Unique Partition Research Articles

Making Bipartite Graphs DM-Irreducible

The Dulmage--Mendelsohn decomposition (or the DM-decomposition) gives a unique partition of the vertex set of a bipartite graph reflecting the structure of all the maximum matchings therein. A bipa...

Material distributions

The concept of material distribution is introduced as describing the geometric material structure of a general non-uniform body. Any smooth constitutive law is shown to give rise to a unique smooth integrable singular distribution. Ultimately, the material distribution and its associated singular foliation result in a rigorous and unique subdivision of the material body into strictly smoothly uniform components. Thus, the constitutive law induces a unique partition of the body into smoothly uniform sub-bodies, laminates, filaments and isolated points.

Distribution and risk assessment of heavy metals in overlying water, porewater, and sediments of Yongding River in a coal mine brownfield

Heavy metals from coal mines migrate into rivers and cause a severe impact on the health of local residents utilizing these rivers as a water source. However, current studies on coal mine brownfields have yet to thoroughly investigate river-associated heavy metals produced by coal mining. Therefore, this study aims to explore the distribution and risk assessment of heavy metals, namely Cr, Cu, Mn, Ni, Pb, and Zn, in overlying water, porewater, and sediments of the Yongding River, China. Overlying water, porewater, and sediments were sampled at 11 sites along the Yongding River. Heavy metal concentrations were determined through ICP-MS, and sediments were subjected to a three-step sequential extraction procedure to verify the behavior of heavy metals through principal component analysis (PCA). Sediments were separated into seven size fractions (< 4, 4–8, 8–16, 16–75, 75–150, 150–250, and 250–840 μm) with sedimentation method based on Stokes’ law. The characteristics of heavy metal distribution in different particle sizes were also explored using the Tucker 3 method. Interstitial water criteria toxic units (IWCTUs) were used to assess porewater heavy metal pollution. Risk assessment code (RAC), sediment quality guidelines (SQGs), and pollution load index (PLI) were applied to evaluate the ecological risk of heavy metals in sediments. Heavy metal concentrations increase in the following sequence: overlying water < porewater < sediment. Lead and Zn in porewater are higher than the criteria maximum concentrations (CMCs). The IWCTUs of Pb and Zn indicate their possible risk of toxicity. Principal component analysis reveals that Mn in sediments is preferentially associated with the exchangeable fraction, Cu and Zn are mainly associated with the reducible fraction, and Cr, Ni, and Pb are strongly related to the residual fractions. Heavy metals in sediments mainly accumulate in the fine fractions (< 4, 4–8, and 8–16 μm). Nickel, Pb, and Zn in sediments occasionally cause adverse biological effects. Overall, PLIs of sediments at sites 1, 4, and 10 in the Yongding River show moderate pollution due to Cr, Cu, Mn, Ni, Pb, and Zn. Different heavy metals possess unique partition coefficients between overlying water and sediments because these metals exhibit various properties and mobility. The chemical fractions of heavy metals also confirm the partition coefficients. Heavy metals in fine sediment fractions also enhance their migration and occasionally cause severe biological effects. Therefore, drinking water security in these areas should be ensured.

Stepping community detection algorithm based on label propagation and similarity

Community or module structure is one of the most common features in complex networks. The label propagation algorithm (LPA) is a near linear time algorithm that is able to detect community structure effectively. Nevertheless, when labeling a node, the LPA adopts the label belonging to the majority of its neighbors, which means that it treats all neighbors equally in spite of their different effects on the node. Another disadvantage of LPA is that the results it generates are not unique. In this paper, we propose a modified LPA called Stepping LPA-S, in which labels are propagated by similarity. Furthermore, our algorithm divides networks using a stepping framework, and uses an evaluation function proposed in this paper to select the final unique partition. We tested this algorithm on several artificial and real-world networks. The results show that Stepping LPA-S can obtain accurate and meaningful community structure without priori information.

