Real Spectrum Research Articles

A high-throughput framework for lattice dynamics

AbstractWe develop an automated high-throughput workflow for calculating lattice dynamical properties from first principles including those dictated by anharmonicity. The pipeline automatically computes interatomic force constants (IFCs) up to 4th order from perturbed training supercells, and uses the IFCs to calculate lattice thermal conductivity, coefficient of thermal expansion, and vibrational free energy and entropy. It performs phonon renormalization for dynamically unstable compounds to obtain real effective phonon spectra at finite temperatures and calculates the associated free energy corrections. The methods and parameters are chosen to balance computational efficiency and result accuracy, assessed through convergence testing and comparisons with experimental measurements. Deployment of this workflow at a large scale would facilitate materials discovery efforts toward functionalities including thermoelectrics, contact materials, ferroelectrics, aerospace components, as well as general phase diagram construction.

Enhancing Prediction Stability and Performance in LIBS Analysis Using Custom CNN Architectures

LIBS-based analysis has experienced an ever-increasing interest in recent years as a well-suited technique for chemical analysis tasks relying on elemental fingerprinting. This method stands out for its ability to offer rapid, simultaneous multi-element analysis with the advantage of portability. In the context of this research, our aim is to bridge the gap between the analysis of simulated and real data to better account for variations in plasma temperature and electron density, which are typically not considered in LIBS analysis. To achieve this, we employ two distinct methodologies, PLS and CNNs, to construct predictive models for predicting the concentration of the 24 elements within each LIBS spectrum. The initial phase of our investigation concentrates on the training and testing of these models using simulated LIBS data, with results evaluated through RMSEP values. The IQR and median RMSEP values for all the elements demonstrate that CNNs consistently achieved values below 0.01, while PLS results ranged from 0.01 to 0.05, highlighting the superior stability and predictive accuracy of CNNs model. In the next phase, we applied the pre-trained models to analyze the real LIBS spectra, consistently identifying Aluminum (Al), Iron (Fe), and Silicon (Si) as having the highest predicted concentrations. The overall predicted values were approximately 0.5 for Al, 0.6 for Si, and 0.04 for Fe. In the third phase, deliberate adjustments are made to the training parameters and architecture of the proposed CNNs model to force the network to emphasize specific elements, prioritizing them over other components present in each real LIBS spectrum. The generation of the three modified versions of the initially proposed CNNs allows us to explore the impact of regularization, sample weighting, and a customized loss function on prediction outcomes. Some elements emerge during the prediction phase, with Calcium (Ca), Magnesium (Mg), Zinc (Zn), Titanium (Ti), and Gallium (Ga) exhibiting more pronounced patterns.

Consideration of spectral interference in total reflection X-ray fluorescence analysis using a limited number of calibration samples: Case study of ocean polymetallic nodules

A novel approach for quantitative analysis in total-reflection X-ray fluorescence (TXRF) which can be employed for the samples with complex matrices inducing overlapping spectral signals in the scenario of limited availability of standard samples is proposed. The approach is based on the least squares (LS) decomposition of real spectra into the weighted sum of simulated individual element subspectra and requires only a single sample of matrix-matched reference material. Oceanic polymetallic nodules and crusts, having complex mineral matrix and containing high concentrations of overlapping elements, were chosen to demonstrate the feasibility of the approach in quantification of 11 elements. The comparison of the results obtained with LS decomposition, internal standard and linear calibration with three standard samples was performed. It was demonstrated that the performance of the proposed approach and of the linear calibration was better than that of internal standard for most of the elements; however, the proposed method requires only a single standard sample.

Pursuing Truth: Improving Retrievals on Mid-infrared Exo-Earth Spectra with Physically Motivated Water Abundance Profiles and Cloud Models

Atmospheric retrievals are widely used to constrain exoplanet properties from observed spectra. We investigate how the common nonphysical retrieval assumptions of vertically constant molecule abundances and cloud-free atmospheres affect our characterization of an exo-Earth (an Earth-twin orbiting a Sun-like star). Specifically, we use a state-of-the-art retrieval framework to explore how assumptions for the H2O profile and clouds affect retrievals. In the first step, we validate different retrieval models on a low-noise simulated 1D mid-infrared (MIR) spectrum of Earth. Thereafter, we study how these assumptions affect the characterization of Earth with the Large Interferometer For Exoplanets (LIFE). We run retrievals on LIFE mock observations based on real disk-integrated MIR Earth spectra. The performance of different retrieval models is benchmarked against ground truths derived from remote sensing data. We show that assumptions for the H2O abundance and clouds directly affect our characterization. Overall, retrievals that use physically motivated models for the H2O profile and clouds perform better on the empirical Earth data. For observations of Earth with LIFE, they yield accurate estimates for the radius, pressure–temperature structure, and the abundances of CO2, H2O, and O3. Further, at R = 100, a reliable and bias-free detection of the biosignature CH4 becomes feasible. We conclude that the community must use a diverse range of models for temperate exoplanet atmospheres to build an understanding of how different retrieval assumptions can affect the interpretation of exoplanet spectra. This will enable the characterization of distant habitable worlds and the search for life with future space-based instruments.

