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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 469 records · Page 26
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Measurement of $\mathrm{t\bar t}$ normalised multi-differential cross sections in pp collisions at $\sqrt s=13$ TeV, and simultaneous determination of the strong coupling strength, top quark pole mass, and parton distribution functions

Normalised multi-differential cross sections for top quark pair ($\mathrm{t\overline{t}}$) production are measured in proton-proton collisions at a centre-of-mass energy of 13 TeV using events containing two oppositely charged leptons. The analysed data were recorded with the CMS detector in 2016 and correspond to an integrated luminosity of 35.9 fb$^{-1}$. The double-differential $\mathrm{t\overline{t}}$ cross section is measured as a function of the kinematic properties of the top quark and of the $\mathrm{t\overline{t}}$ system at parton level in the full phase space. A triple-differential measurement is performed as a function of the invariant mass and rapidity of the $\mathrm{t\overline{t}}$ system and the multiplicity of additional jets at particle level. The data are compared to predictions of Monte Carlo event generators that complement next-to-leading-order (NLO) quantum chromodynamics (QCD) calculations with parton showers. Together with a fixed-order NLO QCD calculation, the triple-differential measurement is used to extract values of the strong coupling strength $\alpha_S$ and the top quark pole mass ($m_\mathrm{T}^\text{pole}$) using several sets of parton distribution functions (PDFs). Furthermore, a simultaneous fit of the PDFs, $\alpha_S$, and $m_\mathrm{T}^\text{pole}$ is performed at NLO, demonstrating that the new data have significant impact on the gluon PDF, and at the same time allow an accurate determination of $\alpha_S$ and $m_\mathrm{T}^\text{pole}$.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

scikit-SUNDAE ((SUN)DIALS Differential Algebraic Equations) [SWR-24-137]

Scikit-SUNDAE provides Python bindings to SUNDIALS integrators. The implicit differential algebraic (IDA) solver and C-based variable-coefficient ordinary differential equations (CVODE) solver are both included. The name SUNDAE combines (SUN)DIALS and DAE, which stands for differential algebraic equations. Solvers specific to DAE problems are not frequently available in Python. An ordinary differential equation (ODE) solver is also included for completeness. ODEs can be categorized as a subset of DAEs (i.e., DAEs with no algebraic constraints). https://pypi.org/project/scikit-sundae

Randall, Corey↗
DOE OSTI2021

Global transcriptome profile of the developmental principles of in vitro iPSC-to-motor neuron differentiation

Abstract Background Human induced pluripotent stem cells (iPSC) have opened new avenues for regenerative medicine. Consequently, iPSC-derived motor neurons have emerged as potentially viable therapies for spinal cord injuries and neurodegenerative disorders including Amyotrophic Lateral Sclerosis. However, direct clinical application of iPSC bears in itself the risk of tumorigenesis and other unforeseeable genetic or epigenetic abnormalities. Results Employing RNA-seq technology, we identified and characterized gene regulatory networks triggered by in vitro chemical reprogramming of iPSC into cells with the molecular features of motor neurons (MNs) whose function in vivo is to innervate effector organs. We present meta-transcriptome signatures of 5 cell types: iPSCs, neural stem cells, motor neuron progenitors, early motor neurons, and mature motor neurons. In strict response to the chemical stimuli, along the MN differentiation axis we observed temporal downregulation of tumor growth factor-β signaling pathway and consistent activation of sonic hedgehog, Wnt/β-catenin, and Notch signaling. Together with gene networks defining neuronal differentiation (neurogenin 2, microtubule-associated protein 2, Pax6, and neuropilin-1), we observed steady accumulation of motor neuron-specific regulatory genes, including Islet-1 and homeobox protein HB9. Interestingly, transcriptome profiling of the differentiation process showed that Ca 2+ signaling through cAMP and LPC was downregulated during the conversion of the iPSC to neural stem cells and key regulatory gene activity of the pathway remained inhibited until later stages of motor neuron formation. Pathways shaping the neuronal development and function were well-represented in the early motor neuron cells including, neuroactive ligand-receptor interactions, axon guidance, and the cholinergic synapse formation. A notable hallmark of our in vitro motor neuron maturation in monoculture was the activation of genes encoding G-coupled muscarinic acetylcholine receptors and downregulation of the ionotropic nicotinic acetylcholine receptors expression. We observed the formation of functional neuronal networks as spontaneous oscillations in the extracellular action potentials recorded on multi-electrode array chip after 20 days of differentiation. Conclusions Detailed transcriptome profile of each developmental step from iPSC to motor neuron driven by chemical induction provides the guidelines to novel therapeutic approaches in the re-construction efforts of muscle innervation.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2016

