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T. N. Sherry - One of the best experts on this subject based on the ideXlab platform.
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Spinors and Supersymmetry in Four-Dimensional Euclidean Space
Annals of Physics, 2001Co-Authors: D. G. C. Mckeon, T. N. SherryAbstract:Abstract Spinors in four-Dimensional Euclidean Space are treated using the decomposition of the Euclidean Space SO (4) symmetry group into SU (2)× SU (2). Both 2- and 4-spinor representations of this SO (4) symmetry group are shown to differ significantly from the corresponding spinor representations of the SO (3, 1) symmetry group in Minkowski Space. The simplest self conjugate supersymmetry algebra allowed in four-Dimensional Euclidean Space is demonstrated to be an N =2 supersymmetry algebra which resembles the N =2 supersymmetry algebra in four-Dimensional Minkowski Space. The differences between the two supersymmetry algebras gives rise to different representations; in particular an analysis of the Clifford algebra structure shows that the momentum invariant is bounded above by the central charges in 4 dE , while in 4 dM the central charges bound the momentum invariant from below. Dimensional reduction of the N =1 SUSY algebra in six-Dimensional Minkowski Space (6 dM ) to 4 dE reproduces our SUSY algebra in 4 dE . This Dimensional reduction can be used to introduce additional generators into the SUSY algebra in 4 dE . Well known interpolating maps are used to relate the N =2 SUSY algebra in 4 dE derived in this paper to the N =2 SUSY algebra in 4 dM . The nature of the spinors in 4 dE allows us to write an axially gauge invariant model which is shown to be both Hermitian and anomaly-free. No equivalent model exists in 4 dM . Useful formulae in 4 dE are collected together in two appendixes.
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Extended Supersymmetry in Four-Dimensional Euclidean Space
Annals of Physics, 2000Co-Authors: D. G. C. Mckeon, T. N. SherryAbstract:Abstract Since the generators of the two SU (2) groups which comprise SO (4) are not Hermitian conjugates of each other, the simplest supersymmetry algebra in four-Dimensional Euclidean Space more closely resembles the N =2 than the N =1 supersymmetry algebra in four-Dimensional Minkowski Space. An extended supersymmetry algebra in four-Dimensional Euclidean Space is considered in this paper; its structure resembles that of N =4 supersymmetry in four-Dimensional Minkowski Space. The relationship of this algebra to the algebra found by Dimensionally reducing the N =1 supersymmetry algebra in ten-Dimensional Euclidean Space to four-Dimensional Euclidean Space is examined. The Dimensional reduction of N =1 super Yang–Mills theory in ten-Dimensional Minkowski Space to four-Dimensional Euclidean Space is also considered.
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Aspects of the Supersymmetry Algebra in Four Dimensional Euclidean Space
arXiv: High Energy Physics - Theory, 1998Co-Authors: D. G. C. Mckeon, T. N. SherryAbstract:The simplest supersymmetry (SUSY) algebra in four Dimensional Euclidean Space ($4dE$) has been shown to closely resemble the $N = 2$ SUSY algebra in four Dimensional Minkowski Space ($4dM$). The structure of the former algebra is examined in greater detail in this paper. We first present its Clifford algebra structure. This algebra shows that the momentum Casimir invariant of physical states has an upper bound which is fixed by the central charges. Secondly, we use reduction of the $N = 1$ SUSY algebra in six Dimensional Minkowski Space ($6dM$) to $4dE$; this reproduces our SUSY algebra in $4dE$. Moreover, this same reduction of supersymmetric Yang-Mills theory (SSYM) in $6dM$ reproduces Zumino's SSYM in $4dE$. We demonstrate how this Dimensional reduction can be used to introduce additional generators into the SUSY algebra in $4dE$.