Bulk-local dS3 holography: the matter with TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document} + Λ2

We propose an algorithm which builds a concrete dual for large-radius 3d de Sitter with a timelike York boundary for both gravity and bulk effective fields. This generalizes the solvable TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document} + Λ2 deformation, whose finite real spectrum accounts for the refined Gibbons-Hawking entropy as a microstate count while reproducing the radial static patch geometry. The required generalization to produce approximately local boundary conditions for bulk quantum fields requires a scheme for defining double-trace operators dual to deformed boundary conditions to realize the finite timelike boundary, valid at finite N. By starting with a small stint of a pure TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document} trajectory, the theory becomes finite, enabling well-defined subtractions to define the double-trace deformation so as to match the large-N prescription of Hartman, Kruthoff, Shaghoulian, and Tajdini to good approximation. We incorporate the matter effecting an uplift from negative to positive cosmological constant, and analyze the effect of matter on the energy spectrum of the theory arising from time-dependent bulk excitations. This validates the cosmic horizon dS3 microstate count for large-radius dS3 holography, embedding TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document} + Λ2 concretely into a larger theory consistent with bulk locality for matter fields. We comment briefly on potential upgrades to four dimensions and other future directions.

Direct Identification of the Continuous Relaxation Time and Frequency Spectra of Viscoelastic Materials

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined.

Hilbert 17th property and central cones

This paper is dedicated to the study of the converse implication in Hilbert 17th problem for a general commutative ring. We introduce the notions of central and precentral prime cones which generalize the notion of central real points of irreducible real algebraic varieties. We study these families of prime cones which both live in the real spectrum of the ring and allow to state new Positivstellensätze and to obtain an equivalence in Hilbert 17th problem.

On the real spectrum of differential operators with PT‐symmetric periodic matrix coefficients

AbstractWe study the spectrum of the operator generated by the differential expression of order with the Parity‐Time (PT)‐symmetric periodic matrix coefficients. The case when and are the odd numbers was investigated in [18]. In this paper, we consider the all remained cases: (a) is an odd number and is an even number, (b) is an even number and is an arbitrary positive integer. We find conditions on the coefficients under which in the cases (a) and (b) the spectrum of contains the sets and respectively for some .

Rapid elimination of scattering in three-dimensional fluorescence spectra via deep learning

Three-dimensional fluorescence spectra are often affected by scattering effects, traditional scattering elimination methods rely excessively on parameter settings and cannot automatically eliminate scattering in batches, thereby limiting the application of fluorescence spectroscopy technology in rapid online monitoring and analysis of samples. In this study, we have developed a model based on a deep learning CycleGAN to rapidly eliminate scattering from three-dimensional fluorescence spectra. The proposed model efficiently eliminates scattering by simply inputting single or batches of contaminated fluorescent spectra. By training the CycleGAN using a large dataset of simulated three-dimensional fluorescence spectra and employing data augmentation, to the model can transform fluorescence spectra with scattering into ones without scattering. To validate the effectiveness of the proposed methed, we confirmed its generalization and reliability by eliminating scattering from two sets of previously unseen real experimental three-dimensional fluorescence spectra. We evaluated the effectiveness of scattering elimination across various noise levels and scattering widths, using metrics such as the mean absolute error, peak signal-to-noise ratio, structural similarity and cosine similarity. Furthermore, we conducted a component analysis using PARAFAC on the spectra post-scattering elimination, yielding correlation coefficients of >0.97 when compared to that in case of actual components. Finally, we compared the proposed model with traditional mathematical methods, such as blank subtraction and Delaunay triangulation. Results showed that the proposed model can automatically and efficiently eliminate scattering from fluorescence spectra in batches, substantially improving the efficiency of scattering elimination.

Restoring Adiabatic State Transfer in Time-Modulated Non-Hermitian Systems.

