Representation Of Partition Research Articles

The Concepts of Pseudo Compound Poisson and Partition Representations in Discrete Probability

The mathematical/statistical concepts of pseudo compound Poisson and partition representations in discrete probability are reviewed and clarified. A combinatorial interpretation of the convolution of geometric distributions in terms of a variant of Newton’s identities is obtained. The practical use of the twofold convolution leads to an improved goodness-of-fit for a data set from automobile insurance that was up to now not fitted satisfactorily.

Proof of the Andrews–Dyson–Rhoades conjecture on the spt-crank

The spt-crank of a vector partition, or an S-partition, was introduced by Andrews, Garvan and Liang. Let NS(m,n) denote the net number of S-partitions of n with spt-crank m, that is, the number of S-partitions (π1,π2,π3) of n with spt-crank m such that the length of π1 is odd minus the number of S-partitions (π1,π2,π3) of n with spt-crank m such that the length of π1 is even. Andrews, Dyson and Rhoades conjectured that {NS(m,n)}m is unimodal for any n, and they showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions. They obtained an asymptotic formula for the difference between the rank and crank of ordinary partitions, which implies NS(m,n)≥NS(m+1,n) for sufficiently large n and fixed m. In this paper, we introduce a representation of an ordinary partition, called the m-Durfee rectangle symbol, which is a rectangular generalization of the Durfee symbol introduced by Andrews. We give a proof of the conjecture of Andrews, Dyson and Rhoades. We also show that this conjecture implies an inequality between the positive rank and crank moments obtained by Andrews, Chan and Kim.

On the partition dimension of a class of circulant graphs

For a vertex v of a connected graph G(V,E) and a subset S of V, the distance between a vertex v and S is defined by d(v,S)=min{d(v,x):x∈S}. For an ordered k-partition π={S1,S2…Sk} of V, the partition representation of v with respect to π is the k-vector r(v|π)=(d(v,S1),d(v,S2)…d(v,Sk)). The k-partition π is a resolving partition if the k-vectors r(v|π), v∈V(G) are distinct. The minimum k for which there is a resolving k-partition of V is the partition dimension of G. Salman et al. [1] in their paper which appeared in Acta Mathematica Sinica, English Series proved that partition dimension of a class of circulant graph G(n,±{1,2}), for all even n⩾6 is four. In this paper we prove that it is three.

Comparison of soundscapes across the Bering Sea shelf, a biological perspective

Selectively decomposing and visualizing different aspects of an acoustic time series provides a greater understanding of the sources and environmental dynamics contributing to and shaping the temporal and spatial patterns of the measured sound. Source contributions to the overall soundscape vary in space and time, and the biological component is highly dependent on temporal, geographic, and oceanographic factors. The biological contributions to soundscapes along the eastern Bering Sea shelf are tightly coupled to the presence of sea ice and have strong annual patterns. The differences between locations reflect the relative contribution of different species to the soundscape. Using soundscape plots to display the acoustic environment during the overlap of peak vocal activity by different marine mammal species in the winter/spring season produces a visual representation of acoustic niche partitioning by Arctic species.

On the partition dimension of trees

Given an ordered partition Π={P1,P2,…,Pt} of the vertex set V of a connected graph G=(V,E), the partition representation of a vertex v∈V with respect to the partition Π is the vector r(v|Π)=(d(v,P1),d(v,P2),…,d(v,Pt)), where d(v,Pi) represents the distance between the vertex v and the set Pi. A partition Π of V is a resolving partition of G if different vertices of G have different partition representations, i.e., for every pair of vertices u,v∈V, r(u|Π)≠r(v|Π). The partition dimension of G is the minimum number of sets in any resolving partition of G. In this paper we obtain several tight bounds on the partition dimension of trees.

Characterization of process-oriented hydrologic model behavior with temporal sensitivity analysis for flash floods in Mediterranean catchments

Abstract. This paper presents a detailed analysis of 10 flash flood events in the Mediterranean region using the distributed hydrological model MARINE. Characterizing catchment response during flash flood events may provide new and valuable insight into the dynamics involved for extreme catchment response and their dependency on physiographic properties and flood severity. The main objective of this study is to analyze flash-flood-dedicated hydrologic model sensitivity with a new approach in hydrology, allowing model outputs variance decomposition for temporal patterns of parameter sensitivity analysis. Such approaches enable ranking of uncertainty sources for nonlinear and nonmonotonic mappings with a low computational cost. Hydrologic model and sensitivity analysis are used as learning tools on a large flash flood dataset. With Nash performances above 0.73 on average for this extended set of 10 validation events, the five sensitive parameters of MARINE process-oriented distributed model are analyzed. This contribution shows that soil depth explains more than 80% of model output variance when most hydrographs are peaking. Moreover, the lateral subsurface transfer is responsible for 80% of model variance for some catchment-flood events' hydrographs during slow-declining limbs. The unexplained variance of model output representing interactions between parameters reveals to be very low during modeled flood peaks and informs that model-parsimonious parameterization is appropriate to tackle the problem of flash floods. Interactions observed after model initialization or rainfall intensity peaks incite to improve water partition representation between flow components and initialization itself. This paper gives a practical framework for application of this method to other models, landscapes and climatic conditions, potentially helping to improve processes understanding and representation.

