Статті в журналах: "Lattice theory"

1

Day, Alan. "Doubling Constructions in Lattice Theory." Canadian Journal of Mathematics 44, no. 2 (April 1, 1992): 252–69. http://dx.doi.org/10.4153/cjm-1992-017-7.

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AbstractThis paper examines the simultaneous doubling of multiple intervals of a lattice in great detail. In the case of a finite set of W-failure intervals, it is shown that there in a unique smallest lattice mapping homomorphically onto the original lattice, in which the set of W-failures is removed. A nice description of this new lattice is given. This technique is used to show that every lattice that is a bounded homomorphic image of a free lattice has a projective cover. It is also used to give a sufficient condition for a fintely presented lattice to be weakly atomic and shows that the problem of which finitely presented lattices are finite is closely related to the problem of characterizing those finite lattices with a finite W-cover.

2

Harremoës, Peter. "Entropy Inequalities for Lattices." Entropy 20, no. 10 (October 12, 2018): 784. http://dx.doi.org/10.3390/e20100784.

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We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice.

3

McCulloch, Ryan. "Finite groups with a trivial Chermak–Delgado subgroup." Journal of Group Theory 21, no. 3 (May 1, 2018): 449–61. http://dx.doi.org/10.1515/jgth-2017-0042.

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Abstract The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of G. The least element of the Chermak–Delgado lattice of G is known as the Chermak–Delgado subgroup of G. This paper concerns groups with a trivial Chermak–Delgado subgroup. We prove that if the Chermak–Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak–Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak–Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak–Delgado subgroup. We establish lattice theoretic properties of Chermak–Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain p-group lattices play a major role in the author’s constructions.

4

Ballal, Sachin, and Vilas Kharat. "Zariski topology on lattice modules." Asian-European Journal of Mathematics 08, no. 04 (November 17, 2015): 1550066. http://dx.doi.org/10.1142/s1793557115500667.

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Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text] and [Formula: see text] be the set of all prime elements in lattice modules [Formula: see text]. In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of [Formula: see text] and algebraic properties of [Formula: see text].

5

Ježek, J., P. PudláK, and J. Tůma. "On equational theories of semilattices with operators." Bulletin of the Australian Mathematical Society 42, no. 1 (August 1990): 57–70. http://dx.doi.org/10.1017/s0004972700028148.

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In 1986, Lampe presented a counterexample to the conjecture that every algebraic lattice with a compact greatest element is isomorphic to the lattice of extensions of an equational theory. In this paper we investigate equational theories of semi-lattices with operators. We construct a class of lattices containing all infinitely distributive algebraic lattices with a compact greatest element and closed under the operation of taking the parallel join, such that every element of the class is isomorphic to the lattice of equational theories, extending the theory of a semilattice with operators.

6

Bronzan, J. B. "Hamiltonian lattice gauge theory: wavefunctions on large lattices." Nuclear Physics B - Proceedings Supplements 30 (March 1993): 916–19. http://dx.doi.org/10.1016/0920-5632(93)90356-b.

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7

Futa, Yuichi, та Yasunari Shidama. "Lattice of ℤ-module". Formalized Mathematics 24, № 1 (1 березня 2016): 49–68. http://dx.doi.org/10.1515/forma-2016-0005.

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Summary In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

8

JANSEN, KARL. "LATTICE FIELD THEORY." International Journal of Modern Physics E 16, no. 09 (October 2007): 2638–79. http://dx.doi.org/10.1142/s0218301307008355.

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Starting with the example of the quantum mechanical harmonic oscillator, we develop the concept of euclidean lattice field theory. After describing Wilson's formulation of quantum chromodynamics on the lattice, we will introduce modern lattice QCD actions which greatly reduce lattice artefacts or are even chiral invariant. The substantial algorithmic improvements of the last couple of years will be shown which led to a real breakthrough for dynamical Wilson fermion simulations. Finally, we will present some results of present simulations with dynamical quarks and demonstrate that nowadays even at small values of the quark mass high precision simulation results for physical quantities can be obtained.

9

Martinelli, G. "Lattice field theory." Nuclear Physics B - Proceedings Supplements 16 (August 1990): 16–29. http://dx.doi.org/10.1016/0920-5632(90)90456-5.

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10

Capitani, S. "Lattice perturbation theory." Physics Reports 382, no. 3-5 (July 2003): 113–302. http://dx.doi.org/10.1016/s0370-1573(03)00211-4.

