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Explainable Machine Learning for Functional Data

Black-box machine learning models are recognized as useful tools for prediction applications, but the algorithmic complexity of some models causes interpretation challenges. Explainability methods have been proposed to provide insight into these models, but there is little research focused on supervised modeling with functional data inputs. We argue that, especially in applications of high consequence, it is important to explicitly model the functional dependence in a black-box analysis to not obscure or misrepresent patterns in explanations. As such, we propose the V ariable importance E xplainable E lastic S hape A nalysis (VEESA) pipeline for training supervised machine learning models with functional inputs. The pipeline is an analysis process that includes the data preprocessing, modeling, and post-hoc explanations. The preprocessing is done using elastic functional principal components analysis, which accounts for vertical and horizontal variability in functional data and, ultimately, allows for explanations in the original data space that identify the important functional variability without bias due to correlated variables. Here, we demonstrate the pipeline on two high-consequence applications: explosives classification for national security and inkjet printer identification in forensic science. The applications exhibit the VEESA pipeline’s ability to provide an understanding of the characteristics of the functional data useful for prediction. Code for implementing the pipeline is available in the veesa R package (and supplemental python code).

Elastic Shape Analysis

Machine learning inversion from scattering for mechanically driven polymers

A machine learning inversion method is developed for analyzing scattering functions of mechanically driven polymers and extracting the corresponding feature parameters, which include energy parameters and conformation variables. The polymer is modeled as a chain of fixed-length bonds constrained by bending energy, and it is subject to external forces such as stretching and shear. We generate a data set consisting of random combinations of energy parameters, including bending modulus, stretching and shear force, along with Monte Carlo-calculated scattering functions and conformation variables such as end-to-end distance, radius of gyration and off-diagonal component of the gyration tensor. The effects of the energy parameters on the polymer are captured by the scattering function, and principal component analysis ensures the feasibility of the machine learning inversion. Finally, we train a Gaussian process regressor using part of the data set as a training set and validate the trained regressor for inversion using the rest of the data. The regressor successfully extracts the feature parameters.

Gaussian process regressors

Design Basis Document / Owner’s Technical Specification for Nitrate Salt Systems in CSP Projects: Volume 1 Specifications for Parabolic Trough Projects, Volume 2 - Specifications for Central Receiver Projects, Volume 3 - Narrative

Design Basis Documents / Owner’s Technical Specifications are developed for parabolic trough and central receiver power plants using nitrate salt as the heat transport fluid and the thermal storage medium. The goals were to 1) distill the successful experience with nitrate salt systems from as many commercial projects as possible, 2) provide technical bases for equipment design/selection that an owner can impose on an EPC contractor, 3) compile this information in one location to provide guidance on salt systems that is as broadly applicable to as many projects as possible, and 4) work toward an industry consensus on a Design Basis Document that reflects the lessons learned from earlier commercial projects. The document allows an owner to provide design guidance, and to impose a minimum set of requirements, above and beyond those in the normal Codes and Standards, on an EPC contractor in an effort to avoid a repeat of past mistakes. If successful, this would allow salt systems to achieve the same levels of reliability and availability as commercial parabolic trough plants using diphenyl oxide / biphenyl (i.e., Dowtherm A, Therminol VP-1) as the heat transfer fluid. The report consists of 3 volumes: • Volume 1 - Specifications for Parabolic Trough Projects • Volume 2 - Specifications for Central Receiver Projects • Volume 3 - Narrative Volumes 1 and 2 include discussions of the following topics: • Plant functional requirements for the salt systems • Plant operating states, and transitions between states, for the salt systems • Risk analysis of the principal salt components • Plant requirements for the salt systems to meet the functional and the operating requirements • Type of specification for the principal salt components: functional; or prescriptive • Current state of the art for salt systems. Volume 3 discusses a number of cases in which the salt equipment at commercial parabolic trough and central receiver projects has not met the projected levels of reliability and availability. Possible reasons for the sources of the problems are discussed, as is a range of possible alternate designs that could avoid the known problems.

14 SOLAR ENERGY

Possible Realization of Hyperbolic Plasmons in Few-Layered Rhenium Disulfide

Hyperbolic plasmons are a highly desired property in optoelectronics and biomolecular sensing. The necessary condition to realize hyperbolic plasmons is a significant anisotropy of the principal components of the dielectric function, such that at a certain frequency range, one component is negative and the other is positive, i.e., one component is metallic, and the other one is dielectric. Here, we study the effect of anisotropy in ReS 2 , and our theory shows that ReS 2 can host hyperbolic plasmons in the ultraviolet frequency range. The operating frequency range of the hyperbolic plasmons can be tuned with the number of ReS 2 layers. However, we note that the significantly large imaginary part of the macroscopic dielectric response in all layered variants of ReS 2 can result in substantial losses for the hyperbolic plasmons, as in the case with other known hyperbolic materials, with the exception of MoOCl 2 . We also note that ReS 2 hosts ultraviolet hyperbolic plasmons while ZrSiSe, WTe 2 , and CuS nanocrystals host infrared plasmons, providing a novel platform for optoelectronics in the ultraviolet range.

36 MATERIALS SCIENCE

SpecDis: Value Added Distance Catalog for 4 Million Stars from DESI Year-1 Data

We present the SpecDis value-added stellar distance catalog accompanying DESI Data Release 1. SpecDis trains a feed-forward neural network (NN) with Gaia parallaxes and gets the distance estimates. To build up an unbiased training sample, we do not apply selections on parallax error or signal-to-noise (S/N) of the stellar spectra, and instead, we incorporate parallax error into the loss function. Moreover, we employ principal component analysis to reduce the noise and dimensionality of stellar spectra. Validated by independent external samples of member stars with precise distances from globular clusters, dwarf galaxies, stellar streams, combined with blue horizontal branch stars, we demonstrate that our distance measurements show no significant bias up to 100 kpc, and are much more precise than Gaia parallax beyond 7 kpc. The median distance uncertainties are 23%, 19%, 11%, and 7% for S/N < 20, 20 ≤ S/N < 60, 60 ≤ S/N < 100, and S/N ≥ 100. Selecting stars with ${\mathrm{log}}\,g\lt 3.8$ and distance uncertainties smaller than 25%, we have more than 74,000 giant candidates within 50 kpc of the Galactic center and 1500 candidates beyond this distance. Additionally, we develop a Gaussian mixture model to identify unresolvable equal-mass binaries by modeling the discrepancy between the NN-predicted and the geometric absolute magnitudes from Gaia parallaxes and identify 120,000 equal-mass binary candidates. Our final catalog provides distances and distance uncertainties for >4 million stars, offering a valuable resource for Galactic astronomy.

astronomy data analysis

Birefringence axis rotation and nonlinear response of a side-hole fiber pressure sensor

The pressure-induced birefringence of side-hole fibers has been widely explored for hydrostatic pressure measurement. Here, this paper studies the effects of the initial birefringence of a side-hole fiber on the pressure response of polarimetric sensors based on side-hole fibers. A mathematical model, where the stress field induced by the pressure is assumed to be uniform in the core region of the fiber, was established to calculate the birefringence and optical axis rotation as a function of the differences between the two principal stress components of the stress field in the core region. Simulation results show that when there is misalignment between the initial birefringence axes and the principal axes of the stress field, pressure changes cause rotations of the birefringence axis, leading to complicated and nonlinear changes of birefringence as a function of pressure. Experiments were conducted using fiber-Bragg-grating-based resonators on side-hole fibers with different initial birefringence, and the results agree well with the simulation. The results underscore the importance of the control of magnitude and direction of the initial birefringence of the side-hole fiber and provide guidelines for the design and fabrication of side-hole fiber pressure sensors.

42 ENGINEERING

Using the optimal combined index weight ratio to improve the probability of anomaly detection in big area additive manufacturing

Big Area Additive Manufacturing (BAAM) of composites requires significant time, energy, and material, so it is critical to reduce production inefficiencies to make functional parts without multiple iterations. Statistical process control coupled with Principal Component Analysis (PCA) is a powerful technique that provides a quick, computationally inexpensive, and intuitive way for operators to detect defects that form in a manufacturing process without massive datasets. Recently, a combined index that is a weighted sum of the Hotelling's T 2 and squared residual error statistics has been proposed that can be monitored in one chart, improving interpretation accuracy and simplicity. However, the literature does not offer a formal method to optimise the weights. Here, we introduce two new approaches to the traditional weight selection approach using simulated and BAAM image data. Approach 1 uses a theoretically motivated optimum inspired by probabilistic principal component analysis. Approach 2 systematically varies the ratio of the weights to find the optimum. We show that approach 1 delivers optimal anomaly detection performance in select cases while approach 2 fares better in practice. Surprisingly, we also show that choosing a more complex PCA model has a minimal negative impact on anomaly detection performance compared to a more simplistic model.

