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Journal articles on the topic 'Lattice path'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Friedman, Jane, Ira Gessel, and Doron Zeilberger. "Talmudic lattice path counting." Journal of Combinatorial Theory, Series A 68, no. 1 (October 1994): 215–17. http://dx.doi.org/10.1016/0097-3165(94)90100-7.

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2

BRLEK, S., G. LABELLE, and A. LACASSE. "PROPERTIES OF THE CONTOUR PATH OF DISCRETE SETS." International Journal of Foundations of Computer Science 17, no. 03 (June 2006): 543–56. http://dx.doi.org/10.1142/s012905410600398x.

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We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of non-crossing closed paths, generalizing in this way a result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points. Moreover we obtain a similar result for hexagonal lattices and show that there is no other regular lattice having that property.

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3

Maier, Robert S., and Robert S. Bernard. "Accuracy of the Lattice-Boltzmann Method." International Journal of Modern Physics C 08, no. 04 (August 1997): 747–52. http://dx.doi.org/10.1142/s0129183197000631.

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The accuracy of the lattice-Boltzmann method (LBM) is moderated by several factors, including Mach number, spatial resolution, boundary conditions, and the lattice mean free path. Results obtained with 3D lattices suggest that the accuracy of certain two-dimensional (2D) flows, such as Poiseuille and Couette flow, persist even when the mean free path between collisions is large, but that of the 3D duct flow deteriorates markedly when the mean free path exceeds the lattice spacing. Accuracy in general decreases with Knudsen number and Mach number, and the product of these two quantities is a useful index for the applicability of LBM to 3D low-Reynolds-number flow. The influence of boundary representations on LBM accuracy is captured by the proposed index, when the accuracy of the prescribed boundary conditions is consistent with that of LBM.

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4

Chen, Yu, Jinguo You, Benyuan Zou, Guoyu Gan, Ting Zhang, and Lianyin Jia. "Exploring Structural Characteristics of Lattices in Real World." Complexity 2020 (January 21, 2020): 1–11. http://dx.doi.org/10.1155/2020/1250106.

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There are two important models for data analysis and knowledge system: data cube lattices and concept lattices. They both essentially have lattice structures, which are actually irregular in our real world. However, their structural characteristics and relationship are not yet clear. To the best of our knowledge, no work has paid enough attention to this challenging issue from the perspective of graph data, in spite of the importance of structures in lattice data. In this paper, we first tackle the structural statistics of lattice data from three aspects: the degree distribution, clustering coefficient, and average path length. We demonstrated by various datasets that data cube lattices and concept lattices share similarities underlying their topology, which are, in general, different from random networks and complex networks. Specifically, lattice data follow the Poisson distribution and have smaller clustering coefficient and greater average path length. We further discuss and explain these characteristics intrinsically by building the analytical model and the generating mechanism.

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5

Callan, David, and Kenneth Bernstein. "A Lattice Path Equality: 11106." American Mathematical Monthly 113, no. 7 (August 1, 2006): 656. http://dx.doi.org/10.2307/27642018.

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6

Quesney, Alexandre. "The relative lattice path operad." Algebraic & Geometric Topology 18, no. 3 (April 3, 2018): 1753–98. http://dx.doi.org/10.2140/agt.2018.18.1753.

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7

Wagner, Carl G. "The Carlitz lattice path polynomials." Discrete Mathematics 222, no. 1-3 (July 2000): 291–98. http://dx.doi.org/10.1016/s0012-365x(00)00058-3.

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8

Bonin, Joseph E., and Anna de Mier. "Lattice path matroids: Structural properties." European Journal of Combinatorics 27, no. 5 (July 2006): 701–38. http://dx.doi.org/10.1016/j.ejc.2005.01.008.

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9

Deutsch, Emeric, David Callan, M. Beck, D. Beckwith, W. Bohm, R. F. McCoart, and GCHQ Problems Group. "Another Type of Lattice Path: 10658." American Mathematical Monthly 107, no. 4 (April 2000): 368. http://dx.doi.org/10.2307/2589192.

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10

Delucchi, Emanuele, and Martin Dlugosch. "Bergman Complexes of Lattice Path Matroids." SIAM Journal on Discrete Mathematics 29, no. 4 (January 2015): 1916–30. http://dx.doi.org/10.1137/130944242.

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11

Kornilovitch, P. E. "Path-integral approach to lattice polarons." Journal of Physics: Condensed Matter 19, no. 25 (May 30, 2007): 255213. http://dx.doi.org/10.1088/0953-8984/19/25/255213.

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12

Bonin, Joseph E. "Lattice path matroids: The excluded minors." Journal of Combinatorial Theory, Series B 100, no. 6 (November 2010): 585–99. http://dx.doi.org/10.1016/j.jctb.2010.05.001.

