Stairs of Natural Set Theories
Issue:
Volume 3, Issue 3, June 2014
Pages:
49-65
Received:
15 May 2014
Accepted:
3 June 2014
Published:
10 June 2014
DOI:
10.11648/j.pamj.20140303.11
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Abstract: All contemporary set theories have intersected classes. We have built the stairs of set theories with disjoint classes. We call such theories natural. We numerate these theories by ordinals. The first set theory is T_0. We build the theory from the set N of natural numbers by using the operations of direct products and of power set by finite times. The theory contains all results of Cantor’s theory. We argue that the theory can satisfy all needs of applied mathematics. We build theory T_1 by using the universe set of all sets of T_0 and by using the operations of direct products and of power set by finite times. We build theory T_α+1 from the set of previous by using the operations of direct products and of power set by finite times, too. We build theory T_ω from the set of all sets of T_α with α < ω again by using the operations of direct products and of power set by finite times. And so on for every theory T_α, if theory T_α-1 does not exists. We use the join of all these sets to build theory T_On without operation of power set. We call members of T_On families, members of families sets, families, which are not members of families, up-sets. Families are an analog of classes of the MK set theory and up-sets are an analog of proper classes of MK theory. The theory T_On is more strong than MK theory because we use more strong axiom of comprehension. The last theory T_On+1 is external to T_On. We use T_On+1 to prove those theorems of T_On that are unproved in T_On.
Abstract: All contemporary set theories have intersected classes. We have built the stairs of set theories with disjoint classes. We call such theories natural. We numerate these theories by ordinals. The first set theory is T_0. We build the theory from the set N of natural numbers by using the operations of direct products and of power set by finite times....
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On Expressive Construction of Solitons from Physiological Wave Phenomena
Issue:
Volume 3, Issue 3, June 2014
Pages:
70-77
Received:
29 May 2014
Accepted:
30 June 2014
Published:
20 July 2014
DOI:
10.11648/j.pamj.20140303.13
Downloads:
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Abstract: Physiological waves, much like the waves of some other physical phenomena, consist of non-linear and dispersive terms. In studies involving patho-physiology, models on arterial pulse waves indicate that the waveforms behave like solitons. The Korteweg-deVrie (KdV) equation, which is known to admit soliton solutions, is seen to hold well for arterial pulse waves. The foregoing underpins the need for detailed knowledge of the construction of solitons. In the light of this, plane wave solution would fail to yield the desired goal, let alone where arterial pulse waves are physiological waves that decompose into a travelling wave representing fast transmission phenomena during systolic phase and a windkessel term representing slow transmission phenomena during diastolic phase. This paper elucidates the construction of the solitons that arise from the so called KdV equation. The goal is to enhance an authentic analysis of soliton-based clinical details.
Abstract: Physiological waves, much like the waves of some other physical phenomena, consist of non-linear and dispersive terms. In studies involving patho-physiology, models on arterial pulse waves indicate that the waveforms behave like solitons. The Korteweg-deVrie (KdV) equation, which is known to admit soliton solutions, is seen to hold well for arteria...
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