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Volume 3, Number 6 - March 1923
Volume 3, Number 6 - March 1923. 38 pages including covers and advertisements.
Contents Eldy, Francis, The Silver Crown Keliher, J. F., The Indubitable Thomas Gibbon, Charles A., Try It Dwyer, Francis L., Unusual Boppell, Leo J., A Real Short Story K., J. F., Spring of Life Lynch, James H., 135 Bedside Eldy, Francis, Silver Plated Said the Walrus to the Carpenter K., J. F., Filio Dominici Editorial K., J. F., Youth Mitchell, J., College Chronicle Simpson, V. F., Eternal Promise Olivier, L., Exchang
Mathematical Theory of Exchange-driven Growth
Exchange-driven growth is a process in which pairs of clusters interact and
exchange a single unit of mass. The rate of exchange is given by an interaction
kernel which depends on the masses of the two interacting clusters. In
this paper we establish the fundamental mathematical properties of the mean
field kinetic equations of this process for the first time. We find two
different classes of behaviour depending on whether is symmetric or
not. For the non-symmetric case, we prove global existence and uniqueness of
solutions for kernels satisfying . This result is optimal in
the sense that we show for a large class of initial conditions with kernels
satisfying ( the solutions cannot exist. On
the other hand, for symmetric kernels, we prove global existence of solutions
for (
while existence is lost for
( In the intermediate regime we can only show
local existence. We conjecture that the intermediate regime exhibits
finite-time gelation in accordance with the heuristic results obtained for
particular kernels.Comment: Mistakes in the uniqueness proofs are fixed. Some typos are
corrected. Some references are adde
Near-colorings: non-colorable graphs and NP-completeness
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned
into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of
V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus
on complexity aspects of such colorings when l=2,3. More precisely, we prove
that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3,
either every planar graph with girth at least g is (k,j)-colorable or it is
NP-complete to determine whether a planar graph with girth at least g is
(k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine
whether a planar graph that is either (0,0,0)-colorable or
non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit
non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar
graphs with girth 7
Overscreened Single Channel Kondo Problem
We consider the single channel Kondo problem with the Kondo coupling between
a spin impurity and conduction electrons with spin . These problems
arise as multicritical points in the parameter spaces of two- and higher-level
tunneling systems, and some impurity models of heavy fermion compounds. In
contrast to the previous Bethe-anstaz conjectures, it turns out that the
dynamics of the spin sector is the same as that of a spin impurity coupled
to channels of spin electrons with . As a
result, for , the system shows non-Fermi liquid behavior with the
same exponents for the thermodynamic quantities as those of channel
Kondo problem. However, both the finite-size spectrum and the operator content
are different due to the presence of the other sectors and can be obtained by
conformal field theory techniques.Comment: 4 pages, revtex, no figures. Revised Versio
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