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11

Morningstar, Colin J. "Lattice perturbation theory." Nuclear Physics B - Proceedings Supplements 47, no. 1-3 (March 1996): 92–99. http://dx.doi.org/10.1016/0920-5632(96)00035-7.

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12

Ježek, Jaroslav, and George F. McNulty. "The existence of finitely based lower covers for finitely based equational theories." Journal of Symbolic Logic 60, no. 4 (December 1995): 1242–50. http://dx.doi.org/10.2307/2275885.

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By an equational theory we mean a set of equations from some fixed language which is closed with respect to logical consequences. We regard equations as universal sentences whose quantifier-free parts are equations between terms. In our notation, we suppress the universal quantifiers. Once a language has been fixed, the collection of all equational theories for that language is a lattice ordered by set inclusion The meet in this lattice is simply intersection; the join of a collection of equational theories is the equational theory axiomatized by the union of the collection. In this paper we prove, for languages with only finitely many fundamental operation symbols, that any nontrivial finitely axiomatizable equational theory covers some other finitely axiomatizable equational theory. In fact, our result is a little more general.There is an extensive literature concerning lattices of equational theories. These lattices are always algebraic. Compact elements of these lattices are the finitely axiomatizable equational theories. We also call them finitely based. The largest element in the lattice is compact; it is the equational theory based on the single equation x ≈ y. The smallest element of the lattice is the trivial theory consisting of tautological equations. For all but the simplest languages, the lattice of equational theories is intricate. R. McKenzie in [6] was able to prove in essence that the underlying language can be recovered from the isomorphism type of this lattice.

13

Luo, Congwen. "S-Lattice Congruences of S-Lattices." Algebra Colloquium 19, no. 03 (July 5, 2012): 465–72. http://dx.doi.org/10.1142/s1005386712000326.

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In this paper, the S-lattices are introduced as a representation of lattice-ordered monoids. The smallest S-lattice congruence induced by a relation on an S-lattice is characterized and the correspondence between the S-lattice congruences and S-ideals in an S-distributive lattice is discussed. These generalize some recent results of lattices and lattice-ordered semigroups.

14

Horváth, Eszter K., Sándor Radeleczki, Branimir Šešelja, and Andreja Tepavčević. "A Note on Cuts of Lattice-Valued Functions and Concept Lattices." Mathematica Slovaca 73, no. 3 (June 1, 2023): 583–94. http://dx.doi.org/10.1515/ms-2023-0043.

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ABSTRACT Motivated by applications of lattice-valued functions (lattice-valued fuzzy sets) in the theory of ordered structures, we investigate a special kind of posets and lattices induced by these mappings. As a framework, we use the Formal Concept Analysis in which these ordered structures can be naturally observed. We characterize the lattice of cut sets and the Dedekind-MacNeille completion of the set of images of a lattice valued function by suitable concept lattices and we give necessary and sufficient conditions under which these lattices coincide. In addition, we give conditions under which the lattice of cuts is completely distributive.

15

Borcherds, Richard E. "Lattices like the Leech lattice." Journal of Algebra 130, no. 1 (April 1990): 219–34. http://dx.doi.org/10.1016/0021-8693(90)90110-a.

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16

Pardo-Guerra, Sebastián, Hugo Alberto Rincón-Mejía, and Manuel Gerardo Zorrilla-Noriega. "Some isomorphic big lattices and some properties of lattice preradicals." Journal of Algebra and Its Applications 19, no. 07 (July 24, 2019): 2050140. http://dx.doi.org/10.1142/s0219498820501406.

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According to Albu and Iosif, [2, Definition 1.1] a lattice preradical is a subfunctor of the identity functor on the category [Formula: see text] of linear modular lattices, whose objects are the complete modular lattices and whose morphisms are linear morphisms. In this paper, we describe some big lattices which are isomorphic to the big lattice of lattice preradicals and we study the four classical operations that occur in the lattice of preradicals of modules over a ring [Formula: see text], namely, the join, the meet, the product and the coproduct. We show that some results about the lattice of module preradicals can be extended to the lattice of lattice preradicals. In particular, we show the existence of the equalizer, the annihilator, the coequalizer and the totalizer for a lattice preradical [Formula: see text], as well as some of their properties.

