Notes on Elementary Linear Algebra

A set of new and refined principles and definitions in Real Numbers and Vectors are presented. What is a Vector? What is the meaning of the Addition of two Vectors? What is a Real Number? What is the meaning of their signs? What is the meaning of the Addition of two Real Numbers? What is the Summation Principle in Addition Operation? What is the Cancellation Principle in Addition Operation? What is the Meaning of the Multiplication of two Real Numbers? Is Field Theory a law? Can it be proved? All these issues are addressed in this paper. With better pictures and graphical presentations, proof of Field Theories in Real Numbers and Vectors including Commutativity, Associativity and Distributivity are also proposed.

ArXiv, 2016

Axiomatizing mathematical structures is a goal of Mathematical Logic. Axiomatizability of the theories of some structures have turned out to be quite difficult and challenging, and some remain open. However axiomatization of some mathematical structures are now classical theorems in Logic, Algebra and Geometry. In this paper we will study the axiomatizability of the theories of multiplication in the domains of natural, integer, rational, real, and complex numbers. We will review some classical theorems, and will give some new proofs for old results. We will see that some structures are missing in the literature, thus leaving it open whether the theories of that structures are axiomatizable (decidable) or not. We will answer one of those open questions in this paper.

Theoretical Computer Science, 1976

We demonstrate the usefulness of Petri nets for treating problems about vector addition systems by giving a simple exposition of Rabin's proof of the undecidability of the inclusion problem for v+ ctor addition system reachability sets, and then proceed to show that the inclusion problem can be reduced to the equality problem for reachability sets. * This relationship is also discussed in [ 11).

Demonstratio Mathematica, 1994

2010

In previous work, we introduced a framework for proof nets of the multiplicative fragment of Linear Logic, where partially sequentialized nets are allowed. In this paper we extend this result to include additives, using a definition of proof net, called J-proof net, which corresponds to a typed version of an L-net of Faggian and Maurel. In J-proof nets, sequentiality constraints are represented using the proof-net notion of jumps; by gradual insertion or removal of jumps then it is possible to characterize nets with different degrees of sequentiality. As a byproduct, we obtain then a proof of the sequentialization theorem. Moreover, we provide a denotational model for J-proof nets which is an extension of the relational one, and we show that the “degree” of sequentiality of a J-proof net can be read-off from its semantics, by proving that the model is injective with respect to J-proof nets.

We are familiar with the binary operations of addition and multiplication among many objects: complex numbers, square matrices with entries in a field, real or complex valued functions defined on a set, polynomials and power series with coefficients in a field, integers with modular arithmetic, etc. Ring theory deals with such objects. Groups are sets of objects with one binary operation while rings are sets of objects with two binary operations, addition and multiplication, inter related to each other by distributive laws. The theorems obtained as a result of abstract study of rings apply to these diverse objects mentioned above which can then be used to solve problems arising in number theory, geometry and many other fields. Definition 1.1. A nonempty set R is called a ring, if it has two binary operations called addition denoted by a + b and multiplication denoted by ab for a, b ∈ R satisfying the following axioms: (1) (R, +) is an abelian group. (2) Multiplication is associative, i.e. a(bc) = (ab)c for all a, b, c ∈ R. (3) Distributive laws hold: a(b + c) = ab + ac and (b + c)a = ba + ca for all a, b, c ∈ R. Definition 1.2. Let R be a ring. (1) If multiplication in R is commutative, it is called a commutative ring. (2) If there is an identity for multiplication, then R is said to have identity. Notes for the lectures on basic ring theory in the

2012

Let K ⊂ L be a field extension. Given K-subspaces A,B of L, we study the subspace 〈AB 〉 spanned by the product set AB = {ab | a ∈ A,b ∈ B}. We obtain some lower bounds on dimK〈AB 〉 and dimK〈B n 〉 in terms of dimK A, dimK B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson. 1

Springer Undergraduate Mathematics Series, 2002