Sum Of Roots Research Articles

Some Design Aspects of a Two-Stage Rail-to-Rail CMOS Op Amp

A two-stage low-voltage CMOS op amp with rail-to-rail input and output voltage ranges is presented. The circuit uses complementary differential input pairs to achieve the rail-to-rail common-mode input voltage range. The differential pairs operate in strong inversion, and the constant transconductance is obtained by keeping the sum of the square roots of the tail currents constant. Such an input stage has an offset voltage which depends on the common input voltage level, resulting in a poor common-mode rejection ratio. Therefore, special attention has been given to the reduction of the op amp’s systematic offset voltage. Gain-boost amplifiers are connected in a special way to provide not only an increase of the low-frequency open-loop gain, but also to provide a significant reduction of the systematic offset voltage.

The Influence of VAR Dimensions on Estimator Biases

Vector AutoRegressions (VARs) have now become the most popular tool of Time Series analysis amongst econometricians. Unfortunately, little is known about the analytic finite-sample properties of parameter estimators for such systems. The asymptotic analysis of VARs published to date does not address questions regarding the influence of the number and nature of the system's variates on parameter estimates. Clearly, both questions will have repercussions on the way VARs are used, and we intend to address them here.We consider the implications of varying the dimensions of VARs on the biases of Maximum Likelihood and Least Squares Estimators (MLE and LSE, respectively). In the purely nonstationary case (k-dimensional random walk), estimator biases are approximately equal to the dimension of the system (k) times the univariate bias, even when the variates are generated independently of each other. We show that the variance too increases with the dimension of the system, hence also raising the Mean Squared Error (MSE) of the estimator. When some stable linear combinations exist, the biases are generally smaller and are asymptotically proportional to the sum of the characteristic roots of the VAR. One source of such combinations is meaningful economic relations that are represented by the cointegration of some of the components of the VAR. Adding economically-irrelevant variables to a VAR is thus shown to have more serious negative consequences in integrated time series than in classical ergodic or cross section analyses. The findings strengthen the case for parsimonious modelling and for the reduction step of the general-to-specific marginalization method. They also support the use of seasonally unadjusted data whenever possible.

Below-ground interactions in dryland agroforestry

This paper discusses the effects of intercropping and tree pruning on root distribution and soil water depletion in an alley cropping system with Acacia saligna and Sorghum bicolor in northern Kenya. Root distribution was determined by destructive sampling, and the soil water suction was measured with tensiometers and gypsum blocks, both up to 150 cm depth. The root systems of the intercropped trees and crops were distinguished using the natural 13 C discrimination between C 3 and C 4 plants. The root carbohydrate contents were used to estimate plant water stress integrated over time. The highest root length density was always measured in the topsoil, regardless of season or cropping system. In the dry season, the proportion of roots under the tree row compared to the alley was higher than during the wet season; the same was found for the proportion of roots in the subsoil compared to the topsoil. Pruning decreased the total root length density of sole cropped trees by 47%. The highest root length density was found when the pruned trees were intercropped with Sorghum. If the trees were not pruned, combining trees and crops did not increase root length density. Intercropping resulted in a spatial separation of the root systems of trees and crops between the hedgerows, Sorghum having more roots in the topsoil and the trees having more roots in the subsoil under alley cropping than in monoculture. At the hedgerow of the agroforestry system, however, the root systems of trees and crop overlapped and more roots were found than the sum of roots of sole cropped trees and crops. Soil water depletion was higher under the tree row than in the alley and higher in alley cropping than in monocultural systems. Water competition between tree and crop was confirmed by the carbohydrate analyses showing lower sugar contents of roots in agroforestry than in monoculture. The agroforestry combination used the soil water between the hedgerows more efficiently than the sole cropped trees or crops, as water uptake of the trees reached deeper and started earlier after the flood irrigation than of the Sorghum, whereas the crop could better utilize topsoil water. Under the experimental conditions, the root system of the alley cropped Acacia and Sorghum exploited a larger soil volume utilizing soil resources more efficiently than the respective monocultures.

A Weyl group generating function that ought to be better known

Let W be a finite Weyl group with root system Φ, simple roots α 1,…, α n , exponents e 1,…, e n , and index of connection ƒ. Let b 1,…, b n denote the simple root coordinates for the sum of all positive roots, and for w ϵ W let ℓ( w) denote the length and D( w) = { i: w −1 α i < 0} the descent set of w. By analyzing the structure of the corresponding affine Weyl group, we prove that ∑ wϵWq σ(w)−ℓ(w) = ƒ · Π n i = 1 (1 − q bi) (1 − q ei) , where σ( w) = ∑ iϵD( w) bi .

