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All sources NASA NTRS DOE OSTI

DOE OSTI2014

A physical model for the evolving ultraviolet luminosity function of high redshift galaxies and their contribution to the cosmic reionization

We present a physical model for the evolution of the ultraviolet (UV) luminosity function of high-redshift galaxies, taking into account in a self-consistent way their chemical evolution and the associated evolution of dust extinction. Dust extinction is found to increase fast with halo mass. A strong correlation between dust attenuation and halo/stellar mass for UV selected high-z galaxies is thus predicted. The model yields good fits of the UV and Lyman-α (Lyα) line luminosity functions at all redshifts at which they have been measured. The weak observed evolution of both luminosity functions between z = 2 and z = 6 is explained as the combined effect of the negative evolution of the halo mass function; of the increase with redshift of the star formation efficiency due to the faster gas cooling; and of dust extinction, differential with halo mass. The slope of the faint end of the UV luminosity function is found to steepen with increasing redshift, implying that low luminosity galaxies increasingly dominate the contribution to the UV background at higher and higher redshifts. The observed range of the UV luminosities at high z implies a minimum halo mass capable of hosting active star formation M {sub crit} ≲ 10{sup 9.8} M {sub ☉}, which is consistent with the constraints from hydrodynamical simulations. From fits of Lyα line luminosity functions, plus data on the luminosity dependence of extinction, and from the measured ratios of non-ionizing UV to Lyman-continuum flux density for samples of z ≅ 3 Lyman break galaxies and Lyα emitters, we derive a simple relationship between the escape fraction of ionizing photons and the star formation rate. It implies that the escape fraction is larger for low-mass galaxies, which are almost dust-free and have lower gas column densities. Galaxies already represented in the UV luminosity function (M {sub UV} ≲ –18) can keep the universe fully ionized up to z ≅ 6. This is consistent with (uncertain) data pointing to a rapid drop of the ionization degree above z ≅ 6, such as indications of a decrease of the comoving emission rate of ionizing photons at z ≅ 6, a decrease of sizes of quasar near zones, and a possible decline of the Lyα transmission through the intergalactic medium at z > 6. On the other hand, the electron scattering optical depth, τ{sub es}, inferred from cosmic microwave background (CMB) experiments favor an ionization degree close to unity up to z ≅ 9-10. Consistency with CMB data can be achieved if M {sub crit} ≅ 10{sup 8.5} M {sub ☉}, implying that the UV luminosity functions extend to M {sub UV} ≅ –13, although the corresponding τ{sub es} is still on the low side of CMB-based estimates.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2015

Better alternatives to “astronomical silicate”: Laboratory-based optical functions of chondritic/solar abundance glass with application to HD 161796

“Astronomical” or “circumstellar” silicate optical functions (real and imaginary indices of refraction n(λ) and k(λ)) have previously been derived from compositionally and structurally disparate samples; past values were compiled from different sources in the literature, and are essentially kluges of observational, laboratory, and extrapolated or interpolated values. These synthetic optical functions were created because astronomers lack the quantitative data on amorphous silicates at all wavelengths needed for radiative transfer modeling. This paper provides optical functions that (1) are created with a consistent methodology, (2) use the same sample across all wavelengths, and (3) minimize interpolation and extrapolation wherever possible. We present electronic data tables of optical functions derived from mid-ultraviolet to far-infrared (FIR) laboratory transmission spectra for two materials: iron-free glass with chondritic/solar atmospheric abundances, and metallic iron. We compare these optical functions to other popular n, k data used to model amorphous silicates (e.g., “astronomical” or “circumstellar” silicate), both directly and in application to a simple system: the dust shell of the post-AGB star HD 161796. Using the new optical functions, we find that the FIR profile of model spectral energy distributions are significantly affected by the ratio of glass to iron. Our case study on HD 161796 shows that in modeling with our new optical functions, the mineralogy is markedly different from that derived using synthetic optical functions and suggests a new scenario of crystalline silicate formation.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Quantum simulation of molecules without fermionic encoding of the wave function

