and the area of the Kambin’s triangle was calculated;Using Kirschner wire,pull and fix the traveling nerve root to make the Kambin’s triangle into a rectangle,measure the length of the bottom edge and height again,calculate the area,and compare the two groups of data.Results:the average height of the Kambin’s triangle was 11.20mm±2.10mm,and the average height of the improved four corners was 11.19mm±1.93mm.The height of the improved four corners was slightly shorter than that of the Kambin’s triangle.There was a significant correlation between the two,but the difference was not statistically significant.The average bottom of Kambin’s triangle is 10.78mm±1.95mm,and the average bottom of improved four corners is 12.14mm±1.78mm.The length of the bottom edge of improved four corners is greater than that of Kambin’s triangle.There is a significant correlation between them,and the difference is statistically significant;The average area of Kambin’s triangle is 61.79mm^(2)±20.71mm^(2),and the area of improved four corners is 137.71mm^(2)±38.20mm^(2).The area of improved four corners is significantly larger than that of Kambin’s triangle.There is a significant correlation between the two,and the difference is statistically significant.Conclusion:there is a narrow right angle triangle area surrounded by traveling nerve root,dural sac and superior endplate of lower vertebral body in the lumbar intervertebral foramen.If the traveling nerve root is pulled and fixed to turn the traditional Kambin’s triangle into a quadrilateral,the bottom edge of the Kambin’s triangle area can be significantly longer and the area can be significantly expanded,which can be operated more safely.展开更多
Over the years the defense industry has become a de facto participant in the policy-making process. As in other areas dominated by big business interests, a policy sub-government of "iron triangle" has emerg...Over the years the defense industry has become a de facto participant in the policy-making process. As in other areas dominated by big business interests, a policy sub-government of "iron triangle" has emerged. In the view of some American scholars, such an "iron triangle" as a political relationship that brings together .展开更多
In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical...In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .展开更多
The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the hist...The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the historical backgrounds of these concepts, we propose an algorithm that will generate the elements of these matrices. In fact, we will show that the indicated algorithm can be used to construct the elements of any power series matrix generated by any polynomial (see Definition 1), and hence, it is a generalization of the specific algorithms that give us the Pascal and the Fibonacci matrices.展开更多
Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I ex...Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.展开更多
After posing the axiom of linear algebra, the author develops how this allows the calculation of arbitrary base powers, which provides an instantaneous calculation of powers in a particular base such as base ten;first...After posing the axiom of linear algebra, the author develops how this allows the calculation of arbitrary base powers, which provides an instantaneous calculation of powers in a particular base such as base ten;first of all by developing the any base calculation of these powers, then by calculating triangles following the example of the “arithmetical” triangle of Pascal and showing how the formula of the binomial of Newton is driving the construction. The author also develops the consequences of the axiom of linear algebra for the decimal writing of numbers and the result that this provides for the calculation of infinite sums of the inverse of integers to successive powers. Then the implications of these new forms of calculation on calculator technologies, with in particular the storage of triangles which calculate powers in any base and the use of a multiplication table in a very large canonical base are discussed.展开更多
We demonstrate how to extract the Planck length from hydrostatic pressure without relying on any knowledge of Newton’s gravitational constant, G. By measuring the pressure from a water column, we can determine the Pl...We demonstrate how to extract the Planck length from hydrostatic pressure without relying on any knowledge of Newton’s gravitational constant, G. By measuring the pressure from a water column, we can determine the Planck length without requiring knowledge of either G or the Planck constant. This experiment is simple to perform and cost-effective, making it not only of interest to researchers studying gravity but also suitable for low-budget educational settings. Despite its simplicity, this has never been demonstrated to be possible before, and it is achievable due to new theoretical insights into gravity and its connection to quantum gravity and the Planck scale. This provides new insights into fluid mechanics and the Planck scale. We are also exploring initial concepts related to what we are calling “Planck fluid”, which could potentially play a central role in quantum gravity and quantum fluid mechanics.展开更多
文摘The closed forms of two kinds of generating functions for the Pascal-type triangle are derived.A certain of properties like the Pascal triangle are found.Explicit expression of any element and closed forms of several kinds of sums are given.A number of combinatorial identities associated with the triangle are constructed.
