Polyhedra Circuits and Their Applications

To better compute the volume and count the lattice points in geometric objects, we propose polyhedral circuits. Each polyhedral circuit characterizes a geometric region in Rd . They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of polyhedron. They can be also used to approximate a large class of d-dimensional manifolds in Rd . Barvinok [3] developed polynomial time algorithms to compute the volume of a rational polyhedron, and to count the number of lattice points in a rational polyhedron in Rd with a fixed dimensional number d. Let d be a fixed dimensional number, TV(d,n) be polynomial time in n to compute the volume of a rational polyhedron, TL(d,n) be polynomial time in n to count the number of lattice points in a rational polyhedron, where n is the total number of linear inequalities from input polyhedra, and TI(d,n) be polynomial time in n to solve integer linear programming problem with n be the total number of input linear inequalities. We develop algorithms to count the number of lattice points in geometric region determined by a polyhedral circuit in O(nd⋅rd(n)⋅TV(d,n)) time and to compute the volume of geometric region determined by a polyhedral circuit in O(n⋅rd(n)⋅TI(d,n)+rd(n)TL(d,n)) time, where rd(n) is the maximum number of atomic regions that n hyperplanes partition Rd . The applications to continuous polyhedra maximum coverage problem, polyhedra maximum lattice coverage problem, polyhedra (1−β) -lattice set cover problem, and (1−β) -continuous polyhedra set cover problem are discussed. We also show the NP-hardness of the geometric version of maximum coverage problem and set cover problem when each set is represented as union of polyhedra

Prolonged Depression-Like Behavior Caused by Immune Challenge: Influence of Mouse Strain and Social Environment

Immune challenge by bacterial lipopolysaccharide (LPS) causes short-term behavioral changes indicative of depression. The present study sought to explore whether LPS is able to induce long-term changes in depression-related behavior and whether such an effect depends on mouse strain and social context. LPS (0.83 mg/kg) or vehicle was administered intraperitoneally to female CD1 and C57BL/6 mice that were housed singly or in groups of 4. Depression-like behavior was assessed with the forced swim test (FST) 1 and 28 days post-treatment. Group-housed CD1 mice exhibited depression-like behavior 1 day post-LPS, an effect that leveled off during the subsequent 28 days, while the behavior of singly housed CD1 mice was little affected. In contrast, singly housed C57BL/6 mice responded to LPS with an increase in depression-like behavior that was maintained for 4 weeks post-treatment and confirmed by the sucrose preference test. Group-housed C57BL/6 mice likewise displayed an increased depression-like behavior 4 weeks post-treatment. The behavioral changes induced by LPS in C57BL/6 mice were associated with a particularly pronounced rise of interleukin-6 in blood plasma within 1 day post-treatment and with changes in the dynamics of the corticosterone response to the FST. The current data demonstrate that immune challenge with LPS is able to induce prolonged depression-like behavior, an effect that depends on genetic background (strain). The discovery of an experimental model of long-term depression-like behavior after acute immune challenge is of relevance to the analysis of the epigenetic and pathophysiologic mechanisms of immune system-related affective disorders

FDR-controlled metabolite annotation for high-resolution imaging mass spectrometry

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