Identities on q-Harmonic Numbers

With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.

On Prime Numbers between kn and (k + 1) n

In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.

Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

The Theoretical Significance and Reality of Imaginary Number

It is a fact that imaginary numbers do not have practical significance. But the role of imaginary numbers is very broad and enormous, due to the existence of Euler’s formula. Due to Euler’s formula, imaginary numbers have been applied in many theoretical theories. One of the biggest functions of imaginary numbers is to represent changes in phase, which is indispensable in signal analysis theory. The imaginary numbers in quantum mechanics pose a greater mystery: do the imaginary numbers really exist? This question still needs further scientific development to be answered.

Non-Darcy Flow in Molding Sands

Darcy’s law is widely used to describe the flow in porous media in which there is a linear relationship between fluid velocity and pressure gradient. However, it has been found that for high numbers of Reynolds this law ceases to be valid. In this work, the Ergun equation is employed to consider the non-linearity of air velocity with the pressure gradient in casting sands. The contribution of non-linearity to the total flow in terms of a variable defined as a non-Darcy flow fraction is numerically quantified. In addition, the influence of the shape factor of the sand grains on the non-linear flow fraction is analyzed. It is found that for values of the Reynolds number less or equal than 1, the contribution of non-linearity for spherical particles is around 1.15%.

Reciprocity Relation between Alternative Gravity Formulas

We compare Newton’s force law of universal gravitation with a corrected simple approach based on Bhandari’s recently presented work, where the gravitation constant G is maintained. A reciprocity relation exists between both alternative gravity formulas with respect to the distances between mass centers. We conclude a one-to-one mapping of the two gravitational formulas. We don’t need Einstein’s construct of spacetime bending by matter.

雷诺数和侧风对S型垂直轴风力机气动特性的影响

相较于工作在郊外良好风场的大型风力机,小型风力机易受城市风环境如低风速、高湍流度以及时变风速风向的影响。采用数值模拟手段研究了雷诺数和来流倾角对小型S型垂直轴风力机气动性能的影响。研究结果表明:S型风力机的气动性能对雷诺数不敏感,功率系数几乎不随雷诺数变化;侧风导致S型风力机的气动性能变差,静扭矩系数和功率系数降低,并且随着侧风来流倾角的增大,气动性能越差,侧风影响的流动机理一方面是来流倾角降低了水平速度分量,另一个方面是竖直速度分量减小了前进叶片的压力差,增加回转叶片的压力差,进一步加剧了气动性能的恶化。

“S”型隧道近距离上穿既有盾构隧道的变形控制方法

依托呼和浩特市新建“S”型地下连接通道近距离上穿既有地铁2号线工程,采用有限元数值模拟和现场监测相结合的方法,对砂袋压重范围、砂袋压重高度以及中隔壁拆除方式进行了对比分析,得出了对既有盾构隧道变形控制的最优方案。研究表明:(1)既有盾构隧道各阶段最大变形位置随新建隧道的掘进而动态变化,施工完毕后最大变形出现在既有盾构隧道远端,这是由于新建隧道开挖至盾构正上方时,该位置受到的施工扰动影响最大,是变形控制的重点区域;(2)采用砂袋压重对既有盾构隧道的变形控制效果较为明显,变形控制效果基本在25%~30%之间,但增加压重范围和压重高度对减小隆起变形幅度影响不大,基本在4%~6%之间,考虑到施工经济性,选用压重范围为49~87榀,压重高度为2.0 m的压重措施方案;(3)采用不同的中隔壁拆除方式对隧道隆起变形有较大影响,为了便于及时形成稳定的受力体系,选用了“隔一拆二”的拆除方式。通过采取合理的砂袋压重措施和中隔壁拆除方式,在上穿施工中有效控制了既有盾构隧道的隆起变形,施工完毕后既有盾构隧道道床变形为2.68 mm,接近道床实测数据2.51 mm,满足规范要求(4.00 mm),可为类似工程提供相关借鉴经验。

Strong Law of Large Numbers and Complete Convergence for Sequences of -Mixing Random Variables

We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann＇s strong law of large numbers and Teicher＇s strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.

Detecting a Regularity in the Generation and Utilization of Primes in the Multiplicative Number Theory

If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.

