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Journal articles on the topic 'At n- dimensional torus'

Author: Grafiati

Published: 5 June 2025

Last updated: 24 June 2025

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1

Miller, David. "Noncharacteristic Embeddings of the n-Dimensional Torus in the (n + 2)-Dimensional Torus." Transactions of the American Mathematical Society 342, no. 1 (1994): 215. http://dx.doi.org/10.2307/2154691.

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2

Miller, David. "Noncharacteristic embeddings of the $n$-dimensional torus in the $(n+2)$-dimensional torus." Transactions of the American Mathematical Society 342, no. 1 (1994): 215–40. http://dx.doi.org/10.1090/s0002-9947-1994-1179398-7.

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3

Andujar-Munoz, Francisco J., Juan A. Villar-Ortiz, Jose L. Sanchez, Francisco Jose Alfaro, and Jose Duato. "N-Dimensional Twin Torus Topology." IEEE Transactions on Computers 64, no. 10 (2015): 2847–61. http://dx.doi.org/10.1109/tc.2014.2378267.

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4

Baladi, V., D. Rockmore, N. Tongring, and C. Tresser. "Renormalization on the n-dimensional torus." Nonlinearity 5, no. 5 (1992): 1111–36. http://dx.doi.org/10.1088/0951-7715/5/5/005.

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5

Hu, Xiaomin, Yingzhi Tian, Xiaodong Liang, and Jixiang Meng. "Matching preclusion for n -dimensional torus networks." Theoretical Computer Science 687 (July 2017): 40–47. http://dx.doi.org/10.1016/j.tcs.2017.05.002.

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6

Li, Jing, Yuxing Yang, and Xiaohui Gao. "Hamiltonicity of the Torus Network Under the Conditional Fault Model." International Journal of Foundations of Computer Science 28, no. 03 (2017): 211–27. http://dx.doi.org/10.1142/s0129054117500149.

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Low-dimensional Tori are regularly used as interconnection networks in distributed-memory parallel computers. This paper investigates the fault-Hamiltonicity of two-dimensional Tori. A sufficient condition is derived for the graph Row-Torus(m, 2n + 1) with two faulty edges to have a Hamiltonian cycle, where m ≥ 3 and n ≥ 1. By applying the fault-Hamiltonicity of Row-Torus to a two-dimensional torus, we show that Torus(m, n), m, n ≥ 5, with at most four faulty edges is Hamiltonian if the following two conditions are satisfied: (1) the degree of every vertex is at least two, and (2) there do not exist a pair of nonadjacent vertices in a 4-cycle whose degrees are both two after faulty edges are removed.

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7

DE MARCO, GIANLUCA, and ADELE A. RESCIGNO. "TIGHTER TIME BOUNDS ON BROADCASTING IN TORUS NETWORKS IN PRESENCE OF DYNAMIC FAULTS." Parallel Processing Letters 10, no. 01 (2000): 39–49. http://dx.doi.org/10.1142/s0129626400000068.

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We consider the problem of broadcasting in a network G under the hypothesis that each vertex can inform in one unit of time all of its neighbours and that any number of message transmissions, less than the edge-connectivity of G, may fail during each time unit. In particular, we study broadcasting in the n-dimensional k-ary torus, a popular topology for link connections in communication networks. We prove that under the above strong fault-assumption, if k is even and polynomially limited in n, and n is sufficient large, broadcasting in the n-dimensional k-ary torus can be accomplished in time [Formula: see text], where [Formula: see text] is the diameter of the n-dimensional k-ary torus.

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Hu, Xiaomin, Yingzhi Tian, Xiaodong Liang, and Jixiang Meng. "Strong matching preclusion for n-dimensional torus networks." Theoretical Computer Science 635 (July 2016): 64–73. http://dx.doi.org/10.1016/j.tcs.2016.05.008.

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9

Neeb, Karl-Hermann. "On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups." Canadian Mathematical Bulletin 51, no. 2 (2008): 261–82. http://dx.doi.org/10.4153/cmb-2008-027-7.

