... The authors gratefully acknowledge the contributions of DH and Emma Lehmer, whose ideas forme... more ... The authors gratefully acknowledge the contributions of DH and Emma Lehmer, whose ideas formed the basis for the proof of the theorem. REFERENCES 1. DA Burgess, On character sums and primitive roots, Proc. London Math. Soc.(3) 12 (1962), 179-192. ...
The function F(x) = (1/2-x) (1-x2)1/2+x(1+(1-(1/2+x)2)1/2) has a maximum at about x = .343771, wh... more The function F(x) = (1/2-x) (1-x2)1/2+x(1+(1-(1/2+x)2)1/2) has a maximum at about x = .343771, where it attains the value of approximately .674981. This value is the root of an irreducible polynomial of tenth degree over the integers; the problem is to find this polynomial. The obvious way of proceeding is as follows:(1) Differentiate F(x), set it equal to zero, and clear radicals. The result is a tenth degree polynomial P(x) over the integers which has a root at about x = .343771.
Given a simple polygon with rational coordinates having one vertex at the origin and an adjacent ... more Given a simple polygon with rational coordinates having one vertex at the origin and an adjacent vertex on the x-axis, we look at the problem of the location of the vertices for a tiling of the polygon using lattice triangles (i.e., triangles which are congruent to a triangle with the coordinates of the vertices being integer). We show that the
Suppose we are given some xed (but unknown) subset X of a set , and our object is to learn as muc... more Suppose we are given some xed (but unknown) subset X of a set , and our object is to learn as much as possible about the elements of X by asking binary questions. Specically, each question is just a function F : !f 0; 1g ,a nd the answer to F is just the value F (Xi )f orsome Xi2
In recent years, a variety of graph optimization problems have arisen in which the graphs involve... more In recent years, a variety of graph optimization problems have arisen in which the graphs involved are much too large for the usual algorithms to be effective. In these cases, even though we are not able to examine the entire graph (which may be changing dynamically), we would still like to deduce various properties of it, such as the size
A problem arising in taste testing, medical, and parapsychology experiments can be modeled as fol... more A problem arising in taste testing, medical, and parapsychology experiments can be modeled as follows. A deck of $n$ cards contains $c_i$ cards labeled $i, 1 \leq i \leq r$. A subject guesses at the cards sequentially. After each guess the subject is told the card just guessed (or at least if the guess was correct or not). We determine
The Electronic Journal of Combinatorics - Electr. J. Comb., 1997
We analyze a certain random walk on the cartesian product Gn of a flnite group G which is often u... more We analyze a certain random walk on the cartesian product Gn of a flnite group G which is often used for generating random elements from G. In particular, we show that the mixing time of the walk is at most crn2 logn where the constant cr depends only on the order r of G.
We introduce a large class of tournament properties, all of which are shared by almost all random... more We introduce a large class of tournament properties, all of which are shared by almost all random tournaments. These properties, which we term "quasi-random:' have the property that tournaments possessing any one of the properties must of necessity possess them all. In contrast to random tournaments, however, it is often very easy to verify that a particular family of tournaments satisfies one of the quasi-random properties, thereby giving explicit tournaments with "random-like" behavior. This paper continues an approach initiated in several earlier papers of the authors where analogous results for graphs (with R. M. Wilson) and hypergraphs are proved.
Abstract We consider a class of routing problems on connected graphs G. Initially, each vertex v ... more Abstract We consider a class of routing problems on connected graphs G. Initially, each vertex v of G is occupied by a pebble which has a unique destination T (V) in G, so that mis a permutation of the vertices of G. It is required to route all the pebbles to their respective ...
The fractional covering number r* of a hypergraph H (V, E) is defined to be the minimum possible ... more The fractional covering number r* of a hypergraph H (V, E) is defined to be the minimum possible value of ,, v t(x) where ranges over all functions t: V which satisfy ,xe t(x) >= for all edges e e E. In the case of ordinary graphs G, it is known that 2r*(G) is always an integer. By contrast, it is shown (among other things) that for any rational p/q >= 1, there is a 3-uniform hypergraph H with -*(H) p/q.
Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms. [SIAM Journal on Com... more Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms. [SIAM Journal on Computing 3, 299 (1974)]. DS Johnson, A. Demers, JD Ullman, MR Garey, RL Graham. Abstract. The following abstract problem ...
