SummaryThere are alternative definitions for the Riemann integral, many of which avoid some of the unpleasant computations that arise when using Riemann sums. In this version a simple distance function for step functions is used and the Riemann integral is defined and developed by employing exclusively “convergent” sequences of step functions. While only modestly different from the standard presentation it might have some extra intuitive appeal. This is a common device in advanced theories of integration and can be introduced at this elementary level.

When students first encounter an example of a continuous nowhere differentiable function in a real analysis course, what do they visualize? It is not easy to visualize an infinite sum of functions when the partial sums get increasingly complicated. We offer a geometric approach via the graph of such a function. Our theme is based on the fact that, if the graph of a function intersects all non-vertical lines in a special way, then that function cannot have a derivative at any point. We will require that at each point P on the graph G of the function there must be many lines L through P having P as a limit point of the intersection G & cap; L . This picture is easy to imagine but impossible to represent graphically: it avoids all of the computational complications that faced nineteenth-century mathematicians when they first attempted to describe these functions.

The Kupol epithermal Au-Ag vein district is located in the northern part of the Okhotsk-Chukotka volcanic belt, a Late Cretaceous subduction-related continental volcanic arc exposed for >3,000 km along the eastern coast of Russia. High-grade veins are hosted in the Kupol andesite sequence, a 300-to 1,000-m-thick, subhorizontal, layered sequence of andesite flows, sills, and ash tuffs, dated at 97 to 96 Ma (Cenomanian). The Kupol andes-ite sequence is underlain by mixed mafic-felsic volcanic units plus sedimentary rocks ("older volcanics") and overlain by a >1-km-thick "upper felsic" sequence of dacitic-rhyolitic tuffs and associated dikes and flow domes, dated at 95 to 85 Ma, with local sequences of fluvio-lacustrine sedimentary rocks.The epithermal veins occupy N-striking, steeply dipping normal faults that cut thick coherent andesite flows and sills in the central-upper part of the Kupol andesite sequence. The district is dominated by the large Kupol vein (180.7 tonnes (t) Au and 1,986 t Ag produced to 2020), hosted by the 5.5-km-long Kupol fault, which accommodates normal, east-side-down displacement of up to 190 m. The Moroshka and Providence veins, 5 km east-southeast of Kupol, occupy shorter faults (1-to 2-km strike) with smaller vertical displacements (to 70 m). The Moroshka vein is dated at 93.5 +/- 1.5 Ma (Turonian; 40Ar/39Ar method on adularia), and the timing of vein mineralization here and at Kupol overlaps with the early stage of upper felsic sequence magmatism. Veins contain subhorizontal ore shoots, controlled by the intersection of the steep faults with flat-lying Kupol andesite sequence stratigraphy and by steepening of the faults to a more dilational orientation as the inferred paleosurface is approached. Local structural controls are also evident, reflecting a component of oblique slip on the Kupol fault, with the thickest vein segments at steeply pitching jogs and relays. Main-stage veins grew via repeated encrustation by quartz-chalcedony +/- amethyst +/- lattice bladed calcite (replaced by quartz), with Au-Ag-bearing crustiform adularia +/- clays +/- sulfides/sulfosalts/electrum +/- chlorite +/- hematite bands. The main controls on Au grade are inferred to have been boiling, resulting in sharp vertical limits to high metal grades typical of epithermal veins, coupled with optimal dilation of the vein system where the hosting normal fault steepens near surface with decreasing differential stress. Although much of the displacement on the controlling faults is pre-mineralization in timing, lithified cataclastic breccia, coeval with some vein stages, and vein geom-etry patterns indicate that some vein development occurred contemporaneously during late normal displace-ment along the fault system. Waning of the hydrothermal system is marked by late carbonate fill, initially Fe dolomite, then coarse calcite as veins, matrix to vein breccia, and central vein cavity fill. The Kupol district veins have proximal adularia-quartz alteration (haloes meters wide), within an extensive (hundreds of meters in scale) clay alteration halo. Clays are zoned both vertically and laterally with respect to veins, with inner illite-chlorite that was magnetite-destructive (at highest paleotemperature; >220 degrees C), grading outward and upward to illite/interlayered illite-smectite with kaolinite, then to an outer zone (or upper blan-ket) of smectite, at lowest paleo-temperature (<150 degrees C). The boundary between the illite and smectite zones is interpreted to mark the interaction limit of paleo-hydrothermal systems with cooler groundwater. District-scale pathfinder element zonation correlates with clays, with S-Te-Bi-As in the illite-chlorite core and Sb-Cs-Tl(-As-Li) in the smectite blanket. Pathfinder zonation patterns at Kupol point to a magmatic source at depth or, more likely given the scale of the anomalies, multiple magmatic sources, with the surface clay zonation indicating the extent of coalesced paleo-hydrothermal systems associated with upflow plumes. This is the best-defined altera-tion record with geochemical signature for a complete district hosting a large, high-grade vein deposit. Early definition of clay and pathfinder element patterns across an entire epithermal district can be carried out at low cost to provide useful constraints on vein targeting.

