SEARCH · Engineering Papers

Results for “MATRICES”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 199 records · Page 11

All sources NASA NTRS DOE OSTI

DOE OSTI2022

Trimethyloxonium-mediated methylation strategies for the rapid and simultaneous analysis of chlorinated phenols in various soils by electron impact gas chromatography–mass spectrometry

Abstract The efficient methylation of a panel of five industrial and environmentally-relevant chlorophenols (CPs) employing trimethyloxonium tetrafluoroborate (TMO) for their qualitative detection and identification by electron impact gas chromatography–mass spectrometry (EI-GC–MS) is presented. The protocol’s execution is simple and smoothly converts the phenols into their O-methylated counterparts conveniently at ambient temperature. The efficiency of two versions of the protocol was successfully tested in their ability to simultaneously derivatize five CPs (2-chlorophenol, 2,4-dichlorophenol, 2,4,6-trichlorophenol, pentachlorophenol and triclosan) in six distinct, separate soil matrices (Nebraska EPA standard soil, Virginia Type A soil, Ottawa sand, Baker sand, Silt and Georgia EPA standard soil) when present at low levels (~ 10 μgg −1 ). The first version involves the direct derivatization of the spiked soils with the methylating salt while the second one involves an initial soil extraction step of the CPs followed by methylation. The MDL values for each methylated CP were determined and lower values were found (4.1–13.2 ng . mL −1 ) for both sand matrices (Ottawa and Baker) as well as for the Georgia EPA standard soil, while larger values (8.2–21.8 ng . mL −1 ) were found for the Virginia Type soil, Nebraska EPA standard soil and Silt. The presented protocol offers a safer and more practical alternative to the universally employed diazomethane method and can be readily applicable to matrices other than soils. Furthermore, the protocols described herein may find applicability to the methylation of other analytes bearing acidic protons.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

A universal variational quantum eigensolver for non-Hermitian systems

Abstract Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional $$O(log_{2}{N})$$ O ( l o g 2 N ) quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

DESI mock challenge: Halo and galaxy catalogues with the bias assignment method

We present a novel approach to the construction of mock galaxy catalogues for large-scale structure analysis based on the distribution of dark matter halos obtained with effective bias models at the field level. We aim to produce mock galaxy catalogues capable of generating accurate covariance matrices for a number of cosmological probes that are expected to be measured in current and forthcoming galaxy redshift surveys (e.g. two- and three-point statistics). The construction of the catalogues shown in this paper is part of a mock-comparison project within the Dark Energy Spectroscopic Instrument (DESI) collaboration. We use the bias assignment method ( BAM ) to model the statistics of halo distribution through a learning algorithm using a few detailed N-body simulations, and approximated gravity solvers based on Lagrangian perturbation theory. We introduce cosmic-web-dependent corrections to modelling redshift-space distortions at the N-body level – both in the halo and galaxy distributions –, as well as a multi-scale approach for accurate assignment of halo properties. Using specific models of halo occupation distributions to populate halos, we generate galaxy mocks with the expected number density and central-satellite fraction of emission-line galaxies, which are a key target of the DESI experiment. BAM generates mock catalogues with per cent accuracy in a number of summary statistics, such as the abundance, the two- and three-point statistics of halo distributions, both in real and redshift space. In particular, the mock galaxy catalogues display ~3%-10% accuracy in the multipoles of the power spectrum up to scales of k ~ 0.4 h -1 Mpc. We show that covariance matrices of two- and three-point statistics obtained with BAM display a similar structure to the reference simulation. BAM offers an efficient way to produce mock halo catalogues with accurate two- and three-point statistics and is able to generate a variety of multi-tracer catalogues with precise covariance matrices of several cosmological probes. We discuss future developments of the algorithm towards mock production in DESI and other galaxy-redshift surveys.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green’s function methods: Theory and application to GW and GF2

