Beth definability in infinitary languages

1974 ◽  
Vol 39 (1) ◽  
pp. 22-26 ◽  
Author(s):  
John Gregory
Keyword(s):  
Unpublished Result ◽  
Negative Results ◽  
Beth Definability ◽  
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Some negative results will be proved concerning the following for certain infinitary languages ℒ1 and ℒ2.Definition. Beth(ℒ1, ℒ2) iff, for every sentence ϕ(R) of ℒ1, and n-place relation symbols R and S such that S does not occur in ϕ(R), ifthen there is an ℒ2 formula θ(x1, …, xn) such thatand θ is built up using only those constant and relation symbols of ϕ other than R.That is, Beth(ℒ1, ℒ2) iff for every implicit ℒ1 definition ϕ(R) of relations, there is a corresponding explicit ℒ2 definition θ. Beth(ℒωω, ℒωω) was proved by Beth.Malitz proved that not Beth(ℒω1 ω1, ℒ∞∞) (hence not Beth (ℒ∞∞, ℒ∞∞)), but Beth (ℒ∞ω, ℒ∞∞). In §1, it is shown that Beth(ℒ∞ω, ℒ∞ω) is false. In §2, we strengthen this by showing that, for every cardinal κ, not Beth(ℒ∞ω, ℒ∞κ). In fact, not Beth (ℒκ+ω, ℒ∞κ) follows from property A(κ) defined in §2, and A(κ) is known for regular κ > ω (unpublished result of Morley).More information on infinitary Beth and Craig theorems is given in [2] and [3]. We assume that the reader is acquainted with the languages ℒκλ which allow conjunctions over ≺κ formulas and quantifiers over ≺λ variables. Thus, we assume that the reader is acquainted with the back and forth argument for showing that two structures are ≡∞κ (ℒ∞κ-elementarily equivalent). Our notation is fairly standard.

EXPRESSIVE LIMITATIONS OF NAÏVE SET THEORY IN LP AND MINIMALLY INCONSISTENT LP.

2014 ◽  
Vol 7 (2) ◽  
pp. 341-350 ◽  
Author(s):  
NICK THOMAS
Keyword(s):  
Set Theory ◽  
Element Model ◽  
Linear Orders ◽  
Negative Results ◽  
Naive Set Theory ◽  
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AbstractWe give some negative results on the expressiveness of naïve set theory (NS) in LP and in the four variants of minimally inconsistent LP defined in Crabbé (2011): ${\rm{L}}{{\rm{P}}_m},{\rm{L}}{{\rm{P}}_ = },{\rm{L}}{{\rm{P}}_ \subseteq }$, and ${\rm{L}}{{\rm{P}}_ \supseteq }$. We show that NS in LP cannot prove the existence of sets that behave like singleton sets, Cartesian pairs, or infinitely ascending linear orders. We show that NS is close to trivial in ${\rm{L}}{{\rm{P}}_m}$ and ${\rm{L}}{{\rm{P}}_ \subseteq }$, in the sense that its only minimally inconsistent model is a one-element model. We show that NS in ${\rm{L}}{{\rm{P}}_ = }$ and ${\rm{L}}{{\rm{P}}_ \supseteq }$ has the same limitations we give for NS in LP.

On a result of Szemerédi

2008 ◽  
Vol 73 (3) ◽  
pp. 953-956 ◽  
Author(s):  
Albin L. Jones
Keyword(s):  
Regular Cardinal ◽  
Short Proof ◽  
Unpublished Result ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
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AbstractWe provide a short proof that if κ is a regular cardinal with κ < c, thenfor any ordinal α < min{, κ}. In particular,for any ordinal α < . This generalizes an unpublished result of E. Szemerédi that Martin's axiom implies thatfor any cardinal κ with κ < c.

Second nilpotent BFC groups

1970 ◽  
Vol 11 (1) ◽  
pp. 9-18 ◽  
Author(s):  
Iain. M. Bride
Keyword(s):  
Upper Bound ◽  
Nilpotent Groups ◽  
Conjugacy Classes ◽  
Unpublished Result ◽  
Special Cases ◽  
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The BFC number of a group G is defined to be the least upper bound n of the cardinals of the conjugacy classes of G, provided this is finite, and we then say that G is n-BFC. It was shown by B. H. Neumann [2] that the derived group G′ of such a group is finite, and J. Wiegold [5] proved that.This bound was sharpened by I. D. Macdonald [1] to, and P. M. Neumann has recently communicated the (unpublished) result that G′ ≦ nq(n) with q(n) a quadratic in log2w, an immense improvement on the above. J. A. H. Shepperd and J. Wiegold [4] improved the bound in two special cases, showing that if G is soluble, G′ ≦ np(n) with p(n) a quintic in Iog2n, and that if G is nilpotent of class 2, , It is conjectured that for any n-BFC group G, , Wiegold [5] having shown that this bound is attained by certain nilpotent groups of class 2.

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

X-ray microanalysis of thin specimens in the transmission electron microscope at voltages up to 1000 Kv.

1978 ◽  
Vol 36 (1) ◽  
pp. 540-541
Author(s):  
G. Cliff ◽  
M.J. Nasir ◽  
G.W. Lorimer ◽  
N. Ridley
Keyword(s):  
Electron Microscope ◽  
Transmission Electron Microscope ◽  
Weight Fraction ◽  
Present Series ◽  
Constant Independent ◽  
X Ray ◽  
Transmission Electron ◽  
Series Of Experiments ◽  
Image Position ◽  
X Ray Absorption

In a specimen which is transmission thin to 100 kV electrons - a sample in which X-ray absorption is so insignificant that it can be neglected and where fluorescence effects can generally be ignored (1,2) - a ratio of characteristic X-ray intensities, I1/I2 can be converted into a weight fraction ratio, C1/C2, using the equationwhere k12 is, at a given voltage, a constant independent of composition or thickness, k12 values can be determined experimentally from thin standards (3) or calculated (4,6). Both experimental and calculated k12 values have been obtained for K(11<Z>19),kα(Z>19) and some Lα radiation (3,6) at 100 kV. The object of the present series of experiments was to experimentally determine k12 values at voltages between 200 and 1000 kV and to compare these with calculated values.The experiments were carried out on an AEI-EM7 HVEM fitted with an energy dispersive X-ray detector.

SEM analysis of the change in the reaction rates with deposit structures in the copper-iron cementation system

1978 ◽  
Vol 36 (1) ◽  
pp. 456-457
Author(s):  
V. Annamalai ◽  
L.E. Murr
Keyword(s):  
Structural Characteristics ◽  
Secondary Electron Emission ◽  
Copper Deposit ◽  
Reaction Rates ◽  
Sem Analysis ◽  
Experimental Conditions ◽  
Major Drawback ◽  
Emission Mode ◽  
Scrap Iron ◽  
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Economical recovery of copper metal from leach liquors has been carried out by the simple process of cementing copper onto a suitable substrate metal, such as scrap-iron, since the 16th century. The process has, however, a major drawback of consuming more iron than stoichiometrically needed by the reaction.Therefore, many research groups started looking into the process more closely. Though it is accepted that the structural characteristics of the resultant copper deposit cause changes in reaction rates for various experimental conditions, not many systems have been systematically investigated. This paper examines the deposit structures and the kinetic data, and explains the correlations between them.A simple cementation cell along with rotating discs of pure iron (99.9%) were employed in this study to obtain the kinetic results The resultant copper deposits were studied in a Hitachi Perkin-Elmer HHS-2R scanning electron microscope operated at 25kV in the secondary electron emission mode.

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

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