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DOE OSTI2022

Inferring the Contribution of Microbial Taxa and Organic Matter Molecular Formulas to Ecological Assembly

Understanding the mechanisms underlying the assembly of communities has long been the goal of many ecological studies. While several studies have evaluated community wide ecological assembly, fewer have focused on investigating the impacts of individual members within a community or assemblage on ecological assembly. Here, we adapted a previous null model β-nearest taxon index (βNTI) to measure the contribution of individual features within an ecological community to overall assembly. This new metric, called feature-level βNTI (βNTI feat ), enables researchers to determine whether ecological features (e.g., individual microbial taxa) contribute to divergence, convergence, or have insignificant impacts across spatiotemporally resolved metacommunities or meta-assemblages. Using βNTI feat , we revealed that unclassified microbial lineages often contributed to community divergence while diverse groups (e.g., Crenarchaeota, Alphaproteobacteria, and Gammaproteobacteria) contributed to convergence. We also demonstrate that βNTI feat can be extended to other ecological assemblages such as organic molecules comprising organic matter (OM) pools. OM had more inconsistent trends compared to the microbial community though CHO-containing molecular formulas often contributed to convergence, while nitrogen and phosphorus-containing formulas contributed to both convergence and divergence. A network analysis was used to relate βNTI feat values from the putatively active microbial community and the OM assemblage and examine potentially common contributions to ecological assembly across different communities/assemblages. This analysis revealed that P-containing formulas often contributed to convergence/divergence separately from other ecological features and N-containing formulas often contributed to assembly in coordination with microorganisms. Additionally, members of Family Geobacteraceae were often observed to contribute to convergence/divergence in conjunction with both N- and P-containing formulas, suggesting a coordinated ecological role for family members and the nitrogen/phosphorus cycle. Overall, we show that βNTI feat offers opportunities to investigate the community or assemblage members, which shape the phylogenetic or functional landscape, and demonstrate the potential to evaluate potential points of coordination across various community types.

59 BASIC BIOLOGICAL SCIENCES↗

NASA NTRS1972

Classical eight- and lower-order Runge-Kutta-Nystroem formulas with stepsize control for special second-order differential equations

The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the truncation error in x. The formulas require fewer evaluations per stop than other Runge-Kutta-Nystrom formulas if the latter are operated by using the standard procedure for stepsize control. An example is presented. With results being of the same accuracy, Runge-Kutta-Nystrom formulas discussed save 50 percent or more computer time compared with other Runge-Kutta-Nystrom formulas.

Fehlberg, E.↗

NASA NTRS1973

Classical eighth- and lower-order Runge-Kutta-Nystroem formulas with a new stepsize control procedure for special second-order differential equations

New Runge-Kutta-Nystrom formulas of the eighth, seventh, sixth, and fifth order are derived for the special second-order (vector) differential equation x = f (t,x). In contrast to Runge-Kutta-Nystrom formulas of an earlier NASA report, these formulas provide a stepsize control procedure based on the leading term of the local truncation error in x. This new procedure is more accurate than the earlier Runge-Kutta-Nystrom procedure (with stepsize control based on the leading term of the local truncation error in x) when integrating close to singularities. Two central orbits are presented as examples. For these orbits, the accuracy and speed of the formulas of this report are compared with those of Runge-Kutta-Nystrom and Runge-Kutta formulas of earlier NASA reports.

Fehlberg, E.↗

NASA NTRS1990

Formula for the rms blur circle radius of Wolter telescope based on aberration theory

A formula for the rms blur circle for Wolter telescopes has been derived using the transverse ray aberration expressions of Saha (1985), Saha (1984), and Saha (1986). The resulting formula for the rms blur circle radius over an image plane and a formula for the surface of best focus based on third-, fifth-, and seventh-order aberration theory predict results in good agreement with exact ray tracing. It has also been shown that one of the two terms in the empirical formula of VanSpeybroeck and Chase (1972), for the rms blur circle radius of a Wolter I telescope can be justified by the aberration theory results. Numerical results are given comparing the rms blur radius and the surface of best focus vs the half-field angle computed by skew ray tracing and from analytical formulas for grazing incidence Wolter I-II telescopes and a normal incidence Cassegrain telescope.

