Formally, let G be a Coxeter group with reduced root system R and k < sub > v </ sub > a multiplicity function on R ( so k < sub > u </ sub > = k < sub > v </ sub > whenever the reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to the roots u and v are conjugate in G ).

The last two correspond to the B < sub > 2 </ sub > and A < sub > 2 </ sub > Coxeter planes.

* King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker and Company, 2006.

* 1907 – Donald Coxeter, British-born Canadian geometer ( d. 2003 )

Image: Tutte eight cage. svg | The Tutte – Coxeter graph ( Tutte eight cage ) has a girth of 8

The last two correspond to the A < sub > 2 </ sub > and H < sub > 2 </ sub > Coxeter planes.

Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane.

In 1959, Coxeter published his finding that these works were extraordinarily accurate: " Escher got it absolutely right to the millimeter.

The second and third correspond to the B < sub > 2 </ sub > and A < sub > 2 </ sub > Coxeter planes.

Some sources use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an ( n − 1 )- dimensional element.

The last two correspond to the B < sub > 2 </ sub > and A < sub > 2 </ sub > Coxeter planes.

It is important to realize that the word problem is in fact solvable for many groups G. For example, polycyclic groups have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable ; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the Todd – Coxeter algorithm and the Knuth – Bendix completion algorithm.

Coxeter notation is another notation, denoting the reflectional dihedral symmetry as, order 2n, and rotational dihedral symmetry as < sup >+</ sup >, order n. Orbifold notation gives the reflective symmetry as * nn and rotational symmetry as n.

For instance, in 1961 Coxeter wrote Introduction to Geometry without mention of cross-ratio.

The snub cube has two special orthogonal projections, centered, on two types of faces: triangles, and squares, correspond to the A < sub > 2 </ sub > and B < sub > 2 </ sub > Coxeter planes.

Only five vertex-transitive graphs with no Hamiltonian cycles are known: the complete graph K < sub > 2 </ sub >, the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.

Harold Scott MacDonald " Donald " Coxeter, ( February 9, 1907 – March 31, 2003 ) was a British-born Canadian geometer.

Coxeter is regarded as one of the great geometers of the 20th century.

In his youth, Coxeter composed music and was an accomplished pianist at the age of 10.

Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics.

* D < sub > n </ sub >, a Coxeter – Dynkin diagram

The A, D, E nomenclature also yields the simply laced finite Coxeter groups, by the same diagrams: in this case the Dynkin diagrams exactly coincide with the Coxeter diagrams, as there are no multiple edges.

He earned his Ph. D. from the University of Toronto in 1966 with a dissertation title of The Theory of Uniform Polytopes and Honeycombs under the supervision of H. S. M. Coxeter.

orthographic projection of the truncated 120-cell, in the H < sub > 3 </ sub > Coxeter plane ( D < sub > 10 </ sub > symmetry ).

A second order Fuchsian equation with n singular points has a group of symmetries acting ( projectively ) on its solutions, isomorphic to the Coxeter group D < sub > n </ sub > of order 2 < sup > n-1 </ sup >× n !.

Heun's equation has a group of symmetries of order 192, isomorphic to the Coxeter group of the Coxeter diagram D < sub > 4 </ sub >, analogous to the 24 symmetries of the hypergeometric differential equations obtained by Kummer.

According to Coxeter, the transformation by inversion in circle was invented by L. I. Magnus in 1831.

* Coxeter, H. S. M .; Greitzer, S. L .: Geometry Revisited.

* Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, 1974.

As our radius of penetration, R, increases, the area of new knowledge increases by Af, and the total of human knowledge becomes measured in terms of Af.

For United States expenditures under subsections ( A ), ( B ), ( D ), ( E ), ( F ), ( H ) through ( R ) of Section 104 of the Act or under any of such subsections, the rupee equivalent of $200 million.

Companies of all types have made great advances in production capabilities and efficiencies -- in modern equipment and new processes, enlarged R & D facilities, faster new product development.

M & R Dietetic Laboratories, Inc., Columbus, gives all its workers a facsimile checkbook -- each check showing the amount the company spends on a particular fringe.

Many companies have systems, particularly in R & D, which work more or less well, depending upon size and actual belief in the policy on the part of administration, as will be abundantly apparent in subsequent quotations.

We have set up a central R & D department, as well as engineering-management departments -- about 80 people working on problems related to those of our plants.

`` Above these jobs we have chief engineer for the company and vice-president of Engrg, R & Aj.

Paper electrophoresis was carried out on the concentrated samples in a Spinco model R cell using barbital buffer, pH 8.6, ionic strength 0.075, at room temperature on Whatman 3MM filter paper.

We say that N is nilpotent if there is some positive integer R such that Af.

for, using the fact that N and N' commute Af and so when R is sufficiently large every term in this expression for Af will be 0.

Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

Some predictions had been made concerning factors R, N, I and Co on these inventories which appeared to be directly related to control and security aspects of personality functioning which were hypothesized as being of importance in differential Kohnstamm reactivity.

The action and reaction forces are R and Af, respectively, and are equal and opposite in direction.

An analysis of the vector relationships shows that the rake angle **yc and the friction angle **yt determine the vector direction Af of the force resultants R and Af.

It is difficult to measure the direction and magnitude of R directly.

This is not a constant value like Af, but varies with the thickness of the coating and the direction and magnitude of the resultants R and Af of Fig. 6.

If the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.

A sufficiently good approximation for determining the end reflection losses R can be obtained from the angle independent Fresnel formula: Af.

Moreover, by systematically varying the operating conditions in the first stage and always using the optimal Af-stage policy for the remaining stages, we shall eventually find the optimal policy for all R stages.

If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.

But even more important than this is the fact that the direct search by simultaneously varying all operating conditions has produced only one optimal policy, namely, that for the given feed state and R stages.

But Af is vastly larger than R.

Consider the process illustrated in Fig. 2.1, consisting of R distinct stages.

These will be numbered in the direction opposite to the flow of the process stream, so that stage R is the T stage from the end.

Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.