High-grade epithermal Au–Ag veins of the Vodorazdelnaya district in far eastern Russia are hosted by Early Cretaceous volcanic rocks of the Tytylveem belt. The largest deposit is Zone 37, a 1-km long, northeast-striking vein that is up to 35-m wide, containing at least 32 t Au. The main shoot consists of crustiform—colloform banded quartz-chalcedony-adularia-sulfide veins, of low-sulfidation style. The vein is cut by late-mineral rhyolite sills and dikes. Zone 37 vein is dated at 117 ± 2 Ma, overlapping within error with the cross-cutting rhyolites (121–119 Ma) and with intrusions of the nearby Ilirney granitic pluton (119–117 Ma). The volcanic host rocks are dated at 121–115 Ma. At September Northeast, 15 km west-northwest of Zone 37, mineralisation consists of veins up to 4 m wide, with crustiform banded quartz ± chalcedony ± adularia ± chlorite, and Zn-Pb-Cu sulfides, of intermediate sulfidation style, in a 1-km-long, north-northeast striking trend. High grades are associated with gold dendrites in fine-grained quartz bands and with minor late-stage tellurides. The veins were disrupted by intrusion of rhyolite dikes and associated phreatic breccias. Some breccias host mineralisation as vein clasts and minor matrix sulfides. The Tytylveem belt is preserved below a regional unconformity that forms the base to the overlying Late Cretaceous (106–77 Ma) Okhotsk-Chukotka Volcanic Belt (OCVB). Intrusive rocks attributed to OCVB magmatism were emplaced at about 96 Ma and are associated with overprinting hydrothermal alteration, sub-economic veins and thermal resetting of some vein adularia ages at Zone 37, to 91–97 Ma.

We revisit the nineteenth-century version of the bounded convergence theorem formulated by C. Arzela in 1885 for Riemann integrable functions and, independently, by W. F. Osgood in 1897 for continuous functions.

This work offers detailed coverage of every important aspect of symmetric structures in function of a single real variable, providing a historical perspective, proofs and useful methods for addressing problems. It provides assistance for real analysis problems involving symmetric derivatives, symmetric continuity and local symmetric structure of sets or functions.

The derivative dy/dt of a function y(t) is the slope of the tangent line to that function at time t:

The Paracatu deposit in Brazil is a shallowly dipping, bulk-tonnage, low-grade, vein-style orogenic Au orebody hosted in very strongly deformed Neoproterozoic carbonaceous phyllite of the southern Brasília fold belt. At regional to district scales, the gold orebody lies along the eastern, hanging-wall edge of a major thrust of the ~630 Ma Brasiliano orogeny. This thrust cuts through a facies transition between clastic-dominated rocks of the Canastra Group and carbonate-dominant rocks of the Vazante Group, deposited at ~1000 Ma in a rift to passive-margin environment on the flank of the São Francisco craton. At the same scales, the footwall of this major thrust system hosts numerous structurally controlled zinc deposits including Vazante and Morro Agudo. At Paracatu, ore genesis occurred primarily by the formation of early tectonic quartz sulfide-carbonate veins, prior to substantial ductile deformation (boudinage), local physico-chemical reworking of these veins, and redistribution of some gold. Structural, geochemical, and isotopic data indicate a strong influence of the local rocks (cm to 100-m scales) on many ore ingredients, and the quartz and carbonate in ore veins were most likely derived locally (cm to m scales). However, the coassociation of gold and arsenic with the boudinaged veins and a major thrust, and the absence of metal enrichments normally associated with syngenetic metalliferous black shales, supports a model of far-field derivation of gold within this metasedimentary package (km to 10-km scales). Transport of metal-bearing fluids toward a favorable structural and chemical site during thrusting and orogenesis was possibly focused, during precipitation to ore grades, by the position of transverse structures in the basement, which also influenced deposition of the adjacent zinc deposits. Successful mining of the low-grade resource was initially favored by the subhorizontal orebody geometry and weathering characteristics, and subsequently by high production rates from the 100-m-thick mineralized zone.

