Shellable graphs and sequentially Cohen-Macaulay bipartite graphs

Associated to a simple undirected graph G is a simplicial complex whose faces correspond to the independent sets of G. We call a graph G shellable if this simplicial complex is a shellable simplicial complex in the non-pure sense of Bjorner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we classify all the shellable bipartite graphs; they are precisely the sequentially Cohen-Macaulay bipartite graphs. We also give an recursive procedure to verify if a bipartite graph is shellable. Because shellable implies that the associated Stanley-Reisner ring is sequentially Cohen-Macaulay, our results complement and extend recent work on the problem of determining when the edge ideal of a graph is (sequentially) Cohen-Macaulay. We also give a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.Comment: 16 pages; more detail added to some proofs; Corollary 2.10 was been clarified; the beginning of Section 4 has been rewritten; references updated; to appear in J. Combin. Theory, Ser.

CAPABILTY FOR PROCESSING POWER REACTOR FUELS

The reprocessing of commercial power reactor fuels in existing or modified facilities has been examined for four different reprocessing rates. Existing capabilities in the 325A hot cells will permit processing up to 2 kg of fuel per day using an 8-stage centrifugal contactor and discarding the uranium-plutonium stream in the organic phase. The principal problems with this system are feed clarification, equipment reliability, and waste disposal. The cost is anticipated to be about $30,000 per run for runs up to 10 kg each. Higher assurance of continued operation could be obtained by fabricating a spare contactor and replacement parts. An increased capacity of up to about 50 kg per day could be achieved by installation of two inch pulse columns in 325A followed by separating the plutonium from the uranium by anion exchange. The existing dissolver system would be nearly adequate. The plutonium would then be sent to ARHCO for further purification and recovery. The first cycle waste would be collected and shipped to 324 without further concentration. The cost will be about$2,000,000 over a nine month period for equipment installation and checkout. After installation and checkout, operating costs will be about $1,500,000 per year. The annual throughput expected in this system will depend on the operating efficiency. At an operating efficiency of 75% the average throughput would be slightly over one tonne per month. One disadvantage of this method. of operation is that it will occupy over half of the 325 hot cells, and could conflict with other work in the cells. The major uncertainties in this processing are possible zirconium fires during shearing, offgas cleanup and waste disposal problems. An increase to 200 kg per day could be achieved by installing 4 inch diameter pulse columns in the 324 hot cells. Columns of this size are possible in 324 but not in 325A because of the higher operating space available in 324. The major drawback to this arrangement is that installation of a major dissolution and solvent extraction capability in 324 would severely limit the operation of the waste solidification program. An alternate combination of 324 and 325A would be to provide dissolution in 324 and solvent extraction in 325A. However, this would cause solvent extraction in 325A to be the limiting step of the system. Throughput of about 100 kg per day might be achieved in a 2 inch column system, and perhaps 150 kg/day in 2 1/2 inch columns. Two parallel systems of 2 to 2 1/2 inch columns could achieve 200 kg/day. This alternative would still result in some interference with the waste programs because fuel receipt, shearing, dissolving, and waste solidification would all be done in the same facility. The Hot Semiworks which was established as a pilot plant for the Redox and Purex plants at Hanford is potentially available for use in commercial fuels reprocessing. The capacity could approach one tonne per day. The cost for refurbishing the Hot Semiworks for satisfactory operation has not been evaluated accurately. It will cost at least tens of millions and perhaps a hundred million to prepare the facility for operation. Likewise, operating costs would be high, and the need for such large quantities of waste is not clear

SAFETY OF ORGANIC SOLVENTS IN WASTE TANKS

This report addresses flash points and flammability limits of flammable gases found in waste tanks, primarily hydrocarbon mixtures derived from waste solvents. The effect of vapor pressure is discussed. Particular attention is given to Purex solvent. The pertinent facts are then applied to the safety of the waste tanks of concern

RESULTS OF RESEARCH TO EVALUATE SOLID PLUTONIUM NITRATE AS A SAFE SHIPPING FORM

The three compounds chosen for evaluation were Pu (NO{sub 3}){sub 4}*XH{sub 20}O, K{sub 2}Pu (NO{sub 3}){sub 6} and (NH{sub 4}){sub 2}Pu(NO{sub 3}){sub 6}. The nitrates were selected since they would not add an extraneous anion to the process streams. The halides were ruled out since they would not be compatible with stainless steel, and organics were not felt to be sufficiently stable for use. A systematic study of the radiolytic and thermal stability, solubility, and ease of preparation of each of the three nitrate compounds selected was then accomplished. These results allowed each compound to be evaluated as a potential solid shipping form as a guide to shipping package requirements. This report presents the results of the chemical development on a laboratory scale

Symptom complexes in patients with seropositive arthralgia and in patients newly diagnosed with rheumatoid arthritis: a qualitative exploration of symptom development

Objective: The aim of this study was to explore symptoms and symptom development during the earliest phases of rheumatoid arthritis (RA) in patients with seropositive arthralgia and patients newly diagnosed with RA

Assessment of intergenomic recombination through GISH analysis of F1, BC1 and BC2 progenies of Tulipa gesneriana and T. fosteriana

Using 23 F1 hybrids, 14 BC1 and 32 BC2 progenies, the genome composition of Darwin hybrid tulips was analysed through genomic in situ hybridisation (GISH) of somatic chromosomes. All plants were diploids (2n = 2x = 24) with the exception of one tetraploid BC1 (2n = 4x = 48) and one aneuploid BC2 (2n = 2x + 1 = 25) hybrid. Morphometric analysis in F1 hybrids revealed a difference in the total length of chromosomes representing genomes of T. gesneriana and T. fosteriana, where the percentage of each genome equaled 55.18 ± 0.8 and 44.92 ± 0.6% respectively. GISH distinguished chromosomes from both parent genomes although there was a lack of consistent chromosome labelling in some cases. In both T. gesneriana and T. fosteriana chromosomes some segments of heterochromatin in the telomeric and intercalary regions exhibited a higher intensity of fluorescence. In situ hybridisation with 5S rDNA and 45S rDNA probes to metaphase chromosomes of F1 hybrids showed that these regions are rich in rDNA. A notable feature was that, despite genome differences, there was a considerable amount of intergenomic recombination between the parental chromosomes of the two species as estimated in both BC1 and BC2 offspring. The number of recombinant chromosomes ranged from 3 to 8 in BC1 and from 1 to 7 in BC2 progenies. All recombinant chromosomes possessed mostly a single recombinant segment derived from either a single crossover event or in a few cases double crossover events. This explains the fact that, unlike the situation in most F1 hybrids of other plant species, certain genotypes of Darwin hybrid tulips behave like normal diploid plants producing haploid gametes and give rise to mostly diploid sporophytes

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.Comment: 26 pages. This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop "Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems" held in February 2015. Comments are welcome. Revised version corrects some typos, updates the references, and clarifies some hypotheses. To appear in the Journal of Algebraic Combinatoric

Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to appea

Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure

Applying science in practice: the optimization of biological therapy in rheumatoid arthritis

Most authorities recommend starting biological agents upon failure of at least one disease-modifying agent in patients with rheumatoid arthritis. However, owing to the absence of head-to-head studies, there is little guidance about which biological to select. Still, the practicing clinician has to decide. This review explores the application of published evidence to practice, discussing the goals of treatment, the (in) ability to predict individual responses to therapy, and the potential value of indirect comparisons. We suggest that cycling of biological agents, until remission is achieved or until the most effective agent for that individual patient is determined, deserves consideration in the current stage of knowledge