scholarly journals THE POLARISED PARTITION RELATION FOR ORDER TYPES

2020 ◽  
Vol 71 (3) ◽  
pp. 823-842
Author(s):  
L D Klausner ◽  
T Weinert
Keyword(s):  
Natural Numbers ◽  
Partition Relation ◽  
Cardinal Characteristics ◽  
Order Types ◽  
The Given ◽  
The Continuum

Abstract We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal characteristics of the continuum. We build on work by Erd̋s, Garti, Jones, Orr, Rado, Shelah and Szemerédi. In particular, we show that a theorem of Jones extends from the natural numbers to the rational ones, but consistently extends only to three further equimorphism classes of countable orderings. This is made possible by applying a 13-year-old theorem of Orr about embedding a given order into a sum of finite orders indexed over the given order.

scholarly journals On the cofinality of ultrapowers

1999 ◽  
Vol 64 (2) ◽  
pp. 727-736 ◽  
Author(s):  
Andreas Blass ◽  
Heike Mildenberger
Keyword(s):  
Natural Numbers ◽  
Cardinal Characteristics ◽  
Splitting Number ◽  
The Continuum

AbstractWe prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve three cardinal characteristics of the continuum, the splitting number s, the unsplitting number r, and the groupwise density number g. We also prove some related results for reduced powers with respect to filters other than ultrafilters.

Converse dual cardinals

2006 ◽  
Vol 71 (1) ◽  
pp. 22-34 ◽  
Author(s):  
Jörg Brendle ◽  
Shuguo Zhang
Keyword(s):  
Natural Numbers ◽  
Cardinal Invariants ◽  
The Continuum

AbstractWe investigate the set (ω) of partitions of the natural numbers ordered by ≤* where A ≤* B if by gluing finitely many blocks of A we can get a partition coarser than B. In particular, we determine the values of a number of cardinals which are naturally associated with the structure ((ω), ≥*), in terms of classical cardinal invariants of the continuum.

Power Withstanding Capability and Transient Temperature of Carbon Nanotube-Based Nano Electrical Interconnects

2021 ◽  
Author(s):  
Femi Robert
Keyword(s):  
Carbon Nanotube ◽  
Transient Analysis ◽  
Electronic Devices ◽  
Transient Temperature ◽  
Nanoelectronic Devices ◽  
Cu Interconnects ◽  
Electrical Interconnects ◽  
Steady State Temperature ◽  
The Given ◽  
The Continuum

Abstract This paper exhibits the electrothermal modelling and evaluation of Carbon Nanotube (CNT) based electrical interconnects for electronic devices. The continuum model of the CNT is considered and the temperature across interconnect is predicted for the given power. Finite element modelling software COMSOL Multiphysics is used to carry out the simulations. The results are compared with Al and Cu interconnects. An electrothermal analysis is also carried out to obtain the temperature for the given power for Single-Walled CNT, Double-Walled CNT, Triple-Walled CNT, and Multi-Walled CNT. Results show that the CNT interconnects performs better when compared to Al and Cu interconnects. The power withstanding capability of CNT is 68.75 times more than Al and 32.35 times more than Cu. Based on the transient analysis, the time taken by the CNT interconnects to reach a steady temperature is obtained as 0.007 ns. On the application of power, Cu and Al interconnects takes 0.1 ns to reach the steady-state temperature. The nanostructured CNT based electrical interconnects would play a considerable role in replacing Cu and Al electrical interconnect applications for micro and nanoelectronic devices.

Some remarks on the partition calculus of ordinals

1999 ◽  
Vol 64 (2) ◽  
pp. 436-442 ◽  
Author(s):  
Péter Komjáth
Keyword(s):  
Regular Cardinal ◽  
Continuum Hypothesis ◽  
Alternative Proof ◽  
Partition Relation ◽  
Negative Relation ◽  
Partition Calculus ◽  
Counter Example ◽  
The Continuum

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

The nature of flux variations in the continua and broad-line regions of selected active galactic nuclei

2020 ◽  
Vol 496 (1) ◽  
pp. 784-800
Author(s):  
A Bewketu Belete ◽  
L J Goicoechea ◽  
B L Canto Martins ◽  
I C Leão ◽  
J R De Medeiros
Keyword(s):  
Active Galactic Nuclei ◽  
Moving Average ◽  
Sampling Rate ◽  
Galactic Nuclei ◽  
Small Sample ◽  
Light Curves ◽  
Ngc 4151 ◽  
The Given ◽  
Ngc 7469 ◽  
The Continuum

