Weak Diamond Research Articles

Weak diamond, weak projectivity, and transfinite extensions of simple artinian rings

We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. We prove that if R is small, then the Weak Diamond implies that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.

Double weakness

We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $\lambda$ for which the sequence $\langle 2^\theta:\theta<\lambda\rangle$ is not eventually constant and the weak diamond fails at $\lambda$. We also prove that consistently diamond fails but a parametrized version of weak diamond holds at some strongly inaccessible $\lambda$.

Weak prediction principles

We prove the consistency of the failure of the weak diamond $\Phi _\lambda $ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $\Psi _\lambda $ is equivalent to the statement $2^{ \lt \lambda } \lt 2^\lambda $ and h

Spectroscopic Investigation for Purity Evaluation of Detonation Nanodiamonds: Experimental Approach in Absorbance and Scattering

Nanodiamond powders have excellent mechanical, chemical, physical, and optical properties. In this research, two non-pure groups of nanodiamond particles labeled as ND1 and ND2 have been selected for purity evaluation by means of absorbance and scattering analysis. The nanodiamond powders have been used for Raman and dynamic light scattering (DLS), Fourier transform infrared (FTIR) and UV–Vis absorbance approaches. The Raman spectra show a weak diamond signal in ND2 as purity and 4 basic set of bands. FTIR absorbance spectroscopy was used in the spectral range of 400–3600 cm−1. The results show that ND2 is almost non-absorbance with wavelength. It means that the diamond phase purity of ND2 is greater than ND1, and ND2 shows a better structure of diamond. UV–Vis absorption spectra of the sample have been recorded in the spectral range of 200–800 nm by means of nanodiamond suspended in methanol (NDM). The UV–Vis absorbance of NDM1 is stronger than NDM2 at the same concentration. Therefore the diamond phase purity of ND2 is greater than ND1. Particle size distribution and zeta potential of DNDs were investigated by DLS method. Finally the structure and phase of samples have been evaluated by X-ray diffraction (XRD) for confirmation.

Weak diamond and Galvin’s property

We prove that the Devlin-Shelah weak diamond implies Galvin's property. On the other hand, Galvin's property is consistent with the negation of the weak diamond, and even with Martin's axiom. We show that the proper forcing axiom implies a relative to the negation of Galvin's property for $\aleph_1$.

De Bruijn’s weak diamond property revisited

In this paper we revisit an unpublished but influential technical report from 1978 by N.G. de Bruijn, written in the framework of the Automath project. This report describes a technique for proving confluence of abstract reduction systems, called the weak diamond property. It paved the way for the powerful technique developed by Van Oostrom to prove confluence of abstract reduction systems, called decreasing diagrams.We first revisit in detail De Bruijn’s old proof, providing a few corrections and hints for understanding. We find that this original criterion and proof technique are still worthwhile. Next, we establish that De Bruijn’s confluence criterion can be used to derive the decreasing diagrams theorem (the reverse was already known). We also provide a short proof of decreasing diagrams in the spirit of De Bruijn. We finally address the issue of completeness of this method.

Kingdom of Lesotho: Fifth Review Under the Three-Year Arrangement Under the Extended Credit Facility and Request for Extension of the Arrangement and Rephasing of Disbursement

The Extended Credit Facility (ECF) program extended to Lesotho after a sharp fall in revenues remained broadly on track. Lesotho maintained positive growth supported by expansion of mining and construction. Fiscal consolidation efforts have helped to strengthen international reserves, despite weak diamond prices. Executive Directors welcomed the government’s continued efforts to improve the business climate and promote private sector development. They also suggested the need to maintain fiscal consolidation efforts, while safeguarding priority social and growth-promoting capital spending.

Covering properties which, under weak diamond principles, constrain the extents of separable spaces

We show that separable, locally compact spaces with property (a) necessarily have countable extent — i.e., have no uncountable closed, discrete subspaces — if the effective weak diamond principle ⋄(ω,ω,<) holds. If the stronger, non-effective, diamond principle Φ(ω,ω,<) holds then separable, countably paracompact spaces also have countable extent. We also give a short proof that the latter principle implies there are no small dominating families in ω1ω.

Specializing Aronszajn Trees and Preserving Some Weak Diamonds

We show that ◊(ℝ, , ∈) together with CH and “all Aronszajn trees are special” is consistent relative to ZFC. The weak diamond for the covering relation of Lebesgue null sets was the only weak diamond in the Cichoń diagramme for relations whose consistency together with “all Aronszajn trees are special” was not yet settled. Our forcing proof gives also new proofs to the known consistencies of several other weak diamonds stemming from the Cichoń diagramme together with “all Aronszajn trees are special” and CH. The main part of our work is an application [Shelah, Proper and Improper Forcing, Springer-Velag, 1998, Chapter V, §§ 1–7] for a special completeness system, such that we have a genericity game. Thus we show new preservation properties of the known forcings.

