scholarly journals A SEPARATION RESULT FOR COUNTABLE UNIONS OF BOREL RECTANGLES

2019 ◽  
Vol 84 (02) ◽  
pp. 517-532
Author(s):  
DOMINIQUE LECOMTE
Keyword(s):  
Countable Union ◽  
Binary Relations ◽  
Separation Result ◽  
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AbstractWe provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of ${\bf{\Sigma }}_1^0 \times {\bf{\Sigma }}_\xi ^0$ sets, or by a ${\bf{\Pi }}_1^0 \times {\bf{\Pi }}_\xi ^0$ set.

The McKinsey axiom is not canonical

1991 ◽  
Vol 56 (2) ◽  
pp. 554-562 ◽  
Author(s):  
Robert Goldblatt
Keyword(s):  
Relational Semantics ◽  
Binary Relations ◽  
Order Condition ◽  
Modal Formula ◽  
First Order ◽  
Canonical Frame ◽  
Historical Importance ◽  
Definition Of ◽  
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Normal Modal Logics

The logic KM is the smallest normal modal logic that includes the McKinsey axiomIt is shown here that this axiom is not valid in the canonical frame for KM, answering a question first posed in the Lemmon-Scott manuscript [Lemmon, 1966].The result is not just an esoteric counterexample: apart from interest generated by the long delay in a solution being found, the problem has been of historical importance in the development of our understanding of intensional model theory, and is of some conceptual significance, as will now be explained.The relational semantics for normal modal logics first appeared in [Kripke, 1963], where a number of well-known systems were shown to be characterised by simple first-order conditions on binary relations (frames). This phenomenon was systematically investigated in [Lemmon, 1966], which introduced the technique of associating with each logic L a canonical frame which invalidates every nontheorem of L. If, in addition, each L-theorem is valid in , then L is said to be canonical. The problem of showing that L is determined by some validating condition C, meaning that the L-theorems are precisely those formulae valid in all frames satisfying C, can be solved by showing that satisfies C—in which case canonicity is also established. Numerous cases were studied, leading to the definition of a first-order condition Cφ associated with each formula φ of the formwhere Ψ is a positive modal formula.

scholarly journals DISCRETE METRIC SPACES: STRUCTURE, ENUMERATION, AND 0-1 LAWS

2019 ◽  
Vol 84 (4) ◽  
pp. 1293-1325 ◽  
Author(s):  
DHRUV MUBAYI ◽  
CAROLINE TERRY
Keyword(s):  
Metric Spaces ◽  
Extremal Combinatorics ◽  
Binary Relations ◽  
Easy Consequence ◽  
First Order ◽  
Colored Graphs ◽  
Long Line ◽  
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Almost All ◽  
Continuous Setting

AbstractFix an integer $r \ge 3$. We consider metric spaces on n points such that the distance between any two points lies in $\left\{ {1, \ldots ,r} \right\}$. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces is $\left\lceil {{{r + 1} \over 2}} \right\rceil ^{\left( {\matrix{ n \cr 2 \cr } } \right) + o\left( {n^2 } \right)} .$Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [34]. When r is even, our structural characterization is more precise and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when r is even, we improve the error term above from $o\left( {n^2 } \right)$ to $o\left( 1 \right)$, and also show a labeled first-order 0-1 law in the language ${\cal L}_r $, consisting of r binary relations, one for each element of $[r]$ . In particular, we show the almost sure theory T is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\left\{ {r/2, \ldots ,r} \right\}$.Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical 0-1 laws.

On the elementary theory of restricted elementary functions

1988 ◽  
Vol 53 (3) ◽  
pp. 796-808 ◽  
Author(s):  
Lou van den Dries
Keyword(s):  
Real Number ◽  
Elementary Theory ◽  
Elementary Function ◽  
Sine Function ◽  
Binary Relations ◽  
Elementary Functions ◽  
Closed Intervals ◽  
Existential Formula ◽  
Classical Work ◽  
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As a contribution to definability theory in the spirit of Tarski's classical work on (R, <, 0, 1, +, ·) we extend here part of his results to the structureHere exp ∣[0, 1] and sin ∣[0, π] are the restrictions of the exponential and sine function to the closed intervals indicated; formally we identify these restricted functions with their graphs and regard these as binary relations on R. The superscript “RE” stands for “restricted elementary” since, given any elementary function, one can in general only define certain restrictions of it in RRE.Let (RRE, constants) be the expansion of RRE obtained by adding a name for each real number to the language. We can now formulate our main result as follows.Theorem. (RRE, constants) is strongly model-complete.This means that every formula ϕ(X1, …, Xm) in the natural language of (RRE, constants) is equivalent to an existential formulawith the extra property that for each x ∈ Rm such that ϕ(x) is true in RRE there is exactly one y ∈ Rn such that ψ(x, y) is true in RRE. (Here ψ is quantifier free.)

