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The Round Hill gold-scheelite deposit is located three kilometres from Macraes Flat and approximately 60 kilometres due north of Dunedin, New Zealand. A mining feasibility study is currently being undertaken on the deposit by BHP Minerals (New Zealand) Limited, who are the managers and operators of a Joint Venture with Home Reef Mining Limited. BHP has a 70 per cent equity in the project area.

Jan 1, 1987

By Woodcock J. T

During the dissolution of gold in fully aerated cyanide solutions the CN ion is the active agent, and consequently the usual method of evaluating the dissolving power of such solutions is to determine the CN- concentration analytically. The presence in the solutions of copper and/or zinc which themselves form complex cyanides, may result in erroneous conclusions as to the free cyanide ion concentration determined by titration.In this paper the theory of titration of such solutions has been developed, and potentiometric methods have been used to follow the behaviour of complex cyanides of copper and zinc. The results for cuprocyanides show Cu (CN)s to be the most important copper complex, and in addition they explain the reported low rates of solution of gold in copperbearing solutions which when titrated in the usual way, appear to have adequate free cyanide. An explanation of the misleading value of the titration is given, and suggestions for recognition of the condition are offered. Preliminary results for zinc-bearing solutions show that there is no satisfactory end-point for the titration of CN- in excess of that in the complex Zn (CN) 4=, which dissociates to CN and Zn(CN)g-.

Jan 1, 1950

By Stitt P. H

In Australia, the term "heavy mineral deposits" is used to denote sands which contain variable but economic quantities of the minerals rutile, ilmenite and zircon, often with minor amounts of monazite and xenotime as co-products. Australia has been a significant producer of rutile, ilmenite and zircon from such sands for fifty years and has long been the leading producer of rutile and zircon. The industry had its beginnings with the mining of present day beaches in northern NSW in 1934. Early work was small scale and in some cases involved hand winning of naturally concen- trated mineral for direct export. During World War II and immediately after the industry became increasingly mechanised and by the 1970s on the east coast operated between Newcastle, NSW north to Fraser Island in Queensland. Deposits worked included present day beaches, fossil strand lines up to 5 km or more inland from the present coast and aeolian concentrations. In the mid 1950s, operations commenced on the west coast of WA, south of Perth. Western Australian mineral sands tend to be rich in ilmenite as compared to the east coast. However the WA ilmenites are lower in chrome oxide and make a better syn- thetic rutile feedstock.

Jan 1, 1988

By Jillian Spindura

This work investigates the carbothermic reduction of the complex sulphide mineral, pentlandite, in the presence of lime. The effects of change in temperature and the use of different carbon sources are studied and the role of these variables on the reducibility of pentlandite mineral is discussed. The microstructures of the heat-treated pellets are examined using X-ray diffraction and scanning electron microscopic techniques. The latter analytical method also allows any variation in the morphology and composition of the metallic phases formed, with temperature, to be considered. The kinetics of the reduction process is studied and the magnitude of the overall activation energy is determined from the plots of percentage reduction versus time.

