Local Foliation Research Articles

Archean Phlogopite Peridotite from Gneissic Enderbites of the Bug Granulite–Gneiss Terrane (Ukrainian Shield): Compositional Features and Possible Mechanisms of its Formationn

The paper reports mineral chemistry, whole-rock major and trace element composition, structure, and geological setting of the phlogopite peridotite lens within the Bug granulite–gneiss terrane of the Dniester–Bug Province, Ukrainian Shield. Geochemical features of the studied peridotites and minerals indicate a complex evolution of the lens. The early stage is marked by the crystallization of olivine + spinel (Al-chromite) cumulus from melt of supposably picritic composition, and further crystallization of clinopyroxene. The melt shows signs of hybridization by host gneissic enderbite. Orthopyroxene crystallized later and frequently replaces clinopyroxene. The phlogopite cystallization is likely related to the fluid activity and the growth of potassium potential in the fluid. Granulite-facies metamorphism accompanied by ductile deformations affected the mineral asseblage and chemical compositions of peridotites and orthopyroxenites. At the final stage, the rocks and minerals experienced retrograde metamorphism and local foliation of the rocks.

Local space time constant mean curvature and constant expansion foliations

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in [23] to study local constant mean curvature foliations we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18].

Local foliations by critical surfaces of the Hawking energy and small sphere limit

Local foliations of area constrained Willmore surfaces on a 3-dimensional Riemannian manifold were constructed by Lamm et al (2020 Ann. Inst. Fourier 70 1639–62) and Ikoma et al (2020 Int. Math. Res. Not. 70 6538–68), the leaves of these foliations are in particular critical surfaces of the Hawking energy in case they are contained in a totally geodesic spacelike hypersurface. We generalize these foliations to the general case of a non-totally geodesic spacelike hypersurface, constructing an unique local foliation of area constrained critical surfaces of the Hawking energy. A discrepancy when evaluating the so called small sphere limit of the Hawking energy was found by Friedrich (2020 arXiv:1909.02388v2 [math.DG]), he studied concentrations of area constrained critical surfaces of the Hawking energy and obtained a result that apparently differs from the well established small sphere limit of the Hawking energy of Horowitz and Schmidt (1982 Proc. R. Soc. A 381 215–24), this small sphere limit in principle must be satisfied by any quasi local energy. We independently confirm the discrepancy and explain the reasons for it to happen. We also prove that these surfaces are suitable to evaluate the Hawking energy in the sense of Lamm et al (2011 Math. Ann. 350 1–78), and we find an indication that these surfaces may induce an excess in the energy measured.

Local foliation of manifolds by surfaces of Willmore type

We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constant mean curvature.

More of the bulk from extremal area variations

It was shown recently in (Bao N et al 2019 Class. Quantum Grav. 36 185002), building on work of Alexakis, Balehowksy, and Nachman (Alexakis S et al 2017 arXiv:1711.09379), that the geometry of (some portion of) a manifold with boundary is uniquely fixed by the areas of a foliation of two-dimensional disk-shaped surfaces anchored to the boundary. In the context of AdS/CFT, this implies that (a portion of) a four-dimensional bulk geometry can be fixed uniquely from the entanglement entropies of disk-shaped boundary regions, subject to several constraints. In this note, we loosen some of these constraints, in particular allowing for the bulk foliation of extremal surfaces to be local and removing the constraint of disk topology; these generalizations ensure uniqueness of more of the deep bulk geometry by allowing for e.g. surfaces anchored on disconnected asymptotic boundaries, or HRT surfaces past a phase transition. We also explore in more depth the generality of the local foliation requirement, showing that even in a highly dynamical geometry like AdS-Vaidya it is satisfied.