D. G. C. Mckeon - One of the best experts on this subject based on the ideXlab platform.
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Spinors and Supersymmetry in Four-Dimensional Euclidean Space
Annals of Physics, 2001Co-Authors: D. G. C. Mckeon, T. N. SherryAbstract:Abstract Spinors in four-Dimensional Euclidean Space are treated using the decomposition of the Euclidean Space SO (4) symmetry group into SU (2)× SU (2). Both 2- and 4-spinor representations of this SO (4) symmetry group are shown to differ significantly from the corresponding spinor representations of the SO (3, 1) symmetry group in Minkowski Space. The simplest self conjugate supersymmetry algebra allowed in four-Dimensional Euclidean Space is demonstrated to be an N =2 supersymmetry algebra which resembles the N =2 supersymmetry algebra in four-Dimensional Minkowski Space. The differences between the two supersymmetry algebras gives rise to different representations; in particular an analysis of the Clifford algebra structure shows that the momentum invariant is bounded above by the central charges in 4 dE , while in 4 dM the central charges bound the momentum invariant from below. Dimensional reduction of the N =1 SUSY algebra in six-Dimensional Minkowski Space (6 dM ) to 4 dE reproduces our SUSY algebra in 4 dE . This Dimensional reduction can be used to introduce additional generators into the SUSY algebra in 4 dE . Well known interpolating maps are used to relate the N =2 SUSY algebra in 4 dE derived in this paper to the N =2 SUSY algebra in 4 dM . The nature of the spinors in 4 dE allows us to write an axially gauge invariant model which is shown to be both Hermitian and anomaly-free. No equivalent model exists in 4 dM . Useful formulae in 4 dE are collected together in two appendixes.
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Harmonic superSpace with four-Dimensional Euclidean Space
Canadian Journal of Physics, 2000Co-Authors: D. G. C. MckeonAbstract:The supersymmetry algebra in four-Dimensional Euclidean Space is formulated in such a way that an SU(2) invariance of the algebra is apparent. This leads to a superfield formalism for representations of the algebra that can be adapted to harmonic superSpace. Models in this harmonic superSpace are quite similar to N = 2 models in four-Dimensional Minkowski Space.PACS Nos.: 11.30Pb, 03.65Fd
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Extended Supersymmetry in Four-Dimensional Euclidean Space
Annals of Physics, 2000Co-Authors: D. G. C. Mckeon, T. N. SherryAbstract:Abstract Since the generators of the two SU (2) groups which comprise SO (4) are not Hermitian conjugates of each other, the simplest supersymmetry algebra in four-Dimensional Euclidean Space more closely resembles the N =2 than the N =1 supersymmetry algebra in four-Dimensional Minkowski Space. An extended supersymmetry algebra in four-Dimensional Euclidean Space is considered in this paper; its structure resembles that of N =4 supersymmetry in four-Dimensional Minkowski Space. The relationship of this algebra to the algebra found by Dimensionally reducing the N =1 supersymmetry algebra in ten-Dimensional Euclidean Space to four-Dimensional Euclidean Space is examined. The Dimensional reduction of N =1 super Yang–Mills theory in ten-Dimensional Minkowski Space to four-Dimensional Euclidean Space is also considered.
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Aspects of the Supersymmetry Algebra in Four Dimensional Euclidean Space
arXiv: High Energy Physics - Theory, 1998Co-Authors: D. G. C. Mckeon, T. N. SherryAbstract:The simplest supersymmetry (SUSY) algebra in four Dimensional Euclidean Space ($4dE$) has been shown to closely resemble the $N = 2$ SUSY algebra in four Dimensional Minkowski Space ($4dM$). The structure of the former algebra is examined in greater detail in this paper. We first present its Clifford algebra structure. This algebra shows that the momentum Casimir invariant of physical states has an upper bound which is fixed by the central charges. Secondly, we use reduction of the $N = 1$ SUSY algebra in six Dimensional Minkowski Space ($6dM$) to $4dE$; this reproduces our SUSY algebra in $4dE$. Moreover, this same reduction of supersymmetric Yang-Mills theory (SSYM) in $6dM$ reproduces Zumino's SSYM in $4dE$. We demonstrate how this Dimensional reduction can be used to introduce additional generators into the SUSY algebra in $4dE$.
Masafumi Yamashita - One of the best experts on this subject based on the ideXlab platform.
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brief announcement pattern formation problem for synchronous mobile robots in the three Dimensional Euclidean Space
Principles of Distributed Computing, 2016Co-Authors: Yukiko Yamauchi, Taichi Uehara, Masafumi YamashitaAbstract:We investigate the pattern formation problem that requires a swarm of autonomous mobile robots to form a given target pattern in the three-Dimensional Euclidean Space. We show a necessary and sufficient condition for synchronous robots to form a given target pattern from an initial configuration. We give a pattern formation algorithm for solvable instances that does not need any local memory at each robot.