Surface energy partitioning over four dominant vegetation types across the United States in a coupled regional climate model (Weather Research and Forecasting Model 3–Community Land Model 3.5)

Accurate representation of surface energy partitioning is crucial for studying land surface processes and the climatic influence of land cover and land use change using coupled climate‐land surface models. A critical question for these models, especially for newly coupled ones, is whether they can adequately distinguish differences in surface energy partitioning among different vegetation types. We evaluated 3 years (2004–2006) of surface energy partitioning and surface climate over four dominant vegetation types (cropland, grassland, needleleaf evergreen forest, and broadleaf deciduous forest) across the United States in a recently coupled regional climate model, Weather Research and Forecasting Model 3–Community Land Model 3.5 (WRF3‐CLM3.5), by comparing model output to observations (AmeriFlux, Clouds and the Earth's Radiant Energy System (CERES), and Parameter‐elevation Regressions on Independent Slopes Model (PRISM) data) and to standard WRF model output. We found that WRF3‐CLM3.5 can capture the seasonal pattern in energy partitioning for needleleaf evergreen forest but needs improvements in cropland, grassland, and broadleaf deciduous forest. Correcting the leaf area index representation for cropland and grassland could immediately improve the simulation of latent heat flux and hence the energy partitioning. Adding an irrigation scheme is especially important for cropland in the Midwest, where the strongly coupled soil moisture and precipitation can form a positive feedback that reduces latent heat flux and increases the warm bias. For deciduous forest, the simulated excess latent heat flux before leaf emergence is mainly from soil evaporation, requiring further improvement in the soil evaporation scheme. Finally, the domain‐wide overestimated net radiation contributes to positive biases in sensible, latent, and ground heat flux, as well as surface temperature. The standard WRF simulation has a similar warm bias, implicating errors in modules other than the land surface code. A sensitivity test suggests that improved simulation of downward solar radiation could reduce the energy flux and temperature biases. After adding irrigation process and correcting the leaf area index, WRF3‐CLM3.5 appears reliable for studying conversions between natural grassland and irrigated cropland and between needleleaf evergreen forest and grassland.

A new class of lattice paths and partitions with n copies of n

Agarwal and Bressoud (Pacific J. Math.136(2) (1989) 209–228) defined a class of weighted lattice paths and interpreted several q-series combinatorially. Using the same class of lattice paths, Agarwal (Utilitas Math.53 (1998) 71–80; ARS Combinatoria76 (2005) 151–160) provided combinatorial interpretations for several more q-series. In this paper, a new class of weighted lattice paths, which we call associated lattice paths is introduced. It is shown that these new lattice paths can also be used for giving combinatorial meaning to certain q-series. However, the main advantage of our associated lattice paths is that they provide a graphical representation for partitions with n + t copies of n introduced and studied by Agarwal (Partitions with n copies of n, Lecture Notes in Math., No. 1234 (Berlin/New York: Springer-Verlag) (1985) 1–4) and Agarwal and Andrews (J. Combin. TheoryA45(1) (1987) 40–49).

Average Values of some Z-Parameters in a Random Set Partition

We find exact and asymptotic formulas for the average values of several statistics on set partitions: of Carlitz's $q$-Stirling distributions, of the numbers of crossings in linear and circular representations of set partitions, of the numbers of overlappings and embracings, and of the numbers of occurrences of a 2-pattern.

An Intelligent Model For Detection Of Post-Operative Infections

An effective intelligent diagnosis model is aiming to provide a comprehensive ANALYSIS to form optimal partitioning REPRESENTATION of patient data, and extracts the most significant features for each partition which raise the accuracy of diagnosis process. Optimal Clustering for support feature machine (OCSFM) is proposed to improve the feature selection in medical data classification comprises clustering, feature selection, and classification concepts which is based on fuzzy C-means, max-min, and support feature machine (SFM) models. Experiments have been conducted on database of surgical patients to detect postoperative infections. The performance of the method is evaluated using classification sensitivity, specificity, overall accuracy, and Matthew's correlation coefficient. The results show that the highest classification performance is obtained for the OCSFM model, and this is very promising compared to NaiveBayes, Linear Support Vector Machine (Linear SVM), Polykernal SVM, artificial neural network (ANN), and SFM models.

