Ph. D.

Ferromagnetic abelian lattice systems are the topic of this paper. Namely, at each site of ZV-invariant lattice is placed a finite abelian group. The interaction is given by any real, negative definite, and translation invariant function on the space of configurations.Algebraic structure of the system is investigated. This allows a complete · description of the family of equilibrium states for given. interaction at low temperatures. At the same time it is proven that the low temperature expansion for Gibbs free energy is analytic. It is also shown that it is not necessary to consider gauge models in the case of Zm on ZV lattice. / Ph. D.

The change in the number of phase transitions for perturbations of finite range interactions is studied. A Monte-Carlo simulation was performed for a translation invariant spin 1/2 ferromagnetic model in Z² with fundamental bonds A = {(0,0);(0,1)} B = {(0,0);(2,0)} C = {(0,0);(0,1);(1,1);(1,0)} The model exhibits one phase transition if the coupling constant J(A) is zero, but two phase transitions were found when J(A) is non zero and small enough. The generalization of this situation is provided by a construction, due to J. Slawny, which through a sequence of progressively smaller perturbations yields models with an arbitrary minimum number of phase transitions. However, such construction requires the existence of interactions with one fundamental bond such that for all values of the coupling constants the Gibbs state is unique even when the interaction is perturbed by an arbitrary finite range perturbation of small enough norm. In this work it is proven that such property is exhibited by some translation invariant systems in Z^ν with finite state space at each point. The proof applies to models with real interactions and whose fundamental bonds are all multiple of a single bond which is of prime order and which is obtained as the product—in the group ring structure of the dual space—of one dimensional bonds whose non trivial projections at each lattice site are unique. The proof is based on the Dobrushin-Pecherski criterion concerning the uniqueness of Gibbs states under perturbations. Such criterion is restated so that only transition functions on sets of simple geometry are involved. In addition, an algebraic characterization is presented for the set of Gibbs states for ferromagnetic systems for which the state space at each lattice site is a compact abelian group. This is a generalization of the theory originally introduced by Slawny for spin 1/2 ferromagnetic models and later extended by Pfister to ferromagnetic models for which the state space at each point is a finite product of tori and finite abelian groups. / Ph. D.

Call number: LD2668 .T4 1985 E39 / Master of Science

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The main objective of our study is enumerating spanning trees (G) and perfect matchings PM(G) on graphs G and lattices L. We demonstrate two methods of enumerating spanning trees of any connected graph, namely the matrix-tree theorem and as a special value of the Tutte polynomial T(G; x; y). We present a general method for counting spanning trees on lattices in d 2 dimensions. In particular we apply this method on the following regular lattices with d = 2: rectangular, triangular, honeycomb, kagomé, diced, 9 3 lattice and its dual lattice to derive a explicit formulas for the number of spanning trees of these lattices of finite sizes. Regarding the problem of enumerating of perfect matchings, we prove Cayley’s theorem which relates the Pfaffian of a skew symmetric matrix to its determinant. Using this and defining the Pfaffian orientation on a planar graph, we derive explicit formula for the number of perfect matchings on the following planar lattices; rectangular, honeycomb and triangular. For each of these lattices, we also determine the bulk limit or thermodynamic limit, which is a natural measure of the rate of growth of the number of spanning trees (L) and the number of perfect matchings PM(L). An algorithm is implemented in the computer algebra system SAGE to count the number of spanning trees as well as the number of perfect matchings of the lattices studied. / AFRIKAANSE OPSOMMING: Die hoofdoel van ons studie is die aftelling van spanbome (G) en volkome afparings PM(G) in grafieke G en roosters L. Ons beskou twee metodes om spanbome in ’n samehangende grafiek af te tel, naamlik deur middel van die matriks-boom-stelling, en as ’n spesiale waarde van die Tutte polinoom T(G; x; y). Ons behandel ’n algemene metode om spanbome in roosters in d 2 dimensies af te tel. In die besonder pas ons hierdie metode toe op die volgende reguliere roosters met d = 2: reghoekig, driehoekig, heuningkoek, kagomé, blokkies, 9 3 rooster en sy duale rooster. Ons bepaal eksplisiete formules vir die aantal spanbome in hierdie roosters van eindige grootte. Wat die aftelling van volkome afparings aanbetref, gee ons ’n bewys van Cayley se stelling wat die Pfaffiaan van ’n skeefsimmetriese matriks met sy determinant verbind. Met behulp van hierdie stelling en Pfaffiaanse oriënterings van planare grafieke bepaal ons eksplisiete formules vir die aantal volkome afparings in die volgende planare roosters: reghoekig, driehoekig, heuningkoek. Vir elk van hierdie roosters word ook die “grootmaat limiet” (of termodinamiese limiet) bepaal, wat ’n natuurlike maat vir die groeitempo van die aantaal spanbome (L) en die aantal volkome afparings PM(L) voorstel. ’n Algoritme is in die rekenaaralgebra-stelsel SAGE geimplementeer om die aantal spanboome asook die aantal volkome afparings in die toepaslike roosters af te tel.