17

NEBE, GABRIELE. "ON AUTOMORPHISMS OF EXTREMAL EVEN UNIMODULAR LATTICES." International Journal of Number Theory 09, no. 08 (December 2013): 1933–59. http://dx.doi.org/10.1142/s179304211350067x.

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The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are determined using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with φ(m) > 24. In particular the lattice P48nis the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.

18

Pliev, M. A. "Каждая латеральная полоса является ядром положительного ортогонально аддитивного оператора". Владикавказский математический журнал, № 4 (23 грудня 2021): 115–18. http://dx.doi.org/10.46698/e4075-8887-4097-s.

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{In this paper we continue a study of relationships between the lateral partial order $\sqsubseteq$ in a vector lattice (the relation $x \sqsubseteq y$ means that $x$ is a fragment of $y$) and the theory of orthogonally additive operators on vector lattices. It was shown in~\cite{pMPP} that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice $E$ and a lateral band $G$ of~$E$, there exists a vector lattice~$F$ and a positive, disjointness preserving orthogonally additive operator $T \colon E \to F$ such that ${\rm ker} \, T = G$. As a consequence, we partially resolve the following open problem suggested in \cite{pMPP}: Are there a vector lattice~$E$ and a lateral ideal in $E$ which is not equal to the kernel of any positive orthogonally additive operator $T\colon E\to F$ for any vector lattice $F$?

19

Decker, Karsten M., and Philippe de Forcrand. "Pure SU(2) lattice gauge theory on 324 lattices." Nuclear Physics B - Proceedings Supplements 17 (September 1990): 567–70. http://dx.doi.org/10.1016/0920-5632(90)90315-l.

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20

Sinclair, R. "Calculations on infinite lattices applied to lattice gauge theory." Physical Review D 42, no. 12 (December 15, 1990): 4182–85. http://dx.doi.org/10.1103/physrevd.42.4182.

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21

Frapolli, Nicolò, Shyam Chikatamarla, and Ilya Karlin. "Theory, Analysis, and Applications of the Entropic Lattice Boltzmann Model for Compressible Flows." Entropy 22, no. 3 (March 24, 2020): 370. http://dx.doi.org/10.3390/e22030370.

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The entropic lattice Boltzmann method for the simulation of compressible flows is studied in detail and new opportunities for extending operating range are explored. We address limitations on the maximum Mach number and temperature range allowed for a given lattice. Solutions to both these problems are presented by modifying the original lattices without increasing the number of discrete velocities and without altering the numerical algorithm. In order to increase the Mach number, we employ shifted lattices while the magnitude of lattice speeds is increased in order to extend the temperature range. Accuracy and efficiency of the shifted lattices are demonstrated with simulations of the supersonic flow field around a diamond-shaped and NACA0012 airfoil, the subsonic, transonic, and supersonic flow field around the Busemann biplane, and the interaction of vortices with a planar shock wave. For the lattices with extended temperature range, the model is validated with the simulation of the Richtmyer–Meshkov instability. We also discuss some key ideas of how to reduce the number of discrete speeds in three-dimensional simulations by pruning of the higher-order lattices, and introduce a new construction of the corresponding guided equilibrium by entropy minimization.

22

Hartung, Tobias, Karl Jansen, Frances Y. Kuo, Hernan Leövey, Dirk Nuyens, and Ian H. Sloan. "Lattice meets lattice: Application of lattice cubature to models in lattice gauge theory." Journal of Computational Physics 443 (October 2021): 110527. http://dx.doi.org/10.1016/j.jcp.2021.110527.

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23

Han, Bao Chuan, Ya Jun Du, Chang Wang, and Jing Xu. "A Concept Lattice Merger Approach for Ontology Construction." Advanced Materials Research 181-182 (January 2011): 667–72. http://dx.doi.org/10.4028/www.scientific.net/amr.181-182.667.

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The method of merging concept lattice in domain ontology construction can describe the implicit concepts and relationships between concepts more appropriately for semantic representation and query match. In order to enrich semantic query, the paper intends to apply the theory of Formal Concept Analysis (FCA) to establish source concept lattices, through which the domain concepts are extracted from source concept lattices to generate the optimized concept lattice. Then, the ontology tree is generated by lattice mapping ontology algorithm (LMOA) combing some hierarchical relations in the optimized concept lattice. The experiment proves that the domain ontology can be achieved effectively by merging concept lattices and provide the semantic relations more precisely.