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13

Goldman, Jay R., and Thomas Sundquist. "Lattice path enumeration by formal schema." Advances in Applied Mathematics 13, no. 2 (June 1992): 216–51. http://dx.doi.org/10.1016/0196-8858(92)90010-t.

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14

Mendelson, Haim. "Lattice path combinatorics and linear probing." Journal of Statistical Planning and Inference 14, no. 1 (May 1986): 79–93. http://dx.doi.org/10.1016/0378-3758(86)90013-3.

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15

Melczer, Stephen, and Marni Mishna. "Asymptotic Lattice Path Enumeration Using Diagonals." Algorithmica 75, no. 4 (September 14, 2015): 782–811. http://dx.doi.org/10.1007/s00453-015-0063-1.

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16

Sen, P., and P. Ray. "Longest path in percolating hierarchical lattice." Journal of Statistical Physics 59, no. 5-6 (June 1990): 1573–80. http://dx.doi.org/10.1007/bf01334764.

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17

Ericksen, Larry. "Lattice path combinatorics for multiple product identities." Journal of Statistical Planning and Inference 140, no. 8 (August 2010): 2213–26. http://dx.doi.org/10.1016/j.jspi.2010.01.017.

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18

Mansfield, Marc L. "Unbiased sampling of lattice Hamilton path ensembles." Journal of Chemical Physics 125, no. 15 (October 21, 2006): 154103. http://dx.doi.org/10.1063/1.2357935.

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19

Pergola, E., R. Pinzani, S. Rinaldi, and R. A. Sulanke. "Lattice path moments by cut and paste." Advances in Applied Mathematics 30, no. 1-2 (February 2003): 208–18. http://dx.doi.org/10.1016/s0196-8858(02)00532-8.

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20

Kauers, Manuel, Christoph Koutschan, and Doron Zeilberger. "Proof of Ira Gessel's lattice path conjecture." Proceedings of the National Academy of Sciences 106, no. 28 (June 25, 2009): 11502–5. http://dx.doi.org/10.1073/pnas.0901678106.

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21

Diep, Madeline M., Matthew B. Dwyer, and Sebastian Elbaum. "Lattice-Based Sampling for Path Property Monitoring." ACM Transactions on Software Engineering and Methodology 21, no. 1 (December 2011): 1–43. http://dx.doi.org/10.1145/2063239.2063244.

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22

Stokke, Anna, and Terry Visentin. "Lattice path constructions for orthosymplectic determinantal formulas." European Journal of Combinatorics 58 (November 2016): 38–51. http://dx.doi.org/10.1016/j.ejc.2016.05.002.

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23

Buzeţeanu, Şerban N., and Virgil Domocoş. "Polynomial identities from weighted lattice path counting." Discrete Mathematics 150, no. 1-3 (April 1996): 421–25. http://dx.doi.org/10.1016/0012-365x(95)00209-f.

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24

Roux, Stéphane, Alex Hansen, Luciano R. da Silva, Liacir S. Lucena, and Ras B. Pandey. "Minimal path on the hierarchical diamond lattice." Journal of Statistical Physics 65, no. 1-2 (October 1991): 183–204. http://dx.doi.org/10.1007/bf01329855.

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25

Gessel, Ira M. "A probabilistic method for lattice path enumeration." Journal of Statistical Planning and Inference 14, no. 1 (May 1986): 49–58. http://dx.doi.org/10.1016/0378-3758(86)90009-1.

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26

An, Suhyung, JiYoon Jung, and Sangwook Kim. "Facial structures of lattice path matroid polytopes." Discrete Mathematics 343, no. 1 (January 2020): 111628. http://dx.doi.org/10.1016/j.disc.2019.111628.

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27

Johnson, Samuel, Marni Mishna, and Karen Yeats. "A combinatorial understanding of lattice path asymptotics." Advances in Applied Mathematics 92 (January 2018): 144–63. http://dx.doi.org/10.1016/j.aam.2017.08.001.

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28

Rajan, Dayanand S., and Anil M. Shende. "Root Lattices are Efficiently Generated." International Journal of Algebra and Computation 07, no. 01 (February 1997): 33–50. http://dx.doi.org/10.1142/s0218196797000046.

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We show that root lattices are exactly those lattices whose relevant vectors are all equal in length. We define notions of "compatibility" between the euclidean metric and the shortest path metric in the infinite directed graph induced by a subset, G, of a lattice, i.e., the directed graph whose vertices are the points in the lattice, and whose arcs are the ordered pairs (x, x+g), with x a lattice point and g a point in G. We present some ("easy to check for") criteria that a lattice and a subset of it must satisfy to ensure "compatibility" between the corresponding graphical and the euclidean metrics. We use these criteria to characterize, in more than one way, a set of "economically and efficiently generated" lattices, including root lattices. Our results include a "graph theoretic" characterization of root lattices as well. We also discuss, in brief, certain algorithmic considerations in the simulation of macroscopic physical phenomena in massively parallel computers based on suitable discretizations of euclidean space that led us to our graphical treatment of lattices.

