Linear Relation Research Articles

Acoustically Enhanced Triboelectric Stethoscope for Ultrasensitive Cardiac Sounds Sensing and Disease Diagnosis.

Electronic stethoscope used to detect cardiac sounds that contains essential clinical information is a primary tool for diagnosis of various cardiac disorders. However, the linear electro-mechanical constitutive relation makes conventional piezoelectric sensors rather ineffective to detect low-intensity, low-frequency heart acoustic signal without the assistance of complex filtering and amplification circuits. Herein, we find that triboelectric sensor features superior advantages over piezoelectric one for micro-quantity sensing originated from the fast saturated constitutive characteristic. As a result, the triboelectric sensor shows ultrahigh sensitivity (1215mV/Pa) than the piezoelectric counterpart (21mV/Pa) in the sound pressure range of 50 - 80dB under the same testing condition. By designing a trumpet-shaped auscultatory cavity with a power function cross-section to achieve acoustic energy converging and impedance matching, triboelectric stethoscope delivers 36dB signal-to-noise ratio for human test (2.3 times of that for piezoelectric one). Further combining with machine learning, five cardiac states can be diagnosed at 97% accuracy. In general, the triboelectric sensor is distinctly unique in basic mechanism, provides a novel design concept for sensing micromechanical quantities, and presents significant potential for application in cardiac sounds sensing and disease diagnosis. This article is protected by copyright. All rights reserved.

Deflection of Light by a Reissner-Nordström Black Hole and Painlevé VI equation

Abstract We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner-Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard-Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

Determination of stearic acid and 12-hydroxyoctadecanoic acid in PEG-60 hydrogenated castor oil by HPLC-ELSD after alkaline hydrolysis

AbstractA rapid and simple method for the determination of stearic acid and 12-hydroxystearic acid in PEG-60 hydrogenated castor oil by high performance liquid chromatography with evaporative light scattering detection was established. The oil sample was first pretreated by alkaline hydrolysis. The analysis was performed on a Zhongpu Develop XD-C18 column (250 mm × 4.6 mm, 5 µm) with gradient elution of methanol and 1% acetic acid aqueous solution at a flow rate of 1.2 mL·min−1 and a column temperature of 40 °C. The drift tube temperature of the evaporative light scattering detection system was set at 40 °C, and the pressure of carrier gas (N2) was 337 kPa. The regression equation revealed a good linear relation (r = 0.9993–0.9995) during the test ranges (119.1–1190.7 μg·mL−1 for 12-hydroxystearic acid, 10.7–107.4 μg·mL−1 for stearic acid). The detection limits of 12-hydroxystearic acid and stearic acid were 1.1 and 2.5 μg·mL−1, the limits of quantitation were 3.2 and 7.4 μg·mL−1, respectively. And the mean recoveries were 101.5 and 101.0%, the corresponding relative standard deviations (RSDs) were 2.1 and 2.8%, respectively. The RSDs corresponding to repeatability (n = 6) were both less than 1.7% in terms of precision. As to the stability, the test results remained stable after 8 h at room temperature (RSDs were both less than 2.6%). The developed method showed high sensitivity, recovery, repeatability and stability, which indicated that the method could be applied as a quality evaluation method for the determination of stearic acid and 12-hydroxyoctadecanoic acid in PEG-60 hydrogenated castor oil.

A Zn/Cd/Ni Metal-Organic Framework-Based Solid and Flexible Fluoroprobes for Highly Selective Detection of 4-Nitroaniline in Real Water Samples.

