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All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

A Two-Fold Structural Classification Method for Determining the Accurate Ensemble of Protein Structures

Atomic-level structural characterization of flexible proteins, such as intrinsically disordered proteins and multi-domain proteins connected by flexible linkers, is challenging as they possess distinct conformations in physiological conditions. Significant efforts have been made to develop integrated approaches by combining small angle neutron/X-ray scattering experiments with molecular simulations to reveal the distinct atomic structures and the corresponding populations for these flexible proteins. One widely used method, the basis-set supported ensemble method, classifies the simulation-generated protein conformations into a set of structural basis and then derives the corresponding populations by fitting to the experimental data. This method makes an implicit assumption that protein conformations of similar structures have similar small angle scattering profiles.The present work demonstrates that, for various protein systems ranging from compact globular proteins and flexible multidomain proteins through to intrinsically disordered proteins, this method provides inaccurate assessment of the structural ensemble of the protein molecules due to the breakdown of the assumption made. To alleviate this problem, a two-fold-clustering method is developed to cluster the simulation-generated protein structures using information on both 3D structure and scattering profiles. As benchmarked by both simulation and experimental results, this new method yields much more accurate populations of structural basis of protein molecules.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS1993

Hypersonic blunt body wake computations using DSMC and Navier-Stokes solvers

Numerical results obtained with direct simulation Monte Carlo (DSMC) and Navier-Stokes methods are presented for Mach 20 nitrogen flow about a 70-deg blunted cone. The flow conditions simulated are those that can be obtained in existing low-density hypersonic wind tunnels. Three sets of flow conditions are considered with freestream Knudsen numbers ranging from 0.03 to 0.001. The focus is on the wake structure: how does the wake structure change as a function of rarefaction, what are the afterbody levels of heating, and to what limits are continuum models realistic as rarefaction in the wake is progressively increased. Calculations are made with and without an afterbody sting. Results for the afterbody sting are emphasized in anticipation of an experimental study for the current flow conditions and model configuration. The Navier-Stokes calculations were made with and without slip boundary conditions. Comparisons of the results obtained with the two simulation methodologies are made for both flowfield structure and surface quantities.

Moss, James N.↗
NASA NTRS1994

Direct simulation Monte Carlo and Navier-Stokes simulations of blunt body wake flows

Numerical results obtained with direct simulation Monte Carlo and Navier-Stokes methods are presented for a Mach-20 nitrogen flow about a 70-deg blunted cone. The flow conditions simulated are those that can be obtained in existing low-density hypersonic wind tunnels. Three sets of flow conditions are considered with freestream Knudsen numbers ranging from 0.03 to 0.001. The focus is on the wake structure: how the wake structure changes as a function of rare faction, what the afterbody levels of heating are, and to what limits the continuum models are realistic as rarefunction in the wake is progressively increased. Calculations are made with and without an afterbody sting. Results for the afterbody sting are emphasized in anticipation of an experimental study for the current flow conditions and model configuration. The Navier-Stokes calculations were made with and without slip boundary conditions. Comparisons of the results obtained with the two simulation methodologies are made for both flowfield structure and surface quantities.

WAKE STRUCTURE↗
NASA NTRS1998

Bounded Error Schemes for the Wave Equation on Complex Domains

This paper considers the application of the method of boundary penalty terms ("SAT") to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell's equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g. the staggered Yee scheme) - we achieve a decrease of two orders of magnitude in the level of the L2-error.

Abarbanel, Saul↗
NASA NTRS2004

Inner Radiation Belt Representation of the Energetic Electron Environment: Model and Data Synthesis Using the Salammbo Radiation Belt Transport Code and Los Alamos Geosynchronous and GPS Energetic Particle Data