One-particle Green’s function methods can model molecular and solid spectra at zero or non-zero temperatures. One-particle Green’s functions directly provide electronic energies and one-particle properties, such as dipole moment. However, the evaluation of two-particle properties, such as $\langle$S 2 $\rangle$ and $\langle$N 2 $\rangle$, can be challenging because they require a solution of the computationally expensive Bethe–Salpeter equation to find two-particle Green’s functions. We demonstrate that the solution of the Bethe–Salpeter equation can be completely avoided. Applying the thermodynamic Hellmann–Feynman theorem to self-consistent one-particle Green’s function methods, we derive expressions for two-particle density matrices in a general case and provide explicit expressions for GF2 and GW methods. Such density matrices can be decomposed into an antisymmetrized product of correlated one-electron density matrices and the two-particle electronic cumulant of the density matrix. Cumulant expressions reveal a deviation from ensemble representability for GW, explaining its known deficiencies. We analyze the temperature dependence of $\langle$S 2 $\rangle$ and $\langle$N 2 $\rangle$ for a set of small closed-shell systems. Interestingly, both GF2 and GW show a non-zero spin contamination and a non-zero fluctuation of the number of particles for closed-shell systems at the zero-temperature limit.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Real-world CMB lensing quadratic estimator power spectrum response

I describe a method to estimate response matrices of Cosmic Microwave Background (CMB) lensing power spectra estimators to the true sky power under realistic conditions. Applicable to all lensing reconstruction pipelines based on quadratic estimators (QE), it uses a small number of Gaussian CMB Monte-Carlos and specially designed QE's in order to obtain sufficiently accurate matrices with little computational effort. This method may be used to improve the modelling of CMB lensing band-powers by incorporating at least some of the non-idealities encountered in CMB lensing reconstruction. These non-idealities always include masking, and often inhomogeneous filtering, either in the harmonic domain or pixel space. I obtain these matrices for Planck latest lensing reconstructions, and then show that the residual couplings induced by masking explain very well the residual multiplicative bias seen on the Planck simulations, removing the need for an empirical correction.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Computational properties of finite PNmatrices

Abstract Recent compositionality results in logic have highlighted the advantages of enlarging the traditional notion of logical matrix semantics, namely by incorporating non-determinism and partiality. Still, several important properties which are known to be computable for finite logical matrices have not been studied in the wider context of partial non-deterministic matrices (PNmatrices). In this paper, we study how incorporating non-determinism and/or partiality in logical matrices impacts on the computational properties of some natural problems regarding their induced logics and concretely their sets of theorems. We show that, while for some of these problems there is no relevant computational impact, there are problems whose computational complexity increases and still other problems that simply become undecidable. In particular, we show that the problem of checking whether the logics characterized by two finite PNmatrices have the same set of theorems is not decidable. This undecidability result explores the connection between PNmatrices and term-DAG-automata, where the universality problem is known to be undecidable. This link also motivates a final contribution, in the form of a pumping-like lemma, which can be used, in some cases, to show that a given logic cannot be characterized by a finite PNmatrix.

Filipe, Pedro↗

DOE OSTI2020

Polaritons and excitons: Hamiltonian design for enhanced coherence

The primary questions motivating this report are: Are there ways to increase coherence and delocalization of excitation among many molecules at moderate electronic coupling strength? Coherent delocalization of excitation in disordered molecular systems is studied using numerical calculations. The results are relevant to molecular excitons, polaritons, and make connections to classical phase oscillator synchronization. In particular, it is hypothesized that it is not only the magnitude of electronic coupling relative to the standard deviation of energetic disorder that decides the limits of coherence, but that the structure of the Hamiltonian—connections between sites (or molecules) made by electronic coupling—is a significant design parameter. Inspired by synchronization phenomena in analogous systems of phase oscillators, some properties of graphs that define the structure of different Hamiltonian matrices are explored. The report focuses on eigenvalues and ensemble density matrices of various structured, random matrices. Some reasons for the special delocalization properties and robustness of polaritons in the single-excitation subspace (the star graph) are discussed. The key result of this report is that, for some classes of Hamiltonian matrix structure, coherent delocalization is not easily defeated by energy disorder, even when the electronic coupling is small compared to disorder.

Science & Technology - Other Topics↗

DOE OSTI2023

Bell inequalities in 2-2 scattering

We consider Bell inequalities in 2-2 scattering of photons, gravitons, fermions, and pions. We choose measurement settings that give maximum Bell violation for maximally entangled states and calculate the relevant Bell inequalities for these processes. For photon scattering at low energies, quantum electrodynamics exhibits Bell violation for all scattering angles except for a small transverse region. This leads to a fine-tuning problem. Incorporating a light axion/axionlike particle removes the fine-tuning problem and constrains the axion-coupling–axion-mass parameters. Allowing for graviton exchange and demanding Bell violation in photon scattering, we find that the weak-gravity conjecture is satisfied. Quantum gravity effects on axion coupling is discussed. For 2-2 graviton scattering, we find that CEMZ bounds allow for, at most, small Bell violations. Restriction on the Weinberg angle is found by demanding Bell violation in Bhabha scattering. We use recent $S$-matrix bootstrap data for pions and photons to study the Bell parameter in the space of allowed $S$-matrices. In the photon case, we study the Bell parameters as a function of energy and find support for the effective field theory observations. We discuss Bell parameter for pion $S$-matrices, which are qutrits. For pions, we find that there is a minimization of a suitable Bell parameter for $S$-matrices which exhibit Regge behavior.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

