scholarly journals Asymptotic density in quasi-logarithmic additive number systems

2008 ◽  
Vol 144 (2) ◽  
pp. 267-287
Author(s):  
BRUNO NIETLISPACH
Keyword(s):  
Counting Function ◽  
Second Order ◽  
Asymptotic Density ◽  
Additive Number ◽  
Relational Structures ◽  
Number Systems ◽  
Density Theorems ◽  
Irreducible Elements

AbstractWe show that in quasi-logarithmic additive number systems$\mycal{A}$all partition sets have asymptotic density, and we obtain a corresponding monadic second-order limit law for adequate classes of relational structures. Our conditions on the local counting functionp(n) of the set of irreducible elements of$\mycal{A}$allow situations which are not covered by the density theorems of Compton [6] and Woods [15]. We also give conditions onp(n) which are sufficient to show the assumptions of Compton's result are satisfied, but which are not necessarily implied by those of Bell and Burris [2], Granovsky and Stark [8] or Stark [14].

scholarly journals Local Normal Forms for First-Order Logic with Applications to Games and Automata

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Thomas Schwentick ◽  
Klaus Barthelmann
Keyword(s):  
Normal Forms ◽  
Winning Strategy ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Relational Structures ◽  
Second Order Logic ◽  
Monadic Second Order Logic

International audience Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures

2020 ◽  
Vol 32 (1) ◽  
pp. 121-138
Author(s):  
Lenon Alexander Minorics
Keyword(s):  
Borel Measure ◽  
Differential Operators ◽  
Counting Function ◽  
Second Order ◽  
Spectral Asymptotics ◽  
Compact Interval ◽  
Limiting Behavior ◽  
Convergence Behavior ◽  
Neumann Eigenvalue

AbstractWe study the limiting behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second-order differential operators {\frac{\mathop{}\!d}{\mathop{}\!d\mu}\frac{\mathop{}\!d}{\mathop{}\!dx}}, where μ is a finite atomless Borel measure on some compact interval {[a,b]}. Therefore, we firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we make a proposition about the convergence behavior for so-called random V-variable Cantor measures.

Fusion in relational structures and the verification of monadic second-order properties

2002 ◽  
Vol 12 (2) ◽  
pp. 203-235 ◽  
Author(s):  
B. COURCELLE ◽  
J. A. MAKOWSKY
Keyword(s):  
Second Order ◽  
Algebraic Expression ◽  
Order Property ◽  
Unary Predicate ◽  
Free Graph ◽  
Relational Structures ◽  
Clear Cut ◽  
Algebraic Expressions ◽  
Common Framework ◽  
Fusion Operation

Relational structures offer a common framework for handling graphs and hypergraphs of various kinds. Operations like disjoint union, the creation of new relations by means of quantifier-free formulas, and relabellings of relations make it possible to denote them using algebraic expressions. It is known that every monadic second-order property of a structure is verifiable in time proportional to the size of such an algebraic expression defining it. We prove here that this result remains true if we also use in these algebraic expressions a fusion operation that fuses all elements of the domain satisfying some unary predicate. The value mapping from these algebraic expressions to the structures they denote is a monadic second-order definable transduction, which means that the structure is definable inside the tree representing the algebraic expression by monadic second-order formulas. It follows (by using results of other articles) that, with this fusion operation, we cannot generate more graph families, but we can generate them with less unary auxiliary predicates. We also obtain clear-cut characterizations of Vertex Replacement and Hyperedge Replacement context-free graph grammars in terms of four types of operations, amongst which is the fusion of vertices satisfying a specified predicate.

QUADRATIC DYNAMICS IN MATRIX RINGS: TALES OF TERNARY NUMBER SYSTEMS

Fractals ◽  
2005 ◽  
Vol 13 (02) ◽  
pp. 147-156 ◽  
Author(s):  
PAUL E. FISHBACK ◽  
MATTHEW D. HORTON
Keyword(s):  
Julia Sets ◽  
Second Order ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Matrix Rings ◽  
The Real ◽  
Number Systems ◽  
Quadratic Family ◽  
Real Quadratic ◽  
Real Matrices

We describe the quadratic dynamics in certain three-component number systems, which like the complex numbers, can be expressed as rings of real matrices. This description is accomplished using the properties of the real quadratic family and its various first- and second-order phase and parameter derivatives. We demonstrate that the fundamental dichotomy of defining the Mandelbrot set either in terms of filled Julia sets or in terms of the orbit of the origin extends to these ternary number systems.

ON ${\omega _1}$-STRONGLY COMPACT CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 266-278 ◽  
Author(s):  
JOAN BAGARIA ◽  
MENACHEM MAGIDOR
Keyword(s):  
Measurable Cardinal ◽  
Second Order ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Relational Structures ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Strongly Compact Cardinals ◽  
Order Reflection

Abstract An uncountable cardinal κ is called ${\omega _1}$ -strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$ -complete ultrafilter on I. We show that the first ${\omega _1}$ -strongly compact cardinal, ${\kappa _0}$ , cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$ . We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$ . Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$ .

Disappearance Voltage Determinations Using a Convergent Beam

1971 ◽  
Vol 29 ◽  
pp. 184-185
Author(s):  
W. L. Bell
Keyword(s):  
Second Order ◽  
Objective Method ◽  
Uniform Thickness ◽  
Rocking Curve ◽  
Crystal Perfection ◽  
Kikuchi Line ◽  
Central Maximum ◽  
Diffraction Patterns ◽  
Convergent Beam ◽  
Beam Technique

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.

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