Grain size normalization: a case for post extraction normalization and inclusion of selective extraction procedures

The spatial mapping of trace metal and/or nutrient elemental concentrations in surface is often aided by normalization approaches that allow comparison of texturally, geochemically, and mineralogically dissimilar sites. Frequently, this is a grain‐size normalization at < 62.5 µm before chemical analysis. However, material that is > 62.5 µm (generally geochemically inert) may have fine‐grained coatings that bind components of interest and/or may consist of aggregates of fine‐grained sediment. The misplacement of these essential components within the coarse fraction of size‐normalized sediments may skew data and subsequent geochemical interpretations, especially when these components make up a substantial proportion of the sediment metal and/or nutrient binding capacity. In this article, sedimentary contaminant data were normalized using size normalization (wet sieving < 62.5 µm) and a post‐extraction normalization (PEN). PEN is a size‐based normalization procedure devised to minimize the misplacement of fine‐grained coatings/aggregates by applying a size and mineralogical normalization to inert digestion residuals, post‐chemical extraction. The data show that PEN provides a more ecologically and diagenetically interpretable representation of partitioning of elements between sediments in a freshwater reservoir. In addition, PEN samples are more rapidly collected, and consequently, PEN provides a cheaper alternative to traditional size normalization. In addition, application of the PEN allows the collection of data not only relevant to ecological health, but also relevant sedimentary diagenetic mobility data using a single sediment sample.

Dual Mechanism of Neuronal Ensemble Inhibition in Primary Auditory Cortex

Inhibition plays an essential role in shaping and refining the brain's representation of sensory stimulus attributes. In primary auditory cortex (A1), so-called "sideband" inhibition helps to sharpen the tuning of local neuronal responses. Several distinct types of anatomical circuitry could underlie sideband inhibition, including direct thalamocortical (TC) afferents, as well as indirect intracortical mechanisms. The goal of the present study was to characterize sideband inhibition in A1 and to determine its mechanism by analyzing laminar profiles of neuronal ensemble activity. Our results indicate that both lemniscal and nonlemniscal TC afferents play a role in inhibitory responses via feedforward inhibition and oscillatory phase reset, respectively. We propose that the dynamic modulation of excitability in A1 due to the phase reset of ongoing oscillations may alter the tuning of local neuronal ensembles and can be regarded as a flexible overlay on the more obligatory system of lemniscal feedforward type responses.

Front Representation of Set Partitions

Let $\pi$ be a set partition of $[n]=\{1,2,\ldots,n\}$. The standard representation of $\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block which does not contain any integer between i and j. The front representation of $\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block whose smallest integer is i. Using the front representation, we find a recurrence relation for the number of $12\cdots k12$-avoiding partitions for $k\geq2$. Similarly, we find a recurrence relation for the number of k-distant noncrossing partitions for $k=2,3$. We also prove that the front representation has several joint symmetric distributions for crossings and nestings as the standard representation does.

Symbolic partition refinement with automatic balancing of time and space

State space lumping is one of the classical means to fight the state space explosion problem in state-based performance evaluation and verification. Particularly when numerical algorithms are applied to analyze a Markov model, one often observes that those algorithms do not scale beyond systems of moderate size. To alleviate this problem, symbolic lumping algorithms have been devised to effectively reduce very large–but symbolically represented–Markov models to moderate size explicit representations. This lumping step partitions the Markov model in such a way that any numerical analysis carried out on the lumped model is guaranteed to produce exact results for the original system. But even this lumping preprocessing may fail due to time or memory limitations. This paper discusses the two main approaches to symbolic lumping, and combines them to improve on their respective limitations. The algorithm automatically converts between known symbolic partition representations in order to provide a trade-off between memory consumption and runtime. We show how to apply this algorithm for the lumping of Markov chains, but the same techniques can be adapted in a straightforward way to other models like Markov reward models, labeled transition systems, or interactive Markov chains.

Experimental Determination of LSER Parameters for a Set of 76 Diverse Pesticides and Pharmaceuticals

Linear solvation energy relationships (LSERs) have more recently been proposed as the method of choice to describe and/or predict the partitioning behavior of neutral organic compounds over a large range of environmental matrices and for a broad variety of compounds in a consistent manner. However, when dealing with more complex, polar compounds with multiple functional groups, it has also been noted that there is a severe lack of substance descriptors quantifying the different intermolecular interactions that these compounds may undergo. In this study, we used a system of eight reversed phase, normal phase, and hydrophilic interaction HPLC systems to determine the substance descriptors for H-bond donor (A) and acceptor (B) interactions and for polarizability and dipolarity (S) for a set of 76 complex compounds containing multiple functional groups, comprising mainly pesticides and some pharmaceuticals. The obtained substance descriptors for most compounds are unique in that values of A, S, and B are high and lie at the very upper end of the numerical range of currently known substance descriptors. The substance descriptors have been cross-compared against literature values of the octanol-water (Kow) and air-water (Kaw) partition coefficients and against a set of heptan-methanol partition coefficients (Khm) experimentally determined with a consistent methodology herein. The comparison confirmed plausibility of the substance descriptors of all except six tested compounds. The newly determined substance descriptors promise to be highly valuable in chemicalfate modeling, allowing, in conjunction with available phase descriptors, for a better representation of partitioning of polar compounds in those models. The results also reveal a systematic deviation of the log Kow values predicted with our substance descriptors from the literature values. The deviation points toward a possible problem when existing LSER equations are applied to polar, multifunctional compounds with high values of A, S, and B. Hence, the substance descriptors determined herein should also be helpful in revisiting the validity of existing LSERs for complex, polar compounds.