The combination of diazabicyclic compound i.e. 1,4-diazabicyclo[2.2.2]octane (DABCO) and dicarboxylate linker 1,4-dicarboxylic acid are employed to self-assemble with three divalent metal ions such as Zn2+, Cd2+, Ni2+ led to the formation of highly stable pillar layered luminescent metal-organic frameworks. The emissive response of as-synthesized MOFs was studied in organic solvents with different functionalities resulting in the development of fluoroprobes having excellent recognition ability for 4-nitroaniline with detection limit upto micromolar. All the MOFs display a good linear relation between the fluorescence intensity and concentration of 4-NA in the range of 10-90µmol L-1. The values of Ksv for Zn-MOF, Cd-MOF, and Ni-MOF are found to be 1.75 × 104mol-1L, 1.25 × 104mol-1L, and 28.0 × 104mol-1L, respectively. The calculated detection limit values are 19.7µmol L-1, 27.6µmol L-1, and 1.19µmol L-1 respectively. Moreover, the study was further extended to fabricate the solid membrane-based fluoroprobe using a linear chitosan polysaccharide and act as an efficient solid fluoroprobe for the detection of 4-NA. This proposed synthesis of polymeric membrane facilitates the MOF@chitosan fluorophore to transfer its fluorescence-emissive nature to a solid state.

Identification of Novel Regulators of Leaf Senescence Using a Deep Learning Model.

Deep learning has emerged as a powerful tool for investigating intricate biological processes in plants by harnessing the potential of large-scale data. Gene regulation is a complex process that transcription factors (TFs), cooperating with their target genes, participate in through various aspects of biological processes. Despite its significance, the study of gene regulation has primarily focused on a limited number of notable instances, leaving numerous aspects and interactions yet to be explored comprehensively. Here, we developed DEGRN (Deep learning on Expression for Gene Regulatory Network), an innovative deep learning model designed to decipher gene interactions by leveraging high-dimensional expression data obtained from bulk RNA-Seq and scRNA-Seq data in the model plant Arabidopsis. DEGRN exhibited a compared level of predictive power when applied to various datasets. Through the utilization of DEGRN, we successfully identified an extensive set of 3,053,363 high-quality interactions, encompassing 1430 TFs and 13,739 non-TF genes. Notably, DEGRN's predictive capabilities allowed us to uncover novel regulators involved in a range of complex biological processes, including development, metabolism, and stress responses. Using leaf senescence as an example, we revealed a complex network underpinning this process composed of diverse TF families, including bHLH, ERF, and MYB. We also identified a novel TF, named MAF5, whose expression showed a strong linear regression relation during the progression of senescence. The mutant maf5 showed early leaf decay compared to the wild type, indicating a potential role in the regulation of leaf senescence. This hypothesis was further supported by the expression patterns observed across four stages of leaf development, as well as transcriptomics analysis. Overall, the comprehensive coverage provided by DEGRN expands our understanding of gene regulatory networks and paves the way for further investigations into their functional implications.

Uniqueness of the Eigenvalue Inverse Problem for a Class of X-type Matrices with Linear Relations

Constraining a matrix by adding additional conditions to determine the matrix if given eigenpairs is called the eigenvalue inverse problem for matrices. The eigenvalue inverse problem of a matrix can be based on a given combination of different eigenvectors and real numbers to inverse a matrix method, and the method of inversion varies for different types of matrices. In this paper, we investigate the eigenvalue inverse problem for a class of X-type matrices characterized by linear relations, Invert the matrix based on its characteristics, determine eigenvalues and eigenvectors, and finally, prove that the solution of the problem exists and is unique, derive a series of expressions and recursive formulas, and we also check the algorithm’s accuracy correctness by giving different instances.

