Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry ? are solvable in time g(d,?) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and ?. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth. We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width

Finitely forcible graph limits are universal

The theory of graph limits represents large graphs by analytic objects called graphons. Graph limits determined by finitely many graph densities, which are represented by finitely forcible graphons, arise in various scenarios, particularly within extremal combinatorics. Lovasz and Szegedy conjectured that all such graphons possess a simple structure, e.g., the space of their typical vertices is always finite dimensional; this was disproved by several ad hoc constructions of complex finitely forcible graphons. We prove that any graphon is a subgraphon of a finitely forcible graphon. This dismisses any hope for a result showing that finitely forcible graphons possess a simple structure, and is surprising when contrasted with the fact that finitely forcible graphons form a meager set in the space of all graphons. In addition, since any finitely forcible graphon represents the unique minimizer of some linear combination of densities of subgraphs, our result also shows that such minimization problems, which conceptually are among the simplest kind within extremal graph theory, may in fact have unique optimal solutions with arbitrarily complex structure

Finitely forcible graph limits are universal

The theory of graph limits represents large graphs by analytic objects called graphons. Graph limits determined by finitely many graph densities, which are represented by finitely forcible graphons, arise in various scenarios, particularly within extremal combinatorics. Lovasz and Szegedy conjectured that all such graphons possess a simple structure, e.g., the space of their typical vertices is always finite dimensional; this was disproved by several ad hoc constructions of complex finitely forcible graphons. We prove that any graphon is a subgraphon of a finitely forcible graphon. This dismisses any hope for a result showing that finitely forcible graphons possess a simple structure, and is surprising when contrasted with the fact that finitely forcible graphons form a meager set in the space of all graphons. In addition, since any finitely forcible graphon represents the unique minimizer of some linear combination of densities of subgraphs, our result also shows that such minimization problems, which conceptually are among the simplest kind within extremal graph theory, may in fact have unique optimal solutions with arbitrarily complex structure

Measuring atomic NOON-states and using them to make precision measurements

A scheme for creating NOON-states of the quasi-momentum of ultra-cold atoms has recently been proposed [New J. Phys. 8, 180 (2006)]. This was achieved by trapping the atoms in an optical lattice in a ring configuration and rotating the potential at a rate equal to half a quantum of angular momentum . In this paper we present a scheme for confirming that a NOON-state has indeed been created. This is achieved by spectroscopically mapping out the anti-crossing between the ground and first excited levels by modulating the rate at which the potential is rotated. Finally we show how the NOON-state can be used to make precision measurements of rotation.Comment: 14 preprint pages, 7 figure

Modeling Life as Cognitive Info-Computation

This article presents a naturalist approach to cognition understood as a network of info-computational, autopoietic processes in living systems. It provides a conceptual framework for the unified view of cognition as evolved from the simplest to the most complex organisms, based on new empirical and theoretical results. It addresses three fundamental questions: what cognition is, how cognition works and what cognition does at different levels of complexity of living organisms. By explicating the info-computational character of cognition, its evolution, agent-dependency and generative mechanisms we can better understand its life-sustaining and life-propagating role. The info-computational approach contributes to rethinking cognition as a process of natural computation in living beings that can be applied for cognitive computation in artificial systems.Comment: Manuscript submitted to Computability in Europe CiE 201

3-Acetyl­benzoic acid

In the crystal structure of the title compound, C9H8O3, essentially planar mol­ecules [the carboxyl group makes a dihedral angle of 4.53 (7)° with the plane of the ring, while the acid group forms a dihedral angle of 3.45 (8)° to the ring] aggregate by centrosymmetric hydrogen-bond pairing of ordered carboxyl groups. This yields dimers which have two orientations in a unit cell, creating a herringbone pattern. In addition, two close C—H⋯O inter­molecular contacts exist: one is between a methyl H atom and the ketone of a symmetry-related mol­ecule and the other involves a benzene H atom and the carboxyl group O atom of another mol­ecule. The crystal studied was a non-merohedral twin with twin law [100, 00, 0] and a domain ratio of 0.8104(14): 0.1896(14)