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Abstract:: We consider the gelation of particles which are permanently connected by random crosslinks, drawn from an ensemble of finite-dimensional continuum percolation. To average over the randomness, we apply the replica trick, and interpret the replicated and crosslink-averaged model as an effective molecular fluid. A Mayer-cluster expansion for moments of the local static density fluctuations is set up. The simplest non-trivial contribution to this series leads back to mean-field theory. The central quantity of mean-field theory is the distribution of localization lengths, which we compute for all connectivities. The highly crosslinked gel is characterized by a one-to-one correspondence of connectivity and localization length. Taking into account higher contributions in the Mayer-cluster expansion, systematic corrections to mean-field can be included. The sol-gel transition shifts to a higher number of crosslinks per particle, as more compact structures are favored. The critical behavior of the model remains unchanged as long as finite truncations of the cluster expansion are considered. To complete the picture, we also discuss various geometrical properties of the crosslink network, e.g. connectivity correlations, and relate the studied crosslink ensemble to a wider class of ensembles, including the Deam-Edwards distribution.

Abstract:: In this paper, the dynamics of heuristic algorithms for constructing small vertex covers (or independent sets) of finite-connectivity random graphs is analysed. In every algorithmic step, a vertex is chosen with respect to its vertex degree. This vertex, and some environment of it, is covered and removed from the graph. This graph reduction process can be described as a Markovian dynamics in the space of random graphs of arbitrary degree distribution. We discuss some solvable cases, including algorithms already analysed using different techniques, and develop approximation schemes for more complicated cases. The approximations are corroborated by numerical simulations.

Abstract:: A variational approach to finite connectivity spin-glass-like models is developed and applied to describe the structure of optimal solutions in random satisfiability problems. Our variational scheme accurately reproduces the known replica symmetric results and also allows for the inclusion of replica symmetry breaking effects. For the 3-SAT problem, we find two transitions as the ratio α of logical clauses per Boolean variables increases. At the first one $${a_s} \simeq 3.96$$, a non-trivial organization of the solution space in geometrically separated clusters emerges. The multiplicity of these clusters as well as the typical distances between different solutions are calculated. At the second threshold $${a_c} \simeq 4.48$$, satisfying assignments disappear and a finite fraction $${B_o} \simeq 0.13$$ of variables are overconstrained and take the same values in all optimal (though unsatisfying) assignments. These values have to be compared to $${a_c} \simeq 4.27$$, $${B_o} \simeq 0.4$$ obtained from numerical experiments on small instances. Within the present variational approach, the SAT-UNSAT transition naturally appears as a mixture of a first and a second order transition. For the mixed 2+p-SAT with p<2/5, the behavior is as expected much simpler: a unique smooth transition from SAT to UNSAT takes place at $${a_c} = 1/(1 - p)$$.

Experiments were performed in the isolated perfused kidney of K+ adaptedRana pipiens to investigate the relationship between luminal K+ conductance and H+ transport in cells of the diluting segment. Inhibition of luminal Na+/H+ exchange by amiloride or by omission of luminal Na+ blocked luminal K+ conductance. Acidification of the kidney perfusate by elevation of pCO2 also reduced luminal K+ conductance. This effect could be prevented by furosemide. Since the steepest transcellular Na+ potential difference, directed from the lumen into the cell, is found when luminal Na+/Cl-/K+ cotransport is inhibited by furosemide, we conclude that luminal Na+/H+ exchange is most efficient at these conditions and thus could attenuate intracellular acidification.

The hypothesis was tested if the mineralocorticoid hormone aldosterone stimulates Na+/H+ exchange in "giant cells" fused from individual target cells of the distal nephron of the frog kidney. By means of microelectrodes, steady-state intracellular pH (pHi) and pHi recovery from an acid load were recorded continuously while the fused cells were exposed to aldosterone. Twenty minutes after addition of the hormone, pHi started to rise and reached a new steady state after about 60 min (delta pHi = 0.28 +/- 0.01). After hormone treatment, pHi recovered significantly faster in response to an intracellular acid load. The diuretic drug amiloride blocked pHi recovery. Experiments in intact tubules showed that aldosterone induces H+ and K+ secretion. Thus, intracellular alkalosis, mediated by Na+/H+ exchange, could serve as a signal that activates pH-sensitive K+ channels of the luminal cell membrane.