Unveiling very young O stars

Context. O-type stars are known to significantly contribute to both the dynamics and evolution of galaxies. Massive and luminous, they probably control and regulate the galaxies star formation rates. The stellar feedback generated by such cosmic beasts can strongly affect the local star formation rate, with effects in the current (and future) generations of low and intermediate mass stars, and possibly also in the disruption process of the galaxies’ giant gas reservoirs. Aims. For this work I performed a redetermination of the spectral types and effective temperatures of the Galactic O-type stars MSP182, MSP183, MSP199, VPHAS-01338, and VPHAS-01273. Methods. From a careful examination of the spectral features present in the blue optical spectral region, it was possible to identify several nitrogen lines usually only seen in the blue optical spectra of O2–O3 stars. From the nitrogen ionic equivalent width ratios measured in the spectra of MSP182, MSP183, MSP199, VPHAS-01338, and VPHAS-01273, and in those of standard stars of the O2–O4 spectral types, earlier spectral types and hotter effective temperature values were derived. Results. Two O2V((f*))z, together with three new O3 V stars are now firmly identified in the Westerlund 2 region. Besides RFS1 in NGC 3603, the O2 V stars found in Westerlund 2 are the only other exemplars known to date in the Milky Way. From the nitrogen equivalent width line ratios measured in the spectra of standard stars of the O2–O4 spectral types, linear relations between the N IVλ4058/N IIIλ4640 ratio and the effective temperature in the 47 000–51 000 K range were derived. Based on my spectroscopic analysis of the science targets and the use of a HRD, a mean heliocentric distance of 5 kpc to Westerlund 2 was computed, a result that is in line with the mean heliocentric distance of 5.3 ± 1.5 kpc obtained from the associated Gaia DR3 parallaxes and distances. Conclusions. The Westerlund 2 massive stars studied in this work probably share a common evolutionary process that might be representative of the evolutionary ages of a large fraction of the cluster’s O-type stellar population, which seems to be much younger than 1 Myr.

Fluid simulations of resistive drift-wave turbulence with diamagnetic flow in ZPED experiments

We derive a diamagnetic resistive fluid model (DRF) and develop an associated two-dimensional fluid simulation code (DRF-2D) to explore the dynamics of resistive drift modes within the plasmas of the Zheda Plasma Experiment Device (ZPED). The validation of the linear dispersion relation for the DRF-2D code revealed a harmonious agreement between analytical theory and linear numerical simulations. Leveraging plasma parameters obtained from the ZPED experiments, we conducted a comprehensive series of nonlinear simulations using the DRF-2D code. Our simulations successfully replicate the nonlinear trends in turbulent fluctuations and transport observed in the ZPED experiments, particularly demonstrating a remarkably accurate alignment of the turning point in the magnetic field. Notably, the DRF model sheds light on the observed frequency sign reversal from the electron diamagnetic direction to the ion diamagnetic direction in the ZPED experiments. This is demonstrated through well-matched turning points in the confining magnetic field between the nonlinear simulations and ZPED experiments. The fidelity of our model in capturing these phenomena underscores its efficacy in providing valuable insights and predictive capabilities for the intricate dynamics observed in the ZPED plasmas.

Knockoffs-SPR: Clean Sample Selection in Learning With Noisy Labels.

A noisy training set usually leads to the degradation of the generalization and robustness of neural networks. In this article, we propose a novel theoretically guaranteed clean sample selection framework for learning with noisy labels. Specifically, we first present a Scalable Penalized Regression (SPR) method, to model the linear relation between network features and one-hot labels. In SPR, the clean data are identified by the zero mean-shift parameters solved in the regression model. We theoretically show that SPR can recover clean data under some conditions. Under general scenarios, the conditions may be no longer satisfied; and some noisy data are falsely selected as clean data. To solve this problem, we propose a data-adaptive method for Scalable Penalized Regression with Knockoff filters (Knockoffs-SPR), which is provable to control the False-Selection-Rate (FSR) in the selected clean data. To improve the efficiency, we further present a split algorithm that divides the whole training set into small pieces that can be solved in parallel to make the framework scalable to large datasets. While Knockoffs-SPR can be regarded as a sample selection module for a standard supervised training pipeline, we further combine it with a semi-supervised algorithm to exploit the support of noisy data as unlabeled data. Experimental results on several benchmark datasets and real-world noisy datasets show the effectiveness of our framework and validate the theoretical results of Knockoffs-SPR.

A Nonlinear Relation between Body Mass Index and Long-Term Poststroke Functional Outcome-The Importance of Insulin Resistance, Inflammation, and Insulin-like Growth Factor-Binding Protein-1.