Origin of the high-grade Early Jurassic Brucejack epithermal Au-Ag deposits, Sulphurets Mining Camp, northwestern British Columbia

The high-grade (8.1 million ounces of gold; 15.6 Mt grading 16.1 g/t Au; Pretium, 2016) Brucejack epithermal Au-Ag deposits are located in the Canadian Cordillera of northwestern British Columbia, and formed in association with extensive early Mesozoic island arc magmatism. Porphyry-type Cu-Au-Mo mineralization occurs nearby at the Kerr-Sulphurets-Mitchell (196–190 Ma; Febbo et al., 2015), Bridge Zone (191.7 ± 0.8 Ma, 191.5 ± 0.8 Ma), and West Zone (188.9 ± 0.9 Ma) prospects. Gold-silver vein-type mineralization at Brucejack is hosted by variably altered and deformed Early Jurassic porphyritic latite lava flows, volcaniclastic rocks, and volcanic-derived sandstones, siltstones, and conglomerates. This study focuses on the Valley of the Kings Zone at Brucejack, where host rocks have been dated at 188–184 Ma (U-Pb, zircon). Post-mineralization (but not post-alteration) basaltic and trachybasaltic dikes that cut the veins are geochemically similar to the host volcanic sequence and follow the same structural trends of the mineralized vein corridors. We suggest these dikes are broadly coeval with mineralization, and that ore formation at the Valley of the Kings formed during the overall Jurassic tectonomagmatic event.Gold-silver mineralization is hosted by quartz-carbonate veins that cut sericitized and pyritized volcaniclastic rocks. Six stages of veining are defined, with Au-Ag mineralization (electrum) focused in stages III–V: stage I–III veins consist of quartz with minor carbonate, chlorite, sericite, and sulfide minerals; stage IV quartz veins contain more abundant base metal sulfides (pyrite, sphalerite, and galena with minor sulfosalts); stage V veins are dominantly calcitic; late post-mineralization stage VI veins are quartz-calcite and contain sparse pyrite and chlorite, but no electrum. The veins are interpreted to be syn- to late-tectonic, with deformation decreasing from locally penetrative in the host-rocks prior to veining (resulting in local foliation of sericite and pressure shadows around pyrite), to dismemberment and shearing of early stage I–II quartz veins, brittle disruption of stage III–IV quartz and stage V carbonate veins, and minimal deformation of post-mineralization stage VI quartz-carbonate veins. A much younger set of extensional muscovite veins locally cuts the deposit with Late Cretaceous apparent ages, and appears to be associated with a weak thermal overprint that has reset K-Ar and 40Ar/39Ar ages in sericite throughout the district.The deformation experienced by the vein system, particularly in the later stages (III–V), was not uniform, and original undeformed vein textures and fluid inclusion assemblages are locally preserved. We report fluid inclusion microthermometric data from carefully selected primary fluid inclusion assemblages from quartz, calcite, and sphalerite from vein stages III–IV, many of which show evidence for boiling (coexisting liquid- and vapor-rich primary fluid inclusions). Liquid-rich fluid inclusions from stage III and V veins have moderate homogenization temperatures (∼170 °C and ∼160 °C respectively) and salinities of 2–8 wt% NaCl equiv., whereas inclusions from base-metal-sulfide-bearing stage IV veins show evidence of mixing with a cooler, more saline brine (∼140 °C, ∼10–15 wt% NaCl equiv.). Carbon dioxide was observed as clathrate during cooling in some fluid inclusions, suggesting that minor amounts of CO2 were present in the fluids.Calculated oxygen isotopic compositions of fluids in equilibrium with quartz and calcite from vein stages III–V range from δ18Ofluid = −10.7 to +1.8‰, whereas δ13CCO2 ranges from −9.5 to −4.5‰, and δ34Spyrite ranges from −1.7 to +0.6‰. Taken together, these data suggest a magmatic source for S and some C, carried by a fluid of evolved or diluted magmatic origin, which variably mixed with meteoric-derived groundwater or seawater containing carbon of sedimentary (organic) origin. Mixing is supported by large salinity ranges of vein stage IV fluid inclusions, where individual assemblages range from ∼1 to 15 wt% NaCl equiv.These fluid temperatures, salinities, and isotopic compositions are typical of epithermal deposits where distal magmatic fluids mix with local heated groundwaters. The low fluid temperatures but evidence of boiling suggest formation at shallow crustal depths. This is despite the evidence for penetrative deformation in the sericitized volcanosedimetary host rocks, which we attribute to rapid uplift immediately prior to or during the earliest stages of mineralization. The vein system is therefore interpreted to have formed during the later stages of a deformational event related to arc accretion.