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AbstractAn n-dimensional quantum torus is a twisted group algebra of the group ℤn. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for n = 2 the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

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10

Rieffel, Marc A. "Projective Modules over Higher-Dimensional Non-Commutative Tori." Canadian Journal of Mathematics 40, no. 2 (1988): 257–338. http://dx.doi.org/10.4153/cjm-1988-012-9.

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The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus Tn with its algebra, C(Tn), of continuous complex-valued functions under pointwise multiplication. But C(Tn) is the universal C*-algebra generated by n commuting unitary operators. By definition, [15, 16, 50], a non-commutative n-torus is the universal C*-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [8, 10], these algebras have a natural differentiable structure, defined by a natural ergodic action of Tn as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [37] that the K-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK-equivalent to ordinary tori by Corollary 7.5 of [52].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [17].)

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11

ANISHCHENKO, V. S., and S. M. NIKOLAEV. "TRANSITION TO CHAOS FROM QUASIPERIODIC MOTIONS ON A FOUR-DIMENSIONAL TORUS PERTURBED BY EXTERNAL NOISE." International Journal of Bifurcation and Chaos 18, no. 09 (2008): 2733–41. http://dx.doi.org/10.1142/s0218127408021956.

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We propose a new autonomous dynamical system of dimension N = 4 that demonstrates the regime of stable two-frequency motions. It is shown that the system of two generators of quasiperiodic oscillations with symmetric coupling can realize motions on four-dimensional torus with resonant structures on it in the form of three- and two-dimensional torus. We show that with increase of noise intensity when the dimension of torus is higher it is destroyed faster.

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12

YAMAGUCHI, YOSHIKAZU. "Higher even dimensional Reidemeister torsion for torus knot exteriors." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 2 (2013): 297–305. http://dx.doi.org/10.1017/s0305004113000248.

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AbstractWe study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal–Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of 1/(2N)2) log | Tor(EK; ρ2N)| converges to zero when N goes to infinity where TorEK; ρ2N is the higher dimensional Reidemeister torsion of a torus knot exterior and an acyclic SL2N(ℂ)-representation of the torus knot group. We also give a classification for SL2(ℂ)-representations of torus knot groups, which induce acyclic SL2N(ℂ)-representations.

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13

SIVAKOFF, DAVID. "Site Percolation on the d-Dimensional Hamming Torus." Combinatorics, Probability and Computing 23, no. 2 (2013): 290–315. http://dx.doi.org/10.1017/s096354831300059x.

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The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a1n × a2n × ⋅⋅⋅ × adn box in $\mathbb{R}^d$ (for constants a1, . . ., ad > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1 − p. We show that if p = λ/n, then there exists λc > 0, which is the positive root of a degree d polynomial whose coefficients depend on a1, . . ., ad, such that for λ < λc the largest component has O(log n) vertices (w.h.p. as n → ∞), and for λ > λc the largest component has $(1-q) \lambda \bigl(\prod_i a_i \bigr) n^{d-1} + o (n^{d-1})$ vertices and the second largest component has O(log n) vertices w.h.p. An implicit formula for q < 1 is also given. The value of λc that we find is distinct from the critical value for the emergence of a giant component in bond percolation on the Hamming torus.

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14

Hu, Xiaomin, Yingzhi Tian, Jixiang Meng, and Weihua Yang. "Conditional fractional matching preclusion of n-dimensional torus networks." Discrete Applied Mathematics 293 (April 2021): 157–65. http://dx.doi.org/10.1016/j.dam.2021.01.011.

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15

Rushton, Brian. "A finite subdivision rule for the n-dimensional torus." Geometriae Dedicata 167, no. 1 (2012): 23–34. http://dx.doi.org/10.1007/s10711-012-9802-5.

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16

Hu, Xiaomin, Xiangyu Ren, and Weihua Yang. "Conditional k-matching preclusion for n-dimensional torus networks." Discrete Applied Mathematics 353 (August 2024): 181–90. http://dx.doi.org/10.1016/j.dam.2024.04.026.