Non-Hermitian systems have attracted much interest in recent decades, driven partly by the existence of exotic spectral singularities, known as exceptional points (EPs), where the dimensionality of the system evolution operator is reduced. Among various intriguing applications, the discovery of EPs has suggested the potential for implementing a symmetric mode switch, when encircling them in a system parameter space. However, subsequent theoretical and experimental works have revealed that dynamical encirclement of EPs invariably results in asymmetric mode conversion; namely, the mode switching depends only on the winding direction but not on the initial state. This chirality arises from the failure of adiabaticity due to the complex spectrum of non-Hermitian systems. Although the chirality revealed has undoubtedly made a significant impact in the field, a realization of the originally sought symmetric adiabatic passage in non-Hermitian systems with EPs has since been elusive. In this work, we bridge this gap and theoretically demonstrate that adiabaticity, and therefore a symmetric state transfer, is achievable when dynamically winding around an EP. This becomes feasible by specifically choosing a trajectory in the system parameter space along which the corresponding evolution operator attains a real spectrum. Our findings, thus, offer a promise for advancing various wave manipulation protocols in both quantum and classical domains.

Uniqueness of real ring spectra up to higher homotopy

Uniqueness of real ring spectra up to higher homotopy

Robust baseline correction for Raman spectra by constrained Gaussian radial basis function fitting

Accurate baseline correction is a fundamental requirement for extracting meaningful spectral information and enabling precise quantitative analysis using Raman spectroscopy. Although numerous baseline correction techniques have been developed, they often require meticulous parameter adjustments and yield inconsistent results. To address these challenges, we have introduced a novel approach, namely constrained Gaussian radial basis function fitting (CGF). Our method involves solving a curve-fitting problem using Gaussian radial basis functions under specific constraints. To ensure stability and efficiency, we developed a linear programming algorithm for the proposed approach. We evaluated the performance of CGF using simulated Raman spectra and demonstrated its robustness across various scenarios, including changes in data length and noise levels. In contrast to standard methods, which frequently require complicated parameter adjustments and may exhibit varying errors, our approach provides a simple parameter search and consistently achieves low errors. We further assessed CGF using real Raman spectra, leading to enhanced accuracy in the quantitative analysis of the Raman spectra of chemical warfare agents. Our results emphasize the potential of CGF as a valuable tool for Raman spectroscopy data analysis, significantly advancing sophisticated analytical techniques.

Supersymmetric spectrum for vector multiplet on Euclidean AdS2

Quantum study of supersymmetric theories on Euclidean two dimensional anti-de Sitter space (EAdS2) requires complexified spectrum. For a chiral multiplet, we showed that the spectrum of the Dirac operator acquires a universal shift of i/2 from the real spectrum to make the supersymmetry between boson and fermion manifest, where both the bosonic and fermionic eigenfunctions are normalizable using an appropriate definition of Euclidean inner product. We extend this analysis to the vector multiplet, where we show that the gaugino requires both +i/2 and i/2 shift from the real spectrum, and there is additional isolated point at vanishing spectral parameter which is mapped by supersymmetry to the boundary zero modes of the vector field. Furthermore, this spectral analysis shows that not every bosonic fields in the vector multiplet can satisfy normalizable boundary condition. Nevertheless, aided by a reorganization of fields into a cohomological form, we find the supersymmetry mapping between bosons and fermions in terms of the expansion coefficients with respect to the newly constructed basis.

A Fast Matrix Solver for Linear Boundary-Value Problem in the PEM Fuel Cell Impedance Model

A core physics-based model for PEM fuel cell impedance reduces to the numerical solution of a boundary value problem for two coupled, linear, complex–valued ordinary differential Eqs. with variable coefficients. We report a matrix formulation of a previously developed recurrent method for the fast numerical solution of this problem. This formulation does not require iterations and it utilizes only elementary functions and 4 × 4 matrix products, resulting in one to two orders of magnitude faster code compared to the standard Python boundary–value problem solver. This significantly improves the computational speed of the physics-based impedance model, making it competitive with the popular equivalent circuit modeling approach. The numerical calculation of cell impedance is demonstrated based on the model that includes proton transport in the catalyst layer and oxygen transport in the catalyst and gas-diffusion layers. Least–squares fitting of a real experimental spectrum shows that the matrix solver returns the cell transport parameters 50 times faster than the classic boundary–value problem solver.

Robust Recovery of Optimally Smoothed Polymer Relaxation Spectrum from Stress Relaxation Test Measurements.