Differential geometry-based solvation and electrolyte transport models for biomolecular modeling: a review

This chapter reviews the differential geometry-based solvation and electrolyte transport for biomolecular solvation that have been developed over the past decade. A key component of these methods is the differential geometry of surfaces theory, as applied to the solvent-solute boundary. In these approaches, the solvent-solute boundary is determined by a variational principle that determines the major physical observables of interest, for example, biomolecular surface area, enclosed volume, electrostatic potential, ion density, electron density, etc. Recently, differential geometry theory has been used to define the surfaces that separate the microscopic (solute) domains for biomolecules from the macroscopic (solvent) domains. In these approaches, the microscopic domains are modeled with atomistic or quantum mechanical descriptions, while continuum mechanics models (including fluid mechanics, elastic mechanics, and continuum electrostatics) are applied to the macroscopic domains. This multiphysics description is integrated through an energy functional formalism and the resulting Euler-Lagrange equation is employed to derive a variety of governing partial differential equations for different solvation and transport processes; e.g., the Laplace-Beltrami equation for the solvent-solute interface, Poisson or Poisson-Boltzmann equations for electrostatic potentials, the Nernst-Planck equation for ion densities, and the Kohn-Sham equation for solute electron density. Extensive validation of these models has been carried out over hundreds of molecules, including proteins and ion channels, and the experimental data have been compared in terms of solvation energies, voltage-current curves, and density distributions. We also propose a new quantum model for electrolyte transport.

chemistry↗
DOE OSTI2014

Detection of Cervical Cancer Biomarker Patterns in Blood Plasma and Urine by Differential Scanning Calorimetry and Mass Spectrometry

Improved methods for the accurate identification of both the presence and severity of cervical intraepithelial neoplasia (CIN) and extent of spread of invasive carcinomas of the cervix (IC) are needed. Differential scanning calorimetry (DSC) has recently been shown to detect specific changes in the thermal behavior of blood plasma proteins in several diseases. This methodology is being explored to provide a complementary approach for screening of cervical disease. The present study evaluated the utility of DSC in differentiating between healthy controls, increasing severity of CIN and early and advanced IC. Significant discrimination was apparent relative to the extent of disease with no clear effect of demographic factors such as age, ethnicity, smoking status and parity. Of most clinical relevance, there was strong differentiation of CIN from healthy controls and IC, and amongst patients with IC between FIGO Stage I and advanced cancer. The observed disease-specific changes in DSC profiles (thermograms) were hypothesized to reflect differential expression of disease biomarkers that subsequently bound to and affected the thermal behavior of the most abundant plasma proteins. The effect of interacting biomarkers can be inferred from the modulation of thermograms but cannot be directly identified by DSC. To investigate the nature of the proposed interactions, mass spectrometry (MS) analyses were employed. Quantitative assessment of the low molecular weight protein fragments of plasma and urine samples revealed a small list of peptides whose abundance was correlated with the extent of cervical disease, with the most striking plasma peptidome data supporting the interactome theory of peptide portioning to abundant plasma proteins. The combined DSC and MS approach in this study was successful in identifying unique biomarker signatures for cervical cancer and demonstrated the utility of DSC plasma profiles as a complementary diagnostic tool to evaluate cervical cancer health.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Differentiable Predictions for Large Scale Structure with SHAMNet

