2,178 research outputs found
On a problem of Erd\H{o}s and Rothschild on edges in triangles
Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by
H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which
is contained in at least one triangle, must contain an edge that is in at least
H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether
for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all
sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed
C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best
possible in terms of the range for C, as it is known that every N-vertex graph
with more than (N^2)/4 edges contains an edge that is in at least N/6
triangles.Comment: 8 page
Mathematical Structure of Relativistic Coulomb Integrals
We show that the diagonal matrix elements where
are the standard Dirac matrix operators
and the angular brackets denote the quantum-mechanical average for the
relativistic Coulomb problem, may be considered as difference analogs of the
radial wave functions. Such structure provides an independent way of obtaining
closed forms of these matrix elements by elementary methods of the theory of
difference equations without explicit evaluation of the integrals. Three-term
recurrence relations for each of these expectation values are derived as a
by-product. Transformation formulas for the corresponding generalized
hypergeometric series are discussed.Comment: 13 pages, no figure
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