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17

Tsukamoto, Yuki. "Prescribed mean curvature equation on torus." Analysis 41, no. 2 (2021): 69–77. http://dx.doi.org/10.1515/anly-2020-0054.

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Abstract Prescribed mean curvature problems on the torus have been considered in one dimension. In this paper, we prove the existence of a graph on the n-dimensional torus 𝕋 n {\mathbb{T}^{n}} , the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.

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18

Kim, Jong-Seok, Hyeong-Ok Lee, Mihye Kim, and Sung Won Kim. "Broadcasting Algorithms of Three-Dimensional Petersen-Torus Network." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/935737.

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The three-dimensional Petersen-torus network 3PT is based on the Petersen graph and has recently been proposed as an interconnection network. 3PT is better than the well-known 3D torus and 3D honeycomb mesh in terms of diameter and network cost. In this paper, we propose one-to-all and all-to-all broadcasting algorithms for 3PT(l;m;n) under SLA (single-link available) and MLA (multiple-link available) models.

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19

Connelly, Robert, and William Dickinson. "Periodic planar disc packings." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, no. 2008 (2014): 20120039. http://dx.doi.org/10.1098/rsta.2012.0039.

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Several conditions are given when a packing of equal discs in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of any collectively jammed packing, whose graph does not consist of all triangles, and the torus lattice is the standard triangular lattice, is at most , where n is the number of packing discs in the torus. Several classes of collectively jammed packings are presented where the conjecture holds.

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20

SOUSA JR., LUIZ A. M. "Rigidity theorems of Clifford Torus." Anais da Academia Brasileira de Ciências 73, no. 3 (2001): 327–32. http://dx.doi.org/10.1590/s0001-37652001000300003.

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Let M be an n-dimensional closed minimally immersed hypersurface in the unit sphere Sn + 1. Assume in addition that M has constant scalar curvature or constant Gauss-Kronecker curvature. In this note we announce that if M has (n - 1) principal curvatures with the same sign everywhere, then M is isometric to a Clifford Torus <img src="http:/img/fbpe/aabc/v73n3/03ab.gif" alt="03ab.gif (725 bytes)" align="middle">.

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21

Shchukin, M. V. "n-Homogeneous C*-algebras." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 57, no. 2 (2021): 185–89. http://dx.doi.org/10.29235/1561-2430-2021-57-2-185-189.

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The classical results by J. Fell, J. Tomiyama, M. Takesaki describe n-homogeneous С*-algebras as algebras of all continuous sections for an appropriate algebraic bundle. By using this realization, several authors described the set of n-homogeneous С*-algebras with different spaces of primitive ideals. In 1974 F. Krauss and T. Lawson described the set of all n-homogeneous С*-algebras whose space Prim of primitive ideals is homeomorphic to the sphere S2. Suppose the space PrimA of primitive ideals is homeomorphic to the sphere S3 for some n-homogeneous С*-algebra A. In this case, these authors showed that the algebra A is isomorphic to the algebra C(S3,Cn×n). If n ≥ 2 then there are countably many pairwise non-isomorphic n-homogeneous С*-algebras A such that PrimA ≅ S 4. Further, let n ≥ 3. There is only one n-homogeneous С*-algebra A such that PrimA ≅ S 5. There are two non-isomorphic 2-homogeneous С*-algebras A and B with space PrimA ≅ S 5. On the other hand, algebraic bundles over the torus T 2 are described by a residue class [p] in Z/nZ = π1(PUn). Two such bundles with classes [pi] produce isomorphic С*-algebras if and only if [p1] = ±[p2]. An algebraic bundle over the torus T 3 is determined by three residue classes in Z/nZ. Anatolii Antonevich and Nahum Krupnik introduced some structures on the set of algebraic bundles over the sphere S2. Algebraic bundles over the compact connected two-dimensional oriented manifolds were considered by the author. In this case, the set of non-equivalent algebraic bundles over such space is like the set of algebraic bundles over the torus T2. Further advances could be in describing the set of algebraic bundles over the 3-dimensional manifolds.

