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Ashay Dharwadker

Born January 1, 1967, New Delhi, India

Founder & Director, Institute of Mathematics, Gurgaon

Honorary Professor, University Euroacademy, Tallinn

Address:

Institute of Mathematics, H-501 Palam Vihar, District Gurgaon, Haryana 122017, India.
 
Email: ashay@dharwadker.org
 
Website: http://www.dharwadker.org

Research:

Fundamental research in mathematics and its applications. Algebra, topology, graph theory, computer science and the foundations of physics.

:: Google Scholar Citations , 2000 - .
:: Amazon Book Reviews , 2000 - .

The Cosmological Constant,
:: http://www.dharwadker.org/cosmology , 2011.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466272317
We show how to calculate Einstein's cosmological constant Λ using the Grand Unified Theory. Using the topological properties of the gauge, we calculate the exact percentages of ordinary baryonic matter, dark matter and dark energy in the universe. These values are in perfect agreement with the seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations. Thus dark matter, dark energy and the cosmological constant are intrinsic properties of the gauge in the Grand Unified Theory.

Space, Time and Matter,
:: http://www.dharwadker.org/space_time , 2010.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466403926
Baltic Horizons No. 14 (111), Special Issue on Fundamental Problems in Mathematics, 2010
We show how the grand unified theory based on the proof of the four color theorem, can be obtained entirely in terms of the Poincaré group of isometries of space and time. Electric and gauge charges of all the particles of the standard model can now be interpreted as elements of the Poincaré group. We define the space and time chiralities of all spin 1/2 fermions in agreement with Dirac's relativistic wave equation. All the particles of the standard model now correspond to irreducible representations of the Poincaré group according to Wigner's classification. Finally, we construct the Steiner system of fermions and show how the Mathieu group acts as the group of symmetries of the fundamental building blocks of matter.

Higgs Boson Mass predicted by the Four Color Theorem,
:: http://www.dharwadker.org/khachatryan/higgs , arXiv:0912.5189 , 2009.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466403993
Based on the proof of the four color theorem and the grand unification of the standard model with quantum gravity, we show how to derive the values of the famous Cabibbo angle and CKM matrix, in excellent agreement with experimental observations. We make a precise prediction for the elusive Higgs boson mass MH0 = 125.992 ~ 126 GeV, as a direct consequence of our theory.

The Graph Isomorphism Algorithm,
:: http://www.dharwadker.org/tevet/isomorphism , 2009.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466394374
The Structure Semiotics Research Group, S.E.R.R., Euroacademy, Tallinn, 2009
We present a new polynomial-time algorithm for determining whether two given graphs are isomorphic or not. We prove that the algorithm is necessary and sufficient for solving the Graph Isomorphism Problem in polynomial-time, thus showing that the Graph Isomorphism Problem is in P. The semiotic theory for the recognition of graph structure is used to define a canonical form of the sign matrix of a graph. We prove that the canonical form of the sign matrix is uniquely identifiable in polynomial-time for isomorphic graphs. The algorithm is demonstrated by solving the Graph Isomorphism Problem for many of the hardest known examples. We implement the algorithm in C++ and provide a demonstration program.

Grand Unification of the Standard Model with Quantum Gravity,
:: http://www.dharwadker.org/standard_model , 2008.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466272317
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.dharwadker.org/pirzada/applications , 2007.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466397098
Journal of The Korean Society for Industrial and Applied Mathematics (KSIAM), Vol. 11, No. 4, 2007

The Vertex Coloring Algorithm,
:: http://www.dharwadker.org/vertex_coloring , 2006.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466391324
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.dharwadker.org/clique , 2006.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466391219
Baltic Horizons, No. 8 (107), 270 Years of Graph Theory Conference, Euroacademy, 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.dharwadker.org/independent_set , 2006.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466387696
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.dharwadker.org/vertex_cover , 2006.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466384476
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.dharwadker.org/coset.html , 2005.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466265302
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.dharwadker.org/hamilton , 2004.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 146638137X
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 Diracís 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.dharwadker.org/witt.html , 2002.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466265302
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.dharwadker.org , 2000.
Proceedings of the Institute of Mathematics, Amazon Books :: ISBN 1466265302
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
:: Higgs Boson Mass predicted by the Four Color Theorem
:: 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:

Graph Theory, Ashay Dharwadker and Shariefuddin Pirzada, Amazon Books :: ISBN 1466254998

Software:

Statistics 1.0,
:: http://www.dharwadker.org/statistics.html , 2007 - .
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.dharwadker.org/calculus.html , 2003 - .
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://dharwadker.freehostia.com , 2003 -
A prototype online relational database management system in Boyce-Codd normal form using MySQL, PHP and Apache web server.

Teaching:

:: Today's Lecture


Copyright © by Ashay Dharwadker. All rights reserved.