In simulation-based models of the galaxy-halo connection, theoretical predictions for galaxy clustering and lensing are typically made based on Monte Carlo realizations of a mock universe. In this paper, we use Subhalo Abundance Matching (SHAM) as a toy model to introduce an alternative to stochastic predictions based on mock population, demonstrating how to make simulation-based predictions for clustering and lensing that are both exact and differentiable with respect to the parameters of the model. Conventional implementations of SHAM are based on iterative algorithms such as Richardson-Lucy deconvolution; here we use the JAX library for automatic differentiation to train SHAMNet, a neural network that accurately approximates the stellar-to-halo mass relation (SMHM) defined by abundance matching. In our approach to making differentiable predictions of large-scale structure, we map parameterized PDFs onto each simulated halo, and calculate gradients of summary statistics of the galaxy distribution by using autodiff to propagate the gradients of the SMHM through the statistical estimators used to measure one- and two-point functions. Our techniques are quite general, and we conclude with an overview of how they can be applied in tandem with more complex, higher-dimensional models, creating the capability to make differentiable predictions for the multi-wavelength universe of galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

End-to-End Differentiable Modeling and Management of the Environment

Focal Area: (2) Data acquisition and assimilation enabled by machine learning, AI, and advanced methods including experimental/network design/optimization. We emphasize the importance of leveraging optimization techniques from AI/machine learning (ML) to solve challenging problems in Earth system modeling. Science Challenge: Automatic differentiation has had a transformative effect on ML by allowing the calculation of gradients of arbitrary functions in an incredibly large class of models. We can potentially realize similar improvements in parameter estimation and control for Earth system models (ESMs) by reimplementing them in computational frameworks from ML. Practitioners working with large (>10 7 parameters) models in ML and AI can obtain good predictive performance in a range of spatiotemporal tasks by making use of optimization via stochastic gradient descent and incorporating prior knowledge at multiple levels. We propose writing ESMs in open-source computational frameworks such as Torch, Tensorflow, and JAX to greatly expand the scope of environmental forecasting and management challenges, which can be addressed by leveraging automatic differentiation and gradient descent-like algorithms. We do not call for a wholesale replacement of physical models with data-driven surrogates, but rather advocate for interleaving physical and empirical equations in a manner that is most faithful to the extent of our scientific knowledge and observational data. Central to this topic is the merging of differentiable physical simulations with differentiable optimization layers, which are now both beginning to come to the forefront.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

First Measurement of Differential Charged Current Quasielastic-like $\nu_\mu$–Argon Scattering Cross Sections in Kinematic Imbalance Variables with the MicroBooNE Detector

We report the first measurement of flux-integrated multi-differential cross sections for charged-current events with muon neutrinos scattering on argon with solely a muon and a single proton in the final state as a function of kinematic imbalance variables. The measurement was carried out using the Booster Neutrino Beam at Fermi National Accelerator Laboratory within the MicroBooNE Liquid Argon Time Projection Chamber detector with an exposure of 6.79 × 10 20 protons on target. Events were selected to enhance the contribution of charged-current mesonless interactions with one proton detected in the final state. The data discussed here are reported in terms of multi-differential cross sections in kinematic imbalance variables, which are generally sensitive to nuclear effects. The double-differential results in these variables can provide an excellent handle to disentagle specific nuclear aspects not easily isolated via single differential cross sections. Our results pave a path towards identifying regions of the phase-space where future interaction modeling development and Monte Carlo neutrino generator tuning efforts should concentrate.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

Cyclic tensile strain enhances human mesenchymal stem cell Smad 2/3 activation and tenogenic differentiation in anisotropic collagen-glycosaminoglycan scaffolds