The study of functions of generalized bounded variation (VBG) and generalized absolute continuity (ACG) that appears in Saks’s treatise Theory of the Integral can be thoroughly reworked by using some aspects of the theory of variational measures proposed originally by Ralph Henstock and extended by many others. We present a development of these concepts and use it for a characterization of the Denjoy-Khintchine integral.

Multiscale structural and geochemical studies have been applied to understand the genesis of the Paracatu deposit in Brazil, a shallow-dipping, bulk-tonnage, vein-style low-grade Au(-Ag-Pb-Zn) orebody hosted in ~1000 Ma black phyllites of the Paracatu Formation, associated with intense shearing accompanying thrusting during the ~680 Ma Brasiliano orogeny. Massive to laminated quartz-sulfide-carbonate veins and associated alteration and Au mineralization formed early in the deformation history, and these veins were boudinaged during subsequent progressive increase in shear strain. Geochemical profiles across the ore allow recognition of a relatively homogeneous protolith at 100-m scales with respect to Ti, Al, Zr, V, and rare earth elements, including a footwall with protolith attributes similar to those of the ore. Oxygen, hydrogen, and sulfur isotopes do not reveal a distinctive external fluid source; rather, they reflect fluid-rock equilibration with the host phyllites during greenschist facies regional metamorphism. Detailed geochemical sampling around a prominent population of smaller boudinaged veins shows that Si, Ca, and Sr were removed and Al, K, and Ti were residually concentrated during formation of synboudinage dark selvages. This process of mass transfer occurred at scales more local than the processes responsible for ore genesis. Despite the local processes of chemical equilibration and redistribution, many of the main ore components appear externally derived at scales broader than the orebody. Gold, As, Ag, Sb, Bi and, to some extent, Pb and Zn show a strong spatial and statistical correlation with boudinaged quartz-sulfide ± carbonate veins, and these are surrounded by a broader (10–50-m vertical scale) halo of enriched K, Ba, and volatiles (loss on ignition), with depleted Na and Sr. The majority of gold is present as inclusions in pyrite and arsenopyrite within and immediately surrounding the veins, and gold-bearing pyrite shows mineral chemistry consistent with a hydrothermal origin. The main population of large veins locally preserve preboudinage, complex internal vein textures, including sulfide-rich wall-rock laminae similar to laminated fault-fill veins in many orogenic-style vein deposits. These larger veins contain most of the gold in the Paracatu resource. Metals were most likely sourced distal to the Paracatu Formation and were precipitated in sheeted shear or tensile veins during the early stages of the Brasiliano thrusting. The oblate strain recorded by the boudinaged, mineralized veins is anomalous and distinctive to Paracatu at km scales, even though the thrusting is regional. Although the orebody may have been physically dislocated by thrusting from the place where the early veins formed, the anomalous strains recorded suggest a perturbation in the footwall to the regional thrust or, perhaps, a spatially restricted competency contrast within the stratigraphy, which contributed to the localization of the present mineralization during the thrusting.

The strong derivative is not, without some caution, a useful tool in the study of McShane's (i.e., Lebesgue's) integral. Even so, the underlying structure of that process of derivation is closely connected to the formulation of the Riemann sums definition that McShane gave for his integral. This article discusses some of the features and traps for the study of those connections.

Summary The usual definition of the Riemann integral as a limit of Riemann sums can be strengthened to demand more of the function to be integrated. This super-Riemann integrability has interesting properties and provides an easy proof of a simple change of variables formula and a novel characterization of derivatives. This theory offers teachers and students of elementary integration theory a curious and illuminating detour from the usual Rieman integral.

If a sum of the form \[\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1})\] is used without the familiar requirement that the sequence of points \(a=x_0, x_1, \dots, x_n=b\) is increasing, do we still get a useful approximation to the integral? With a suitable set of hypotheses the answer is yes. We give applications to change of variable formulas and the problem of characterizing derivatives.

If a sum of the formSigma(n)(i=1)f(xi(i))(x(i) - x(i-1))is used without the familiar requirement that the sequence of points a = x0, x1,..., xn = b is increasing, do we still get a useful approximation to the integral? With a suitable set of hypotheses the answer is yes. We give applications to change of variable formulas and the problem of characterizing derivatives.