Lifting a weak Poisson bracket to the algebra of forms

We detail the construction of a weak Poisson bracket over a submanifold Σ of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle Π T M , interpreted as a weak Koszul bracket on differential forms on M . This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian function that generates this structure. Such a construction has direct physical interpretation. For a generic gauge system, the submanifold Σ may be viewed as a stationary surface or a constraint surface, with the foliation given by the foliation of the gauge orbits. Through this interpretation, the lift of the weak Poisson structure is simply a lift of the action generating the corresponding BRST operator of the system.

Twistor Geometry of Null Foliations in Complex Euclidean Space

We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface $\mathcal{Q}^n$ of dimension $n \geq 3$, and its twistor space $\mathbb{PT}$, defined to be the space of all linear subspaces of maximal dimension of $\mathcal{Q}^n$. Viewing complex Euclidean space $\mathbb{CE}^n$ as a dense open subset of $\mathcal{Q}^n$, we show how local foliations tangent to certain integrable holomorphic totally null distributions of maximal rank on $\mathbb{CE}^n$ can be constructed in terms of complex submanifolds of $\mathbb{PT}$. The construction is illustrated by means of two examples, one involving conformal Killing spinors, the other, conformal Killing-Yano $2$-forms. We focus on the odd-dimensional case, and we treat the even-dimensional case only tangentially for comparison.

Polyfolds, cobordisms, and the strong Weinstein conjecture

We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.

Scale-dependent influence of pre-existing basement shear zones on rift faulting: a case study from NE Brazil

Rifting of continental crust initiates faults that are commonly influenced by pre-existing structures. We document newly identified faults cutting Precambrian units in the interior of the NE Brazilian margin to assess the effects of structural inheritance on both rift geometry and fault architecture. Stratigraphic and structural data indicate that the faults were active in the main phase of rifting of Gondwana. The influence of pre-existing structures on the Mesozoic rift faulting is scale dependent. Regionally, the faults trend parallel to subvertical, crustal-scale Brasiliano ( c . 750–540 Ma) shear zones. Mylonitic foliations and broadly distributed low strain in the lower crust indicated by shear-wave splitting controlled the overall orientation and kinematics of the rift faults. However, outcrop observations of the faults show that at scales up to hundreds of metres, mylonitic foliations have little influence on fault architectures. Faults cross-cut shear zones and do not commonly utilize foliation planes as shear fractures. Instead, slip zones and fractures have a range of orientations that form acute angles to the local foliation orientation. This observation explains the range of focal mechanisms associated with seismicity that coincides with ancient shear zones in intra-continental areas.

Local foliations and optimal regularity of Einstein spacetimes

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.

Normal forms for periodic orbits of real vector fields

We consider normal forms of real vector fields near periodic orbits and provide sufficient conditions for their smooth linearization. In addition, the main results also assert the existence of vertical local foliations, whose leaves are all transversal to the periodic orbit.

Mauna Loa's submarine western flank: Landsliding, deep volcanic spreading, and hydrothermal alteration

Four new remotely operated vehicle dives carried out by Monterey Bay Aquarium Research Institute (MBARI) reveal a heterogeneous distribution of lithologies and compositions along a transect across the submarine west flank of Mauna Loa, from the outer scarp of the frontal bench to the upper flank. The frontal bench is composed predominantly of volcaniclastic sediments, ranging from very fine‐grained monomictic hyaloclastites to coarse‐grained, compositionally mixed volcaniclastic breccias. The predominance of subaerially derived clasts suggests accumulations of landslide deposits, probably emplaced along a regional shear plane preserved in cataclastic breccias with local foliations and grain trails. Repeated packages of inversely graded strata are interpreted to reflect thrust imbrication of the resulting volcaniclastic apron during volcanic spreading of Mauna Loa's western flank, similar to that now documented along Kīlauea's south flank. Many of the rocks from the bench show evidence for alteration, ranging from low‐grade burial diagenesis to higher‐grade hydrothermal alteration, including phases never before observed in submarine Hawaiian rocks, including epidote, talc, sphene, and corrensite. Alteration is concentrated in deformed zones, denoting pathways for fluid flow into or out of the volcanic edifice. Formed at depth, the altered rocks were subsequently transported along low‐angle thrust faults into the bench and exposed along high‐angle fractures and faults. The upper submarine flanks are draped by subaerially erupted, submarine emplaced pillow lavas and interbedded hyaloclastites, generated by shoreline‐crossing lava flows. Basalt glasses indicate Mauna Loa origin but imply earlier compositions than present‐day lavas, consistent with Ar‐Ar ages suggesting eruption 0.28 ± 0.10 Ma. Late stage detachment of a nearshore slump produced the 'Ālika 2 debris avalanche that broke through the frontal bench, perhaps portending the evolution of the active Hilina slump now present on Kīlauea volcano's south flank.