ABSTRACT We present a multifractal analysis of the long-term light curves of a small sample of type 1 active galactic nuclei: NGC 4151, Arp 102B, 3C 390.3, E1821+643 and NGC 7469. We aim to investigate how the degrees of multifractality of the continuum and Hβ line vary among the five different objects and to check whether the multifractal behaviours of the continuum and the Hβ line correlate with standard accretion parameters. The backward (θ  = 0) one-dimensional multifractal detrended moving average procedure was applied to light curves covering the full observation period and partial observation periods containing an equal number of epochs for each object. We detected multifractal signatures for the continua of NGC 4151, Arp 102B and 3C 390.3 and for the Hβ lines of NGC 4151 and 3C 390.3. However, we found nearly monofractal signatures for the continua of E1821+643 and NGC 7469, as well as for the Hβ lines of Arp 102B, E1821+643 and NGC 7469. In addition, we did not find any correlations between the degree of multifractality of the Hβ line and accretion parameters, while the degree of multifractality of the continuum seems to correlate with the Eddington ratio (i.e. the smaller the ratio is, the stronger the degree of multifractality). The given method is not robust, and these results should be taken with caution. Future analysis of the sampling rate and other properties of the light curves should help with better constraining and understanding these results.

Monotone inductive definitions over the continuum

1976 ◽  
Vol 41 (1) ◽  
pp. 188-198 ◽  
Author(s):  
Douglas Cenzer
Keyword(s):  
Mathematical Logic ◽  
Recursion Theory ◽  
Axiom System ◽  
Baire Space ◽  
Natural Numbers ◽  
Inductive Definition ◽  
Inductive Definitions ◽  
Definition Of ◽  
The Continuum

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

Simple Cardinal Characteristics of the Continuum

2002 ◽  
Vol 8 (4) ◽  
pp. 552
Author(s):  
Heike Mildenberger ◽  
Andreas Blass ◽  
Haim Judah
Keyword(s):  
Cardinal Characteristics ◽  
The Continuum

Cardinal characteristics on graphs

2011 ◽  
Vol 76 (1) ◽  
pp. 1-33
Author(s):  
Nick Haverkamp
Keyword(s):  
Short Proof ◽  
Property A ◽  
New Approach ◽  
Cardinal Characteristics ◽  
Cardinal Characteristic ◽  
The Given ◽  
Bounded Graph

AbstractA cardinal characteristic can often be described as the smallest size of a family of sequences which has a given property. Instead of this traditional concern for a smallest realization of the given property, a basically new approach, taken in [4] and [5], asks for a realization whose members are sequences of labels that correspond to 1-way infinite paths in a labelled graph. We study this approach as such, establishing tools that are applicable to all these cardinal characteristics. As an application, we demonstrate the power of the tools developed by presenting a short proof of the bounded graph conjecture [4].

Functional Pearls: The Minout problem

1991 ◽  
Vol 1 (1) ◽  
pp. 121-124
Author(s):  
Richard S. Bird
Keyword(s):  
Natural Number ◽  
Linear Time ◽  
Constant Time ◽  
Natural Numbers ◽  
Free Object ◽  
Functional Program ◽  
Practical Applications ◽  
The World ◽  
The Given ◽  
Easy Solution

The problem of computing the smallest natural number not contained in a given set of natural numbers has a number of practical applications. Typically, the given set represents the indices of a class of objects ‘in use’ and it is required to find a ‘free’ object with smallest index. Our purpose in this article is to derive a linear-time functional program for the problem. There is an easy solution if arrays capable of being accessed and updated in constant time are available, but we aim for an algorithm that employs only standard lists. Noteworthy is the fact that, although an algorithm using lists is the result, the derivation is carried out almost entirely in the world of sets.

scholarly journals On Study of Vertex Labeling of Graph Operations

2011 ◽  
Vol 3 (2) ◽  
pp. 291-301
Author(s):  
M. A. Rajan ◽  
N. M. Kembhavimath ◽  
V. Lokesha
Keyword(s):  
Direct Product ◽  
Graph Labeling ◽  
Natural Numbers ◽  
Graph Vertex ◽  
Direct Sum ◽  
Graph Operations ◽  
Subdivision Graph ◽  
Maximal Vertex ◽  
The Given

Vertices of the graphs are labeled from the set of natural numbers from 1 to the order of the given graph. Vertex adjacency label set (AVLS) is the set of ordered pair of vertices and its corresponding label of the graph. A notion of vertex adjacency label number (VALN) is introduced in this paper. For each VLS, VLN of graph is the sum of labels of all the adjacent pairs of the vertices of the graph. is the maximum number among all the VALNs of the  different labeling of the graph and the corresponding VALS is defined as maximal vertex  adjacency label set . In this paper  for different graph operations are discussed. Keywords: Subdivision; Graph labeling; Direct sum; Direct product.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i26222                  J. Sci. Res. 3 (2), 291-301 (2011) 

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