Abstract Molecular simulations generally require fermionic encoding in which fermion statistics are encoded into the qubit representation of the wave function. Recent calculations suggest that fermionic encoding of the wave function can be bypassed, leading to more efficient quantum computations. Here we show that the two-electron reduced density matrix (2-RDM) can be expressed as a unique functional of the unencoded N -qubit-particle wave function without approximation, and hence, the energy can be expressed as a functional of the 2-RDM without fermionic encoding of the wave function. In contrast to current hardware-efficient methods, the derived functional has a unique, one-to-one (and onto) mapping between the qubit-particle wave functions and 2-RDMs, which avoids the over-parametrization that can lead to optimization difficulties such as barren plateaus. An application to computing the ground-state energy and 2-RDM of H 4 is presented.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2018

Hybrid functional pseudopotentials

The consistency between the exchange-correlation functional used in pseudopotential construction and in the actual density functional theory calculation is essential for the accurate prediction of fundamental properties of materials. However, routine hybrid density functional calculations at present still rely on generalized gradient approximation pseudopotentials due to the lack of hybrid functional pseudopotentials. Here, we present a scheme for generating hybrid functional pseudopotentials, and we analyze the importance of pseudopotential density functional consistency for hybrid functionals. For the PBE0 hybrid functional, we benchmark our pseudopotentials for structural parameters and fundamental electronic gaps of the Gaussian-2 (G2) molecular dataset and some simple solids. Our results demonstrate that using our PBE0 pseudopotentials in PBE0 calculations improves agreement with respect to all-electron calculations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Optimized nuclear energy density functionals including long-range pion contributions

Nuclear energy density functionals successfully reproduce properties of nuclei across almost the entire nuclear chart. However, nearly all available functionals are phenomenological in nature and lack a rigorous connection to systematically improvable nuclear forces. This issue might be solved with an energy density functional obtained from first principles. As an intermediate step towards this goal we construct the GUDE family of functionals that is obtained from a hybrid scheme consisting of long-range pion-exchange contributions derived from chiral effective field theory at the Hartree-Fock level and a phenomenological Skyrme part. When including pion contributions beyond next-to-leading order in the chiral expansion, we find significant improvements over a reference Skyrme functional constructed following the same protocol. We analyze the importance of different pion contributions and identify which terms drive the observed improvements. Since pions are incorporated without adding further optimization parameters to the functionals, the improvements can be attributed to the functional form of these terms. Our work therefore suggests that the considered chiral contributions constitute useful ingredients for true energy density functionals. Published by the American Physical Society 2024

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2020

Time Evolution of Correlation Functions in Quantum Many-Body Systems

We give rigorous analytical results on the temporal behavior of two-point correlation functions—also known as dynamical response functions or Green’s functions—in closed many-body quantum systems. We show that in a large class of translation-invariant models the correlation functions factorize at late times $\langle$A ( t ) B$\rangle$ β → $\langle$ A$\rangle$ β $\langle$ B$\rangle$ β , thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that for systems with a generic spectrum the fluctuations around this late-time value are bounded by the purity of the thermal ensemble, which generally decays exponentially with system size. For autocorrelation functions we provide an upper bound on the timescale at which they reach the factorized late time value. Remarkably, this bound is only a function of local expectation values and does not increase with system size. We give numerical examples that show that this bound is a good estimate in nonintegrable models, and argue that the timescale that appears can be understood in terms of an emergent fluctuation-dissipation theorem. Our study extends to further classes of two point functions such as the symmetrized ones and the Kubo function that appears in linear response theory, for which we give analogous results.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

Functional diversity and redundancy across fish gut, sediment and water bacterial communities