Pickle Lake revisited: New structural, geochronological and geochemical constraints on greenstone belt assembly, western Superior Province, Canada

New field work, U–Pb ages, geochemical data, and Sm–Nd isotopic analyses have established the timing and determined the nature of volcanism, deformation, and tectonic assembly of the Pickle Lake greenstone belt in the Uchi subprovince of the western Superior Province of the Canadian Shield. The >2860 Ma Pickle Crow assemblage has been redefined to include the former Northern Pickle assemblage on the basis of stratigraphic continuity and similar volcanic geochemistry between the two units across a previously inferred fault contact. The Pickle Crow assemblage consists of tholeiitic basalt with thin, but laterally extensive, oxide-facies iron formation overlain by alkalic basalts and minor calc-alkaline andesites to dacites with primitive Nd isotopic compositions (εNd2.89 Ga = +2.1 to +2.4) suggestive of deposition in a sediment-starved oceanic basin. The ~2 km thick ~2836 Ma Kaminiskag assemblage (former Woman assemblage) consists of tholeiitic basalt interbedded with intermediate and rare felsic pyroclastic flows with primitive Nd isotopic compositions (εNd2.836 Ga = +2.4). Two samples of intermediate volcanic rocks interbedded with southeast-younging pillowed basalt, previously inferred to be part of the Pickle Crow assemblage, yielded U–Pb zircon ages of 2744 [Formula: see text] Ma and 2729 ± 3 Ma. These rocks are thus part of the younger Confederation assemblage, which consists of intercalated basalt and dacite (εNd2.74 Ga = +0.1 to +0.8) exhibiting diverse compositions probably reflecting eruption in a continental margin arc to back-arc setting. The contact between the Confederation and Kaminiskag assemblages is assumed to be a fault. The greenstone belt is intruded by late syn- to posttectonic plutons including the composite quartz dioritic to gabbroic July Falls stock with a new U–Pb zircon age of 2749 [Formula: see text] Ma, and the ~2741 to 2740 Ma trondhjemitic to granodioritic Ochig Lake pluton and Pickle Lake stock, as well as the ~2697 to 2716 Ma Hooker–Burkoski stock. The earliest recognized deformation (D1) is recorded by a local bedding-parallel foliation in the Pickle Crow assemblage. This foliation is truncated by the ~2735 Ma Albany quartz–feldspar porphyry dyke and is not recognized in the volcanic rocks of the Confederation assemblage. The early deformation event is attributed to overturning of the Pickle Crow assemblage prior to deposition of the ~2744 to 2729 Ma Confederation assemblage. Subsequent deformation and development of a regionally penetrative planar fabric (S2) postdates ~2729 Ma volcanism, pre-dates the intrusion of the ca. <2716 Ma Hooker–Burkoski stock and is host to gold mineralization.

Constant mean curvature hypersurfaces condensing on a submanifold

Constant mean curvature (CMC) hypersurfaces in a compact Riemannian manifold (Mm+1, g) constitute an important class of submanifolds and have been studied extensively. In this paper we are interested in degenerating families of such submanifolds which ‘condense’ to a submanifold Kk ⊂ Mm+1 of codimension greater than 1. It is not hard to see that the closer a CMC hypersurface is (e.g. in the Hausdorff metric) to such a submanifold, the larger its mean curvature must be; in other words, the mean curvatures of the elements of a condensing family of CMC hypersurfaces must tend to infinity. Less obvious is the fact that under fairly mild geometric assumptions, cf. [10], the existence of such a family implies that K is minimal. Two cases have been studied previously: Ye [14], [15] proved the existence of a local foliation by constant mean curvature hypersurfaces condensing to a point (which is required to be a nondegenerate critical point of the scalar curvature function); more recently, the second and third authors [10] proved the existence of a ‘partial foliation’ by constant mean curvature hypersurfaces in a neighborhood of a nondegenerate closed geodesic. In this paper we extend the result and methods of [10] to handle the general case, when K is an arbitrary nondegenerate minimal submanifold (no extra curvature hypotheses are required). As we explain below, this more general problem has a number of new analytic and geometric features, and despite the apparent similarities with the case when K is one-dimensional, is considerably more subtle to analyze. We now describe our result in more detail. Let Kk be a closed (embedded or immersed) submanifold in Mm+1, 1 ≤ k ≤ m− 1; the geodesic tube of radius ρ about K is ∗Email: mahmoudi@univ-paris12.fr †Email: mazzeo@math.stanford.edu. Supported by the NSF under Grant DMS-0204730 ‡Email: pacard@univ-paris12.fr, membre de l’Institut Universitaire de France