Noncrossing Normal Ordering for Functions of Boson Operators

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento, 10(13):565--567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a^{r}(a^{\dag})^{s})^{n}) and (a^{r}+(a^{\dag})^{s})^{n}, plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.

Investigative modeling of new pathways for secondary organic aerosol formation

Abstract. Recent advances in secondary organic aerosol (SOA) research are reviewed and the status of current understanding is investigated using a box model of SOA formation. Benzene and isoprene are newly identified precursors that are included in this SOA model; these precursors form SOA via secondary products. The model is also extended to include some representation of aqueous partitioning and the formation of high molecular weight products via oligomerization. Experimental data and empirical relationships are used where possible, because a detailed representation of SOA formation is not supported by the current state of information. Sensitivity studies are conducted with the SOA model and SOA predictions are found to be very sensitive to the treatment of the interactions between particulate water and organic compounds. While uncertainties due to model formulation are significant, influential model parameters include the aerosol partitioning ratios for several small products of isoprene and the partitioning constants for unidentified products (currently, the partitioning constants are derived by fitting experimental data). The pH value used as the reference for the activation of oligomerization is also a critical parameter. Recommendations for future work needed to improve SOA models include the elucidation of the water-organic relationship, the extent of phase separation, and laboratory experiments conducted under conditions more relevant to ambient studies (e.g., lower concentrations, higher relative humidity).

Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks

Atkin and Garvan introduced the moments of ranks of partitions in their work connecting ranks and cranks. Here we consider a combinatorial interpretation of these moments. This requires the introduction of a new representation for partitions, the Durfee symbol, and subsequent refinements. This in turn leads us to a variety of new congruences for our ‘marked’ Durfee symbols much in the spirit of Dyson’s original conjectures on the ranks of partitions.

Genealogies of Partition; History, History‐Writing and ‘the Troubles’ in Ireland

Contemporary political disputes have a long history of expression and contestation through the genre of history‐writing in Ireland. The role of history writing and political science writing during the nearly 40 years of the so‐called ‘Troubles’ has been no exception to this. Battles between competing versions of what the conflict ‘is about’, mediated through academic and popular texts have themselves in turn become constitutive of it. This builds upon centuries of the representation of the complicated politics of this island as ‘an issue’ in British domestic politics – first ‘the Catholic question’, then ‘the Irish question’. The location of political power outside the island for centuries has created successive battles for the representation of sectional interests in a metropolitan centre. The skills of propaganda, history writing, newspaper writing have consequently been deployed at a remarkable level of skill and intensity. In the recent period one of the consequences of this has been the removal from the debate of the actuality of partition; this builds upon a particular historical representation of partition as an historical inevitability. To seek to restore partition to the debate is not to call for its undoing but to recognise that seeking to circumvent debates about its origins in the key period of democratisation in Irish politics (1880–1920) has been counter‐productive. This essay examines the genealogies of partition in Irish and international contexts in the light of these battles for representation, and aims to return a lost dimension to the debate about the so‐called ‘Troubles’in Ireland. The genealogy of partition is the issue that has been marginalised in academic study and this has affected both policy and politics.

Modeling of the partitioning of membrane protein and phase equilibria for Triton X-100–salt aqueous two-phase systems using a modified generalized multicomponent osmotic virial equation

The partitioning of membrane protein and liquid–liquid equilibria for Triton X-100–salt–water aqueous two-phase systems were measured at 293.15 K. A generalized model was proposed to predict both partitioning of membrane protein and phase behavior of the surfactant aqueous two-phase systems. By using part of liquid–liquid equilibria (LLE) data, Triton X-100–Na 2SO 4 and Triton X-100–Na 2CO 3 aqueous two-phase systems were successfully predicted. The results showed that most of the average absolute deviations were less than 1.0% (w/w). The partition coefficient of membrane protein in Triton X-100–Na 2SO 4 and Triton X-100–Na 2CO 3 aqueous two-phase systems were also successfully predicted. The results showed that most of the average absolute deviations of the two systems were less than 5% and 8%, respectively. The model could give good representation of partitioning of membrane protein and the phase diagrams.