Ashay Dharwadker

Born January 1, 1967, New Delhi, India

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Research:

Grand Unification of the Standard Model with Quantum Gravity,
:: http://www.geocities.com/dharwadker/standard_model , 2008.
We show that the mathematical proof of the four colour theorem directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem, at the most fundamental level. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with t'Hooft's table. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.

Applications of Graph Theory,
:: http://www.geocities.com/dharwadker/pirzada/applications , 2007. 
Journal of The Korean Society for Industrial and Applied Mathematics (KSIAM), Vol. 11, No. 4, 2007.

The Vertex Coloring Algorithm,
:: http://www.geocities.com/dharwadker/vertex_coloring , 2006.
A new polynomial-time algorithm for finding proper m-colorings of the vertices of a graph. We prove that every graph with n vertices and maximum vertex degree Δ must have chromatic number χ(G) less than or equal to Δ+1 and that the algorithm will always find a proper m-coloring of the vertices of G with m less than or equal to Δ+1. Furthermore, we prove that this condition is the best possible in terms of n and Δ by explicitly constructing graphs for which the chromatic number is exactly Δ+1. In the special case when G is a connected simple graph and is neither an odd cycle nor a complete graph, we show that the algorithm will always find a proper m-coloring of the vertices of G with m less than or equal to Δ. In the process, we obtain a new constructive proof of Brooks' famous theorem of 1941. For all known examples of graphs, the algorithm finds a proper m-coloring of the vertices of the graph G for m equal to the chromatic number χ(G). In view of the importance of the P versus NP question, we ask: does there exist a graph G for which this algorithm cannot find a proper m-coloring of the vertices of G with m equal to the chromatic number χ(G)? The algorithm is demonstrated with several examples of famous graphs, including a proper four-coloring of the map of India and two large Mycielski benchmark graphs with hidden minimum vertex colorings. We implement the algorithm in C++ and provide a demonstration program.
:: The Math Forum Review

The Clique Algorithm,
:: http://www.geocities.com/dharwadker/clique , 2006.
Baltic Horizons, No. 8 (107), Special Issue Dedicated to 270 Years of Graph Theory, 2007.
A new polynomial-time algorithm for finding maximal cliques in graphs. It is shown that every graph with n vertices and minimum vertex degree δ must have a maximum clique of size at least ⌈n/(n−δ)⌉ and that this condition is the best possible in terms of n and δ. As a corollary, we obtain new bounds on the famous Ramsey numbers in terms of the maximum and minimum vertex degrees of the corresponding Ramsey graphs. The algorithm finds a maximum clique in all known examples of graphs. In view of the importance of the P versus NP question, we ask if there exists a graph for which the algorithm cannot find a maximum clique. The algorithm is demonstrated by finding maximum cliques for several famous graphs, including two large benchmark graphs with hidden maximum cliques. We implement the algorithm in C++ and provide a demonstration program.
:: The Math Forum Review

The Independent Set Algorithm,
:: http://www.geocities.com/dharwadker/independent_set , 2006.
A new polynomial-time algorithm for finding maximal independent sets in graphs. It is shown that every graph with n vertices and maximum vertex degree Δ must have a maximum independent set of size at least ⌈n/(Δ+1)⌉ and that this condition is the best possible in terms of n and Δ. As a corollary, we obtain new bounds on the famous Ramsey numbers in terms of the maximum and minimum vertex degrees of the corresponding Ramsey graphs. The algorithm finds a maximum independent set in all known examples of graphs. In view of the importance of the P versus NP question, we ask if there exists a graph for which the algorithm cannot find a maximum independent set. The algorithm is demonstrated by finding maximum independent sets for several famous graphs, including two large benchmark graphs with hidden maximum independent sets. We implement the algorithm in C++ and provide a demonstration program.
:: The Math Forum Review

The Vertex Cover Algorithm,
:: http://www.geocities.com/dharwadker/vertex_cover , 2006.
A new polynomial-time algorithm for finding minimal vertex covers in graphs. It is shown that every graph with n vertices and maximum vertex degree Δ must have a minimum vertex cover of size at most n−⌈n/(Δ+1)⌉ and that this condition is the best possible in terms of n and Δ. The algorithm finds a minimum vertex cover in all known examples of graphs. In view of the importance of the P versus NP question, we ask if there exists a graph for which the algorithm cannot find a minimum vertex cover. The algorithm is demonstrated by finding minimum vertex covers for several famous graphs, including two large benchmark graphs with hidden minimum vertex covers. We implement the algorithm in C++ and provide a demonstration program.
:: The Math Forum Review
:: Proceedings of the World Academy of Science, Engineering and Technology

Common Systems of Coset Representatives,
:: http://www.geocities.com/dharwadker/coset.html , 2005.
Using the axiom of choice, we prove that given any group G and a finite subgroup H, there always exists a common system of representatives for the left and right cosets of H in G.

