Mathematics

The Erd?s-Straus conjecture is a renowned problem which describes that for every natural number n (??) , 4 n can be represented as the sum of three unit fractions. The main purpose of this study is to show that the Erd?s-Straus conjecture is true. The study also re-demonstrates Mordell theorem which states that 4 n has a expression as the sum of three unit fractions for every number n except possibly for those primes of the form n?�r (mod 780) with r= 1 2 , 11 2 , 13 2 , 17 2 , 19 2 , 23 2 . For l,r,a?�N ; 4 24l+1 ??1 6l+r = 4r?? (6l+r)(24l+1) with 1?�r??2l , if at least one of the sums in right side of the expression, say, a+(4r?�a??), 1?�a??r?? for at least one of the possible value of r such that a,(4r?�a??) divide (6l+r)(24l+1) ; then the conjecture is valid for the corresponding l . However, in this way the conjecture can not be proved only twelve values of l for l up to l= 10 5 .

In this tutorial survey we recall the basic properties of the special function of the Mittag-Leffler and Wright type that are known to be relevant in processes dealt with the fractional calculus. We outline the major applications of these functions. For the Mittag-Leffler functions we analyze the Abel integral equation of the second kind and the fractional relaxation and oscillation phenomena. For the Wright functions we distinguish them in two kinds. We mainly stress the relevance of the Wright functions of the second kind in probability theory with particular regard to the so-called M-Wright function that generalizes the Gaussian and is related with the time-fractional diffusion equation.

The problem of measuring an unbounded system attribute near a singularity has been discussed. Lenses have been introduced as formal objects to study increasingly precise measurements around the singularity and a specific family of lenses called Exterior probabilities have been investigated. It has been shown that under such probabilities, measurement variance of a measurable function around a 1st order pole on a complex manifold, consists of two separable parts - one that decreases with diminishing scale of the lenses, and the other that increases. It has been discussed how this framework can lend mathematical support to ideas of non-deterministic uncertainty prevalent at a quantum scale. In fact, the aforementioned variance decomposition allows for a minimum possible variance for such a system irrespective of how close the measurements are. This inequality is structurally similar to Heisenberg uncertainty relationship if one considers energy/momentum to be a meromorphic function of a complex spacetime.

This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to convey the gist of the proof. A rigorous version of the proof can be developed using only single-variable calculus and basic properties of complex numbers, but the technical details are somewhat involved. In order to facilitate the reader's intuitive grasp of the proof, we first present the main points of the argument, which can be illustrated by computer experiments. Next we fill in some of the details, using single-variable calculus. Finally, we give a numerical procedure for finding all roots of an n'th degree polynomial by solving 2n differential equations in parallel.

We derive a closed form for the generalized Bernoulli polynomial of order n in terms of Bell polynomials and Stirling numbers of the second kind using the Faà di Bruno's formula.

In this work we show ζ(3)=4 π 2 ln(B) with the Bendersky-Adamchik constant B .

In this paper, we solve the Navier-Stokes equation, one of the seven Millennium Prize Problems suggested by Clay Mathematics Institute(CMI). We prove that, for any initial velocity u0, there has been a force vector f, such that there exists no smooth solution to the respective Navier-Stokes equation. By the way, exact solutions can be explicitly solved as series, where the coefficients are all known functions determined only by u0 abd f. Similar result also holds for the Euler equation.For any n>=2, similar results also hold for dimensional Navier-Stokes equation or Euler equation.

I found a novel class of magic square analogue, magic 24-cell. The problem is to assign the consecutive numbers 1 through 24 to the vertices in a graph, which is composed of 24 octahedra and 24 vertices, to make the sum of the numbers of each octahedron the same. It is known that there are facet-magic and face-magic labelings of tesseract. However, because of 24-cell contains triangle, face-magic labeling to assign different labels to each vertex is impossible. So I tried to make a cell-magic labeling of 24-cell. Linear combination of three binary labeling and one ternary labeling gives 64 different magic labelings of 24-cell. Due to similarity in the number of vertices between 5x5 magic square and magic 24-cell, It might be possible to calculate the number of magic 24-cell. Further analysis would be need to determine the number of magic 24-cell.

In the paper, the author presents a double inequality for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function. This result partially confirms one in a series of conjectures on completely monotonic degrees of remainders of asymptotic expansions for the logarithm of the gamma function and for polygamma functions.

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin Academy of Mathematic. In that paper, he proposed that this function, called Riemann-zeta function takes values 0 on the complex plane when s=0.5+it. This hypothesis has great significance for the world of mathematics and physics. This solutions would lead to innumerable completions of theorems that rely upon its truth. Over a billion zeros of the function have been calculated by computers and shown that all are on this line s = 0.5+it. In this paper, we initially show that Riemann's (Zêta) function and the analytical extension of this function called (Aleph)) are distinct. After extending this function in the complex plane except the point s=1, we will show the existence and then the uniqueness of real part zeros equal to 1/2.