We present the path-sum formulation for OE [H](t ′ , t) = T exp t ′ t H(τ ) dτ , the time-ordered exponential of a time-dependent matrix H(t). The path-sum formulation gives OE[H] as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as selfavoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on graphs and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the timeordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.Keywords: Time-ordered exponential, path-ordered exponential, path-sum, differential equation, finite continued fraction, super-exponential decay ContextThe time-ordered exponential function OE[H](t ′ , t) = T exp t ′ t H(τ ) dτ , also known as path-ordered exponential, is the unique solution of the system of differential equationssuch that OE[H](t ′ , t ′ ) = I is the identity at all times. We take H ∈ C n×n [I], n ∈ N\{0}, to be a matrix depending smoothly on the continuous variable t ∈ I, which we call time without loss of generality. In spite of the importance of the

We introduce the method of path-sums which is a tool for analytically evaluating a function of a square discrete matrix, based on the closed-form resummation of infinite families of terms in the corresponding Taylor series. If the matrix is finite, our approach yields the exact result in a finite number of steps. We achieve this by combining a mapping between matrix powers and walks on a weighted directed graph with a universal graph-theoretic result on the structure of such walks. We present path-sum expressions for a matrix raised to a complex power, the matrix exponential, matrix inverse, and matrix logarithm. We present examples of the application of the path-sum method.

The di-ethanol solvate was characterised by means of a single-crystal X-ray diffraction study and 1 H NMR spectroscopy. The structure consists of an ethanol molecule hydrogenbonded in an asymmetric bifurcated fashion to the {Pt 2 S 2 } group, with the second, disordered ethanol molecule involved in a cooperative hydrogen-bonding interaction with the oxygen of the first ethanol. Thermogravimetric analysis shows that the alcohol is relatively easily lost, regenerating

A multi phase program was undertaken with the stated goal of using advanced design and development tools to create a unique combination of existing technologies to create a powertrain system specification that allowed minimal increase of volumetric fuel consumption when operating on E85 relative to gasoline. Although on an energy basis gasoline / ethanol blends typically return similar fuel economy to straight gasoline, because of its lower energy density (gasoline  31.8MJ/l and ethanol  21.1MJ/l) the volume based fuel economy of gasoline / ethanol blends are typically considerably worse.This project was able to define an initial engine specification envelope, develop specific hardware for the application, and test that hardware in both single and multi-cylinder test engines to verify the ability of the specified powertrain to deliver reduced E85 fuel consumption. Finally, the results from the engine testing were used in a vehicle drive cycle analysis tool to define a final vehicle level fuel economy result.During the course of the project, it was identified that the technologies utilized to improve fuel economy on E85 also enabled improved fuel economy when operating on gasoline. However, the E85 fueled powertrain provided improved vehicle performance when compared to the gasoline fueled powertrain due to the improved high load performance of the E85 fuel. Relative to the baseline comparator engine and considering current market fuels, the volumetric fuel consumption penalty when running on E85 with the fully optimized project powertrain specification was reduced significantly. This result shows that alternative fuels can be utilized in high percentages while maintaining or improving vehicle performance and with minimal or positive impact on total cost of ownership to the end consumer.The justification for this project was two-fold. In order to reduce the US dependence on crude oil, much of which is imported, the US Environmental Protection Agency (EPA) developed the Renewable Fuels Standard (RFS) under the Energy Policy Act of 2005. The RFS specifies targets for the amount of renewable fuel to be blended into petroleum based transportation fuels. The goal is to blend 36 billion gallons of renewable fuels into transportation fuels by 2022 (9 billion gallons were blended in 2008). The RFS also requires that the renewable fuels emit fewer greenhouse gasses than the petroleum fuels replaced. Thus the goal of the EPA is to have a more fuel efficient national fleet, less dependent on petroleum based fuels.The limit to the implementation of certain technologies employed was the requirement to run the developed powertrain on gasoline with minimal performance degradation. The addition of ethanol to gasoline fuels improves the fuels octane rating and increases the fuels evaporative cooling. Both of these fuel property enhancements make gasoline / ethanol blends more suitable than straight gasoline for use in downsized engines or engines with increased compression ratio. The use of engine downsizing and high compressio...