Unique Decomposition Research Articles

Fast factorization of Cartesian products of (directed) hypergraphs

Cartesian products of graphs and hypergraphs have been studied since the 1960s. For (un)directed hypergraphs, unique prime factor decomposition (PFD) results with respect to the Cartesian product are known. However, there is still a lack of algorithms, that compute the PFD of directed hypergraphs with respect to the Cartesian product.In this contribution, we focus on the algorithmic aspects for determining the Cartesian prime factors of a finite, connected, directed hypergraph and present a first polynomial time algorithm to compute its PFD. In particular, the algorithm has time complexity O(|E||V|r2) for hypergraphs H=(V,E), where the rank r is the maximum number of vertices contained in a hyperedge of H. If r is bounded, then this algorithm performs even in O(|E|log2⁡(|V|)) time. Thus, our method additionally improves also the time complexity of PFD-algorithms designed for undirected hypergraphs that have time complexity O(|E||V|r6Δ6), where Δ is the maximum number of hyperedges a vertex is contained in.

A novel approach for nurse rerostering based on a parallel algorithm

This paper addresses the Nurse Rerostering Problem (NRRP) that arises when a roster is disrupted by unexpected circumstances. The objective is to find a feasible roster having the minimal number of changes with respect to the original one. The problem is solved by a parallel algorithm executed on a graphics processing unit (GPU) to significantly accelerate its performance. To the best of our knowledge, this is the first parallel algorithm solving the NRRP on GPU. The core concept is a unique problem decomposition allowing efficient parallelization. Two parallel algorithms, homogeneous and heterogeneous, are proposed (available online), and their performance evaluated on benchmark datasets in terms of quality of the results compared to the state-of-the-art results and speedup. In general, higher acceleration was obtained by the homogeneous model with speedup 12.6 and 17.7 times on the NRRP dataset with 19 and 32 nurses respectively. These results encourage further research on parallel algorithms to solve Operational Research problems.

Unique decomposition of low-order time series

ABSTRACTWe give necessary and sufficient conditions on the parameters of processes ARMA(1, 1) and ARMA(2, 1) for representation of each as unique sums of independent simpler ARMA processes, including deduction from the sum process of the innovation variances of these summands. This work on inversion is motivated by examples in the article of Granger and Morris (1976) and by our earlier article (Ku and Seneta, 1998), to which the present article is a self-contained sequel. The theory is illustrated by the analysis of tree ring data.

Angle distribution of loading subspaces (ADLS) for chemical rank estimation in three-way analysis

Three-way analysis is increasingly popular because of the unique decomposition of the trilinear model, clear physical/chemical interpretation, and “second-order advantage,” which means the possible quantification of the interesting components in complex system with unknown interferents. The estimation of chemical rank is a crucial step in three-way analysis. In this paper, we propose a new method for the estimation of the chemical rank in three-way analysis, namely, angle distribution of loading subspace (ADLS). The stability of the three-way analysis model is tested by the variation of the angle distribution of the loading subspaces using bootstrapping. The largest number of components related to a stable model is selected as a chemical rank. We compared the results of ADLS and core consistency diagnostic (CORCONDIA), which is a well-known method for the estimation of chemical rank, based on two three-way analysis algorithms parallel factor analysis (PARAFAC) and self-weighted alternating trilinear decomposition (SWATLD). Using simulated and real data sets with “non-trilinear” structure, e.g., noise, the coherence and the presence of component with low sensitivity (s), the comparison shows that ADLS is more robust than CORCONDIA, and furthermore, the results of ADLS based on SWATLD are clearer than its results based PARAFAC.

High-Tc Layered Ferrielectric Crystals by Coherent Spinodal Decomposition.

Research in the rapidly developing field of 2D electronic materials has thus far been focused on metallic and semiconducting materials. However, complementary dielectric materials such as nonlinear dielectrics are needed to enable realistic device architectures. Candidate materials require tunable dielectric properties and pathways for heterostructure assembly. Here we report on a family of cation-deficient transition metal thiophosphates whose unique chemistry makes them a viable prospect for these applications. In these materials, naturally occurring ferrielectric heterostructures composed of centrosymmetric In4/3P2S6 and ferrielectrically active CuInP2S6 are realized by controllable chemical phase separation in van der Waals bonded single crystals. CuInP2S6 by itself is a layered ferrielectric with a ferrielectric transition temperature (Tc) just over room temperature, which rapidly decreases with homogeneous doping. Surprisingly, in our composite materials, the ferrielectric Tc of the polar CuInP2S6 phase increases. This effect is enabled by unique spinodal decomposition that retains the overall van der Waals layered morphology of the crystal, but chemically separates CuInP2S6 and In4/3P2S6 within each layer. The average spatial periodicity of the distinct chemical phases can be finely controlled by altering the composition and/or synthesis conditions. One intriguing prospect for such layered spinodal alloys is large volume synthesis of 2D in-plane heterostructures with periodically alternating polar and nonpolar phases.

