The Experts below are selected from a list of 261 Experts worldwide ranked by ideXlab platform
Manuela Heuer - One of the best experts on this subject based on the ideXlab platform.
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Combinatorial Aspects of Root Lattices and Words
2010Co-Authors: Manuela HeuerAbstract:This thesis is concerned with two topics that are of interest for the theory of aperiodic order. In the first part, the similar subLattices and coincidence site Lattices of the Root Lattice A4 are analysed by means of a particular quaternion algebra. Dirichlet series generating functions are derived, which count the number of similar subLattices, respectively coincidence site Lattices, of each index. In the second part, several strategies to derive upper and lower bounds for the entropy of certain sets of powerfree words are presented. In particular, Kolpakov's arguments for the derivation of lower bounds for the entropy of powerfree words are generalised. For several explicit sets we derive very good upper and lower bounds for their entropy. Notably, Kolpakov's lower bounds for the entropy of ternary squarefree, binary cubefree and ternary minimally repetitive words are confirmed exactly.
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csls of the Root Lattice a4
arXiv: Metric Geometry, 2010Co-Authors: Manuela Heuer, Peter ZeinerAbstract:Recently, the group of coincidence isometries of the Root Lattice A4 has been determined providing a classification of these isometries with respect to their coincidence indices. A more difficult task is the classification of all CSLs, since different coincidence isometries may generate the same CSL. In contrast to the typical examples in dimensions d ≤ 3, where coincidence isometries generating the same CSL can only differ by a symmetry operation, the situation is more involved in 4 dimensions. Here, we find coincidence isometries that are not related by a symmetry operation but nevertheless give rise to the same CSL. We indicate how the classification of CSLs can be obtained by making use of the icosian ring and provide explicit criteria for two isometries to generate the same CSL. Moreover, we determine the number of CSLs of a given index and encapsulate the result in a Dirichlet series generating function.
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coincidence rotations of the Root Lattice a4
European Journal of Combinatorics, 2008Co-Authors: Michael Baake, Uwe Grimm, Manuela Heuer, Peter ZeinerAbstract:The coincidence site Lattices of the Root Lattice A"4 are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of A"4 into the icosian ring with its rich arithmetic structure, which recently [M Baake, M. Heuer, R.V. Moody, Similar subLattices of the Root Lattice A"4, preprint arXiv:math.MG/0702448] led to the classification of the similar subLattices of A"4.
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similar subLattices of the Root Lattice a4
Journal of Algebra, 2008Co-Authors: Michael Baake, Manuela Heuer, Robert V MoodyAbstract:Abstract Similar subLattices of the Root Lattice A 4 are possible [J.H. Conway, E.M. Rains, N.J.A. Sloane, On the existence of similar subLattices, Can. J. Math. 51 (1999) 1300–1306] for each index that is the square of a non-zero integer of the form m 2 + m n − n 2 . Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H ( Q ( 5 ) ) and the existence of a particular involution of the second kind, which also provides the actual subLattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.
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similar subLattices and coincidence rotations of the Root Lattice a4 and its dual
arXiv: Metric Geometry, 2008Co-Authors: Manuela HeuerAbstract:A natural way to describe the Penrose tiling employs the projection method on the basis of the Root Lattice A4 or its dual. Properties of these Lattices are thus related to properties of the Penrose tiling. Moreover, the Root Lattice A4 appears in various other contexts such as sphere packings, efficient coding schemes and Lattice quantizers. Here, the Lattice A4 is considered within the icosian ring, whose rich arithmetic structure leads to parametrisations of the similar subLattices and the coincidence rotations of A4 and its dual Lattice. These parametrisations, both in terms of a single icosian, imply an index formula for the corresponding subLattices. The results are encapsulated in Dirichlet series generating functions. For every index, they provide the number of distinct similar subLattices as well as the number of coincidence rotations of A4 and its dual.
Vaughan I. L. Clarkson - One of the best experts on this subject based on the ideXlab platform.
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On the Error Performance of the $A_{n}$ Lattices
IEEE Transactions on Information Theory, 2012Co-Authors: Robby G. Mckilliam, Ramanan Subramanian, Emanuele Viterbo, Vaughan I. L. ClarksonAbstract:We consider the Root Lattice $A_{n}$ and derive explicit recursive formulas for the moments of its Voronoi cell. These formulas enable accurate prediction of the error probability of Lattice codes constructed from $A_{n}$ .
Michael Baake - One of the best experts on this subject based on the ideXlab platform.
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coincidence rotations of the Root Lattice a4
European Journal of Combinatorics, 2008Co-Authors: Michael Baake, Uwe Grimm, Manuela Heuer, Peter ZeinerAbstract:The coincidence site Lattices of the Root Lattice A"4 are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of A"4 into the icosian ring with its rich arithmetic structure, which recently [M Baake, M. Heuer, R.V. Moody, Similar subLattices of the Root Lattice A"4, preprint arXiv:math.MG/0702448] led to the classification of the similar subLattices of A"4.
