The consistency of ZFC + 2ℵ0 > ℵω + ℐ(ℵ2) = ℐ(ℵω)

1997 ◽  
Vol 62 (4) ◽  
pp. 1151-1160 ◽  
Author(s):  
Martin Gilchrist ◽  
Saharon Shelah
Keyword(s):  
Complete Graph ◽  
Color Patterns ◽  
Natural Place ◽  
Uncountable Cardinal ◽  
Compactness Theorems ◽  
Made In

Let κ be an uncountable cardinal and the edges of a complete graph with κ vertices be colored with ℵ0 colors. For the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if , then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when . In particular, we are concerned with the case ℵ1 ≤ κ ≤ ℵω.The study of these color patterns (known as identities) has a history that involves the existence of compactness theorems for two cardinal models [4]. When the graph being colored has size ℵ1, the identities that must occur ((ℵ1)) have been classified by Shelah [6]. If the graph has size greater than or equal to ℵω the identities that must occur ((ℵω)) have also been classified in [5]. This leaves open the question of how the sets (ℵm) (2 ≤ m < ω) fit between (ℵ1) and ⊆ (ℵω). Some progress in this direction has been made in the paper [2]. It is there shown that if ZFC is consistent then so is for each m < ω. The number of colors is fixed at ℵ0 as it is the natural place to start and the results here can be generalized to more colors. We first give some definitions and establish some notation.

Identities on cardinals less than ℵω

1996 ◽  
Vol 61 (3) ◽  
pp. 780-787 ◽  
Author(s):  
M. Gilchrist ◽  
S. Shelah
Keyword(s):  
Complete Graph ◽  
Binary Tree ◽  
Complete Binary Tree ◽  
Color Patterns ◽  
Natural Place ◽  
Uncountable Cardinal ◽  
Compactness Theorems

Let κ be an uncountable cardinal and the edges of a complete graph with κ vertices be colored with ℵ0 colors. For the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if , then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when the edges of the complete graph on vertices are colored with ℵ0 colors. In particular, we are concerned with the case ℵ1 ≤ κ ≤ ℵω.The study of these color patterns (known as identities) has a history that involves the existence of compactness theorems for two cardinal models [2]. When the graph being colored has size ℵ1, the identities that must occur have been classified by Shelah [4]. If the graph has size greater than or equal to ℵω the identities have also been classified in [3]. The number of colors is fixed at ℵ0 as it is the natural place to start and the results here can be generalized to situations where more colors are used.There is one difference that we now make explicit. When countably many colors are used we can define the following coloring of the complete graph on vertices. First consider the branches in the complete binary tree of height ω to be vertices of a complete graph.

A Compact Imbedding Theorem for Functions without Compact Support

1971 ◽  
Vol 14 (3) ◽  
pp. 305-309 ◽  
Author(s):  
R. A. Adams ◽  
John Fournier
Keyword(s):  
Sobolev Space ◽  
Compact Support ◽  
Unbounded Domains ◽  
Bounded Domains ◽  
Complete Continuity ◽  
Compact Imbedding ◽  
Compactness Theorems ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Made In

The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort1to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings2are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case.

scholarly journals Experience herb relocation for preserving its population during reconstruction period

2020 ◽  
Vol 157 ◽  
pp. 02004
Author(s):  
Anton Martsev ◽  
Oleg Selivanov
Keyword(s):  
Selection Methods ◽  
Species Status ◽  
Natural Conditions ◽  
Laboratory Methods ◽  
Natural Place ◽  
Reconstruction Methods ◽  
Vladimir Region ◽  
Anemone Nemorosa ◽  
Reconstruction Period ◽  
Made In

The article is devoted to the experience of transplantation Anemone nemorosa L., listed in the Red book of the Vladimir region as a rare species (status category 3). A feature of this work is transplanting in the autumn period, in connection with the construction of a natural place of growth, in accordance with the conclusion of environmental impact assessment. The aim of the work was to preserve the aboriginal population in natural conditions that are close in their ecological characteristics to the natural place of growth and which do not fall into the zone of reconstruction. Methods. We used the geobotanical description method, soil selection methods, and laboratory methods. Results. The data of the geobotanical description of the initial and recipient habitats of the Anemone nemorosa and soil characteristics are presented. The method of plant transplantation is described in detail. Conclusion. The phenollogical observations made in May 2019 showed that Anemone nemorosa successfully transferred the autumn transplant to a new place. An artificial population of Anemone oakwood was formed, which requires further monitoring studies for adaptation and resistance to transplantation.

