scholarly journals Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem

2017 ◽  
Vol 249 (1187) ◽  
pp. 0-0
Author(s):  
Denis Hirschfeldt ◽  
Karen Lange ◽  
Richard Shore
Keyword(s):  
Homogeneous Model ◽  
Combinatorial Principles

Karakteristik aliran dua fase pada saluran ekspansi tiba-tiba

2018 ◽  
Vol 11 (1) ◽  
pp. 7
Author(s):  
Latif Ngudi Wibawanto ◽  
Budi Santoso ◽  
Wibawa Endra Juwana
Keyword(s):  
Standard Deviation ◽  
Flow Pattern ◽  
Flow Characteristic ◽  
Homogeneous Model ◽  
Flow Equation ◽  
Pressure Recovery ◽  
Sudden Expansion ◽  
Air Velocity ◽  
Model Flow ◽  
Two Phases

This research was conducted to find out the flow characteristic of two phases through the channel with sudden expansion in the form of change of flow pattern and pressure recovery. The test was carried out with variation of superficial velocity of water 0.2-1.3 m / s and superficial air velocity of 0.2-1.9 m / s resulting in pattern of three flow patterns ie bubble, plug, and slug. The expansion channel resulted in some changes to the flow pattern that originally plugs in the upstream channel into bubble in the downstream channel and the slug becomes plug. Pressure recovery experimental results compared with the homogeneous model flow equation and Wadle correlation, both correlations have predictions with standard deviation values of 0.32 and 0.43.

A characterization of the 0-basis homogeneous bounding degrees

2010 ◽  
Vol 75 (3) ◽  
pp. 971-995
Author(s):  
Karen Lange
Keyword(s):  
Homogeneous Model ◽  
Countable Model

AbstractWe say a countable model has a 0-basis if the types realized in are uniformly computable. We say has a (d-)decidable copy if there exists a model ≅ such that the elementary diagram of is (d-)computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous model with a 0-basis but no decidable copy. We extend this result here. Let d ≤ 0′ be any low2 degree. We show that there exists a homogeneous model with a 0-basis but no d-decidable copy. A degree d is 0-basis homogeneous bounding if any homogenous with a 0-basis has a d-decidable copy. In previous work, we showed that the non low2 Δ20 degrees are 0-basis homogeneous bounding. The result of this paper shows that this is an exact characterization of the 0-basis homogeneous bounding Δ20 degrees.

scholarly journals Teaching Combinatorial Principles Using Relations through the Placemat Method

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1825
Author(s):  
Viliam Ďuriš ◽  
Gabriela Pavlovičová ◽  
Dalibor Gonda ◽  
Anna Tirpáková
Keyword(s):  
Graph Theory ◽  
Complexity Theory ◽  
Slovak Republic ◽  
Teaching Mathematics ◽  
Combinatorial Structures ◽  
Algorithmic Approach ◽  
Scientific Disciplines ◽  
Combinatorial Principles ◽  
The Subject ◽  
As Graph

The presented paper is devoted to an innovative way of teaching mathematics, specifically the subject combinatorics in high schools. This is because combinatorics is closely connected with the beginnings of informatics and several other scientific disciplines such as graph theory and complexity theory. It is important in solving many practical tasks that require the compilation of an object with certain properties, proves the existence or non-existence of some properties, or specifies the number of objects of certain properties. This paper examines the basic combinatorial structures and presents their use and learning using relations through the Placemat method in teaching process. The effectiveness of the presented innovative way of teaching combinatorics was also verified experimentally at a selected high school in the Slovak Republic. Our experiment has confirmed that teaching combinatorics through relationships among talented children in mathematics is more effective than teaching by a standard algorithmic approach.

