Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

The analysis of Metarhizium robertsii potential as endophytic plant root coloniser, plant growth enhancer and antagonist to bean root pathogen Fusarium solani f. sp. phaseolis.

The soil-inhabiting insect-pathogenic fungus Metarhizium robertsii also colonizes plant roots endophytically, thus showing potential as a plant symbiont. M robertsii is not randomly distributed in soils but preferentially associates with the plant rhizosphere when applied in agricultural settings. Root surface and endophytic colonization of switchgrass (Panicum virgatum) and haricot beans (Phaseolus vulgaris) by M robertsii were examined after inoculation with fungal conidia. Light and confocal microscopies were used to ascertain this rhizosphere association. Root lengths, root hair density and emergence of lateral roots were also measured. Initially, M robertsii conidia adhered to, germinated on, and colonized, roots. Furthermore, plant roots treated with Metarhizium grew faster and the density of plant root hairs increased when compared with control plants. The onset of plant root hair proliferation was initiated before germination of M robertsii on the root (within 1-2 days). Plants inoculated with M robertsii AMAD2 (plant adhesin gene) took significantly longer to show root hair proliferation than the wild type. Cell free extracts of M robertsii did not stimulate root hair proliferation. Longer term (60 days) associations showed that M robertsii endophytically colonized individual cortical cells within bean roots. Metarhizium appeared as an amorphous mycelial aggregate within root cortical cells as well as between the intercellular spaces with no apparent damage to the plant. These results suggested that not only is M robertsii rhizosphere competent but displays a beneficial endophytic association with plant roots that results in the proliferation of root hairs. The biocontrol of bean (Phaseolis vulgaris) root rot fungus Fusarium solani f. sp. phaseolis by Metarhizium robertsii was investigated in vitro and in vivo. Dual cultures on Petri dishes showed antagonism of M robertsii against F. solani. A relative inhibition of ca. 60% of F. solani growth was observed in these assays. Cell free culture filtrates of M robertsii inhibited the germination of F. solani conidia by 83% and the inhibitory metabolite was heat stable. Beans plants colonized by M robertsii then exposed to F. solani showed healthier plant profiles and lower disease indices compared to plants not colonized by M robertsii. These results suggested that the insect pathogenic/endophytic fungus M robertsii could also be utilized as a biocontrol agent against certain plant pathogens occurring in the rhizosphere.

Report on the Expenditure of the Welland and Burlington Canals, 1830

This copy is signed in the upper left hand corner by Mr. Dickson. Mr. Robert Dickson was one of the directors of the Welland Canal Office. The report covers meetings which were held: January 15-16, and 19 of 1830. The meetings were attended by Messrs. Blacklock, Mackenzie, Woodruff, Longley and Hopkins. Balance sheets are also included within the report. The report of the Welland Canal Company for 1829 is also included within this document, and this is dated December 31, 1829. Names at the end of the 1929 report are members of the Welland Canal Office and they include: John Henry Dunn, president; Henry J. Boulton, vice-president and William Allan, George Keefer, John J. Lefferty and Robert Dickson who were directors The report is dated January 26, 1830, and submitted by Thomas Horner, chairman of the Commons House of Assembly.

Tests of Joint Hypotheses for Time Series Regression with a Unit Root

This Paper Studies Tests of Joint Hypotheses in Time Series Regression with a Unit Root in Which Weakly Dependent and Heterogeneously Distributed Innovations Are Allowed. We Consider Two Types of Regression: One with a Constant and Lagged Dependent Variable, and the Other with a Trend Added. the Statistics Studied Are the Regression \"F-Test\" Originally Analysed by Dickey and Fuller (1981) in a Less General Framework. the Limiting Distributions Are Found Using Functinal Central Limit Theory. New Test Statistics Are Proposed Which Require Only Already Tabulated Critical Values But Which Are Valid in a Quite General Framework (Including Finite Order Arma Models Generated by Gaussian Errors). This Study Extends the Results on Single Coefficients Derived in Phillips (1986A) and Phillips and Perron (1986).

GLS Detrending, Efficient Unit Root Tests and Structural Change

We extend the class of M-tests for a unit root analyzed by Perron and Ng (1996) and Ng and Perron (1997) to the case where a change in the trend function is allowed to occur at an unknown time. These tests M(GLS) adopt the GLS detrending approach of Dufour and King (1991) and Elliott, Rothenberg and Stock (1996) (ERS). Following Perron (1989), we consider two models : one allowing for a change in slope and the other for both a change in intercept and slope. We derive the asymptotic distribution of the tests as well as that of the feasible point optimal tests PT(GLS) suggested by ERS. The asymptotic critical values of the tests are tabulated. Also, we compute the non-centrality parameter used for the local GLS detrending that permits the tests to have 50% asymptotic power at that value. We show that the M(GLS) and PT(GLS) tests have an asymptotic power function close to the power envelope. An extensive simulation study analyzes the size and power in finite samples under various methods to select the truncation lag for the autoregressive spectral density estimator. An empirical application is also provided.

