Interpolation and preservation for pebble logics

1999 ◽  
Vol 64 (2) ◽  
pp. 846-858
Author(s):  
Alexandru Baltag
Keyword(s):  
General Problem ◽  
General Theorem ◽  
Case Analysis ◽  
Interpolation Theorem ◽  
Preservation Theorems ◽  
Closure Conditions ◽  
Pebble Game ◽  
General Method

In Barwise and van Benthem [6], the authors give a general method for obtaining interpolation and preservation theorems for fragments of L∞ω, those for which there is a co-inductive pebble game Γ characterizing equivalence in the logic. The method is exemplified by an analysis of the following fragments: L∞ω itself, its existential fragment , its positive fragment , the k-variable fragment (and its existential and positive subfragments) and the modal fragment (and its existential and positive subfragments).While most of their method is general, there is one part (showing that Γ has the Scott property relative to the fragment) that required a case-by-case analysis. The purpose of our paper is to replace this case-by-case analysis by a general theorem, and to illustrate this method by obtaining their kinds of results for some additional fragments of L∞ω.Our general problem can be stated in the following way: Given a “nice” fragment F of L∞ω (one satisfying some natural closure conditions), find a pebble game characterization Γ of “preservation of F-formulas” and prove that Γ has the Scott property with respect to F. Applying the Abstract Interpolation Theorem from [6], we can conclude that F has Γ-interpolation, and the corresponding preservation result. In this paper, we shall give an answer to this question. (Our answer is “sufficient” but we don't know if our conditions are necessary.)

GENERAL METHOD TO SELECT CONVENIENT LATTICE VECTORS ON A SURFACE WITH ARBITRARY MILLER INDICES

1997 ◽  
Vol 04 (05) ◽  
pp. 1063-1069 ◽  
Author(s):  
K. HERMANN
Keyword(s):  
Single Crystal ◽  
Crystal Lattice ◽  
General Problem ◽  
Computational Algorithms ◽  
Linear Transformations ◽  
Crystal Surfaces ◽  
Single Crystal Surfaces ◽  
Theoretical Methods ◽  
Electronic And Structural Properties ◽  
General Method

Many theoretical methods dealing with electronic and structural properties of single crystal surfaces rely on a convenient description of the surface and bulk periodicity at the same time. This can be achieved by using surface adapted lattice vectors [Formula: see text], where vectors [Formula: see text] define (h k l) net planes parallel to the surface while [Formula: see text] connects adjacent (h k l) net planes. For selected low index (h k l) surfaces of common crystals the construction of appropriate lattice vectors may be trivial. However, the general problem of determining a lattice basis adapted to a surface orientation which is described by Miller indices (h k l) in a general crystal lattice is more involved. In this paper we show that such bases, [Formula: see text], can be uniquely determined by linear transformations from the bulk lattice basis [Formula: see text]. The transformations depend on Miller indices (h k l) but not on the lattice type and can be quantified by number-theoretical methods. Thus, they are numerically stable and can be easily implemented in computational algorithms dealing with surfaces of most general crystals.

Infinitary formulas preserved under unions of models

1972 ◽  
Vol 37 (3) ◽  
pp. 449-465 ◽  
Author(s):  
Bienvenido F. Nebres
Keyword(s):  
Predicate Calculus ◽  
Compactness Theorem ◽  
Mathematical Research ◽  
First Order ◽  
Syntactic Form ◽  
Preservation Theorems ◽  
First Results ◽  
Closure Conditions ◽  
Special Case ◽  
Order Predicate Calculus