Here, this work explores the functional diversity and redundancy in a bacterial metacommunity constituted of three habitats (sediment, water column and fish gut) in a coastal lagoon under anthropogenic pressure. Comprehensive functional gene arrays covering a wide range of ecological processes and stress resistance genes to estimate the functional potential of bacterial communities were used. Then, diversity partitioning was used to characterize functional diversity and redundancy within (α), between (β) and across (γ) habitats. It was showed that all local communities exhibit a highly diversified potential for the realization of key ecological processes and resistance to various environmental conditions, supporting the growing evidence that macro-organisms microbiomes harbour a high functional potential and are integral components of functional gene dynamics in aquatic bacterial metacommunities. Several levels of functional redundancy at different scales of the bacterial metacommunity were observed (within local communities, within habitats and at the metacommunity level). The findings indicated a high potential for the realization of spatial ecological insurance within this ecosystem, that is, the functional compensation among microorganisms for the realization and maintenance of key ecological processes, within and across habitats. Finally, the role of macro-organisms as dispersal vectors of microbes and their potential influence on marine metacommunity dynamics were discussed.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2015

SU-E-J-86: Functional Conformal Planning for Stereotactic Body Radiation Therapy with CT-Pulmonary Ventilation Imaging

Purpose: To evaluate the functional planning using CT-pulmonary ventilation imaging for conformal SBRT. Methods: The CT-pulmonary ventilation image was generated using the Jacobian metric in the in-house program with the NiftyReg software package. Using the ventilation image, the normal lung was split into three lung regions for functionality (high, moderate and low). The anatomical plan (AP) and functional plan (FP) were made for ten lung SBRT patients. For the AP, the beam angles were optimized with the dose-volume constraints for the normal lung sparing and the PTV coverage. For the FP, the gantry angles were also optimized with the additional constraint for high functional lung. The MLC aperture shapes were adjusted to the PTV with the additional 5 mm margin. The dosimetric parameters for PTV, the functional volumes, spinal cord and so on were compared in both plans. Results: Compared to the AP, the FP showed better dose sparing for high- and moderate-functional lungs with similar PTV coverage while not taking care of the low functional lung (High:−12.9±9.26% Moderate: −2.0±7.09%, Low: +4.1±12.2%). For the other normal organs, the FP and AP showed similar dose sparing in the eight patients. However, the FP showed that the maximum doses for spinal cord were increased with the significant increment of 16.4Gy and 21.0Gy in other two patients, respectively. Because the beam direction optimizer chose the unexpected directions passing through the spinal cord. Conclusion: Even the functional conformal SBRT can selectively reduce high- and moderatefunctional lung while keeping the PTV coverage. However, it would be careful that the optimizer would choose unexpected beam angles and the dose sparing for the other normal organs can be worse. Therefore, the planner needs to control the dose-volume constraints and also limit the beam angles in order to achieve the expected dose sparing and coverage.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2016

Different Functions of Phylogenetically Distinct Bacterial Complex I Isozymes

ABSTRACT NADH:quinone oxidoreductase (complex I) is a bioenergetic enzyme that transfers electrons from NADH to quinone, conserving the energy of this reaction by contributing to the proton motive force. While the importance of NADH oxidation to mitochondrial aerobic respiration is well documented, the contribution of complex I to bacterial electron transport chains has been tested in only a few species. Here, we analyze the function of two phylogenetically distinct complex I isozymes in Rhodobacter sphaeroides , an alphaproteobacterium that contains well-characterized electron transport chains. We found that R. sphaeroides complex I activity is important for aerobic respiration and required for anaerobic dimethyl sulfoxide (DMSO) respiration (in the absence of light), photoautotrophic growth, and photoheterotrophic growth (in the absence of an external electron acceptor). Our data also provide insight into the functions of the phylogenetically distinct R. sphaeroides complex I enzymes (complex I A and complex I E ) in maintaining a cellular redox state during photoheterotrophic growth. We propose that the function of each isozyme during photoheterotrophic growth is either NADH synthesis (complex I A ) or NADH oxidation (complex I E ). The canonical alphaproteobacterial complex I isozyme (complex I A ) was also shown to be important for routing electrons to nitrogenase-mediated H 2 production, while the horizontally acquired enzyme (complex I E ) was dispensable in this process. Unlike the singular role of complex I in mitochondria, we predict that the phylogenetically distinct complex I enzymes found across bacterial species have evolved to enhance the functions of their respective electron transport chains. IMPORTANCE Cells use a proton motive force (PMF), NADH, and ATP to support numerous processes. In mitochondria, complex I uses NADH oxidation to generate a PMF, which can drive ATP synthesis. This study analyzed the function of complex I in bacteria, which contain more-diverse and more-flexible electron transport chains than mitochondria. We tested complex I function in Rhodobacter sphaeroides , a bacterium predicted to encode two phylogenetically distinct complex I isozymes. R. sphaeroides cells lacking both isozymes had growth defects during all tested modes of growth, illustrating the important function of this enzyme under diverse conditions. We conclude that the two isozymes are not functionally redundant and predict that phylogenetically distinct complex I enzymes have evolved to support the diverse lifestyles of bacteria.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2022

