scholarly journals THE NUMBER OF ATOMIC MODELS OF UNCOUNTABLE THEORIES

2018 ◽  
Vol 83 (1) ◽  
pp. 84-102
Author(s):  
DOUGLAS ULRICH
Keyword(s):  
Atomic Model ◽  
Complete Theory ◽  
Atomic Models ◽  
Image Position

AbstractWe show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that there is a complete theory in a language of size ${\aleph _1}$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with $ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that for every complete theory T in a language of size ${\aleph _1}$, if T has uncountable atomic models but no constructible models, then T has ${2^{{\aleph _1}}}$ atomic models of size ${\aleph _1}$.

CONSTRUCTING MANY ATOMIC MODELS IN ℵ1

2016 ◽  
Vol 81 (3) ◽  
pp. 1142-1162 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
MICHAEL C. LASKOWSKI ◽  
SAHARON SHELAH
Keyword(s):  
Order Theory ◽  
Atomic Model ◽  
First Order ◽  
Atomic Models ◽  
First Order Theory ◽  
Image Position

AbstractWe introduce the notion of pseudoalgebraicity to study atomic models of first order theories (equivalently models of a complete sentence of ${L_{{\omega _1},\omega }}$). Theorem: Let T be any complete first-order theory in a countable language with an atomic model. If the pseudominimal types are not dense, then there are 2ℵ0 pairwise nonisomorphic atomic models of T, each of size ℵ1.

scholarly journals BINARY PRIMITIVE HOMOGENEOUS SIMPLE STRUCTURES

2017 ◽  
Vol 82 (1) ◽  
pp. 183-207 ◽  
Author(s):  
VERA KOPONEN
Keyword(s):  
Algebraic Closure ◽  
The Other ◽  
Equivalence Relations ◽  
Complete Theory ◽  
Future Research ◽  
Random Structure ◽  
Homogeneous Structures ◽  
Image Position

AbstractSuppose that ${\cal M}$ is countable, binary, primitive, homogeneous, and simple. We prove that the SU-rank of the complete theory of ${\cal M}$ is 1 and hence 1-based. It follows that ${\cal M}$ is a random structure. The conclusion that ${\cal M}$ is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedron-free 3-hypergraph. However, to show that the generic tetrahedron-free 3-hypergraph is 1-based requires some work (it is known that it has the other properties) since this notion is defined in terms of imaginary elements. This is partly why we also characterize equivalence relations which are definable without parameters in the context of ω-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.

scholarly journals Building a pseudo-atomic model of the anaphase-promoting complex

2013 ◽  
Vol 69 (11) ◽  
pp. 2236-2243 ◽  
Author(s):  
Kiran Kulkarni ◽  
Ziguo Zhang ◽  
Leifu Chang ◽  
Jing Yang ◽  
Paula C. A. da Fonseca ◽  
...  
Keyword(s):  
Cell Cycle ◽  
Homology Modelling ◽  
Active Form ◽  
Protein Crystallography ◽  
Hybrid Approach ◽  
Atomic Model ◽  
Anaphase Promoting Complex ◽  
Atomic Models ◽  
Entire Complex ◽  
Cell Cycle Regulatory Proteins

The anaphase-promoting complex (APC/C) is a large E3 ubiquitin ligase that regulates progression through specific stages of the cell cycle by coordinating the ubiquitin-dependent degradation of cell-cycle regulatory proteins. Depending on the species, the active form of the APC/C consists of 14–15 different proteins that assemble into a 20-subunit complex with a mass of approximately 1.3 MDa. A hybrid approach of single-particle electron microscopy and protein crystallography of individual APC/C subunits has been applied to generate pseudo-atomic models of various functional states of the complex. Three approaches for assigning regions of the EM-derived APC/C density map to specific APC/C subunits are described. This information was used to dock atomic models of APC/C subunits, determined either by protein crystallography or homology modelling, to specific regions of the APC/C EM map, allowing the generation of a pseudo-atomic model corresponding to 80% of the entire complex.