scholarly journals XXXV. Account of a new micrometer and megameter

1777 ◽  
Vol 67 ◽  
pp. 789-798
Keyword(s):  
Considerable Improvement ◽  
Circular Motion ◽  
Rock Crystal ◽  
Double Refraction ◽  
Natural Place ◽  
Pass Through ◽  
Made In

Having heard that the Abbé Rochon had exhibited a kind of micrometer, which, by means of a prism of rock crystal whose angles could be varied, gave two images of the same object, and changed their distances by the circular motion of one of the two parts which composed it; I told some of my friends, and among others the celebrated Abbé Fontana, that I saw very well how the thing was done; but that a considerable improvement would be made in it if the distance from the prism to the focus of the eye-glass were made variable. I added that the same effect might be produced without the double refraction of the rock crystal, with a prism made of common glass, only smaller than the aperture of the object-glass. The rays which pass through the prism would them form an image which would be seen out of its natural place; and those which pass without would give another image, in the same place it would have appeared in, if the prism had not been there.

Functional significance of preserved color patterns of mollusks from the Gosport Sand (Eocene) of Alabama

1988 ◽  
Vol 62 (01) ◽  
pp. 83-87 ◽  
Author(s):  
Patricia H. Kelley ◽  
Charles T. Swann
Keyword(s):  
Functional Morphology ◽  
Functional Significance ◽  
Intraspecific Variability ◽  
Color Pattern ◽  
Color Patterns ◽  
Historical Factors ◽  
Life Mode ◽  
Molluscan Fauna ◽  
Excellent Preservation ◽  
Architectural Constraints

The excellent preservation of the molluscan fauna from the Gosport Sand (Eocene) at Little Stave Creek, Alabama, has made it possible to describe the preserved color patterns of 15 species. In this study the functional significance of these color patterns is tested in the context of the current adaptationist controversy. The pigment of the color pattern is thought to be a result of metabolic waste disposal. Therefore, the presence of the pigment is functional, although the patterns formed by the pigment may or may not have been adaptive. In this investigation the criteria proposed by Seilacher (1972) for testing the functionality of color patterns were applied to the Gosport fauna and the results compared with life mode as interpreted from knowledge of extant relatives and functional morphology. Using Seilacher's criteria of little ontogenetic and intraspecific variability, the color patterns appear to have been functional. However, the functional morphology studies indicate an infaunal life mode which would preclude functional color patterns. Particular color patterns are instead interpreted to be the result of historical factors, such as multiple adaptive peaks or random fixation of alleles, or of architectural constraints including possibly pleiotropy or allometry. The low variability of color patterns, which was noted within species and genera, suggests that color patterns may also serve a useful taxonomic purpose.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

Surface Structure in Microtomed Sections Caused by Glass and Diamond Knives

1971 ◽  
Vol 29 ◽  
pp. 456-457
Author(s):  
J. Temple Black ◽  
William G. Boldosser
Keyword(s):  
Plastic Deformation ◽  
Epoxy Resin ◽  
Carbon Coating ◽  
Surface Damage ◽  
Body Angle ◽  
Normal Manner ◽  
Bottom Side ◽  
Cutting Direction ◽  
Made In ◽  
Diamond Knife

Ultramicrotomy produces plastic deformation in the surfaces of microtomed TEM specimens which can not generally be observed unless special preparations are made. In this study, a typical biological composite of tissue (infundibular thoracic attachment) infiltrated in the normal manner with an embedding epoxy resin (Epon 812 in a 60/40 mixture) was microtomed with glass and diamond knives, both with 45 degree body angle. Sectioning was done in Portor Blum Mt-2 and Mt-1 microtomes. Sections were collected on formvar coated grids so that both the top side and the bottom side of the sections could be examined. Sections were then placed in a vacuum evaporator and self-shadowed with carbon. Some were chromium shadowed at a 30 degree angle. The sections were then examined in a Phillips 300 TEM at 60kv.Carbon coating (C) or carbon coating with chrom shadowing (C-Ch) makes in effect, single stage replicas of the surfaces of the sections and thus allows the damage in the surfaces to be observable in the TEM. Figure 1 (see key to figures) shows the bottom side of a diamond knife section, carbon self-shadowed and chrom shadowed perpendicular to the cutting direction. Very fine knife marks and surface damage can be observed.