The highly energetic electron environment in the inner magnetosphere (GEO inward) has received a lot of research attention in resent years, as the dynamics of relativistic electron acceleration and transport are not yet fully understood. These electrons can cause deep dielectric charging in any space hardware in the MEO to GEO region. We use a new and novel approach to obtain a global representation of the inner magnetospheric energetic electron environment, which can reproduce the absolute environment (flux) for any spacecraft orbit in that region to within a factor of 2 for the energy range of 100 KeV to 5 MeV electrons, for any levels of magnetospheric activity. We combine the extensive set of inner magnetospheric energetic electron observations available at Los Alamos with the physics based Salammbo transport code, using the data assimilation technique of "nudging". This in effect input in-situ data into the code and allows the diffusion mechanisms in the code to interpolate the data into regions and times of no data availability. We present here details of the methods used, both in the data assimilation process and in the necessary inter-calibration of the input data used. We will present sample runs of the model/data code and compare the results to test spacecraft data not used in the data assimilation process.

Friedel, R. H. W.↗
NASA NTRS2007

Spacebased Estimation of Moisture Transport in Marine Atmosphere Using Support Vector Regression

An improved algorithm is developed based on support vector regression (SVR) to estimate horizonal water vapor transport integrated through the depth of the atmosphere ((Theta)) over the global ocean from observations of surface wind-stress vector by QuikSCAT, cloud drift wind vector derived from the Multi-angle Imaging SpectroRadiometer (MISR) and geostationary satellites, and precipitable water from the Special Sensor Microwave/Imager (SSM/I). The statistical relation is established between the input parameters (the surface wind stress, the 850 mb wind, the precipitable water, time and location) and the target data ((Theta) calculated from rawinsondes and reanalysis of numerical weather prediction model). The results are validated with independent daily rawinsonde observations, monthly mean reanalysis data, and through regional water balance. This study clearly demonstrates the improvement of (Theta) derived from satellite data using SVR over previous data sets based on linear regression and neural network. The SVR methodology reduces both mean bias and standard deviation comparedwith rawinsonde observations. It agrees better with observations from synoptic to seasonal time scales, and compare more favorably with the reanalysis data on seasonal variations. Only the SVR result can achieve the water balance over South America. The rationale of the advantage by SVR method and the impact of adding the upper level wind will also be discussed.

Support vector regression↗
NASA NTRS2010

The ICARE Method

The ICARE method is a flexible, widely applicable method for systems engineers to solve problems and resolve issues in a complete and comprehensive manner. The method can be tailored by diverse users for direct application to their function (e.g. system integrators, design engineers, technical discipline leads, analysts, etc.). The clever acronym, ICARE, instills the attitude of accountability, safety, technical rigor and engagement in the problem resolution: Identify, Communicate, Assess, Report, Execute (ICARE). This method was developed through observation of Space Shuttle Propulsion Systems Engineering and Integration (PSE&I) office personnel approach in an attempt to succinctly describe the actions of an effective systems engineer. Additionally it evolved from an effort to make a broadly-defined checklist for a PSE&I worker to perform their responsibilities in an iterative and recursive manner. The National Aeronautics and Space Administration (NASA) Systems Engineering Handbook states, engineering of NASA systems requires a systematic and disciplined set of processes that are applied recursively and iteratively for the design, development, operation, maintenance, and closeout of systems throughout the life cycle of the programs and projects. ICARE is a method that can be applied within the boundaries and requirements of NASA s systems engineering set of processes to provide an elevated sense of duty and responsibility to crew and vehicle safety. The importance of a disciplined set of processes and a safety-conscious mindset increases with the complexity of the system. Moreover, the larger the system and the larger the workforce, the more important it is to encourage the usage of the ICARE method as widely as possible. According to the NASA Systems Engineering Handbook, elements of a system can include people, hardware, software, facilities, policies and documents; all things required to produce system-level results, qualities, properties, characteristics, functions, behavior and performance. The ICARE method can be used to improve all elements of a system and, consequently, the system-level functional, physical and operational performance. Even though ICARE was specifically designed for a systems engineer, any person whose job is to examine another person, product, or process can use the ICARE method to improve effectiveness, implementation, usefulness, value, capability, efficiency, integration, design, and/or marketability. This paper provides the details of the ICARE method, emphasizing the method s application to systems engineering. In addition, a sample of other, non-systems engineering applications are briefly discussed to demonstrate how ICARE can be tailored to a variety of diverse jobs (from project management to parenting).