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

Combinatorial principles weaker than Ramsey's Theorem for pairs

2007 ◽  
Vol 72 (1) ◽  
pp. 171-206 ◽  
Author(s):  
Denis R. Hirschfeldt ◽  
Richard A. Shore
Keyword(s):  
Reverse Mathematics ◽  
The Other ◽  
Base System ◽  
Ramsey's Theorem ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Points Of View ◽  
The One ◽  
Infinite Linear

AbstractWe investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs () splits into a stable version () and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is -conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch. Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply (and so does not imply ). This answers a question of Cholak, Jockusch, and Slaman.Our proofs suggest that the essential distinction between ADS and CAC on the one hand and on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.

An homogeneous model for adiabatic capillary tubes

1998 ◽  
Vol 18 (3-4) ◽  
pp. 207-219 ◽  
Author(s):  
P.K. Bansal ◽  
A.S. Rupasinghe
Keyword(s):  
Homogeneous Model ◽  
Capillary Tubes

The spectrum of resplendency

1990 ◽  
Vol 55 (2) ◽  
pp. 626-636
Author(s):  
John T. Baldwin
Keyword(s):  
Homogeneous Model ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Recursive Theory ◽  
Finite Language

AbstractLet T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

Predicting Groundwater Flow and Solute Transport in the Heterogeneous Aquifer Sandbox Using Different Parameter Estimation Methods

2021 ◽  
Author(s):  
Liqun Jiang ◽  
Ronglin Sun ◽  
Xing Liang
Keyword(s):  
Steady State ◽  
Groundwater Flow ◽  
Solute Transport ◽  
Transport Process ◽  
Homogeneous Model ◽  
Estimation Methods ◽  
Aquifer Heterogeneity ◽  
Heterogeneous Aquifer ◽  
Tracer Distribution ◽  
Parameter Estimation Methods

<p>Protection and management of groundwater resources demand high-resolution distributions of hydraulic parameters (e.g., hydraulic conductivity (K) and specific storage (Ss)) of aquifers. In the past, these parameters were obtained by traditional analytical solutions (e.g., Theis (1935) or Cooper and Jacob (1946)). However, traditional methods assume the aquifer to be homogeneous and yield the equivalent parameter, which are averages over a large volume and are insufficient for predicting groundwater flow and solute transport process (Butler & Liu, 1993). For obtaining the aquifer heterogeneity, some scholars have used kriging (e.g., Illman et al., 2010) and hydraulic tomography (HT) (e.g., Yeh & Liu, 2000; Zhu & Yeh, 2005) to describe the K distribution.</p><p>In this study, the laboratory heterogeneous aquifer sandbox is used to investigate the effect of different hydraulic parameter estimation methods on predicting groundwater flow and solute transport process. Conventional equivalent homogeneous model, kriging and HT are used to characterize the heterogeneity of sandbox aquifer. A number of the steady-state head data are collected from a series of single-hole pumping tests in the lab sandbox, and are then used to estimate the K fields of the sandbox aquifer by the steady-state inverse modeling in HT survey which was conducted using the SimSLE algorithm (Simultaneous SLE, Xiang et al., 2009), a built-in function of the software package of VSAFT2. The 40 K core samples from the sandbox aquifer are collected by the Darcy experiments, and are then used to obtain K fields through kriging which was conducted using the software package of Surfer 13. The role of prior information on improving HT survey is then discussed. The K estimates by different methods are used to predict the process of steady-state groundwater flow and solute transport, and evaluate the merits and demerits of different methods, investigate the effect of aquifer heterogeneity on groundwater flow and solute transport.</p><p>According to lab sandbox experiments results, we concluded that compared with kriging, HT can get higher precision to characterize the aquifer heterogeneity and predict the process of groundwater flow and solute transport. The 40 K fields from the K core samples are used as priori information of HT survey can promote the accuracy of K estimates. The conventional equivalent homogeneous model cannot accurately predict the process of groundwater flow and solute transport in heterogeneous aquifer. The enhancement of aquifer heterogeneity will lead to the enhancement of the spatial variability of tracer distribution and migration path, and the dominant channel directly determines the migration path and tracer distribution.</p>

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