Unified Wind-Wave Growth and Spectrum Functions for All Water Depths: Field Observations and Model Results

Abstract Wind-wave development is governed by the fetch- or duration-limited growth principle that is expressed as a pair of similarity functions relating the dimensionless elevation variance (wave energy) and spectral peak frequency to fetch or duration. Combining the pair of similarity functions, the fetch or duration variable can be removed to form a dimensionless function of elevation variance and spectral peak frequency, which is interpreted as the wave energy evolution with wave age. The relationship is initially developed for quasi-neural stability and quasi-steady wind forcing conditions. Further analyses show that the same fetch, duration, and wave-age similarity functions are applicable to unsteady wind forcing conditions, including rapidly accelerating and decelerating mountain gap wind episodes and tropical cyclone (TC) wind fields. Here it is shown that with the dimensionless frequency converted to dimensionless wavenumber using the surface wave dispersion relationship, the same similarity function is applicable in all water depths. Field data collected in shallow to deep waters and mild to TC wind conditions and synthetic data generated by spectrum model computations are assembled to illustrate the applicability. For the simulation work, the finite-depth wind-wave spectrum model and its shoaling function are formulated for variable spectral slopes. Given wind speed, wave age, and water depth, the measured and spectrum-computed significant wave heights and the associated growth parameters are in good agreement in forcing conditions from mild to TC winds and in all depths from deep ocean to shallow lake. Significance Statement This paper presents a growth function and spectrum model to describe wind-wave development in all water depths. Their applicability covers a wide range of wind forcing conditions including steady, accelerating, decelerating, and tropical cyclone events. Support for the unified spectrum model and growth function is presented with field observations and numerical computations.

Hwang, Paul A.↗

DOE OSTI2014

Microbial Community Functional Structures in Wastewater Treatment Plants as Characterized by GeoChip

Background: Biological WWTPs must be functionally stable to continuously and steadily remove contaminants which rely upon the activity of complex microbial communities. However, knowledge is still lacking in regard to microbial community functional structures and their linkages to environmental variables. Aims: To investigate microbial community functional structures of activated sludge in wastewater treatment plants (WWTPs) and to understand the effects of environmental factors on their structure. Methods: 12 activated sludge samples were collected from four WWTPs in Beijing. A comprehensive functional gene array named GeoChip 4.2 was used to determine the microbial functional genes involved in a variety of biogeochemical processes such as carbon, nitrogen, phosphorous and sulfur cycles, metal resistance, antibiotic resistance and organic contaminant degradation. Results: High similarities of the microbial community functional structures were found among activated sludge samples from the four WWTPs, as shown by both diversity indices and the overlapped genes. For individual gene category, such as egl, amyA, lip, nirS, nirK, nosZ, ureC, ppx, ppk, aprA, dsrA, sox and benAB, there were a number of microorganisms shared by all 12 samples. Canonical correspondence analysis (CCA) showed that the microbial functional patterns were highly correlated with water temperature, dissolved oxygen (DO), ammonia concentrations and loading rate of chemical oxygen demand (COD). Based on the variance partitioning analyses (VPA), a total of 53% of microbial community variation from GeoChip data can be explained by wastewater characteristics (25%) and operational parameters (23%), respectively. Conclusions: This study provided an overall picture of microbial community functional structures of activated sludge in WWTPs and discerned the linkages between microbial communities and environmental variables in WWTPs.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2014