Henke, Luke↗
DOE OSTI2012

Anomaly Detection in Multiple Scale for Insider Threat Analysis

We propose a method to quantify malicious insider activity with statistical and graph-based analysis aided with semantic scoring rules. Different types of personal activities or interactions are monitored to form a set of directed weighted graphs. The semantic scoring rules assign higher scores for the events more significant and suspicious. Then we build personal activity profiles in the form of score tables. Profiles are created in multiple scales where the low level profiles are aggregated toward more stable higherlevel profiles within the subject or object hierarchy. Further, the profiles are created in different time scales such as day, week, or month. During operation, the insider s current activity profile is compared to the historical profiles to produce an anomaly score. For each subject with a high anomaly score, a subgraph of connected subjects is extracted to look for any related score movement. Finally the subjects are ranked by their anomaly scores to help the analysts focus on high-scored subjects. The threat-ranking component supports the interaction between the User Dashboard and the Insider Threat Knowledge Base portal. The portal includes a repository for historical results, i.e., adjudicated cases containing all of the information first presented to the user and including any additional insights to help the analysts. In this paper we show the framework of the proposed system and the operational algorithms.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2011

Economic Impacts of Wind Turbine Development in U.S. Counties

The objective is to address the research question using post-project construction, county-level data, and econometric evaluation methods. Wind energy is expanding rapidly in the United States: Over the last 4 years, wind power has contributed approximately 35 percent of all new electric power capacity. Wind power plants are often developed in rural areas where local economic development impacts from the installation are projected, including land lease and property tax payments and employment growth during plant construction and operation. Wind energy represented 2.3 percent of the U.S. electricity supply in 2010, but studies show that penetrations of at least 20 percent are feasible. Several studies have used input-output models to predict direct, indirect, and induced economic development impacts. These analyses have often been completed prior to project construction. Available studies have not yet investigated the economic development impacts of wind development at the county level using post-construction econometric evaluation methods. Analysis of county-level impacts is limited. However, previous county-level analyses have estimated operation-period employment at 0.2 to 0.6 jobs per megawatt (MW) of power installed and earnings at $9,000/MW to $50,000/MW. We find statistically significant evidence of positive impacts of wind development on county-level per capita income from the OLS and spatial lag models when they are applied to the full set of wind and non-wind counties. The total impact on annual per capita income of wind turbine development (measured in MW per capita) in the spatial lag model was $21,604 per MW. This estimate is within the range of values estimated in the literature using input-output models. OLS results for the wind-only counties and matched samples are similar in magnitude, but are not statistically significant at the 10-percent level. We find a statistically significant impact of wind development on employment in the OLS analysis for wind counties only, but not in the other models. Our estimates of employment impacts are not precise enough to assess the validity of employment impacts from input-output models applied in advance of wind energy project construction. The analysis provides empirical evidence of positive income effects at the county level from cumulative wind turbine development, consistent with the range of impacts estimated using input-output models. Employment impacts are less clear.

17 WIND ENERGY↗
DOE OSTI2017

Prediction of Mass Spectral Response Factors from Predicted Chemometric Data for Druglike Molecules

A method is developed for the prediction of mass spectral ion counts of drug-like molecules using in silico calculated chemometric data. Various chemometric data, including polar and molecular surface areas, aqueous solvation free energies, and gas-phase and aqueous proton affinities were computed, and a statistically significant relationship between measured mass spectral ion counts and the combination of aqueous proton affinity and total molecular surface area was identified. In particular, through multilinear regression of ion counts on predicted chemometric data, we find that log{sub 10}(MS ion counts) = –4.824 + c{sub 1}•PA + c{sub 2}•SA, where PA is the aqueous proton affinity of the molecule computed at the SMD(aq)/M06-L/MIDI!//M06-L/MIDI! level of electronic structure theory, SA is the total surface area of the molecule in its conjugate base form, and c{sub 1} and c{sub 2} have values of –3.912 × 10{sup –2} mol kcal{sup –1} and 3.682 × 10{sup –3} Å{sup –2}. On a 66-molecule training set, this regression exhibits a multiple R value of 0.791 with p values for the intercept, c{sub 1}, and c{sub 2} of 1.4 × 10{sup –3}, 4.3 × 10{sup –10}, and 2.5 × 10{sup –6}, respectively. Application of this regression to an 11-molecule test set provides a good correlation of prediction with experiment (R = 0.905) albeit with a systematic underestimation of about 0.2 log units. This method may prove useful for semiquantitative analysis of drug metabolites for which MS response factors or authentic standards are not readily available. .

