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Fin whales are among the largest predators on earth, yet little is known about their foraging behavior at depth. These whales obtain their prey by lunge-feeding, an extraordinary biomechanical event where large amounts of water and prey are engulfed and filtered. This process entails a high energetic cost that effectively decreases dive duration and increases post-dive recovery time. To examine the body mechanics of fin whales during foraging dives we attached high-resolution digital tags, equipped with a hydrophone, a depth gauge and a dual-axis accelerometer, to the backs of surfacing fin whales in the Southern California Bight. Body pitch and roll were estimated by changes in static gravitational acceleration detected by orthogonal axes of the accelerometer, while higher frequency, smaller amplitude oscillations in the accelerometer signals were interpreted as bouts of active fluking. Instantaneous velocity of the whale was determined from the magnitude of turbulent flow noise measured by the hydrophone and confirmed by kinematic analysis. Fin whales employed gliding gaits during descent, executed a series of lunges at depth and ascended to the surface by steady fluking. Our examination of body kinematics at depth reveals variable lunge-feeding behavior in the context of distinct kinematic modes, which exhibit temporal coordination of rotational torques with translational accelerations. Maximum swimming speeds during lunges match previous estimates of the flow-induced pressure needed to completely expand the buccal cavity during feeding.

Marine mammals face the challenge of performing energetic tasks while breath holding (Butler and Jones,1997; Kooyman and Ponganis,1998). Conflicting metabolic demands of locomotor activity with a limited oxygen supply must therefore be balanced in order to maximize foraging efforts at depth (Castellini et al.,1985; Hochachka,1986; Davis et al.,2004). Limits to diving capacity, defined by the ability to aerobically dive deeper and longer, are determined by the magnitude of oxygen stores within the body and the rate at which that oxygen supply is consumed(Scholander, 1940; Snyder, 1983; Kooyman, 1989). As body size increases, oxygen stores increase while mass-specific metabolic rates decrease(Klieber, 1932), suggesting that larger animals should be able to dive longer(Butler and Jones, 1982). In general, marine mammals have indeed been shown to dive longer and deeper with increasing size, although this allometric relationship appears to be effected by ecological, behavioral and physiological differences among species (Shreer and Kovacs, 1997; Noren and Williams,2000; Halsey et al.,2006). Ultimately, oxygen stores must be used wisely during a dive through the implementation of strategies to reduce the cost of locomotion.

Cetaceans and phocid seals use lift to generate thrust by the periodic oscillation of a crescent-shaped hydrofoil(Fish et al., 1988; Fish, 1993b; Fish, 1998; Fish and Rohr, 1999). The number of strokes taken during a dive directly increases the energetic cost of foraging (Davis et al., 1985; Fish et al., 1988; Williams et al., 2004). To reduce this cost and enhance diving capacity, locomotor activity is decreased by taking advantage of changes in buoyancy associated with lunge collapse at depth and employing gliding or stroke-and-glide gaits(Skrovan et al., 1999; Williams et al., 2000; Williams, 2001). Differences in body composition among different marine mammals permit the use of gliding gaits at different stages of a dive. Phocid seals(Sato et al., 2003),bottlenose dolphins (Skrovan et al.,1999) and blue whales(Williams et al., 2000) glide during descent and actively stroke to the surface, whereas more positively buoyant right whales (Nowacek et al.,2001) and sperm whales (Miller et al., 2004) actively stroke to depth and glide more during ascent. Drag forces resist forward motion of the body throughout a dive and pose a considerable energetic cost, but the fusiform body shape characteristic of all accomplished swimmers reduces drag by minimizing the development of pressure gradients along the body and delaying separation of a turbulent boundary layer (Vogel, 1994; Fish, 1993a; Fish and Rohr, 1999).

The fin whale, Balaenoptera physalus (Linnaeus 1758), is a fast,streamlined swimmer and one of the largest animals on earth(Bose and Lien, 1989; Bose et al., 1990). Mysticete cetaceans of the crown group Balaenopteridae (sensuRice, 1998), namely blue and fin whales, exhibit significantly shorter dive durations than would be predicted from their extreme body size(Croll et al., 2001). The rorquals are most notably distinguished from other baleen whales by their lunge-feeding behavior, an extraordinary biomechanical process in which large amounts of water and prey are engulfed and filtered(Brodie, 1993; Pivorunas, 1979; Werth, 2000). This mode of intermittent filter feeding requires that the whale uses inertia of the body to stretch its buccal cavity around a volume of prey-laden water(Orton and Brodie, 1987). Blue whale diving behavior combined with oceanographic data show that feeding efforts are primarily directed towards subsurface aggregations of euphausiid crustaceans associated with steep submarine canyon topography(Croll et al., 1998; Fiedler et al., 1998; Croll et al., 2005). Fin whale tracks are also closely linked to aggregations of krill and capelin situated against similar topographic features(Simard et al., 2002). Dive profiles of blue and fin whales reveal longer recovery time at the surface following foraging dives in comparison with non-foraging dives, suggesting that lunge-feeding is energetically costly and consequently limits maximum dive duration (Acevedo-Gutierrez et al.,2002).

Lunge-feeding is facilitated by a host of remarkable morphological and biomechanical adaptations, most of which have been described post mortem. The throat wall is lined with a series of longitudinal throat grooves that consist of tough ridges connected by furrows of delicate elastic tissue (Brodie, 1977; Orton and Brodie, 1987; deBakker et al., 1997). The ventral groove blubber is reversibly extensible up to several times its resting length to accommodate an expanding buccal cavity during engulfment feeding (Orton and Brodie,1987). From the forces required to stretch this tissue, Orton and Brodie predicted that a swimming speed of 3.0 m s–1 would generate enough hydrodynamic force to completely inflate the buccal cavity(Orton and Brodie, 1987). Opening of the mouth causes a lateral expansion and outward rotation of the mandibles, effectively increasing surface area of the mouth to oncoming flow(Lambersten et al., 1995). Excellent underwater video footage of lunging dwarf minke whales has confirmed this phenomenon of mandible rotation in situ (Arnold et al.,2005). An elastic, weakly muscularized tongue is thought to initiate distension of the ventral pouch and increase the capacity of the mouth through invagination into a hollow sac(Lambertsen, 1983). A well-developed coronoid process of the mandible is mechanically linked to the frontal bone by a fibrous frontomandibular stay that is closely associated with the temporalis muscle (Lambersten et al., 1995). This tendon may act to limit hyperdepression of the lower jaw, provide elastic recoil to reverse the direction of jaw movement, and enhance mechanical power output of the temporalis when acting to elevate the lower jaw (Lambersten et al., 1995). Once the jaws have closed around the volume of prey-laden seawater, a novel articulation between the mandibles and maxillary suborbital plate may provide a hydrodynamic seal of the buccal cavity, thereby maintaining a fusiform body shape in spite of possessing a highly expandable mouth(Lambertsen and Hintz, 2004). The forces to deflate the ventral pouch are suggested to come from the dynamic pressure from oncoming flow, elastic energy stored within the stretched tissues and active muscle shortening beneath the blubber(Orton and Brodie, 1987). Video footage at the sea surface of lunge-feeding rorquals has provided evidence of a rebounding wave within the ventral pouch that is thought to enhance filtration (Kot,2005).

Our knowledge of the lunge-feeding process in situ is limited to aerial or ship observations near the sea surface. Humpback whales lunge-feeding at the sea surface exhibit three kinematic modes that are distinguished by the orientation of the body with respect to the water surface(Jurasz and Jurasz, 1979). Two of these modes were termed lateral lunge feeding' and inverted lunge feeding', which involved rotations about the whale's longitudinal axis (roll)of approximately 90° and 180°, respectively(Jurasz and Jurasz, 1979). Lateral lunge-feeding was also observed for blue and fin whales surface-feeding on euphausiids or schooling fish(Andrews, 1909; Tomilin, 1957; Watkins and Schevill, 1979; Gaskin, 1982; Corkeron, 1999). However, Watkins and Schevill reported that fin whales primarily lunged with their ventral sides down (Watkins and Schevill, 1979). Underwater observations of lunge-feeding include humpbacks executing lateral lunges while bottom feeding on sand lance at 30 m depth (Hain et al., 1995) and Crittercam video of a blue whale performing inverted lunges on pelagic krill aggregations (J.C., personal communication). Fin whales have also been observed to feed in shallow waters on sand lance(Bigelow and Schroeder, 1953; Overholtz and Nicolas, 1979)and herring (Nottestad et al.,2002), but the types of feeding modes used were not reported. Overall, lunge-feeding behavior appears to be modal as well as highly variable across species, but the detailed mechanics of the process remain elusive.

To investigate the swimming kinematics during foraging dives, we attached high-resolution digital tags to the backs of surfacing fin whales in the Southern California Bight. Accelerometer data were used to analyze both body orientation and fluking behavior, while hydrophone-measured flow noise was used to estimate body speeds throughout the dive cycle. We present the first kinematic analysis of a diving rorqual, including lunge-feeding behavior at depth.

### Digital tag

The high-resolution digital tag (Bioacoustic Probe; Greeneridge Sciences,Inc., Goleta, CA, USA; Burgess et al.,1998) included a hydrophone (sampling rate, 1025 Hz), a depth gauge and a two-axis accelerometer (MXA2500GL/ML; MEMSIC Inc., North Andover,MA, USA) encapsulated in a cylindrical polyurethane shell with a hemispheric nose. Data from the depth gauge and accelerometer were digitally recorded at 1 Hz and stored within the tag. The tag was attached to two silicon suction cups with zip-ties and harnessed with a flotation device(Fig. 1). Typically, the tag stayed attached for several hours, fell off during lunges at depth and then floated to the surface. Upon tag recapture, the data were downloaded for analysis via infrared transmission.

### Tagging methodology

Fin whales (Balaenoptera physalus; family Balaenopteridae) were tagged off the Tanner-Cortez banks in the Southern California Bight during a visual and acoustic marine mammal monitoring operation in the summer of 2003(Oleson, 2005). A 5.3 m Rigid-Hulled Inflatable Boat approached whales from behind, and tags were attached using a 4 m fiberglass pole. We aimed to place the tag so that its long axis was largely parallel with the long axis of the animal(Fig. 2). However, as soon as the whale started to dive, it was apparent that flow forces helped to align the tag more parallel with the longitudinal axis of the body. When possible,tagged whales were followed visually or by radio VHF transmission.

### Whale speed estimates

Flow noise has previously been used as a method for estimating flow speed(Finger et al., 1979; Fletcher et al., 1996; Burgess et al., 1998). We utilized this approach to estimate the instantaneous speed of whales for a given level of flow noise recorded by the hydrophone on the tag. To determine the magnitude of flow noise at different flow velocities, the tag was attached to a dihedral wing (V-FIN, Type 166; Endeco/YSI Inc., Marion, MA, USA) and towed at approximately 0, 1.5, 3.0 and 5.0 m s–1 by the R/V Sproul. The flow noise was analyzed at different frequencies at each flow velocity by calculating the root-mean-square sound pressure over 1/3 octave bands. The 50 Hz spectrum exhibited both the highest flow noise level and the most distinct partitioning of this noise level for each flow velocity. Therefore, we used the 50-Hz 1/3 octave band of the flow noise signal in order to determine flow speed. A positive relationship between flow speed and flow noise was obtained through a least-squares curvilinear regression(Fig. 3; r2=0.99). This relationship was used to calculate instantaneous velocity, VS, for a given flow noise level recorded from the tag deployments. As a corollary, whale speed derived from the kinematics of the body, VK, was estimated by dividing the vertical velocity obtained from the depth profile by the sine of the body pitch angle (see Miller et al.,2004). The first derivative of speed with respect to time was determined in order to estimate the instantaneous acceleration of the body throughout the dive cycle.

### Body orientation: theory

The accelerometer measured both static (gravitational acceleration) and dynamic acceleration at 1 Hz, which allowed for the analysis of body orientation and fluking, respectively. Accelerometer signals were low-pass filtered (cutoff frequency, 0.1 Hz) to remove higher frequency oscillations for analysis; the low-frequency signal became the data used to determine orientation while the high-frequency signal was used to analyze fluking patterns (see Fluking analysis section below). A linear scaling adjustment performed at acquisition time accounted for the fact that the accelerometer was being sampled before it completely settled (W. C. Burgess, personal communication).

Body orientation was represented by two kinematic degrees of freedom: pitch and roll. The x-axis was defined as the long axis of the tag, which is parallel to one axis of the accelerometer, while the y-axis was defined as perpendicular to this axis, which extends radially on the tag. Acceleration along each axis was measured in gravitational units (range,±1 g; g=9.8 m s–2). Changes in acceleration detected by the x-axis were used to estimate body pitch or tilt:
$\ {\gamma}=\mathrm{asin}(A_{\mathrm{x}}{/}A_{\mathrm{x}}^{{\ast}}),$
(1)
where Ax is the static acceleration measured along the x-axis of the accelerometer, Ax* is the maximum value recorded by the accelerometer along that axis (1.0 g),and γ is the pitch of the long axis of the animal with respect to horizontal. Equation 1 describes the revolution of the accelerometer axis about the arc of a unit circle and its resulting nonlinear response. Therefore, γ=0° would represent a horizontal body angle andγ=±90° would reflect vertical body orientations.
Rotations about the x-axis signify body roll and will be observed in changes in static acceleration by the y-axis of the accelerometer, Ay. Body roll estimates are affected by different pitch orientations such that progressive degrees of tilt significantly decrease the magnitude of static acceleration measured along the y-axis. Instead of the accelerometer axis revolving about the arc of a circle, its path effectively becomes an arc of a projected ellipse of diminished height onto the plane perpendicular to gravity. The magnitude of reduced height of the projected ellipse is determined by:
$\ {\hat{h}}=\mathrm{cos}({\gamma}).$
(2)
Substituting h into the equation for an ellipse and keeping the other axis of the ellipse perpendicular to gravity (width) equivalent to a unit circle, the angle of revolution, θ, about this ellipse is augmented to become:
$\ {\theta}=\left[\frac{1}{1-(\frac{A_{\mathrm{x}}}{A_{\mathrm{x}}^{{\ast}}})}\right][\mathrm{cos}(\mathrm{asin}{\ }A_{\mathrm{x}})]^{2}\left[\mathrm{asin}(\frac{A_{\mathrm{y}}}{A_{\mathrm{y}}^{{\ast}}})\right],$
(3)
where Ay* is the maximum value recorded by the accelerometer when the y-axis is parallel to gravity.

The three terms of Eqn 3, as denoted by brackets, each describe a particular characteristic of the accelerometer response, which ultimately combine to give a roll estimate, θ, for a given output of the dual axis accelerometer within the tag. Term 1 describes a tilt factor', where high levels of pitch drastically increase the sensitivity of the response and decrease the overall magnitude of the response. Term 2 is related to an ellipse of decreasing height with increasing values of body pitch, while term 3 is analogous to Eqn 1, the revolution about an arc of a unit circle.

### Body orientation: calibration

To experimentally test the validity of Eqn 3, the accelerometer was calibrated in a custom-made device. The apparatus statically held the tag at different degrees of tilt. At each level of tilt, as determined by Eqn 1, the tag was rolled at 5° intervals as measured by a laser pointer and protractor attached parallel to the long axis of the tag. Data from this calibration were used to describe the range of θ in the context of the orthogonal dependence of Ay on Ax(Fig. 4). For static orientations, the maximum value recorded by the accelerometer was 0.9 g, which may be due to the interaction between sampling and settling rates of the accelerometer (W. C. Burgess, personal communication). However, standardizing the response with respect to the maximum value is expected to account for this difference (Eqns 1, 3).

The accelerometer data from the static calibration(Fig. 4) were entered into Eqn 3 to determine what roll angle would be predicted by theory. Roll angles predicted by theory compared with the roll angles measured experimentally showed a strong correlation (Fig. 5; r2=0.99), suggesting that Eqn 3 is a reliable method to estimate body roll at pitch angles less than 65°. At the highest degrees of roll, when the y-axis of the accelerometer is nearly parallel to gravity, we suggest a maximum error of approximately±10°. At low to moderate pitch and roll angles we expect error to be less than ±5°.

The range of θ is limited to 180° if the axis of the accelerometer begins in the plane parallel to gravity, whereas the range ofθ is limited to 90° if the axis begins in the plane perpendicular to gravity. This dependence of Ay on Axdid not allow for an analysis of roll angle during descent or ascent whereγ exceeded 65° and high Ax values were involved.

The position of the tag may move during the course of the dive, which would significantly alter kinematic analysis. To address this potentially confounding factor, tag orientation was examined when the whale was at the surface. Before and after each dive, the mean pitch angle was 2.5±2.7° (N=28) from horizontal when the whale was at the surface, suggesting that the tag was largely parallel with the long axis of the whale's body and maintained this orientation throughout the deployment. The orientation of the y-axis of the accelerometer (radial axis of the tag) was also determined before and after a dive. In this way, sliding of the tag was sometimes observed, which discounted the dive from being included in further analysis. With the assumption that the whale did not roll on average while at the surface, the average value of Ay was recorded at the surface. This served as an indication of when the whale was level at depth and also to what extent the body rolled during lunges or maneuvers.

### Fluking analysis

Dorsal–ventral oscillations of the flukes were detected as small-amplitude oscillations by the x-axis of the accelerometer. These distinct patterns of dynamic acceleration, likely to be a result of recoil forces manifest throughout the body (see Fish et al., 2003), were isolated from the static acceleration profile used to determine body orientation by low-pass filtering at 0.10 Hz. These patterns were so distinct and repeatable in form and fashion that we consider this filtering process to have removed all accelerations due to fluking itself. However, it was not possible to account for surge, heave or sideslip (see fig. 8 in Fish, 2004 for definitions)that may have contributed to the accelerometer signal. Fluking frequencies were calculated by counting the number of acceleration maxima divided by the time of a given bout of fluking. Either through recoil forces detected by the tag in the mid-body region or by actual tilting with the caudal stock,downstrokes produced positive peaks in the fluking profile while upstrokes resulted in negative peaks.

### Dive profiles

Data from the pressure transducer within the tag recorded changes in depth over time and provided a context for which to evaluate other kinematic parameters. Depth profiles were categorized into several phases. Descent was defined as the time between a depth value of zero and the time when maximum velocity was recorded, since each whale continued to accelerate until a preferred depth was attained. Ascent time was defined as the time from the last velocity minimum (end of last lunge) until the time when depth equaled zero again. Lunging time, or foraging time, was defined as the time between descent and ascent. Post-dive time, or recovery time, was the time spent at the surface after a dive, until another foraging dive was recorded.

A series of vertical excursions at the bottom of a dive was assumed to represent a foraging dive. Although we have no direct video evidence to confirm that these whales were actually feeding, previous studies have shown that these vertical excursions occur at the precise location of their preferred prey (Croll et al.,1998; Croll et al.,2005). Crittercam deployments on blue whales also confirm the presence of prey during these types of lunges.

### Statistics

All statistical analyses were performed using Minitab (version 13). If a parameter failed, the Anderson–Darling test for normality, a Mann–Whitney U-test, was used to test whether two given kinematic parameters were significantly different from one another. A P-value less than or equal to 0.05 accepted the hypothesis that the two parameters were significantly different. Sample sizes among individuals did not allow for an effect of individuals to be assessed.

From 13 total attempts, seven fin whales were successfully tagged from 20 to 26 August 2003, resulting in 28 foraging dives recorded for analysis(Table 1). All tagged whales would consistently dive to depths below 200 m, making a concerted effort to dive to a preferred depth where lunges took place and ascend back to the sea surface. Maximum depth attained during each dive in the context of local bathymetry suggests that foraging occurred very close to the sea floor. All feces seen from these fin whales were composed of euphausiids. However, there may have been a bias since feces composed of fish is sometimes darker and more difficult to notice. Dive durations averaged 7.0±1.0 min (minimum, 4.6 min; maximum, 8.2 min) and were responsible for approximately 60% of each whale's total time budget, while the other 40% of the time the whales were closely associated with the sea surface (depth <50 m).

### Kinematics during descent and ascent

A representative foraging dive recorded by the tag is presented in Fig. 6. At the beginning of descent strong fluking was observed, but stopped or decreased dramatically(<0.1 g) at a depth of 21±7 m. At the end of descent, small amplitude oscillations (<0.1 g) were often observed in the fluking profile, but we attributed these vibrations to an increasingly turbulent flow regime associated with high speeds (>4 m s–1). Prolonged gliding was employed during 55±23% of descent durations but ranged widely from 19 to 95%. Strong fluking was observed at the end of each descent. Stroke-and-glide gaits were sometimes observed for many whales on descent (Table 1). By contrast, steady fluking at a frequency of 0.30±0.03 Hz was observed on ascent of every dive recorded. At the end of each ascent,steady fluking discontinued at a depth of 30±5 m as whales glided to the surface.

The highest speeds of the body during the entire dive cycle were recorded on descent. A comparison of the two methods used to estimate speed shows striking similarity, particularly at speeds between 1 and 5 m s–1 (Fig. 7). Since the calculation of VK is extremely sensitive to low values of pitch, a comparison to VS during ascent and descent was appropriate given that changes in pitch were minimal and far from zero. A least-squares linear regression through instantaneous speed values calculated via each method suggests that there is a one-to-one correlation between them (N=4062, r2=0.91, VK=1.0024VS+0.2013). In this way, VS was justifiably used as a method to estimate speed throughout the dive, particularly during lunges, where pitch values were close to zero, which would have resulted in spurious VKestimates.

Whales continually accelerated throughout descent at an average pitch angle of –53±8° to a maximum speed of 5.7±0.3 m s–1 (Fig. 6). As the bottom of the dive was reached, body speed decreased to very low speeds(0.5–1.0 m s–1) in approximately 5 s, representing a large deceleration, which perhaps indicated a lunge-feeding event upon initial descent. Speeds during ascent were significantly lower and relatively more constant at steeper mean pitch angles of 64±7° (Mann–Whitney U-test, P<0.005), reaching maximum speeds of 3.4±0.4 m s–1(Table 1).

The relationship between body acceleration and body pitch during descent and ascent is shown in Fig. 8. At depths greater than 21 m on descent, where fluking had typically stopped,the body experienced net positive acceleration at relatively constant pitch angles, which indicated that the body was negatively buoyant and sinking. During ascent, body accelerations were much closer to zero, suggesting that a relatively constant speed was maintained, but gradually decreased as depth decreased. The highest decelerations were recorded at 30 m, the average depth at which gait transition from steady fluking to gliding took place.

### Kinematics during lunges

A series of vertical excursions was typically observed at the bottom of each foraging dive that was associated with lunge-feeding behavior(Fig. 6). These excursions ranged from less than 5 m to as high as 20 m. Coincident with the depth minimum of each excursion was a distinct maximum in speed. We interpreted these speed maxima to represent lunge-feeding events; as the whale rushes toward a prey patch and opens its jaws, it will incur a massive drag load and decelerate rapidly. We arbitrarily defined a lunge to be a distinct speed maximum that is greater than 2.0 m s–1. With this definition,we recorded 121 lunges during 28 foraging dives(Table 2). Whales executed anywhere from one to seven lunges per dive, but averaged 4.4±1.4 per dive. Maximum speed during each lunge was 3.0±0.5 m s–1 produced by fluking frequencies of 0.27±0.04 Hz. Speed values at one second intervals averaged from 50 lunges were calculated(1.26, 1.27, 1.29, 1.33, 1.44, 1.65, 1.93, 2.28, 2.64, 2.92, 3.03, 2.91, 2.56,2.07, 1.57, 1.14, 0.85, 0.68, 0.60, 0.58, 0.61, 0.65, 0.69, 0.71, 0.70, 0.68 m s–1). Whales glided until a bout of fluking marked another lunge-feeding event. Each bout of fluking lasted 16.2±3.9 s, while durations between consecutive lunges, the time between speed maxima, averaged 44.5±19.1 s (Table 2).

A time series of fluking, body orientation, and translational acceleration during four consecutive lunges reveals a distinct and consistent kinematic mode (Fig. 9). Maxima, minima and zero values of each kinematic variable were superimposed onto depth profiles to determine the body dynamics that occur during a lunge. Before the lunge, the whale approaches a prey patch with a slight downward pitch(γ<30°). Maximum acceleration of the body typically occurred just as the first full stroke cycle was completed. At this moment the body begins to roll. The long axis of the body becomes level, parallel to the sea surface,as maximum velocity is reached. Opening of the mandibles, which is assumed to occur at maximum speed, causes a deceleration at the same time that the body completes a full 90° roll (Fig. 10). Meanwhile, the body begins to tilt upward and roll back as the final fluke stroke is executed. Maximum pitch is attained as the whale slows to a minimum speed.

This kinematic sequence was fundamentally conserved among all individuals,except for changes in body roll (Fig. 11). In these cases rolling was still observed, presumably a reflection of maneuvering, but the degree of roll was typically less than 45° and not temporally associated with other kinematic landmarks. Therefore, we categorized lunges into two modes based on the degree of roll and its temporal association with other kinematic parameters. Specifically,these modes were distinguished by the degree of roll observed at open gape,during maximum body deceleration. If the body was level, with its dorsal side facing the sea surface, the lunges were considered regular lunges'(N=59). The remaining lunges involved body rolls of 87±18°(N=62) and were regarded as lateral lunges'(Table 2). Overall,lunge-feeding behavior was variable, with some individuals performing either lunge type exclusively and some exhibiting both kinematic modes, sometimes within the same dive.