24

Han, Bao Chuan, Ya Jun Du, Chang Wang, and Jing Xu. "A Concept Lattice Merger Approach for Ontology Construction." Advanced Materials Research 181-182 (January 2011): 754–59. http://dx.doi.org/10.4028/www.scientific.net/amr.181-182.754.

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The method of merging concept lattice in domain ontology construction can describe the implicit concepts and relationships between concepts more appropriately for semantic representation and query match. In order to enrich semantic query, the paper intends to apply the theory of Formal Concept Analysis (FCA) to establish source concept lattices, through which the domain concepts are extracted from source concept lattices to generate the optimized concept lattice. Then, the ontology tree is generated by lattice mapping ontology algorithm (LMOA) combing some hierarchical relations in the optimized concept lattice. The experiment proves that the domain ontology can be achieved effectively by merging concept lattices and provide the semantic relations more precisely.

25

Periwal, Vipul. "Improving lattice perturbation theory." Physical Review D 53, no. 5 (March 1, 1996): 2605–9. http://dx.doi.org/10.1103/physrevd.53.2605.

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26

Wetterich, C. "Scalar lattice gauge theory." Nuclear Physics B 876, no. 1 (November 2013): 147–86. http://dx.doi.org/10.1016/j.nuclphysb.2013.08.004.

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27

Wetterich, C. "Linear lattice gauge theory." Nuclear Physics B 884 (July 2014): 44–65. http://dx.doi.org/10.1016/j.nuclphysb.2014.04.002.

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28

Lepage, G. Peter, and Paul B. Mackenzie. "Renormalized lattice perturbation theory." Nuclear Physics B - Proceedings Supplements 20 (May 1991): 173–76. http://dx.doi.org/10.1016/0920-5632(91)90902-q.

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29

MÃÂ¼nster, Gernot. "Lattice quantum field theory." Scholarpedia 5, no. 12 (2010): 8613. http://dx.doi.org/10.4249/scholarpedia.8613.

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30

Iseki, K. "Contribution to lattice theory." Publicationes Mathematicae Debrecen 2, no. 3-4 (July 1, 2022): 194–203. http://dx.doi.org/10.5486/pmd.1952.2.3-4.07.

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31

Dimm, W., G. Peter Lepage, and Paul B. Mackenzie. "Nonperturbative “lattice perturbation theory”." Nuclear Physics B - Proceedings Supplements 42, no. 1-3 (April 1995): 403–5. http://dx.doi.org/10.1016/0920-5632(95)00263-9.

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32

Freese, Ralph. "Directions in lattice theory." Algebra Universalis 31, no. 3 (September 1994): 416–29. http://dx.doi.org/10.1007/bf01221796.

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33

CIACH, A., and G. STELL. "MESOSCOPIC FIELD THEORY OF IONIC SYSTEMS." International Journal of Modern Physics B 19, no. 21 (August 20, 2005): 3309–43. http://dx.doi.org/10.1142/s0217979205032176.

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A mesoscopic field theory for the primitive model of ionic systems with additional, short-range interactions is presented. Generic models in continuum space and with positions of the ions restricted to lattice sites of various lattices are described in detail. We describe briefly the field-theoretic methods and review the foundations of the mesoscopic description. The types of phase diagrams predicted by our theory for different versions of the model are presented and discussed. They all agree with recent simulations. On the quantitative level our theory yields an RPM tricritical-point location on the sc lattice which is in good agreement with the simulation results. Arguments indicating that the critical point in the RPM belongs to the Ising universality class are given.

34

de la Maza, Ana Cecilia, and Remo Moresi. "Hermitean (semi) lattices and Rolf’s lattice." Algebra universalis 66, no. 1-2 (August 30, 2011): 49–62. http://dx.doi.org/10.1007/s00012-011-0141-4.

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35

WEHRUNG, FRIEDRICH. "FROM JOIN-IRREDUCIBLES TO DIMENSION THEORY FOR LATTICES WITH CHAIN CONDITIONS." Journal of Algebra and Its Applications 01, no. 02 (June 2002): 215–42. http://dx.doi.org/10.1142/s0219498802000148.

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For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J (L) of join-irreducible elements of L and the join-dependency relation DL on J (L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form: Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations [Formula: see text] As a consequence of this, we obtain the following results: Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom [Formula: see text] Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A □ B of A and B is join-semidistributive, and the following isomorphism holds: [Formula: see text] where ⊗ denotes the tensor product of commutative monoids.