Logical deduction and abstraction from detail are fundamental, yet distinct aspects of reasoning about programs. This dissertation shows that the combination of logic and abstract interpretation enables a unified and simple treatment of several theoretical and practical topics which encompass the model theory of temporal logics, the analysis of satisfiability solvers, and the construction of Craig interpolants. In each case, the combination of logic and abstract interpretation leads to more general results, simpler proofs, and a unification of ideas from seemingly disparate fields. The first contribution of this dissertation is a framework for combining temporal logics and abstraction. Chapter 3 introduces trace algebras, a new lattice-based semantics for linear and branching time logics. A new representation theorem shows that trace algebras precisely capture the standard trace-based semantics of temporal logics. We prove additional representation theorems to show how structures that have been independently discovered in static program analysis, model checking, and algebraic modal logic, can be derived from trace algebras by abstract interpretation. The second contribution of this dissertation is a framework for proving when two lattice-based algebras satisfy the same logical properties. Chapter 5 introduces functions called subsumption and bisubsumption and shows that these functions characterise logical equivalence of two algebras. We also characterise subsumption and bisubsumption using fixed points and finitary logics. We prove a representation theorem and apply it to derive the transition system analogues of subsumption and bisubsumption. These analogues strictly generalise the well studied notions of simulation and bisimulation. Our fixed point characterisations also provide a technique to construct property preserving abstractions. The third contribution of this dissertation is abstract satisfaction, an abstract interpretation framework for the design and analysis of satisfiability procedures. We show that formula satisfiability has several different fixed point characterisations, and that satisfiability procedures can be understood as abstract interpreters. Our main result is that the propagation routine in modern sat solvers is a greatest fixed point computation involving abstract transformers, and that clause learning is an abstract transformer for a form of negation. The final contribution of this dissertation is an abstract interpretation based analysis of algorithms for constructing Craig interpolants. We identify and analyse a lattice of interpolant constructions. Our main result is that existing algorithms are two of three optimal abstractions of this lattice. A second new result we derive in this framework is that the lattice of interpolation algorithms can be ordered by logical strength, so that there is a strongest and a weakest possible construction.

by Kwok Siu Yu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 48-49). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background of switching networks --- p.1 / Chapter 1.2 --- Nonblocking properties of 3-stage networks --- p.5 / Chapter 1.3 --- Wide-sense nonblocking networks --- p.10 / Chapter 1.4 --- Routing algorithms by packing --- p.12 / Chapter 2 --- The Concept of the Upper Ideals --- p.15 / Chapter 3 --- "Routing algorithm over the network [6x10, 3x3, 10x6]" --- p.26 / Chapter 4 --- Simulation Program (SP) --- p.30 / Chapter 5 --- "Nonexistence of routing algorithm over the network [5x8, 3x3, 8x5]" --- p.35 / Chapter 6 --- Packing algorithms --- p.42 / Chapter 7 --- Summary and directions of further study --- p.47

In the chapter 3, we study ferromagnetic fluctuations on two types of bilayer triangular lattices by the single-band Hubbard model. We start from the tight-binding model to obtain energy spectrum, the density of sates, and the spin susceptibility. With finite Coulomb interaction turned on, we apply the random phase approximation and use the determinant quantum Monte Carlo method to study spin susceptibility for the two bilayer triangular lattices and make comparisons of their magnetic properties. The effects of the interlayer coupling is also examined in detail. / In the chapter 4, we addresses the issue of the ferromagnetism in graphene-based samples. To study magnetic correlations in graphene, we systematically carry out quantum Monte Carlo simulations of the Hubbard model on a honeycomb lattice. In the filling region below the Van Hove singularity, the system shows a short-range ferromagnetic correlation, which is slightly strengthened by the on-site Coulomb interaction and markedly by the next-nearest-neighbor hopping integral. The ferromagnetic properties depend on the electron filling strongly, which may be manipulated by the electric gate. Due to its resultant high controllability of ferromagnetism, graphene-based samples may facilitate the new development of many applications. / In the chapter 5, we examined theoretically the magnetism of impurity adatoms in graphene by quantum Monte Carlo simulation technique based on Hirsch-Fye algorithm. When tuning the Fermi energy of graphene by gate voltage with available experiments, the values of occupancy and local moment for impurity can be changed. Furthermore, with medium and large hybridizations between impurity and graphene atoms, the local moment can be switched on and off by Kondo effects. We also use maximum entropy method to study the spectral density from Green's function for impurity, and we find very unconventional behaviors which are absolutely different from the cases in the normal metal. These signatures of spectral density enlarge the possibility for controlling the impurity magnetism by gate voltage. / In this research thesis, we mainly study three strongly correlated systems: Hubbard model in bilayer triangular lattice which corresponds to the real material of NaxCoO2 · yH 2O, strong-interaction electrons in graphene system and Anderson impurity in graphene. Our numerical method is determinant quantum Monte Carlo method which will be introduced in the chapter 2. / Hu, Feiming = 阻錯系統的量子門特卡洛研究 / 胡飛鳴. / Adviser: Lin Hai-Qing. / Source: Dissertation Abstracts International, Volume: 73-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 107-126). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Hu, Feiming = Zu cuo xi tong de liang zi Mente Kaluo yan jiu / Hu Feiming.