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22

de Carvalho, Edson Donizete, Waldir Silva Soares, and Eduardo Brandani da Silva. "Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori." Entropy 23, no. 8 (2021): 959. http://dx.doi.org/10.3390/e23080959.

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In this work, we show that an n-dimensional sublattice Λ′=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ′=Rn/Λ′, where the group G is isomorphic to the lattice partition Λ/Λ′. As a consequence, we obtain, via this technique, toric codes of parameters [[2m2,2,m]], [[3m3,3,m]] and [[6m4,6,m2]] from the lattices Z2, Z3 and Z4, respectively. In particular, for n=2, if Λ1 is either the lattice Z2 or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell P0′ of each hexagonal sublattice Λ′ of hexagonal lattices Λ, using either the fundamental cell P0 or the Voronoi cell V0. These partitions allow us to present new classes of toric codes with parameters [[3m2,2,m]] and color codes with parameters [[18m2,4,4m]] in the flat torus from families of hexagonal lattices in R2.

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23

Balgimbayeva, Sh A., and A. K. Janabilova. "Gagliardo–Nirenberg type inequalities for smoothness spaces related to Morrey spaces over n-dimensional torus." Bulletin of the Karaganda University-Mathematics 117, no. 1 (2025): 53–62. https://doi.org/10.31489/2025m1/53-62.

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In the paper, the Gagliardo–Nirenberg type inequalities for smoothness spaces Bpqsτ(Tn) of Nikol’skii–Besov type and spaces Fpqsτ(Tn) of Lizorkin–Triebel type both related to Morrey spaces over n-dimensional torus for some range of the parameters s, p, q, τ were proved. These spaces are natural analogues of the spaces Bpqsτ(Rn) and Fpqsτ(Rn) in the case of multidimensional torus Tn. The main results of the article are two theorems, each of which proves the Gagliardo–Nirenberg type inequality for the Lizorkin–Triebel type spaces or the Nikol’skii–Besov type spaces respectively.

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24

YURI, GLIKLIKH. "ON A CERTAIN SYSTEM OF STOCHASTIC EQUATIONS WITH MEAN DERIVATIVES, CONNECTED WITH HYDRODYNAMICS." Global and Stochastic Analysis 9, no. 3 (2022): 1–7. https://doi.org/10.5281/zenodo.7050018.

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We study a system of two special equations with mean derivatives on the group of Sobolev diffeomorphisms of the at n-dimensional torus that leads to a ow on the torus, described by a system of two equations, one of which is the Burgers equation, and the second one is a continuity-type equation. We prove the existence of solution theorem and interpret this flow as a flow of a special viscous fluid.

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25

Albeverio, Sergio, Sergey Dobrokhotov, and Michael Poteryakhin. "On quasimodes of small diffusion operators corresponding to stable invariant tori with nonregular neighborhoods." Asymptotic Analysis 43, no. 3 (2005): 171–203. https://doi.org/10.3233/asy-2005-689.

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Smooth vector fields V in $\[$\mathbb{R}$$ n having a k-dimensional invariant torus are considered and the possibility of constructing the asymptotic eigenfunctions localized in the neighborhood of this torus is analyzed for the case of a corresponding small diffusion operator V·∇−εΔ, ε>0. The asymptotic stability property of the invariant torus is a sufficient condition for the existence of such eigenfunctions. Both the regular case (where the variational system is reducible) and the nonregular case (where the variational system is nonreducible) are considered.

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26

DULAT, SAYIPJAMAL. "THE N = 2 SUPERCONFORMAL ℤ3 ORBIFOLD-PRIME MODEL WITH c = 3". Modern Physics Letters A 18, № 07 (2003): 503–13. http://dx.doi.org/10.1142/s0217732303009381.