36

Liu, Keh-Fei. "Many body theory and lattice gauge theory." Physics Reports 242, no. 4-6 (July 1994): 463–69. http://dx.doi.org/10.1016/0370-1573(94)90179-1.

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37

Estaji, A. A., M. R. Hooshmandasl, and B. Davvaz. "Rough set theory applied to lattice theory." Information Sciences 200 (October 2012): 108–22. http://dx.doi.org/10.1016/j.ins.2012.02.060.

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38

Chashchin, Georgy Sergeevich. "Lattice Boltzmann method: simulation of isothermal low-speed flows." Keldysh Institute Preprints, no. 99 (2021): 1–31. http://dx.doi.org/10.20948/prepr-2021-99.

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In this work, lattice Boltzmann method on standard lattices was descript as one of the modern method of computation fluid dynamics. The article has main theorems, which prove computational algorithm, different type’s boundary conditions and defect in Galilean invariance. Moreover, the paper has some theoretical background about physical kinetic theory, Hermite polynomials and numeric integration. Here has not any new scientist discoveries, but has explanation of basic lattice Boltzmann theory.

39

Symonds, Peter. "Relative characters for H-projective RG-lattices." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 2 (September 1988): 207–13. http://dx.doi.org/10.1017/s0305004100065397.

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If G is a group with a subgroup H and R is a Dedekind domain, then an H-projective RG-lattice is an RG-lattice that is a direct summand of an induced lattice for some RH-lattice N: they have been studied extensively in the context of modular representation theory. If H is the trivial group these are the projective lattices. We define a relative character χG/H on H-projective lattices, which in the case H = 1 is equivalent to the Hattori–Stallings trace for projective lattices (see [5, 8]), and in the case H = G is the ordinary character. These characters can be used to show that the R-ranks of certain H-projective lattices must be divisible by some specified number, generalizing some well-known results: cf. Corollary 3·6. If for example we take R = ℤ, then |G/H| divides the ℤ-rank of any H-projective ℤG-lattice.

40

Kumar, Kamlesh. "Lattice Theory of Fourth Order Elastic Constants of Primitive Lattices." Bulletin of Pure & Applied Sciences- Physics 40d, no. 2 (2021): 129–31. http://dx.doi.org/10.5958/2320-3218.2021.00020.8.

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41

McCune, W., and R. Padmanabhan. "Single identities for lattice theory and for weakly associative lattices." Algebra Universalis 36, no. 4 (December 1996): 436–49. http://dx.doi.org/10.1007/bf01233914.

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42

Goyal, S. C., and K. Kumar. "Lattice Theory of Fourth-Order Elastic Constants of Primitive Lattices." physica status solidi (b) 131, no. 2 (October 1, 1985): 451–57. http://dx.doi.org/10.1002/pssb.2221310206.

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43

MONAHAN, C. J. "THE BEAUTY OF LATTICE PERTURBATION THEORY: THE ROLE OF LATTICE PERTURBATION THEORY IN B PHYSICS." Modern Physics Letters A 27, no. 37 (November 25, 2012): 1230040. http://dx.doi.org/10.1142/s0217732312300406.

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As new experimental data arrive from the LHC the prospect of indirectly detecting new physics through precision tests of the Standard Model grows more exciting. Precise experimental and theoretical inputs are required to test the unitarity of the CKM matrix and to search for new physics effects in rare decays. Lattice QCD calculations of non-perturbative inputs have reached a precision at the level of a few percent; in many cases aided by the use of lattice perturbation theory. This review examines the role of lattice perturbation theory in B physics calculations on the lattice in the context of two questions: how is lattice perturbation theory used in the different heavy quark formalisms implemented by the major lattice collaborations? And what role does lattice perturbation theory play in determinations of non-perturbative contributions to the physical processes at the heart of the search for new physics? Framing and addressing these questions reveals that lattice perturbation theory is a tool with a spectrum of applications in lattice B physics.

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Ge, Mo-Lin, Liangzhong Hu, and Yiwen Wang. "KNOT THEORY, PARTITION FUNCTION AND FRACTALS." Journal of Knot Theory and Its Ramifications 05, no. 01 (February 1996): 37–54. http://dx.doi.org/10.1142/s0218216596000047.

