Discrete mathematics

G. Ramachandran and T. Schlick. Beyond optimization Simulating the dynamics of supercoiled DNA by a macroscopic model. In P. M. Pardalos, D. Shal-loway, and G. Xue, editors. Global Minimization of Nonconvex Energy Functions Molecular Conformation and Protein Folding, volume 23 of DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, pages 215-231, Providence, Rhode Island, 1996. American Mathematical Society. 

PalE85 Palmer, E. M. Graphical Evolution. An introduction to the theory of random graphs. Wiley-Iiherscience Series in Discrete Mathematics, John Wiley and Sons, 1985. 

Cook, W., Cunningham, W., Pulley-blank, W. and Schrijver, A. (1998) Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York. 

Basak, S. C., Mills, D., Mumtaz, M. M. Use of graph invariants in the protection of human and ecological health. In Basak, S. C., Balakrishnan, R., Eds., Lecture Notes of the First Indo-US Lecture Series on Discrete Mathematical Chemistry, 2007. 

Alt73] A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Mathematics 4 (1973) 201-217. 

BGGMTW05] G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discrete Mathematics 305-1 3 (2005) 33-54. 

DeDeOO] O. Delgado-Friedrichs and M. Deza, More icosahedral fulleroids, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 51 (2000) 97-115. 

DDG98] A. Deza, M Deza, and V.P. Grishukhin, Embeddings of fullerenes and coordination polyhedra into half-cubes, Discrete Mathematics 192 (1998) 41-80. 

DeSt02c] M. Deza and M.I. Shtogrin, Mosaics, embeddable into cubic lattices, Discrete Mathematics 244-1 3 (2002) 43-53. 

GrZa74] B. Griinbaum and J. Zaks, The existence of certain planar maps, Discrete Mathematics 10 (1974) 93-115. 

Jen90] S. Jendrol, Convex 3-polytopes with exactly two types of edges, Discrete Mathemat-ics 84 (1990) 143-160. 

Mor97] J. F. Moran, The growth rate and balance of homogeneous tilings in the hyperbolic plane, Discrete Mathematics 173 (1997) 151-186. 

SeSe94] B. Servatius and H. Servatius, Self-dual maps on the sphere, Discrete Mathematics 134-1 3 (1994) 139-150. 

It is clear from the foregoing discussion that r is defined as a necessarily discrete parameter. However, to enable the use of familiar techniques, it will hereafter be supposed that r represents a continuum—with the understanding that a fully consistent treatment will require the use of discrete mathematics throughout. 

A. Schrijver. Theory of linear and integer programming. Wiley-Interscience series in discrete mathematics and optimization. J. Wiley, 1986. 

The mill is described by a discrete mathematical model, developed on the basis of the axial dispersion model... 

Introducing now the notation x/ (i/2) = ihx — hx/2, the discrete mathematical model of the grinding mill-classifier circuit, shown schematically in Figs. 1 and 2, is formed by the following set of recurrence algebraic equations. 

With these assumptions, the discrete mathematical model of the grinding mill-classifier system is as follows. Let the interval [0,F] represent the length of the mill. We subdivide [0,F] into J equal subintervals of length hy, denoted by yj = jhy, j — 1,2,. .., J, the discrete axial coordinate of the mill. Similarly, if x stands for the particle size, the interval [xmin, xmax], representing the total size interval of particles is subdivided into I equal subintervals, and we denote the zth size of particles by x = xmin + ihx, 7=1,2,. ..,/. Then, the rectangular domain [y7 i, > /] x [x i,x ] is called the (/,/)th cell of the model, and as a consequence, the 2D-discrete computational scheme of the mill consists of J columns termed sections and J x I cells as it is illustrated in Fig. 2. 

Cvetkovic DM, Doob M, Gutman I, Torgascv A (1988) Recent results in the theory of graph spectra, North-Holland, Amsterdam (Annals of Discrete Mathematics 36)... 

Partial order ranking (POR) is based on elementary methods of discrete mathematics (e.g., Hasse diagrams) — if A < B and B < C, then A < C in the ranking procedures. POR does not assume linearity or any assumptions about distribution properties such as normality. The disadvantage is that often a preprocessing of data is needed to avoid the effects of stochastic noise. Combining POR with PCA may improve its usefulness. POR can only be applied for interpolation. 

K.A. Ross, C.R.B. Wright, Discrete Mathematics , Prentice-Hall International, Inc., Second Edition, London, 1988, 3 P. Harary, Graph Theory , Addison Wesley, Reading, 1972, chapters 2, 4, and 13,... 

Paul G. Mezey and Nenad Trinajstic, Applied Graph Theory and Discrete Mathematics in Chemistry. Proceedings of the Symposium in Honor of Professor Frank Harary on His 70th Birthday, Saskatoon, Canada, 12-14 September 1991, inJ. Math. Chem., 12 (1-4), Baltzer, Basel, 1993. 

Mathematical chemistry, the new challenging discipline of chemistry has established itself in recent years. Its main goal is to develop formal (mathematical) methods for chemical theory and (to some extent) for data analysis. Its history may be traced back to Caley s attempt, more than 100 years ago, to use the graph theoretical representation and interpretation of the chemical constitution of molecules for the enumeration of acyclic chemical structures. Graph theory and related areas of discrete mathematics are the main tools of qualitative theoretical treatment of chemistry [1,2]. However, attempts to contemplate connections between mathematics and chemistry and to predict new chemical facts with the help of formal mathematics have been scarce throughout the entire history of chemistry. 

However, before mathematical methods could be employed in chemistry, the latter had to reach sufficient theoretical maturity. Thus, the graph theory and some other divisions of discrete mathematics could be applied to the solution of chemical problems only after the chemical structure theory had emerged. In the next section we shall briefly discuss some important landmarks in the development of ideas concerning the structure of chemical compounds. 

Computer simulations of this discrete mathematical model show that after the initial rapid approach to a quasi-steady-state, the activity level a(n) decreases as a second-order process in terms of the remaining activity i.e., for some P ... 

The smaller the squares of the grid, the better the resolution of the representation of D by the animals. By approximately filling up the interior D of J by animals at various levels of resolution, a shape characterization of the continuous Jordan curve J can be obtained by the shape characterization of animals. The animals contain a finite number of square cells, consequently, their shape characterization can be accomplished using the methods of discrete mathematics. As a result, one obtains an approximate, discrete characterization of the shape of the Jordan curve (i.e., the shape of a continuum). The level of resolution can be represented indirectly, by the number of cells of the animals. In particular, one can show [240,243] that the number of cells required to distinguish between two Jordan curves provides a numerical measure of their similarity. 

I. Lukovits, in Topology in Chemistry-Discrete Mathematics of Molecules, eds. D.H. Rouvray and R.B. King, Horwood Publishing Ltd., Chichester, 2002, pp. 327-337. 

D. Veljan, Combinatorics and Discrete Mathematics, Algorithm, Zagreb, 2001. 

Gc d93a Daniel M. Gordon Discrete logarithms in GF(p) using the number field sieve SIAM Journal on Discrete Mathematics 6/1 (1993) 124-138. 

College Mathematics Differential Equations Discrete Mathematics Elementary Algebra Geometry... 

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