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Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space
2015Co-Authors: Yukiko Yamauchi, Taichi Uehara, Shuji Kijima, Masafumi YamashitaAbstract:Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, this paper investigates the plane formation problem that requires a swarm of robots moving in the three Dimensional Euclidean Space to reside in a common plane. The robots are fully synchronous and endowed with visual perception. But they have neither identifiers, access to the global coordinate system, any means of explicit communication with each other, nor memory of past. Though there are plenty of results on the agreement problem for robots in the two Dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three Dimensional Space. This paper presents a necessary and sufficient condition to solve the plane formation problem. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can from every regular polyhedron (except a regular icosahedron), which consists of the same regular polygon faces and the robots on its vertices are “more” symmetric than semi-regular polyhedra.
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pattern formation problem for synchronous mobile robots in the three Dimensional Euclidean Space
arXiv: Distributed Parallel and Cluster Computing, 2015Co-Authors: Yukiko Yamauchi, Taichi Uehara, Masafumi YamashitaAbstract:We consider a swarm of autonomous mobile robots each of which is an anonymous point in the three-Dimensional Euclidean Space (3D-Space) and synchronously executes a common distributed algorithm. We investigate the pattern formation problem that requires the robots to form a given target pattern from an initial configuration and characterize the problem by showing a necessary and sufficient condition for the robots to form a given target pattern. The pattern formation problem in the two Dimensional Euclidean Space (2D-Space) has been investigated by Suzuki and Yamashita (SICOMP 1999, TCS 2010), and Fujinaga et al. (SICOMP 2015). The symmetricity $\rho(P)$ of a configuration (i.e., the positions of robots) $P$ is intuitively the order of the cyclic group that acts on $P$. It has been shown that fully-synchronous (FSYNC) robots can form a target pattern $F$ from an initial configuration $P$ if and only if $\rho(P)$ divides $\rho(F)$. We extend the notion of symmetricity to 3D-Space by using the rotation groups each of which is defined by a set of rotation axes and their arrangement. We define the symmetricity $\varrho(P)$ of configuration $P$ in 3D-Space as the set of rotation groups that acts on $P$ and whose rotation axes do not contain any robot. We show the following necessary and sufficient condition for the pattern formation problem which is a natural extension of the existing results of the pattern formation problem in 2D-Space: FSYNC robots in 3D-Space can form a target pattern $F$ from an initial configuration $P$ if and only if $\varrho(P) \subseteq \varrho(F)$. For solvable instances, we present a pattern formation algorithm for oblivious FSYNC robots. The insight of this paper is that symmetry of mobile robots in 3D-Space is sometimes lower than the symmetry of their positions and the robots can show their symmetry by their movement.
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plane formation by synchronous mobile robots in the three Dimensional Euclidean Space
arXiv: Distributed Parallel and Cluster Computing, 2015Co-Authors: Yukiko Yamauchi, Taichi Uehara, Shuji Kijima, Masafumi YamashitaAbstract:Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three Dimensional Euclidean Space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two Dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three Dimensional Space. This paper presents a necessary and sufficient condition for fully-synchronous robots to solve the plane formation problem that does not depend on obliviousness i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedrdon.
Giri Narasimhan - One of the best experts on this subject based on the ideXlab platform.
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optimally sparse spanners in 3 Dimensional Euclidean Space
Symposium on Computational Geometry, 1993Co-Authors: Gautam Das, Paul J Heffernan, Giri NarasimhanAbstract:Let V be a set of n points in 3-Dimensional Euclidean Space. A subgraph of the complete Euclidean graph is a t -spanner if for any u and v in V , the length of the shortest path from u to v in the spanner is at most t times d(u, v) . We show that for any t > 1, a greedy algorithm produces a t -spanner with O(n) edges, and total edge weight O(1).wt(MST) , where MST is a minimum spanning tree of V .