Jan 1, 1996

By Avnulf Muan, P. V. Riboud

The effect of Cr2O3 on melting relations of iron oxide at oxygen pressures slightly above those prevailing in contact with metallic iron has been determined. Liquidus and solidus temperatures of wüstite are raised by the addition of Cr2O3, from 1385°C for pure iron oxide to 1420°C at the peri-ledie point where wüstite, iron chromite, liquid, and gas coexist in equilibrium under the atmospheric conditions used in the present investigation (CO2/H2 = 1). It has been shown in a previous investigation1 that the liquidus and solidus temperatures of iron oxide are drastically increased by the addition of Cr2O3 under the relatively oxidizing conditions prevailing in air. This is to be expected in view of the easy substitution of cr3+ for Fe3+ in crystalline oxide structures which are stable at high oxygen pressures, as demonstrated for instance by the completeness of the solid solution series Fe2O3-Cr2O3 at high temperatures (>1000°C). The stable phase of iron oxide at oxygen pressures approximating those prevailing when metallic iron is present is wüstite. This phase, although commonly represented by the simplified formula FeO for sake of convenience, in reality has a defect structure in which a part of the iron ions are present as Fe3+. It is to be expected that at least a limited amount of cr3 ions will substitute for Fe in this structure when Cr2O3 is added to wüstite. Depending on the ease of this substitution in comparison with that of cr3 substitution for Fe in iron oxide-Cr2O3 liquids under the same atmos- pheric conditions, the liquidus and solidus temperatures of wüstite may increase or decrease upon addition of Cr2O3. The methods of establishing these melting relations and the results obtained are presented in the following. I) EXPERIMENTAL METHOD 1) General Procedure. The quenching technique was used in this study. Oxide mixtures were heated at chosen constant temperatures in an atmosphere consisting of equal parts of CO2 and H2 until equilibrium was established among gas and condensed phases. The samples were then quenched in mercury to room temperature, and the phases present were determined by microscopic and X-ray investigation. 2) Preparation of the Mixtures. Starting mixtures were prepared by weighing and thoroughly mixing Fe2O3 and Cr2O3, or wüstite and Cr2O3. The Fe2O3 and Cr2O3 used were "Baker Analyzed" reagent grades, whereas wüstite of composition was synthesized by reducing Fe203 at 1325°C in an atmosphere consisting of equal parts of CG andHz. In order to avoid contact with crucibles, the equilibration runs were made with small pellets suspended from thin (-0.004-in.) platinum wires. Mixtures made from FOs and Cr203 were readily pelletized at a pressure of 500 atm in a 9.5-mm die, whereas mixtures made from wüstite and Cr203 when treated identically were too weak for handling. In order to obtain a stronger product, these pellets were sintered by heating for 2 min at -1200°C in the same atmosphere as was subsequently used for the equilibration runs (CG/H2 = 1:1). During the sintering, these pellets were wrapped in thin (0.0004-in.) platinum foil, which was stripped off before the equilibration runs. In order to reduce the time needed for equilibrium to be attained, preliminary equilibration of relatively large pellets of each mixture was carried out at -1200°C for 4 to 12 hr. Smaller pellets

Jan 1, 1964

By D. J. Murdock

Cominco Ltd. has developed a number of innovative technologies for electrowinning. The paper discusses the application of these technologies to the copper industry. Tankhouse and cell design will be discussed in addition to anode cleaning and cathode handling. Specific examples of the application of this technology to the copper industry are cited.

Jan 1, 1987

By K. V. Raj, C. A. Berry, A. T. Weakley, A. L. Miller, P. R. Griffiths, D. A. Parks

Diesel particulate matter (DPM) has been classified as a carcinogen to humans by the International Agency for Research on Cancer. As a result of its potential carcinogenic nature, DPM exposure is regulated by the Mine Safety and Health Administration. Currently, diesel emissions in the workplace are monitored by collecting the aerosol onto filters, which are then sent to a laboratory for thermal-optical analysis using the NIOSH method 5040. This process can take days or even weeks, and workers can potentially be exposed to excessive levels of DPM before the problem is identified. Moreover, the delay involved in getting the loaded filter to the lab inevitably means the loss of some of the more volatile organic carbon. To remedy this delay, researchers from the National Institute for Occupational Safety and Health are seeking to develop a field-portable, real-time method for measuring elemental and organic carbons in DPM aerosols. In the current study, the use of mid-infrared spectrometry was investigated. It is believed that mid-infrared spectroscopy is more suitable for use in a real-time field-portable device than thermo-optical analysis methods. This article presents a method for measuring organic carbon (OC) and elemental carbon (EC) in DPM for a broad range of OC/EC ratios. The method has been successfully applied to laboratory-generated and mine samples.