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In this paper we first provide the open chain and the closed chain method to calculate the partition functions of the typical fractal lattices, i.e. a special kind of Sierpinski carpets(SC) and the triangular Sierpinski gaskets(SG). We then apply knot theory to fractal lattices by changing lattice graphs into link diagrams according to the interaction models, and explicitly obtain the partition functions of a special SC for the edge interaction models. These partition functions are also the knot invariants of the corresponding link diagrams. This is the first time that topology enters into fractals.

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Demonet, Laurent, Osamu Iyama, Nathan Reading, Idun Reiten та Hugh Thomas. "Lattice theory of torsion classes: Beyond 𝜏-tilting theory". Transactions of the American Mathematical Society, Series B 10, № 18 (25 квітня 2023): 542–612. http://dx.doi.org/10.1090/btran/100.

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The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set t o r s A \mathsf {tors} A of torsion classes over a finite-dimensional algebra A A . We show that t o r s A \mathsf {tors} A is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of t o r s A \mathsf {tors} A . In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that t o r s A \mathsf {tors} A is completely congruence uniform. When I I is a two-sided ideal of A A , t o r s ( A / I ) \mathsf {tors} (A/I) is a lattice quotient of t o r s A \mathsf {tors} A which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of t o r s A \mathsf {tors} A that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras Π \Pi , for which t o r s Π \mathsf {tors} \Pi is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between t o r s k Q \mathsf {tors} k Q and the Cambrian lattice when Q Q is a Dynkin quiver. We also prove that, in type A A , the algebraic quotients of t o r s Π \mathsf {tors} \Pi are exactly its Hasse-regular lattice quotients.

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BRAZHNYI, V. A., and V. V. KONOTOP. "THEORY OF NONLINEAR MATTER WAVES IN OPTICAL LATTICES." Modern Physics Letters B 18, no. 14 (June 10, 2004): 627–51. http://dx.doi.org/10.1142/s0217984904007190.

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We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

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Tewary, V. K., and Robb Thomson. "Lattice statics of interfaces and interfacial cracks in bimaterial solids." Journal of Materials Research 7, no. 4 (April 1992): 1018–28. http://dx.doi.org/10.1557/jmr.1992.1018.

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A method for calculating lattice statics Green's function is described for a bimaterial lattice or a bicrystal containing a plane interface. The method involves creation of two half space lattices containing free surfaces and then joining them to form a bicrystal. The two half space lattices may have different structures as in a two-phase bicrystal or may be of the same type but joined at different orientations to form a grain boundary interface. The method is quite general but, in this paper, has been applied only to a simple model bicrystal formed by two simple cubic lattices with nearest neighbor interactions. The bimaterial Green's function is modified to account for an interfacial crack that is used to calculate the displacement field due to an applied external force. It is found that the displacement field, as calculated by using the lattice theory, does not have the unphysical oscillations predicted by the continuum theory.

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Bullivant, Alex, Marcos Calçada, Zoltán Kádár, João Faria Martins, and Paul Martin. "Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry." Reviews in Mathematical Physics 32, no. 04 (November 4, 2019): 2050011. http://dx.doi.org/10.1142/s0129055x20500117.

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Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.

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Grabowski, Adam. "Stone Lattices." Formalized Mathematics 23, no. 4 (December 1, 2015): 387–96. http://dx.doi.org/10.1515/forma-2015-0031.

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Summary The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. The core of the paper is of course the idea of Stone identity $$a^* \sqcup a^{**} = {\rm{T}},$$ which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11].

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NEGRI, SARA, and JAN VON PLATO. "Proof systems for lattice theory." Mathematical Structures in Computer Science 14, no. 4 (August 2004): 507–26. http://dx.doi.org/10.1017/s0960129504004244.

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A formulation of lattice theory as a system of rules added to sequent calculus is given. The analysis of proofs for the contraction-free calculus of classical predicate logic known as G3c extends to derivations with the mathematical rules of lattice theory. It is shown that minimal derivations of quantifier-free sequents enjoy a subterm property: all terms in such derivations are terms in the endsequent.An alternative formulation of lattice theory as a system of rules in natural deduction style is given, both with explicit meet and join constructions and as a relational theory with existence axioms. A subterm property for the latter extends the standard decidable classes of quantificational formulas of pure predicate calculus to lattice theory.

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