Orthopaedic injuries, particularly those involving ligaments and tendons, are some of the most commonly treated ailments in the United States and are associated with both high costs and poor outcomes. Regenerative medicine strategies for tendon injuries could be enhanced by three-dimensional biomaterials that can promote cell alignment and pro-tenogenic differentiation of patientderived MSCs. We have previously described a collagenglycosaminoglycan (CG) scaffold possessing aligned structural features able to promote bone marrow MSC differentiation towards a tenogenic lineage, in the absence of growth factor supplementation. We aimed to employ a bioreactor to enhance MSC tenogenic differentiation within the aligned CG scaffold via cyclic tensile strain (CTS), and further to evaluate the relative effects of strain cycle duration and extended application of repeated cycles of CTS on MSC response. Human MSCs were cultured in CG scaffolds for up to 6 d under static (unloaded) or cyclic tensile strain (1 Hz) for 10 min every 6 h. Time-dependent activation of ERK 1/2 and p38 mechanotransduction pathways was observed within each 6 h strain cycle. MSCs remained viable throughout the experiment and application of CTS robustly upregulated the expression of tendon-specific extracellular matrix proteins and phenotypic markers. Simultaneously, CTS promoted increased phosphorylation of Smad 2/3, suggesting a link between tensile stimulation and TGF-β family growth factor production. Together, we demonstrated the design, fabrication and validation of a high-throughput tensile stimulation bioreactor to increase MSC tenogenic differentiation in porous CG scaffolds.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2023

Differentiated mouse kidney tubuloids as a novel in vitro model to study collecting duct physiology

Kidney tubuloids are cell models that are derived from human or mouse renal epithelial cells and show high similarities with their in vivo counterparts. Tubuloids grow polarized in 3D, allow for long-term expansion, and represent multiple segments of the nephron, as shown by their gene expression pattern. In addition, human tubuloids form tight, functional barriers and have been succesfully used for drug testing. Our knowledge of mouse tubuloids, on the other hand, is only minimal. In this study, we further characterized mouse tubuloids and differentiated them towards the collecting duct, which led to a significant upregulation of collecting duct-specific mRNAs of genes and protein expression, including the water channel AQP2 and the sodium channel ENaC. Differentiation resulted in polarized expression of collecting duct water channels AQP2 and AQP3. Also, a physiological response to desmopressin and forskolin stimulation by translocation of AQP2 to the apical membrane was demonstrated. Furthermore, amiloride-sensitive ENaC-mediated sodium uptake was shown in differentiated tubuloids using radioactive tracer sodium. This study demonstrates that mouse tubuloids can be differentiated towards the collecting duct and exhibit collecting duct-specific function. This illustrates the potential use of mouse kidney tubuloids as novel in vitro models to study (patho)physiology of kidney diseases.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2023

CRISPR/Cas9 Directed Reprogramming of iPSC for Accelerated Motor Neuron Differentiation Leads to Dysregulation of Neuronal Fate Patterning and Function

Neurodegeneration causes a significant disease burden and there are few therapeutic interventions available for reversing or slowing the disease progression. Induced pluripotent stem cells (iPSCs) hold significant potential since they are sourced from adult tissue and have the capacity to be differentiated into numerous cell lineages, including motor neurons. This differentiation process traditionally relies on cell lineage patterning factors to be supplied in the differentiation media. Genetic engineering of iPSC with the introduction of recombinant master regulators of motor neuron (MN) differentiation has the potential to shorten and streamline cell developmental programs. We have established stable iPSC cell lines with transient induction of exogenous LHX3 and ISL1 from the Tet-activator regulatory region and have demonstrated that induction of the transgenes is not sufficient for the development of mature MNs in the absence of neuron patterning factors. Comparative global transcriptome analysis of MN development from native and Lhx-ISL1 modified iPSC cultures demonstrated that the genetic manipulation helped to streamline the neuronal patterning process. However, leaky gene expression of the exogenous MN master regulators in iPSC resulted in the premature activation of genetic pathways characteristic of the mature MN function. Dysregulation of metabolic and regulatory pathways within the developmental process affected the MN electrophysiological responses.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2026

Identification of Differential Equations by Dynamics-Guided Weighted Weak Form with Voting