Weighted pseudo almost periodic functions, convolutions and abstract integral equations

This paper deals with a systematic study of the convolution operator Kf=f⁎k defined on weighted pseudo almost periodic functions space PAP(X,ρ) and with k∈L1(R). Upon making several different assumptions on k,f and ρ, we get five main results. The first two main results establish sufficient conditions on k and ρ such that the weighted ergodic space PAP0(X,ρ) is invariant under the operator K. The third result specifies a sufficient condition on all functions (k,f and ρ) such that the Kf∈PAP0(X,ρ). The fourth result is a sufficient condition on the weight function ρ such that PAP0(X,ρ) is invariant under K. The hypothesis of the convolution invariance results allows to establish a fifth result related to the translation invariance of PAP0(X,ρ). As a consequence of the fifth result, we obtain a new sufficient condition such that the unique decomposition of a weighted pseudo almost periodic function on its periodic and ergodic components is valid and also for the completeness of PAP(X,ρ) with the supremum norm. In addition, the results on convolution are applied to general abstract integral and differential equations.

Unique parallel decomposition in branching and weak bisimulation semantics

We consider the property of unique parallel decomposition modulo branching and weak bisimilarity. First, we show that normed behaviours always have parallel decompositions, but that these are not necessarily unique. Then, we establish that finite behaviours have unique parallel decompositions. We derive the latter result from a general theorem about unique decompositions in partial commutative monoids.

Tunable photocatalytic selectivity and stability of Ba-doped Ag3PO4 hollow nanosheets

A one-step cation exchange process has been applied for the first time to the fabrication of nanoporous Ba-doped Ag3PO4 hollow nanosheets using Ba3(PO4)2 as the starting material. The constituent-dependent changes in phase and morphology were investigated comprehensively to elucidate the nanosheet formation mechanism. The resultant Ba-doped Ag3PO4 hollow nanosheets exhibited much higher photocatalytic performance and photoconversion efficiency than Ag3PO4 cubes and spherical particles under visible light irradiation. More importantly, the photocatalyst exhibited unique preferential decomposition of methyl orange over rhodamine B and high photocatalytic stability under visible light irradiation.

Minimality and fiber sum decompositions of Lefschetz fibrations

We give a short proof of a conjecture of Stipsicz on the minimality of fiber sums of Lefschetz fibrations, which was proved earlier by Usher. We then construct the first examples of genus g > 1 Lefschetz fibrations on minimal symplectic 4-manifolds which, up to diffeomorphisms of the summands, admit unique decompositions as fiber sums.

Chemometrics-assisted excitation–emission fluorescence analytical data for rapid and selective determination of optical brighteners in the presence of uncalibrated interferences

This study describes a novel approach for the simultaneous determination of CBS-X and CXT as widely used optical brighteners in household detergent, by combining the advantage of the high sensitivity of molecular fluorescence, and the selectivity of second-order chemometric methods. The proposed method is assisted by second-order chemometric analyses employing the PARAFAC, SWATLD and APTLD that help us to determine CBS-X and CXT in laundry powders and environmental samples, through the unique decomposition of the three-way data array. Proposed method can provide the extraction of relative concentrations of the analytes, as well as the spectral profiles. This approach achieves the second-order advantage and in principle could be able to overcome the spectral uncalibrated interference problems in the determination of CBS-X and CXT at the ng g−1 level.By spiking the known concentrations of these compounds to the real samples, the accuracy of the proposed methods was validated and recoveries of the spiked values were calculated. High recoveries (90.00%–113.33%) for the spiked laundry powders and real environmental samples indicate the present method successfully faces this complex challenge without the necessity of applying separation and preconcentration steps in environmental contaminations.

