On finite lattices of degrees of constructibility

1977 ◽  
Vol 42 (3) ◽  
pp. 349-371 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Finite Lattice ◽  
Partial Ordering ◽  
Generic Model ◽  
Crucial Point ◽  
Generic Models ◽  
Ordinal Numbers ◽  
Finite Lattices ◽  
Definition Of ◽  
Two Stages ◽  
The Given

We shall prove the following theorem:Theorem. For any finite lattice there is a model of ZF in which the partial ordering of the degrees of constructibility is isomorphic with the given lattice.Let M be a standard countable model of ZF satisfying V = L. Let K be the given finite lattice. We shall extend M by forcing.The paper is divided into two parts. The first part concerns the definition of the set of forcing conditions and some properties of this set expressible without the use of generic filters.We define first a representation of a lattice and then the set of conditions. In Lemmas 1, 2 we show that there are some canonical isomorphisms between some conditions and that a single condition has some canonical automorphisms.In Lemma 3 and Definition 7 we show some methods of defining conditions. We shall use those methods in the second part to define certain conditions with special properties.Lemma 4 gives a connection between the sets P and Pk (see Definitions 4 and 5). It is next employed in the second part in Lemma 10 in an essential way.Indeed, Lemma 10 is necessary for Lemma 13, which is the crucial point of the whole construction. Lemma 5 is also employed in Lemma 13 (exactly in its Corollary).The second part of the paper is devoted to the examination of the structure of degrees of constructibility in a generic model. First, we show that degrees of some “sections” of a generic real (Definition 9) form a lattice isomorphic with K. Secondly, we show that there are no other degrees in the generic model; this is the most difficult property to obtain by forcing. We prove, in two stages, that it holds in our generic models. We first show, by using special properties of the forcing conditions, that sets of ordinal numbers have no other degrees. Then we show that the degrees of sets of ordinals already determine the degrees of other sets.

On finite lattices of degrees of constructibility of reals

1976 ◽  
Vol 41 (2) ◽  
pp. 313-322 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Standard Model ◽  
Finite Lattice ◽  
Partial Ordering ◽  
Turing Degrees ◽  
Upper Semilattice ◽  
Finite Lattices ◽  
Definition Of ◽  
The Universe

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice.The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lerman's paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lerman's representation.Let K be a given finite lattice. Assume that the universe of K is an integer l. Let ≤K be the ordering in K. A sequential representation of K is a sequence Ui ⊆ Ui+1 of finite subsets of ωi such that the following holds:(1) For any s, s′ Є Ui, i Є ω, k, m Є l, k ≤Km & s(m) = s′(m) → s(k) = s′(k).(2) For any s Є Ui, i Є ω, s(0) = 0 where 0 is the least element of K.(3) For any s, s′ Є i Є ω, k,j Є l, if k y Kj = m and s(k) = s′(k) & s(j) = s′(j) → s(m) = s′(m), where vK denotes the join in K.

scholarly journals The Development of Mongolian as a Minority Language in digital spaces

2020 ◽  
Vol 4 (2) ◽  
pp. 73-76
Author(s):  
Kewen Xu
Keyword(s):  
State Level ◽  
Minority Language ◽  
Geographic Region ◽  
The State ◽  
Minority Languages ◽  
South American ◽  
Crucial Point ◽  
The Usa ◽  
Definition Of ◽  
The Given

Chaklader (1981) argues for adopting a definition of minority languages at the state level. A ‘minority language’, in the most straightforward sense, is simply one language spoken by less than 50 percent of a population within a specific geographic region which is different from the language of the majority community and the language of the state. The crucial point is the proportion of speaker population in the given region or country. In other words, a minority language might be only a minority language in this specific region, but a majority language in other region (Grenoble, 2014). For example, Spanish is the majority language in a group of south American countries, however, it is a minority language in the USA in general. Mongolian, is also an example of this kind of minority language.