Testing for a Unit Root in Panels with Dynamic Factors

This paper studies testing for a unit root for large n and T panels in which the cross-sectional units are correlated. To model this cross-sectional correlation, we assume that the data is generated by an unknown number of unobservable common factors. We propose unit root tests in this environment and derive their (Gaussian) asymptotic distribution under the null hypothesis of a unit root and local alternatives. We show that these tests have significant asymptotic power when the model has no incidental trends. However, when there are incidental trends in the model and it is necessary to remove heterogeneous deterministic components, we show that these tests have no power against the same local alternatives. Through Monte Carlo simulations, we provide evidence on the finite sample properties of these new tests.

Beyond Panel Unit Root Tests: Using Multiple Testing to Determine the Non Stationarity Properties of Individual Series in a Panel

Most panel unit root tests are designed to test the joint null hypothesis of a unit root for each individual series in a panel. After a rejection, it will often be of interest to identify which series can be deemed to be stationary and which series can be deemed nonstationary. Researchers will sometimes carry out this classification on the basis of n individual (univariate) unit root tests based on some ad hoc significance level. In this paper, we demonstrate how to use the false discovery rate (FDR) in evaluating I(1)=I(0) classifications based on individual unit root tests when the size of the cross section (n) and time series (T) dimensions are large. We report results from a simulation experiment and illustrate the methods on two data sets.

Salinity Intrusion and Seasonal Water Quality Variations in the Tidal Canals of Cochin

The study conducted on the salinity intrusion and seasonal water quality variations in the tidal canals of cochin. The main objectives are, salinity intrusion profile, water quality variation of the surface water of the canals,hierarchical utility of the water bodies and to understand the non-conservative components in the water body. The parameters monitored werepH,temperature,alkalinity,conductivity,DO(dissolvedoxygen),COD(chemical oxygen demand),BOD(biochemical oxygen demand0,chloride, total hardness, calcium hardness, dissolved phosphate, nitrate, total iron, sulphate, turbidity, total coliform and SUVA at 254nm. The tidal canals of GCDA were found to be creeks extending to the interior, canals inter connecting parts of the estuary or canals with seasonally broken segments. Based on utility the canals could be classified as: canals heavely polluted and very saline,canals polluted by urban waste , canals having fresh water for most part of the year and not much polluted, fresh water bodies heavily polluted. During the rainy months carbon fixation by plankton is nonexistent,and during the dry months Chitrapuzha becomes a sink of phosphate. The study indicated abiotic subrouts for dissolved phosphate and revealed the potential pitfalls in LOICZ modeling exercise on sewage ladentidal canals. It was also found that all canals except for the canals of West cochin and chittoorpuzha have fresh water for some part of the year. The water quality index in the durable fresh water stretches was found to be of below average category.