Algorithmic aspects of switch cographs

This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structures which has a unique linear-sized decomposition tree. We derive an O(n2) decomposition algorithm and we take advantage of the involution modular decomposition tree to obtain several algorithmic results. It is well-known that cographs (or equivalently P4-free graphs) are the graphs that are totally decomposable w.r.t modular decomposition. In a similar way, we introduce the class of switch cographs, as the class of graphs that are totally decomposable w.r.t involution modular decomposition. This class generalizes the class of cographs and appears to be exactly the class of (Bull, Gem, Co-Gem, C5)-free graphs. We use our new decomposition tool to design three practical algorithms for the maximum cut, vertex cover and vertex separator problems. The complexity of these problems was still unknown for this class of graphs. This paper also improves the complexity of the maximum clique, the maximum independent set, the chromatic number and the maximum clique cover problems by giving efficient algorithms, thanks to the decomposition tree. Eventually, we show that this class of graphs has clique-width at most 4 and that a clique-width expression can be computed in linear time.

PARAFAC Decomposition for Ultrasonic Wave Sensing of Fiber Bragg Grating Sensors: Procedure and Evaluation.

Ultrasonic wave-sensing technology has been applied for the health monitoring of composite structures, using normal fiber Bragg grating (FBG) sensors with a high-speed wavelength interrogation system of arrayed waveguide grating (AWG) filters; however, researchers are required to average thousands of repeated measurements to distinguish significant signals. To resolve this bottleneck problem, this study established a signal-processing strategy that improves the signal-to-noise ratio for the one-time measured signal of ultrasonic waves, by application of parallel factor analysis (PARAFAC) technology that produces unique multiway decomposition without additional orthogonal or independent constraints. Through bandpass processing of the AWG filter and complex wavelet transforms, ultrasonic wave signals are preprocessed as time, phase, and frequency profiles, and then decomposed into a series of conceptual three-way atoms by PARAFAC. While an ultrasonic wave results in a Bragg wavelength shift, antiphase fluctuations can be observed at two adjacent AWG ports. Thereby, concentrating on antiphase features among the three-way atoms, a fitting atom can be chosen and then restored to three-way profiles as a final result. An experimental study has revealed that the final result is consistent with the conventional 1024-data averaging signal, and relative error evaluation has indicated that the signal-to-noise ratio of ultrasonic waves can be significantly improved.

Functional and Structural Succession of Soil Microbial Communities below Decomposing Human Cadavers.

The ecological succession of microbes during cadaver decomposition has garnered interest in both basic and applied research contexts (e.g. community assembly and dynamics; forensic indicator of time since death). Yet current understanding of microbial ecology during decomposition is almost entirely based on plant litter. We know very little about microbes recycling carcass-derived organic matter despite the unique decomposition processes. Our objective was to quantify the taxonomic and functional succession of microbial populations in soils below decomposing cadavers, testing the hypotheses that a) periods of increased activity during decomposition are associated with particular taxa; and b) human-associated taxa are introduced to soils, but do not persist outside their host. We collected soils from beneath four cadavers throughout decomposition, and analyzed soil chemistry, microbial activity and bacterial community structure. As expected, decomposition resulted in pulses of soil C and nutrients (particularly ammonia) and stimulated microbial activity. There was no change in total bacterial abundances, however we observed distinct changes in both function and community composition. During active decay (7 - 12 days postmortem), respiration and biomass production rates were high: the community was dominated by Proteobacteria (increased from 15.0 to 26.1% relative abundance) and Firmicutes (increased from 1.0 to 29.0%), with reduced Acidobacteria abundances (decreased from 30.4 to 9.8%). Once decay rates slowed (10 - 23 d postmortem), respiration was elevated, but biomass production rates dropped dramatically; this community with low growth efficiency was dominated by Firmicutes (increased to 50.9%) and other anaerobic taxa. Human-associated bacteria, including the obligately anaerobic Bacteroides, were detected at high concentrations in soil throughout decomposition, up to 198 d postmortem. Our results revealed the pattern of functional and compositional succession in soil microbial communities during decomposition of human-derived organic matter, provided insight into decomposition processes, and identified putative predictor populations for time since death estimation.

Essential tangle decompositions of knots with tunnel number one tangles

It is shown by Ozawa that a knot in the 3-sphere has a unique essential tangle decomposition if it admits an essential free 2-tangle decomposition. We show that Ozawa's result cannot be generalized even if a knot admits an essential 2-tangle decomposition such that one of the decomposed tangles is of tunnel number one.