Shealy, David L.↗

NASA NTRS2001

Synthesizing Dynamic Programming Algorithms from Linear Temporal Logic Formulae

The problem of testing a linear temporal logic (LTL) formula on a finite execution trace of events, generated by an executing program, occurs naturally in runtime analysis of software. We present an algorithm which takes an LTL formula and generates an efficient dynamic programming algorithm. The generated algorithm tests whether the LTL formula is satisfied by a finite trace of events given as input. The generated algorithm runs in linear time, its constant depending on the size of the LTL formula. The memory needed is constant, also depending on the size of the formula.

Rosu, Grigore↗

NASA NTRS2007

A Fast Method of Deriving the Kirchhoff Formula for Moving Surfaces

The Kirchhoff formula for a moving surface is very useful in many wave propagation problems, particularly in the prediction of noise from rotating machinery. Several publications in the last two decades have presented derivations of the Kirchhoff formula for moving surfaces in both time and frequency domains. Here we present a method originally developed by Farassat and Myers in time domain that is both simple and direct. It is based on generalized function theory and the useful concept of imbedding the problem in the unbounded three-dimensional space. We derive an inhomogeneous wave equation with the source terms that involve Dirac delta functions with their supports on the moving data surface. This wave equation is then solved using the simple free space Green's function of the wave equation resulting in the Kirchhoff formula. The algebraic manipulations are minimal and simple. We do not need the Green's theorem in four dimensions and there is no ambiguity in the interpretation of any terms in the final formulas. Furthermore, this method also gives the simplest derivation of the classical Kirchhoff formula which has a fairly lengthy derivation in physics and applied mathematics books. The Farassat-Myers method can be used easily in frequency domain.

Farassat, F.↗

DOE OSTI2026

Leveraging 13C-Labeling to Assign Molecular Formulas to Unknown Yeast Metabolites

Mass spectrometry analyses have identified tens of thousands of unknown small molecule-associated peaks in different biological specimens. Notably, even the simplest and best studied organisms like Escherichia coli and Saccharomyces cerevisiae yield thousands of unknown peaks. A key question is how many of these reflect actual novel endogenous metabolites. To explore this, Mahieu and Patti used complete 13 C -labeling in E. coli to credential peaks as biological. This reduced the number of unknowns by more than 90%. Here, we carry out similar uniform 13 C-labeling in the Baker’s yeast S. cerevisiae and two less-studied bioenergy-relevant yeasts Rhodotorula toruloides (lipid producer) and Issatchenkia orientalis (organic acid producer). Identification of unknown metabolite peaks and their molecular formulas is facilitated through software tailored for 13 C labeling data and resulting knowledge of carbon atom count. A classification model evaluates the plausibility of each candidate formula, with peaks lacking plausible candidate formulas unlikely to reflect metabolite molecular ions. This approach prioritizes about one hundred candidate abundant unknown metabolites with logical molecular formulas. Most of these are species-specific rather than conserved across yeasts, and more are found in the nonmodel yeasts than S. cerevisiae. Thus, 13 C-labeling data on unknown metabolites highlights the potential for discovering new metabolites and pathways in nonmodel yeasts.

Carbon↗

DOE OSTI2021

Proving the 6d Cardy formula and matching global gravitational anomalies

A Cardy formula for 6d superconformal field theories (SCFTs) conjectured by Di Pietro and Komargodski in [1] governs the universal behavior of the supersymmetric partition function on S^1_\beta \times S^5 S β 1 × S 5 in the limit of small \beta β and fixed squashing of the S^5 S 5 . For a general 6d SCFT, we study its 5d effective action, which is dominated by the supersymmetric completions of perturbatively gauge-invariant Chern-Simons terms in the small \beta β limit. Explicitly evaluating these supersymmetric completions gives the precise squashing dependence in the Cardy formula. For SCFTs with a pure Higgs branch (also known as very Higgsable SCFTs), we determine the Chern-Simons levels by explicitly going onto the Higgs branch and integrating out the Kaluza-Klein modes of the 6d fields on S^1_\beta S β 1 . We then discuss tensor branch flows, where an apparent mismatch between the formula in [1] and the free field answer requires an additional contribution from BPS strings. This ``missing contribution’’ is further sharpened by the relation between the fractional part of the Chern-Simons levels and the (mixed) global gravitational anomalies of the 6d SCFT. We also comment on the Cardy formula for 4d \mathcal{N}=2 𝒩 = 2 SCFTs in relation to Higgs branch and Coulomb branch flows.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1972

