ABSTRACT A theoretic and diagrammatic relationship between knots and planar graphs has enabled us... more ABSTRACT A theoretic and diagrammatic relationship between knots and planar graphs has enabled us to establish necessary condition for achirality. It is shown that the regions, crossings and consequently the number of vertices, edges, and faces in the corresponding LR-graph are same and invariant. Establishment of new but pivotal moves such as R*-move, 2π-twist and π-twist enabled us to prove that the black regions can be changed into white regions via Reidemeister moves. Consequently, the equivalence of the companion graphs, necessary and sufficient conditions for achirality.
We have established that tricolourability would be a way of distinguishing some of knots (links) ... more We have established that tricolourability would be a way of distinguishing some of knots (links) by showing that tricolorability is an ambient isotopy invariant. We have extended the notion of tricolorability to colorability of knot(link) and have shown that colorability of knot (link) is also an ambient isotopy invariant. We have shown that no knot is colorable mod 2 but instead every link with more than one component is colorable mod 2. We have also established that bridge number of a knot is always one.
Knot theory is a rich subject because of its many readily available examples. It has undergone dr... more Knot theory is a rich subject because of its many readily available examples. It has undergone dramatic changes during the last 12 years. A connection between knot theory and graph theory has been established by Reidemeister [R]. Graphs of knots (links) have been repeatedly employed in knot theory [Au], [C] and [KT]. In the recent past L. H. Kauffman [K] has established that "Universes of knots (links) are in one-to-one correspondence with planar graphs". In the proof, he has beautifully given the method of constructing corresponding universe from a given graph. With the introduction of LR-Graphs, one can easily extend the one-to-one correspondence to knots (linked links). The pivotal moves in the theory of knots are the Reidemeister moves. I will view these moves as Reidemeister moves of type I, type II and type III as shown in .
This article is devoted to give a self-contained presentation of classification of atoms of proba... more This article is devoted to give a self-contained presentation of classification of atoms of probability space as equivalent or non-equivalent. It will be established that an event, i.e., a member of a σ-field of a probability space can contains uncountable many equivalent atoms. We will show that the relation of being equivalent atoms is an equivalence relation. An independent proof will enable us to state that an event of a probability space with σ-finite probability measure can contains at most countable many non-equivalent atoms. We will also establish that for a purely atomic probability space with σ-finite probability measure, probability measure of every event is equal to the sum of the probability measures of its non-equivalent atoms. We will also justify that in some of the results, the probability space and respective probability measure can be replaced as measure space and respective measure.
A random coincidence point theorem for multifunction under the very mild conditions is establishe... more A random coincidence point theorem for multifunction under the very mild conditions is established. The theorem proved here can be viewed as stochastic version of Ljubomir Ciric (Indian J. pure appl. Math., 24 (1993), 145-149).
The present doldrums position and state of decadence, internal differences, external aggression (... more The present doldrums position and state of decadence, internal differences, external aggression (geographical and ideological), lack of self-confidence and dependence, illiteracy, political instability, economic disaster, lack of knowledge and wisdom, back benchers in science and technology, education, medicine, trade and business, banking system and defensive incapability of Muslim World prompted me to look at our principal sources of inspiration, which are, the Qur’an, Sunnah of the Prophet (SAW), and examples of the “enlightened Caliphs” and see what is Islam’s view about seeking knowledge, technology and inventions in general and mathematics’ education in particular. We will discuss the nature of mathematics and its scientific status. We will highlight the position of mathematics in Islamic classification of knowledge. We will also discuss the current state of mathematics and future suggestions. We have gathered together some of these impressions; these are all tentative, nothin...
This paper describes a Statistical Process Control (SPC) for failure crushing time data using com... more This paper describes a Statistical Process Control (SPC) for failure crushing time data using competing risks model. The model is based on the widely known proportional hazard regression model for a variety of censoring. A competing risks model identifies the set of ...
Uploads
Papers by Mohammad Azram