After Data Ecologies 2014, a group of participants and some others collected in a small villa nea... more After Data Ecologies 2014, a group of participants and some others collected in a small villa near Attersee in order to try and summarise their thoughts on how to "Think out loud about Futures" into a single volume. This four-day booksprint cilminated in the present volume, which summarises who should be thinking about the future, why, as well as how to think about it, express it, explore it and get into it. We anticipate that the book will be revised and superceded in the next few months, with the hope that a printed version will arise in 2015. http://www.timesup.org/Futurish0.99
We study the groups 𝐺 with the curious property that there exists an element k ∈ G k\in G and a f... more We study the groups 𝐺 with the curious property that there exists an element k ∈ G k\in G and a function f : G → G f\colon G\to G such that f ( x k ) = x f ( x ) f(xk)=xf(x) holds for all x ∈ G x\in G . This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a 𝐽-group. Finite 𝐽-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a 𝐽-group if its nilpotency class 𝑐 satisfies c ⩽ 6 c\leqslant 6 . If 𝐺 is a finite 𝑝-group, with p > 2 p>2 and p 2 > 2 c - 1 p^{2}>2c-1 , then we prove that 𝐺 is 𝐽-group. Finally, if p > 2 p>2 and 𝐺 is a regular 𝑝-group or, more generally, a power-closed one (i.e., in each section and for each m ⩾ 1 m\geqslant 1 , the subset of p m p^{m} -th powers is a subgroup), then we prove that 𝐺 is a 𝐽-group.
We introduce a condition on arrays in some way maximally distinct from Latin square condition, as... more We introduce a condition on arrays in some way maximally distinct from Latin square condition, as well as some other conditions on algebras, graphs and 0,1-matrices. We show that these are essentially the same structures, generalising a similar collection of models presented by Knuth in 1970. We find ways in which these structures can be made more specific, relating to existing investigations, then show that they are also extremely general; the groupoids satisfy no nontrivial equations. Some construction methods are presented and some conjectures made as to how certain structures are preserved by these constructions. Finally we investigate to what degree partial arrays satisfying our conditions and partial Latin squares overlap.
It is well-known that the Toffoli gate and the negation gate together yield a universal gate set,... more It is well-known that the Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of {0,1}^n can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every permutation, and it is known that one bit is indeed enough. Without auxiliary bits, all even permutations can be implemented. We generalize these results to non-binary logic: If A is a finite set of odd cardinality then a finite gate set can generate all permutations of A^n for all n, without any auxiliary symbols. If the cardinality of A is even then, by the same argument as above, only even permutations of A^n can be implemented for large n, and we show that indeed all even permutations can be obtained from a finite universal gate set. We also consider the conservative case, that is, those permutations of A^n that preserve the weight of the ...
Looking at the automata defined over a group alphabet as a nearring, we see that they are a highl... more Looking at the automata defined over a group alphabet as a nearring, we see that they are a highly complicated structure. As with ring theory, one method to deal with complexity is to look at semisimplicity modulo radical structures. We find some bounds on the Jacobson 2-radical and show that in certain groups, this radical can be explicitly found and the semisimple image determined.
We generalise clones, which are sets of functions f:A^n → A, to sets of mappings f:A^n → A^m. We ... more We generalise clones, which are sets of functions f:A^n → A, to sets of mappings f:A^n → A^m. We formalise this and develop language that we can use to speak about it. We then look at bijective mappings, which have connections to reversible computation, which is important for physical (e.g. quantum computation) as well as engineering (e.g. heat dissipation) reasons. We generalise Toffoli's seminal work on reversible computation to arbitrary arity logics. In particular, we show that some restrictions he found for reversible computation on alphabets of order 2 do not apply for odd order alphabets. For A odd, we can create all invertible mappings from the Toffoli 1- and 2-gates, demonstrating that we can realise all reversible mappings from four generators. We discuss various forms of closure, corresponding to various systems of permitted manipulations. These correspond, amongst other things, to discussions about ancilla bits in quantum computation.