On the existence of atomic models

1993 ◽  
Vol 58 (4) ◽  
pp. 1189-1194 ◽  
Author(s):  
M. C. Laskowski ◽  
S. Shelah
Keyword(s):  
Atomic Model ◽  
Atomic Models ◽  
Countable Theory

AbstractWe give an example of a countable theory T such that for every cardinal λ ≥ ℵ2 there is a fully indiscernible set A of power λ such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T(A) is a theory of size λ where the principal types are dense, yet T(A) has no atomic model.

VII.—On Eisenstein's Continued Fractions

1877 ◽  
Vol 28 (1) ◽  
pp. 135-143 ◽  
Author(s):  
Thomas Muir
Keyword(s):  
Continued Fractions ◽  
General Equation ◽  
Primitive Root ◽  
Complete Theory ◽  
The Subject ◽  
Image Position ◽  
Cubic Forms

In Crelle's Journal for 1844, at the end of a paper on Cubic Forms, Eisenstein gives the following results:—where m is any odd number and ρ a primitive root of the equation zm = 1;whereNo demonstrations are given of these identities; but they are said by the author to be only particular cases of a very general equation, the complete theory connected with which he hoped to give on another occasion.At p. 193 of the same volume of Crelle he returns to the subject, not however for the purpose of giving the promised theory, but to add several other results similar to those before given.

scholarly journals Near-atomic model of microtubule-tau interactions

Science ◽  
2018 ◽  
Vol 360 (6394) ◽  
pp. 1242-1246 ◽  
Author(s):  
Elizabeth H. Kellogg ◽  
Nisreen M. A. Hejab ◽  
Simon Poepsel ◽  
Kenneth H. Downing ◽  
Frank DiMaio ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Alzheimer’S Disease ◽  
Alzheimer's Disease ◽  
Atomic Model ◽  
Developmentally Regulated ◽  
Computational Approaches ◽  
Atomic Models ◽  
Cryo Electron Microscopy ◽  
Tubulin Binding ◽  
Axonal Protein

Tau is a developmentally regulated axonal protein that stabilizes and bundles microtubules (MTs). Its hyperphosphorylation is thought to cause detachment from MTs and subsequent aggregation into fibrils implicated in Alzheimer’s disease. It is unclear which tau residues are crucial for tau-MT interactions, where tau binds on MTs, and how it stabilizes them. We used cryo–electron microscopy to visualize different tau constructs on MTs and computational approaches to generate atomic models of tau-tubulin interactions. The conserved tubulin-binding repeats within tau adopt similar extended structures along the crest of the protofilament, stabilizing the interface between tubulin dimers. Our structures explain the effect of phosphorylation on MT affinity and lead to a model of tau repeats binding in tandem along protofilaments, tethering together tubulin dimers and stabilizing polymerization interfaces.

An Atomic Model of Cellulose Network in Wood Cell Wall

2008 ◽  
Author(s):  
Ying Wang ◽  
Youping Chen
Keyword(s):  
Hydrogen Bond ◽  
Cell Wall ◽  
Molecular Mechanisms ◽  
Mechanical Performance ◽  
Atomic Model ◽  
Crystalline Cellulose ◽  
First Principle Calculation ◽  
Atomic Models ◽  
Water Solvent ◽  
Assembly Method

Wood is composed of parallel columns of long hollow cells which are made up of layered composite of semi-crystalline cellulose fibrils embedded in an amorphous matrix of hemicellulose and lignin. The extraordinary mechanical performance of wood is believed to result from a molecular mechanism operated through hydrogen bond connection. However, the molecular interactions, the assembly method of cell-wall components, as well as the molecular mechanisms responsible for the deformation of wood, are not well understood yet. Progress in studying the superior mechanical properties of wood cell is severely hindered because of this fact. To overcome this barrier, the foremost step is to build up an atomic model of the native cellulose fibril network, which is the dominant polysaccharide in wood cell walls. Then, in this work, we proposed the atomic models to study the cellulose network which includes a single cellulose microfibril (MF), and a thin film which is built up by first secondary layers (S1) and second secondary layers (S2) composed of cellulose MF with periodic boundary conditions. Additionally, we investigated the length effect of the microfibril and compared the effect of explicit water solvent environment with the vacuum environment. Moreover, the spatial arrangements of these atomic models have been determined by molecular mechanics simulation (energy minimization). The hydrogen bond length of the crystalline part of the inner cellulose was evaluated using first principle calculation.

