[7] **viXra:1910.0653 [pdf]**
*submitted on 2019-10-31 23:04:59*

**Authors:** Mark Anthony Musgrave

**Comments:** 11 Pages.

Design of the ancient 360-degree circle is proposed to be the result of using a scientific length standard, the Egyptian Royal cubit, to define the dimensions of the circle reference frame. The ancient length standard is subdivided into its own sub-units of palms and fingers, but it is the equivalence to other ancient length units (inches, feet) in proportions that match specific circle features that allow underling design aspects to be identified. The available evidence suggests that the original circle design described by Hipparchus, as being based on “a radius of 3438 minutes”, should be interpreted to mean that the design circle radius was actually 3438 Royal cubits. From this framework it is possible to observe direct metrological relationships between the design of the Royal cubit and the design of the 360-degree circle, as well as the origin of the inch and feet units. Multiple shared features between the circle and the Royal cubit suggest a common design principle was involved, and the evidence suggests that an understanding of electromagnetic physics was in place when the sexagesimal circle was created. If verified, the hypothesis presented here infers that a new frequency standard could be implemented in modern metrology that would provide both the time and length units and allow for complete integration with the 360-degree circle reference frame. This step may also then provide deeper insights into astronomical physics as dimensional features are examined under a suitable length unit

**Category:** General Mathematics

[6] **viXra:1910.0567 [pdf]**
*replaced on 2019-12-14 00:10:54*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

Up to now, I have tried to expand this equation and prove Riemann hypothesis with the equation of cos, sin, but the proof was impossible.
However, I realized that a simple formula before expansion can prove it.
The real value is zero only when the real part of s is 1/2.
Non-trivial zeros must always have a real value of zero.
The real part of s being 1/2 is the minimum requirement for s to be a non-trivial zeros.

**Category:** General Mathematics

[5] **viXra:1910.0560 [pdf]**
*submitted on 2019-10-27 07:22:45*

**Authors:** William F. Gilreath

**Comments:** 34 Pages. Published in the General Science Journal

The absolute value function is a fundamental mathematical concept taught in elementary algebra. In differential calculus, the absolute value function has certain well-known mathematical properties that are often used to illustrate such concepts of—a continuous function, differentiability or the existence of a derivative, the limit, and etcetera.
An alternative to the classical definition of absolute value is given to define a new function that is mathematically equivalent to the absolute value, yet the different mathematically. This new mathematical formalism, the cogent value function, does not have the same mathematical properties of the absolute value function. Two other new mathematical functions are used in the definition of the cogent value function—the parabolin function, and the magnum function.
The cogent value function and the absolute value function have the same domain and range, but both are mathematically very different. The cogent value function demonstrates that the same mathematical concept when formally defined by an alternative method has different mathematical properties. The functions by operation are mathematically similar, but in mathematical formalism each is unique.

**Category:** General Mathematics

[4] **viXra:1910.0477 [pdf]**
*submitted on 2019-10-23 19:21:11*

**Authors:** Saburou Saitoh

**Comments:** 4 Pages. In this short note, for the elementary theorem of remainder in polynomials we recall the division by zero calculus that appears naturally in order to show the importance of the division by zero calculus.

In this short note, for the elementary theorem of remainder in polynomials we recall the division by zero calculus that appears naturally in order to show the importance of the division by zero calculus.

**Category:** General Mathematics

[3] **viXra:1910.0361 [pdf]**
*replaced on 2019-10-23 21:26:29*

**Authors:** Steven Kenneth Kauffmann

**Comments:** 4 Pages.

Trigonometry studies the properties of the cosine and sine functions, which relate a contiguous arc of the unit-radius circle centered on the origin of coordinates to the rectangular Cartesian coordinates of the arc's endpoints. Since the Pythagorean theorem underlies the concept of Cartesian coordinates, this tutorial commences with a plane-geometry recapitulation of that theorem. In the non-calculus treatment of the cosine and sine, their demonstrable properties are encompassed by the unit length of unit-radius circle vectors and the "angle-addition formula" which relates the rectangular coordinates of the endpoints of two immediately successive arcs of the unit-radius circle to the rectangular coordinates of the endpoints of the combined contiguous arc. Those properties are insensitive, however, to simultaneous single-parameter rescaling of all of the arc lengths involved, and so don't unambiguously characterize the cosine and sine functions of directed arc length. Unambiguous determination of the cosine and sine hinges on whether their derivatives with respect to directed arc length are well-defined, which presents no issues for arcs of the unit-radius circle. In fact the cosine and sine functions fascinatingly are the real and imaginary parts of the hyper-well-behaved exponential function of imaginary argument.

**Category:** General Mathematics

[2] **viXra:1910.0165 [pdf]**
*replaced on 2019-10-12 16:29:46*

**Authors:** Steven Kenneth Kauffmann

**Comments:** 4 Pages.

This tutorial explores the relation of the local concept of a function's continuity to its global consequences on closed intervals, such as a continuous function's unavoidable boundedness on a closed interval, its attainment of its least upper and greatest lower bounds on that interval, and its unavoidable assumption on that closed interval of all of the values which lie between that minimum and maximum. In a nutshell, continuous functions map closed intervals into closed intervals. It is understandable that verifying this local-to-global fact involves subtle and very intricate manipulation of the least-upper-bound/greatest-lower-bound postulate for the real numbers. In conjunction with the basic inequality properties of integrals, this continuous-function fact immediately implies the integral form of the mean-value theorem, which is parlayed into its differential form by the fundamental theorem of the calculus. Taylor expansion and its error estimation are further developments which are intertwined with these fascinating concepts.

**Category:** General Mathematics

[1] **viXra:1910.0022 [pdf]**
*replaced on 2019-10-08 04:49:08*

**Authors:** Timothy W. Jones

**Comments:** 5 Pages. Further corrections and amplifications.

The rational root test gives a way to solve polynomial equations. We apply the idea to factoring quadratics (and other polynomials). A calculator speeds up the filtering through possible rational roots.

**Category:** General Mathematics