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29

STÄMPFLE, M. "CELLULAR AUTOMATA AND OPTIMAL PATH PLANNING." International Journal of Bifurcation and Chaos 06, no. 03 (March 1996): 603–10. http://dx.doi.org/10.1142/s0218127496000291.

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Cellular automata are deterministic dynamical systems in which time, space, and state values are discrete. Although they consist of uniform elements, which interact only locally, cellular automata are capable of showing complex behavior. This property is exploited for solving path planning problems in workspaces with obstacles. A new automaton rule is presented which calculates simultaneously all shortest paths between a starting position and a target cell. Based on wave propagation, the algorithm ensures that the dynamics settles down in an equilibrium state which represents an optimal solution. Rule extensions include calculations with multiple starts and targets. The method allows applications on lattices and regular, weighted graphs of any finite dimension. In comparison with algorithms from graph theory or neural network theory, the cellular automaton approach has several advantages: Convergence towards optimal configurations is guaranteed, and the computing costs depend only linearly on the lattice size. Moreover, no floating-point calculations are involved.

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30

Böhm, W. "Lattice path counting and the theory of queues." Journal of Statistical Planning and Inference 140, no. 8 (August 2010): 2168–83. http://dx.doi.org/10.1016/j.jspi.2010.01.013.

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31

Castillo, Alejandro, Juan José Ortiz-Servin, Raúl Perusquía, and Yerania Campos Silvestre. "Fuel lattice design with Path Relinking in BWR’s." Progress in Nuclear Energy 53, no. 4 (May 2011): 368–74. http://dx.doi.org/10.1016/j.pnucene.2011.01.009.

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32

Chamizo, F., and A. Córdoba. "A path integral approach to lattice point problems." Journal de Mathématiques Pures et Appliquées 81, no. 10 (October 2002): 957–66. http://dx.doi.org/10.1016/s0021-7824(02)01263-1.

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33

Catterall, Simon, and Eric Gregory. "A lattice path integral for supersymmetric quantum mechanics." Physics Letters B 487, no. 3-4 (August 2000): 349–56. http://dx.doi.org/10.1016/s0370-2693(00)00835-2.

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34

Iyama, Osamu, Idun Reiten, Hugh Thomas, and Gordana Todorov. "Lattice structure of torsion classes for path algebras." Bulletin of the London Mathematical Society 47, no. 4 (July 14, 2015): 639–50. http://dx.doi.org/10.1112/blms/bdv041.

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35

Bonin, Joseph, Anna de Mier, and Marc Noy. "Lattice path matroids: enumerative aspects and Tutte polynomials." Journal of Combinatorial Theory, Series A 104, no. 1 (October 2003): 63–94. http://dx.doi.org/10.1016/s0097-3165(03)00122-5.

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36

Schweig, Jay. "Toric ideals of lattice path matroids and polymatroids." Journal of Pure and Applied Algebra 215, no. 11 (November 2011): 2660–65. http://dx.doi.org/10.1016/j.jpaa.2011.03.010.

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37

Bostan, A., I. Kurkova, and K. Raschel. "A human proof of Gessel’s lattice path conjecture." Transactions of the American Mathematical Society 369, no. 2 (April 14, 2016): 1365–93. http://dx.doi.org/10.1090/tran/6804.

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38

Gianvittorio, R., R. Gambini, and A. Trias. "Path calculus for Z2-Higgs lattice gauge model." Journal of Physics A: Mathematical and General 24, no. 13 (July 7, 1991): 3159–66. http://dx.doi.org/10.1088/0305-4470/24/13/027.

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39

Fox, N. Bradley. "A Lattice Path Interpretation of the Diamond Product." Annals of Combinatorics 20, no. 3 (July 8, 2016): 569–86. http://dx.doi.org/10.1007/s00026-016-0323-z.

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40

Muto, Kiyoshi, Haruo Miyazaki, Yoichi Seki, Yoshihiro Kimura, and Yukio Shibata. "Lattice path counting andM/M/c queueing systems." Queueing Systems 19, no. 1-2 (March 1995): 193–214. http://dx.doi.org/10.1007/bf01148946.

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41

Brak, R., J. W. Essam, and A. L. Owczarek. "Partial difference equation method for lattice path problems." Annals of Combinatorics 3, no. 2-4 (June 1999): 265–75. http://dx.doi.org/10.1007/bf01608787.