Hierarchical image simplification and segmentation based on Mumford–Shah-salient level line selection

Hierarchies, such as the tree of shapes, are popular representations for image simplification and segmentation thanks to their multiscale structures. Selecting meaningful level lines (boundaries of shapes) yields to simplify image while preserving intact salient structures. Many image simplification and segmentation methods are driven by the optimization of an energy functional, for instance the celebrated Mumford–Shah functional. In this paper, we propose an efficient approach to hierarchical image simplification and segmentation based on the minimization of the piecewise-constant Mumford–Shah functional. This method conforms to the current trend that consists in producing hierarchical results rather than a unique partition. Contrary to classical approaches which compute optimal hierarchical segmentations from an input hierarchy of segmentations, we rely on the tree of shapes, a unique and well-defined representation equivalent to the image. Simply put, we compute for each level line of the image an attribute function that characterizes its persistence under the energy minimization. Then we stack the level lines from meaningless ones to salient ones through a saliency map based on extinction values defined on the tree-based shape space. Qualitative illustrations and quantitative evaluation on Weizmann segmentation evaluation database demonstrate the state-of-the-art performance of our method.

Induced Cycle Path Number of Graphs

Let G=(V, E) be a simple connected graph. An induced cycle path partition of G is a partition of V (G) into subsets such that the subsets induce cycles and paths. The minimum order taken over all induced cycle path partitions is called the induced cycle path number of G and is denoted by ρcp(G). In this paper, we dertermine this parameter for some standard graphs and investigate the same for unicyclic graphs and Halin graphs. We also obtain an upper bound for arbitrary graphs and characterize the extremal graphs. The graphs which have unique induced cycle path partition are also discussed

Externality effects in the formation of societies

We study a finite decision model where the utility function is an additive combination of a personal valuation component and an interaction component. Individuals are characterized according to these two components (their valuation type and externality type), and also according to their crowding type (how they influence others). We study how positive externalities lead to type symmetries in the set of Nash equilibria, while negative externalities allow the existence of equilibria that are not type-symmetric. In particular, we show that positive externalities lead to equilibria having a unique partition into a minimum number of societies (similar individuals using the same strategy, see [27]); and negative externalities lead to equilibria with multiple societal partitions, some with the maximum number of societies.

Exploring Fluorous Affinity by Liquid Chromatography.

Terms such as "fluorous affinity" and "fluorophilicity" have been used to describe the unique partition and sorption properties often exhibited by highly fluorinated organic compounds, that is molecules rich in sp(3) carbon-fluorine bonds. In this work, we made use of a highly fluorinated stationary phase and a series of benzene derivatives to study the effect of one single perfluorinated carbon on the chromatographic behavior and adsorption properties of molecules. For this purpose, the adsorption equilibria of α,α,α-trifluorotoluene, toluene, and other alkylbenzenes have been studied by means of nonlinear chromatography in a variety of acetonitrile/water eluents. Our results reveal that one single perfluorinated carbon is already enough to induce a drastic change in the adsorption properties of molecules on the perfluorinated stationary phase. In particular, it has been found that adsorption is monolayer if the perfluoroalkyl carbon is present but that, when this unit is missing, molecules arrange as multilayer stack structures. These findings can contribute to the understanding of molecular mechanisms of fluorous affinity.

Can fuzzy set theory bring complex issues in sizing air quality monitoring network into focus?

This paper presents an objective methodology for determining the optimum number of ambient air quality monitoring stations in a air quality monitoring network that will give maximum information of present air quality. The suggested two fold approach is a combination of fuzzy similarity measures and fuzzy c-mean (FCM) clustering. In the first stage, cosine amplitude—one of the fuzzy similarity measures is used to classify or to regroup monitoring stations for the most critical air pollutant PM10 and NO2 for the defined possibility (α-cut) levels. In the second stage, average values of PM10, NO2, SO2, CO and SPM for the winter months being a worst case scenario are used in FCM clustering. Finally, the optimal number of air quality monitoring locations is selected as those (1) which form single station partition (unique partition) in crisp relational matrix for PM10 and NO2 and (2) have highest membership in fuzzy clustering. The methodology has been demonstrated by applying to case study on Delhi Metro City in India. The air quality data of criteria pollutants was collected for 15 months from the installed 41 stations. The outcome of the study reveals that the city of Delhi needs only 16 monitoring stations which will result into sizable reduction in the capital cost and the recurring expenses in ambient air quality monitoring in Delhi.

On the type(s) of minimum size subspace partitions

Let V=V(kt+r,q) be a vector space of dimension kt+r over the finite field with q elements. Let σq(kt+r,t) denote the minimum size of a subspace partition P of V in which t is the largest dimension of a subspace. We denote by ndi the number of subspaces of dimension di that occur in P and we say [d1nd1,…,dmndm] is the type of P. In this paper, we show that a partition of minimum size has a unique partition type if t+r is an even integer. We also consider the case when t+r is an odd integer, but only give partial results since this case is indeed more intricate.