3-dimensional printing

Image Distinguishability Analysis Testing Through Principal Components and Its Application to Hot Spot Scale Invariance

Hot spots are spatial regions of intense energy localization that govern initiation of secondary high explosives. Studies that characterize or compare simulated hot spots are frequently either qualitatively descriptive or resort to quantitative distribution functions that neglect stochastic variations and spatial correlations—effects that are also neglected in common comparison tests like the Kolmogorov–Smirnov test. To this end, we develop an image distinguishability analysis (IDA) test based on principal component (PC) analysis that makes pixel-by-pixel comparisons between small, for example, O(<10), image data sets. The IDA test makes comparisons through a generalized distance metric in the PC space and a test statistic that is derived to calculate mathematical equation-values. Here, we derive a statistical distribution and criticality criterion to determine whether images are distinguishable from established baselines. We apply the IDA test on images generated from molecular dynamics simulations of hot spots from pore collapse in TATB to assess scale invariance in the complex patterns of hot spots that form in a representative high explosive crystal. The IDA test shows that TATB hot spot spatial temperature fields and their derived temperature histograms exhibit scale-invariant features over specific intervals of shock orientation, strength, and initial pore diameter. However, the IDA test also shows that qualitatively different conclusions regarding invariance can be reached depending on whether the hot spot is treated as a spatially correlated field as opposed to a distribution function that lacks spatial information.

organic

Uncertainty Quantification for Smooth Functional Data with Application to Material Properties

This document outlines a method for processing functional output (i.e., curves) for the ultimate purpose of sampling curves under specified input conditions for use in modeling and simulation uncertainty quantification (UQ) studies. A set of benchmark curves sufficiently representative of the relevant scenario(s) being simulated are provided to the process and formatted as described in Section 1. Principal Component Analysis (PCA) is utilized to discover the components of uncertainty in the benchmark curves and is outlined in Section 2. Section 3 describes the application of uncertainty quantification to the PCA results for the purpose of sampling curves to be used in UQ analysis. Section 4 applies these techniques to an example benchmark dataset. Concluding remarks are provided in the final section.

36 MATERIALS SCIENCE

Global teleconnections influencing large-scale drought in the United States using SVDI

Understanding recent large-scale drought patterns and the mechanisms producing extreme drought events is vital for future drought forecasts and understanding future drought risks. Increasingly, vapor pressure deficit (VPD) has been used as an important measure of evaporative demand and proxy for drought detection. In this study, VPD is used to calculate the new Standardized VPD Drought Index (SVDI) with NASA North American Land Data Assimilation System (NLDAS) data. Previous studies have shown that SVDI accurately identifies the timing and magnitude short-term droughts in the United States (U.S). In the present study, SVDI is now used to identify large-scale drought patterns between 1980 and 2021 and drought variability driven by selected global teleconnections originating in the Pacific and Atlantic Oceans. Spatial drought characteristics were extracted from SVDI using empirical orthogonal function (EOF) analysis. Then a k-means clustering algorithm was applied to both EOF principal components and primary teleconnections, including the El Nino-Southern Oscillation (ENSO) and Pacific Decadal Oscillation (PDO) to identify drought events driven by the Pacific Ocean. Results show that the SVDI is useful in evaluating large-scale drought variability in the U.S. related to global teleconnections, and that mechanisms influencing summer drought patterns in the Western and Southwestern U.S. are driven by a tropical-extratropical interactions originating in the equatorial Pacific Ocean related to ENSO dynamics with interdecadal variability modulated by PDO. The large-scale droughts in the Central and Southern U.S., like those in 2011 and 2012, on the other hand, are driven by the North Pacific Ocean warm pool during a strong negative PDO, which subsequently influenced variability in the Bermuda-Azores High in the Atlantic Ocean. In summer 2011, the Bermuda-Azores High weakened, reducing the onshore winds and moisture transport along the eastern Gulf of Mexico and contributing to ongoing drought in the region. The Northern Pacific and Atlantic Ocean sea surface temperatures (SSTs) have increased between 1980 and 2021. In conclusion, as SSTs continue to rise in the Northern Pacific Ocean, one consequence of the coupled North Pacific warm pool and atmospheric dynamics, is to increase summer drought variability over a large region in the southern and midwestern U.S. under global warming.

54 ENVIRONMENTAL SCIENCES

Unraveling Fundamental Activity–Stability Relationships in Rutile Oxides

The oxygen evolution reaction (OER) is a key anodic half-cell reaction that accompanies several critical electrochemical reduction reactions of interest to a variety of applications. Despite steady advances in understanding and qualitatively predicting OER activity and selectivity trends, a comprehensive description or prediction of material aqueous (in)stability and degradation mechanisms remains elusive, even though these processes critically influence device lifetime and economic feasibility. In this work, we investigate the interplay, or lack thereof, between OER activity and material aqueous stability across rutile oxides, with a particular focus on iridium oxide (IrO 2 ). By applying a Born–Haber cycle, we calculate the thermodynamic driving force for metal dissolution as a function of the applied bias and electrolyte conditions. We apply interpretable machine learning techniques, including principal component analysis and symbolic regression, to analyze trends across rutile oxides and find that key thermodynamic descriptors for OER activity and surface stability are only very weakly correlated. Instead, the local atomic environment─especially electronic structure signatures for interactions between the active site and its neighbors─plays a more important role in predicting material stability. Leveraging these insights, we investigate the impact of doping IrO 2 with a range of transition metals and show that the stability of Ir active sites can be tuned largely independently of its predicted OER activity. These insights lay the foundation for material design to improve stability with respect to corrosion, with the ultimate aim to enhance long-term stability without sacrificing catalytic performance in the OER.

evolution reactions

Structural features of xylan dictate reactivity and functionalization potential for bio-based materials

Plant-based materials have the potential to replace some petroleum-based products, offering compostability and biodegradability as critical advantages. Xylan-rich biomass sources are gaining recognition due to their abundance and underutilization in current industrial applications. Research of potential xylan applications has been complicated by the complex and heterogeneous structure that varies for different xylan feedstocks. Acylation is a broadly used reaction in functionalization of polysaccharides at an industrial scale. However, the efficiency of this reaction varies with the xylan source. To optimize xylan valorization, a systematic understanding of structure–reactivity relationships is essential. This study explores, characterizes, and compares various xylan feedstocks in the acylation process. Xylan feedstocks were analyzed for their chemical composition, degree of polymerization, branching, solubility, and presence of impurities. These features were correlated with xylan glycotypes’ reactivity toward functionalization with succinic anhydride in an optimized DMSO/KOH condition, achieving carboxyl contents of up to 1.46. We used principal component analysis and hierarchical clustering to identify key structural features of xylan that promote its reactivity. Our findings reveal that xylans with higher xylose content and lower degrees of branching exhibit enhanced reactivity, achieving higher carboxyl content and yields. Structural analyses confirmed successful modification, and light scattering analyses showed dramatic changes in the solution properties. Succinylation improves the solubility and film-forming properties of native xylans. This study shows key structure–reactivity relationships in xylan succinylation, establishing that low branching, high xylose content, and reduced lignin impurity enhance chemical functionalization. The results offer a framework for selecting optimal biomass feedstocks and support future efforts in genetic and synthetic biology to design plants with tunable xylan architectures. These findings advance the hemicellulose valorization for applications in coatings and packaging.

Acylation

Accelerating the Structure Exploration of Diverse Bi–Pt Nanoclusters via Physics‐Informed Machine Learning Potential and Particle Swarm Optimization

Bimetallic Bi–Pt nanoclusters exhibit diverse structural motifs, including core-shell, Janus, and mixed alloy configurations, due to the unique bonding characteristics between Bi and Pt atoms. Using density functional theory refinements from ChIMES physically machine-learned potential and CALYPSO particle swarm optimization global searches, 34 Bi20-Pt20 nanoclusters are systematically classified. The results reveal that Bi atoms predominantly occupy surface sites, driven by charge transfer effects. Cohesive energy trends alone prove insufficient for structure differentiation, necessitating a data-driven approach employing principal component analysis and K-means clustering. Furthermore, vibrational, electronic, and infrared spectral analyses provide additional insights into structure-property relationships. The findings offer an original framework for the automated classification and analysis of bimetallic nanoclusters, enhancing the understanding of their stability and functional properties.