Site-Directed Mutagenesis of IRX9, IRX9L and IRX14 Proteins Involved in Xylan Biosynthesis: Glycosyltransferase Activity Is Not Required for IRX9 Function in Arabidopsis

Xylans constitute the main non-cellulosic polysaccharide in the secondary cell walls of plants. Several genes predicted to encode glycosyltransferases are required for the synthesis of the xylan backbone even though it is a homopolymer consisting entirely of b-1,4-linked xylose residues. The putative glycosyltransferases IRX9, IRX14, and IRX10 (or the paralogs IRX9L, IRX14L, and IRX10L) are required for xylan backbone synthesis in Arabidopsis. To investigate the function of IRX9, IRX9L, and IRX14, we identified amino acid residues known to be essential for catalytic function in homologous mammalian proteins and generated modified cDNA clones encoding proteins where these residues would be mutated. The mutated gene constructs were used to transform wild-type Arabidopsis plants and the irx9 and irx14 mutants, which are deficient in xylan synthesis. The ability of the mutated proteins to complement the mutants was investigated by measuring growth, determining cell wall composition, and microscopic analysis of stem cross-sections of the transgenic plants. The six different mutated versions of IRX9 and IRX9-L were all able to complement the irx9 mutant phenotype, indicating that residues known to be essential for glycosyltransferases function in homologous proteins are not essential for the biological function of IRX9/IRX9L. Two out of three mutated IRX14 complemented the irx14 mutant, including a mutant in the predicted catalytic amino acid. A IRX14 protein mutated in the substrate-binding DxD motif did not complement the irx14 mutant. Thus, substrate binding is important for IRX14 function but catalytic activity may not be essential for the function of the protein. The data indicate that IRX9/IRX9L have an essential structural function, most likely by interacting with the IRX10/ IRX10L proteins, but do not have an essential catalytic function. Most likely IRX14 also has primarily a structural role, but it cannot be excluded that the protein has an important enzymatic activity.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2017

Bacterial community and arsenic functional genes diversity in arsenic contaminated soils from different geographic locations

To understand how soil microbial communities and arsenic (As) functional genes respond to soil arsenic (As) contamination, five soils contaminated with As at different levels were collected from diverse geographic locations, incubated for 54 days under flooded conditions, and examined by both MiSeq sequencing of 16S rRNA gene amplicons and functional gene microarray (GeoChip 4.0). The results showed that both bacterial community structure and As functional gene structure differed among geographical locations. The diversity of As functional genes correlated positively with the diversity of 16S rRNA genes (P< 0.05). Higher diversities of As functional genes and 16S rRNA genes were observed in the soils with higher available As. Soil pH, phosphate-extractable As, and amorphous Fe content were the most important factors in shaping the bacterial community structure and As transformation functional genes. Geographic location was also important in controlling both the bacterial community and As transformation functional potential. These findings provide insights into the variation of As transformation functional genes in soils contaminated with different levels of As at different geographic locations, and the impact of environmental As contamination on the soil bacterial community.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2025

Identifying microbial functional guilds performing cryptic organotrophic and lithotrophic redox cycles in anaerobic granular biofilms

Granular biofilms used in anaerobic digester systems contain diverse microbial populations that interact to hydrolyze organic matter and produce methane within controlled environments. Prior research investigated the feasibility of utilizing granular biofilms obtained from an anaerobic digester to remove nitrate without the addition of exogenous electron donors. These granules possessed a unique structure of alternating light and dark iron sulfide and pyrite rich layers that potentially served as both an electron source and sink, linking carbon, nitrogen, sulfur, and iron cycles. To characterize the functional roles of diverse microbial populations enriched within these layered biofilms, we analyzed metagenomes obtained from three different granules. Comparisons between the functional gene content of forty metagenome assembled genomes (MAGs) identified phylogenetically cohesive functional guilds. Each of these functional MAG clusters was assigned to specific steps in anaerobic digestion (hydrolysis, acidogenesis, acetogenesis, and methanogenesis) and anaerobic respiration (denitrification and sulfate reduction). Comparisons with metagenomes derived from a variety of natural and engineered ecosystems confirmed that the enriched denitrifying bacteria were similar to populations typically found in wetlands and biological nitrogen removal systems. Analysis of read alignments to individual genes within the forty MAGs identified conserved genomic features that were representative of the functions that distinguished functional guilds. Overall, this research illustrates the utility of functional based classification of microorganisms for characterizing ecosystem functions and highlights the potential application of engineered ecosystems to serve as experimental models for complex natural ecosystems.