By H. L. Barwood

Luminescence excited by electron bombardment, or cathodoluminescence (CL), is a very useful tool in petrographic examination of rocks, minerals and ores. Differences in luminescence colors and intensities allow detection and delineation of mineral species and composition zones difficult to separate visually by normal polarized light microscopy. The cold cathode CL chamber at Indiana University is used with a binocular microscope that has interchangeable attachments for a normal film camera, a digital camera and a fiber optic spectrometer. This allows imaging of the specimens as well as limited spectrographic analysis of the emitted luminescence. The chamber on this CL unit is large enough to accommodate several thin sections, polished mounts, grain mounts or rock slices at one time. Luminescence petrographic techniques were used to study alkali syenites and carbonatites from the Arkansas. Alkali feldspars in the syenites reveal a history of growth, fenitization and replacement through color and intensity differences. Primary orthoclase luminesces bright blue and is often zoned with a duller blue overgrowth of microcline. Fenitization of the feldspars is demonstrated by dull to bright red luminescence and replacement veins of albite often have a bright yellow green luminescence. Late microcline and perthites from the syenite pegmatites have a dull yellow to orange luminescence. Easily visible on this background of feldspars are grains of apatite (yellow), fluorite (violet), zircon (greenish), sodalite (orange) and mosandrite (green). Complex pseudoleucite syenites from Magnet Cove, Arkansas show similar minerals but also include xenolithic quartz (violet), manganwollastonite (bright orange), benitoite (intense blue) and lorenzenite (green). Carbonatites from Arkansas contain a wide variety of calcite textures and intensities of luminescence. Grains of carbonatite in well cuttings are easily distinguished from sedimentary calcites by their complex “flame” luminescence and the frequent presence of euhedral apatite grains (yellow). Apatite crystals separated from the Arkansas River Valley carbonatites by acid digestion show complex zoning. The cores of these apatites are yellow luminescing due to CL activation by Mn enrichment while the rims are violet luminescing due to REE enrichment. Complex zonation between the cores and rims is easily visible using CL.

Jan 1, 2001

Jan 1, 1880

By C. Heath

US coal consumption reached new heights in 1985, but production was constrained by excess consumer inventories. Domestic consumption of coal reached a record 742 Mt (818 million st) during the year. This is a 3.4% gain over the previous consumption high 718 Mt (791 million st) in 1984. An additional 84 Mt (92.7 million st) of US coal were shipped abroad in 1985, up 13.7% from exports a year earlier. Coal production in 1985 totaled 804 Mt (886 million st), 1.1% below the record 813 Mt (896 million st) produced in 1984. The slight decline in coal output is attributed primarily to a 13% drawdown of large consumer stockpiles built up in 1984 in anticipation of a miners strike that never occurred. Coal demand, and hence production, were also restrained by slower economic growth in 1985 and additional gains in nuclear generation of electricity. Domestic market requirements for coal in 1985 were mixed. Electric utility consumption rose 4.4%; coke plants consumed 6.7% less coal; coal requirements in the other industrial sector expanded by 2.1% and consumption in the residential and commercial sector fell 14.8%. The 2.2% increase in electric power generation in 1985 resulted in a 4.4% rise in utility coal consumption, totaling 630 Mt (694 million st) for the year. Twenty-nine of 45 states where coal-fired electric power plants were operating shared in the 26.7 Mt (29.5 million st) increase in 1985. Coal receipts at utilities fell by 16 Mt (18 million st) as these facilities worked off 21 Mt (23.4 million st) from stockpiles during the year. Electricity generation at coal-fired power plants similarly climbed 4.4% to 1.4 trillion kWh in 1985. This gave coal a 56.8% share of electricity generated from all energy sources compared with 55.5% in 1984. The average price of coal delivered to these facilities remained essentially unchanged during the year. It declined on 1.5% from $38.26/t ($34.71 per st) during the fourth quarter of 1984 to $37.70/t ($34.20 per st) in the fourth quarter of 1985. Coal consumption at coke plants continue to reflect a depressed steel industry. After rebounding 18.9% in 1984 for the first time in five years, coal consumption at coke plants again declined 6.7% to 37 Mt (41.1 million st) in 1985. Coke consumption 26.5 Mt (29.3 million st) exceeded production, 26 Mt (28.7 million st), in 1985. This resulted in a 1-Mt (1.2-million st) decline in coke stocks at producer and distributor levels during the year. The average price of coal delivered to coke plants decreased 7.6% from $64.30/t ($58.34 per st) during the final quarter of 1984 to $59.42/t (53.91 per st) in the last quarter of 1985. Coal consumption in other industries, including all manufacturing and industrial consumers except coke plants, totaled 68.3 Mt (75.3 million st) in 1985. This was the highest level attained in more than a decade. But the lower percentage rise last year (2.1% compared to 11.8% in 1984) reflected slower growth in the nation's economy in 1985. The average price of coal delivered to industrial firms declined 4.5% from $42.32/t ($38.39 per st) in the fourth quarter of 1984 to $40.39/t ($36.65 per st) in the last quarter of 1985. Coal-producing states in Appalachia mined 9.8 Mt (10.8 million st) tons less in 1985 than in 1984 for a 2.4% loss. The largest contributors to the region's production decrease to 393 Mt (433 million st) were Pennsylvania, down 14.7% to 60 Mt (66 million st); Ohio, down 8.3% to 32.6 Mt (36 million st); and West Virginia, down 2.9% to 115 Mt (127 million st). Conversely, eastern Kentucky production was up 4.8% to 111 Mt (122.8 million st) in 1985, while production in Tennessee climbed 9.7% to 40 Mt (44 million st). No coal was mined in Georgia last year. Coal production in the interior region in 1985 declined 3.9% to 173 Mt (190.4 million st). Substantial production losses occurred in Indiana, down 8% to 31 Mt (34.5 million st); and in western Kentucky, down 7.9% to 36 Mt (40 million st). Coal production in Texas rose 5.7% to 39 Mt (43.5 million st) in 1985. Louisiana began mining lignite 140 kt (151,000 st) during the year. Production in the western region increased 3.4% to 238 Mt (262.5 million st) in 1985. Wyoming, up 5.3% to 125 Mt (140 million st); and North Dakota, up 16.5% to 23.5 Mt (26 million st) contributing the largest amounts. Production gains were also shown in Washington, Utah and, notably, in Alaska where coal output climbed 72.9% to 1.3 Mt (1.5 million st) in 1985. Most of the additional coal produced in Alaska last year was shipped abroad. Coal stocks held by consumers totaled 154 Mt (170 million st) at the end of the year. This was 27 Mt (30 million st), or 13.7% below the inventory level at the end of 1984. Stocks held by producers and distributors declined by 868 kt (957,000 st), or 2.8% in 1985 to 30 Mt (33 million st). A drawdown in stockpiles occurred in each consumer sector during 1985. Stocks held by electric utilities totaled 142 Mt (156 million st) at the year's end, 13% lower than stock levels at the end of 1984.