A new bionic MAV's flapping motion based on fruit fly hovering at low Reynolds number

On the basis of the studies on the high unsteady aerodynamic mechanisms of the fruit fly hovering the aerodynamic advantages and disadvantages of the fruit fly flapping motion were analyzed. A new bionic flapping motion was proposed to weaken the disadvantages and maintain the advantages, it may be used in the designing and manufacturing of the micro air vehicles （MAV＇s）. The translation of the new bionic flapping motion is the same as that of fruit fly flapping motion. However, the rotation of the new bionic flapping motion is different. It is not a pitching-up rotation as the fruit fly flapping motion, but a pitching-down rota- tion at the beginning and the end of a stroke. The numerical method of 3rd-order Roe scheme developed by Rogers was used to study these questions. The correctness of the numerical method and the computational program was justified by comparing the present CFD results of the fruit fly flapping motion in three modes, i.e., the advanced mode, the symmetrical mode and the delayed mode, with Dickinson＇s experimental results. They agreed with each other very well. Subsequently, the aerodynamic characteristics of the new bionic flapping motion in three modes were also numerically simulated, and were compared with those of the fruit fly flap- ping. The conclusions could be drawn that the high unsteady lift mechanism of the fruit fly hovering is also effectively utilized by this new bionic flapping. Compared with the fruit fly flapping, the unsteady drag of the new flapping decreases very much and the ratio of lift to drag increases greatly. And the great discrepancies among the mean lifts of three flapping modes of the fruit fly hovering are effectively smoothed in the new flapping. On the other hand, this new bionic flapping motion should be realized more easily. Finally, it must be pointed out that the above conclusions were just drawn for the hovering flapping motion. And the aerodynamic characteristics of the new bionic flapping motion in forward flight are going to be studied in the next step.

Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

Mass Constituents of a Flat Lattice Multiverse: Conclusion from Similarity between Two Universal Numbers, the Rocksalt-Type 2<i>D</i>Madelung Constant and the Golden Mean

In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean &phi;=0.6180339887, yielding for dark energy to matter mass fractions .?Assuming the baryonic matter to be only 4.432%, the ratio of matter to baryonic matter would be , and further the ratio of dark matter to baryonic one . If one subtracts from the dark matter a contribution of antimatter with the same mass of baryonic matter, according to the antigravity theories of Villata respectively Hajdukovic, the remaining mass ratio would yield . Replacing the “Madelung” constant α of Villata’s “lattice universe” by &phi;, one reaches again 1 + &phi;as the ratio of the repulsive mass contribution to the attractive one. Assuming instead of a 3D lattice a flat 2D one of rocksalt type, the numerical similarity between the Madelung constant and φ&minus;1 could not be just coincidence. The proposed scaling of the cosmological mass fractions with the square of the most irrational universal number &phi;may indicate that the chaotic cosmological processes have reached a quite stable equilibrium. This may be confirmed by another, but similar representation of the mass constituents by the Archimedes’ constant &pi;, giving for respectively for the dark components . However, the intimate connection of φ with its reciprocal may ignite the discussion whether our universe is intertwined with another universe or even part of a multiverse with the dark constituents contributed from there.

Circle of Willis atherosclerosis, Alzheimer's disease and the Dean number

The important role of atherosclerosis in pathophysiology of Alzheimer's Disease has become evident.Mechanisms such as hyperlipidemia,inflammation,abdominal obesity and insulin resistance are important yet they may not fully explain the specific involvement of the Circle of Willis in these pathologies.The Circle of Wills is a complex geometrical structure which has several areas with different curvature as well as various branching angles of vessels composing the circle.The hemodynamics in this region should take into account the Dean number which indicates the influence of curvature on the resistance to blood flow.Thus,areas with various curvature and angles may have different hemodynamics and there are certain areas in the Circle of Willis that are more likely to develop atherosclerotic changes.Therefore,this could suggest the novel pathophysiological pathway resulting from the geometric peculiarities of the Circle of Willis.One of the directions of future research is to examine whether specific areas of the Circle of Willis are more likely to develop atherosclerotic changes compared to other ones.Selective areas of the Circle of Willis affected by atherosclerotic changes could indicate the primary role of atherosclerosis promoting Alzheimer's disease although other pathophysiological mechanisms suggesting the opposite direction should be also examined in prospective studies.

