[8] **viXra:1807.0497 [pdf]**
*submitted on 2018-07-29 18:52:13*

**Authors:** James A. Smith

**Comments:** 12 Pages.

Because the Chain Rule can confuse students as much as it helps them solve real problems, we put ourselves in the shoes of the mathematicians who derived it, so that students may understand the motivation for the rule; its limitations; and why textbooks present it in its customary form. We begin by finding the derivative of sin2x without using the Chain Rule. That exercise, having shown that even a comparatively simple compound function can be bothersome to differentiate using the definition of the derivative as a limit, provides the motivation for developing our own formula for the derivative of the general compound function g[f(x)]. In the course of that development, we see why the function f must be continuous at any value of x to which the formula is applied. We finish by comparing our formula to that which is commonly given.

**Category:** General Mathematics

[7] **viXra:1807.0477 [pdf]**
*replaced on 2018-07-29 09:52:37*

**Authors:** Waldemar Zieliński

**Comments:** 3 Pages.

This paper is a supplement to the previous "Union of two arithmetic sequences - Basic calculation formula (1)” (viXra:1712.0636). We will derive a simpler version of formula for the union of two arithmetics progressions.

**Category:** General Mathematics

[6] **viXra:1807.0415 [pdf]**
*submitted on 2018-07-23 11:15:19*

**Authors:** Martin Peter Neuenhofen

**Comments:** 13 Pages. Proof of convergence for a direct transcription method for OCP

In the arXiv paper [arXiv:1712.07761] from December 2017 we presented a convergent direct transcription method for optimal control problems. In the present paper we present a significantly generalized convergence theory in succinct form. Therein, we replace strong assumptions that we had formerly made on local uniqueness of the solution, and on differentiability of a particular functional. These assumptions are removed now.

**Category:** General Mathematics

[5] **viXra:1807.0399 [pdf]**
*submitted on 2018-07-24 08:41:02*

**Authors:** Edgar Valdebenito

**Comments:** 6 Pages.

This note presents some sequences related with the cosine fixed point constant.

**Category:** General Mathematics

[4] **viXra:1807.0197 [pdf]**
*submitted on 2018-07-09 09:58:18*

**Authors:** Thinh D. Nguyen

**Comments:** 3 Pages.

In this work, we will take one problem, namely Packing Triangles as an example of combinatorial optimization problems. We show that if one has ever loved reading Prasolov’s books, then one should not try to ﬁnd efﬁcient algorithm for various restricted cases of this problem.

**Category:** General Mathematics

[3] **viXra:1807.0125 [pdf]**
*submitted on 2018-07-05 14:59:41*

**Authors:** Thinh Nguyen

**Comments:** 19 Pages.

We prove that for every d≥2, deciding if a pure, d-dimensional, simplicial complex is
shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj
and Klee in 1978. Our reduction also yields that for every d≥2 and k≥0, deciding if a pure,
d-dimensional, simplicial complex is k-decomposable is NP-hard. For d≥ 3, both problems
remain NP-hard when restricted to contractible pure d-dimensional complexes. Another
simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-
shellable.

**Category:** General Mathematics

[2] **viXra:1807.0098 [pdf]**
*submitted on 2018-07-03 06:11:19*

**Authors:** Thinh Nguyen

**Comments:** 4 Pages.

The Erd ̋os-Szekeres theorem states that, for every $k$, there is a number $n_k$ such that every set of $n_k$ points in general position in the plane contains a subset of $k$ points in convex position. If we ask the same question for subsets whose convex hull does not contain any other point from the set, this is not true: as shown by Horton, there are sets of arbitrary size that do not contain an empty 7-gon.
These questions have also been studied extensively
from a computational point of view, and polynomial time algorithms for finding the largest (empty) convex set have been given for the planar case. In higher dimension, it is not known how to compute such a set efficiently. In this paper, we show that already in 3 dimensions no polynomial time algorithm exists for determining the largest (empty) convex set (unless $P$=$NP$), by proving that the corresponding decision problem is $NP$-hard. This answers a question by Dobkin, Edelsbrunner and Overmars from 1990.
As a corollary, we derive a similar result for the closely related problem of testing weak $ε$-nets in $R^3$ . Answering a question by Chazelle et al. from 1995, our reduction shows that the problem is $co-NP$-hard.
Finally, we make several suggestions for further research on the subject.

**Category:** General Mathematics

[1] **viXra:1807.0031 [pdf]**
*replaced on 2018-07-02 07:45:23*

**Authors:** August Lau

**Comments:** 6 Pages.

Mathematics has been highly effective in its application of formalism to the real world. It spans from physical science to data science. Mathematical algorithm affects our everyday life. Mathematization (converting data to equations or mathematical forms) has been very successful. It has led to comments like “the unreasonable effectiveness of mathematics,” but we might have “unreasonable expectation of mathematics.”

**Category:** General Mathematics