Uniquely monopolar-partitionable block graphs

Special issue PRIMA 2013 As a common generalization of bipartite and split graphs, monopolar graphs are defined in terms of the existence of certain vertex partitions. It has been shown that to determine whether a graph has such a partition is NP-complete for general graphs and polynomial for several classes of graphs. In this paper, we investigate graphs that admit a unique such partition and call them uniquely monopolar-partitionable graphs. By employing a tree trimming technique, we obtain a characterization of uniquely monopolar-partitionable block graphs. Our characterization implies a polynomial time algorithm for recognizing them.

SAR Image Classification Through Information-Theoretic Textural Features, MRF Segmentation, and Object-Oriented Learning Vector Quantization

Segmentation of optical images may be obtained through algorithms based on image prior models that exploit the spatial dependencies of land covers. In synthetic aperture radar (SAR) images, speckle conceals such spatial dependencies and segmentation algorithms suitable for optical images may become ineffective. Textural features may be used to emphasize spatial dependencies in the data and hence to improve segmentation. Once segmentation has been accomplished, a number of shapes is available. In this paper, the problem is tackled through the joint use of information-theoretic (IT) SAR features, of a segmentation algorithm based on tree structured Markov random fields (TS-MRFs), and of object-oriented classification achieved through learning vector quantization (LVQ). The proposed system works with one or more coregistered images, not necessarily all SAR, and one or more spatial maps of pixel features derived from each input image. A unique partition into connected regions, or segments, is achieved from the plurality of input channels, either images or feature maps. From each segment, representing a shape, geometric, radiometric, and textural parameters are extracted and fed to an LVQ classifier, trained through a partial reference ground truth (GT) of the scene. Classification results on a textured SAR image of a city and its surroundings validate the proposed object-oriented approach. Good performances can be achieved with small sizes of training sets, but they can be improved by using a decision fusion through majority voting (MV) of the outcomes of several experiments.

A Limit Law of Almost l-partite Graphs

AbstractFor integers l ≥ 1, d ≥ 0 we study (undirected) graphs with vertices 1, …, n such that the vertices can be partitioned into l parts such that every vertex has at most d neighbours in its own part. The set of all such graphs is denoted Pn (l, d). We prove a labelled first-order limit law, i.e., for every first-order sentence φ, the proportion of graphs in Pn (l, d) that satisfy φ converges as n → ∞. By combining this result with a result of Hundack, Prömel and Steger [12] we also prove that if 1 ≤ s1 ≤ … ≤ sl are integers, then Forb() has a labelled first-order limit law, where Forb() denotes the set of all graphs with vertices 1, …, n, for some n, in which there is no subgraph isomorphic to the complete (l + 1 )-partite graph with parts of sizes 1, s1, …, sl. In the course of doing this we also prove that there exists a first-order formula ξ, depending only on l and d, such that the proportion of ∈ Pn (l, d) with the following property approaches 1 as n → ∞: there is a unique partition of {1, …, n} into l parts such that every vertex has at most d neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by ξ.

Extended Reed-Solomon Codes for Optical CDMA

In this paper, the extended Reed-Solomon codes are modified to construct a new family of 2-D codes for asynchronous optical code-division multiple access (O-CDMA). In addition of having expanded and asymptotically optimal cardinality, these 2-D asynchronous optical codes can be partitioned into multiple tree structures of code subsets, in which code cardinality is a function of the (periodic) cross-correlation value assigned to the subset. The performance of these 2-D optical codes is analyzed and compared with that of the multilevel prime codes. Our results show that the unique partition property of the new optical codes supports a trade-off between code cardinality and performance for meeting different system requirements, such as user capacity and throughput. In addition, the multiple tree structures of the new codes potentially support applications that require rapid switching of many codewords, such as in O-CDMA-network gateway or in strategic environments where code obscurity is essential.