So-called “preservation theorems” relate the (possible) syntactic form of the axioms of a theory to certain closure conditions on its class of models. Such results are well known for the first-order predicate calculus, Lω, ω, and there are various expositions; e.g., Keisler [14], [15]. For the language , the first results were the theorems of Lopez-Escobar on sentences preserved under homomorphic images and of Malitz on formulas preserved under substructures. More recently, Feferman added a result on formulas preserved under (or persistent for) ∈-extensions. Some of these theorems will be considered in subsequent sections. A more thorough treatment may be found in Makkai [17]. The main new preservation result obtained here characterizes the sentences preserved under ω-unions. This notion and the statement of the theorem will be explained shortly.It is a familiar experience in mathematical research that concepts which are equivalent in a special case diverge in general. In the case at hand, one must expect to consider different possible statements for , which generalize a known result for Lω, ω. Moreover, diverse proofs may yield the same result in the special case, not all of which can be extended to the general case. Again, since the compactness theorem fails for , one cannot expect to extend the arguments from Lω, ω which use this in an essential way.

Ω-estimates for a class of arithmetic error terms

2007 ◽  
Vol 142 (3) ◽  
pp. 385-394 ◽  
Author(s):  
JERZY KACZOROWSKI ◽  
KAZIMIERZ WIERTELAK
Keyword(s):  
Harmonic Functions ◽  
Remainder Term ◽  
General Theorem ◽  
Classical Case ◽  
Divisor Function ◽  
Boundary Values ◽  
Number Formula ◽  
Upper Half Plane ◽  
Error Terms ◽  
General Method

AbstractThe main aim of this paper is to present a general method of proving Ω-estimates for a class of arithmetic error terms. We assume that error terms in question are boundary values of harmonic functions on the upper half-plane satisfying certain subsidiary conditions. We prove a general theorem for an axiomatically defined class of such functions and then we show how this result can be used to give statements in concrete situations. As examples we treat the classical case of the remainder term in the prime number formula obtaining a new proof of the well-known result of J. E. Littlewood, and the case of the remainder term in the asymptotic formula for the summatory function of the square-free divisor function. In the latter case our result is new.

scholarly journals Universal graphs at the successor of a singular cardinal

2003 ◽  
Vol 68 (2) ◽  
pp. 366-388 ◽  
Author(s):  
Mirna Džamonja ◽  
Saharon Shelah
Keyword(s):  
General Problem ◽  
Supercompact Cardinal ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Consistency Results ◽  
Universal Graphs ◽  
General Method

AbstractThe paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are μ++ graphs on μ+ that taken jointly are universal for the graphs on μ+, while .The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.

scholarly journals Some theorems on generalized polars with arbitrary weight

1987 ◽  
Vol 10 (4) ◽  
pp. 757-776
Author(s):  
Neyamat Zaheer ◽  
Aijaz A. Khan
Keyword(s):  
Critical Points ◽  
General Problem ◽  
General Theorem ◽  
Rational Functions ◽  
Homogeneous Polynomials ◽  
Special Cases ◽  
Arbitrary Weight ◽  
Complex Valued ◽  
Sufficiency Conditions

The present paper, which is a continuation of our earlier work in Annali di Mathematica [1] and Journal Math. Seminar [2] (EγEUθPIA), University of Athens, Greece, deals with the problem of determining sufficiency conditions for the nonvanishing of generalized polars (with a vanishing or nonvanishing weight) of the product of abstract homogeneous polynomials in the general case when the factor polynomials have been preassigned independent locations for their respective null-sets. Our main theorems here fully answer this general problem and include in them, as special cases, all the results on the topic known to date and established by Khan, Marden and Zaheer (see Pacific J. Math. 74 (1978), 2, pp. 535-557, and the papers cited above). Besides, one of the main theorems leads to an improved version of Marden's general theorem on critical points of rational functions of the formf1f2…fp/fp+1…fq,fibeing complex-valued polynomials of degreeni.