Rooting Softwood Cuttings from Forced Stem Segments of Adult White Ash

On 10 Apr. 1997, the smaller branch on forked Fraxinus americana L. (white ash) trees in an 8-year-old clonal plantation were removed, cut into 25-cm-long stem sections, and placed horizontally in perlite under seven different forcing regimes. Sprouts from latent buds were excised 67 and 99 days later and trimmed to retain the apical pair of leaves and terminal bud. The basal 2 cm of each cuttings was set for 30 to 60 min in one of six aqueous dilutions made from a stock solution of 1% IBA and 0.5% NAA (Dip `n Grow) before placing in a 1 perlite: 1 vermiculite medium. Eight weeks after treating, cuttings data were collected. Most cuttings treated with 3200 mg/L IBA plus 1600 mg/L NAA quickly died. Survival (85%) and rooting percentage (86%) were similar for the remaining auxin-treated cuttings and controls with water only. Cuttings treated with 1600 mg/L IBA plus 800 mg/L NAA produced the most adventitious roots (6.2 roots), the longest adventitious root (22 cm), and longest combined sum of all adventitious roots (51 cm); however, these rooted cuttings died when transplanted to soil. Cuttings treated with 100 mg/L IBA plus 50 mg/L NAA or 400 mg/L IBA plus 200 mg/L NAA had more adventitious roots (3.2 roots) and total length of adventitious roots (370 mm/cutting) than cuttings treated with 40 mg/L IBA plus 20 mg/L NAA or controls with water only (2.2 roots, 220 mm long). Results indicate softwood cuttings forced on stem segments of adult white ash readily root under mist following a brief soak in a 400 mg/L IBA plus 200 mg/L NAA solution.

Distribution of Transferrin Saturations in the African-American Population

To determine if transferrin saturations in African Americans may reflect the presence of a gene that influences iron metabolism, we analyzed the distribution of these values in 808 African Americans from the second National Health and Nutrition Survey. We tested for a mixture of three normal distributions consistent with population genetics for a major locus effect in which the proportion of normal homozygotes is p2; of heterozygotes, 2pq; of affected homozygotes, q2; and in which p+q= 1. Three subpopulations based on transferrin saturation were present (P &lt; .0001) and the fit to a mixture of three normal distributions was good (P = .2). A proportion of .009 was included in a subpopulation with a mean ± standard deviation transferrin saturation of 63.4% ± 5.7% (postulated homozygotes for a gene that influences iron metabolism), while a proportion of .136 had an intermediate saturation of 38.0% ± 5.7% (postulated heterozygotes) and .856 a saturation of 24.6% ± 5.7% (postulated normal homozygotes). These proportions were consistent with population genetics because the sum of the square roots of the proportions with the lowest mean transferrin saturation (P = .925) and the highest (q = 0.093) was approximately 1 (1.018). The results are consistent with the presence in African Americans of a common locus that influences iron metabolism.

Distribution of Transferrin Saturations in the African-American Population

AbstractTo determine if transferrin saturations in African Americans may reflect the presence of a gene that influences iron metabolism, we analyzed the distribution of these values in 808 African Americans from the second National Health and Nutrition Survey. We tested for a mixture of three normal distributions consistent with population genetics for a major locus effect in which the proportion of normal homozygotes is p2; of heterozygotes, 2pq; of affected homozygotes, q2; and in which p+q= 1. Three subpopulations based on transferrin saturation were present (P< .0001) and the fit to a mixture of three normal distributions was good (P= .2). A proportion of .009 was included in a subpopulation with a mean ± standard deviation transferrin saturation of 63.4% ± 5.7% (postulated homozygotes for a gene that influences iron metabolism), while a proportion of .136 had an intermediate saturation of 38.0% ± 5.7% (postulated heterozygotes) and .856 a saturation of 24.6% ± 5.7% (postulated normal homozygotes). These proportions were consistent with population genetics because the sum of the square roots of the proportions with the lowest mean transferrin saturation (P = .925) and the highest (q = 0.093) was approximately 1 (1.018). The results are consistent with the presence in African Americans of a common locus that influences iron metabolism.