The Stopping Power of Hydrogen Atoms for α-Particles according to the New Quantum Theory

1927 ◽  
Vol 23 (6) ◽  
pp. 732-754 ◽  
Author(s):  
J. A. Gaunt
Keyword(s):  
Quantum Theory ◽  
Satisfactory Agreement ◽  
Average Energy ◽  
Single Electron ◽  
Atomic Model ◽  
Stopping Power ◽  
Hydrogen Atoms ◽  
Natural Period ◽  
Image Position ◽  
Α Particles

An α-particle passing through a gas loses velocity because it gives up energy to the gaseous atoms. If we can calculate the average energy transferred to an atom, the stopping power of the gas follows immediately. The earlier theories, by Thomson and Darwin, took account of only the close collisions, in which the α-particle actually passes through the atom. Bohr, however, in 1913, took into account the transfers of energy to atoms at distances from the track considerably larger than atomic dimensions. His calculation was purely classical, and dealt with an atomic, model in which the electrons were capable of simple harmonic motions. The result was in very satisfactory agreement with experiment. For atoms containing each a single electron, with mass μ, charge ε, and natural period ω, Bohr's formula iswhere − dT/dx is the rate of loss of energy by an α-particle whose charge is E, and velocity v; N is the number of atoms per c.c.; and γ = 1·123.

Prime and atomic models

1978 ◽  
Vol 43 (3) ◽  
pp. 385-393 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Model Theory ◽  
Existence Results ◽  
Atomic Model ◽  
Elementary Embedding ◽  
Prime Model ◽  
Atomic Models ◽  
Large Model

This paper gives some simple existence results on prime and atomic models over sets. It also contains an example in which there is no prime model over a certain set even though there is an atomic model over the set. The existence results are “local” in that they deal with just one set rather than all sets contained in models of some theory. For contrast, see the “global” results in [6] or [7, p. 200].Throughout the paper, L is a countable language, and T is a complete L-theory with infinite models. There is a “large” model of T that contains the set X and any other sets and models to be used in a particular construction of a prime or atomic model over X.A model is said to be prime over X if and every elementary monomorphism on X can be extended to an elementary embedding on all of . This notion is used in a variety of ways in model theory. It aids in distinguishing between models that are not isomorphic, as in Vaught [10]. It also aids in showing that certain models are isomorphic, as in Baldwin and Lachlan [1].

The generalised RK-order, orthogonality and regular types for modules

1985 ◽  
Vol 50 (1) ◽  
pp. 202-219 ◽  
Author(s):  
Mike Prest
Keyword(s):  
Model Theory ◽  
Algebraic Structure ◽  
Structural Information ◽  
Complete Theory ◽  
Saturated Model ◽  
Injective Modules ◽  
Complete Theories ◽  
Image Position

I characterise various model-theoretic properties of types, in complete theories of modules, in terms of the algebraic structure of pure-injective modules. More specifically, I consider the generalised RK-order, and the relation of domination between types, orthogonality of types, and regular types. It will be seen that, essentially, it is the stationary types over pure-injective models which bear the algebraic structural information.For background in the model theory of modules, see [4], [5], [13], [17], [19], and for the model-theoretic background [10], [11], [12], [18]. More specific references will be given in the text. I summarise some principal definitions and results below. First, though, let me describe the main results.Throughout this paper, R will be a ring with 1; the language will be that for (right R-) modules. All types will be complete types in a complete theory, T, of R-modules. The notation is reserved for an extremely saturated model of T, inside which all “small” situations may be found. Thus, sets of parameters are just subsets of , and all models will be elementary substructures of . A positive primitive, or pp, formula is one which is equivalent to a formula of the formwith the rij ∈ R.Suppose that p and q are 1-types over a pure-injective model M.

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