Conditions for duality between fluxes and concentrations in biochemical networks

Mathematical and computational modelling of biochemical networks is often done in terms of either the concentrations of molecular species or the fluxes of biochemical reactions. When is mathematical modelling from either perspective equivalent to the other? Mathematical duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one manner. We present a novel stoichiometric condition that is necessary and sufficient for duality between unidirectional fluxes and concentrations. Our numerical experiments, with computational models derived from a range of genome-scale biochemical networks, suggest that this flux-concentration duality is a pervasive property of biochemical networks. We also provide a combinatorial characterisation that is sufficient to ensure flux-concentration duality. That is, for every two disjoint sets of molecular species, there is at least one reaction complex that involves species from only one of the two sets. When unidirectional fluxes and molecular species concentrations are dual vectors, this implies that the behaviour of the corresponding biochemical network can be described entirely in terms of either concentrations or unidirectional fluxes

Efficient algorithms for conditional independence inference

The topic of the paper is computer testing of (probabilistic) conditional independence (CI) implications by an algebraic method of structural imsets. The basic idea is to transform (sets of) CI statements into certain integral vectors and to verify by a computer the corresponding algebraic relation between the vectors, called the independence implication. We interpret the previous methods for computer testing of this implication from the point of view of polyhedral geometry. However, the main contribution of the paper is a new method, based on linear programming (LP). The new method overcomes the limitation of former methods to the number of involved variables. We recall/describe the theoretical basis for all four methods involved in our computational experiments, whose aim was to compare the efficiency of the algorithms. The experiments show that the LP method is clearly the fastest one. As an example of possible application of such algorithms we show that testing inclusion of Bayesian network structures or whether a CI statement is encoded in an acyclic directed graph can be done by the algebraic method

The Convex Hull Problem in Practice : Improving the Running Time of the Double Description Method

The double description method is a simple but widely used algorithm for computation of extreme points in polyhedral sets. One key aspect of its implementation is the question of how to efficiently test extreme points for adjacency. In this dissertation, two significant contributions related to adjacency testing are presented. First, the currently used data structures are revisited and various optimizations are proposed. Empirical evidence is provided to demonstrate their competitiveness. Second, a new adjacency test is introduced. It is a refinement of the well known algebraic test featuring a technique for avoiding redundant computations. Its correctness is formally proven. Its superiority in multiple degenerate scenarios is demonstrated through experimental results. Parallel computation is one further aspect of the double description method covered in this work. A recently introduced divide-and-conquer technique is revisited and considerable practical limitations are demonstrated

On Computability of Equilibria in Markets with Production

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 of extending his complementary pivot algorithm to markets with production. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.Comment: An extended abstract will appear in SODA 201

Computation Reuse in Statics and Dynamics Problems for Assemblies of Rigid Bodies

The problem of determining the forces among contacting rigid bodies is fundamental to many areas of robotics, including manipulation planning, control, and dynamic simulation. For example, consider the question of how to unstack an assembly, or how to find stable regions of a rubble pile. In considering problems of this type over discrete or continuous time, we often encounter a sequence of problems with similar substructure. The primary contribution of our work is the observation that in many cases, common physical structure can be exploited to solve a sequence of related problems more efficiently than if each problem were considered in isolation. We examine three general problems concerning rigid-body assemblies: dynamic simulation, assembly planning, and assembly stability given limited knowledge of the structure\u27s geometry. To approach the dynamic simulation and assembly planning applications, we have optimized a known method for solving the system dynamics. The accelerations of and forces among contacting rigid bodies may be computed by formulating the dynamics equations and contact constraints as a complementarity problem. Dantzig\u27s algorithm, when applicable, takes n or fewer major cycles to find a solution to the linear complementarity problem corresponding to an assembly with n contacts. We show that Dantzig\u27s algorithm will find a solution in n - k or fewer major cycles if the algorithm is initialized with a solution to the dynamics problem for a subassembly with k internal contacts. Finally, we show that if we have limited knowledge of a structure\u27s geometry, we can still learn about stable regions of its surface by physically pressing on it. We present an approach for finding stable regions of planar assemblies: sample presses on the surface to identify a stable cone in wrench space, partition the space of applicable wrenches into stable and unstable regions, and map these back to the surface of the structure