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42

Knauer, Kolja, Leonardo Martínez-Sandoval, and Jorge Luis Ramírez Alfonsín. "A Tutte polynomial inequality for lattice path matroids." Advances in Applied Mathematics 94 (March 2018): 23–38. http://dx.doi.org/10.1016/j.aam.2016.11.008.

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43

Jiang, Yunjiang, and Weijun Xu. "On the Number of Turns in Reduced Random Lattice Paths." Journal of Applied Probability 50, no. 2 (June 2013): 499–515. http://dx.doi.org/10.1239/jap/1371648957.

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We consider the tree-reduced path of a symmetric random walk on ℤd. It is interesting to ask about the number of turns Tn in the reduced path after n steps. This question arises from inverting the signatures of lattice paths: Tn gives an upper bound of the number of terms in the signature needed to reconstruct a ‘random’ lattice path with n steps. We show that, when n is large, the mean and variance of Tn in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for Tn, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.

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44

Seeker, Wolfgang, and Özlem Çetinoğlu. "A Graph-based Lattice Dependency Parser for Joint Morphological Segmentation and Syntactic Analysis." Transactions of the Association for Computational Linguistics 3 (December 2015): 359–73. http://dx.doi.org/10.1162/tacl_a_00144.

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Space-delimited words in Turkish and Hebrew text can be further segmented into meaningful units, but syntactic and semantic context is necessary to predict segmentation. At the same time, predicting correct syntactic structures relies on correct segmentation. We present a graph-based lattice dependency parser that operates on morphological lattices to represent different segmentations and morphological analyses for a given input sentence. The lattice parser predicts a dependency tree over a path in the lattice and thus solves the joint task of segmentation, morphological analysis, and syntactic parsing. We conduct experiments on the Turkish and the Hebrew treebank and show that the joint model outperforms three state-of-the-art pipeline systems on both data sets. Our work corroborates findings from constituency lattice parsing for Hebrew and presents the first results for full lattice parsing on Turkish.

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45

Jiang, Yunjiang, and Weijun Xu. "On the Number of Turns in Reduced Random Lattice Paths." Journal of Applied Probability 50, no. 02 (June 2013): 499–515. http://dx.doi.org/10.1017/s0021900200013528.

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We consider the tree-reduced path of a symmetric random walk on ℤ d . It is interesting to ask about the number of turns T n in the reduced path after n steps. This question arises from inverting the signatures of lattice paths: T n gives an upper bound of the number of terms in the signature needed to reconstruct a ‘random’ lattice path with n steps. We show that, when n is large, the mean and variance of T n in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for T n, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.

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46

Wheelwright, A. V., and C. A. Glasbey. "Distances between censored intersections between a square lattice and a random smooth path." Journal of Applied Probability 30, no. 1 (March 1993): 269–74. http://dx.doi.org/10.2307/3214640.

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A random smooth path of infinite length crossed a square lattice. Intersections with the lattice were censored if they lay within a threshold distance of a preceding uncensored intersection, defined by tracking along the path in one direction. The distribution of distances between consecutive uncensored intersections is derived.

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47

Wheelwright, A. V., and C. A. Glasbey. "Distances between censored intersections between a square lattice and a random smooth path." Journal of Applied Probability 30, no. 01 (March 1993): 269–74. http://dx.doi.org/10.1017/s0021900200044181.

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A random smooth path of infinite length crossed a square lattice. Intersections with the lattice were censored if they lay within a threshold distance of a preceding uncensored intersection, defined by tracking along the path in one direction. The distribution of distances between consecutive uncensored intersections is derived.

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48

BEIRL, W., H. MARKUM, and J. RIEDLER. "TWO-DIMENSIONAL LATTICE GRAVITY AS A SPIN SYSTEM." International Journal of Modern Physics C 05, no. 02 (April 1994): 359–61. http://dx.doi.org/10.1142/s0129183194000490.

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Quantum gravity is studied in the path integral formulation applying the Regge calculus. Restricting the quadratic link lengths of the originally triangular lattice the path integral can be transformed to the partition function of a spin system with higher couplings on a Kagomé lattice. Various measures acting as external field are considered. Extensions to matter fields and higher dimensions are discussed.

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49

Wen-chang, Chu. "On the lattice path method in convolution-type combinatorial identities (II)—The weighted counting function method on lattice paths." Applied Mathematics and Mechanics 10, no. 12 (December 1989): 1131–35. http://dx.doi.org/10.1007/bf02016301.

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50

Wang, Qian, and Shuang Liang Tian. "The Adjacent Vertex Distinguishing Incidence Coloring of Some Infinite Planar Graph." Applied Mechanics and Materials 543-547 (March 2014): 1769–72. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.1769.

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An adjacent vertex distinguishing incidence coloring of graph of G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors. We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of triangular lattice and infinite path, and hexagonal lattice and infinite path.

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