Jan 5, 1986

By H. A. Watson

The introduction and use of synthetic materials in various applications may lead to potential hazards from toxic, gaseous, decomposition products if these materials are subjected to conditions abnormal to their intended use, such as high temperatures or burning from accidental causes. This report covers an investigation of such possible hazards from toxic gases produced by the burning or thermal decomposition of some low-density, expanded, synthetic materials, three of which consisted of polyvinylchloride and one of acrylonitrile. In the production of certain expanded plastic materials, Chemical blowing agents frequently are used to produce a cellular structure. One of these blowing or foaming agents contains azo-bis-isobutyronitrile, which is designated in this report as PN. Information indicates that a product of decomposition of this blowing agent, formed in the blowing operation, is tetra-methyl-succino-nitrile, designated as TSN, Which is significantly toxic. Decomposition of the blowing agent, with release of its azo nitrogen, is represented by the following reaction:

Jan 1, 1951

By R. A. Gee, G. B. Ashton, S. T. Warren, P. N. Groves

Northside Storage Tunnel provides 500 megalitres (Ml) (130 million gallons) of storage capacity to capture stormwater/sewage overflows into Sydney Harbour. The6.6 m diameter TBM driven tunnel traverses Middle Harbour 85 m (280 ft) below sea level and through a sandstone/shale rock mass, highly disturbed by stress relief, 20 m(65 ft) beneath the base of the sediment filled palaeochannel. Target limits for inflow, groundwater drawdown and settlement of the valley beaches were achieved by systematic forward grouting from the TBM. Support included rockbolts, sets and shotcrete to carry rock loads, control block instability and spalling and to provide durability and serviceability. This paper describes the tunnelling and grouting works and their effectiveness.

Jan 1, 2001

By W. Lin, M. J. McPherson

When a large area of open gob collapses suddenly, a windblast is produced that can cause considerable damage throughout the infrastructure of a mine. In a few cases, the windblast has been accompanied by ignitions of methane and/or coal dust. This paper describes the conduct and results of laboratory tests involving the compression of mixtures of air, methane and coal dust under a falling weight and while allowing controlled escape of the mixture. The transient responses on pressure and temperature sensors were recorded. In addition to an analysis of those records, the paper highlights those conditions in which ignitions occurred.

