Mathematics

In any triangle, the perpendicular side bisectors meet the corresponding internal angle bisectors on the circumcircle. If we take those three points as the vertices of a new triangle and repeat the operation indefinitly, we end up in the limit with a par of equilateral triangles whose sides are parallel to the sides of the Morley triangle of the initial triangle.

In this paper, new improvement of celebrated Hölder inequality by means of isotonic linear functionals is established. An important feature of the new inequality obtained in here is that many existing inequalities related to the Hölder inequality can be improved via new improvement of Hölder inequality. We also show this in an application.

The main purpose of the present paper is to define and study the notion of quasi bi-slant submanifolds of almost contact metric manifolds. We mainly concerned with quasi bi-slant submanifolds of cosymplectic manifolds as a generalization of slant, semi-slant, hemi-slant, bi-slant and quasi hemi-slant submanifolds. First, we give non-trivial examples in order to demostrate the method presented in this paper is effective and investigate the geometry of distributions. Moreover, We study these types of submanifolds with parallel canonical structures.

The meromorphic function W(s) introduced in the Riemann-Zeta function ζ(s)=W(s)ζ(1−s) maps the line of s=1/2+it onto the unit circle in W -space. |W(s)|=0 gives the trivial zeroes of the Riemann-Zeta function ζ(s) . In the range: 0<|W(s)|≠1 , ζ(s) does not have nontrivial zeroes. |W(s)|=1 is the necessary condition for the nontrivial zeros of the Riemann-Zeta function. Writing s=σ+it , in the range: 0≤σ≤1 , but σ≠1/2 , even if |W(s)|=1 , the Riemann-Zeta function ζ(s) is non-zero. Based on these arguments, the nontrivial zeros of the Riemann-Zeta function ζ(s) can only be on the s=1/2+it critical line. Therefore a proof of the Riemann Hypothesis is presented.

The Sendovs conjecture asserts that if all the zeros of a polynomial p(z) lie in the closed unit disk, then there must be a critical point of p(z) within unit distance of each zero. The conjecture has been proved to be true for many special cases.Here we give a proof of the conjecture.

We remark a variant of the existence part of the fundamental theorem of calculus, which, together with the Lebesgue differentiation theorem, constitute a new proof that every Riemann-integrable function on a compact interval having limit everywhere on the interior is almost everywhere continuous with respect to Lebesgue measure. The proof is intended as a new connection between Lebesgue differentiation theorem and Lebesgue criterion of Riemann integrability.

In this short note, we show that both Riemann--Liouville and Hilfer fractional differential operators coincide on the space of continuous functions.

We present a short and simple proof of the Riemann's Hypothesis (RH) where only undergraduate mathematics is needed.

The four-color theorem states that no more than four colors are required to color all nodes in planar graphs such that no two adjacent nodes are of the same color. The theorem was first propounded by Francis Guthrie in 1852. Since then, scholars have either failed to solve this theorem or required computer assistance to prove it. Hence, the goal of this paper is to provide the first correct proof of this 170-year-old mathematical problem composed with the human brain and without computer assistance in only five pages.

Carathéodory's well-known conjecture states that every sufficiently smooth, closed convex surface in three dimensional Euclidean space admits at least two umbilic points. It has been established that the conjecture is true for all rotationally symmetric surfaces; in this paper, we investigate the umbilic points of two families of surfaces without rotational symmetry, and compute their indices. In particular, we find that the family of surfaces of the form a x 2k +b y 2k +c z 2k =1 with a,b,c>0 , k∈ Z >1 admit 14 umbilic points: six of one known form and eight of another. For many tested values of a,b,c,k , such umbilic points have indices −1/2 and 1 , respectively. We also explore the dependence of the umbilic points on the parameter ϵ of the surface a x 2 +ϵ x 4 +a y 2 +ϵ y 4 +b z 2 =1 . In particular, for both a<b and a>b, there exist exactly two umbilic points with index 1 for ϵ smaller than certain critical values. For larger ϵ, surfaces with a>b admit exactly ten umbilic points; for many tested values of a,b,ϵ, these points have indices 1/2 and -1. For larger ϵ, surfaces with a<b admit eighteen umbilic points; for many tested values of a,b,ϵ, these points have indices -1/2 and 1.