Both high serum insulin-like growth factor-binding protein-1 (s-IGFBP-1) and insulin resistance (IR) are associated with poor functional outcome poststroke, whereas overweight body mass index (BMI; 25-30) is related to fewer deaths and favorable functional outcome in a phenomenon labeled "the obesity paradox". Furthermore, IGFBP-1 is inversely related to BMI, in contrast to the linear relation between IR and BMI. Here, we investigated s-IGFBP-1 and IR concerning BMI and 7-year poststroke functional outcome. We included 451 stroke patients from the Sahlgrenska Study on Ischemic Stroke (SAHLSIS) with baseline measurements of s-IGFBP1, homeostasis model assessment of IR (HOMA-IR), BMI (categories: normal-weight (8.5-25), overweight (25-30), and obesity (>30)), and high-sensitivity C-reactive protein (hs-CRP) as a measure of general inflammation. Associations with poor functional outcome (modified Rankin scale [mRS] score: 3-6) after 7 years were evaluated using multivariable binary logistic regression, with overweight as reference due to the nonlinear relationship. Both normal-weight (odds-ratio [OR] 2.32, 95% confidence interval [CI] 1.30-4.14) and obese (OR 2.25, 95% CI 1.08-4.71) patients had an increased risk of poor functional outcome, driven by deaths only in the normal-weight. In normal-weight, s-IGFBP-1 modestly attenuated (8.3%) this association. In the obese, the association was instead attenuated by HOMA-IR (22.4%) and hs-CRP (10.4%). Thus, a nonlinear relation between BMI and poor 7-year functional outcome was differently attenuated in the normal-weight and the obese.

Elasticity of the Cervix in Relation to Uterus Position.

Strain elastography allows the evaluation of tissue elasticity. Background/Objectives: Tissue elasticity depends on the content and distribution of collagen fibers and is shaped by the applied tensile forces that may differ in uteri with a different angle of flexion of the corpus on the cervix. The objective was to investigate whether the angle of uterine flexion is related to cervical tissue elasticity. Methods: The anterior angle between the longitudinal axis of the uterus corpus and that of the cervix was measured in 275 non-pregnant young women by transvaginal ultrasonography and considered both as an absolute value or categorized as ≤150°, between >150° and ≤210°, and >210°. Strain elastography was used to assess tissue elasticity by placing the probe in the anterior vaginal fornix. Tissue elasticity was evaluated in the middle of the anterior cervical compartment (ACC), in the middle of the posterior cervical compartment (PCC), in the middle portion of the cervical canal (MCC), and at the internal cervical os (ICO). In a sagittal plane MCC was evaluated across the cervical canal, and ACC and PCC at a distance equal between the cervical canal and the outer anterior or posterior part of the cervix. MCC, ACC and PCC were evaluated at equal distance between the ICO and the external cervical os. Elasticity was expressed as a color score ranging from 0.1 (low elasticity) to 3 (high elasticity). Results: The angle of uterine flexion show a negative linear relation with the elasticity of the ACC (p = 0.001) and MCC (p = 0.002) and a positive relation with the elasticity of the PCC (p = 0.054). In comparison to uteri with an angle of flexion of <150°, those with an angle of flexion of >210° had lower elasticity of the ACC (p = 0.001) and MCC (p = 0.001) and higher elasticity of the PCC (p = 0.004). The ACC/PCC and PCC/MCC elasticity ratios were also significantly different (p = 0.001). Conclusions: The angle of uterine flexion is associated with changes in cervix elasticity. Retroflexion is associated with stiffer ACC and MCC and a more elastic PCC. Differences in tissue elasticity suggest structural changes of the cervix that may have implication in variate obstetric and gynecological conditions.

Multiple Beads on a String: Dark-matter-deficient Galaxy Formation in a Mini-Bullet Satellite–Satellite Galaxy Collision