Ladder systems on trees

We formulate the notion of uniformization of colorings of ladder systems on subsets of trees. We prove that Suslin trees have this property and also Aronszajn trees in the presence of Martin's Axiom. As an application we show that if a tree has this property, then every countable discrete family of subsets of the tree can be separated by a family of pairwise disjoint open sets. Such trees are then normal and hence countably paracompact. As a dual result for special Aronszajn trees we prove that the weak diamond, Φ ω , implies that no special Aronszajn tree can be countably paracompact.

Decompositions of reflexive modules

In a recent paper [11] we answered to the negative a question raised in the book by Eklof and Mekler [8, p. 455, Problem 12] under the set theoretical hypothesis of $ \diamondsuit _{\aleph _1} $ which holds in many models of set theory. The Problem 12 in [8] reads as follows: If A is a dual (abelian) group of infinite rank, is $ A \cong A \oplus { \Bbb Z} $ ? The set theoretic hypothesis we made is the axiom $ \diamondsuit _{\aleph _1} $ which holds in particular in Godel's universe as shown by R. Jensen, see [8]. Here we want to prove a stronger result under the special continuum hypothesis (CH). The question in [8] relates to dual abelian groups. We want to find a particular example of a dual group, which will provide a negative answer to the question in [8]. In order to derive a stronger and also more general result we will concentrate on reflexive modules over countable principal ideal domains R. Following H. Bass [3] an R-module G is reflexive if the evaluation map $ \sigma : G \,\longrightarrow\, G^{**} $ is an isomorphism. Here $ G^* = {\rm Hom}(G, R) $ denotes the dual group of G. Guided by classical results the question about the existence of a reflexive R-module G of infinite rank with $ G \cong\hskip -8.5dd/\hskip 3dd G \oplus R $ is natural, see [8, p. 455]. We will use a theory of bilinear forms on free R-modules which strengthens our algebraic results in [11]. Moreover we want to apply a model theoretic combinatorial theorem from [14] which allows us to avoid the weak diamond principle. This has the great advantage that the used prediction principle is still similar to $ \diamondsuit _{\aleph _1} $ but holds under CH. This will simplify algebraic argument for $ G \cong\hskip -8.5dd/\hskip 3dd G \oplus R $ .

The property optimization of diamond-cutting tools with the help of micro-structural characterization

In this study, a general defect characterization in diamond-cutting tools used in marble, granite and natural stone cutting has been investigated. An important factor determining the performance of the PCD tool and the quality of the cutting operation is the bond characteristic obtained during hot pressing as a function of the pressure, sintering temperature and time. Successful diamond tool design starts with the appropriate choice of high-quality matrix powders and diamonds. The matrix should support the diamond particles without damaging them. During cutting the matrix is expected to wear to some extent provided the tips of the diamond particles run easily and also chip flow smoothly. The bond reaction of the diamond surface and surrounding metal matrix is effective for efficient tool performance. With this type of cutting/abrasive material produced by the PM techniques the primary important factor for their properties is cleanness (cleanness of operation media, cleanness of powders and diamonds used). In general, a weak diamond–matrix bond as well as oxide films and inclusions in the matrix overtake the role of crack initiation. Also the diamond particles can act as defects through their size, form and area distribution (clustering). In this study for a micro-structural optimization, the defect characterization should be realized primarily. Transverse rupture tests which are generally used for hard material were carried out with different matrix and diamond compositions. The toughness of the tested materials were determined and the defects leading to crack formation are classified.

Diamond deposition on as-anodized porous silicon: some nucleation aspects

We have investigated diamond deposition on porous silicon (PS) substrates without any pretreatment for nucleation enhancement. We find that the widely held expectation that, the highly rough nature of the PS surface would necessarily lead to a significant enhancement in the diamond nucleation density, does not hold. For the diamond deposition conditions used in this study, the nucleation density on PS (10 6 cm −2) is similar to that on untreated c-Si substrate. This indicates an absence of extra high-energy nucleation sites in spite of the micro-columnar surface of PS. As a result, the incubation time was often much longer on PS than on heavily scratched c-Si for the same diamond deposition conditions. Under such conditions, the XRD patterns of deposit on PS showed silicon carbide-related peaks in addition to weak diamond peaks. The results are compared with the earlier observations regarding diamond deposition on PS pretreated with diamond abrasive or negative DC bias. The comparative study, which needs further explanation, indicates a synergistic relation between the negative DC bias pretreatment and the effect of porosity of the substrate on diamond nucleation on PS substrate.

More on the weak diamond

We prove that e.g., 2 ℵ 1<2 ℵ 2 does not imply the weak diamond for {δ<ℵ 2:cf δ=ℵ 0 (even if CH holds).