文摘Thousands of landslide data being taken as the nation wide statistics of sampling and the two state variables of landslide being processed with two methods described in the references, the main types of lithological groups of landslides in China have been sieved and selected.On the other hand, through the displacement table of Pascal Yanghui triangle used in the information encoding theory, the mark weight of sampling can be calculated and the main lithological groups which have close relationship with landslide occurrence can be gained.In comparison with the both results, the characteristics of main sliding lithological groups are determinated, and the main distribution regions of landslides can be prognosticated.
文摘Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal triangle elements and pascal matrix formation, etc. This paper explains about its functions and properties of N-Summet-k. The result of variation between N and k is shown in tabulation.
基金Hainan Provincial Natural Science Foundation(No.819QN365)National Natural Science Foundation of China(No.81902270)。
文摘Objective:We through the anatomy of cadavers to study the"Kambin’s triangle"in the safe working area of lumbar intervertebral foramen and to provide anatomical reference for clinical lumbar fusion through Kambin’s triangle approach.Methods:five complete cadaveric specimens were taken,the soft tissue of the lumbar back was removed,the transverse process,upper and lower articular processes and part of the vertebral lamina were bitten,the Kambin’s triangle area of the lumbar spine was completely exposed,the bottom edge and height of the Kambin’s triangle were measured,and the area of the Kambin’s triangle was calculated;Using Kirschner wire,pull and fix the traveling nerve root to make the Kambin’s triangle into a rectangle,measure the length of the bottom edge and height again,calculate the area,and compare the two groups of data.Results:the average height of the Kambin’s triangle was 11.20mm±2.10mm,and the average height of the improved four corners was 11.19mm±1.93mm.The height of the improved four corners was slightly shorter than that of the Kambin’s triangle.There was a significant correlation between the two,but the difference was not statistically significant.The average bottom of Kambin’s triangle is 10.78mm±1.95mm,and the average bottom of improved four corners is 12.14mm±1.78mm.The length of the bottom edge of improved four corners is greater than that of Kambin’s triangle.There is a significant correlation between them,and the difference is statistically significant;The average area of Kambin’s triangle is 61.79mm^(2)±20.71mm^(2),and the area of improved four corners is 137.71mm^(2)±38.20mm^(2).The area of improved four corners is significantly larger than that of Kambin’s triangle.There is a significant correlation between the two,and the difference is statistically significant.Conclusion:there is a narrow right angle triangle area surrounded by traveling nerve root,dural sac and superior endplate of lower vertebral body in the lumbar intervertebral foramen.If the traveling nerve root is pulled and fixed to turn the traditional Kambin’s triangle into a quadrilateral,the bottom edge of the Kambin’s triangle area can be significantly longer and the area can be significantly expanded,which can be operated more safely.
文摘Over the years the defense industry has become a de facto participant in the policy-making process. As in other areas dominated by big business interests, a policy sub-government of "iron triangle" has emerged. In the view of some American scholars, such an "iron triangle" as a political relationship that brings together .
文摘In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .
文摘The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the historical backgrounds of these concepts, we propose an algorithm that will generate the elements of these matrices. In fact, we will show that the indicated algorithm can be used to construct the elements of any power series matrix generated by any polynomial (see Definition 1), and hence, it is a generalization of the specific algorithms that give us the Pascal and the Fibonacci matrices.
文摘Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.
文摘After posing the axiom of linear algebra, the author develops how this allows the calculation of arbitrary base powers, which provides an instantaneous calculation of powers in a particular base such as base ten;first of all by developing the any base calculation of these powers, then by calculating triangles following the example of the “arithmetical” triangle of Pascal and showing how the formula of the binomial of Newton is driving the construction. The author also develops the consequences of the axiom of linear algebra for the decimal writing of numbers and the result that this provides for the calculation of infinite sums of the inverse of integers to successive powers. Then the implications of these new forms of calculation on calculator technologies, with in particular the storage of triangles which calculate powers in any base and the use of a multiplication table in a very large canonical base are discussed.
文摘We demonstrate how to extract the Planck length from hydrostatic pressure without relying on any knowledge of Newton’s gravitational constant, G. By measuring the pressure from a water column, we can determine the Planck length without requiring knowledge of either G or the Planck constant. This experiment is simple to perform and cost-effective, making it not only of interest to researchers studying gravity but also suitable for low-budget educational settings. Despite its simplicity, this has never been demonstrated to be possible before, and it is achievable due to new theoretical insights into gravity and its connection to quantum gravity and the Planck scale. This provides new insights into fluid mechanics and the Planck scale. We are also exploring initial concepts related to what we are calling “Planck fluid”, which could potentially play a central role in quantum gravity and quantum fluid mechanics.