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2013

Passive Neutron Non-Destructive Assay for Remediation of Radiological Waste at Hanford Burial Grounds- 13189

The Hanford burial grounds contains a broad spectrum of low activity radioactive wastes, transuranic (TRU) wastes, and hazardous wastes including fission products, byproduct material (thorium and uranium), plutonium and laboratory chemicals. A passive neutron non-destructive assay technique has been developed for characterization of shielded concreted drums exhumed from the burial grounds. This method facilitates the separation of low activity radiological waste containers from TRU waste containers exhumed from the burial grounds. Two identical total neutron counting systems have been deployed, each consisting of He-3 detectors surrounded by a polyethylene moderator. The counts are processed through a statistical filter that removes outliers in order to suppress cosmic spallation events and electronic noise. Upon completion of processing, a 'GO / NO GO' signal is provided to the operator based on a threshold level equivalent to 0.5 grams of weapons grade plutonium in the container being evaluated. This approach allows instantaneous decisions to be made on how to proceed with the waste. The counting systems have been set up using initial on-site measurements (neutron emitting standards loaded into surrogate waste containers) combined with Monte Carlo modeling techniques. The benefit of this approach is to allow the systems to extend their measurement ranges, in terms of applicable matrix types and container sizes, with minimal interruption to the operations at the burial grounds. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2024

Direct pulse-level compilation of arbitrary quantum logic gates on superconducting qutrits

Advanced simulations and calculations on quantum computers require high-fidelity implementations of quantum operations. The universal gateset approach builds complex unitaries from a small set of primitive gates, often resulting in a long gate sequence, which is typically a leading factor in the total accumulated error. Compiling a complex unitary for processors with higher-dimensional logical elements, such as qutrits, exacerbates the accumulated error per unitary, since an even longer gate sequence is required. Optimal control methods promise time- and resource-efficient compact gate sequences and, therefore, higher fidelity. These methods generate pulses that can directly implement any complex unitary on a quantum device. In this work, we demonstrate that any arbitrary qubit and qutrit gate can be realized with high fidelity, which can significantly reduce the length of a gate sequence. We generate and test pulses for a large set of randomly selected arbitrary unitaries on several quantum processing units (QPUs): the Lawrence Livermore National Laboratory Quantum Device and Integration Testbed’s (QuDIT’s) standard QPU and three of Rigetti’s QPUs: Ankaa-2, Ankaa-9Q-1, and Aspen-M-3. On the QuDIT platform’s standard QPU, the average fidelity of random qutrit gates is 97.9 ± 0.5% measured with conventional QPT and 98.8 ± 0.6% from QPT with gate folding. Rigetti’s Ankaa-2 achieves random qubit gates with an average fidelity of 98.4 ± 0.5% (conventional QPT) and 99.7 ± 0.1% (QPT with gate folding). On Ankaa-9Q-1 and Aspen-M-3, the average fidelities with conventional qubit QPT measurements were higher than 99% (see Appendix). Here we show that optimal control gates are robust to drift for at least 3 h and that the same calibration parameters can be used for all implemented gates. Our work promises that the calibration overheads for optimal control gates can be made small enough to enable efficient quantum circuits based on this technique.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

High-Accuracy Heats of Formation for Alkane Oxidation: From Small to Large via the Automated CBH-ANL Method