In the identification of differential equations from data, significant progresses have been made with the weak/integral formulation. In this paper, we explore the direction of finding more efficient and robust test functions adaptively given the observed data. While this is a difficult task, we propose weighting a collection of localized test functions for better identification of differential equations from a single trajectory of noisy observations on the differential equation. We find that using high dynamic regions is effective in finding the equation as well as the coefficients, and propose a dynamics indicator per differential term and weight the weak form accordingly. For stable identification against noise, we further introduce a voting strategy to identify the active features from an ensemble of recovered results by selecting the features that frequently occur in different weighting of test functions. Systematic numerical experiments are provided to demonstrate the robustness of our method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Dynamics and heterogeneity of a fate determinant during transition towards cell differentiation

Yan is an ETS-domain transcription factor responsible for maintaining Drosophila eye cells in a multipotent state. Yan is at the core of a regulatory network that determines the time and place in which cells transit from multipotency to one of several differentiated lineages. Using a fluorescent reporter for Yan expression, we observed a biphasic distribution of Yan in multipotent cells, with a rapid inductive phase and slow decay phase. Transitions to various differentiated states occurred over the course of this dynamic process, suggesting that Yan expression level does not strongly determine cell potential. Consistent with this conclusion, perturbing Yan expression by varying gene dosage had no effect on cell fate transitions. However, we observed that as cells transited to differentiation, Yan expression became highly heterogeneous and this heterogeneity was transient. Signals received via the EGF Receptor were necessary for the transience in Yan noise since genetic loss caused sustained noise. Since these signals are essential for eye cells to differentiate, we suggest that dynamic heterogeneity of Yan is a necessary element of the transition process, and cell states are stabilized through noise reduction.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2015

Exact-Differential Large-Scale Traffic Simulation

Analyzing large-scale traffics by simulation needs repeating execution many times with various patterns of scenarios or parameters. Such repeating execution brings about big redundancy because the change from a prior scenario to a later scenario is very minor in most cases, for example, blocking only one of roads or changing the speed limit of several roads. In this paper, we propose a new redundancy reduction technique, called exact-differential simulation, which enables to simulate only changing scenarios in later execution while keeping exactly same results as in the case of whole simulation. The paper consists of two main efforts: (i) a key idea and algorithm of the exact-differential simulation, (ii) a method to build large-scale traffic simulation on the top of the exact-differential simulation. In experiments of Tokyo traffic simulation, the exact-differential simulation shows 7.26 times as much elapsed time improvement in average and 2.26 times improvement even in the worst case as the whole simulation.

Hanai, Masatoshi↗
DOE OSTI2017

Differential and Integral Data Evaluation for Titanium: An Application to Criticality Safety

The intent of this work is to highlight the role of differential and integral data evaluation to address issues in connection with criticality safety applications. Demonstration is made by using, as an example, differential data measurements and evaluation, and the benchmark integral experiments for titanium. Energy differential data are measured analyzed and evaluated to produce nuclear data libraries for criticality safety applications. Alternatively, integral experiments are performed at critical facilities, small experimental reactors, and play an important part in the validation of the differential nuclear data. The demonstration provided here for titanium gives its importance to criticality safety. Titanium is an effective neutron absorber that serves as baseline material for chemical separation in high-activity waste solutions in US. Titanium has not been considered for use in nuclear applications such as reactor design and analysis. Rather, it appears as a structural material that may be present in fuel cycle facilities or canisters for transport and disposition of nuclear waste. Criticality safety evaluations of systems in which titanium is present require an understanding of the nuclear data and its uncertainty.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2022

Fully differential rail-to-rail output amplifier with inverter-based input pair

A fully differential rail-to-rail-output amplifier includes a differential input inverter pair, folded cascode pair, class AB control pair, and class AB output rail-to-rail pair. A drain associated with the folded cascode pair is operatively coupled to the class AB control pair, and the drain associated with the folded cascode pair is unconnected to the current source associated with the class AB control pair. A method of providing fully differential rail-to-rail-output amplification includes coupling a folded cascode pair operatively to a differential input inverter pair, coupling a drain associated with the folded cascode pair operatively to a class AB control pair, and coupling a class AB output rail-to-rail pair operatively to the class AB control pair.