Iron storage sites in the myocardium as revealed by cytohistochemistry

1988 ◽  
Vol 46 ◽  
pp. 394-395
Author(s):  
M. Ashraf ◽  
F. Thompson ◽  
S. Miki ◽  
P. Srivastava
Keyword(s):  
Cardiac Tissue ◽  
Coronary Perfusion ◽  
Intracellular Iron ◽  
Ether Anesthesia ◽  
Intense Staining ◽  
Iron Storage ◽  
Thin Sections ◽  
Rat Hearts ◽  
Cacodylate Buffer ◽  
Made In

Iron is believed to play an important role in the pathogenesis of ischemic injury. However, the sources of intracellular iron in myocytes are not yet defined. In this study we have attempted to localize iron at various cellular sites of the cardiac tissue with the ferrocyanide technique.Rat hearts were excised under ether anesthesia. They were fixed with coronary perfusion with 3% buffered glutaraldehyde made in 0.1 M cacodylate buffer pH 7.3. Sections, 60 μm in thickness, were cut on a vibratome and were incubated in the medium containing 500 mg of potassium ferrocyanide in 49.5 ml H2O and 0.5 ml concentrated HC1 for 30 minutes at room temperature. Following rinses in the buffer, tissues were dehydrated in ethanol and embedded in Spurr medium.The examination of thin sections revealed intense staining or reaction product in peroxisomes (Fig. 1).

Microanalysis of precipitates in a ferritic steel

1978 ◽  
Vol 36 (1) ◽  
pp. 618-619
Author(s):  
J.M. Titchmarsh
Keyword(s):  
Ferritic Steel ◽  
Particle Identification ◽  
Energy Dispersive ◽  
Objective Lens ◽  
Ferritic Steels ◽  
Replica Technique ◽  
X Ray ◽  
Large Numbers ◽  
Scanning Transmission ◽  
Made In

The advances in recent years in the microanalytical capabilities of conventional TEM's fitted with probe forming lenses allow much more detailed investigations to be made of the microstructures of complex alloys, such as ferritic steels, than have been possible previously. In particular, the identification of individual precipitate particles with dimensions of a few tens of nanometers in alloys containing high densities of several chemically and crystallographically different precipitate types is feasible. The aim of the investigation described in this paper was to establish a method which allowed individual particle identification to be made in a few seconds so that large numbers of particles could be examined in a few hours.A Philips EM400 microscope, fitted with the scanning transmission (STEM) objective lens pole-pieces and an EDAX energy dispersive X-ray analyser, was used at 120 kV with a thermal W hairpin filament. The precipitates examined were extracted using a standard C replica technique from specimens of a 2¼Cr-lMo ferritic steel in a quenched and tempered condition.

AEM investigation of ZrO2 solid solution formation in ZrO2/mullite composites

1984 ◽  
Vol 42 ◽  
pp. 408-409
Author(s):  
T. R. Dinger
Keyword(s):  
Solid Solution ◽  
Ion Beam ◽  
Analytical Electron Microscopy ◽  
Transformation Toughening ◽  
Solid Solution Formation ◽  
Solution Formation ◽  
Propagating Crack ◽  
Analytical Electron ◽  
Microcrack Toughening ◽  
Made In

Zirconia (ZrO2) is often added to ceramic compacts to increase their toughness. The mechanisms by which this toughness increase occurs are generally accepted to be those of transformation toughening and microcracking. The mechanism of transformation toughening is based on the presence of metastable tetragonal ZrO2 which transforms to the monoclinic allotrope when stressed by a propagating crack. The decrease in volume which accompanies this transformation effectively relieves the applied stress at the crack tip and toughens the material; microcrack toughening arises from the deflection of a propagating crack around sharply angular inclusions.These mechanisms, however, do not explain the toughness increases associated with the class of composites investigated here. Analytical electron microscopy (AEM) has been used to determine whether solid solution effects could be the cause of this increased toughness. Specimens of a mullite (3Al2O3·2SiO2) + 15 vol. % ZrO2 were prepared by the usual technique of mechanical thinning followed by ion beam milling. All observations were made in a Philips EM400 TEM/STEM microscope fitted with EDXS and EELS spectrometers.

Sign in / Sign up

Export Citation Format

Share Document