Monadic binary relations and the monad systems at near-standard points

1987 ◽  
Vol 52 (3) ◽  
pp. 689-697
Author(s):  
Nader Vakil
Keyword(s):  
Topological Space ◽  
Binary Relation ◽  
Hausdorff Space ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Completely Regular ◽  
General Topological Space ◽  
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AbstractLet (*X, *T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in *X is defined as m{x) = μT(st(x)). Consider the relationFrank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of Rns to the whole of *X. Wattenberg's extensions of Rns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of Rns to *X should be possible in any topological space. A study of such extensions of Rns is the purpose of the present paper. We call a binary relation W ⊆ *X × *X an infinitesimal on *X if it is monadic and reflexive on *X. We prove, among other things, that the existence of an infinitesimal on *X that extends Rns is equivalent to the condition that the space (X, T) be regular.

scholarly journals COMPLEXITY OF EQUIVALENCE RELATIONS AND PREORDERS FROM COMPUTABILITY THEORY

2014 ◽  
Vol 79 (3) ◽  
pp. 859-881 ◽  
Author(s):  
EGOR IANOVSKI ◽  
RUSSELL MILLER ◽  
KENG MENG NG ◽  
ANDRÉ NIES
Keyword(s):  
Complexity Theory ◽  
Polynomial Time ◽  
Equivalence Relation ◽  
Computability Theory ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Exponential Time ◽  
Relative Complexity ◽  
Turing Reducibility ◽  
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AbstractWe study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relationsR,S, a componentwise reducibility is defined byR≤S⇔ ∃f∀x, y[x R y↔f(x)S f(y)].Here,fis taken from a suitable class of effective functions. For us the relations will be on natural numbers, andfmust be computable. We show that there is a${\rm{\Pi }}_1^0$-complete equivalence relation, but no${\rm{\Pi }}_k^0$-complete fork≥ 2. We show that${\rm{\Sigma }}_k^0$preorders arising naturally in the above-mentioned areas are${\rm{\Sigma }}_k^0$-complete. This includes polynomial timem-reducibility on exponential time sets, which is${\rm{\Sigma }}_2^0$, almost inclusion on r.e. sets, which is${\rm{\Sigma }}_3^0$, and Turing reducibility on r.e. sets, which is${\rm{\Sigma }}_4^0$.

On partitions of the real line into compact sets

1987 ◽  
Vol 52 (2) ◽  
pp. 353-359 ◽  
Author(s):  
Ludomir Newelski
Keyword(s):  
Dense Subset ◽  
Point Of View ◽  
Generic Model ◽  
Countable Union ◽  
Compact Sets ◽  
Random Real ◽  
Small Cardinality ◽  
Sacks Forcing ◽  
Standard Set ◽  
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The problem mentioned in the title has already been investigated by J. Baumgartner, J. Stern, A. Miller and many others (see [2] and [5]). We prove here some generalizations of theorems of Miller and Stern from [2] and [5]. We use standard set-theoretical notation. LetOne can check that in the above definition we can replace “compact subset of ωω” by “closed nowhere dense subset of ω2” or “Fσ and meager subset of ω2” (as any Fσ subset of ω2 can be presented as a disjoint countable union of compact sets).For functions f, g ϵ ωω we define f ≼ g if for all but finitely many n ϵ ω we have f(n) ≤ g(n). Let denote the least cardinality of a family A ⊆ ωω such that for any f ϵ ωω there is g ϵ A for which f ≼ g. It is easy to see that ≤ κω ≤ κ1. If f ϵ ωω then let ≼(f) = {h ϵωω: h ≼ f}.We find an axiom which implies = ω1 → κ1 = ω1, and which can be preserved by any ccc notion of forcing of “small cardinality”. We construct also in a generic model many partitions of ωω into compact sets preserved not only by any random real extension, but also by Sacks' notion of forcing. This shows that from some point of view Miller's modification of Sacks' forcing (from [2]) is the “minimal” one able to destroy a partition of ωω into compact sets.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
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Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
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A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

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