One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation

2009 ◽  
Vol 228 (6) ◽  
pp. 1815-1829 ◽  
Author(s):  
C.M. Linton ◽  
I. Thompson
Keyword(s):  
Helmholtz Equation ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
Lattice Sums

scholarly journals DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (39) ◽  
pp. 3591-3600 ◽  
Author(s):  
HIROSI OOGURI ◽  
NAOKI SASAKURA
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Analogue Model ◽  
Gauge Invariant ◽  
Invariant Functions ◽  
Dimensional Lattice ◽  
Chern Simons ◽  
Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

A COMPARISON BETWEEN THE TWO-DIMENSIONAL AND THREE-DIMENSIONAL LATTICES

2004 ◽  
Vol 18 (25) ◽  
pp. 1301-1309 ◽  
Author(s):  
ANDREI DOLOCAN ◽  
VOICU OCTAVIAN DOLOCAN ◽  
VOICU DOLOCAN
Keyword(s):  
Lattice Constant ◽  
Cohesive Energy ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Coupling Constants ◽  
Two Dimensional ◽  
Coupling Constant ◽  
Dimensional Lattice ◽  
Analytical Formulae ◽  
Three Dimensional Space

By using a new Hamiltonian of interaction we have calculated the interaction energy for two-dimensional and three-dimensional lattices. We present also, approximate analytical formulae and the analytical formulae for the constant of the elastic force. The obtained results show that in the three-dimensional space, the two-dimensional lattice has the lattice constant and the cohesive energy which are smaller than that of the three-dimensional lattice. For appropriate values of the coupling constants, the two-dimensional lattice in a two-dimensional space has both the lattice constant and the cohesive energy, larger than that of the two-dimensional lattice in a three-dimensional space; this means that if there is a two-dimensional space in the Universe, this should be thinner than the three-dimensional space, while the interaction forces should be stronger. On the other hand, if the coupling constant in the two-dimensional lattice in the two-dimensional space is close to zero, the cohesive energy should be comparable with the cohesive energy from three-dimensional space but this two-dimensional space does not emit but absorbs radiation.

(TP-EDOT)2PF6—Two-dimensional spin array in a three-dimensional lattice

2006 ◽  
Vol 142 (3-4) ◽  
pp. 285-290
Author(s):  
H. Yamochi ◽  
M. Soeda ◽  
J. Hagiwara ◽  
G. Saito
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Lattice

(TP-EDOT)2PF6 — Two-Dimensional Spin Array in a Three-Dimensional Lattice

2007 ◽  
Vol 142 (3-4) ◽  
pp. 289-294
Author(s):  
H. Yamochi ◽  
M. Soeda ◽  
J. Hagiwara ◽  
G. Saito
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Lattice

COUNTING d-DIMENSIONAL POLYCUBES AND NONRECTANGULAR PLANAR POLYOMINOES

2009 ◽  
Vol 19 (03) ◽  
pp. 215-229 ◽  
Author(s):  
GADI ALEKSANDROWICZ ◽  
GILL BAREQUET
Keyword(s):  
Hexagonal Lattice ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
Connected Set ◽  
Dimensional Face

A planar polyomino of size n is an edge-connected set of n squares on a rectangular two-dimensional lattice. Similarly, a d-dimensional polycube (for d ≥ 2) of size n is a connected set of n hypercubes on an orthogonal d-dimensional lattice, where two hypercubes are neighbors if they share a (d - 1)-dimensional face. There are also two-dimensional polyominoes that lie on a triangular or hexagonal lattice. In this paper we describe a generalization of Redelmeier's algorithm for counting two-dimensional rectangular polyominoes, which counts all the above types of polyominoes. For example, our program computed the number of distinct three-dimensional polycubes of size 18. To the best of our knowledge, this is the first tabulation of this value.

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