An analysis of mixed integer linear sets based on lattice point free convex sets

Split cuts are cutting planes for mixed integer programs whose validity is derived from maximal lattice point free polyhedra of the form $S:=\{x : \pi_0 \leq \pi^T x \leq \pi_0+1 \}$ called split sets. The set obtained by adding all split cuts is called the split closure, and the split closure is known to be a polyhedron. A split set $S$ has max-facet-width equal to one in the sense that $\max\{\pi^T x : x \in S \}-\min\{\pi^T x : x \in S \} \leq 1$. In this paper we consider using general lattice point free rational polyhedra to derive valid cuts for mixed integer linear sets. We say that lattice point free polyhedra with max-facet-width equal to $w$ have width size $w$. A split cut of width size $w$ is then a valid inequality whose validity follows from a lattice point free rational polyhedron of width size $w$. The $w$-th split closure is the set obtained by adding all valid inequalities of width size at most $w$. Our main result is a sufficient condition for the addition of a family of rational inequalities to result in a polyhedral relaxation. We then show that a corollary is that the $w$-th split closure is a polyhedron. Given this result, a natural question is which width size $w^*$ is required to design a finite cutting plane proof for the validity of an inequality. Specifically, for this value $w^*$, a finite cutting plane proof exists that uses lattice point free rational polyhedra of width size at most $w^*$, but no finite cutting plane proof that only uses lattice point free rational polyhedra of width size smaller than $w^*$. We characterize $w^*$ based on the faces of the linear relaxation

Screened Perturbation Theory to Three Loops

The thermal physics of a massless scalar field with a phi^4 interaction is studied within screened perturbation theory (SPT). In this method the perturbative expansion is reorganized by adding and subtracting a mass term in the lagrangian. We consider several different mass prescriptions that generalize the one-loop gap equation to two-loop order. We calculate the pressure and entropy to three-loop order and the screening mass to two-loop order. In contrast to the weak-coupling expansion, the SPT-improved approximations appear to converge even for rather large values of the coupling constant.Comment: 30 pages, 10 figure

CALL BROADCASTING AND AUTOMATED RECORDERS AS TOOLS FOR ANURAN SURVEYS IN A SUBARCTIC TUNDRA LANDSCAPE

Relatively little is known about population ecology of anurans in arctic and subarctic tundra regions, in part because it is difficult to survey anurans in these landscapes. Anuran survey protocols developed for temperate regions have limited applicability in arctic and subarctic tundra landscapes, which may lack roads and vehicle access, and experience variable and inclement weather during short anuran breeding seasons. To evaluate approaches to address some of the limitations of surveying anurans in tundra landscapes, we assessed the effectiveness of using breeding call broadcasts to increase detection of Boreal Chorus Frogs (Pseudacris maculata) and Wood Frogs (Lithobates sylvaticus) near Cape Churchill, Manitoba, Canada. We also evaluated how counts of anurans derived from automated audio recorders compared with those obtained simultaneously by observers. We detected on average 0.4 additional Wood Frogs per survey when we broadcasted calls (x = 0.82, SD = 1.38), an increase of > 40% compared to surveys without broadcasts (x = 1.24, SD = 1.51; Wilcoxon test; Z = 2.73, P = 0.006). In contrast, broadcasting Boreal Chorus Frog calls did not increase the number of chorus frog detections (Wilcoxon test; Z < 0.001, P > 0.90). Detections of Wood Frogs in a 100-m radius were lower via automated recorders (x = 0.60, SD = 0.87 SD) than by observers during simultaneous surveys (x = 0.96, SD = 1.27 Z = 2.07, P = 0.038), but those of Boreal Chorus Frogs were not different (x = 1.72, SD = 1.31;x = 1.44, SD = 1.5; Z = 1.55, P > 0.121). Our results suggest that broadcasting calls can increase detection of Wood Frogs, and that automated recorders are useful in detecting both Wood Frogs and Boreal Chorus Fogs in arctic and subarctic tundra landscapes

Enumeration of chord diagrams on many intervals and their non-orientable analogs

Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the former case, we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter, we instead record the number of boundary cycles of each fixed length. Second order, non-linear, algebraic partial differential equations are derived which are satisfied by these generating functions in each case giving efficient enumerative schemes. Moreover, these generating functions provide multi-parameter families of solutions to the KP hierarchy. For each model, there is furthermore a non-orientable analog, and each such model likewise has its own associated differential equation. The enumerative problems we solve are interpreted in terms of certain polygon gluings. As specific applications, we discuss models of several interacting RNA molecules. We also study a matrix integral which computes numbers of chord diagrams in both orientable and non-orientable cases in the model with bivalent vertices, and the large-N limit is computed using techniques of free probability.Comment: 23 pages, 7 figures; revised and extended versio

W Plus Multiple Jets at the LHC with High Energy Jets

We study the production of a W boson in association with n hard QCD jets (for n>=2), with a particular emphasis on results relevant for the Large Hadron Collider (7 TeV and 8 TeV). We present predictions for this process from High Energy Jets, a framework for all-order resummation of the dominant contributions from wide-angle QCD emissions. We first compare predictions against recent ATLAS data and then shift focus to observables and regions of phase space where effects beyond NLO are expected to be large.Comment: 19 pages, 9 figure

Comment on "Mean First Passage Time for Anomalous Diffusion"

We correct a previously erroneous calculation [Phys. Rev. E 62, 6065 (2000)] of the mean first passage time of a subdiffusive process to reach either end of a finite interval in one dimension. The mean first passage time is in fact infinite.Comment: To appear in Phys. Rev.

Solution to the 3-Loop Φ\Phi-Derivable Approximation for Massless Scalar Thermodynamics

We develop a systematic method for solving the 3-loop $\Phi$-derivable approximation to the thermodynamics of the massless $\phi^4$ field theory. The method involves expanding sum-integrals in powers of $g^2$ and m/T, where g is the coupling constant, m is a variational mass parameter, and T is the temperature. The problem is reduced to one with the single variational parameter m by solving the variational equations order-by-order in $g^2$ and m/T. At the variational point, there are ultraviolet divergences of order $g^6$ that cannot be removed by any renormalization of the coupling constant. We define a finite thermodynamic potential by truncating at $5^{th}$ order in g and m/T. The associated thermodynamic functions seem to be perturbatively stable and insensitive to variations in the renormalization scale.Comment: 57 pages, 10 figure