The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary

We consider the dimer-monomer problem for the rectangular lattice. By mapping the problem into one of close-packed dimers on an extended lattice, we rederive the Tzeng-Wu solution for a single monomer on the boundary by evaluating a Pfaffian. We also clarify the mathematical content of the Tzeng-Wu solution by identifying it as the product of the nonzero eigenvalues of the Kasteleyn matrix.Comment: 4 Pages to appear in the Physical Review E (2006

Spanning Trees on Graphs and Lattices in d Dimensions

The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees $N_{ST}$ and establish inequalities relating the numbers of spanning trees of different graphs or lattices. A general formulation is presented for the enumeration of spanning trees on lattices in $d\geq 2$ dimensions, and is applied to the hypercubic, body-centered cubic, face-centered cubic, and specific planar lattices including the kagom\'e, diced, 4-8-8 (bathroom-tile), Union Jack, and 3-12-12 lattices. This leads to closed-form expressions for $N_{ST}$ for these lattices of finite sizes. We prove a theorem concerning the classes of graphs and lattices ${\cal L}$ with the property that $N_{ST} \sim \exp(nz_{\cal L})$ as the number of vertices $n \to \infty$, where $z_{\cal L}$ is a finite nonzero constant. This includes the bulk limit of lattices in any spatial dimension, and also sections of lattices whose lengths in some dimensions go to infinity while others are finite. We evaluate $z_{\cal L}$ exactly for the lattices we considered, and discuss the dependence of $z_{\cal L}$ on d and the lattice coordination number. We also establish a relation connecting $z_{\cal L}$ to the free energy of the critical Ising model for planar lattices ${\cal L}$.Comment: 28 pages, latex, 1 postscript figure, J. Phys. A, in pres

Exposing errors related to weak memory in GPU applications

© 2016 ACM.We present the systematic design of a testing environment that uses stressing and fuzzing to reveal errors in GPU applications that arise due to weak memory effects. We evaluate our approach on seven GPUS spanning three NVIDIA architectures, across ten CUDA applications that use fine-grained concurrency. Our results show that applications that rarely or never exhibit errors related to weak memory when executed natively can readily exhibit these errors when executed in our testing environment. Our testing environment also provides a means to help identify the root causes of such errors, and automatically suggests how to insert fences that harden an application against weak memory bugs. To understand the cost of GPU fences, we benchmark applications with fences provided by the hardening strategy as well as a more conservative, sound fencing strategy

Theory of impedance networks: The two-point impedance and LC resonances

We present a formulation of the determination of the impedance between any two nodes in an impedance network. An impedance network is described by its Laplacian matrix L which has generally complex matrix elements. We show that by solving the equation L u_a = lambda_a u_a^* with orthonormal vectors u_a, the effective impedance between nodes p and q of the network is Z = Sum_a [u_{a,p} - u_{a,q}]^2/lambda_a where the summation is over all lambda_a not identically equal to zero and u_{a,p} is the p-th component of u_a. For networks consisting of inductances (L) and capacitances (C), the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of lambda_a. This curious result suggests the possibility of practical applications to resonant circuits. Our formulation is illustrated by explicit examples.Comment: 21 pages, 3 figures; v4: typesetting corrected; v5: Eq. (63) correcte

Uniform tiling with electrical resistors

The electric resistance between two arbitrary nodes on any infinite lattice structure of resistors that is a periodic tiling of space is obtained. Our general approach is based on the lattice Green's function of the Laplacian matrix associated with the network. We present several non-trivial examples to show how efficient our method is. Deriving explicit resistance formulas it is shown that the Kagom\'e, the diced and the decorated lattice can be mapped to the triangular and square lattice of resistors. Our work can be extended to the random walk problem or to electron dynamics in condensed matter physics.Comment: 22 pages, 14 figure