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Symposium on Computational Geometry - Optimally sparse spanners in 3-Dimensional Euclidean Space
Proceedings of the ninth annual symposium on Computational geometry - SCG '93, 1993Co-Authors: Gautam Das, Paul J Heffernan, Giri NarasimhanAbstract:Let V be a set of n points in 3-Dimensional Euclidean Space. A subgraph of the complete Euclidean graph is a t -spanner if for any u and v in V , the length of the shortest path from u to v in the spanner is at most t times d(u, v) . We show that for any t > 1, a greedy algorithm produces a t -spanner with O(n) edges, and total edge weight O(1).wt(MST) , where MST is a minimum spanning tree of V .
Yukiko Yamauchi - One of the best experts on this subject based on the ideXlab platform.
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brief announcement pattern formation problem for synchronous mobile robots in the three Dimensional Euclidean Space
Principles of Distributed Computing, 2016Co-Authors: Yukiko Yamauchi, Taichi Uehara, Masafumi YamashitaAbstract:We investigate the pattern formation problem that requires a swarm of autonomous mobile robots to form a given target pattern in the three-Dimensional Euclidean Space. We show a necessary and sufficient condition for synchronous robots to form a given target pattern from an initial configuration. We give a pattern formation algorithm for solvable instances that does not need any local memory at each robot.
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Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space
2015Co-Authors: Yukiko Yamauchi, Taichi Uehara, Shuji Kijima, Masafumi YamashitaAbstract:Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, this paper investigates the plane formation problem that requires a swarm of robots moving in the three Dimensional Euclidean Space to reside in a common plane. The robots are fully synchronous and endowed with visual perception. But they have neither identifiers, access to the global coordinate system, any means of explicit communication with each other, nor memory of past. Though there are plenty of results on the agreement problem for robots in the two Dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three Dimensional Space. This paper presents a necessary and sufficient condition to solve the plane formation problem. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can from every regular polyhedron (except a regular icosahedron), which consists of the same regular polygon faces and the robots on its vertices are “more” symmetric than semi-regular polyhedra.
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pattern formation problem for synchronous mobile robots in the three Dimensional Euclidean Space
arXiv: Distributed Parallel and Cluster Computing, 2015Co-Authors: Yukiko Yamauchi, Taichi Uehara, Masafumi YamashitaAbstract:We consider a swarm of autonomous mobile robots each of which is an anonymous point in the three-Dimensional Euclidean Space (3D-Space) and synchronously executes a common distributed algorithm. We investigate the pattern formation problem that requires the robots to form a given target pattern from an initial configuration and characterize the problem by showing a necessary and sufficient condition for the robots to form a given target pattern. The pattern formation problem in the two Dimensional Euclidean Space (2D-Space) has been investigated by Suzuki and Yamashita (SICOMP 1999, TCS 2010), and Fujinaga et al. (SICOMP 2015). The symmetricity $\rho(P)$ of a configuration (i.e., the positions of robots) $P$ is intuitively the order of the cyclic group that acts on $P$. It has been shown that fully-synchronous (FSYNC) robots can form a target pattern $F$ from an initial configuration $P$ if and only if $\rho(P)$ divides $\rho(F)$. We extend the notion of symmetricity to 3D-Space by using the rotation groups each of which is defined by a set of rotation axes and their arrangement. We define the symmetricity $\varrho(P)$ of configuration $P$ in 3D-Space as the set of rotation groups that acts on $P$ and whose rotation axes do not contain any robot. We show the following necessary and sufficient condition for the pattern formation problem which is a natural extension of the existing results of the pattern formation problem in 2D-Space: FSYNC robots in 3D-Space can form a target pattern $F$ from an initial configuration $P$ if and only if $\varrho(P) \subseteq \varrho(F)$. For solvable instances, we present a pattern formation algorithm for oblivious FSYNC robots. The insight of this paper is that symmetry of mobile robots in 3D-Space is sometimes lower than the symmetry of their positions and the robots can show their symmetry by their movement.
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plane formation by synchronous mobile robots in the three Dimensional Euclidean Space
arXiv: Distributed Parallel and Cluster Computing, 2015Co-Authors: Yukiko Yamauchi, Taichi Uehara, Shuji Kijima, Masafumi YamashitaAbstract:Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three Dimensional Euclidean Space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two Dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three Dimensional Space. This paper presents a necessary and sufficient condition for fully-synchronous robots to solve the plane formation problem that does not depend on obliviousness i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedrdon.