In a landmark paper published in 1957, Kneser [39] introduced a method for enumerating classes in the genus of a definite, integral quadratic form. This method has been deeply influential, on account of its theoretical importance as well as its practicality. In this survey, we exhibit Kneser's method of neighbors and indicate some of its applications in number theory.

The simple relation OmegaOmega-alpha = 0, where Omega is a measurable quantity such as strain and A and alpha are empirical constants, describes the behavior of materials in terminal stages of failure under conditions of approximately constant stress and temperature. Applicable to metals and alloys, ice, concrete, polymers, rock, and soil, the relation may be extended to conditions of variable and multiaxial stress and may be used to predict time to failure.

Dome growth at the Soufriere Hills volcano (1996 to 1998) was frequently accompanied by repetitive cycles of earthquakes, ground deformation, degassing, and explosive eruptions. The cycles reflected unsteady conduit flow of volatile-charged magma resulting from gas exsolution, rheological stiffening, and pressurization. The cycles, over hours to days, initiated when degassed stiff magma retarded flow in the upper conduit. Conduit pressure built with gas exsolution, causing shallow seismicity and edifice inflation. Magma and gas were then expelled and the edifice deflated. The repeat time-scale is controlled by magma ascent rates, degassing, and microlite crystallization kinetics. Cyclic behavior allows short-term forecasting of timing, and of eruption style related to explosivity potential.

Following about two months of intense outward movement and strength deterioration associated with magmatic intrusion, seismicity and gravitational creep, an earthquake on 18 May 1980 marked the collapse of the hot, fluid-rich, north sector of Mount St Helens. Pressure release associated with slide movements resulted in hydrothermal and magmaiic explosions. These explosions produced a lateral blast that partly overran the first slide pulse and devastated a landscape of over 550 km2. Disruption of the sliding masses resulted in the formation of an enormous avalanche of debris that travelled for about 10 min, as far as 23 km. Average velocity was about 35 m/s, and peak velocity about 70 m/s. An area of 60 km2 was buried with 2·8 km3 of hummocky, poorly sorted debris to an average depth of 45 m, and levees to 30 m high were plastered against valley walls and impounded tributaries. One avalanche lobe entered a lake and caused wave run-up to 260 m. Limiting equilibrium analyses and laboratory testing of slide debris suggest that initial failure occurred in a material with c <6 bar, &hi&i;40°, with significant pore fluid pressures and transient shear stresses from a trigger earthquake. Early motion can be characterized by block sliding with an apparent basal friction coefficient of about 0·1. Disintegration of the slide blocks then led to fully developed avalanche flow, to a first approximation involving a Bingham material with strength of about 0·1–1 bar and viscosity 105–106P. The high mobility of the Mount St Helens avalanche, typical for volcanic avalanches, was induced by hot fluids of the depressurized magmatic-hydrothermal system. À la suite d'environ deux mois de poussée intense vers l'extérieur et d'une réduction de résistance due à des intrusions magmatiques accompagnant des événements séismiques et au fluage par gravité, l'affaissement du secteur nord du Mont St Helens, chaud et riche en fluides, fut marqué par un tremblement de terre qui eut lieu le 18 mai 1980. Une chute de pression associée à des mouvements glissants causa des explosions hydrothermales et magmatiques. Ces explosions créèrent un souffle latéral qui dépassa en partie la première pulsation de glissement et dévasta un paysage de plus de 550km2. La rupture des masses glissantes créa une coulée énorme de débris qui se déplaça pendant environ 10 min jusqu'à une distance de 23 km à une vitesse moyenne d'environ 35 m/s, la vitesse maximale étant d'environ 70 m/s. Un terrain couvrant 60 km2 fur enseveli sur une profondeur moyenne de 45 m sous 2·8 km3 de débris accidentés et mal assortis, et des levées jusqu'à une hauteur de 30m furent plaquées contre les côtés des vallées et bloquèrent les affluents. Une branche de l'avalanche pénétra dans un lac, causant des vagues jusqu'à une hauteur de 260 m. Des analyses de l'équilibre limite et des essais de laboratoire des débris de glissement donne l'impression que la rupture initiale a eu lieu dans une matière de c′<6 bar, ø&i;40°, avec des pressions interstitielles considérables et des contraintes transitoires de cisaillement à partir d'un tremblement de terre causateur. Les premiers mouvements peuvent être qualifiés de glissements de blocs avec un coefficient de frottement évident à a la base d'environ 0-1. La désintégration des blocs glissants entraîna alors à une véritable coulée d'une matière du type Bingham ayant en première approximation une résistance de l'ordre de 0·1 — 1 bar et une viscosité de 105—106P. La grande mobilité du Mont St Helens typique des coulées volcaniques fut provoquée par des fluides chauds du système magmatique-hydrothermal décomprimé.

In 1997 Soufriére Hills Volcano on Montserrat produced 88 Vulcanian explosions: 13 between 4 and 12 August and 75 between 22 September and 21 October. Each episode was preceded by a large dome collapse that decompressed the conduit and led to the conditions for explosive fragmentation. The explosions, which occurred at intervals of 2.5 to 63 hours, with a mean of 10 hours, were transient events, with an initial high-intensity phase lasting a few tens of seconds and a lower-intensity, waning phase lasting 1 to 3 hours. In all but one explosion, fountain collapse during the first 10-20 seconds generated pyroclastic surges that swept out to 1-2 km before lofting, as well as high-concentration pumiceous pyroclastic flows that travelled up to 6 km down all major drainages around the dome. Buoyant plumes ascended 3-15 km into the atmosphere, where they spread out as umbrella clouds. Most umbrella clouds were blown to the north or NW by high-level (8-18 km) winds, whereas the lower, waning plumes were dispersed to the west or NW by low-level (<5 km) winds. Exit velocities measured from videos ranged from 40 to 140 ms-1 and ballistic blocks were thrown as far as 1.7 km from the dome. Each explosion discharged on average 3 x 105m3 of magma, about one-third forming fallout and two-thirds forming pyroclastic flows and surges, and emptied the conduit to a depth of 0.5-2 km or more. Two overlapping components were distinguished in the explosion seismic signals: a low-frequency (c. 1 Hz) one due to the explosion itself, and a high-frequency (>2 Hz) one due to fountain collapse, ballistic impact and pyroclastic flow. In many explosions a delay between the explosion onset and start of the pyroclastic flow signal (typically 10-20 seconds) recorded the time necessary for ballistics and the collapsing fountain to hit the ground. The explosions in August were accompanied by cyclic patterns of seismicity and edifice deformation due to repeated pressurization of the upper conduit. The angular, tabular forms of many fallout pumices show that they preserve vesicularities and shapes acquired upon fragmentation, and suggest that the explosions were driven by brittle fragmentation of overpressured magmatic foam with at least 55 vol% bubbles present in the upper conduit prior to each event.

Magmatic intrusions can initiate and sustain massive and catastrophic volcano collapse. Their role is twofold, involving both driving and resisting forces. First, flank stability is diminished by magmastatic and magma overpressures, and steepened slopes, that accompany intrusion. Second, excess pore pressures in potential failure zones can be generated as a result of intrusion-related mechanical or thermal straining of the rock-fluid medium, pressurized retrograde boiling in high level magma chambers, or hydrothermal fluid circulation. Also, earthquakes may aid collapse through inertial forces and shaking-induced pore pressure generation. These excess pore pressures reduce the sliding resistance, as shown for wedge-shaped slide blocks for selected cases. The destabilizing influence of mechanically induced pore pressures is maximized as the intruded width, or corresponding overpressure, of the intrusion is increased. The destabilizing influence of thermally induced pore pressures is conditioned by the severity of thermal forcing, ratios of thermal and hydraulic diffusivities, and the time required for the fluid pressure disturbance to propagate outwards from the intrusion. Retrograde boiling and hydrothermal circulation overpressure mechanisms may be evaluated by similar models. Failure initiation does not imply sustained failure; in some cases, enhancement of pore pressures through deviatoric shearing, frictional heating, or runout over compressible saturated alluvium or marine sediments may be necessary following slide initiation to maintain the impetus of flank failure for long runout. Models are examined for oceanic volcanoes of shallow flank inclination and for terrestrial composite volcanoes with considerably steeper flanks. Les intrusions magmatiques peuvent provoquer et entretenir l'effondrement massif et catastro-phique d'un volcan. Elles le font en faisant intervenir à la fois des forces d'entraînement et des forces de résistance. Pour commencer, les surpressions magmastatiques et magmatiques et l'accentuation des pentes qui accompagnent l'intrusion réduisent la stabilitè des versants. Deuxièmement, la pression des pores pent devenir excessive dans les zones d'effondrement potentiel sous l'effet des contraintes mécaniques ou thermiques imposées au milieu roche-fluide par l'intrusion, du bouillonnement rétrograde sous pression dans les réservoirs de magma de haut niveau, ou de la circulation du liquide hydrothermal. Les séismes peuvent, eux aussi, contribuer à l'effondrement en produisant des forces d'inerde et des secousses qui font augmenter la pression des pores. L'excès de pression réduit la résistance au glissement, comme le montrent les exemples de blocs de glissement en coin illustrés dans Particle. Plus l'intrusion est large ou plus la surpression correspondante est forte, plus Paugmentation mécanique de la pression des pores a un effet déstabilisant. L'effet déstabilisant de Paugmentation thermique de la pression des pores dépend de l'importance du refoulement thermique, des ratios de diffusivité thermique et hydraulique, et du temps de propagation de Paugmentation de pression. On pent utiliser des modéles semblables pour évaluer le bouillonnement rétrograde et les mécanismes de surpression de la circulation hydrothermale. L'amorce d'un effondrement n'est pas nécessairement suivie d'un effondrement entretenu: dans certains cas, Peffondrement des versants, une foisamorcé, ne pent se poursuivre que si la pression des pores augmente sous Peffet d'un cisaillement déviateur, d'un réchauffement produit par frottement, ou d'un écoulement sur des alluvions ou des sédiments marins saturés et compressibles. Let auteurs examinent des modèles pour des volcans océaniques à pentes faibles et pour des volcans terrestres composites à versants bien plus raides.

Abstract. Telemetered high-resolution tiltmeters were installed in Montserrat in summer of 1995, in December 1996, and in May 1997. The 1995 installations, several km from the Soufriere Hills vent, were too distant to yield useful data. However, the 1996 and 1997 installations on the crater rim revealed 6-14 h inflation cycles caused by magma pressurization at shallow depths (< 0.6 km below the base of dome). The tilt data correlated with seismicity, explosions, and pyroclastic flow activity, and were used to forecast times of increased volcanic hazard to protect scientific field workers and the general public.

Significant fracture zone deformation has been imparted to Tertiary lavas on the peninsula (Flateyjarskagi) located between Eyjafjördur and Skjálfandi in north central Iceland. The region lies immediately south of the Flatey fault, one of a series of WNW trending, left‐stepping faults comprising the right‐lateral, oblique slip Húsavík‐Flatey fault system (HFFS). The HFFS defines the present southern margin of the 70 to 80‐km‐wide Tjörnes Fracture Zone (TFZ). Lava bedding, dikes, and faults in the western half of Flateyjarskagi display a progressive 0°–110° clockwise change of trend and steepening of lava dip angles to 15°–45° over an 11‐km‐wide zone nearest the Flatey fault. Accompanying this structural curvature is an increase in fault and dike frequency and amygdule/vein mineralization. Northeast striking faults develop predominant downthrows to the east approaching the TFZ, accommodating the steep lava dips via repetition of stratigraphic section and normal fault‐bounded block rotations. Although the possibility cannot yet be conclusively ruled out that structural trends formed in curved orientations, the simplest interpretation suggests that the lava pile was tectonically rotated by heterogeneous simple shear. Maximum on‐land shear strain has been crudely estimated at γ = 3.5; estimates of shear displacement suggest that the TFZ was active for at least 2 m.y. sometime prior to 6–7 Ma, at a strain rate of approximately 5.5×10−14 s−1. Antithetic faulting and associated flexural bending may have been the means by which a great part of the proposed simple shear rotations were accomplished. WNW extension produced by rotational normal faulting is interpreted to have accommodated systematic variations in shear strain parallel to the direction of shear. Paleomagnetic and structural evidence suggest that rocks of Tertiary age on Tjörnes peninsula immediately to the east also have been tectonically rotated, implying an original shear zone width of at least 25 km.