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similar subLattices of the Root Lattice a4
Journal of Algebra, 2008Co-Authors: Michael Baake, Manuela Heuer, Robert V MoodyAbstract:Abstract Similar subLattices of the Root Lattice A 4 are possible [J.H. Conway, E.M. Rains, N.J.A. Sloane, On the existence of similar subLattices, Can. J. Math. 51 (1999) 1300–1306] for each index that is the square of a non-zero integer of the form m 2 + m n − n 2 . Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H ( Q ( 5 ) ) and the existence of a particular involution of the second kind, which also provides the actual subLattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.
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similar subLattices of the Root Lattice a_4
arXiv: Metric Geometry, 2007Co-Authors: Michael Baake, Manuela Heuer, Robert V MoodyAbstract:Similar subLattices of the Root Lattice $A_4$ are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra $\mathbb{H} (\mathbb{Q} (\sqrt{5}))$ and the existence of a particular involution of the second kind, which also provides the actual subLattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.
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Similar subLattices of the Root Lattice $A_4$
arXiv: Metric Geometry, 2007Co-Authors: Michael Baake, Manuela Heuer, Robert V MoodyAbstract:Similar subLattices of the Root Lattice $A_4$ are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra $\mathbb{H} (\mathbb{Q} (\sqrt{5}))$ and the existence of a particular involution of the second kind, which also provides the actual subLattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.
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Symmetry Structure of the Elser-Sloane Quasicrystal
arXiv: Condensed Matter, 1998Co-Authors: Michael Baake, Franz GählerAbstract:The 4D quasicrystal of Elser and Sloane, obtained from the Root Lattice E8 by the cut-and-project method, can be parametrized by the points of an 8D torus. This allows for an explicit analysis of its point and inflation symmetry structure.
Helena Verrill - One of the best experts on this subject based on the ideXlab platform.
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On modularity of rigid and nonrigid Calabi-Yau varieties associated to the Root Lattice $A_{4}$
Nagoya Mathematical Journal, 2020Co-Authors: Klaus Hulek, Helena VerrillAbstract:AbstractWe prove the modularity of four rigid and three nonrigid Calabi-Yau threefolds associated with the A4 Root Lattice.
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on modularity of rigid and nonrigid calabi yau varieties associated to the Root Lattice a_4
arXiv: Algebraic Geometry, 2003Co-Authors: Klaus Hulek, Helena VerrillAbstract:We prove the modularity of four rigid and three nonrigid Calabi-Yau threefolds associated with the Root Lattice A_4.
Keijiro Takahashi - One of the best experts on this subject based on the ideXlab platform.
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Three family GUT-like models from heterotic string
AIP Conference Proceedings, 2008Co-Authors: Keijiro TakahashiAbstract:We construct three‐family SU(5) and SO(10) GUT‐like models, based on an orbifold in the E8×E8 heterotic string theory [28]. We recently classified orbifolds on the E6 Root Lattice. Interestingly, we found that some of the twisted sectors from the Z3×Z3 orbifold on the E6 Root Lattice have just three fixed tori respectively, and it leads to three degenerate massless states. These models also include strongly coupled sectors in the low energy and messenger states charged with both hidden and visible sectors. We present the massless spectra of the models, and consider their interactions.
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three family models from a heterotic orbifold on the e6 Root Lattice
Progress of Theoretical Physics, 2008Co-Authors: Keijiro TakahashiAbstract:We classify N = 1 orbifolds on the E6 Root Lattice and investigate explicit model constructions on the Z3×Z3 orbifold in heterotic string theory. Interestingly some of the twisted sectors from the Z3 × Z3 orbifold on the E6 Root Lattice have just three fixed tori respectively, and generate three degenerate massless states. We also found three point functions with flavor mixing terms. We assume only non-standard gauge embeddings and find that they lead to three-family SU(5) and SO(10) GUT-like models. These models also include strongly coupled sectors in the low energy and messenger states charged with both hidden and visible sectors.
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three family gut models from heterotic orbifold on e_6 Root Lattice
arXiv: High Energy Physics - Theory, 2007Co-Authors: Keijiro TakahashiAbstract:We construct three-family supersymmetric GUT models from heterotic string. We investigate compactifications on non-factorizable torus defined by E_6 Root Lattice, which admits automorphism of Z_3 x Z_3. Interestingly some of the twisted sectors from the Z_3 x Z_3 orbifold on E_6 Root Lattice have just three fixed tori respectively, and generate three degenerate massless states. We assume only nontrivial gauge embeddings and find that it leads to SU(5) and SO(10) GUT models. These models also include strongly coupled sectors in the low energy and messenger states charged with both hidden and visible sectors.
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three family models from a heterotic orbifold on the e_6 Root Lattice
arXiv: High Energy Physics - Theory, 2007Co-Authors: Keijiro TakahashiAbstract:We classify N=1 orbifolds on the E_6 Root Lattice and investigate explicit model constructions on the Z_3xZ_3 orbifold in heterotic string theory. Interestingly some of the twisted sectors from the Z_3xZ_3 orbifold on the E_6 Root Lattice have just three fixed tori respectively, and generate three degenerate massless states. We also found three point functions with flavor mixing terms. We assume only non-standard gauge embeddings and find that they lead to three-family SU(5) and SO(10) GUT-like models. These models also include strongly coupled sectors in the low energy and messenger states charged with both hidden and visible sectors.