Diffusion Processes on Small-World Networks with Distance-Dependent Random-Links

We considered diffusion-driven processes on small-world networks with distance-dependent random links. The study of diffusion on such networks is motivated by transport on randomly folded polymer chains, synchronization problems in task-completion networks, and gradient driven transport on networks. Changing the parameters of the distance-dependence, we found a rich phase diagram, with different transient and recurrent phases in the context of random walks on networks. We performed the calculations in two limiting cases: in the annealed case, where the rearrangement of the random links is fast, and in the quenched case, where the link rearrangement is slow compared to the motion of the random walker or the surface. It has been well-established that in a large class of interacting systems, adding an arbitrarily small density of, possibly long-range, quenched random links to a regular lattice interaction topology, will give rise to mean-field (or annealed) like behavior. In some cases, however, mean-field scaling breaks down, such as in diffusion or in the Edwards-Wilkinson process in "low-dimensional" small-world networks. This break-down can be understood by treating the random links perturbatively, where the mean-field (or annealed) prediction appears as the lowest-order term of a naive perturbation expansion. The asymptotic analytic results are also confirmed numerically by employing exact numerical diagonalization of the network Laplacian. Further, we construct a finite-size scaling framework for the relevant observables, capturing the cross-over behaviors in finite networks. This work provides a detailed account of the self-consistent-perturbative and renormalization approaches briefly introduced in two earlier short reports.Comment: 36 pages, 27 figures. Minor revisions in response to the referee's comments. Furthermore, some typos were fixed and new references were adde

The two-point resistance of a resistor network: A new formulation and application to the cobweb network

We consider the problem of two-point resistance in a resistor network previously studied by one of us [F. Y. Wu, J. Phys. A {\bf 37}, 6653 (2004)]. By formulating the problem differently, we obtain a new expression for the two-point resistance between two arbitrary nodes which is simpler and can be easier to use in practice. We apply the new formulation to the cobweb resistor network to obtain the resistance between two nodes in the network. Particularly, our results prove a recently proposed conjecture on the resistance between the center node and a node on the network boundary. Our analysis also solves the spanning tree problem on the cobweb network.Comment: 14 pages, 1 figure, adding one references and discussions of special case

Probabilistic Bisimulation: Naturally on Distributions

In contrast to the usual understanding of probabilistic systems as stochastic processes, recently these systems have also been regarded as transformers of probabilities. In this paper, we give a natural definition of strong bisimulation for probabilistic systems corresponding to this view that treats probability distributions as first-class citizens. Our definition applies in the same way to discrete systems as well as to systems with uncountable state and action spaces. Several examples demonstrate that our definition refines the understanding of behavioural equivalences of probabilistic systems. In particular, it solves a long-standing open problem concerning the representation of memoryless continuous time by memory-full continuous time. Finally, we give algorithms for computing this bisimulation not only for finite but also for classes of uncountably infinite systems

3D visualizations of nano-scale phase separation and ultrafast dynamic correlation between phases in (Na0.32K0.68)0.95Fe1.75Se2

Phase separation of metallic and antiferromagnetic (AFM) insulating phases in alkaline iron selenides (AxFe2-ySe2) continues to attract intense interest because the relationship between two peculiar features probably is a key to clarifying the pairing mechanism of AxFe2-ySe2 superconductors. Here we report that the 3D visualizations of nano-scale phase separation in (Na0.32K0.68)0.95Fe1.75Se2 single crystals are revealed by hybrid focused-ion-beam scanning electron microscopy and the superconducting paths are fully percolative in 3D. Moreover, the phase-related ultrafast dynamics in (Na0.32K0.68)0.95Fe1.75Se2 is studied by dual-color pump-probe spectroscopy. The anomalous changes in the electron and acoustic phonon components of transient reflectivity change ({\Delta}R/R) identify two characteristic temperatures T*~100 K (the onset temperature of coupling between nano-metallic and AFM phases) and TH~230 K (the onset temperature of metallic-interface-phase). An energy-transfer channel between the nano-metallic and AFM phases is inferred. This proposed channel provides a new insight into the pairing mechanism of alkaline iron selenide superconductors.Comment: 17 pages, 5 figures; Physical Review Materials, in pres

Spanning trees on the Sierpinski gasket

We obtain the numbers of spanning trees on the Sierpinski gasket $SG_d(n)$ with dimension $d$ equal to two, three and four. The general expression for the number of spanning trees on $SG_d(n)$ with arbitrary $d$ is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are also obtained.Comment: 20 pages, 8 figures, 1 tabl