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We consider N = 2 superconformal field theories on a two-dimensional torus with central charge c = 3. In particular, we present the partition function for this theory. Furthermore, to generate new theories, we recall general orbifold prescription. Finally, we construct the modular invariant ℤ3 orbifold-prime model.

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27

Hartley, Michael I., Peter McMullen, and Egon Schulte. "Symmetric Tessellations on Euclidean Space-Forms." Canadian Journal of Mathematics 51, no. 6 (1999): 1230–39. http://dx.doi.org/10.4153/cjm-1999-055-6.

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AbstractIt is shown here that, for n ⩾ 2, the n-torus is the only n-dimensional compact euclidean space-form which can admit a regular or chiral tessellation. Further, such a tessellation can only be chiral if n = 2.

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28

Xia, Zhihong. "Existence of invariant tori in volume-preserving diffeomorphisms." Ergodic Theory and Dynamical Systems 12, no. 3 (1992): 621–31. http://dx.doi.org/10.1017/s0143385700006969.

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AbstractIn this paper we consider certain volume-preserving diffeomorphisms on I × Tn, where I ∈ ℝ is a closed interval and Tn is an n-dimensional torus. We show that under certain non-degeneracy conditions, all of the maps sufficiently close to the integrable maps preserve a large set of n-dimensional invariant tori.

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29

Aϊssiou, Tayeb. "Semiclassical Limits of Eigenfunctions on Flat n-Dimensional Tori." Canadian Mathematical Bulletin 56, no. 1 (2013): 3–12. http://dx.doi.org/10.4153/cmb-2011-152-9.

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AbstractWe provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an n-dimensional flat torus 𝕋n, and the Fourier transform of squares of the eigenfunctions |φ λ|2 of the Laplacian have uniform ln bounds that do not depend on the eigenvalue λ. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on 𝕋n+2. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an n-dimensional sphere Sn(λ) of radius √λ, and we use it in the proof.

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30

Dani, S. G. "On orbits of endomorphisms of tori and the Schmidt game." Ergodic Theory and Dynamical Systems 8, no. 4 (1988): 523–29. http://dx.doi.org/10.1017/s0143385700004673.

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AbstractWe show that there exists a subset F of the n-dimensional torus n such that F has Hausdorff dimension n and for any x∈F and any semisimple automorphism σ of n the closure of the σ-orbit of x contains no periodic points.

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31

DATT, Gopal, and Shesh Kumar PANDEY. "Slant Toeplitz operators on the Lebesgue space of $n$-dimensional torus." Hokkaido Mathematical Journal 49, no. 3 (2020): 363–89. http://dx.doi.org/10.14492/hokmj/1607936533.

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32

Narayanan, R., H. Neuberger, and F. Reynoso. "Phases of three-dimensional large N QCD on a continuum torus." Physics Letters B 651, no. 2-3 (2007): 246–52. http://dx.doi.org/10.1016/j.physletb.2007.06.016.

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33

Yuan, Jun, Aixia Liu, Hongmei Wu, and Jing Li. "Panconnectivity of n-dimensional torus networks with faulty vertices and edges." Discrete Applied Mathematics 161, no. 3 (2013): 404–23. http://dx.doi.org/10.1016/j.dam.2012.08.027.

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34

Grines, Vyacheslav Z., Dmitrii I. Mints, and Ekaterina E. Chilina. "On perturbations of algebraic periodic automorphisms of a two-dimensional torus." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 24, no. 2 (2022): 141–50. http://dx.doi.org/10.15507/2079-6900.24.202202.141-150.

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According to the results of V. Z. Grines and A. N. Bezdenezhnykh, for each gradient-like diffeomorphism of a closed orientable surface M2 there exist a gradient-like flow and a periodic diffeomorphism of this surface such that the original diffeomorphism is a superposition of a diffeomorphism that is a shift per unit time of the flow and the periodic diffeomorphism. In the case when M2 is a two-dimensional torus, there is a topological classification of periodic maps. Moreover, it is known that there is only a finite number of topological conjugacy classes of periodic diffeomorphisms that are not homotopic to identity one. Each such class contains a representative that is a periodic algebraic automorphism of a two-dimensional torus. Periodic automorphisms of a two-dimensional torus are not structurally stable maps, and, in general, it is impossible to predict the dynamics of their arbitrarily small perturbations. However, in the case when a periodic diffeomorphism is algebraic, we constructed a one-parameter family of maps consisting of the initial periodic algebraic automorphism at zero parameter value and gradient-like diffeomorphisms of a twodimensional torus for all non-zero parameter values. Each diffeomorphism of the constructed one-parameter families inherits, in a certain sense, the dynamics of a periodic algebraic automorphism being perturbed.

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35

Bregman, Corey. "Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles." International Mathematics Research Notices 2019, no. 13 (2017): 4004–46. http://dx.doi.org/10.1093/imrn/rnx243.

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AbstractGiven a matrix $A\in SL(N,\mathbb{Z})$, form the semidirect product $G=\mathbb{Z}^N\rtimes_A \mathbb{Z}$ where the $\mathbb{Z}$-factor acts on $\mathbb{Z}^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this article, we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.

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36

Kurenkov, Evgeniy D., and Dmitriy I. Mints. "On periodic points of torus endomorphisms." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 21, no. 4 (2019): 480–87. http://dx.doi.org/10.15507/2079-6900.21.201904.480-487.

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It is a well-known fact that Anosov endomorphisms of n-torus which are different from automorphisms and expanding endomorphisms are not structurally stable and, in general, are not conjugated to algebraic endomorphisms. Nevertheless, hyperbolic algebraic endomorphisms of torus are conjugated with their C1 perturbations on the set of periodic points. Therefore the study of algebraic toral endomorphisms is very important. This paper is devoted to study of the structure of the sets of periodic and pre-periodic points of algebraic toral endomorphisms. Various group properties of this sets of points are studied. The density of periodic points for algebraic endomorphisms of n-torus is proved; it is clarifief how the number of periodic and pre-periodic points with a fixed denominator depends on the properties of the characteristic polynomial. The Theorem 1.1 is the main result of this paper. It contains an algorithm that allows to split the sets of periodic and pre-periodic points of a given algebraic endomorphism of two-dimensional torus.

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37

Kaneyama, Tamafumi. "Torus-equivariant vector bundles on projective spaces." Nagoya Mathematical Journal 111 (September 1988): 25–40. http://dx.doi.org/10.1017/s0027763000000982.

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For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

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38

Pandey, Vipul Kumar. "Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold II." Advances in High Energy Physics 2022 (August 23, 2022): 1–17. http://dx.doi.org/10.1155/2022/6410245.

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We have previously developed the BRST quantization on the hypersurface V N − 1 embedded in N -dimensional Euclidean space R N in both Hamiltonian and Lagrangian formulation. We generalize the formalism in the case of L -dimensional manifold V L embedded in R N with 1 ≤ L < N . The result is essentially the same as the previous one. We have also verified the results obtained here using a simple example of particle motion on a torus knot.

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39

YANG, YUXING, and LINGLING ZHANG. "Prescribed Hamiltonian Connectedness of 2D Torus." Journal of Interconnection Networks 20, no. 01 (2020): 2050001. http://dx.doi.org/10.1142/s0219265920500012.

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Tori are important fundamental interconnection networks for multiprocessor systems. Hamiltonian paths are important in information communication of multiprocessor systems, and Hamiltonian path embedding capability is an important aspect to determine if a network topology is suitable for a real application. In real systems, some links may have better performance. Therefore, when embedding Hamiltonian path into interconnection networks, it is desirable that these Hamiltonian paths would pass through the links with better performance. Given a two two-dimensional torus T (m, n) with m, n ≥ 5 odd, let L be a linear forest with at most two edges in T (m, n) and let u and v be two distinct vertices in T (m, n) such that none of the paths in L has u or v as internal node or both of them as end nodes. In this paper, we construct a hamiltonian path of T (m, n) between u and v passing through L.

