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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.
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.
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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.
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.
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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.
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].
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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.
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.
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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.
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].
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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.
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.
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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.
Race, David M. (David Michael). "Consistency in Lattices." Thesis, North Texas State University, 1986. https://digital.library.unt.edu/ark:/67531/metadc331688/.
Let L be a lattice. For x ∈ L, we say x is a consistent join-irreducible if x V y is a join-irreducible of the lattice [y,1] for all y in L. We say L is consistent if every join-irreducible of L is consistent. In this dissertation, we study the notion of consistent elements in semimodular lattices.
The thesis analyzes some aspects of the theory of distributive lattices, particularly two representation theorems: Birkhoff's representation theorem for finite distributive lattices and Stone's representation theorem for infinite distributive lattices.
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Endres, Michael G. "Topics in lattice field theory /." Thesis, Connect to this title online; UW restricted, 2007. http://hdl.handle.net/1773/9713.
Michels, Amanda Therese. "Aspects of lattice gauge theory." Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.297217.
The use of computers to produce a user-friendly safe environment is an important area of research in computer science. This dissertation investigates how computers can be used to create an interactive environment for lattice theory. The dissertation is divided into three parts. Chapters two and three discuss mathematical aspects of lattice theory, chapter four describes methods of representing and displaying distributive lattices and chapters five, six and seven describe a definitive based environment for lattice theory. Chapter two investigates lattice congruences and pre-orders and demonstrates that any lattice congruence or pre-order can be determined by sets of join-irreducibles. By this correspondence it is shown that lattice operations in a quotient lattice can be calculated by set operations on the join-irreducibles that determine the congruence. This alternative characterisation is used in chapter three to obtain closed forms for all replacements of the form "h can replace g when computing an element f", and hence extends the results of Beynon and Dunne into general lattices. Chapter four investigates methods of representing and displaying distributive lattices. Techniques for generating Hasse diagrams of distributive lattices are discussed and two methods for performing calculations on free distributive lattices and their respective advantages are given. Chapters five and six compare procedural and functional based notations with computer environments based on definitive notations for creating an interactive environment for studying set theory. Chapter seven introduces a definitive based language called Pecan for creating an interactive environment for lattice theory. The results of chapters two and three are applied so that quotients, congruences and homomorphic images of lattices can be calculated efficiently.
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Craig, Andrew Philip Knott. "Lattice-valued uniform convergence spaces the case of enriched lattices." Thesis, Rhodes University, 2008. http://hdl.handle.net/10962/d1005225.
Using a pseudo-bisymmetric enriched cl-premonoid as the underlying lattice, we examine different categories of lattice-valued spaces. Lattice-valued topological spaces, uniform spaces and limit spaces are described, and we produce a new definition of stratified lattice-valued uniform convergence spaces in this generalised lattice context. We show that the category of stratified L-uniform convergence spaces is topological, and that the forgetful functor preserves initial constructions for the underlying stratified L-limit space. For the case of L a complete Heyting algebra, it is shown that the category of stratified L-uniform convergence spaces is cartesian closed.
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Pugh, David John Rhydwyn. "Topological structures in lattice gauge theory." Thesis, University of Oxford, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.279896.
Thesis (Ph.D.)--Boston University In this dissertation I use lattice gauge theory to study models of electroweak symmetry breaking that involve new strong dynamics.
Electroweak symmetry breaking (EWSB) is the process by which elementary particles acquire mass. First proposed in the 1960s, this process has been clearly established by experiments, and can now be considered a law of nature. However, the physics underlying EWSB is still unknown, and understanding it remains a central challenge in particle physics today. A natural possibility is that EWSB is driven by the dynamics of some new, strongly-interacting force. Strong interactions invalidate the standard analytical approach of perturbation theory, making these models difficult to study.
Lattice gauge theory is the premier method for obtaining quantitatively-reliable, nonperturbative predictions from strongly-interacting theories. In this approach, we replace spacetime by a regular, finite grid of discrete sites connected by links. The fields and interactions described by the theory are likewise discretized, and defined on the lattice so that we recover the original theory in continuous spacetime on an infinitely large lattice with sites infinitesimally close together. The finite number of degrees of freedom in the discretized system lets us simulate the lattice theory using high-performance computing.
Lattice gauge theory has long been applied to quantum chromodynamics, the theory of strong nuclear interactions. Using lattice gauge theory to study dynamical EWSB, as I do in this dissertation, is a new and exciting application of these methods. Of particular interest is non-perturbative lattice calculation of the electroweak S parameter. Experimentally S ~ -0.15(10), which tightly constrains dynamical EWSB. On the lattice, I extract S from the momentum-dependence of vector and axial-vector current correlators. I created and applied computer programs to calculate these correlators and analyze them to determine S. I also calculated the masses and other properties of the new particles predicted by these theories.
I find S > 0.1 in the specific theories I study. Although this result still disagrees with experiment, it is much closer to the experimental value than is the conventional wisdom S > 0.3. These results encourage further lattice studies to search for experimentally viable strongly-interacting theories of EWSB.
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Schenk, Stefan. "Density functional theory on a lattice." kostenfrei, 2009. http://d-nb.info/998385956/34.
