#
Ashay Dharwadker
####
Born January 1, 1967, New Delhi, India
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Address:
Institute of Mathematics, H-501 Palam Vihar, District Gurgaon, Haryana 122017, India.
**Email:** ashay@dharwadker.org
**Website:** http://www.dharwadker.org
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Research:
Fundamental research in mathematics and its applications. Algebra, topology, graph theory, computer science and the foundations of physics.
**::** Mathematical Genealogy , 1850 - .
**::** 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 *M*_{H}^{0} = 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 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* = *z*^{n}.
**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
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Textbook:
Graph Theory, Ashay Dharwadker and Shariefuddin Pirzada, Amazon Books **::** ISBN 1466254998
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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.
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Teaching:
**::** Today's
Lecture |