The relaxation spectrum is a fundamental viscoelastic characteristic from which other material functions used to describe the rheological properties of polymers can be determined. The spectrum is recovered from relaxation stress or oscillatory shear data. Since the problem of the relaxation spectrum identification is ill-posed, in the known methods, different mechanisms are built-in to obtain a smooth enough and noise-robust relaxation spectrum model. The regularization of the original problem and/or limit of the set of admissible solutions are the most commonly used remedies. Here, the problem of determining an optimally smoothed continuous relaxation time spectrum is directly stated and solved for the first time, assuming that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. The relaxation time spectrum model that reproduces the relaxation modulus measurements and is the best smoothed in the class of continuous square-integrable functions is sought. Based on the Hilbert projection theorem, the best-smoothed relaxation spectrum model is found to be described by a finite sum of specific exponential-hyperbolic basis functions. For noise-corrupted measurements, a quadratic with respect to the Lagrange multipliers term is introduced into the Lagrangian functional of the model's best smoothing problem. As a result, a small model error of the relaxation modulus model is obtained, which increases the model's robustness. The necessary and sufficient optimality conditions are derived whose unique solution yields a direct analytical formula of the best-smoothed relaxation spectrum model. The related model of the relaxation modulus is given. A computational identification algorithm using the singular value decomposition is presented, which can be easily implemented in any computing environment. The approximation error, model smoothness, noise robustness, and identifiability of the polymer real spectrum are studied analytically. It is demonstrated by numerical studies that the algorithm proposed can be successfully applied for the identification of one- and two-mode Gaussian-like relaxation spectra. The applicability of this approach to determining the Baumgaertel, Schausberger, and Winter spectrum is also examined, and it is shown that it is well approximated for higher frequencies and, in particular, in the neighborhood of the local maximum. However, the comparison of the asymptotic properties of the best-smoothed spectrum model and the BSW model a priori excludes a good approximation of the spectrum in the close neighborhood of zero-relaxation time.

Improvements in precision and accuracy of complex- relative to real-domain linear combination model spectral fitting not necessarily recovered by zero filling.

Although the information obtained from in vivo proton magnetic resonance spectroscopy (1H MRS) presents a complex-valued spectrum, spectral quantification generally employs linear combination model (LCM) fitting using the real spectrum alone. There is currently no known investigation comparing fit results obtained from LCM fitting over the full complex data versus the real data and how these results might be affected by common spectral preprocessing procedure zero filling. Here, we employ linear combination modeling of simulated and measured spectral data to examine two major ideas: first, whether use of the full complex rather than real-only data can provide improvements in quantification by linear combination modeling and, second, to what extent zero filling might influence these improvements. We examine these questions by evaluating the errors of linear combination model fits in the complex versus real domains against three classes of synthetic data: simulated Lorentzian singlets, simulated metabolite spectra excluding the baseline, and simulated metabolite spectra including measured in vivo baselines. We observed that complex fitting provides consistent improvements in fit accuracy and precision across all three data types. While zero filling obviates the accuracy and precision benefit of complex fitting for Lorentzian singlets and metabolite spectra lacking baselines, it does not necessarily do so for complex spectra including measured in vivo baselines. Overall, performing linear combination modeling in the complex domain can improve metabolite quantification accuracy relative to real fits alone. While this benefit can be similarly achieved via zero filling for some spectra with flat baselines, this is not invariably the case for all baseline types exhibited by measured in vivo data.

Universal Quantum Fisher Information and Simultaneous Occurrence of Landau‐Class and Topological‐Class Transitions in Non‐Hermitian Jaynes‐Cummings Models

AbstractLight‐matter interactions provide an ideal testground for interplay of critical phenomena, topological transitions, quantum metrology, and non‐Hermitian physics with high controllability and tunability. The present work considers two fundamental non‐Hermitian Jaynes‐Cummings models in light‐matter interactions that possess real energy spectra in parity‐time (PT) symmetry and anti‐PT symmetry. The quantum Fisher information is shown to be critical around the transitions at the exceptional points and exhibit a super universality, with respect to different parameters, all energy levels, both models, symmetric phases, and symmetry‐broken phases, which guarantees a universally high measurement precision in quantum metrology. In particular, the transitions are found to be both symmetry‐breaking Landau‐class transitions (LCTs) and symmetry‐protected topological‐class transitions (TCTs), thus realizing a simultaneous occurrence of critical LCTs and TCTs that are conventionally incompatible due to contrary symmetry requirements. Besides establishing a paradigmatic case to break the incompatibility of the LCTs and the TCTs in non‐Hermitian systems, the both availabilities of the sensitive critical feature and the robust topological feature can also provide more potential for designing quantum devices or sensors.