Paleoproterozoic high-sulfidation mineralization in the Tapajós gold province, Amazonian Craton, Brazil: geology, mineralogy, alunite argon age, and stable-isotope constraints

The Brazilian Tapajós gold province contains the first evidence of high-sulfidation gold mineralization in the Amazonian Craton. The mineralization appears to be in large nested calderas. The Tapajós–Parima (or Ventuari–Tapajós) geological province consists of a metamorphic, igneous, and sedimentary sequence formed during a 2.10 to 1.87 Ga ocean−continent orogeny. The high-sulfidation mineralization with magmatic-hydrothermal alunite is related to hydrothermal breccias hosted in a rhyolitic volcanic ring complex that contains granitic stocks ranging in age from 1.89 to 1.87 Ga. Cone-shaped hydrothermal breccias, which flare upward, contain vuggy silica and have an overlying brecciated cap of massive silica; the deposits are located in the uppermost part of a ring-structure volcanic cone. Drill cores of one of the hydrothermal breccias contain alunite, natroalunite, pyrophyllite, andalusite, quartz, rutile, diaspore, woodhouseite–svanbergite, kaolinite, and pyrite along with inclusions of enargite–luzonite, chalcopyrite, bornite, and covellite. The siliceous core of this alteration center is surrounded by advanced argillic and argillic alteration zones that grade outward into large areas of propylitically altered rocks with sericitic alteration assemblages at depth. Several occurrences and generations of alunite are observed. Alunite is disseminated in the advanced argillic haloes that envelop massive and vuggy silica or that underlie the brecciated silica cap. Coarse-grained alunite also occurs in branching veins and locally is partly replaced by a later generation of fine-grained alunite. Silicified hydrothermal breccias associated with the alunite contain an estimated reserve of 30 tonnes of gold in rock that grades up to 4.5 g t −1 Au. Seven alunite samples gave 40Ar/ 39Ar ages of 1.869 to 1.846 Ga, with various degrees of apparent minor Ar loss. Stable isotopic data require a magmatic-hydrothermal origin for the alunite, typical for high-sulfidation mineralization. The δ 34S values of most samples of alunite range from 14.0‰ to 36.9‰. Sulfur isotopic alunite–pyrite and oxygen isotopic alunite SO 4−OH temperatures range from 130 to 420 °C. The δD H 2O and δ 18O H 2O values for alunite-forming hydrothermal fluids suggest a predominance of magmatic water, with a small meteoric contribution. A rare sample of supergene alunite has a δ 34S value of 4.1‰ and an 40Ar/ 39Ar age of 51.3±0.1 Ma. Other than local foliation in the volcanic rocks and recrystallization of alunite near faults, the mineralization and associated alteration appears to have been remarkably undisturbed by later metamorphism and by supergene alteration. The Au mineralization was preserved because of burial by sediments and tuffs in taphrogenic basins that probably developed shortly after mineralization and were probably first exhumed at about 60 Ma. Because high-sulfidation mineralization forms at relatively shallow crustal levels, the discoveries in Tapajós province provide new perspectives for mineral exploration for the Amazonian and perhaps for other Precambrian cratons.

Feuilletages holomorphes locaux et analyticité partielle d'applications C ∞

Let f : M → M′ be a smooth CR mapping between a generic real analytic submanifold M ⊂ ℂn, n > 1, and a real analytic subset M′ ⊂ ℂn′. We prove that if M is minimal and if M′ does not contain any complex curves, then f is analytic on a dense open subset of M. More generally, we establish an upper estimate of the partial analyticity of f, which depends on the maximal dimension of local holomorphic foliations contained in M′.