Field dependence of gaseous-ion mobility - Theoretical tests of approximate formulas.

The approximate formulas considered include relations based on the Wannier free-flight theory, the Kihara medium-field expansion, and the Frost-Patterson interpolation formulas. A few accurate theoretical results are available for testing the foregoing formulas. Cases concerning high fields, intermediate fields, and resonant charge transfer are examined. It is found that of the formulas tested, the one based on the Wannier free-flight theory is the most flexible, since it can be used for all fields and all ion-neutral force laws and mass ratios.

Hahn, H.↗

NASA NTRS1978

A simple formula for estimating Stark widths of neutral lines

A simple formula for the prediction of Stark widths of neutral lines similar to the semiempirical method of Griem (1968) for ion lines is presented. This formula is a simplification of the quantum-mechanical classical path impact theory and can be used for complicated atoms for which detailed calculations are not readily available, provided that the effective position of the closest interacting level is known. The expression does not require the use of a computer. The formula has been applied to a limited number of neutral lines of interest, and the width obtained is compared with the much more complete calculations of Bennett and Griem (1971). The agreement generally is well within 50% of the published value for the lines investigated. Comparisons with other formulas are also made. In addition, a simple estimate for the ion-broadening parameter is given.

Freudenstein, S. A.↗

NASA NTRS1979

Explicit numerical formulas of improved stability and accuracy for the solution of parabolic equations

A class of explicit numerical formulas which involve next nearest neighbor as well as nearest neighbor points are explored in this paper. These formulas are formal approximations to the linear parabolic partial-differential equation of first order in time and second order in distance. It was found that some of these formulas can employ time steps as much as four times that for the conventional explicit technique without becoming unstable. Others showed improved accuracy for a given time step and spatial grid spacing. One formula achieved a steady-state solution of specified accuracy for an example problem in less than 4 percent of the total computational time required by the conventional explicit technique.

Olstad, W. B.↗

NASA NTRS1981

A Single-Expression Formula for Inverting Strain-Life and Stress-Strain Relationships

Starting with the basic fatigue lift formula, an inversion formula is derived. The inversion formula is valid over the entire life range of engineering interest for all materials examined. Conformity between the two equations is extremely close, suitable for all engineering problems. The approach used to invert the life relation is also suitable for the inversion of other formulas involving the sum of two power-law terms.

Manson, S. S.↗

NASA NTRS1981

Improved formula for continuous-wave measurements of ultrasonic phase velocity

An improved formula for continuous-wave ultrasonic phase velocity measurements using contact transducers is derived from the transmission line theory. The effect of transducer-sample coupling bonds is considered for measurements of solid samples even though it is often neglected because of the difficulty of accurately determining the bond thickness. Computer models show that the present formula is more accurate than previous expressions. Laboratory measurements using contacting transducers with the present formula are compared to measurements using noncontacting (hence effectively correction-free) capacitive transducers. The results of the experiments verify the validity and accuracy of the new formula.

Chern, E. J.↗

NASA NTRS1983

Asymptotic unbounded root loci - Formulas and computation

A new geometric way of computing the asymptotic behavior of unbounded root loci of a strictly proper linear time-invariant control system as loop gain goes to infinity is presented. Properties of certain restricted linear maps and nested restrictions of linear maps are developed, and formulas are obtained for the leading coefficient of the asymptotic values of the unbounded multivariable root loci are obtained in terms of eigenvalues of those maps. Published results and a certain simple null structure assumption are used to relate these asymptotic values to the structure at infinity of the Smith-McMillan form of the open loop transfer function. Explicit matrix formulas for the more abstract derived formulas are given and additional geometric insights are developed with orthogonal projections and singular value decomposition. Formulas for the pivots of the unbounded root loci are calculated and shown to have the same form as the coefficients of the unbounded asymptotic root loci.