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Velasco Fuentes, Oscar, and Angélica Romero Arteaga. "Quasi-steady linked vortices with chaotic streamlines." Journal of Fluid Mechanics 687 (October 14, 2011): 571–83. http://dx.doi.org/10.1017/jfm.2011.394.

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AbstractThis paper describes the motion and the flow geometry of two or more linked ring vortices in an otherwise quiescent, ideal fluid. The vortices are thin tubes of near-circular shape which lie on the surface of an immaterial torus of small aspect ratio. Since the vortices are assumed to be identical and evenly distributed on any meridional section of the torus, the flow evolution depends only on the number of vortices ($n$) and the torus aspect ratio (${r}_{1} / {r}_{0} $, where ${r}_{0} $ is the centreline radius and ${r}_{1} $ is the cross-section radius). Numerical simulations based on the Biot–Savart law showed that a small number of vortices ($n= 2, 3$) coiled on a thin torus (${r}_{1} / {r}_{0} \leq 0. 16$) progressed along and rotated around the symmetry axis of the torus in an almost uniform manner while each vortex approximately preserved its shape. In the comoving frame the velocity field always possesses two stagnation points. The transverse intersection, along $2n$ streamlines, of the stream tube emanating from the front stagnation point and the stream tube ending at the rear stagnation point creates a three-dimensional chaotic tangle. It was found that the volume of the chaotic region increases with increasing torus aspect ratio and decreasing number of vortices.

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RIEFFEL, MARC A., and ALBERT SCHWARZ. "MORITA EQUIVALENCE OF MULTIDIMENSIONAL NONCOMMUTATIVE TORI." International Journal of Mathematics 10, no. 02 (1999): 289–99. http://dx.doi.org/10.1142/s0129167x99000100.

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One can describe an n-dimensional noncommutative torus by means of an antisymmetric n×n matrix θ. We construct an action of the group SO(n,n|Z) on the space of n×n antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched.

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Bossard, Antoine, and Keiichi Kaneko. "Torus Pairwise Disjoint-Path Routing." Sensors 18, no. 11 (2018): 3912. http://dx.doi.org/10.3390/s18113912.

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Modern supercomputers include hundreds of thousands of processors and they are thus massively parallel systems. The interconnection network of a system is in charge of mutually connecting these processors. Recently, the torus has become a very popular interconnection network topology. For example, the Fujitsu K, IBM Blue Gene/L, IBM Blue Gene/P, and Cray Titan supercomputers all rely on this topology. The pairwise disjoint-path routing problem in a torus network is addressed in this paper. This fundamental problem consists of the selection of mutually vertex disjoint paths between given vertex pairs. Proposing a solution to this problem has critical implications, such as increased system dependability and more efficient data transfers, and provides concrete implementation of green and sustainable computing as well as security, privacy, and trust, for instance, for the Internet of Things (IoT). Then, the correctness and complexities of the proposed routing algorithm are formally established. Precisely, in an n-dimensional k-ary torus ( n < k , k ≥ 5 ), the proposed algorithm connects c ( c ≤ n ) vertex pairs with mutually vertex-disjoint paths of lengths at most 2 k ( c − 1 ) + n ⌊ k / 2 ⌋ , and the worst-case time complexity of the algorithm is O ( n c 4 ) . Finally, empirical evaluation of the proposed algorithm is conducted in order to inspect its practical behavior.

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Agapov, S. V. "High-Degree Polynomial Integrals of a Natural System on the Two-Dimensional Torus." Siberian Mathematical Journal 64, no. 2 (2023): 261–68. http://dx.doi.org/10.1134/s0037446623020015.

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AbstractWe study a natural mechanical system on the two-dimensional torus which admits an additional first integral polynomial in momenta of an odd degree $ N $ and independent of the energy integral. For $ N=5,7 $, we obtain the estimates on the number of straight lines in the spectrum of the potential.