Ecosystem engineering↗

DOE OSTI2016

WEC Farm Functions: Defining the Behaviors of the Farm

Capabilities and functions are hierarchical structures (i.e. taxonomies) that are used in a systems engineering framework to identify complimentary requirements for the system: what the system must do to achieve what it must be. In the case of capabilities, the taxonomy embodies the list of characteristics that are desired, from the perspective of the stakeholders, for the system to be successful. In terms of the functions, the hierarchy represents the solution agnostic (i.e. independent of specific design embodiments) elements that are needed to meet the stakeholder requirements. This paper will focus on the development of the functions. The functions define the fundamental elements of the solution that must be provided in order to achieve the mission and deliver the capabilities. They identify the behaviors the farm must possess, i.e. the farm must be able to generate and deliver electricity from wave power. High-level functions are independent of the technology or design used to implement the function. However, detailed functions may begin to border on specific design choices. Hence a strong effort has been made to maintain functions that are design agnostic.

13 HYDRO ENERGY↗

DOE OSTI2024

Shorter function summaries for finite state machine-based high consequence systems using logic synthesis and tautologies (Final Report LDRD 24-1302)

Computer programs are often viewed as collections of functions – each function has parameters (inputs) and computes a return value, and each has potential side effects that modify program state (outputs). In this research, a Sandia symbolic execution tool designed to support “human-in-the-loop” analysis was modified to automatically create “function summaries,” and a new tool, “diaboolical,” was created to support enhancing readability of the summary using a novel approach to bit-vector simplification that leverages logic synthesis and tautologies. For this effort, students at Auburn University created several finite state machines (FSMs) to serve as exemplars for high-consequence systems. Function summaries for each of the machines were obtained, and then portions of the summaries were simplified using both diaboolical and the simplification procedure of a popular SMT solver. A comparison of the results shows that diaboolical can often produce smaller function summaries, with expression length improvements over the unsimplified function summaries ranging from 0% to 90% for diaboolical and 0% to 65% for the SMT solver, though diaboolical had a significantly greater cost in time. Diaboolical was evaluated against a collection of “arbitrary” C-code as well as FSM exemplars, and for both datasets it achieved an approximately 10% improvement in expression length compared to simplifications that could be obtained using existing techniques. Function summaries can assist assurance efforts that evaluate existing systems and their executable code. A smaller function summary is likely easier for humans to understand and could thus increase the ability and efficacy of assurance practices centered around the analysis of executable artifacts.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation

Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nev- ertheless, the majority of these studies examine how well deep neural networks can model functions with uniform regularities. In this paper, we explore a different angle: how deep neural networks can adapt to varying degrees of smoothness in functions and nonuni- form data distributions across different locations and scales. More precisely, we focus on a broad class of functions defined by nonlinear tree-based approximation methods. This class encompasses a range of function types, such as functions with uniform regularities and discontinuous functions. We develop nonparametric approximation and estimation theories for this class using deep ReLU networks. Our results show that deep neural networks are adaptive to the nonuniform smoothness of functions and nonuniform data distributions at different locations and scales. We apply our results to several function classes, and derive the corresponding approximation and generalization errors. The validity of our results is demonstrated through numerical experiments.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1979

On one-dimensional stretching functions for finite-difference calculations

The class of one-dimensional stretching functions used in finite-difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two-sided stretching function, for which the arbitrary slopes at the two ends of the one-dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent.

Vinokur, M.↗