Nitrogen-15 labeling effectiveness of two tropical legumes

Nitrogen-15 foliar applications for the production of field-labeled plant tissues may achieve more effective labeling of plant shoot and root tissues and minimize directly labeling the soil N fraction as occurs when15 N is soil applied. Consequently, foliar-labeled plant tissues should be better suited for subsequent 15N mineralization studies. A field experiment was conducted to determine the effectiveness of 15N-labeling and the accumulation of 15N in various plant parts of two tropical legumes. Desmodium ovalifolium Guillemin and Perrottet and Pueraria phaseoloides (Roxb.) Benth., grown in 0.5 m2 microplots, were labeled with foliar-applied urea containing 99 atom% 15N. Plants in each microplot received a total of 0.1698 g 15N that was applied all at once or split equally into two, three or four applications. Legume shoots and roots and soil were destructively harvested and analyzed for total 15N content. Averaged over both legumes and foliar application rates, total plant (shoots, flowers, leaf litter, and roots) recovery was approximately 79% of the 15N applied. The soil contained 3% of the 15N applied, of which 2.5 and 0.5% were in the inorganic and organic fractions, respectively. Nitrogen-15 recovery in shoots (76%) was sixty-five fold greater than in roots (1%) and about nineteen fold greater than the sum of roots and soil (4.1%), a much greater percent recovery than observed in other foliar labeling studies. Averaged over all four foliar split-application rates, 15N recovery by Desmodium shoots was greater than Pueraria. Results demonstrate that 15N foliar application to legumes is an effective method for labeling, resulting in atom% excess 15N levels and 15N recoveries comparable to those reported with the more traditional soil-labeling approach. Another advantage of this method is a nondestructive, in situ labeling method that permits separation of shoot and root residual N contribution to subsequent crops in N tracer studies.

Integral Bases for Subfields of Cyclotomic Fields

Integral bases of cyclotomic fields are constructed that allow to determine easily the smallest cyclotomic field in which a given sum of roots of unity lies. For subfields of cyclotomic fields integral bases are constructed that consist of orbit sums of Galois groups on roots of unity. These bases are closely related to the bases of the enveloping cyclotomic fields mentioned above. In both situations bases over the rationals and over cyclotomic fields are treated.

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rank n defined over the field k, where n⩾3. Let Φ be the root system associated with G(k) and let Π={α1, α2, …, αn} be a set of fundamental roots of Φ, with Φ+ being the set of positive roots of Φ with respect to Π. For α∈Π and γ∈Φ+, let nα(γ) be the coefficient of α in the expression of γ as a sum of fundamental roots; so γ=∑α∈Πnα(γ)α. Also we recall that ht(γ), the height of γ, is given by ht(γ)=∑α∈Πnα(γ). The highest root in Φ+ will be denoted by ρ. We additionally assume that the Dynkin diagram of G(k) is connected.

On a question of A. John coleman

dim [Lα,L−α]≧1 for every root α of a symrnetrizable Kac- Moody Lie algebra L =L(A). Let α be the sum of the fundamental roots of an arbitrary Kac-Moody algebra L. If there is some nonrepeating sequence il,…,ikwith then dim [Lα,L−α]≧ k. In May 1993 A. John Coleman asked me what would happen if there were two such sequences. This note contains a partial solution. In special cases, if there are k such (disjoint) sequences, of lengths slthrough sk, then dim .

Parallelogram-law-type identities

Identities generalizing the well-known formula relating the lengths of the sides and the diagonals of a parallelogram in the plane are given. These generalizations all have the flavor of the parallelogram law, and specialize to formulas involving sums of roots of unity, trigonometric functions, binomial coefficients, and permutations over the symmetric and the alternating groups.

𝑒-invariants and finite covers. II

Let M ~ ↓ M \widetilde {M} \downarrow M be a finite covering of closed framed manifolds. By the Pontrijagin-Thom construction both M ~ \widetilde {M} and M M define elements in the stable homotopy ring of spheres π ∗ s \pi _*^s . Associated to M ~ \widetilde {M} and M M are their e e invariants e L ( M ~ ) {e_L}(\widetilde {M}) , e L ( M ) ∈ Q / Z {e_L}(M) \in \mathbb {Q}/\mathbb {Z} . If N ~ ↓ N \widetilde {N} \downarrow N is a finite covering of closed oriented manifolds, then there is a related invariant I Δ ( N ~ ↓ N ) ∈ Q {I_\Delta }(\widetilde {N} \downarrow N) \in \mathbb {Q} of the diffeomorphism class of the covering. In a previous paper we examined the relation between these invariants. We reduced the determination of e L ( M ~ ) − p e L ( M ) {e_L}(\widetilde {M}) - p{e_L}(M) , as well as I Δ ( N ~ ↓ N ) {I_\Delta }(\widetilde {N} \downarrow N) , for a p p -fold cover, to the evaluation of certain sums of roots of unity. In this sequel we show how the invariant theory of the cyclic group Z / p \mathbb {Z}/p may be used to evaluate these rums. For example we obtain \[ ∑ ζ p = 1 ζ ≠ 1 ( 1 + ζ ) ( 1 + ζ − 1 ) ( 1 − ζ ) ( 1 − ζ − 1 ) = ( p − 1 ) ( p − 2 ) 3 \sum _{\substack {\zeta ^p = 1\\\zeta \ne 1}} \frac {(1 + \zeta )(1 + \zeta ^{-1})}{(1 - \zeta )(1 - \zeta ^{-1})} = \frac {(p - 1)(p - 2)}{3} \] which may be used to determine the value of I Δ {I_\Delta } in degrees congruent to 3 3 mod 2 ( p − 1 ) \mod 2(p - 1) for odd primes p p .