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S.V., Filchakov. "Some Properties of Force Fields on the Groups of Diffeomorphisms of the flat n-Dimensional Torus, connected with the Notion of Parallelism." Global and Stochastic Analysis 1, no. 1 (2013): 41–47. https://doi.org/10.5281/zenodo.7673384.

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Some existence of solution theorems are proved for second order differential inclusions on the groups of diffeomorphisms of a flat n- dimensional torus. The technical tool for the proofs is the use of the notion of parallelism on those groups

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TRINCHERO, R. "SCALAR FIELD ON NON-INTEGER-DIMENSIONAL SPACES." International Journal of Geometric Methods in Modern Physics 09, no. 08 (2012): 1250070. http://dx.doi.org/10.1142/s0219887812500703.

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Deformations of the canonical spectral triples over the n-dimensional torus are considered. These deformations have a discrete dimension spectrum consisting of non-integer values less than n. The differential algebra corresponding to these spectral triples is studied. No junk forms appear for non-vanishing deformation parameter. The action of a scalar field in these spaces is considered, leading to non-trivial extra structure compared to the integer-dimensional cases, which does not involve a loss of covariance. One-loop contributions are computed leading to finite results for non-vanishing deformation.

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Albanese, A. A., and P. Popivanov. "On the global solvability in Gevrey classes on the n-dimensional torus." Journal of Mathematical Analysis and Applications 297, no. 2 (2004): 659–72. http://dx.doi.org/10.1016/j.jmaa.2004.04.033.

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Dasgupta, Jyoti, Bivas Khan, and Vikraman Uma. "Cohomology of torus manifold bundles." Mathematica Slovaca 69, no. 3 (2019): 685–98. http://dx.doi.org/10.1515/ms-2017-0257.

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Abstract Let X be a 2n-dimensional torus manifold with a locally standard T ≅ (S1)n action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T-bundle p : E → B and let π : E(X) → B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as a H*(B)-algebra and the topological K-ring of E(X) as a K*(B)-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt. These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety.

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Belova, Olga, Josef Mikes, and Karl Strambach. "About almost geodesic curves." Filomat 33, no. 4 (2019): 1013–18. http://dx.doi.org/10.2298/fil1904013b.

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We determine in Rn the form of curves C for which also any image under an (n-1)-dimensional algebraic torus is an almost geodesic with respect to an affine connection ? with constant coefficients and calculate the components of ?.

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SUH, YOUNG JIN, and HAE YOUNG YANG. "THE SCALAR CURVATURE OF MINIMAL HYPERSURFACES IN A UNIT SPHERE." Communications in Contemporary Mathematics 09, no. 02 (2007): 183–200. http://dx.doi.org/10.1142/s021919970700237x.

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In this paper, we study n-dimensional compact minimal hypersurfaces in a unit sphere Sn+1(1) and give an answer for S. S. Chern's conjecture. We have shown that [Formula: see text] if S > n, and prove that an n-dimensional compact minimal hypersurface with constant scalar curvature in Sn+1(1) is a totally geodesic sphere or a Clifford torus if [Formula: see text], where S denotes the squared norm of the second fundamental form of this hypersurface.

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Day, Khaled, and Mohammad H. Al-Towaiq. "An Efficient Parallel Gauss-Seidel Algorithm on a 3D Torus Network-on-Chip." Sultan Qaboos University Journal for Science [SQUJS] 20, no. 1 (2015): 29. http://dx.doi.org/10.24200/squjs.vol20iss1pp29-38.

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Network-on-chip (NoC) multi-core architectures with a large number of processing elements are becoming a reality with the recent developments in technology. In these modern systems the processing elements are interconnected with regular NoC topologies such as meshes and tori. In this paper we propose a parallel Gauss-Seidel (GS) iterative algorithm for solving large systems of linear equations on a 3-dimensional torus NoC architecture. The proposed parallel algorithm is O(Nn2/k3) time complexity for solving a system with a matrix of order n on a k×k×k 3D torus NoC architecture with N iterations assuming n and N are large compared to k. We show that under these conditions the proposed parallel GS algorithm has near optimal speedup.

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