Dynamics of the destruction of an armored polymer with allowance for an interfacial layer

A model for numerical simulation of the dynamics and dynamic destruction (foliation) of armored polymers is proposed with allowance for the interfacial layer. A method of numerical simulation is developed and realized. The role of the interfacial layer in the local dynamic foliation of a polymeric composite is demonstrated. At present, polymeric composites with a highly developed interfacial surface are widely used. In these materials, an interfacial layer is formed in the polymeric matrix at the interphase boundary. This layer has a rather complicated microstructure, and its micromechanical characteristics differ from matrix properties [1]. For the interfacial layer consisting of polymeric macromolecules sewn at one end to the surface of the armor, equations were proposed describing the stressstrain state under large deformations [2]. Here, these equations are employed as initial equations in constructing a mathematical model for the case of small strains. In the simulation, we use a method developed for

Singular relative integral invariants and adiabatic processes of thermodynamics

UDC 517.987.5; 517.958 The vector field is rectified in the neighborhood of a nonsingular point. By means of local foliation into nonsingular hypersurfaces containing phase curves of the vector field, we obtain a two-dimensional vector quotient field. A quotient field in the phase plane defines the rectified phase pattern and the module (over the ring of first integrals) of invariant measures. The pair that is composed of the phase pattern and the invariant measure of the vector field in the plane has a stationary subaigebra in the algebra of all vector fields. The triple consisting of the initial vector field, its invariant measure, and an element of the stationary subalgebra of the pairs, admits a relative integral invariant, namely, the integral of the measure over sets that are bounded by the phase curves of two vector fields. The relative integral invariant is generated (over the ring of first integrals) by the action of an element of the stationary subalgebra (the product of the Hamiltonian or the potential by the time of the element of the subalgebra). This explicitly asymmetric construction relative to the pair of vector fields becomes symmetric if we consider singular invariant measures and smooth vector fields. Suppose that an element of the subalgebra is a nonsingular vector field at the point being considered. Then we can take it as the initial element for the next triple. Obviously, the iteration of this procedure generates a chain of triples and, respectively, a chain of measures. We are interested in the convergence of the arising measures (the sense of convergence is defined in the article). We say that a chain of triples is subjected to the weight k > 0 (say, k is a positive integer) if, in each triple, the invariant measure generated by the action of the element of the subaigebra is generated by the first integral of the first element of the triple of degree k (The invariant measure of a triple makes it possible to determine two numbers, namely, the successive weight, i.e., the order of singularity of the measure generated by the action of the element of the subalgebra over the ring of the first integrals of the subalgebra and the preceding weight, which is the same as above, only with the replacement of the element of the subalgebra by the initial vector field.). The order of singularity of an invariant measure is the weight of the measure relative to the action for the element of the subalgebra. Theorem. The orders of singularity of the invariant measures of a chain of triples subjected to the weight k correspond to the orbits of the Feigenbaum mapping.

A computer-based system for supporting decisions for tunneling in rock under conditions of uncertainty

The clustering of orientations of hydraulically conductive fractures in bedrock at the Mirror Lake, New Hampshire fractured rock study site was investigated by comparing the orientations of fracture populations in two subvertical borehole arrays with those mapped on four adjacent subvertical roadcuts. In the boreholes and the roadcuts, the orientation of fracture populations appears very similar after borehole data are compensated for undersampling of steeply dipping fractures. Compensated borehole and pavement fracture data indicate a northeast-striking population of fractures with varying dips concentrated near that of the local foliation in the adjacent rock. The data show no correlation between fracture density (fractures/linear meter) and distance from lithologic contacts in both the boreholes and the roadcuts. The population of water-producing borehole fractures is too small (28 out of 610 fractures) to roadcuts. The population comparisons. However, the orientation of large aperture fractures (which contains all the producing fractures) contains two or three subsidiary clusters in orientation frequency that are not evident in stereographic projections of the entire population containing all aperture sizes. Further, these subsidiary orientation clusters do not coincide with the dominant (subhorizontal and subvertical) regional fracture orientations.