We generalise clones, which are sets of functions f:A^n → A, to sets of mappings f:A^n → A^m. We ... more We generalise clones, which are sets of functions f:A^n → A, to sets of mappings f:A^n → A^m. We formalise this and develop language that we can use to speak about it. We then look at bijective mappings, which have connections to reversible computation, which is important for physical (e.g. quantum computation) as well as engineering (e.g. heat dissipation) reasons. We generalise Toffoli's seminal work on reversible computation to arbitrary arity logics. In particular, we show that some restrictions he found for reversible computation on alphabets of order 2 do not apply for odd order alphabets. For A odd, we can create all invertible mappings from the Toffoli 1- and 2-gates, demonstrating that we can realise all reversible mappings from four generators. We discuss various forms of closure, corresponding to various systems of permitted manipulations. These correspond, amongst other things, to discussions about ancilla bits in quantum computation.
Planar nearrings play an important role in nearring theory, both from the structural side as bein... more Planar nearrings play an important role in nearring theory, both from the structural side as being close to generalised nearfields, as well as from an applications perspective, in geometry and designs. We investigate the distributive elements of planar nearrings. If a planar nearring has nonzero distributive elements, then it is an extension of its zero multiplier part by an abelian group. In the case that there are distributive elements that are not zero multipliers, then this extension splits, giving an explicit description of the nearring. This generalises the structure of planar rings. We provide a family of examples where this does not occur, the distributive elements being precisely the zero multipliers. We apply this knowledge to the question of determining the generalized center of planar nearrings as well as finding new proofs of other older results.
Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. ... more Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families. Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference. We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that we obtain quasigroups. These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups. This result will be of ...
This article takes up the concept of performativity prevalent in the humanities and applies it to... more This article takes up the concept of performativity prevalent in the humanities and applies it to the design of installation arts in mixed reality mode. Based on the design, development and public access to two specific works, the concept is related to a form of research by design. We argue that the concept of performativity may be further usefully employed in investigations (design and research, artistic and public) into digital arts where complex intersections between concepts, technologies, dramaturgy, media and participant actions are in flux and together constitute the emergence and experience of a work. Theories of performativity are related to these two works in an argument that further suggests there is room in research by design to also include ‘performative design’. The article is the result of a wide-ranging interdisciplinary collaboration and aims to convey some sense of that in its reporting style, content and analysis.
Abstract. Difference families are traditionally built using groups as their basis. This paper loo... more Abstract. Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalised difference family constructions could be made, using the standard basis of translation and difference. The main result is that minimal requirements on the structure force nothing that groups cannot give, at least in the finite case. Thus all difference families arise from groups. 1.
ABSTRACT After Data Ecologies 2014, a group of participants and some others collected in a small ... more ABSTRACT After Data Ecologies 2014, a group of participants and some others collected in a small villa near Attersee in order to try and summarise their thoughts on how to "Think out loud about Futures" into a single volume. This four-day booksprint cilminated in the present volume, which summarises who should be thinking about the future, why, as well as how to think about it, express it, explore it and get into it. We anticipate that the book will be revised and superceded in the next few months, with the hope that a printed version will arise in 2015. http://www.timesup.org/Futurish0.99
We study the finite groups G with the curious property that there exists an element k ∈ G and a f... more We study the finite groups G with the curious property that there exists an element k ∈ G and a function f : G → G such that f(xk) = xf(x) holds for all x ∈ G. This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a J-group. Finite J-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a J-group if its nilpotency class c satisfies c 6 6. If G is a finite p-group, with p > 2 and p > 2c−1, then we prove that G is J-group. Finally, if p > 2 and G is a regular p-group or, more generally, a power-closed one (i.e., in each section and for each m > 1 the subset of p-th powers is a subgroup), then we prove that G is a J-group.