The development of algorithms in mathematical programming

This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel University.In this thesis some problems in mathematical programming have been studied. Chapter 1 contains a brief review of the problems studied and the motivation for choosing these problems for further investigation. The development of two algorithms for finding all the vertices of a convex polyhedron and their applications are reported in Chapter 2. The linear complementary problem is studied in Chapter 3 and an algorithm to solve this problem is outlined. Chapter 4 contains a description of the plant location problem (uncapacited). This problem has been studied in some depth and an algorithm to solve this problem is presented. By using the Chinese representation of integers a new algorithm has been developed for transforming a nonsingular integer matrix into its Smith Normal Form; this work is discussed in Chapter 5. A hybrid algorithm involving the gradient method and the simplex method has also been developed to solve the linear programming problem. Chapter 6 contains a description of this method. The computer programs written in FORTRAN IV for these algorithms are set out in Appendices Rl to R5. A report on study of the group theory and its application in mathematical programming is presented as supplementary material. The algorithms in Chapter 2 are new. Part one of Chapter 3 is a collection of published material on the solution of the linear complementary problem; however the algorithm in Part two of this Chapter is original. The formulation of the plant location problem (uncapacited) together with some simplifications are claimed to be original. The use of Chinese representation of integers to transform an integer matrix into its Smith Normal Form is a new technique. The algorithm in Chapter 6 illustrates a new approach to solve the linear programming problem by a mixture of gradient and simplex method

NASA Tech Briefs, January 2007

Topics covered include: Flexible Skins Containing Integrated Sensors and Circuitry; Artificial Hair Cells for Sensing Flows; Video Guidance Sensor and Time-of-Flight Rangefinder; Optical Beam-Shear Sensors; Multiple-Agent Air/Ground Autonomous Exploration Systems; A 640 512-Pixel Portable Long-Wavelength Infrared Camera; An Array of Optical Receivers for Deep-Space Communications; Microstrip Antenna Arrays on Multilayer LCP Substrates; Applications for Subvocal Speech; Multiloop Rapid-Rise/Rapid Fall High-Voltage Power Supply; The PICWidget; Fusing Symbolic and Numerical Diagnostic Computations; Probabilistic Reasoning for Robustness in Automated Planning; Short-Term Forecasting of Radiation Belt and Ring Current; JMS Proxy and C/C++ Client SDK; XML Flight/Ground Data Dictionary Management; Cross-Compiler for Modeling Space-Flight Systems; Composite Elastic Skins for Shape-Changing Structures; Glass/Ceramic Composites for Sealing Solid Oxide Fuel Cells; Aligning Optical Fibers by Means of Actuated MEMS Wedges; Manufacturing Large Membrane Mirrors at Low Cost; Double-Vacuum-Bag Process for Making Resin- Matrix Composites; Surface Bacterial-Spore Assay Using Tb3+/DPA Luminescence; Simplified Microarray Technique for Identifying mRNA in Rare Samples; High-Resolution, Wide-Field-of-View Scanning Telescope; Multispectral Imager With Improved Filter Wheel and Optics; Integral Radiator and Storage Tank; Compensation for Phase Anisotropy of a Metal Reflector; Optical Characterization of Molecular Contaminant Films; Integrated Hardware and Software for No-Loss Computing; Decision-Tree Formulation With Order-1 Lateral Execution; GIS Methodology for Planning Planetary-Rover Operations; Optimal Calibration of the Spitzer Space Telescope; Automated Detection of Events of Scientific Interest; Representation-Independent Iteration of Sparse Data Arrays; Mission Operations of the Mars Exploration Rovers; and More About Software for No-Loss Computing