Compatible pairs of commuting isometries

The paper is devoted to pairs of commuting isometries. A unique decomposition of such pairs into a compatible and a completely non-compatible part is constructed. The compatible part is fully described.

Skewon modification of the light cone structure

Electromagnetic media with generic linear response provide a rich class of Lorentz violation models. In the framework of a general covariant metric-free approach, we study electromagnetic wave propagation in these media. We define the notion of an optic tensor and present its unique canonical irreducible decomposition into the principle and skewon parts. The skewon contribution to the Minkowski vacuum is a subject that does not arise in the ordinary models of Lorentz violation based on a modified Lagrangian. We derive the covector parametrization of the skewon optic tensor and discuss its $U(1)$-gauge symmetry. We obtain several compact expressions for the contribution of the principle and skewon optic tensor to the dispersion relation. As an application of the technique proposed here, we consider the case of a generic skewon tensor contributed to a simple metric-type principle part. Our main result: Every solution of the skewon modified Minkowski dispersion relation is necessary spacelike or null. It provides an extreme violation of the Lorentz symmetry. The case of the antisymmetric skewon is studied in detail and some new special cases (electric, magnetic, and degenerate) are discovered. In the case of a skewon represented by a symmetric matrix, we observe a parametric gap that has some similarity to the Higgs model. We worked out a set of specific examples that justify the generic properties of the skewon models and demonstrate the different types of the Lorentz violation phenomena.

Uniqueness of the direct decomposition of toric manifolds

In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as algebraic varieties is unique up to order of the factors. An algebraically indecomposable toric manifold happens to decompose as smooth manifold and no criterion is known for two toric manifolds to be diffeomorphic, so the unique decomposition problem for toric manifolds as smooth manifolds is highly nontrivial and nothing seems known for the problem so far. We prove that this problem is affirmative if the complex dimension of each factor in the decomposition is less than or equal to two. A similar argument shows that the direct decomposition of a smooth manifold into copies of $\mathbb{C}P^{1}$ and simply connected closed smooth 4-manifolds with smooth actions of $(S^{1})^{2}$ is unique up to order of the factors.

Zappa-Szép Products of Semigroups

The internal Zappa-Szep products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from actions of two semigroups on one another satisfying axiom derived by G. Zappa. We illustrate the correspondence between the two versions internal and the external of Zappa-Szep products of semigroups. We consider the structure of the internal Zappa-Szep product as an enlargement. We show how rectangular band can be described as the Zappa-Szep product of a left-zero semigroup and a right-zero semigroup. We find necessary and sufficient conditions for the Zappa-Szep product of regular semigroups to again be regular, and necessary conditions for the Zappa-Szep product of inverse semigroups to again be inverse. We generalize the Billhardt λ-semidirect product to the Zappa-Szep product of a semilattice E and a group G by constructing an inductive groupoid.

On the Uniqueness of Sparse Time-Frequency Representation of Multiscale Data

In this paper, we analyze the uniqueness of the sparse time-frequency decomposition and investigate the efficiency of the nonlinear matching pursuit method. Under the assumption of scale separation, we show that the sparse time-frequency decomposition is unique up to an error that is determined by the scale separation property of the signal. We further show that the unique decomposition can be obtained approximately by the sparse time-frequency decomposition using nonlinear matching pursuit.

Convergence Rate of Overlapping Domain Decomposition Methods for the Rudin--Osher--Fatemi Model Based on a Dual Formulation

This paper is concerned with overlapping domain decomposition methods (DDMs), based on successive subspace correction (SSC) and parallel subspace correction (PSC), for the Rudin--Osher--Fatemi (ROF) model in image restoration. In contrast to recent attempts, we work with a dual formulation of the ROF model, where one significant difficulty resides in the decomposition of the global constraint of the dual variable. We introduce a stable “unity decomposition” using a set of “partition of unity functions,” which naturally leads to overlapping DDMs based on the dual formulation. The main objective of this paper is to rigorously analyze the convergence of the SSC and PSC algorithms and derive the rate of convergence $O(n^{-1/2})$, where $n$ is the number of iterations. Moreover, we characterize the explicit dependence of the convergence rate on the subdomain overlapping size and other important parameters. To the best of our knowledge, such a convergence rate has not yet been claimed for domain decomposition related algorithms for the ROF model.