Safety is a well-known and important class of property of software programs, and of systems in general. The basic notion that informs this work is that the time to think about safety is when it still exists but could be lost. The notion is not just to analyse safety as existing or not with a given system state, but also in the sense that a system is presently safe but becoming less so. Safety as considered here is not restricted to one type of property, and indeed for much of the discussion it does not matter what types of measures are used to assess safety. The work done here is for the purpose of laying a theoretical and mathematical foundation for allowing static analyses of systems to further safety. This is done using tools from lattice theory applied to the poset of system states partially ordered by reachability. Such analyses are common (e.g., with abstract interpretations of software functioning) with respect to other kinds of systems, but there does not seem to exist a formalism that permits them specifically for safety. Using the basic analytical tools developed, a study is made of the problem of composing systems from components. Three types of composition: direct sum, direct product, and exponentiation---are noted, and the first two are treated in some depth. It is shown that the set of all systems formed with the direct sum and direct product operators can be specified by a BNF grammar. A three-valued ``safety logic'' is specified, using which the safety and fault-tolerance of composed systems can be computed given the system composition. It is also shown that the set of all systems also forms separate monoids (in the sense familiar to mathematicians), and that other monoids can be derived based on equivalence classes of systems. The example of a train approaching a railroad crossing, where a gate must be closed prior to the train's arrival and opened after its exit, is considered and analysed as an example.

Electromagnetic properties of the octet and decuplet baryons are calculated in quenched QCD on a 20 ³ x40 lattice with a lattice spacing of 0.128 fm using the fat - link irrelevant clover ( FLIC ) fermion action. FLIC fermions enable simulations to be performed efficiently at quark masses as low as 300 MeV. By combining FLIC fermions with an improved conserved vector current we ensure that discretization errors occur only at Ο ( α ² ) while maintaining current conservation. Magnetic moments, charge radii and magnetic radii are extracted from the electric and magnetic form factors for each individual quark sector. From these the corresponding baryon properties are constructed. Our results for the octet baryons are compared with the predictions of Quenched Chiral Perturbation Theory ( Q χ PT ) and experimental values where available. Results for the charge radii and magnetic moments of the octet baryons are in accord with the predictions of the Q χ PT, suggesting that the sum of higher order terms makes only a small contribution to the chiral expansion. The regime where chiral physics dominates remains to be explored. We establish the non - analytic behavior of the charge radii and magnetic moment in the case of octet baryons. The neutron charge radius suggests that the chiral regime is still far away. We establish substantial environment sensitivity in the quark behavior in the low mass region. We establish that the u and d quarks make substantial and important contribution to the magnetic moment of the Λ contradicting the predictions of the Simple Quark Model. We present the E0 and M1 form factors of the decuplet baryons and the charge radii and magnetic moments. We compare the decuplet baryon results with the lattice calculation of charge radii and magnetic moments of octet baryons. We establish that the environment sensitivity is far less pronounced in the case of the decuplet baryons compared to that in the octet baryons. A surprising result is that the charge radii of the decuplet baryons are generally smaller than that of the octet baryons. Magnetic moment of the Δ + shows a turn over in the low quark mass region, making it smaller than the proton magnetic moment. This is consistent with the expectations of the Quenched Chiral Perturbation Theory. A similar turn over is also noticed in the magnetic moment of the ∑ * [superscript 0], but not for Ξ * where only kaon loops can appear in Quenched QCD. We present results for the higher order moments of the decuplet baryons, i.e., the electric quadrupole moment E2 and the magnetic octupole moment M3. With these results we provide the first conclusive analysis which shows that decuplet baryons are deformed. The electric quadrupole moment of the The electric quadrupole moment of the Ω ‾ baryon is postive when the negative charge factor is included, and is equal to 0.014 ± 0.0005 fm ², indicating an oblate shape. / Thesis (Ph.D.)--School of Chemistry and Physics, 2006.