Sastry, S. S.↗

NASA NTRS1992

An analytic formula for H-infinity norm sensitivity with applications to control system design

An analytic formula for the sensitivity of singular value peak variation with respect to parameter variation is derived. As a corollary, the derivative of the H-infinity norm of a stable transfer function with respect to a parameter is presented. It depends on some of the first two derivatives of the transfer function with respect to frequency and the parameter. For cases when the transfer function has a linear system realization whose matrices depend on the parameter, analytic formulas for these first two derivatives are derived, and an efficient algorithm for calculating them is discussed. Examples are given which provide numerical verification of the H-infinity norm sensitivity formula and which demonstrate its utility in designing control systems satisfying H-infinity norm constraints. In the appendix, derivative formulas for singular values are paraphrased.

Giesy, Daniel P.↗

NASA NTRS1947

Formulas for additional mass corrections to the moments of inertia of airplanes

Formulas are presented for the calculation of the additional mass corrections to the moments of inertia of airplanes. These formulas are of particular value in converting the virtual moments of inertia of airplanes or models experimentally determined in air to the true moments of inertia. A correlation of additional moments of inertia calculated by these formulas with experimental additional moments of inertia obtained from vacuum chamber tests of 40 spin-tunnel models indicates that formulas give satisfactory estimations of the additional moments of inertia.

MOMENTS OF INERTIA↗

NASA NTRS1997

Exact Integrations of Polynomials and Symmetric Quadrature Formulas over Arbitrary Polyhedral Grids

This paper is concerned with two important elements in the high-order accurate spatial discretization of finite volume equations over arbitrary grids. One element is the integration of basis functions over arbitrary domains, which is used in expressing various spatial integrals in terms of discrete unknowns. The other consists of quadrature approximations to those integrals. Only polynomial basis functions applied to polyhedral and polygonal grids are treated here. Non-triangular polygonal faces are subdivided into a union of planar triangular facets, and the resulting triangulated polyhedron is subdivided into a union of tetrahedra. The straight line segment, triangle, and tetrahedron are thus the fundamental shapes that are the building blocks for all integrations and quadrature approximations. Integrals of products up to the fifth order are derived in a unified manner for the three fundamental shapes in terms of the position vectors of vertices. Results are given both in terms of tensor products and products of Cartesian coordinates. The exact polynomial integrals are used to obtain symmetric quadrature approximations of any degree of precision up to five for arbitrary integrals over the three fundamental domains. Using a coordinate-free formulation, simple and rational procedures are developed to derive virtually all quadrature formulas, including some previously unpublished. Four symmetry groups of quadrature points are introduced to derive Gauss formulas, while their limiting forms are used to derive Lobatto formulas. Representative Gauss and Lobatto formulas are tabulated. The relative efficiency of their application to polyhedral and polygonal grids is detailed. The extension to higher degrees of precision is discussed.

Liu, Yen↗

DOE OSTI2021

"Spectrally gapped" random walks on networks: a Mean First Passage Time formula

We derive an approximate but explicit formula for the Mean First Passage Time of a random walker between a source and a target node of a directed and weighted network. The formula does not require any matrix inversion, and it takes as only input the transition probabilities into the target node. It is derived from the calculation of the average resolvent of a deformed ensemble of random sub-stochastic matrices H=\langle H\rangle +\delta H H = ⟨ H ⟩ + δ H , with \langle H\rangle ⟨ H ⟩ rank- 1 1 and non-negative. The accuracy of the formula depends on the spectral gap of the reduced transition matrix, and it is tested numerically on several instances of (weighted) networks away from the high sparsity regime, with an excellent agreement.

97 MATHEMATICS AND COMPUTING↗