Sets of nonnegative matrices without positive products

For an arbitrary irreducible set of nonnegative d×d-matrices, we consider the following problem: does there exist a strictly positive product (with repetitions permitted) of those matrices? Under some general assumptions, we prove that if it does not exist, then there is a partition of the set of basis vectors of Rd, on which all given matrices act as permutations. Moreover, there always exists a unique maximal partition (with the maximal number of parts) possessing this property, and the number of parts is expressed by eigenvalues of matrices. This generalizes well-known results of Perron–Frobenius theory on primitivity of one matrix to families of matrices. We present a polynomial algorithm to decide the existence of a positive product for a given finite set of matrices and to build the maximal partition. Similar results are obtained for scrambling products. Applications to the study of Lyapunov exponents, inhomogeneous Markov chains, etc. are discussed.

A generalized flux function for three-dimensional magnetic reconnection

The definition and measurement of magnetic reconnection in three-dimensional magnetic fields with multiple reconnection sites is a challenging problem, particularly in fields lacking null points. We propose a generalization of the familiar two-dimensional concept of a magnetic flux function to the case of a three-dimensional field connecting two planar boundaries. In this initial analysis, we require the normal magnetic field to have the same distribution on both boundaries. Using hyperbolic fixed points of the field line mapping, and their global stable and unstable manifolds, we define a unique flux partition of the magnetic field. This partition is more complicated than the corresponding (well-known) construction in a two-dimensional field, owing to the possibility of heteroclinic points and chaotic magnetic regions. Nevertheless, we show how the partition reconnection rate is readily measured with the generalized flux function. We relate our partition reconnection rate to the common definition of three-dimensional reconnection in terms of integrated parallel electric field. An analytical example demonstrates the theory and shows how the flux partition responds to an isolated reconnection event.

Trends in Counter-Current Chromatography: Applications to Natural Products Purification

Counter-current chromatography (CCC), a unique continuous liquid-liquid partition chromatographic technique, has been widely used in the separation and purification of natural products. This technique has made great progress in recent years. In this paper, the different possible separation methods developed in CCC are described. It includes classical CCC, pH-zone-refining, pH-gradient, ion-pairing, multi-channel and two-dimensional CCC, dual-flow, liquid-liquid-liquid three-phase systems, elution-extrusion, gradient elution, powder direct injection, with some words for the various techniques used in pre- and post-CCC procedures, and detection modes. The basic mechanisms of each method are presented and applications are summarized. The future of CCC developments, including its application in protein purification and enrichment, and new platforms, are also discussed. This review paper aims to aid scientists who are interested in the field of CCC research for natural product isolation.

Counter-Current Chromatography for High Throughput Analysis of Natural Products

Counter-current chromatography (CCC) is a unique support-free liquid-liquid partition chromatography winning wide applications in the separation of various components from natural or synthetic mixtures. It has been one of the prime methods for isolating compounds from Traditional Chinese Medicines (TCM) and other comprehensive natural products. Although early CCC models produced a long-standing false image that CCC is a time-consuming technique, rapid and high-performance CCC devices and methods for high-throughput analysis of natural mixtures have been advanced. For instances, multi-channel CCC, dual CCC, elution-extrusion CCC, and solvent simplification protocols can provide high-throughput CCC analysis and produce high purity of compounds or large natural product libraries for drug discovery. This review summarizes the recent advancements of CCC in the high-throughput analysis of natural product with an emphasis on the developments of instruments and methods.

Multilevel Prime Codes for Optical CDMA Systems

In this paper, we study a new class of two-dimensional codes, here called multilevel prime codes, with expanded code cardinality by relaxing the maximum cross-correlation function to any arbitrary positive integer. Besides having asymptotically optimal cardinality and zero autocorrelation sidelobes, these multilevel prime codes can be partitioned into a tree structure of multiple levels of subsets of code matrices. In each level, the number of subsets, the number of code matrices per subset, and the cross-correlation function of each subset are related to the level number. The performance of the new codes in an optical code-division multiple-access system with hard-limiting detection is analyzed. Our results show that the unique partition property of the multilevel prime codes supports a trade-off between code cardinality and performance for meeting different system requirements, such as user capacity and throughput.

A GENERIC APPROACH TO COALITION FORMATION

We propose an abstract approach to coalition formation that focuses on simple merge and split rules transforming partitions of a group of players. We identify conditions under which every iteration of these rules yields a unique partition. The main conceptual tool is a specific notion of a stable partition. The results are parametrized by a preference relation between partitions of a group of players and naturally apply to coalitional TU-games, hedonic games and exchange economy games.