Dark-matter-deficient galaxies (DMDGs) discovered in the survey of ultra-diffuse galaxies (UDGs), in apparent conflict with standard cold dark matter, may be produced by high-velocity galaxy–galaxy collisions, the so-called Mini-Bullet scenario. Recent observations of an aligned trail of 7–11 UDGs near NGC 1052, including DMDGs DF2 and DF4, suggesting a common formation event, ∼8.9 ± 1.5 Gyr ago, provide a test.Hydro/N-body simulations, supplemented by galaxy orbit integrations, demonstrate that satellite–satellite collisions outside the host-galaxy virial radius can reproduce the observed UDGs in the NGC 1052 group. A trail of ∼10 DMDGs is shown to form, including 2 massive ones that replicate the observed motions of DF2 and DF4. The linear relation, v = A x + v 0, conjectured previously to relate positions ( x ) and velocities ( v ) of the aligned DMDGs as a signature of the collision event, is approximately obeyed, but individual DMDGs can deviate significantly from it. The progenitors whose collision spawned the trail of DMDGs survive the collision without themselves becoming DMDGs. We predict that one progenitor is located at the end of the trail, which can be tested by observing the difference between its stars, formed pre-collision, from those of the DMDGs, formed post-collision. By contrast, stellar ages and metallicities of the DMDGs are nearly identical. We further offer a hint that the tidal field of host NGC 1052 may contribute to making DMDGs diffuse. ΛCDM simulation in a 100 cMpc box finds our required initial conditions ∼10 times at z < 3. These results indicate current observations are consistent with the Mini-Bullet scenario.

Boussinesq-type equations of hydroelastic waves in shallow water

Accurate computation of hydroelastic waves in shallow water is critical because many hydroelastic wave applications are nearshores, such as sea-ice and floating infrastructures. In this paper, Boussinesq assumptions for shallow water are employed to derive nonlinear Boussinesq-type equations of hydroelastic waves, in which non-uniform distribution of structural stiffness and varying water depth are considered rigorously. Application of Boussinesq assumptions enables complicated three-dimensional problems to be reduced and formulated on the two-dimensional horizontal plane, therefore the proposed Boussinesq-type models are straightforward and versatile for a wide range of hydroelastic wave applications. Two configurations, a floating plate and a submerged plate, are studied. The first-order linear governing equations are solved analytically with periodic conditions assuming constant depth and uniform stiffness, and the linear dispersion relations are subsequently derived for both configurations. For flexural-gravity waves of a floating plate, unique behaviours of flexural-gravity waves different from shallow-water waves are discussed, and a generalized solitary wave solution is investigated. A nonlinear numerical solver is developed, and nonlinear flexural-gravity waves are found to have smaller wavelength and celerity than their linear counterparts. For hydroelastic waves of a submerged plate, dual-mode analytical solutions are discovered for the first time. Numerical computation has demonstrated that a plate with decreasing submerged depth is able to transfer wave energy from the deeper water to the surface layer.

Experimental studies on the interfacial separation and stiffness of rough elastic-plastic solids

The direct measurement of interfacial separation between rough solids is hitherto one of the most challenging tasks in contact mechanics. This paper develops an optical interferometric technique for in-situ measuring interfacial separation with nanoscale precision. For the contact of metallic samples, the evolutions of interfacial separation and stiffness with load are measured. In the loading process, the dependence of interfacial stiffness on load can be described by a power law with the exponent smaller than unity at low loads and larger than unity at large loads. Similar relations have been found for unloading but interrupted with a linear relation at intermediate loads. These characteristics are significantly different from those known for rough elastic solids, and provide new insights in tribology.

Tuning Electrocatalytic Activities of Dealloyed Nanoporous Catalysts by Macroscopic Strain Engineering.

It is technically challenging to quantitatively apply strains to tune catalysis because most heterogeneous catalysts are nanoparticles, and lattice strains can only be applied indirectly via core-shell structures or crystal defects. Herein, we report quantitative relations between macroscopic strains and hydrogen evolution reaction (HER) activities of dealloyed nanoporous gold (NPG) by directly applying macroscopic strains upon bulk NPG. It was found that macroscopic compressive strains lead to a decrease, while macroscopic tensile strains improve the HER activity of NPG, which is in line with the d-band center model. The overpotential and onset potential of HER display approximately a linear relation with applied macroscopic strains, revealing an ∼2.9 meV decrease of the binding energy per 0.1% lattice strains from compressive to tensile. The methodology with the high strain sensitivity of electrocatalysis, developed in this study, paves a new way to investigate the insights of strain-dependent electrocatalysis with high precision.