GCoD: Graph Convolutional Network Acceleration via Dedicated Algorithm and Accelerator Co-Design

Graph Convolutional Networks (GCNs) have emerged as the state-of-the-art graph learning model. However, it remains notoriously challenging to inference GCNs over large graph datasets, limiting their application to large real-world graphs and hindering the exploration of deeper and more sophisticated GCN graphs. This is because real-world graphs can be extremely large and sparse. Furthermore, the node degree of GCNs tends to follow the power-law distribution and therefore have highly irregular adjacency matrices, resulting in prohibitive inefficiencies in both data processing and movement and thus substantially limiting the achievable GCN acceleration efficiency. To this end, this paper proposes the first GCN algorithm and accelerator Co-Design framework dubbed GCoD which can largely alleviate the aforementioned GCN irregularity and boost GCNs' inference efficiency. Specifically, on the algorithm level, GCoD integrates a divide and conquer GCN training strategy that polarizes the graphs to be either denser or sparser in local neighborhoods without compromising the model accuracy, resulting in graph adjacency matrices that (mostly) have merely two levels of workload and enjoys largely enhanced regularity and thus ease of acceleration. On the hardware level, we further develop a dedicated two-pronged accelerator with a separated engine to process each of the aforementioned workloads, further boosting the overall utilization and acceleration efficiency. Extensive experiments and ablation studies validate that our GCoD consistently outperforms state-of-the-art designs in terms of accelerator efficiency while maintaining or even improving the task accuracy. Additionally, we visualize GCoD trained graph adjacency matrices to better understand its advantages. All codes and pre-trained models will be released upon acceptance.

You, Haoran↗

DOE OSTI2024

Impact of Reordering on the LU Factorization Performance of Bordered Block-Diagonal Sparse Matrix

Power engineers rely on computer-based simulation tools to assess grid performance and ensure security. At the core of these tools are solvers for sparse linear equations. When transformed into a bordered block-diagonal (BBD) structure, part of the sparse linear equation solving can be parallelized. This work focuses on using the Schur-complement-based method for LU factorization on BBD matrices, specifically, Jacobian matrices from large-scale systems. Our findings show that the natural ordering method outperforms the default ordering method in computational performance for each block of the BBD matrix. This observation is validated using synthetic 25k-bus and 70k-bus cases, showing a speedup of up to 38% when using natural ordering without permutation. Additionally, the impact of the number of partitions is studied, and the result shows that computational performance improves with more, smaller partitions in the BBD matrices.

BBD matrix↗

DOE OSTI2026

Quantum Time-Space Tradeoffs for Matrix Problems

We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices 𝐴, including the discrete Fourier transform matrix, we prove that quantum circuits with at most 𝑇 input queries and 𝑆 qubits of memory require 𝑇 = Ω⁢(𝑛 2 /𝑆) to compute matrix-vector product 𝐴⁢𝑥 for 𝑥 ∈{0,1 𝑛 . We similarly prove that matrix multiplication for 𝑛 ×𝑛 binary matrices requires 𝑇 = Ω⁢(𝑛 3 /$\sqrt{𝑆}$). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity—the sum of the space per layer of a circuit. We also consider Boolean (i.e., AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for 𝑛 × 𝑛 Boolean matrix multiplication to 𝑇 = Ω⁢(𝑛 2.5 /𝑆 1/4 ) from 𝑇 = Ω⁢(𝑛 2.5 /𝑆 1/2 ). Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.

lower bounds↗

DOE OSTI2025

Accelerating GNNs on GPU Sparse Tensor Cores through N:M Sparsity-Oriented Graph Reordering