Dabrowski, Mieczyslaw M.↗
DOE OSTI2020

Whose Gas is it anyway? Differentiating the Source of a Large Soil Vapor Plume beneath Two Adjacent Waste Sites - 20487

DOE contractor CH2M Hill Plateau Remediation Company is currently responsible for conducting groundwater contamination monitoring at several RCRA treatment, storage, and disposal units located on the Hanford Site in Richland, Washington State. The Nonradioactive Dangerous Waste Landfill treatment, storage, and disposal unit presents a distinct groundwater monitoring problem because of a large multi-contaminant soil vapor plume beneath it that is a likely source of low-level volatile organic compound groundwater contamination. Adjacent to Nonradioactive Dangerous Waste Landfill is the Solid Waste Landfill. Volatile organic compounds are inventory components of both the Nonradioactive Dangerous Waste Landfill and the Solid Waste Landfill. Therefore, it is possible that both sites could be contributing to the soil vapor plume. For regulatory purposes, it is important to differentiate which site is the primary contributor of volatile organic compounds to the plume. An approach was developed to identify the primary volatile organic compound source of the soil vapor plume beneath Nonradioactive Dangerous Waste Landfill and Solid Waste Landfill. The site conceptual model hypothesis of vapor-phase volatile organic compound transport to the dissolved phase in groundwater was tested by a simple mathematical model of vapor/liquid equilibrium concentrations at the groundwater/air interface. Once it was shown that vapor-phase volatile organic compound transport to groundwater was a valid conceptual model for Nonradioactive Dangerous Waste Landfill and Solid Waste Landfill, spatial and statistical methods were used to determine the primary site contributing to the majority of volatile organic compounds to the soil vapor plume. Average groundwater chloroform, tetrachloroethene, and trichloroethene concentrations from Nonradioactive Dangerous Waste Landfill and Solid Waste Landfill monitoring network wells were plotted on maps of the facilities and immediate vicinities and compared to soil vapor sampling probe locations. Principal component analysis and mixing ratios were used to identify source contributions of each treatment, storage, and disposal unit to the plume. Results of the vapor/liquid equilibrium concentrations mathematical model showed that transport phenomena outweigh steady-state equilibria. Estimated vapor/liquid equilibrium concentrations were considerably lower than soil vapor measurements. The results indicate that dynamic vadose zone and groundwater factors such as decreased vapor concentrations with depth, vapor dilution from dispersion in the vadose zone, and advective and diffusional volatile organic compound dilution in groundwater result in groundwater volatile organic compound concentrations much less than would be measured under steady-state equilibrium conditions. Site source contribution differentiation by principal component analysis and mixing ratios was inconclusive using actual soil gas data because of the similarity in concentration values in both datasets for Nonradioactive Dangerous Waste Landfill and Solid Waste Landfill. Similar data populations suggest mixing of the vapor contributions from both sites by dispersion through the soil matrix pore spaces. However, when groundwater volatile organic compound data were compared between the Nonradioactive Dangerous Waste Landfill and Solid Waste Landfill monitoring networks, Solid Waste Landfill mean concentrations were higher, suggesting more vapor-phase volatile organic compound transport to groundwater at those locations. Simulated volatile organic compound soil vapor and groundwater datasets created to test the methods developed for this study show that the method can be successful in source differentiation when significantly different datasets are compared. This paper will describe a method of testing a conceptual model for vapor-phase contaminant transport to groundwater and for differentiating site sources of contaminants comprising a mixed-constituent soil vapor plume. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2015

Stochastic Differential Games for Fully Coupled FBSDEs with Jumps

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. The upper and the lower value functions are defined by the doubly controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in Buchdahn (Stocha Process Appl 121:2715–2750, 2011), we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower second order integral-partial differential equations of Isaacs’ type combined with an algebraic equation, respectively. Furthermore, for a special case (when σ and h do not depend on (y,z,k)), under the Isaacs’ condition, we get the existence of the value of the game.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