After Data Ecologies 2014, a group of participants and some others collected in a small villa nea... more After Data Ecologies 2014, a group of participants and some others collected in a small villa near Attersee in order to try and summarise their thoughts on how to "Think out loud about Futures" into a single volume. This four-day booksprint cilminated in the present volume, which summarises who should be thinking about the future, why, as well as how to think about it, express it, explore it and get into it. We anticipate that the book will be revised and superceded in the next few months, with the hope that a printed version will arise in 2015. http://www.timesup.org/Futurish0.99
We study the groups 𝐺 with the curious property that there exists an element k ∈ G k\in G and a f... more We study the groups 𝐺 with the curious property that there exists an element k ∈ G k\in G and a function f : G → G f\colon G\to G such that f ( x k ) = x f ( x ) f(xk)=xf(x) holds for all x ∈ G x\in G . This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a 𝐽-group. Finite 𝐽-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a 𝐽-group if its nilpotency class 𝑐 satisfies c ⩽ 6 c\leqslant 6 . If 𝐺 is a finite 𝑝-group, with p > 2 p>2 and p 2 > 2 c - 1 p^{2}>2c-1 , then we prove that 𝐺 is 𝐽-group. Finally, if p > 2 p>2 and 𝐺 is a regular 𝑝-group or, more generally, a power-closed one (i.e., in each section and for each m ⩾ 1 m\geqslant 1 , the subset of p m p^{m} -th powers is a subgroup), then we prove that 𝐺 is a 𝐽-group.
We introduce a condition on arrays in some way maximally distinct from Latin square condition, as... more We introduce a condition on arrays in some way maximally distinct from Latin square condition, as well as some other conditions on algebras, graphs and 0,1-matrices. We show that these are essentially the same structures, generalising a similar collection of models presented by Knuth in 1970. We find ways in which these structures can be made more specific, relating to existing investigations, then show that they are also extremely general; the groupoids satisfy no nontrivial equations. Some construction methods are presented and some conjectures made as to how certain structures are preserved by these constructions. Finally we investigate to what degree partial arrays satisfying our conditions and partial Latin squares overlap.
It is well-known that the Toffoli gate and the negation gate together yield a universal gate set,... more It is well-known that the Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of {0,1}^n can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every permutation, and it is known that one bit is indeed enough. Without auxiliary bits, all even permutations can be implemented. We generalize these results to non-binary logic: If A is a finite set of odd cardinality then a finite gate set can generate all permutations of A^n for all n, without any auxiliary symbols. If the cardinality of A is even then, by the same argument as above, only even permutations of A^n can be implemented for large n, and we show that indeed all even permutations can be obtained from a finite universal gate set. We also consider the conservative case, that is, those permutations of A^n that preserve the weight of the ...
Looking at the automata defined over a group alphabet as a nearring, we see that they are a highl... more Looking at the automata defined over a group alphabet as a nearring, we see that they are a highly complicated structure. As with ring theory, one method to deal with complexity is to look at semisimplicity modulo radical structures. We find some bounds on the Jacobson 2-radical and show that in certain groups, this radical can be explicitly found and the semisimple image determined.
We generalise clones, which are sets of functions f:A^n → A, to sets of mappings f:A^n → A^m. We ... more We generalise clones, which are sets of functions f:A^n → A, to sets of mappings f:A^n → A^m. We formalise this and develop language that we can use to speak about it. We then look at bijective mappings, which have connections to reversible computation, which is important for physical (e.g. quantum computation) as well as engineering (e.g. heat dissipation) reasons. We generalise Toffoli's seminal work on reversible computation to arbitrary arity logics. In particular, we show that some restrictions he found for reversible computation on alphabets of order 2 do not apply for odd order alphabets. For A odd, we can create all invertible mappings from the Toffoli 1- and 2-gates, demonstrating that we can realise all reversible mappings from four generators. We discuss various forms of closure, corresponding to various systems of permitted manipulations. These correspond, amongst other things, to discussions about ancilla bits in quantum computation.
