[4] **viXra:1303.0163 [pdf]**
*replaced on 2013-03-22 09:27:22*

**Authors:** Germán Paz

**Comments:** 6 Pages. Draft version.

In this simple Math exercise we show a property of Pascal's Triangle. More precisely, we show that if $a$ is any positive odd integer, then $\binom{a}{1}-\binom{a}{2}+\binom{a}{3}-\binom{a}{4}+\dots+\binom{a}{a}=1$. Moreover, we prove that if $b$ is any positive even integer, then $\binom{b}{1}-\binom{b}{2}+\binom{b}{3}-\binom{b}{4}+\dots+\binom{b}{b-1}-\binom{b}{b}=1$.

**Category:** General Mathematics

[3] **viXra:1303.0147 [pdf]**
*submitted on 2013-03-20 05:22:21*

**Authors:** Elemer E Rosinger

**Comments:** 17 Pages.

A more careful consideration of the recently introduced "Grossone Theory" of Yaroslav Sergeev, [1], leads to a considerable enlargement of what can constitute possible legitimate mathematical theories by the introduction here of what we may call the {\it Syntactic - Semantic Axiomatic Theories in Mathematics}. The usual theories of mathematics, ever since the ancient times of Euclid, are in fact axiomatic, [1,2], which means that they are {\it syntactic} logical consequences of certain assumed axioms. In these usual mathematical theories {\it semantics} can only play an {\it indirect} role which is
restricted to the inspiration and motivation that may lead to the formulation of axioms, definitions, and of the proofs of theorems. In a significant contradistinction to that, and as manifestly inspired and motivated by the mentioned Grossone Theory, here a {\it direct} involvement of {\it semantics} in the construction of axiomatic mathematical theories is presented, an involvement which gives semantics the possibility to act explicitly, effectively, and altogether directly upon the usual syntactic process of constructing the logical consequences of axioms. Two immediate objections to what appears to be an unprecedented and massive expansion of what may now become legitimate mathematical theories given by the {\it syntactic - semantic axiomatic theories} introduced here can be the following : the mentioned direct role of semantics may, willingly or not, introduce in mathematical theories one, or both of the "eternal taboo-s" of {\it
inconsistency} and {\it self-reference}. Fortunately however, such concerns can be alleviated due to recent developments in both inconsistent and self-referential mathematics, [1,2]. Grateful recognition is acknowledged here for long and most useful ongoing related disccussions with Yaroslav Sergeev.

**Category:** General Mathematics

[2] **viXra:1303.0136 [pdf]**
*submitted on 2013-03-19 02:44:31*

**Authors:** Elemer E Rosinger

**Comments:** 36 Pages.

Physics depends on ”physical intuition”, much of which is formulated in terms of Mathematics. Mathematics itself depends on Logic. The paper presents three latest novelties in Logic which have major consequences in Mathematics. Further, it presents two possible significant departures in Mathematics itself. These five departures can have major implications in Physics. Some of them are indicated, among them in Quantum Mechanics and Relativity.

**Category:** General Mathematics

[1] **viXra:1303.0099 [pdf]**
*replaced on 2013-03-14 10:35:37*

**Authors:** K. Raja Rama Gandhi

**Comments:** 24 Pages.

This is first part of eight parts of lecture notes on Real Analysis. This notes is well designed and useful to all Undergraduate, Graduate and postgraduate in their regular study. Apart from this, the problems discussed in exercise will increase the readability of readers and they love Number Theory as well as analysis without any doubts. Also, some problems presented in the exercises of this part as well as coming parts will create motivation towards research and development.

**Category:** General Mathematics