Recent advancements in GPU hardware support have introduced the capability to leverage N:M sparse patterns for substantial performance gains. Graphs in Graph Neural Networks (GNNs) are typically sparse, but the sparsity is often irregular, not conforming to such sparse patterns. In this paper, we propose a novel graph reordering algorithm, the first of its kind, to reshape irregular graph data into the N:M structured sparse pattern at the tile level, allowing linear-algebra-based graph operations in GNNs to benefit from the N:M sparse hardware. The optimization is lossless, maintaining the accuracy of GNN. It can remove 98-100\% violations of the N:M sparse patterns at the vector level, and increase the proportion of conforming graphs in SuiteSparse collection from 5-9\% to 88.7-93.5\%. On A100 GPUs, the optimization accelerates Sparse Matrix Matrix (SpMM) by up to 43X (2.3X -- 7.5X on average) and speeds up the key graph operations in GNNs on real graphs by as much as 8.6X (3.5X on average).

artificial intelligence, graph neural networks↗

DOE OSTI2021

The Attraction Indian Buffet Distribution

We propose the attraction Indian buffet distribution (AIBD), a distribution for binary feature matrices influenced by pairwise similarity information. Binary feature matrices are used in Bayesian models to uncover latent variables (i.e., features) that explain observed data. The Indian buffet process (IBP) is a popular exchangeable prior distribution for latent feature matrices. In the presence of additional information, however, the exchangeability assumption is not reasonable or desirable. The AIBD can incorporate pairwise similarity information, yet it preserves many properties of the IBP, including the distribution of the total number of features. Thus, much of the interpretation and intuition that one has for the IBP directly carries over to the AIBD. A temperature parameter controls the degree to which the similarity information affects feature-sharing between observations. Unlike other nonexchangeable distributions for feature allocations, the probability mass function of the AIBD has a tractable normalizing constant, making posterior inference on hyperparameters straight-forward using standard MCMC methods. A novel posterior sampling algorithm is proposed for the IBP and the AIBD. We demonstrate the feasibility of the AIBD as a prior distribution in feature allocation models and compare the performance of competing methods in simulations and an application.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Sempervirens: A Fast Reconstruction Algorithm for Noisy and Incomplete Binary Matrix Representations of Trees

Applications such as reconstructing cell lineage trees (represented as phylogenetic trees) from single-cell sequencing data require reconstructing a {0,1}-matrix that has many errors and missing entries. We introduce Sempervirens, a very fast matrix reconstruction algorithm for noisy and incomplete matrix representations of phylogenetic trees. Sempervirens uses an iterative maximum-likelihood approach to determine the topology tree represented by the corrupted data. We show that Sempervirens is at least three orders of magnitude faster than other methods on thousand by thousand matrices, with the speed gap widening with larger matrices. We also show that Sempervirens matches state-of-the-art methods in reconstruction accuracy. The speed of Sempervirens enables it to be tractably applied to reconstructing much larger matrices than those that other methods can reconstruct. In addition to experimental results, we justify the algorithm with a mathematical treatment of its subprocedures.

algorithms↗

DOE OSTI2022

HERO WEC V1.0 - WEC-Sim Detailed Simulation Runs and Summary Data

Data sets from simulation runs include 144 wave cases that were run based on the WEC-Sim model (https://mhkdr.openei.org/submissions/483). The 144 wave cases represent waves with the following wave height and wave period ranges: - Significant Wave Height: 0.25 - 4.0m in 0.25m increments - Wave Period: 5 - 13 sec in 1 sec increments Each run was simulated using a Pierson-Moskowitz irregular wave spectrum with a 200 second ramp time and a total simulation time of 2,200 seconds. Summary data set includes a spreadsheet and image files with matrices that are associated with data from simulation runs. All matrices cover the same significant wave height and wave periods from the sim runs, in the same increments. The following matrices are included: - Mechanical Power Input: Absorbed power calculated using linear input velocity at PTO (power take-off) multiplied by winch tension. - Mechanical Power Output: The hydraulic power calculated at the pump output prior to the pressure relief valve dump. - Feed Pressure: Average pressure at RO (reverse osmosis) input prior to pre-filters - RO Power Input: The hydraulic power calculated at the RO system inlet, accounts for flow loss due to pressure relief valve - Water Production: Average clean water flow at RO output