Feb 24, 1997

The mineralogical composition of the ore was determined and the mineral phases with which nickel is associated were identified. The ore was prepared by grinding to -420 micrometres and classification into granulometric fractions. Prior to the main series of experiments the effect of agitation rate on nickel recovery was investigated. The influence of sulphuric acid concentration, temperature and particle size on the kinetics of nickel extraction was then studied, both quantitatively and qualitatively, over fixed value intervals. Application of the initial rate method enabled determination both of the order of reaction with respect to sulphuric acid concentration and the activation energy for the dissolution of nickel. The results indicate that a physical mechanism can be ruled out as the rate-determining step in the extraction of nickel from the ore and that a chemical mechanism is more likely

Dec 1, 1991

By R. August Heindl

A process for recovering manganese from open-hearth steel-plant slags was investigated. The proposed method of accomplishing this is indicated by the flow diagram in figure 1. Mixtures of open-hearth, steel-plant flush slags and limestone, so proportioned as to have molar [CaO -3P205] ratios of approximately 2, were fired at approximately 2,400° F. in an oxidizing atmosphere and then cooled slowly. This treatment produced a clinker consisting principally of two solid phases comprised of (1) various combinations of the oxides of manganese, iron, aluminum, and magnesium, probably in spinel complexes, and (2) a dicalcium silicate phase, which contains the phosphorus. Reduction of these clinkers with hydrogen or with gaseous mixtures containing CO, CH4, and other hydrocarbons at about 1,300° F. formed manganous oxide as well as varying amounts of ferrous oxide and iron. Carbon monoxide and mixtures of hydrogen and carbon monoxide effectively reduced the clinkers but also gave higher proportions of soluble iron when the manganese was extracted from the reduced clinkers by leaching at about 80° F. with ammonium carbonate solutions containing 26 to 28 percent ammonia and 15 to 26 percent carbon dioxide. It is believed that the manganese is in solution as an ammonia-manganese complex. Concentrations of about 90 grams of manganese per liter were obtained by batch countercurrent procedures. By distilling 50 percent of the ammonia, the ammonia-manganese complex was decomposed, and approximately 90 percent of the manganese was precipitated as the carbonate with the simultaneous precipitation of 15 to 30 percent of the iron, the larger amounts of iron precipitating in the continuous tests. The manganese-iron ratio in the product could be increased either by recycling more manganese from the distillation to the extraction or by aerating the liquor to purge it of iron before, rather than after, the distillation step. Calcination of the pelletized manganese carbonate in an oxidizing stmosphere at 2,000° F. or above gave a final product, Mn304, resistant to crushing and apparently possessing the other physical properties essential in a ferromanganese blast-furnace feed.

Jan 1, 1955

By Tetsuro Ohashi, Tooru Matsumiya, Yoshio Abe

"The three-dimensional deformation of a continuously cast slab was modeled into two-dimensional deformation by developing the solidified shell of the slab. The developed solidified shell was divided into rectangular elements. The thermal strain produced as the slab is wi thdrawn over one roll spacing in the casting direction was calculated from the temperature change during that time and the increment of the geometrical unbending strain during the time was calculated from the change in the radius of curvature of the slab by assuming that the neutral axis of bending is located at the mid-thickness of the slab and that the transverse section of the slab is kept in a plane during unbending. These strains were imparted to each element as initial strains. The interaction between the elements and the overall deformation of the solidified shell were analyzed by the finite element method.The analysis was performed on a slab being cast on a machine of the circular arc type with progressive unbending. The following findings were obtained as a result. The effect of unbending is spread in the upstream and downstream directions of the slab by the shear deformation of the narrow and wide faces. As the shear deformation increases, the strain in the unbending zone decreases and the strain outside of the unbending zone increases. The effects of secondary cooling conditions, slab size, radius of caster circular arc section, compression casting, etc., were also analyzed. IntroductionTo develop a method for preventing the internal cracking of continuously cast slabs, it is important to acurately analyze the deformation of the slab and this subject has been extensively investigated. Many researchers, including the present authors, have analyzed slab bulging. (1)- (11) Analysis has also been made of thermal deformation that assumes importance in continuous casting of small-section blooms.(12)-(18) However, unbending deformation, an extremely important subject for continuous casters of the circular arc type, has not been fully analyzed."