Biased Random Number Generator Based on Bell's Theorem

We propose a biased random number generation protocol whose randomness is based on the violation of the Clauser Home inequality. Non-maximally entangled state is used to maximize the Bell violation. Due to the rotational asymmetry of the quantum state, the ratio of Os to ls varies with the measurement bases. The experimental partners can then use their measurement outcomes to generate the biased random bit string. The bias of their bit string can be adjusted by altering their choices of measurement bases. When this protocol is implemented in a device-independent way, we show that the bias of the bit string can still be ensured under the collective attack.

Changing Trend and Mutation Analysis of Number of Fog and Haze Days in Baoji City during 1981-2013

Based on data of number of fog and haze days in Baoji City from 1981 to 2013,the changing trends and mutation of number of fog and haze days in Baoji over the past 33 years were analyzed by using trend coefficient,tendency rate,linear regression analysis,anomaly percentage,Mann-Kendall mutation test and sliding t test.The results showed that during the 33 years,the number of fog and haze days in Baoji declined by 16.253d/10 a,and there was a cyclical turbulence every 6,15 or 28years.The frequency of fog and haze weather was the highest in winter,followed by spring and autumn,while it was the lowest in summer.According to the anomaly percentage of the number of fog and haze days in 12 months during 1981-2013,the anomaly percentage changed most greatly in July,followed by September,October,April,May,June,August,February and March,but it fluctuated less greatly in January.The number of fog and haze days from 1981 to 2013had obvious mutation trends in a single year and a single season,and mutation types are different.

The Ordinal Interpretation of the Integers and Its Use in Number Theory

The author recently published a paper which claimed that an ordinal interpretation of numbers had limited applicability for cryptography. A further examination of this subject, in particular to what extent an ordinal interpretation is useful for recurrence sequences, is needed. Hilbert favored an interpretation of the natural numbers that placed their ordinal properties prior to their cardinal properties [1] [2]. The author examines ordinal uses of the integers in number theory in order to discuss the possibilities and limitations of this approach. The author hopes this paper will be useful in clarifying or even correcting some matters that were discussed in his paper of January of 2018. I was trained informally in philosophical realism, and while I think idealism too has a place, at this time in my life I believe that the weight of evidence and usefulness is more on the side of philosophical materialism. I hope this discussion will help supplement for my readers the material in Number in Mathematical Cryptography. I still maintain that a lack of clarity on these matters has hindered progress in cryptography;and it has taken time for me to better understand these things. I hope others who have interest and ability will assist in making these matters clearer. My intention was to work in pure mathematics, and the transition to an applied mindset was difficult for me. As a result, I feel most comfortable in a more middle-of-the road attitude, but have had to slowly move to a more precise analysis of the physical quantities involved. I hope my readers will be patient with my terminology, which is still evolving, and my discussion of things which are more indirectly related, and which are necessary for my expression. These are important things for the mathematical community to understand, and I hope smarter and more knowledgeable people will address my errors, and improve upon the things I might have correct. I am discussing sequences which are sometimes a use of both ordinal and cardinal numbers.

Generalized Eulerian Numbers

We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.

Extensions of the Constructivist Real Number System

The paper reviews the most consequential defects and rectification of traditional mathematics and its foundations. While this work is only the tip of the iceberg, so to speak, it gives us a totally different picture of mathematics from what we have known for a long time. This journey started with two teasers posted in SciMath in 1997: 1) The equation 1 = 0.99… does not make sense. 2) The concept ?does not exist. The first statement sparked a debate that raged over a decade. Both statements generated a series of publications that continues to grow to this day. Among the new findings are: 3) There does not exist nondenumerable set. 4) There does not exist non-measurable set. 5) Cantor’s diagonal method is flawed. 6) The real numbers are discrete and countable. 7) Formal logic does not apply to mathematics. The unfinished debate between logicism, intuitionism-constructivism and formalism is resolved. The resolution is the constructivist foundations of mathematics with a summary of all the rectification undertaken in 2015, 2016 and in this paper. The extensions of the constructivist real number system include the complex vector plane and transcendental functions. Two important results in the 2015 are noted: The solution and resolution of Hilbert’s 23 problems that includes the resolution of Fermat’s last theorem and proof Goldbach’s conjecture.

The Proof of Goldbach’s Conjecture on Prime Numbers

Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.