A New Algorithm for finding Hamiltonian Circuits,
:: http://www.geocities.com/dharwadker/hamilton , 2004.
A new polynomial-time algorithm for finding Hamiltonian circuits in certain graphs. It is shown that the algorithm always finds a Hamiltonian circuit in graphs that have at least three vertices and minimum degree at least half the total number of vertices. In the process, we also obtain a constructive proof of Dirac’s famous theorem of 1952, for the first time. The algorithm finds a Hamiltonian circuit (respectively, tour) in all known examples of graphs that have a Hamiltonian circuit (respectively, tour). In view of the importance of the P versus NP question, we ask: does there exist a graph that has a Hamiltonian circuit (respectively, tour) but for which this algorithm cannot find a Hamiltonian circuit (respectively, tour)? The algorithm is implemented in C++ and the program is demonstrated with several examples.
:: The Math Forum Review
:: University of Rome - Computing Large Square Loops

Heptahedron and Roman Surface,
:: http://www.eg-models.de , Electronic Geometry Models, Model 2003.05.001, 2004. 
Using Hilbert's definition of a heptahedron we show how to construct Steiner's Roman surface as a model of the projective plane.
:: MathWorld - Roman Surface
:: MathWorld - Heptahedron

Riemann Surfaces,
:: http://www.eg-models.de , Electronic Geometry Models, Model 2002.05.001, 2003. 
Riemann surfaces were first studied by Bernhard Riemann in his Inauguraldissertation at Göttingen in 1851. This paper shows the construction of the surfaces w = zⁿ.

The Witt Design, 
:: http://www.geocities.com/dharwadker/witt.html , 2002.
The Steiner system S(5,8,24) with a C++ program to generate the Witt design, Golay code and projective plane PG(2,4).
:: Design Resources at Queen Mary, University of London
:: The Math Forum Review
:: Rose-Hulman Math Journal - Tight Subdesigns of The Higman-Sims Design

A New Proof of The Four Colour Theorem,
:: http://www.geocities.com/dharwadker , 2000.
A new proof of the famous Four Colour Theorem using Steiner systems, Eilenberg modules, Hall matchings and Riemann surfaces.
:: Canadian Mathematical Society Award
:: The Math Forum Review
:: Tölvunot Fréttahorn
:: History of the Four-Color Conjecture
:: Four Color Theorem: A Brief Historical Insight
:: Teorema dei Quattro Colori e la Teoria dei Grafi
:: Maximum Degree Chromatic Number of a Graph - Stellenbosch University
:: Yahoo! - Famous Mathematics Problems

Split Extensions and Representations of Moufang Loops,
Communications in Algebra 23(11), 4245-4255, 1995.
A representation theory of Moufang loops generalizing the traditional representation theory of groups.
:: European Mathematical Society Review
 

Textbook:

Software:

Statistics 1.0,
:: http://www.geocities.com/dharwadker/statistics.html , 2007 - 2008.
Software for Windows: Descriptive statistics, statistical inference, quality control, acceptance sampling, regression and correlation, time series and trends, analysis of variance (ANOVA), probability distributions with moment generating functions and random samples.

Calculus 1.0,
:: http://www.geocities.com/dharwadker/calculus.html , 2003 - 2008.
Software for Windows: Compute and graph functions, derivatives, integrals, tangents, arc lengths, areas, roots, maxima/minima, points of inflection, Taylor series and Fourier series, areas and volumes of surfaces of revolution, estimate limits of functions, sequences and series.
:: The Math Forum - Single Variable Calculus

My Students Database,
:: http://members.lycos.co.uk/dharwadker , 2003. 
A prototype online relational database management system in Boyce-Codd normal form using MySQL, PHP and Apache web server.

Teaching: