Truncation errors in natural frequencies as computed by the method of component mode synthesis
Truncation errors in natural frequencies as computed by method of component mode synthesis
TRUNCATION ERROR↗
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Flutter, vibration, and buckling of truncated orthotropic conical shells with generalized elastic edge restraint
A theoretical investigation has been made of the flutter, vibration, and buckling of truncated conical shells with generalized elastic edge restraint. The shell analysis is of the classical Donnell type, in-plane inertias and structural damping are neglected, and the aerodynamic loading is represented by the inviscid two-dimensional quasi-steady approximation. An approximate solution is obtained by the generalized Galerkin method. The accuracy and limitations of the analysis are illustrated by comparing numerical results for buckling an vibration with results of other investigations for various boundary conditions, applied loads, and shell geometries and stiffness. Sufficient numerical results are presented to permit the determination of the flutter condition for simply supported isotropic conical shells for a wide range of cone angle, length-radius ration, and radius-thickness ratio. Results are also presented to indicate some effects of variations in edge restraint, applied loads, and ring or stringer stiffening.
Flutter↗
Sparse matrix‐vector and matrix‐multivector products for the truncated SVD on graphics processors
Summary Many practical algorithms for numerical rank computations implement an iterative procedure that involves repeated multiplications of a vector, or a collection of vectors, with both a sparse matrix and its transpose. Unfortunately, the realization of these sparse products on current high performance libraries often deliver much lower arithmetic throughput when the matrix involved in the product is transposed. In this work, we propose a hybrid sparse matrix layout, named CSRC, that combines the flexibility of some well‐known sparse formats to offer a number of appealing properties: (1) CSRC can be obtained at low cost from the popular CSR (compressed sparse row) format; (2) CSRC has similar storage requirements as CSR; and especially, (3) the implementation of the sparse product kernels delivers high performance for both the direct product and its transposed variant on modern graphics accelerators thanks to a significant reduction of atomic operations compared to a conventional implementation based on CSR. This solution thus renders considerably higher performance when integrated into an iterative algorithm for the truncated singular value decomposition (SVD), such as the randomized SVD or, as demonstrated in the experimental results, the block Golub–Kahan–Lanczos algorithm.
Aliaga, José I.↗
Comparative Hydrophobic Core Dynamics Between Wild‐Type Amyloid‐β Fibrils, Glutamate‐3 Truncation, and Serine‐8 Phosphorylation
Abstract Post‐translational modifications (PTMs) of amyloid‐β (Aβ) species are implicated in the modulation of overall toxicities and aggregation propensities. We investigated the internal dynamics in the hydrophobic core of the truncated ΔE3 mutant fibrils of Aβ 1–40 and compared them with prior and new data for wild‐type fibrils as well as with phosphorylated S8 fibrils. Deuteron static solid‐state NMR techniques, spanning line‐shape analysis, longitudinal relaxation, and chemical exchange saturation transfer methods, were employed to assess the rotameric jumps of several methyl‐bearing and aromatic groups in the core of the fibrils. Taken together, the results indicate the rather significant influence of the PTMs on the hydrophobic core dynamics, which propagates far beyond the local site of the chemical modification. The phosphorylated S8 fibrils display an overall rigidifying of the core based on the higher activation barriers of motions than the wild‐type fibrils, whereas the ΔE3 fibrils induce a broader variety of changes, some of which are thermodynamic in nature rather than the kinetic ones.