bimetallic nanoparticles

MODE: A Web Application for Interactive Visualization and Exploration of Omics Data

Studies generating transcriptomics, proteomics, lipidomics, and metabolomics (colloquially referred to as “omics”) data allow researchers to find biomarkers or molecular targets, or understand complex biological structures and functions by identifying changes in biomolecule abundance and expression between experimental conditions. Omics data is multi-dimensional and oftentimes summarization techniques such as principal component analysis (PCA) are used to identify high-level patterns in data. Though useful, these summaries don’t allow exploration of detailed patterns in omics data that may have biological relevance. The use of interactive HTML displays with plots allows researchers to interact with omics data at a detailed level, but building these displays requires significant coding expertise. To overcome this barrier, the software MODE was built to empower users to build their own interactive HTML displays to support scientific discovery. These displays are easily shareable, do not depend on a specific operating system, and allow users to effortlessly sort and filter plots by categorical or numerical variables. MODE allows users to build and share these displays with several options for plot design and meta selection. In conclusion, the MODE web application and its capabilities are presented and then demonstrated on lipidomics data from a leaf wounding study.

lipidomics

Novel Cell-Type-Specific Drought-Responsive Proteins in Root Tips of Field-Grown Perennial Switchgrass

The root-tip region of plants, including the root cap, forms the most basal terminal of the root and exhibits a high degree of cellular complexity in terms of morphology, cytological function, and interaction with environmental cues in the soil. Cells in this region follow a developmental trajectory, transitioning from stem cells to meristematic cells, and ultimately to fully differentiated cell types. However, our understanding of root-tip cell-type specific proteomic responses to abiotic stresses, such as drought, particularly under field conditions, remains limited. This study aimed to identify spatially resolved, cell type-specific proteomes in switchgrass (Panicum virgatum) root tips under drought stress. Root tips were collected from seven-year-old, field-grown switchgrass ‘Alamo’ plants excavated under both well-watered and long-term drought conditions. Cell type-specific proteins were identified using laser capture microdissection (LCM) coupled with nanoPOTS (Nanodroplet Processing in One Pot for Trace Samples) and nano-LC-MS proteomics analysis. Five distinct cell types were targeted: (1) cells in the quiescent center and stem cell niche (QuC), (2) protodermal epidermal cells (PEC) in the meristematic zone, (3) epidermal cells in the transition and elongation zones above the root cap (Epi), (4) peripheral root cap cells (PRC), forming 2–3 layers below the PEC and 1–2 layers above the root border cells, and (5) columella root cap cells (Col) comprising of the columella initials and a single underlying layer of cells undergoing active growth. Principal component analysis (PCA) revealed clear separation among the five targeted cell types, confirming distinct proteomic profiles. Proteins predominantly enriched in each cell type were linked to distinct cellular functions, with QuC cells showing involvement in chromosomal behavior, DNA replication, and mitosis—key processes for stem cell niche regulation. Drought stress resulted in alterations of proteostasis, as evidenced by significant decreases in ribosomal proteins and increases in protein synthesis inhibitors. Moreover, drought stress induced unique cell-type–specific proteins involved in phytohormone biosynthesis and signaling pathways, including auxin, cytokinin, and jasmonic acid. In particular, QuC cells were more highly enriched in proteins associated with DNA repair and mitotic processes. Metabolic pathways related to amino acids, carbohydrates, and lipids were differentially affected in a cell-type–dependent manner, whereas general stress-responsive proteins exhibited consistent changes across all five cell types. Overall, this study provides unique spatially resolved, cell-type-specific proteomic profiles in root tips, representing a significant advancement in our understanding of the cellular mechanisms underlying plant responses to drought stress in natural field conditions.

perennial grass

The relationship between below average cognitive ability at age 5 years and the child’s experience of school at age 9