Simulated LC-MS Data Set for Assessing the Metabolomics Data Processing Pipeline Implemented into MVAPACK.

Metabolomics commonly relies on using one-dimensional (1D) 1H NMR spectroscopy or liquid chromatography-mass spectrometry (LC-MS) to derive scientific insights from large collections of biological samples. NMR and MS approaches to metabolomics require, among other issues, a data processing pipeline. Quantitative assessment of the performance of these software platforms is challenged by a lack of standardized data sets with "known" outcomes. To resolve this issue, we created a novel simulated LC-MS data set with known peak locations and intensities, defined metabolite differences between groups (i.e., fold change > 2, coefficient of variation ≤ 25%), and different amounts of added Gaussian noise (0, 5, or 10%) and missing features (0, 10, or 20%). This data set was developed to improve benchmarking of existing LC-MS metabolomics software and to validate the updated version of our MVAPACK software, which added gas chromatography-MS and LC-MS functionality to its existing 1D and two-dimensional NMR data processing capabilities. We also included two experimental LC-MS data sets acquired from a standard mixture andMycobacterium smegmatiscell lysates since a simulated data set alone may not capture all the unique characteristics and variability of real spectra needed to assess software performance properly. Our simulated and experimental LC-MS data sets were processed with the MS-DIAL and XCMSOnline software packages and our MVAPACK toolkit to showcase the utility of our data sets to benchmark MVAPACK against community standards. Our results demonstrate the enhanced objectivity and clarity of software assessment that can be achieved when both simulated and experimental data are employed since distinctly different software performances were observed with the simulated and experimental LC-MS data sets. We also demonstrate that the performance of MVAPACK is equivalent to or exceeds existing LC-MS software programs while providing a single platform for processing and analyzing both NMR and MS data sets.

Rings of formal power series and symmetric real semigroups

Sections 1–6 of this paper deal with the rings ▪ of (generalized) power series over a formally real field F of coefficients and (non-negative) exponents in an arbitrary totally ordered abelian group G. We determine explicitly the nature of the elements constituting the real spectra of these rings, its structure under the partial order of specialization, and the structure of their associated real semigroups (RS). Theorem A is a corollary of these results. Starting with section 7 we introduce a new class of symmetric real semigroups by requiring that their spaces of characters satisfy analogs to some of the basic laws holding in the real spectra of the rings ▪. These spaces have, amongst others, remarkable symmetry properties that justify their name (see 7.16 and 7.17). In §8 we prove that every finite symmetric RS is isomorphic to the RS associated to some ring of formal power series, but that such representation fails, in general, for symmetric RSs of suitably large infinite cardinality. Our study leads to showing, in §10, that the inclusion map G⁎∪{0}↪G of the reduced special group G⁎ of invertible elements of a symmetric RS G has a retract. As a consequence, every symmetric RS, G, is isomorphic to the extension of G⁎ by a 3-semigroup Δ defined in terms of that retract. Extensions of this form are introduced and studied in §9. Finally, in §11 we present a first order axiomatisation of the class of symmetric RSs in the natural language for real semigroups.

Detecting relativistic Doppler in galaxy clustering with tailored galaxy samples

We present a method to obtain a high-significance detection of relativistic effects on cosmological scales. Measurements of such effects would be instrumental for our understanding of the Universe, as they would provide a further confirmation of the validity of general relativity as the correct description of the gravitational interaction, in a regime very far from that of strong gravity, where it has been tested to exquisite accuracy. Despite its relevance, the detection of relativistic effects has hitherto eluded us, mainly because they are stronger on the largest cosmic scales, plagued by cosmic variance. Our work focuses on the cosmological probe of galaxy clustering, describing the excess probability of finding pairs of galaxies at a given separation due to them being part of the same underlying cosmic large-scale structure. We focus on the two-point correlation function of the distribution of galaxies in Fourier space — the power spectrum — where relativistic effects appear as an imaginary contribution to the real power spectrum. By carefully tailoring cuts in magnitude/luminosity, we are able to obtain two samples (bright and faint) of the same galaxy population, whose cross-correlation power spectrum allows for a detection of the relativistic contribution. In particular, we optimise the definition of the samples to maximise the detection significance of the relativistic Doppler term for both a low-z Bright Galaxy Sample and a high-z Hα emission line galaxy population.