Vugmeyster, Liliya↗
Factorial growth at low orders in perturbative QCD: control over truncation uncertainties
A method, known as “minimal renormalon subtraction”, relates the factorial growth of a perturbative series (in QCD) to the power p of a power correction Λ p /Q p . (Λ is the QCD scale, Q some hard scale.) Here, the derivation is simplified and generalized to any p , more than one such correction, and cases with anomalous dimensions. Strikingly, the well-known factorial growth is seen to emerge already at low or medium orders, as a consequence of constraints on the Q dependence from the renormalization group. The effectiveness of the method is studied with the gluonic energy between a static quark and static antiquark (the “static energy”). Truncation uncertainties are found to be under control after next-to-leading order, despite the small exponent of the power correction ( p = 1) and associated rapid growth seen in the first four coefficients of the perturbative series.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Truncating 2D Framework Materials Down to a Single Pore: Synthetic Approaches and Opportunities
Here, in this Accounts article, we summarize our recent work on truncating conjugated two-dimensional framework materials down to a single pore, or a single macrocycle. Conjugated 2D architectures have emerged as one of the most synthetically adaptable motifs for coupling semiconductivity and porosity in metal–organic frameworks (MOFs) and covalent organic frameworks (COFs). However, despite their prevalence, 2D architectures have several limitations. In particular, the strong interlayer π–π stacking can limit both processability and the accessibility of internal active sites. We have found that simple macrocycles preserve key aspects of 2D framework structure and function, including porosity and out-of-plane electrical conductivity, while providing improved processability, surface tunability, and mass transport properties. In this article, we first describe our synthetic approach and general design considerations. Specifically, we show how ditopic analogues of the tritopic ligands commonly found in the synthesis of 2D MOFs and COFs can be used to achieve a diverse library of conjugated macrocycles that resemble fragments of semiconducting frameworks in both form and function. The length of the peripheral side chains, the size of the aromatic core, and the solubility of intermediates are all key variables in favoring selective macrocycle formation over undesired linear polymers and oligomers. Next, we highlight the unique advantages that macrocycles provide, including improved processability, atomically precise surface tunability, and greater active site accessibility. In particular, the identity of the peripheral side chains dramatically impacts both solubility and colloidal stability as well as crystal size and morphology. We further show how the solution processability and nanoscale dimensions of macrocycles can simplify electronic device fabrication and improve electrochemical performance. Finally, we end with a forward-looking discussion on how macrocycles offer a unique bridge between conjugated molecules and extended frameworks, enabling new application areas and fundamental science.
charge transport↗
Structure of the Bastnäsite (001) Surface by Crystal Truncation Rod X-ray Diffraction and Ab Initio Molecular Dynamics: Implications for Separations of a Rare Earth Ore Mineral
Bastnäsite ((Ce,La)FCO 3 ) is the primary mineral source of light rare earth elements, but its surface structure is not well understood. This presents a major challenge in improving beneficiation strategies. In this work, a synergistic combination of X-ray scattering and ab initio molecular dynamics (AIMD) was used to gain atomistic insight into the interfacial structure of bastnäsite. Surface X-ray scattering was used to measure crystal truncation rods (CTRs) of the bastnäsite (001) surface, a significant crystal face with a previously unknown termination. The best-fit atomic-scale model of the CTR data features a carbonate layer at the surface, which is stabilized by the relaxation of carbonate groups from their bulk structural positions. AIMD simulations predict similar surface relaxations, which are shown to be influenced by the protonation of oxygen atoms at the surface. Evidence of ordered water at the interface is also observed in the best-fit model and AIMD simulations. Further, the presence of a carbonate layer at this dominant crystal surface is significant for improving separation technologies because most commonly used ligands utilize anionic functional groups to chelate metal cations at particle surfaces. Without modification, anionic ligands are expected to have poor affinity for the carbonate-terminated (001) surface.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Pt–Co truncated octahedral nanocrystals: a class of highly active and durable catalysts toward oxygen reduction
We report a facile and scalable synthesis of Pt–Co truncated octahedral nanocrystals (TONs) by employing Pt(acac) 2 and Co(acac) 2 as precursors, together with CO molecules and Mn atoms derived from the decomposition of Mn 2 (CO) 10 as a reductant and a {111} facet-directing agent, respectively. Both the composition and yield of the Pt–Co TONs could be varied through the introduction of CHCl 3 . When tested at 80 °C using membrane electrode assembly (MEA), the 4 nm Pt 2.6 Co TONs gave a mass activity of 294 A g Pt –1 at beginning-of-life (BOL) and it increased to 384 A g Pt –1 during recovery cycles. The mass activity at BOL only dropped by 24% after 30 000 voltage cycles at end-of-life (EOL) in a metal dissolution accelerated stress test. The Pt 2.6 Co/C catalyst outperformed the commercial TKK Pt 3 Co/C (230 A g Pt –1 at BOL and 40% loss after 30 000 cycles at EOL) in terms of both activity and durability. Our systematic analysis suggested that the enhancement in activity can be attributed to the combination of small, uniform size and well-defined {111} facets. This new class of catalysts holds promise for applications in proton-exchange membrane fuel cells.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Perturbation theory under the truncated Wigner approximation: How system-environment entanglement formation drives quantum decoherence
In discrete quantum systems, quantum decoherence is the disappearance of simple phase relations as a result of interactions with an environment. For many applications, the question is not necessarily how to avoid (inevitable) system-environment interactions, but rather how to design environments that optimally preserve a system's phase relations in spite of such interactions. The formation of system-environment entanglement is a major driving mechanism for decoherence, and a detailed understanding of this process could inform strategies for conserving coherence optimally. This requires scalable, flexible, and systematically improvable quantum dynamical methods that retain detailed information about the entanglement properties of the environment, yet very few current methods offer this combination of features. Here, in this work, we address this need by introducing a theoretical framework wherein we combine the truncated Wigner approximation with standard time-dependent perturbation theory, allowing for computing expectation values of operators in the combined system-environment Hilbert space. We demonstrate the utility of this framework by applying it to the spin-boson model, representative of qubits and simple donor-acceptor systems. For this model, our framework provides an analytical description of perturbative contributions to expectation values. We monitor how quantum decoherence at zero temperature is accompanied by entanglement formation with individual environmental degrees of freedom. Based on this entanglement behavior, we find that the selective suppression of low-frequency environmental modes is particularly effective for mitigating quantum decoherence.
