Perfect Paths Research Articles

The 50th anniversary of reverse transcriptase-and its ironic legacy in the time of coronavirus.

The 50th anniversary of reverse transcriptase-and its ironic legacy in the time of coronavirus.

The Gorenstein-projective modules over a monomial algebra

We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

Solutions to the T-Systems with Principal Coefficients

The $A_\infty$ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016).

Revision proofness

We analyze an equilibrium concept called revision-proofness for infinite-horizon games played by a dynasty of players. Revision-proofness requires strategies to be robust to joint deviations by multiple players and is a refinement of sub-game perfection. Sub-game perfect paths that can only be sustained by reversion to paths with payoffs below those of an alternative path are not revision-proof. However, for the important class of quasi-recursive games careful construction of off-equilibrium play can render many, and in some cases all, sub-game perfect paths revision-proof.

Perfect fractionation in multicomponent, multiphase systems

Perfect fractionation models help infer the conditions under which molten rock material travels from the planet’s interior to the Earth’s surface where it cools and crystallizes. Quantitative models of melt crystallization, perfect fractionation paths through P–T-composition space, return calculated values of predicted quantities that can be compared to glass, rock, and mineral compositions measured in lava flows. Perfect fractionation models are based on thermodynamics and material balance constraints. Linear combinations of chemical potentials define equilibrium positions. The composition of the melt follows a path directly away from the composition of the phases at saturation, a material balance criterion. The number of adjustable parameters in a perfect fraction model is limited to two by Duhem’s theorem. Perfect fractionation models with one or two phases at saturation require only one mole fraction and one adjustable parameter be specified to calculate the fractionation path. The temperature and compositions of the saturating phases are determined by the equilibrium equations for one- or two-phase saturation in a melt of known compositions. With three or more phases at saturation, the composition of the melt has to be adjusted in the general case. Fractionation paths can also be inferred from a sequence of thermodynamic states calculated by minimizing a thermodynamic potential. The minimization procedure produces a fraction path that is conceptually different from the perfect fractionation path. Perfect fractionation models can be constructed to conserve a particular thermodynamic potential or variable such as enthalpy or density.

Superdiffusion in a honeycomb billiard

We investigate particle transport in the honeycomb billiard which consists of connected channels placed on the edges of a honeycomb structure. The spreading of particles is superdiffusive due to the existence of ballistic trajectories which we term perfect paths. Simulations give a time exponent of 1.72 for the mean-square displacement and a starlike, i.e., anisotropic, particle distribution. We present an analytical treatment based on the formalism of continuous-time random walks and explain the anisotropic distribution under the assumption that the perfect paths follow the directions of the six lattice axes. Furthermore, we derive a relation between the time exponent and the exponent of the distribution function for trajectories close to a perfect path. In billiards with randomly distributed channels, conventional diffusion is always observed in the long-time limit, although for small disorder transient superdiffusional behavior exists. Our simulation results are again supported by an analytical analysis.

Optimistic stability in games of perfect information

Abstract I apply the ‘Theory of Social Situations’ (Greenberg, 1990) to a special class of extensive form games of perfect information and without chance moves. I prove the existence of an ‘Optimistic Stable Standard of Behavior’ (OSSB) which is associated with such games. This class includes all n -person discounted sequential bargaining games where a finite set of strictly positive offers can be made at each stage. The derived OSSB yields an appealing selection from among the subgame perfect paths. As is always the case, OSSB is strongly related to the von Neumann and Morgenstern abstract stable sets.

Planning Tactile Recognition Paths in Two and Three Dimensions

A general problem in model-based object recognition is how the system should proceed when the available data are ambiguous (i.e., when either the object or its location in space are not uniquely identified), We show that this "best next view" problem can be solved for two-dimensional polygonal objects solely by touch. The solution is a low-order polynominal algorithm that in corporates uncertainty from the data-analysis level to the path-planning level. It has been tested both by simulation and by implementation on an experimental robot system. In three dimensions, the problem becomes inherently nonlinear and quite difficult. We describe the three-dimensional problem, discuss possible approaches, present a taxonomy that subsumes previous attempts, and display simulation results that describe the efficacy of our approach. We show that the heuristic of attempting to decompose the three-dimensional, four-degree-of-freedom, non linear problem into a pair of two-degree-of-freedom linear problems produces perfect paths about half of the time and is the most efficient complete decomposition method that has been implemented to date. 1. We can rotate and translate an arbitrary starting line so that it coincides with the x-axis, and transform the edges accordingly. 2. These bounds are derived from physical properties of the sensor, such as the angle at which it will skid off a smooth surface. 3. If the sets do not intersect, then there is no line that obeys the constraints and passes through both edges. 4. Some of the critical points in this example have the same α value, so there are fewer dashes marked on the a-axis than dots in the figure. 5. For typical sensor uncertainty values propagated through the interpre tation process, rotational uncertainty vastly dominates translational uncer tainty. Once rotational uncertainty is managed, translational uncertainty can usually be incorporated in a straightforward manner, by offsetting in the appropriate direction. 6. The reader is reminded that the coordinate system is located and oriented for convenience of calculation.

A Property of Superconducting Metals

MR. BARTLETT brings up in his letter a very interesting view to explain the disappearance of the residual resistance at the threshold temperature in supraconductors. As this residual resistance is produced by impurities or structural imperfections, it is suggested that it can be short-circuited by the perfect (healthy) paths of the conducting metal, which suddenly acquire an abnormally high conductivity of quite a different order from that observed in ordinary metals.