... The authors gratefully acknowledge the contributions of DH and Emma Lehmer, whose ideas forme... more ... The authors gratefully acknowledge the contributions of DH and Emma Lehmer, whose ideas formed the basis for the proof of the theorem. REFERENCES 1. DA Burgess, On character sums and primitive roots, Proc. London Math. Soc.(3) 12 (1962), 179-192. ...
The function F(x) = (1/2-x) (1-x2)1/2+x(1+(1-(1/2+x)2)1/2) has a maximum at about x = .343771, wh... more The function F(x) = (1/2-x) (1-x2)1/2+x(1+(1-(1/2+x)2)1/2) has a maximum at about x = .343771, where it attains the value of approximately .674981. This value is the root of an irreducible polynomial of tenth degree over the integers; the problem is to find this polynomial. The obvious way of proceeding is as follows:(1) Differentiate F(x), set it equal to zero, and clear radicals. The result is a tenth degree polynomial P(x) over the integers which has a root at about x = .343771.
Given a simple polygon with rational coordinates having one vertex at the origin and an adjacent ... more Given a simple polygon with rational coordinates having one vertex at the origin and an adjacent vertex on the x-axis, we look at the problem of the location of the vertices for a tiling of the polygon using lattice triangles (i.e., triangles which are congruent to a triangle with the coordinates of the vertices being integer). We show that the
Suppose we are given some xed (but unknown) subset X of a set , and our object is to learn as muc... more Suppose we are given some xed (but unknown) subset X of a set , and our object is to learn as much as possible about the elements of X by asking binary questions. Specically, each question is just a function F : !f 0; 1g ,a nd the answer to F is just the value F (Xi )f orsome Xi2
In recent years, a variety of graph optimization problems have arisen in which the graphs involve... more In recent years, a variety of graph optimization problems have arisen in which the graphs involved are much too large for the usual algorithms to be effective. In these cases, even though we are not able to examine the entire graph (which may be changing dynamically), we would still like to deduce various properties of it, such as the size
A problem arising in taste testing, medical, and parapsychology experiments can be modeled as fol... more A problem arising in taste testing, medical, and parapsychology experiments can be modeled as follows. A deck of $n$ cards contains $c_i$ cards labeled $i, 1 \leq i \leq r$. A subject guesses at the cards sequentially. After each guess the subject is told the card just guessed (or at least if the guess was correct or not). We determine
The Electronic Journal of Combinatorics - Electr. J. Comb., 1997
We analyze a certain random walk on the cartesian product Gn of a flnite group G which is often u... more We analyze a certain random walk on the cartesian product Gn of a flnite group G which is often used for generating random elements from G. In particular, we show that the mixing time of the walk is at most crn2 logn where the constant cr depends only on the order r of G.
We introduce a large class of tournament properties, all of which are shared by almost all random... more We introduce a large class of tournament properties, all of which are shared by almost all random tournaments. These properties, which we term "quasi-random:' have the property that tournaments possessing any one of the properties must of necessity possess them all. In contrast to random tournaments, however, it is often very easy to verify that a particular family of tournaments satisfies one of the quasi-random properties, thereby giving explicit tournaments with "random-like" behavior. This paper continues an approach initiated in several earlier papers of the authors where analogous results for graphs (with R. M. Wilson) and hypergraphs are proved.
Abstract We consider a class of routing problems on connected graphs G. Initially, each vertex v ... more Abstract We consider a class of routing problems on connected graphs G. Initially, each vertex v of G is occupied by a pebble which has a unique destination T (V) in G, so that mis a permutation of the vertices of G. It is required to route all the pebbles to their respective ...
The fractional covering number r* of a hypergraph H (V, E) is defined to be the minimum possible ... more The fractional covering number r* of a hypergraph H (V, E) is defined to be the minimum possible value of ,, v t(x) where ranges over all functions t: V which satisfy ,xe t(x) >= for all edges e e E. In the case of ordinary graphs G, it is known that 2r*(G) is always an integer. By contrast, it is shown (among other things) that for any rational p/q >= 1, there is a 3-uniform hypergraph H with -*(H) p/q.
Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms. [SIAM Journal on Com... more Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms. [SIAM Journal on Computing 3, 299 (1974)]. DS Johnson, A. Demers, JD Ullman, MR Garey, RL Graham. Abstract. The following abstract problem ...
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Papers by Ronald Graham