16 TIDAL AND WAVE POWER↗

DOE OSTI2024

HERO WEC V1.0 2024 - WEC-Sim Detailed Simulation Runs and Summary Data

This dataset includes results from simulations of NREL's hydraulic and electric reverse osmosis wave energy converter (HEREO WEC). Simulation runs include 135 wave cases that were based on the updated WEC-Sim model, which is linked below. The data represented in this repository is based on an updated WEC-Sim model using laboratory data to tune and refine the original WEC-Sim model for the V1.0 HERO WEC. The 135 wave cases represent waves with the following wave height and wave period ranges: - Significant Wave Height: 0.25 - 3.75m in 0.25m increments - Wave Period: 5 - 13 sec in 1 sec increments Each run was simulated using a Pierson-Moskowitz irregular wave spectrum with a 100 second ramp time, a total simulation time of 3,100 seconds, and a simulation time-step of 0.005s. A reference table has been included to map each multi condition run (MCR) case with each wave condition. Summary data set includes a spreadsheet and image files with matrices that are associated with data from simulation runs. All matrices cover the same significant wave height and wave periods from the simulation runs, in the same increments. The following matrices are included: - Power Abs: The average absorbed power from the WEC (calculated from anchor reaction force and heave velocity) - Power Hyd: The average hydraulic power output at pump (calculated from pump output flow and pressure) - Power - Hyd ROi: The average hydraulic power measured at the RO system inlet (calculated from RO system pressure and flow (pre-accumulator)) - Flow - Pump out: The average flowrate measured at the pump outlet - Flow - Perm: The average permeate (clean water) production - Flow - RO (pre): The average flowrate measured at the inlet of the RO system before the accumulators - Flow - RO (post): The average flowrate measured after the accumulator bank in the RO system - Pressure - RO: The average pressure measured at the inlet of the RO system This data set has been developed by the National Renewable Energy Laboratory, operated by the Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Water Power Technologies Office.

16 TIDAL AND WAVE POWER↗

DOE OSTI2021

Experimental Study of Microgel Conformance-Control Treatment for a Polymer-Flooding Reservoir Containing Superpermeable Channels

Summary Polymer flooding has been widely used to improve oil recovery. However, its effectiveness would be diminished when channels (e.g., fractures, fracture-like channels, void-space conduits) are present in a reservoir. In this study, we designed a series of particular sandwich-like channel models and tested the effectiveness and applicable conditions of micrometer-sized preformed particle gels (PPGs, or microgels) in improving the polymer-flooding efficiency. We studied the selective penetration and placement of the microgel particles, and their abilities for fluid diversion and oil-recovery improvement. The results suggest that polymer flooding alone would be inefficient to achieve a satisfactory oil recovery as the heterogeneity of the reservoir becomes more serious (e.g., permeability contrast kc/km > 50). The polymer solution would vainly flow through the channels and leave the majority of oil in the matrices behind. Additional conformance-treatment efforts are required. We tried to inject microgels in an attempt to shut off the channels. After the microgel treatment, impressive improvement of the polymer-flooding performance was observed in some of our experiments. The water cut could be reduced significantly by as high as nearly 40%, and the sweep efficiency and overall oil recovery of the polymer flood were improved. The conditions under which the microgel-treatment strategy was effective were further explored. We observed that the microgels form an external impermeable cake at the very beginning of microgel injection and prevent the gel particles from entering the matrices. Instead, the microgel particles could selectively penetrate and shut off the superpermeable channels under proper conditions. Our results suggest that the 260-µm microgel particles tested in this study are effective to attack the excessive-water-production problem and improve the oil recovery when the channel has a high permeability (>50 darcies). The gels are unlikely to be effective for channels that are less than 30 darcies because of the penetration/transport difficulties. After the gels effectively penetrate and shut off the superpermeable channel, the subsequent polymer solution is diverted to the matrices (i.e., the unswept oil zones) to displace the bypassed oil. Overall, this study provides important insights to help achieve successful polymer-flooding applications in reservoirs with superpermeable channels.

Engineering↗

DOE OSTI2022

Applying Quantum Tomography to Hadronic Interactions

A proper description of inclusive reactions is expressed with density matrices. Quantum tomography reconstructs density matrices from experimental observables. We review recent work that applies quantum tomography to practical experimental data analysis. Almost all field-theoretic formalism and modeling used in a traditional approach is circumvented with great efficiency. Tomographically-determined density matrices can express information about quantum systems which cannot in principle be expressed with distributions defined by classical probability. Topics such as entanglement and von Neumann entropy can be accessed using the same natural language where they are defined. A deep relation exists between separability, as defined in quantum information science, and factorization, as defined in high energy physics. Factorization acquires a non-perturbative definition when expressed in terms of a conditional form of separability. An example illustrates how to go from data for momentum 4-vectors to a density matrix while bypassing almost all the formalism of the Standard Model.

Martens, J. C.↗