Operation Experiences with Biological Phosphorus Removal at the Sewage Treatment Plants of Berlin (West)

1991 ◽  
Vol 24 (7) ◽  
pp. 133-148 ◽  
Author(s):  
A. Peter ◽  
F. Sarfert
Keyword(s):  
Phosphorus Removal ◽  
Sewage Treatment ◽  
Chemical Precipitation ◽  
Biological Phosphorus Removal ◽  
Treatment Results ◽  
Sewage Treatment Plants ◽  
Research Activities ◽  
Two Stages ◽  
The Given ◽  
Biological Phosphorus

In investigations concerning sludge bulking in Berlin enhanced biological phosphorus removal was first observed unexpectedly. Because since 1986 an officially preset limit of 2 mg TP/l must be kept in all Berlin wastewater discharges it was decided to explore the capabilities of the observed mechanism under the specific circumstances of the exciting two large treatment plants in Ruhleben (240,000 m3/d) and Marienfelde (100,000 m3/d). For this purpose some of the existing units at both plants were equipped with anaerobic zones which were generated mainly by process modifications. Additionally stage one of the Ruhleben plant was altered completely in order to investigate the combination of biological phosphorus and nitrogen removal as a special pilot study in three parallel trains. The research activities and treatment results gained in each of the two stages of the Ruhleben and in the Marienfelde plant are reported in detail. For example BOD-related phosphorus removal rates were obtained ranging from 2.3-4.5 mg TP per 100 mg BOD removed. It must be stressed that all examinations were performed on full-scale conditions. At present the given limit of 2 mg TP/l in the Ruhleben plant is met without any chemical precipitation at least on average. From the beginning biological phosphorus removal will be integrated into further projected extensions.

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

scholarly journals Eigenvalue spectrum and scaling dimension of lattice $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Georg Bergner ◽  
David Schaich
Keyword(s):  
Anomalous Dimension ◽  
Finite Lattice ◽  
Continuum Theory ◽  
Lattice Volume ◽  
Yang Mills ◽  
Lattice Regularization ◽  
Perturbative Renormalization ◽  
Zero Values ◽  
Definition Of ◽  
Group Flow

Abstract We investigate the lattice regularization of $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory, by stochastically computing the eigenvalue mode number of the fermion operator. This provides important insight into the non-perturbative renormalization group flow of the lattice theory, through the definition of a scale-dependent effective mass anomalous dimension. While this anomalous dimension is expected to vanish in the conformal continuum theory, the finite lattice volume and lattice spacing generically lead to non-zero values, which we use to study the approach to the continuum limit. Our numerical results, comparing multiple lattice volumes, ’t Hooft couplings, and numbers of colors, confirm convergence towards the expected continuum result, while quantifying the increasing significance of lattice artifacts at larger couplings.

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

On a Definition of Ordinal Numbers

1962 ◽  
Vol 69 (5) ◽  
pp. 381-386 ◽  
Author(s):  
Maurice Sion ◽  
Richard Willmott
Keyword(s):  
Ordinal Numbers ◽  
Definition Of

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

On a Definition of Ordinal Numbers

1962 ◽  
Vol 69 (5) ◽  
pp. 381 ◽  
Author(s):  
Maurice Sion ◽  
Richard Willmott
Keyword(s):  
Ordinal Numbers ◽  
Definition Of

Underground Built Heritage (UBH) as Valuable Resource in China, Japan and Italy

Heritage ◽  
2021 ◽  
Vol 4 (4) ◽  
pp. 3208-3237
Author(s):  
Roberta Varriale ◽  
Laura Genovese
Keyword(s):  
Research Council ◽  
Methodological Approach ◽  
National Research Council ◽  
Functional Classification ◽  
Static And Dynamic Analysis ◽  
Built Heritage ◽  
Living Space ◽  
Definition Of ◽  
The Given ◽  
National Systems

Recent research about the theoretical approach to elements of cultural heritage that can be included in the newly born class Underground Built Heritage (UBH), has provided several instruments for the functional classification and the static and dynamic analysis of all artefacts coherent with the given definition, while introducing several criteria for their reuse and the evaluation of connected enhancement processes as well. These guidelines can be adopted to analyze single artefacts, groups of homogenous or heterogeneous elements, and also selected territorial assets or national systems, even at a comparative level. With reference to this potential, research results from the application of this new methodological approach to the outputs of three ongoing projects by the National Research Council of Italy, all focusing on UBH, in three countries: China, Japan and Italy, are presented. With reference to the above-mentioned geographical contests, the research introduces a comparative study focusing on selected examples of artefacts that have been historically built underground to manage three functions: living space, religion and economy. This study, carried out based on data collected during onsite visits by the authors, consists in three steps: selection and analysis of case studies, definition of level of reuses on the basis of a given scale, and analysis of the different tools adopted for their conservation and enhancement. In the conclusions, possible future implementations of reuses of the analyzed elements are pointed out.

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