A Computer-graphing-based Approach to Solving Inequalities

1989 ◽  
Vol 82 (5) ◽  
pp. 327-331
Author(s):  
Bert K. Waits ◽  
Franklin Demana
Keyword(s):  
High School ◽  
Learning Experience ◽  
Case Analysis ◽  
Geometric Representation ◽  
Traditional Methods ◽  
School Textbook ◽  
Method Of Solution ◽  
General Method

Virtually every algebra or precalculus high school textbook uses either a “case by case” analysis or a “sign chart” method to solve inequalities. Both methods are very limited because they require that the expressions involved be easily factored or given in factored form. Developing noncontrived applications is difficult because of the limitations of these traditional methods. Furthermore, some students do not understand the logic involved in using these traditional methods, nor do they understand the meaning of a solution to an inequality. This lack of understanding about the method and the context of the problem often leads to an ineffective learning experience. Establishing a geometric representation for solving an inequality leads to two major results. First, it serves as a foundation from which the algebraic techniques can be understood. Second, it offers a more general method of solution than the signchart method because it is applicable to expressions that do not factor.

Influence of Magnetic Fields on Solar Hydrodynamics: Experimental Results

1977 ◽  
Vol 36 ◽  
pp. 143-180 ◽  
Author(s):  
J.O. Stenflo
Keyword(s):  
Solar Activity ◽  
Energy Balance ◽  
Magnetic Fields ◽  
Solar Atmosphere ◽  
General Problem ◽  
Solar Rotation ◽  
Active Regions ◽  
Physical Conditions ◽  
Dynamic Phenomena

It is well-known that solar activity is basically caused by the Interaction of magnetic fields with convection and solar rotation, resulting in a great variety of dynamic phenomena, like flares, surges, sunspots, prominences, etc. Many conferences have been devoted to solar activity, including the role of magnetic fields. Similar attention has not been paid to the role of magnetic fields for the overall dynamics and energy balance of the solar atmosphere, related to the general problem of chromospheric and coronal heating. To penetrate this problem we have to focus our attention more on the physical conditions in the ‘quiet’ regions than on the conspicuous phenomena in active regions.

A General Method for Spectrometer Design in the Scoff Approximation

1976 ◽  
Vol 34 ◽  
pp. 538-539
Author(s):  
J. R. Fields
Keyword(s):  
Electron Microscope ◽  
Transmission Electron Microscope ◽  
Energy Analysis ◽  
Incident Beam ◽  
Scanning Transmission Electron Microscope ◽  
Chemical Identification ◽  
Scanning Transmission Electron ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
General Method

The energy analysis of electrons scattered by a specimen in a scanning transmission electron microscope can improve contrast as well as aid in chemical identification. In so far as energy analysis is useful, one would like to be able to design a spectrometer which is tailored to his particular needs. In our own case, we require a spectrometer which will accept a parallel incident beam and which will focus the electrons in both the median and perpendicular planes. In addition, since we intend to follow the spectrometer by a detector array rather than a single energy selecting slit, we need as great a dispersion as possible. Therefore, we would like to follow our spectrometer by a magnifying lens. Consequently, the line along which electrons of varying energy are dispersed must be normal to the direction of the central ray at the spectrometer exit.

Phase behavior of cationic and anionic surfactants at low concentrations

1992 ◽  
Vol 50 (1) ◽  
pp. 694-695
Author(s):  
E. Naranjo
Keyword(s):  
Phase Behavior ◽  
Structural Characterization ◽  
Dynamic Systems ◽  
Anionic Surfactants ◽  
Intermolecular Forces ◽  
Stable Form ◽  
Surfactant Mixtures ◽  
Low Concentrations ◽  
Cationic And Anionic Surfactants ◽  
General Method

Equilibrium vesicles, those which are the stable form of aggregation and form spontaneously on mixing surfactant with water, have never been demonstrated in single component bilayers and only rarely in lipid or surfactant mixtures. Designing a simple and general method for producing spontaneous and stable vesicles depends on a better understanding of the thermodynamics of aggregation, the interplay of intermolecular forces in surfactants, and an efficient way of doing structural characterization in dynamic systems.

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