Benjamin Fine
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View article: Groups of F-Type
Groups of F-Type Open
We consider a class of groups, called groups of F-type, which includes some known and important classes like Fuchsian groups of geometric rank $\ge 3$, surface groups of genus $\ge 2$, cyclically pinched one-relator groups and torus-knot g…
View article: Checklist for Artificial Intelligence in Medical Imaging (CLAIM): 2024 Update
Checklist for Artificial Intelligence in Medical Imaging (CLAIM): 2024 Update Open
To address the rapid evolution of artificial intelligence in medical imaging, the authors present the Checklist for Artificial Intelligence in Medical Imaging (CLAIM) 2024 Update.
View article: Groups of F-Type
Groups of F-Type Open
We consider a class of groups, called groups of F-type, which includes some known and important classes like Fuchsian groups of geometric rank $\ge 3$, surface groups of genus $\ge 2$, cyclically pinched one-relator groups and torus-knot g…
View article: Preface
Preface Open
To many students, as well as to many teachers, mathematics may seem like a mundane discipline, filled with rules and algorithms and devoid of beauty, culture and art.However, the world of mathematics is populated with true gems and results…
View article: The Axiomatics of Free Group Rings
The Axiomatics of Free Group Rings Open
In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a …
View article: The Class of Orderable Groups Is a Quasi-Variety with Undecidable Theory
The Class of Orderable Groups Is a Quasi-Variety with Undecidable Theory Open
Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying hg1 ≤ hg2 whenever g1 ≤ g2. G is orderable provided it admits a total order ≤ satisfying both: hg1 ≤ hg2 whenever g1 ≤ g2 and g1h ≤ g2h whenever g1 ≤ g2. …
View article: Faithful real representations of groups of $F$-type
Faithful real representations of groups of $F$-type Open
Groups of $F$-type were introduced in [B. Fine and G. Rosenberger, Generalizing Algebraic Properties of Fuchsian Groups, \\emph{London Math. Soc. Lecture Note Ser.}, \\textbf{159} (1991) 124--147.] as a natural algebraic generali…
View article: On the Universal Group PSL(2, C)
On the Universal Group PSL(2, C) Open
The group Γ = PSL(2, C) arises in a wide variety of contexts; hyperbolic geometry, automorphic function theory, number theory and group theory. Much of combinatorial group theory arose out of the study of discrete subgroups of Γ = PSL(2, C…
View article: Geometry and Discrete Mathematics: A Selection of Highlights
Geometry and Discrete Mathematics: A Selection of Highlights Open
This two-volume set collects and presents many fundamentals of mathematics in an enjoyable and elaborating fashion. The idea behind the two books is to provide substantials for assessing more modern developments in mathematics and to prese…
View article: Secure Passwords Using Combinatorial Group Theory
Secure Passwords Using Combinatorial Group Theory Open
Password security is a crucial component of modern internet security. In this paper, we present a provably secure method for password verification using combinatorial group theory. This method relies on the group randomizer system, a subse…
View article: The Tarski Theorems and Elementary Equivalence of Group Rings
The Tarski Theorems and Elementary Equivalence of Group Rings Open
The Tarski theorems, proved by Myasnikov and Kharlampovich and inde-pendently by Sela say that all nonabelian free groups satisfy the same first-order or elementary theory. Kharlampovich and Myasnikov also prove that the elementary theory …
View article: Cryptographic Protocols Based on Nielsen Transformations
Cryptographic Protocols Based on Nielsen Transformations Open
We introduce in this paper cryptographic protocols which use combinatorial group theory. Based on a combinatorial distribution of shares we present secret sharing schemes and cryptosystems using Nielsen transformations. Nielsen transformat…