Background At age 5, while only embarking on their educational journey, substantial differences in children’s cognitive ability will already exist. The aim of this study was to examine the causal association between below average cognitive ability at age 5 years and child-reported experience of school and self-concept, and teacher-reported class engagement and emotional-behavioural function at age 9 years. Methods This longitudinal cohort study used data from 7,392 children in the Growing Up in Ireland Infant Cohort, who had completed the Picture Similarities and Naming Vocabulary subtests of the British Abilities Scales at age 5. Principal components analysis was used to produce a composite general cognitive ability score for each child. Children with a general cognitive ability score more than 1 standard deviation (SD) below the mean at age 5 were categorised as ‘Below Average Cognitive Ability’ (BACA), and those scoring above this as ‘Typical Cognitive Development’ (TCD). The outcomes of interest, measured at age 9, were child-reported experience of school, child’s self-concept, teacher-reported class engagement, and teacher-reported emotional behavioural function. Binary and multinomial logistic regression models were used to examine the association between BACA and these outcomes. Results Compared to those with TCD, those with BACA had significantly higher odds of never liking school [Adjusted odds ratio (AOR) 1.82, 95% CI 1.37–2.43, p < 0.001], of being picked on (AOR 1.27, 95% CI 1.09–1.48) and of picking on others (AOR 1.53, 95% CI 1.27–1.84). They had significantly higher odds of experiencing low self-concept (AOR 1.20, 95% CI 1.02–1.42) and emotional-behavioural difficulties (AOR 1.34, 95% CI 1.10–1.63, p = 0.003). Compared to those with TCD, children with BACA had significantly higher odds of hardly ever or never being interested, motivated and excited to learn (AOR 2.29, 95% CI 1.70–3.10). Conclusion Children with BACA at school-entry had significantly higher odds of reporting a negative school experience and low self-concept at age 9. They had significantly higher odds of having teacher-reported poor class engagement and problematic emotional-behavioural function at age 9. The findings of this study suggest BACA has a causal role in these adverse outcomes. Early childhood policy and intervention design should be cognisant of the important role of cognitive ability in school and childhood outcomes.

Bowe, Andrea K.

Design, Processing, and Properties of WTaCrV-Hf Multi-principal Element Alloys

Refractory multi-principal element alloys are candidates for high-temperature structural components due, in part, to their high strength and high melting points. Single-phase materials are initially preferred for isotropic material properties as a function of time and temperature in service conditions. This work outlines a computational rank-ordering and experimental validation methodology for single-phase body-centered-cubic phase stability in WTaCrV-Hf alloys using order–disorder transition temperature. Eight compositions were fabricated by arc-melting and heat-treated at 1400 °C for 24 hrs. X-ray diffraction, energy-dispersive x-ray spectroscopy, and Vickers hardness testing showed alloys with order–disorder transition temperatures below 600 °C formed a single-phase body-centered-cubic structure during solidification and remained single-phase after heat-treatment. The sample possessing the lowest order–disorder transition temperature exhibited slip traces suggestive of room-temperature plastic deformation under Vickers indentation, with both heat-treated single-phase samples exhibiting hardnesses over 800 HV with little cracking compared to tungsten. These results establish order–disorder transition temperature as a viable predictive parameter for multi-principal element alloy phase stability. The methodology outlined in this work provides a framework for future design, fabrication, and characterization of high-temperature structural multi-principal element alloys.

CALPHAD

Dimensional Reduction for Sampled Priors and Application to Photometric Redshift Distributions

A typical Bayesian inference on the values of some parameters of interest q from some data D involves running a Markov Chain (MC) to sample from the posterior $p$($q$,$n$|$D$) $\propto$ $\mathcal{L}$($D$|$q$,$n$)$p$(q)$p$($n$), where n are some nuisance parameters with a separable prior. In some cases, the nuisance parameters are high-dimensional, and their prior p(n) is itself defined only by a set of samples that have been drawn from some other MC. The MC for the posterior will typically require evaluation of p(n) at arbitrary values of n, i.e., one needs to provide a density estimator over the full n space from the provided samples. But the high dimensionality of n hinders both the density estimation and the efficiency of the MC for the posterior. We describe a solution to this problem: a linear compression of the n space into a much lower-dimensional space u, which projects away directions in n space that cannot appreciably alter $\mathcal{L}$. The algorithm for doing so is a slight modification to principal components analysis, and is less restrictive on p(n) than other proposed solutions to this issue. We demonstrate this “mode projection” technique using the analysis of 2-point correlation functions of weak lensing fields and galaxy density in the Dark Energy Survey, where n is a binned representation of the redshift distribution n(z) of the galaxies.

79 ASTRONOMY AND ASTROPHYSICS