74 ATOMIC AND MOLECULAR PHYSICS↗
Crystal truncation rods from miscut surfaces with alternating terminations
Miscut surfaces of layered crystals can exhibit a stair-like sequence of terraces having periodic variation in their atomic structure. For hexagonal close-packed and related crystal structures with an αβαβ stacking sequence, there have been long-standing questions regarding how the differences in adatom attachment kinetics at the steps separating the terraces affect the fractional coverage of α vs. β termination during crystal growth. To demonstrate how surface X-ray scattering can help address these questions, we develop a general theory for the intensity distributions along crystal truncation rods (CTRs) for miscut surfaces with a combination of two terminations. We consider half-unit-cell-height steps, and variation of the coverages of the terraces above each step. Example calculations are presented for the GaN (0001) surface with various reconstructions. These show which CTR positions are most sensitive to the fractional coverage of the two terminations. We compare the CTR profiles for exactly oriented surfaces to those for vicinal surfaces having a small miscut angle, and investigate the circumstances under which the CTR profile for an exactly oriented surface is equal to the sum of the intensities of the corresponding family of CTRs for a miscut surface.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Assessing correlated truncation errors in modern nucleon-nucleon potentials
We test the BUQEYE model of correlated effective field theory (EFT) truncation errors on Reinert, Krebs, and Epelbaum's semilocal momentum-space implementation of the chiral EFT (𝜒EFT ) expansion of the nucleon-nucleon (NN) potential. This Bayesian model hypothesizes that dimensionless coefficient functions extracted from the order-by-order corrections to NN observables can be treated as draws from a Gaussian process (GP). We combine a variety of graphical and statistical diagnostics to assess when predicted observables have a 𝜒EFT convergence pattern consistent with the hypothesized GP statistical model. Our conclusions are that, first, the BUQEYE model is generally applicable to the potential investigated here, which enables statistically principled estimates of the impact of higher EFT orders on observables. Second, parameters defining the extracted coefficients such as the expansion parameter 𝑄 must be well chosen for the coefficients to exhibit a regular convergence pattern—a property we exploit to obtain posterior distributions for such quantities. Third, the assumption of GP stationarity across lab energy and scattering angle is not generally met; this necessitates adjustments in future work. We provide a workflow and interpretive guide for our analysis framework, and show what can be inferred about probability distributions for 𝑄, the EFT breakdown scale Λ 𝑏 , the scale associated with soft physics in the 𝜒EFT potential 𝑚 eff , and the GP hyperparameters. All our results can be reproduced using a publicly available Jupyter notebook, which can be straightforwardly modified to analyze other 𝜒EFT NN potentials.
Bayesian methods↗
The 1.3 Å resolution structure of the truncated group Ia type IV pilin from Pseudomonas aeruginosa strain P1
The type IV pilus is a diverse molecular machine capable of conferring a variety of functions and is produced by a wide range of bacterial species. The ability of the pilus to perform host-cell adherence makes it a viable target for the development of vaccines against infection by human pathogens such as Pseudomonas aeruginosa . Here, the 1.3 Å resolution crystal structure of the N-terminally truncated type IV pilin from P. aeruginosa strain P1 (ΔP1) is reported, the first structure of its phylogenetically linked group (group I) to be discussed in the literature. The structure was solved from X-ray diffraction data that were collected 20 years ago with a molecular-replacement search model generated using AlphaFold ; the effectiveness of other search models was analyzed. Examination of the high-resolution ΔP1 structure revealed a solvent network that aids in maintaining the fold of the protein. On comparing the sequence and structure of P1 with a variety of type IV pilins, it was observed that there are cases of higher structural similarities between the phylogenetic groups of P. aeruginosa than there are between the same phylogenetic group, indicating that a structural grouping of pilins may be necessary in developing antivirulence drugs and vaccines. These analyses also identified the α–β loop as the most structurally diverse domain of the pilins, which could allow it to serve a role in pilus recognition. Studies of ΔP1 in vitro polymerization demonstrate that the optimal hydrophobic catalyst for the oligomerization of the pilus from strain K122 is not conducive for pilus formation of ΔP1; a model of a three-start helical assembly using the ΔP1 structure indicates that the α–β loop and the D-loop prevent in vitro polymerization.