Root effects on the turnover of grain legume residues in soil

Das Ziel dieser Arbeit war, die Einflüsse von Wurzeln und Rhizodeposition auf den Umsatz von Körnerleguminosenresiduen und damit verknüpfte mikrobielle Prozesse zu untersuchen. In einem integrierten Versuch wurden Ackerbohne (Vicia faba L.), Erbse (Pisum sativum L.) und Weiße Lupine (Lupinus albus L.) untersucht. Der Versuch bestand aus drei Teilen, zwei Gefäß-Experimenten und einem Inkubationsexperiment, in denen ausgehend von einem Gefäß-Experiment derselbe Boden und dasselbe Pflanzenmaterial verwendet wurden. In Experiment I wurde die Stickstoff-Rhizodeposition der Körnerleguminosenarten, definiert als wurzelbürtiger N nach dem Entfernen aller sichtbaren Wurzeln im Boden, gemessen und der Verbleib des Rhizodepositions-N in verschiednenen Bodenpools untersucht. Dazu wurden die Leguminosen in einem Gefäßversuch unter Verwendung einer in situ 15N-Docht-Methode mit einer 15N Harnstofflösung pulsmarkiert. In Experiment II wurde der Umsatz der N-Rhizodeposition der Körnerleguminosen und der Einfluss der Rhizodeposition auf den anschließenden C- und N-Umsatz der Körnerleguminosenresiduen in einem Inkubationsexperiment untersucht. In Experiment III wurde der N-Transfer aus den Körnerleguminosenresiduen einschließlich N-Rhizodeposition in die mikrobielle Biomasse und die Folgefrüchte Weizen (Triticum aestivum L.) und Raps (Brassica napus L.) in einem Gewächshaus-Gefäßversuch ermittelt. Die in situ 15N Docht-Markierungs-Methode wies hohe 15N Wiederfindungsraten von ungefähr 84 Prozent für alle drei Leguminosenarten auf und zeigte eine vergleichsweise homogene 15N Verteilung zwischen verschiedenen Pflanzenteilen zur Reife. Die Wurzeln zeigten deutliche Effekte auf die N-Dynamik nach dem Anbau von Körnerleguminosen. Die Effekte konnten auf die N-Rhizodeposition und deren anschließenden Umsatz, Einflüsse der Rhizodeposition von Körnerleguminosen auf den anschließenden Umsatz ihrer Residuen (Stängel, Blätter, erfassbare Wurzeln) und die Wirkungen nachfolgender Nichtleguminosen auf den Umsatzprozess der Residuen zurückgeführt werden: Die N-Rhizodeposition betrug zur Reife der Pflanzen bezogen auf die Gesamt-N- Aufnahme 13 Prozent bei Ackerbohne und Erbse und 16 Prozent bei Weißer Lupine. Bezogen auf den Residual N nach Ernte der Körner erhöhte sich der relative Anteil auf 35 - 44 Prozent. Die N-Rhizodeposition ist daher ein wesentlicher Pool für die N-Bilanz von Körnerleguminosen und trägt wesentlich zur Erklärung positiver Fruchtfolgeeffekte nach Körnerleguminosen bei. 7 - 21 Prozent des Rhizodepositions-N wurden als Feinwurzeln nach Nasssiebung (200 µm) wiedergefunden. Nur 14 - 18 Prozent des Rhizodepositions-N wurde in der mikrobiellen Biomasse und ein sehr kleiner Anteil von 3 - 7 Prozent in der mineralischen N Fraktion gefunden. 48 bis 72 Prozent der N-Rhizodeposition konnte in keinem der untersuchten Pools nachgewiesen werden. Dieser Teil dürfte als mikrobielle Residualmasse immobilisiert worden sein. Nach 168 Tagen Inkubation wurden 21 bis 27 Prozent des Rhizodepositions-N in den mineralisiert. Der mineralisierte N stammte im wesentlichen aus zwei Pools: Zwischen 30 Prozent und 55 Prozent wurde aus der mikrobiellen Residualmasse mineralisiert und eine kleinere Menge stammte aus der mikrobielle Biomasse. Der Einfluss der Rhizodeposition auf den Umsatz der Residuen war indifferent. Durch Rhizodeposition wurde die C Mineralisierung der Leguminosenresiduen nur in der Lupinenvariante erhöht, wobei der mikrobielle N und die Bildung von mikrobieller Residualmasse aus den Leguminosenresiduen in allen Varianten durch Rhizodepositionseinflüsse erhöht waren. Das Potential des residualen Körnerleguminosen-N für die N Ernährung von Folgefrüchten war gering. Nur 8 - 12 Prozent des residualen N wurden in den Folgenfrüchten Weizen und Raps wiedergefunden. Durch die Berücksichtigung des Rhizodepositions-N war der relative Anteil des Residual-N bezogen auf die Gesamt-N-Aufnahme der Folgefrucht hoch und betrug zwischen 18 und 46 Prozent. Dies lässt auf einen höheren N-Beitrag der Körnerleguminosen schließen als bisher angenommen wurde. Die residuale N-Aufnahme von Weizen von der Blüte bis zur Reife wurde durch den Residual-N gespeist, der zur Blüte in der mikrobiellen Biomasse immobilisiert worden war. Die gesamte Poolgröße, Residual-N in der mikrobiellen Biomasse und in Weizen, veränderte sich von der Blüte bis zur Reife nicht. Jedoch konnte ein Rest von 80 Prozent des Residual-N in keinem der untersuchten Pools nachgewiesen werden und dürfte als mikrobielle Residualmasse immobilisiert worden sein oder ist noch nicht abgebaut worden. Die zwei unterschiedlichen Folgefrüchte - Weizen und Raps - zeigten sehr ähnliche Muster bei der N-Aufnahme, der Residual-N Wiederfindung und bei mikrobiellen Parametern für die Residuen der drei Körnerleguminosenarten. Ein differenzierender Effekt auf den Umsatz der Residuen bzw. auf das Residual-N-Aneignungsvermögen der Folgefrüchte konnte nicht beobachtet werden.