It is generally challenging to obtain high-accuracy predictions for the heat of formation for species with more than a handful of heavy atoms, such as those of importance in standard combustion mechanisms. To this end, we construct the CBH-ANL approach and illustrate that, for a set of 194 alkane oxidation species, it can be used to produce ΔH f (o K) values with 2σ uncertainties of 0.2-0.5 kcal mol -1 . This set includes the alkanes, hydroperoxides, and alkyl, peroxy, and hydroperoxyalkyl radicals for 17 representative hydrocarbon fuels containing up to 10 heavy atoms with various degrees of branching in the alkane backbone. The CBH-ANL approach, automated in the QTC and AutoMech software suites, builds balanced chemical equations for the calculation of ΔH f (o K), in which the reference species may be up to five heavy atoms. The high-level ANLo and ANL1 reference ΔH f (o K) values are further refined for even the largest of these reference species with a novel laddering approach. We perform a comprehensive quantification of the uncertainties for both the individual reference species (the largest of which is 0.15 kcal mol -1 ) and the propagation of those uncertainties when used in the calculation of ΔH f (o K) for the 194 target species. Here we examine the sensitivity of the predicted ΔH f (o K) values to (i) electronic energies from various methods, including ωB97X-D/cc-pVTZ, B2PLYP-D3/cc-pVTZ, CCSD(T)-F12b/cc-pVDZ-F12//B2PLYP-D3/cc-pVTZ, and CCSD(T)-F12b/cc-pVTZ-F12//B2PLYP-D3/cc-pVTZ; (ii) the zero-point vibrational energies (ZPVEs), where we consider harmonic ZPVEs as well as two scaling-based estimates of the anharmonic ZPVEs, all implemented for both ωB97X-D/cc-pVTZ and B2PLYP-D3/cc-pVTZ calculations; (iii) the particular CBH-ANL scheme employed; and (iv) the procedure for choosing the reference conformer for the analyses. The discussion concludes with a summary of the estimated overall uncertainty in the predictions and a validation of the predictions for the alkane subset.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Comparison between simulated and observed LHC beam backgrounds in the ATLAS experiment at E beam =4 TeV

Results of dedicated Monte Carlo simulations of beam-induced background (BIB) in the ATLAS experiment at the Large Hadron Collider (LHC) are presented and compared with data recorded in 2012. During normal physics operation this background arises mainly from scattering of the 4 TeV protons on residual gas in the beam pipe. Methods of reconstructing the BIB signals in the ATLAS detector, developed and implemented in the simulation chain based on the \textsc{Fluka} Monte Carlo simulation package, are described. The interaction rates are determined from the residual gas pressure distribution in the LHC ring in order to set an absolute scale on the predicted rates of BIB so that they can be compared quantitatively with data. Through these comparisons the origins of the BIB leading to different observables in the ATLAS detectors are analysed. The level of agreement between simulation results and BIB measurements by ATLAS in 2012 demonstrates that a good understanding of the origin of BIB has been reached.

47 OTHER INSTRUMENTATION↗
DOE OSTI2018

Evidence for associated production of a Higgs boson with a top quark pair in final states with electrons, muons, and hadronically decaying τ leptons at $ \sqrt{s}=13 $ TeV

Results of a search for the standard model Higgs boson produced in association with a top quark pair ($\mathrm{t\overline{t}}$H) in final states with electrons, muons, and hadronically decaying $\tau$ leptons are presented. The analyzed data set corresponds to an integrated luminosity of 35.9 fb$^{-1}$ recorded in proton-proton collisions at $\sqrt{s} =$ 13 TeV by the CMS experiment in 2016. The sensitivity of the search is improved by using matrix element and machine learning methods to separate the signal from backgrounds. The measured signal rate amounts to 1.23$^{+0.45}_{-0.43}$ times the production rate expected in the standard model, with an observed (expected) significance of 3.2$\sigma$ (2.8$\sigma$), which represents evidence for $\mathrm{t\overline{t}}$H production in those final states. In conclusion, an upper limit on the signal rate of 2.1 times the standard model production rate is set at 95% confidence level.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2014