Deciphering air quality index through sample entropy: A nonlinear time series analysis

Pollution and its impacts on human health have become a crisis in regions with poor air quality index (AQI), which is an indicator of concentrations of pollutants, prompting the United Nations (UN) to set sustainable development goals (SDG). The present study proposes a surrogate sample entropy-based method in tune with UN’s SDG, to assess AQI from the time series of any of the pollutants. New Delhi is one of the world's most polluted state capital, with a higher level of particulate matter (PM). The temporal data of the pollutants in New Delhi, recorded in the one-hour interval during the years 2016 and 2017, are subjected to time series analysis. The data collected from the Central Pollution Control Board of India are analyzed with special reference to PM and compared with the World Air Quality Report 2021 and University of Washington data. The dependence of PM2.5 concentration on humidity and rain is also studied. The study reveals the increase in complexity with the concentration of pollutants through the phase portrait. The sample entropy analysis of the nonlinear time series of the pollutants exhibits a linear relation with AQI suggesting the possibility of using sample entropy as a surrogate measure of AQI.

Effects of temperature and water vapour on Cr-species volatilisation, oxidation and scale adhesion of a Type 409L stainless steel for application as interconnect of low temperature solid oxide fuel cells

Though the classical transport theory does not mathematically express the Arrhenius-like relation between Cr-loss flux due to volatilisation and absolute temperature, such relation was experimentally observed for 409L stainless steel exposed at 600–750 °C. Volatilisation calculation in the present cases can be estimated giving such relation because of the nearly linear relation between mass transfer coefficient and temperature. Cr-loss fluxes in O2 with 20 and 40 % H2O are close to or coincided with the fluxes calculated from MnCr2O4 evaporation. Increasing water vapour could enhance volatilisation contributed from the raised CrO2(OH)2 pressure, reduce oxidation rate, and worsen scale adhesion.

Effect of the kinematic viscosity on liquid flow hydrodynamics in vortex mixers

The effect of the kinematic viscosity of similar fluids on liquid flow hydrodynamics and the onset of the instability were investigated to acquire information on the mixing behavior of vortex mixers. To assess the influence of the kinematic viscosity, water and two glycerin aqueous solutions with varying weight percentages (25 wt % and 50 wt %) were used. The critical Reynolds number of the flow regime transitions was found using PLIF and PIV flow-field techniques. The liquid flow hydrodynamics and gas flow dynamics in the proposed vortex mixer geometry were compared, as well. It was observed that the kinematic viscosity has a linear relation with the critical Reynolds number; however, the coefficient is different for liquids and gases. Additionally, kinematic viscosity dictates the minimum Reynolds number where the fluids have a full turn inside the chamber. It was observed that fluids having higher kinematic viscosities have a full turn at lower Reynolds numbers, and the critical Reynolds number value decreases with increasing kinematic viscosity.

Enhanced sonocatalytic degradation of Acid Red 27 with Fe2O3 catalyst: a kinetic study

Abstract This study is focused on elucidating the potential effectiveness of degradation as a method to eliminate dyes from aqueous systems. Specifically, it delves into the influence of ultrasound energy on the degradation kinetics of a dye. The research findings underscore the notable impact of ultrasound energy in accelerating the reaction rate constant (k p), with the degradation kinetics exhibiting a conformity to first-order kinetics. An integral aspect of the study involves the establishment of a robust relationship between time and concentration by integrating the equation governing the degradation of the dye. Additionally, the determination of the rate constant, derived from the gradients of the graphs, attests to the model’s fitting accuracy. Intriguingly, the outcomes of this analysis reveal no discernible structural changes in the dye. The accuracy of the model is further underscored by the establishment of linear relations derived from experimental data. Summarily, this kinetic study provides invaluable insights into the multifaceted impact of ultrasound energy and the Fe2O3 catalytic influence on both the degradation kinetics of the dye. The comprehensive nature of the investigation enhances our understanding of the intricate processes involved, contributing significantly to the broader field of water treatment and dye removal from aqueous environments.

Complementation and Lebesgue‐type decompositions of linear operators and relations

AbstractIn this paper, a new general approach is developed to construct and study Lebesgue‐type decompositions of linear operators or relations in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue‐type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of . The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue‐type decomposition for pairs of quadratic forms. The approach built in this paper uses so‐called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak.