We describe a method for computing the number of Hamilton cycles in cubic polyhedral graphs. The Hamilton cycle counts are expressed in terms of a finite-state machine, and can be written as a matrix expression. In the special case of polyhedral graphs with repeating layers, the state machines become cyclic, greatly simplifying the expression for the exact Hamilton cycle counts, and let us calculate the exact Hamilton cycle counts for infinite series of graphs that are generated by repeating the layers. For some series, these reduce to closed form expressions, valid for the entire infinite series. When this is not possible, evaluating the number of Hamiltonian cycles admitted by the series' k-layer member is found by computing a (k - 1)th matrix power, requiring O(log(2)(k)) matrix-matrix multiplications. We demonstrate our technique for the two infinite series of fullerene nanotubes with the smallest caps. In addition to exact closed form and matrix expressions, we provide approximate exponential formulas for the number of Hamilton cycles.

The synthesis path of the C60-Buckyball fullerene from a planar precursor developed by Scott et al. [Science, 2002, 295, 5559] is investigated with density functional theory (DFT) methods. Various theoretically possible closing paths are analysed with respect to structural and energetic properties. The initial geometries were obtained by geometric interpolation of a cardboard-like model comprising rigid rings connected by hinges, which were then fully optimized with a selection of DFT-functionals. Analysis of the fully optimised geometries shows remarkable stability of face planarity, bond lengths and bond angles for all studied geometries, indicating soundness of the "cardboard with hinges"-model for approximating reaction paths for molecules of this type. This raises hope for development of a force field description of fullerene precursor molecules that can aid in discovery and analysis of good precursor candidates for rational synthesis of new fullerenes.

This work builds an automated anomaly detection method for chaotic time series, and more concretely for turbulent, high-dimensional, ocean simulations. We solve this task by extending the Echo State Network by spatially aware input maps, such as convolutions, gradients, cosine transforms, et cetera, as well as a spatially aware loss function. The spatial ESN is used to create predictions which reduce the detection problem to thresholding of the prediction error. We benchmark our detection framework on different tasks of increasing difficulty to show the generality of the framework before applying it to raw climate model output in the region of the Japanese ocean current Kuroshio, which exhibits a bimodality that is not easily detected by the naked eye. The code is available as an open source Python package, Torsk, available at https://github.com/nmheim/torsk, where we also provide supplementary material and programs that reproduce the results shown in this paper.

The first discussion of basis sets consisting of exponentially decaying Coulomb Sturmian functions for modelling electronic structures is presented. The proposed basis set construction selects Coulomb Sturmian functions using separate upper limits to their principle, angular momentum and magnetic quantum numbers. Their common Coulomb Sturmian exponent is taken as a fourth parameter. The convergence properties of such basis sets are investigated for second and third row atoms at the Hartree-Fock level. Thereby important relations between the values of the basis set parameters and the physical properties of the electronic structure are recognised. For example, an unusually large limit for the angular momentum quantum number in unrestricted Hartree-Fock calculations can be linked to the breaking of spherical symmetry in such cases. Furthermore, a connection between the optimal, i.e. minimum-energy, Coulomb Sturmian exponent and the average Slater exponents values obtained by Clementi and Raimondi (E. Clementi and D. L. Raimondi, J. Chem. Phys. 38, 2686 (1963)) is made. These features of Coulomb Sturmian basis sets emphasise their ability to correctly reproduce the physical features of Hartree-Fock wave functions.

A barrier to efficient array programming, for example in Python/NumPy, is that algorithms written as pure array operations completely without loops, while most efficient on small input, can lead to explosions in memory use. The present paper presents a solution to this problem using array streaming, implemented in the automatic parallelisation high-performance framework Bohrium. This makes it possible to use array programming in Python/NumPy code directly, even when the apparent memory requirement exceeds the machine capacity, since the automatic streaming eliminates the temporary memory overhead by performing calculations in per-thread registers. Using Bohrium, we automatically fuse, stream, JIT-compile, and execute NumPy array operations on GPGPUs without modification to the user programs. We present performance evaluations of three benchmarks, all of which show dramatic reductions in memory use from streaming, yielding corresponding improvements in speed and utilisation of GPGPU-cores. The fusion step is implemented using the theoretical framework presented in Kristensen et al. (2016), using a streaming-maximising cost function. The streaming-enabled Bohrium effortlessly runs programs on input sizes several orders of magnitude beyond sizes that crash on pure NumPy due to exhausting system memory.

We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$ and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is always exactly $\kappa$; (II) there are maximal chains in $\Pi_\kappa$ of cardinality $> \kappa$; (III) if, for every cardinal $\lambda < \kappa$, we have $2^{\lambda} < 2^\kappa$, there exists a maximal chain of cardinality $< 2^{\kappa}$ (but $\ge \kappa$) in $\Pi_{2^\kappa}$; (IV) every non-trivial maximal antichain in $\Pi_\kappa$ has cardinality between $\kappa$ and $2^{\kappa}$, and these bounds are realized. Moreover we can construct maximal antichains of cardinality $\max(\kappa, 2^{\lambda})$ for any $\lambda \le \kappa$; (V) all cardinals of the form $\kappa^\lambda$ with $0 \le \lambda \le \kappa$ occur as the number of complements to some partition $\mathcal{P} \in \Pi_\kappa$, and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterization.

We present the design of a flexible quantum-chemical method development framework, which supports employing any type of basis function. This design has been implemented in the light-weight program package molsturm, yielding a basis-function-independent self-consistent field scheme. Versatile interfaces, making use of open standards like python, mediate the integration of molsturm with existing third-party packages. In this way, both rapid extension of the present set of methods for electronic structure calculations as well as adding new basis function types can be readily achieved. This makes molsturm well-suitable for testing novel approaches for discretising the electronic wave function and allows comparing them to existing methods using the same software stack. This is illustrated by two examples, an implementation of coupled-cluster doubles as well as a gradient-free geometry optimisation, where in both cases, arbitrary basis functions could be used. molsturm is open-sourced and can be obtained from http://molsturm.org.

We present a method for evaluating 4-center electron repulsion integrals (ERI) for Slater-type orbitals by way of expansions in terms of Coulomb Sturmians. The ERIs can then be evaluated using our previously published methods for rapid evaluation of Coulomb Sturmians through hyperspherical harmonics. Numerical investigations are made of the efficiency in 1- and 2-center cases where the exact integrals can be evaluated. technology. This is due to the ease with which many-center interelectron repulsion integrals may be evaluated when Gaussians are used as a basis. In the present chapter, as in a number of our previous papers, we hope to contribute to a growing body of work whose aim is to increase the speed, ease of use, and popularity of ETO calculations. Success in this effort will open the doors to more accurate molecular calculations than those that can achieved using Gaussians as a basis.

Can we solve electronic wave equations absent a coordinate system? The question arises from the wish to treat polyhedral molecules such as fullerenes as two-dimensional closed surfaces. This would allow us to study electronic structure on intrinsic surface manifolds, which can be derived directly from the bond structure. The wave equation restricted to the (non-Euclidean) surface could then be solved without reference to any three-dimensional geometry of the molecule, and hence without the need for quantum chemical geometry optimization. The resulting 2D system can potentially be solved several orders of magnitude faster than the full wave equation. However, because these curved surfaces do not admit any simple coordinate system, we must devise methods that can do without. In this paper, I describe how surface geometries can be derived from fullerene bond graphs as combinatorial objects, and how electronic structure may be studied by solving wave equations directly on these intrinsic surface manifolds, without needing to find three-dimensional geometries. The goal is approximation methods that are rapid enough to systematically analyze entire isomer spaces consisting of millions of molecules, so as to identify structures with desired properties.

Many fundamental properties of graph C *-algebras may be determined directly from the structure of the underlying graph, and because of this, they have been celebrated as C *-algebras that can be seen. This paper shows how permutative endomorphisms of graph C *-algebras can be represented by labelled directed multigraphs that give visual representations of the endomorphisms and facilitate computations. This formalism provides a useful calculus for permutative automorphisms and allows efficient exhaustive construction of such automorphisms.

We present a general face-spiral algorithm for cubic polyhedral graphs (including fullerenes and fulleroids), and extend it to the full class of all polyhedral graphs by way of the leapfrog transform. This yields compact canonical representations of polyhedra with a simple and intuitive geometrical interpretation, well suited for use by both computers and humans. Based on the algorithm, we suggest a unique, unambiguous, and simple notation for canonical naming of polyhedral graphs, up to automorphism, from which the graph is easily reconstructed. From this, we propose a practical nomenclature for all polyhedral molecules, and an especially compact form for the special class of fullerenes. A unique numbering of vertices is obtained as a byproduct of the spiral algorithm. This is required to denote modifications of the parent cage in IUPAC naming schemes. Similarly, the symmetry group of the molecule can be found together with the canonical general spiral at negligible cost. The algorithm is fully compatible with the classical spiral algorithm developed by Manolopoulos for fullerenes, i.e., classical spirals are accepted as input, and spiralable graphs lead to identical output. We prove that the algorithm is correct and complete.The worst case runtime complexity is for general N-vertex polyhedral graphs, with J the sum of all jump lengths. When the number of faces of any particular size is bounded by a constant, such as the case for fullerenes, this reduces to . We have calculated canonical general spirals for all 2,157,751,423 fullerene isomers from C-20 to C-200, as well as for all fullerene graphs that require jumps up to C-400. Further, we have calculated canonical general spirals for large fullerenes with few or no classical spirals: all the Goldberg-Coxeter transforms up to C-50,C-000 of the the non-spiralable chiral T-C-380, D-3-C-384, D-3-C-440, and D-3-C-672 fullerenes, and for assorted fullerenes with no pentagon spiral starts. We verify exhaustively that the algorithm is linear for all the 2.7x10(12) fullerene isomers up to C-400, and show that this holds also for 11,413 large GC-transform fullerenes up to C-50,C-000. On the used hardware, each single general spiral took about Nx200ns to produce for a C-N fullerene, and the canonical general spiral was found in Nx22s-32s. Hence, we claim the algorithm to be efficient even for very large polyhedra.The algorithm is implemented in our program package Fullerene. In addition, the source code for a reference implementation of our proposed nomenclature for polyhedral molecules can be downloaded from http://erda.ku.dk/vgrid/Polyhedra/spirals/.

