A NEW EXAMPLE FOR MINIMALITY OF MONOIDS

By considering the split extension of a free abelian monoid having finite rank by a finite monogenic monoid, the main purposes of this paper are to present examples of efficient monoids and, also, minimal but inefficient monoids. Although results presented in this paper seem as in the branch of pure mathematics, they are actually related to applications of Combinatorial and Geometric Group-Semigroup Theory, especially computer science, network systems, cryptography and physics etc., which will not be handled here.

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Foreword to the special issue of the International Conference on Innovative Network Systems and Applications held under the Federated Conference on Computer Science and Information Systems

Concurrency and Computation Practice and Experience ◽

10.1002/cpe.4356 ◽

2017 ◽

Vol 29 (23) ◽

pp. e4356

Author(s):

Michal Hodon ◽

Hacene Fouchal

Keyword(s):

Information Systems ◽

Computer Science ◽

Special Issue ◽

Network Systems ◽

International Conference

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Calabi-Yau Spaces in the String Landscape

Oxford Research Encyclopedia of Physics ◽

10.1093/acrefore/9780190871994.013.60 ◽

2020 ◽

Author(s):

Yang-Hui He

Keyword(s):

Machine Learning ◽

Computer Science ◽

Data Science ◽

Theoretical Physics ◽

Superstring Theory ◽

Data Set ◽

Pure Mathematics ◽

String Landscape ◽

Natural Solution ◽

Vacuum Solutions

Calabi-Yau spaces, or Kähler spaces admitting zero Ricci curvature, have played a pivotal role in theoretical physics and pure mathematics for the last half century. In physics, they constituted the first and natural solution to compactification of superstring theory to our 4-dimensional universe, primarily due to one of their equivalent definitions being the admittance of covariantly constant spinors. Since the mid-1980s, physicists and mathematicians have joined forces in creating explicit examples of Calabi-Yau spaces, compiling databases of formidable size, including the complete intersecion (CICY) data set, the weighted hypersurfaces data set, the elliptic-fibration data set, the Kreuzer-Skarke toric hypersurface data set, generalized CICYs, etc., totaling at least on the order of 1010 manifolds. These all contribute to the vast string landscape, the multitude of possible vacuum solutions to string compactification. More recently, this collaboration has been enriched by computer science and data science, the former in bench-marking the complexity of the algorithms in computing geometric quantities, and the latter in applying techniques such as machine learning in extracting unexpected information. These endeavours, inspired by the physics of the string landscape, have rendered the investigation of Calabi-Yau spaces one of the most exciting and interdisciplinary fields.

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Relations in the semigroup of 2 × 2 upper-triangular matrices

International Journal of Algebra and Computation ◽

10.1142/s0218196720500113 ◽

2020 ◽

Vol 30 (03) ◽

pp. 567-584

Author(s):

Henri-Alex Esbelin ◽

Marin Gutan

Keyword(s):

Number Theory ◽

Computer Science ◽

Semigroup Theory ◽

Theoretical Computer Science ◽

Multiplicative Semigroup ◽

Triangular Matrices ◽

Theoretical Computer ◽

Upper Triangular Matrices ◽

Matrix Semigroups

Let [Formula: see text] with [Formula: see text] be [Formula: see text] upper-triangular matrices with rational entries. In the multiplicative semigroup generated by these matrices, we check if there are relations of the form [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] We give algorithms to find relations of the previous form. Our results are extensions of some theorems obtained by Charlier and Honkala in [The freeness problem over matrix semigroups and bounded languages, Inf. Comput. 237 (2014) 243–256]. Our paper is at the interface between algebra, number theory and theoretical computer science. While the main results concern decidability and semigroup theory, the methods for obtaining them come from number theory.

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A theorem on numbers of the form 10^x

10.31219/osf.io/ku45y ◽

2019 ◽

Author(s):

Ravin Kumar

Keyword(s):

Number Theory ◽

Computer Science ◽

Natural Numbers ◽

The Core ◽

Pure Mathematics ◽

Major Application

Number theory is one of the core branches of pure mathematics. It has played an important role in the study of natural numbers. In this paper, we are presenting a theorem on the numbers of form 10^x , where x ∊ Z+ . The proposed theorem have a major application in computer science. It can be used to predict ‘n’ bits which will always represent more than 10^x total numbers. We proved that the nature of the ‘n’ bits is always one of the forms 10i, 10i + 4, or 10i + 7, where i ∊ Z+ .

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Automorphism sequences of free nilpotent groups of class two

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100052269 ◽

1976 ◽

Vol 79 (2) ◽

pp. 271-279 ◽

Cited By ~ 9

Author(s):

Joan L. Dyer ◽

Edward Formanek

Keyword(s):

Automorphism Group ◽

Nilpotent Group ◽

Finite Rank ◽

Nilpotent Groups ◽

Split Extension ◽

Finitely Generated ◽

Free Nilpotent Group ◽

Image Position

In this paper we prove that the automorphism group A(N) of a free nilpotent group N of class 2 and finite rank n is complete, except when n is 1 or 3. Equivalently, the centre of A(N) is trivial and every automorphism of A(N) is inner, provided n ≠ 1 or 3. When n = 3, A(N) has an our automorphism of order 2, so A(A(N)) is a split extension of A(N) by . In this case, A(A(N)) is complete. These results provide some evidence supporting a conjecture of Gilbert Baumslag that the sequencebecomes periodic if N is a finitely generated nilpotent group.

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