Reverse Chv\'atal-Gomory rank

We introduce the reverse Chv\'atal-Gomory rank r*(P) of an integral polyhedron P, defined as the supremum of the Chv\'atal-Gomory ranks of all rational polyhedra whose integer hull is P. A well-known example in dimension two shows that there exist integral polytopes P with r*(P) equal to infinity. We provide a geometric characterization of polyhedra with this property in general dimension, and investigate upper bounds on r*(P) when this value is finite.Comment: 21 pages, 4 figure

Subtropical Real Root Finding

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients. Then we employ linear programming to heuristically find roots. There is a specialized variant for roots with exclusively positive coordinates, which is of considerable interest for applications in chemistry and systems biology. An implementation of our method combining the computer algebra system Reduce with the linear programming solver Gurobi has been successfully applied to input data originating from established mathematical models used in these areas. We have solved several hundred problems with up to more than 800000 monomials in up to 10 variables with degrees up to 12. Our method has failed due to its incompleteness in less than 8 percent of the cases

Sparsity of integer solutions in the average case

We examine how sparse feasible solutions of integer programs are, on average. Average case here means that we fix the constraint matrix and vary the right-hand side vectors. For a problem in standard form with m equations, there exist LP feasible solutions with at most m many nonzero entries. We show that under relatively mild assumptions, integer programs in standard form have feasible solutions with O(m) many nonzero entries, on average. Our proof uses ideas from the theory of groups, lattices, and Ehrhart polynomials. From our main theorem we obtain the best known upper bounds on the integer Carathéodory number provided that the determinants in the data are small

Integrality gaps of integer knapsack problems

We obtain optimal lower and upper bounds for the (additive) integrality gaps of integer knapsack problems. In a randomised setting, we show that the integrality gap of a “typical” knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario

A magnetic wormhole

Wormholes are fascinating cosmological objects that can connect two distant regions of the universe. Because of their intriguing nature, constructing a wormhole in a lab seems a formidable task. A theoretical proposal by Greenleaf et al. presented a strategy to build a wormhole for electromagnetic waves. Based on metamaterials, it could allow electromagnetic wave propagation between two points in space through an invisible tunnel. However, an actual realization has not been possible until now. Here we construct and experimentally demonstrate a magnetostatic wormhole. Using magnetic metamaterials and metasurfaces, our wormhole transfers the magnetic field from one point in space to another through a path that is magnetically undetectable. We experimentally show that the magnetic field from a source at one end of the wormhole appears at the other end as an isolated magnetic monopolar field, creating the illusion of a magnetic field propagating through a tunnel outside the 3D space. Practical applications of the results can be envisaged, including medical techniques based on magnetism

Markedly Divergent Tree Assemblage Responses to Tropical Forest Loss and Fragmentation across a Strong Seasonality Gradient

We examine the effects of forest fragmentation on the structure and composition of tree assemblages within three seasonal and aseasonal forest types of southern Brazil, including evergreen, Araucaria, and deciduous forests. We sampled three southernmost Atlantic Forest landscapes, including the largest continuous forest protected areas within each forest type. Tree assemblages in each forest type were sampled within 10 plots of 0.1 ha in both continuous forests and 10 adjacent forest fragments. All trees within each plot were assigned to trait categories describing their regeneration strategy, vertical stratification, seed-dispersal mode, seed size, and wood density. We detected differences among both forest types and landscape contexts in terms of overall tree species richness, and the density and species richness of different functional groups in terms of regeneration strategy, seed dispersal mode and woody density. Overall, evergreen forest fragments exhibited the largest deviations from continuous forest plots in assemblage structure. Evergreen, Araucaria and deciduous forests diverge in the functional composition of tree floras, particularly in relation to regeneration strategy and stress tolerance. By supporting a more diversified light-demanding and stress-tolerant flora with reduced richness and abundance of shade-tolerant, old-growth species, both deciduous and Araucaria forest tree assemblages are more intrinsically resilient to contemporary human-disturbances, including fragmentation-induced edge effects, in terms of species erosion and functional shifts. We suggest that these intrinsic differences in the direction and magnitude of responses to changes in landscape structure between forest types should guide a wide range of conservation strategies in restoring fragmented tropical forest landscapes worldwide

Antiferromagnetic spintronics

Antiferromagnetic materials are magnetic inside, however, the direction of their ordered microscopic moments alternates between individual atomic sites. The resulting zero net magnetic moment makes magnetism in antiferromagnets invisible on the outside. It also implies that if information was stored in antiferromagnetic moments it would be insensitive to disturbing external magnetic fields, and the antiferromagnetic element would not affect magnetically its neighbors no matter how densely the elements were arranged in a device. The intrinsic high frequencies of antiferromagnetic dynamics represent another property that makes antiferromagnets distinct from ferromagnets. The outstanding question is how to efficiently manipulate and detect the magnetic state of an antiferromagnet. In this article we give an overview of recent works addressing this question. We also review studies looking at merits of antiferromagnetic spintronics from a more general perspective of spin-ransport, magnetization dynamics, and materials research, and give a brief outlook of future research and applications of antiferromagnetic spintronics.Comment: 13 pages, 7 figure

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Enhancing the sensitivity of magnetic sensors by 3D metamaterial shells

Magnetic sensors are key elements in our interconnected smart society. Their sensitivity becomes essential for many applications in fields such as biomedicine, computer memories, geophysics, or space exploration. Here we present a universal way of increasing the sensitivity of magnetic sensors by surrounding them with a spherical metamaterial shell with specially designed anisotropic magnetic properties. We analytically demonstrate that the magnetic field in the sensing area is enhanced by our metamaterial shell by a known factor that depends on the shell radii ratio. When the applied field is non-uniform, as for dipolar magnetic field sources, field gradient is increased as well. A proof-of-concept experimental realization confirms the theoretical predictions. The metamaterial shell is also shown to concentrate time-dependent magnetic fields upto frequencies of 100 kHz