Jan 1, 1986

By Tristan Worsey, Nathan Rouse, Kyle Hall

Drilling accuracy is the foundation to any successful blast operation. Even the best blasting practices cannot overcome poor drilling and still obtain desired results. Inaccuracies in boreholes can result in flyrock, increased vibration, uneven floor, and other undesired effects. As depth increases, variation in actual burden and spacing is expected even if collars are located precisely. Problems arise when this deviation reaches a point where it adversely affects the function of the blast design. Not only can unsafe conditions such as insufficient burden at the face develop, but poor fragmentation can compound adverse effects. As with any troubleshooting, problems cannot truly be fixed if they cannot be measured. As a result, significant work has been done in the past on this subject. Currently, there are both cheap and expensive methods used to quantify deviation for a given borehole. However, implementation of these techniques is limited. In general production blasts, little is known about the variance in designed versus true borehole geometry. This is especially true if the blaster does not see telltale signs of drilling deviation. However, losses in blast performance may exist and provide an opportunity to obtain better fragmentation and reduce fines. As drill and blast operations strive for better optimization, greater focus is being placed on drilling accuracy and solutions to this problem. The following paper details the use of several current methods for deviation measurement in order to help prevent compromises of safety and improve overall blast performance.

Jan 1, 2018

By T. Gupta, T. Ghosh

Rare earth elements (REEs) and their compounds are essential in a multitude of mundane and sophisticated applications such as renewable energy technologies, catalysts, medicine, and defense applications. The global supply deficit could be attributed to the destabilization of REE market that has, as a result, incentivized indigenous production of REEs. Cost-effective and environmentally benign opportunities are being explored for the production of REEs from non-conventional mineral sources including coal and coal by-products to stabilize the supply chain. Coal from Healy coal mine and Wishbone Hill deposits in Alaska have been quantified for REEs and were found to possess elevated total REE concentrations as compared to world average levels (60– 70 ppm). The objective of the study was to concentrate the REEs in the froth phase by froth-flotation of the fine size fraction (< 75 μm). A Box-Behnken design of experiment was employed to ascertain the main and interaction effects of operational parameters on response variable by analyzing 3D response surfaces. Experiments were carried out to maximize REE recovery and concentration by modeling and optimization of three operating variables (frother dosage, pulp density (% solids), and collector dosage). Based on the results, the optimum flotation conditions for maximum recovery at an elevated concentration of REEs in the froth fraction was obtained at 4.2% solids and 32.7 ppm of frother dosage for Healy sample and 10% solids and 37.9 ppm of frother dosage for Wishbone Hill coal sample. The quadratic relationship between the input and response variable was found to be independent of collector dosage suggesting good surface hydrophobicity and floatability of the two coal samples.

By H. M. Urban

This paper, one of a series being prepared for and published by the United States Bureau of Mines on methods and costs of mining at various mica mines in the United States, describes the methods now in use in the principal mica mines of the Southern Appalachian States. Those papers are designed to disseminate technical information regarding the methods used. The cost tabulations represent operating expenditures only and not total costs. It is recognized that publication of total costs might in some instances cause embarrassment to individual producers as well as to the industry as a whole. On the other hand, operating costs are essential to the technical discussion and study of methods employed. The attention of the reader is specifically called to this differentiation in order that no misunderstanding of the scope of the cost tabulations shall ensue.

Jan 1, 1932

By J. Killian, S. Cox

Many questions exist within the geotechnical industry as to the applicability of Strength Reduction Factor (SRF) values to the design acceptance criteria (DAC) of mining operations. There are many factors that contribute to changes in the SRF estimate, some of which are numerical (e.g. dilation angle) and some of which are geometric (e.g. model grid size)[1]. Numerous studies have confirmed equivalent values of SRF and FOS are found for soil slopes of limited height as presented in, [2] and [3]. However, significant deviation of the SRF value was observed for highly cohesive materials (rock slopes) of heights typically encountered in mining, [4]. This paper builds on the work presented in [4], providing a more rigorous examination of the modeling parameters that control the SRF estimate. The failure mechanism assumed for this study was a simple circular shear surface in a uniform material. Both LEM and numerical methods are suited well for analysis of this failure mechanism. A parametric study was completed to evaluate and test the relationships of key numerical variables that affect the estimated value of the SRF using Itasca’s Fast Lagrangian Analysis of Continua (FLACTM) software. The effect of groundwater has been tested and may be significant but is not presented as part of this body of work for brevity and simplicity of presentation.

Mar 2, 2022