Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by $n$ generators and $\frac {n(n-1)}{2}$ relations for $n \leq 7$. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.

Hamiltonian Tomography in an Access-Limited Setting without State Initialization

We propose a scheme for the determination of the coupling parameters in a chain of interacting spins. This requires only time-resolved measurements over a single particle, simple data postprocessing and no state initialization or prior knowledge of the state of the chain. The protocol fits well into the context of quantum-dynamics characterization and is efficient even when the spin chain is affected by general dissipative and dephasing channels. We illustrate the performance of the scheme by analyzing explicit examples and discuss possible extensions.

The outflow ranking method for weighted directed graphs

A ranking method assigns to every weighted directed graph a (weak) ordering of the nodes. In this paper we axiomatize the ranking method that ranks the nodes according to their outflow using four independent axioms. Besides the well-known axioms of anonymity and positive responsiveness we introduce outflow monotonicity – meaning that in pairwise comparison between two nodes, a node is not doing worse in case its own outflow does not decrease and the other node’s outflow does not increase – and order preservation – meaning that adding two weighted digraphs such that the pairwise ranking between two nodes is the same in both weighted digraphs, then this is also their pairwise ranking in the ‘sum’ weighted digraph. The outflow ranking method generalizes the ranking by outdegree for directed graphs, and therefore also generalizes the ranking by Copeland score for tournaments.

SoC memory hierarchy derivation from dataflow graphs

Hardware synthesis from dataflow graphs of signal processing systems is a growing research area as focus shifts to high level design methodologies. For data intensive systems, dataflow based synthesis can lead to an inefficient usage of memory due to the restrictive nature of synchronous dataflow and its inability to easily model data reuse. This paper explores how dataflow graph changes can be used to drive both the on-chip and off-chip memory organisation and how these memory architectures can be mapped to a hardware implementation. By exploiting the data reuse inherent to many image processing algorithms and by creating memory hierarchies, off-chip memory bandwidth can be reduced by a factor of a thousand from the original dataflow graph level specification of a motion estimation algorithm, with a minimal increase in memory size. This analysis is verified using results gathered from implementation of the motion estimation algorithm on a Xilinx Virtex-4 FPGA, where the delay between the memories and processing elements drops from 14.2 ns down to 1.878 ns through the refinement of the memory architecture. Care must be taken when modeling these algorithms however, as inefficiencies in these models can be easily translated into overuse of hardware resources.

On the treatment of singularities of the Watson Hamiltonian for nonlinear molecules

The applicability of the Watson Hamiltonian for the description of nonlinear molecules—especially triatomic ones—has always been questioned, as the Jacobian of the transformation that leads to the Watson Hamiltonian, vanishes at the linear configuration. This results in singular behavior of the Watson Hamiltonian, giving rise to serious numerical problems in the computation of vibrational spectra, with unphysical, spurious vibrational states appearing among the physical vibrations, especially in the region of highly excited states. In this work, we analyze the problem and propose a simple way to confine the nuclear wavefunction in such a way that the spurious solutions are eliminated. We study the water molecule and observe an improvement compared with previous results. We also apply the method to the van der Walls molecule XeHe2.

A similarity measure for graphs with low computational complexity

We present and analyze an algorithm to measure the structural similarity of generalized trees, a new graph class which includes rooted trees. For this, we represent structural properties of graphs as strings and define the similarity of two Graphs as optimal alignments of the corresponding property stings. We prove that the obtained graph similarity measures are so called Backward similarity measures. From this we find that the time complexity of our algorithm is polynomial and, hence, significantly better than the time complexity of classical graph similarity methods based on isomorphic relations. (c) 2006 Elsevier Inc. All rights reserved.

Information theoretic measures of UHG graphs with low computational complexity

We introduce a novel graph class we call universal hierarchical graphs (UHG) whose topology can be found numerously in problems representing, e.g., temporal, spacial or general process structures of systems. For this graph class we show, that we can naturally assign two probability distributions, for nodes and for edges, which lead us directly to the definition of the entropy and joint entropy and, hence, mutual information establishing an information theory for this graph class. Furthermore, we provide some results under which conditions these constraint probability distributions maximize the corresponding entropy. Also, we demonstrate that these entropic measures can be computed efficiently which is a prerequisite for every large scale practical application and show some numerical examples. (c) 2007 Elsevier Inc. All rights reserved.

BYPASSING STATE INITIALIZATION IN HAMILTONIAN TOMOGRAPHY ON SPIN-CHAINS

We provide an extensive discussion on a scheme for Hamiltonian tomography of a spin-chain model that does not require state initialization [Phys. Rev. Lett. 102 ( 2009) 187203]. The method has spurred the attention of the physics community interested in indirect acquisition of information on the dynamics of quantum many-body systems and represents a genuine instance of a control-limited quantum protocol.

Detecting Anomalies in Graphs with Numeric Labels

This paper presents Yagada, an algorithm to search labelled graphs for anomalies using both structural data and numeric attributes. Yagada is explained using several security-related examples and validated with experiments on a physical Access Control database. Quantitative analysis shows that in the upper range of anomaly thresholds, Yagada detects twice as many anomalies as the best-performing numeric discretization algorithm. Qualitative evaluation shows that the detected anomalies are meaningful, representing a com- bination of structural irregularities and numerical outliers.