An evaluation of experimental decadal predictions using CCSM4

This study assesses retrospective decadal prediction skill of sea surface temperature (SST) variability in initialized climate prediction experiments with the Community Climate System Model version 4 (CCSM4). Ensemble forecasts initialized with two different historical ocean and sea-ice states are evaluated and compared to an ensemble of uninitialized coupled simulations. Both experiments are subject to identical twentieth century historical radiative forcings. Each forecast consists of a 10-member ensemble integrated over a 10-year period. One set of historical ocean and sea-ice conditions used for initialization comes from a forced ocean-ice simulation driven by the Coordinated Ocean-ice Reference Experiments interannually varying atmospheric dataset. Following the Coordinated Model Intercomparison Project version 5 (CMIP5) protocol, these forecasts are initialized every 5 years from 1961 to 1996, and every year from 2000 to 2006. A second set of initial conditions comes from historical ocean state estimates obtained through the assimilation of in-situ temperature and salinity data into the CCSM4 ocean model. These forecasts are only available over a limited subset of the CMIP5 recommended start dates. Both methods result in retrospective SST prediction skill over broad regions of the Indian Ocean, western Pacific Ocean and North Atlantic Ocean that are significantly better than reference skill levels from a spatio-temporal auto-regressive statistical model of SST. However the subpolar gyre region of the North Atlantic stands out as the only region where the CCSM4 initialized predictions outperform uninitialized simulations. Some features of the ocean state estimates used for initialization and their impact on the forecasts are discussed.

Meteorology & Atmospheric Sciences↗
DOE OSTI2022

Insights into Methodologies and Operational Details of Resource Adequacy Assessment: A Case Study with Application to a Broader Flexibility Framework

Assessing and maintaining resource adequacy (RA) is a core pillar of power systems. However, recent changes in the physical makeup of these systems and the conditions under which these systems must operate have yielded a renewed interest in the methods, metrics, and assumptions that underpin RA assessments. In this paper, we systematically explore a wide range of RA modeling dimensions, including: the objective function and level of operational detail in the underlying model formulation; the quantity (look-ahead) and quality (accuracy) of data that is available for making operational decisions within those models; and the physical configuration of solar photovoltaics (PV) with battery storage hybrid resources. We apply a set of probabilistic RA tools and production cost modeling tools to a realistic test system based loosely on a future Electric Reliability Council of Texas power system dominated by solar PV resources. Under the assumptions of our system and models, we find that multi-stage probabilistic assessments may provide a more robust evaluation of RA by capturing a wider range of operational and system interactions, but this comes at a computational cost of 1-2 orders of magnitude longer run time depending on the specific configuration. In addition, the information on thermal generator availability impacts RA performance by an order of magnitude more than solar resource forecasts, which is driven by the comparatively larger magnitude of thermal outages than solar forecast errors within our test system. Lastly, the flexibility provided by hybrid and other resources can help reduce system load-shedding event frequencies and enable the system to be more robust to inaccurate forecast information, and alternative hybrid inverter sizes can impact RA levels by 1-2 orders of magnitude. Our results point to the importance of a broader flexibility framework to describe the interaction between (1) flexibility "supply" from both physical resource capabilities and operational constraints considered in the modeling, and (2) flexibility "demand" from forecast errors, thermal generator outages, and other sources of uncertainty, as well as their RA impacts. Results are likely sensitive to the system buildout explored; future work could consider additional system configurations and conditions.

ENERGY PLANNING, POLICY, AND ECONOMY,POWER TRANSMI↗
DOE OSTI2025

The latent variable proximal point algorithm for variational problems with inequality constraints

The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. At the continuous level, the two formulations are equivalent, but the saddle point formulation is more amenable to discretization because it introduces a structure-preserving transformation between a latent function space and the feasible set. Working in this latent space is much more convenient for enforcing inequality constraints than the feasible set, as discretizations can employ general linear combinations of suitable basis functions, and nonlinear solvers can involve general additive updates. LVPP yields numerical methods with observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides; in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge–Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. Here, in this paper, we describe the LVPP algorithm in a general form and apply it to ten problems from across mathematics.

Inequality constraints↗