We address the problem of fusing array operations based on criteria such as shape compatibility, data reuse, and minimizing communication. We formulate the problem as a partitioning problem (WSP) that is general enough to handle loop fusion, combinator fusion, and other types of fusion analysis. Traditionally, when optimizing for data reuse, the fusion problem has been formulated as a static weighted graph partitioning problem (known as the Weighted Loop Fusion problem). We show that this scheme cannot accurately track data reuse between multiple independent loops, since it overestimates total data reuse of certain cases. Our formulation in terms of partitions allows use of realistic cost functions that can track resource usage accurately. We give correctness proofs, and prove that WSP can maximize data reuse in programs exactly, in contrast to prior work. For the exact optimal solution, which is NP-hard to find, we present a branch-and-bound algorithm together with a polynomial-time preconditioner that reduces the problem size significantly in practice. We further present a polynomialtime greedy approximation that is fast enough to use for JIT-compilation and gives near-optimal results in practice. All algorithms have been implemented in the automatic parallelization platform Bohrium, run on a set of benchmarks, and compared to existing methods from the literature.

We address the problem of fusing array operations based on criteria such as shape compatibility, data reusability, and communication. We formulate the problem as a graph partition problem that is general enough to handle loop fusion, combinator fusion, and other types of subroutines.

A barrier to efficient array programming, for example in Python/NumPy, is that algorithms written as pure array operations completely without loops, while most efficient on small input, can lead to explosions in memory use. The present paper presents a solution to this problem using array streaming, implemented in the automatic parallelization high-performance framework Bohrium. This makes it possible to use array programming in Python/NumPy code directly, even when the apparent memory requirement exceeds the machine capacity, since the automatic streaming eliminates the temporary memory overhead by performing calculations in per-thread registers. Using Bohrium, we automatically fuse, JIT-compile, and execute NumPy array operations on GPGPUs without modification to the user programs. We present performance evaluations of three benchmarks, all of which show dramatic reductions in memory use from streaming, yielding corresponding improvements in speed and utilization of GPGPU-cores. The streaming-enabled Bohrium effortlessly runs programs on input sizes much beyond sizes that crash on pure NumPy due to exhausting system memory.

Jacobi coordinates have often been used to treat few-particle quantum systems. In this paper, we propose an alternative coordinate system which is very easily generalized as the number of particles becomes larger. We make use of forms of the Laplace–Beltrami operator that are invariant under general coordinate transformations, and use Coulomb Sturmian basis sets to solve the N -body wave equation.

In this article, we discuss a way in which the theory of hyperspherical harmonics may be used for rapid evaluation of difficult molecular integrals when exponential‐type orbitals (ETOs) are used as a basis. One of us (J.E.A.) has implemented the method, and programs are available for general use. As a byproduct of this work, we are also able to evaluate generalized scattering factors for ETOs which allow first‐order density matrices to be measured experimentally using high‐quality X‐ray diffraction data. © 2015 Wiley Periodicals, Inc.

Exponential-type orbitals are better suited to calculations of molecular electronic structure than are Gaussians, since ETO's can accurately represent the behavior of molecular orbitals near to atomic nuclei, as well as their long-distance exponential decay. Orbitals based on Gaussians fail in both these respects. Nevertheless, Gaussian technology continues to dominate computational quantum chemistry, because of the ease with which difficult molecular integrals may be evaluated when Gaussians are used as a basis.In the present chapter, we hope to contribute to a new movement in quantum chemistry, in which ETO's will not only be able to produce more accurate results than could be obtained using Gaussians, but also will compete with Gaussian technology in the speed of integral evaluation. The method presented here makes use of V. Fock's projection of three-dimensional momentum-space onto a four-dimensional hypersphere. Using this projection, Fock was able to show that the Fourier transforms of Coulomb Sturmian basis functions are very simply related to four-dimensional hyperspherical harmonics.With the help of Fock's relationships and the theory of hyperspherical harmonics we are able to evaluate molecular integrals based on Coulomb Sturmians both rapidly and accurately. The method is then extended to Slater-Type Orbitals by using a closed-form expression for expanding STO's in terms of Coulomb Sturmians. A general theorem is presented for the rapid evaluation of the necessary angular and hyperangular integrals. The general methods are illustrated by a few examples.

The use of Slater type orbitals in molecular calculations is hindered by the slowness of integral evaluation. In the present paper, we introduce a method for overcoming this problem by expanding STO’s in terms of Coulomb Sturmians, for which the problem of evaluating molecular integrals rapidly has been satisfactorily solved using methods based on the theory of hyperspherical harmonics.