Huygens' principle for wave equations on symmetric spaces

Let X = G K be a symmetric space, L X the Laplace-Betrami operator on X, and 2ϱ the sum of the positive restricted roots of X (with multiplicity). Using the Fourier transform on X we write down the solution of the modified wave equation ∂ 2u ∂t 2 = (L X + ¦ϱ¦ 2)u on X . It is proved, using the Paley-Wiener theorem for this Fourier transform ( S. Helgason, Ann. of Math. 98 (1973), 451–480) that the equation satisfies Huygens' principle if dim X is odd and all Cartan subgroups of G are conjugate.

A convergence theorem for symmetric functionals of random partitions

This paper gives general conditions under which symmetric functionals of random partitions of the integer m converge in distribution as m → ∞. The main result is used to settle a conjecture of Donnelly et al. (1991) to the effect that the mean of the sum of the square roots of the relative sizes of the components of a random mapping of m integers converges to π/2 as m → ∞.

A convergence theorem for symmetric functionals of random partitions

This paper gives general conditions under which symmetric functionals of random partitions of the integer m converge in distribution as m → ∞. The main result is used to settle a conjecture of Donnelly et al. (1991) to the effect that the mean of the sum of the square roots of the relative sizes of the components of a random mapping of m integers converges to π/2 as m → ∞.

Asymptotic finite propagation speed for heat diffusion on certain Riemannian manifolds

Using pointwise upper bounds recently obtained by the first author, we first show that the heat kernel h t ( x, y) on a Riemannian symmetric space of the noncompact type G K is asymptotically concentrated in an annulus centered at y and moving to infinity with finite speed 2 ¦ϱ¦, ϱ being as usual the half sum of all positive roots of G K . In the higher rank case we prove moreover that heat not only concentrates in an annulus but also along the ( K-orbit) of the ϱ-axis. By applying wave equation techniques developed by M. E. Taylor, we partially extend the above results to Riemannian manifolds with exponential volume growth and L 2 spectrum of the Laplacian bounded away from 0. Some extensions to the vector bundle case are also considered.

Exact power comparison of three criteria to assess the independence between two sets of variates

Powers of the three criteria are evaluated for testing the hypothesis of the independence between a -set and a q-set of variates in a (p + q) -variate normal population. They are: (1) the likelihood ratio type criterion, Wt (2) the largest root criterion, r1, and (3) criterion of the sum of roots, V. For p= 2, Pillai and Jayachandran, and others have studied for the restricted range of the alternative hypothesis. Recently the power of the largest root was investigated in detail by Sugiyama and %%. In this paper, their power functions are compared in a wide range of the alternative hypotheses. The powers of rl and V are locally optimum, but the W shows a large power in a wide range.

Canonical bases for cyclotomic fields

It is shown how the use of a certain integral basis for cyclotomic fields enables one to perform the basic operations in their ring of integers efficiently. In particular, from the representation with respect to this basis, one obtains immediately the smallest possible cyclotomic field in which a given sum of roots of unity lies. This is of particular interest when computing with the ordinary representations of a finite group.

Blocks in weight spaces in the root lattice

It is useful to think of the p-restricted modular Weyl problem in eigenvalueeeigenvector terms. This problem asks for the character of the irreducible p-version of an irreducible module V(A) for a complex semisimple Lie algebra, cf. Humphreys [6]. The eigenvalue results concern the weight lattice of the root system and especially n(A), the saturated set of weights occuring in the module. The eigenvector results are better indexed by 17,(A), the root weights. If p E D(A), then ,~l= ,I T, where r E A:, the positive cone of the root lattice of the root system. So we have p E Z7(A) iff T E n,(n). Consider, for instance, the problem of finding p-modular maximal vectors, cf. [4]. The main eigenvalue result, Kac-Weisfeiler [S], finds that a necessary condition for p E n(A) to be the weight of such a vector is that ,u be p-linked to A. This condition is ultimately descended from the shape of the Weyl character formula. The main p-restricted eigenvector result, [4], yields that a necessary condition for r E n,(A) to be the root weight of such a vector is that r be in the Kostant cone of A,‘, i.e., that r can be written as a sum of positive nonsimple roots. This condition comes from some combinatorial properties of Kostant’s partition function and Chevalley bases. For another kind of eigenvalue-eigenvector result, see Franklin [lo]. This paper, a sequel to [3], provides further foundational results for the eigenvector approach to this restricted modular Weyl problem, as well as deep insights into the combinatorial and geometrical structure of P(z), Kostant’s partition function, and its relations with structures on weight spaces in the root lattice. Let R be a root system and A:, the positive cone of the root lattice with