We generalise clones, which are sets of functions f:A^n → A, to sets of mappings f:A^n → A^m. We ... more We generalise clones, which are sets of functions f:A^n → A, to sets of mappings f:A^n → A^m. We formalise this and develop language that we can use to speak about it. We then look at bijective mappings, which have connections to reversible computation, which is important for physical (e.g. quantum computation) as well as engineering (e.g. heat dissipation) reasons. We generalise Toffoli's seminal work on reversible computation to arbitrary arity logics. In particular, we show that some restrictions he found for reversible computation on alphabets of order 2 do not apply for odd order alphabets. For A odd, we can create all invertible mappings from the Toffoli 1- and 2-gates, demonstrating that we can realise all reversible mappings from four generators. We discuss various forms of closure, corresponding to various systems of permitted manipulations. These correspond, amongst other things, to discussions about ancilla bits in quantum computation.
Planar nearrings play an important role in nearring theory, both from the structural side as bein... more Planar nearrings play an important role in nearring theory, both from the structural side as being close to generalised nearfields, as well as from an applications perspective, in geometry and designs. We investigate the distributive elements of planar nearrings. If a planar nearring has nonzero distributive elements, then it is an extension of its zero multiplier part by an abelian group. In the case that there are distributive elements that are not zero multipliers, then this extension splits, giving an explicit description of the nearring. This generalises the structure of planar rings. We provide a family of examples where this does not occur, the distributive elements being precisely the zero multipliers. We apply this knowledge to the question of determining the generalized center of planar nearrings as well as finding new proofs of other older results.
Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. ... more Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families. Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference. We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that we obtain quasigroups. These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups. This result will be of ...
This article takes up the concept of performativity prevalent in the humanities and applies it to... more This article takes up the concept of performativity prevalent in the humanities and applies it to the design of installation arts in mixed reality mode. Based on the design, development and public access to two specific works, the concept is related to a form of research by design. We argue that the concept of performativity may be further usefully employed in investigations (design and research, artistic and public) into digital arts where complex intersections between concepts, technologies, dramaturgy, media and participant actions are in flux and together constitute the emergence and experience of a work. Theories of performativity are related to these two works in an argument that further suggests there is room in research by design to also include ‘performative design’. The article is the result of a wide-ranging interdisciplinary collaboration and aims to convey some sense of that in its reporting style, content and analysis.
Abstract. Difference families are traditionally built using groups as their basis. This paper loo... more Abstract. Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalised difference family constructions could be made, using the standard basis of translation and difference. The main result is that minimal requirements on the structure force nothing that groups cannot give, at least in the finite case. Thus all difference families arise from groups. 1.
ABSTRACT After Data Ecologies 2014, a group of participants and some others collected in a small ... more ABSTRACT After Data Ecologies 2014, a group of participants and some others collected in a small villa near Attersee in order to try and summarise their thoughts on how to "Think out loud about Futures" into a single volume. This four-day booksprint cilminated in the present volume, which summarises who should be thinking about the future, why, as well as how to think about it, express it, explore it and get into it. We anticipate that the book will be revised and superceded in the next few months, with the hope that a printed version will arise in 2015. http://www.timesup.org/Futurish0.99
We study the finite groups G with the curious property that there exists an element k ∈ G and a f... more We study the finite groups G with the curious property that there exists an element k ∈ G and a function f : G → G such that f(xk) = xf(x) holds for all x ∈ G. This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a J-group. Finite J-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a J-group if its nilpotency class c satisfies c 6 6. If G is a finite p-group, with p > 2 and p > 2c−1, then we prove that G is J-group. Finally, if p > 2 and G is a regular p-group or, more generally, a power-closed one (i.e., in each section and for each m > 1 the subset of p-th powers is a subgroup), then we prove that G is a J-group.
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