Bragagnolo, Nicholas↗
Local Truncation Error-Informed Code Verification
The method of manufactured solutions (MMS) has become increasingly popular in conducting code verification studies on predictive codes, such as nuclear power system codes and computational fluid dynamic codes. The reason for the popularity of this approach is that it can be used when an analytical solution is not available. Using MMS, code developers are able to verify that their code is free of coding errors that impact the observed order of accuracy. While MMS is still an excellent tool for code verification, it does not identify coding errors that are of the same order as the numerical method. This paper presents a method that combines MMS with modified equation analysis (MEA), which calculates the local truncation error (LTE) to identify coding error up to and including the order of the numerical method. This method is referred to as modified equation analysis methd of manufactured solutions (MEAMMS). MEAMMS is then applied to a custom-built code, which solves the shallow water equations, to test the performance of the code verification method. MEAMMS is able to detect all coding errors that impact the implementation of the numerical scheme. To show how MEAMMS is different than MMS, they are both applied to the same first-order numerical method test problem with a first-order coding error. When there are first-order coding errors, only MEAMMS is able to identify them. Finally, this shows that MEAMMS is able to identify a larger set of coding errors while still being able to identify the coding errors MMS is able to identify.
42 ENGINEERING↗
Distributed phase sensing using two-mode squeezed states in a truncated SU(1,1) interferometer
We demonstrate distributed phase sensing with a truncated SU(1,1) interferometer through the measurement of a linear combination of two phases distributed between the two beams of a two-mode squeezed state. We theoretically analyze the sensitivity enhancement with respect to the corresponding classical strategy and experimentally demonstrate a 2 dB quantum noise reduction in the measurement of a linear combination of two phases.
Hong, Seongjin↗
cppTPSA/pyTPSA: a C++/Python package for truncated power series algebra
The truncated power series algebra (TPSA), also referred to as differential algebra (DA), is a well-established and widely used method in particle accelerator physics and astronomy. The most straightforward usage of TPSA/DA is to calculate the Taylor expansion of a given function at a specific point up to order ?. In recent years, as the application of TPSA/TA has been extended to other fields, a reusable implementation of TPSA/DA as a modern C++ library or other high level programming language like Python has become desirable. The cppTPSA package implements TPSA/DA in C++11 and provides developers a convenient library with which to build advanced TPSA/DA-based methods. A Python 3 library, pyTPSA, has also been developed based on the C++ lib.
97 MATHEMATICS AND COMPUTING↗
Chiral rank-$k$ truncations for the multigrid preconditioner of Wilson fermions in lattice QCD
We present a modification to the setup algorithm for the multigrid preconditioner of Wilson fermions in lattice QCD. A larger number of test vectors than that used in conventional multigrid is generated by the smoother. This set of test vectors is then truncated by a singular value decomposition on the chiral components of the test vectors, which are subsequently used to form the prolongation and restriction matrices of the multigrid hierarchy. This modification is demonstrated to improve the convergence of linear equations on an anisotropic lattice with 𝑚𝜋 ≈ 280 MeV from the Hadron Spectrum Collaboration and an isotropic lattice with 𝑚𝜋 ≈ 220 MeV from the MILC Collaboration. The lattice volume dependence of the method is also examined.
Whyte, Travis [Jülich Supercomputing Center, Jülic↗
TRUNCATED REACTION OPERATORS AND THE HELIUM PROBLEM
Use of truncated reaction operator to obtain approximate solutions to schroedinger equation
SCHROEDINGER EQUATION↗
The effect of truncation on tests of hypo- theses for normal populations
Loss of hypothesis testing power for truncated normal populations when complete normal distribution theory is used
TEST METHOD↗