AbstractIn this chapter, we introduce the basic random theory of lattice, including Fourier transform, discrete Gauss measure, smoothing parameter and some properties of discrete Gauss distribution. Random lattice is a new research topic in lattice theory. However, only a special class of random lattices named Gauss lattice has been defined and studied. We will introduce Gauss lattice, define the smoothing parameter on Gauss lattice, and calculate the statistical distance based on the smoothing parameter
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Al-Haj Baddar, Sherenaz W., and Kenneth E. Batcher. "Lattice Theory." In Designing Sorting Networks, 61–71. New York, NY: Springer New York, 2011. http://dx.doi.org/10.1007/978-1-4614-1851-1_10.
Ritter, Gerhard X., and Gonzalo Urcid. "Lattice Theory." In Introduction to Lattice Algebra, 81–109. Boca Raton: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003154242-3.
Stone, Michael. "Lattice Field Theory." In Graduate Texts in Contemporary Physics, 185–200. New York, NY: Springer New York, 2000. http://dx.doi.org/10.1007/978-1-4612-0507-4_15.
Grätzer, George. "Congruences and Ideals." In General Lattice Theory, 169–210. Basel: Birkhäuser Basel, 2003. http://dx.doi.org/10.1007/978-3-0348-9326-8_3.
Тези доповідей конференцій з теми "Lattice theory":
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Monahan, Christopher. "Automated Lattice Perturbation Theory." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Trieste, Italy: Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0021.
Lambrou, Eliana, Luigi Del Debbio, R. D. Kenway, and Enrico Rinaldi. "Searching for a continuum 4D field theory arising from a 5D non-abelian gauge theory." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Trieste, Italy: Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0107.
Bursa, F., and Michael Kroyter. "Lattice String Field Theory." In The XXVIII International Symposium on Lattice Field Theory. Trieste, Italy: Sissa Medialab, 2011. http://dx.doi.org/10.22323/1.105.0047.
Kieburg, Mario, Jacobus Verbaarschot, and Savvas Zafeiropoulos. "A classification of 2-dim Lattice Theory." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Trieste, Italy: Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0337.
Shao, Yingchao, Li Fu, Fei Hao, and Keyun Qin. "Rough Lattice: A Combination with the Lattice Theory and the Rough Set Theory." In 2016 International Conference on Mechatronics, Control and Automation Engineering. Paris, France: Atlantis Press, 2016. http://dx.doi.org/10.2991/mcae-16.2016.23.
Bietenholz, Wolfgang, Ivan Hip, and David Landa-Marban. "Spectral Properties of a 2d IR Conformal Theory." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Trieste, Italy: Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0486.
Zubkov, Mikhail. "Gauge theory of Lorentz group on the lattice." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Trieste, Italy: Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0095.
Veernala, Aarti, and Simon Catterall. "Four Fermion Interactions in Non Abelian Gauge Theory." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Trieste, Italy: Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0108.
Bergner, Georg, Jens Langelage, and Owe Philipsen. "Effective lattice theory for finite temperature Yang Mills." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Trieste, Italy: Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0133.
Hesse, Dirk, Stefan Sint, Francesco Di Renzo, Mattia Dalla Brida, and Michele Brambilla. "The Schrödinger Functional in Numerical Stochastic Perturbation Theory." In 31st International Symposium on Lattice Field Theory LATTICE 2013. Trieste, Italy: Sissa Medialab, 2014. http://dx.doi.org/10.22323/1.187.0325.
McCune, W., and R. Padmanabhan. Single identities for lattice theory and for weakly associative lattices. Office of Scientific and Technical Information (OSTI), March 1995. http://dx.doi.org/10.2172/510566.
Yee, Ken. Lattice gaugefixing and other optics in lattice gauge theory. Office of Scientific and Technical Information (OSTI), June 1992. http://dx.doi.org/10.2172/10156563.
Yee, Ken. Lattice gaugefixing and other optics in lattice gauge theory. Office of Scientific and Technical Information (OSTI), June 1992. http://dx.doi.org/10.2172/5082303.
Becher, Thomas G. Continuum methods in lattice perturbation theory. Office of Scientific and Technical Information (OSTI), November 2002. http://dx.doi.org/10.2172/808671.
Hasslacher, B. Lattice gas hydrodynamics: Theory and simulations. Office of Scientific and Technical Information (OSTI), January 1993. http://dx.doi.org/10.2172/6441616.
Hasslacher, B. Lattice gas hydrodynamics: Theory and simulations. Office of Scientific and Technical Information (OSTI), January 1993. http://dx.doi.org/10.2172/6590163.
Brower, Richard C. National Computational Infrastructure for Lattice Gauge Theory. Office of Scientific and Technical Information (OSTI), April 2014. http://dx.doi.org/10.2172/1127446.
Negele, John W. National Computational Infrastructure for Lattice Gauge Theory. Office of Scientific and Technical Information (OSTI), June 2012. http://dx.doi.org/10.2172/1165874.
Reed, Daniel, A. National Computational Infrastructure for Lattice Gauge Theory. Office of Scientific and Technical Information (OSTI), May 2008. http://dx.doi.org/10.2172/951263.
Creutz, M. Lattice gauge theory and Monte Carlo methods. Office of Scientific and Technical Information (OSTI), November 1988. http://dx.doi.org/10.2172/6530895.