As the largest animals on earth, blue and fin whales face extraordinary consequences of an extreme body size(Calder, 1984). Mechanical principles predict that large body size will decrease agility and maneuverability (Webb and de Buffrenil,1990). To circumvent these effects, balaenopterid foraging behavior incorporates the selection of dense aggregations of small prey(Weihs and Webb, 1983) and increased attacking speed during lunges(Fig. 9; Table 2). A large body size is accompanied by a high energetic demand such that B. physalus is predicted to require one metric ton of krill daily(Brodie, 1975). Physiological scaling laws suggest that increasing body size will increase oxygen stores and decrease mass-specific metabolic rate (Klieber, 1932). Thus, blue and fin whales appear well-equipped to aerobically dive longer and deeper to exploit prey patches at depth. However, myoglobin concentration, the primary oxygen carrier in skeletal muscles, is relatively low for fin whales, indicating that rorquals have been subject to different selective pressures in terms of physiological adaptations to enhance dive capacity(Noren and Williams, 2000). Also, due to differences in body composition related to oxygen storage,cetaceans have considerably less oxygen per unit mass available compared with phocid seals and penguins (Kooyman,1989).

Blue and fin whales typically dive much shorter (<17 min) and shallower(<200 m) than would be predicted by their large body size (Shreer and Kovacs, 1997; Croll et al.,2001). Although fin whales have been reported to dive as deep as 470 m, which is still somewhat shallow for their body size, dive durations during these excursions were less than 13 min(Panigada et al., 1999). Optimality models of dive behavior based on depth profiles of blue and fin whales suggest that lunge-feeding is energetically costly and thus responsible for limiting dive capacity(Acevedo-Gutierrez et al.,2002). Foraging dives in the Weddell seal are associated with an increased energetic cost compared with non-foraging dives of the same duration and such costs can be estimated from the number of strokes taken during a particular dive (Williams et al.,2004). Different types of locomotor activity, particularly involving rapid changes in translational and rotational acceleration, may significantly increase the energetic costs incurred during a dive(Weihs, 1981). While the energetic cost for each stroke does not change with body size among phocid seals (Williams et al., 2004),complex maneuvers executed by the largest whales may prove to be much more energetically expensive.

Here we show that fin whale foraging dives are characterized by a gliding descent, a series of lunges at depth and an ascent to the surface powered by steady fluking (Fig. 6). Other negatively buoyant marine mammals show similar patterns of reduced locomotor activity during descent (Skrovan et al.,1999; Sato et al.,2003), a behavior that is associated with a decrease in oxygen consumption, which in turn enhances diving capacity (Willliams et al., 2000). Fin whales were observed to accelerate primarily while gliding at high descent angles (Fig. 8A), suggesting that buoyant forces are more effective when vertically directed drag forces are minimized. When the body is oriented more vertically, pressure drag is relatively lower because projected area is significantly decreased compared with when the body is broadside to vertically acting buoyant forces. In this way, whales accelerated to the highest velocities recorded over the dive cycle(Fig. 6). Similarly, sperm whales reached maximum speeds near the end of each descent, but such speeds were accompanied by fluking (Miller et al., 2004). Our data suggest that fin whales should be practically neutrally buoyant or slightly positively buoyant at depths shallower than 30 m, as indicated by the depth at which gait transition occurs during ascent and descent (Fig. 8; Table 1). This change in buoyancy is attributed to gradual lunge collapse with depth in other diving marine mammals (Skrovan, 1999; Williams et al., 2000), given that complete lunge collapse is suggested to occur at a depth of about 100 m(Scholander, 1940; Ridgway et al., 1969; Ridgway and Howard, 1979). Williams et al. also observed gait transition at similar depths for the blue whale, Balaenoptera musculus(Williams et al., 2000). Negative buoyancy may be counteracted by hydrodynamic lift provided by the pectoral flippers as they are abducted and extended away from the body(Fish and Battle, 1995; Miklosovic et al., 2004). Accordingly, tethered minke and sei whales were observed to sink while the flippers were held against the body and pitch toward the surface when they were extended (Williamson,1972). Balaenoptera are also reported to be negatively buoyant and typically sink when killed(Slijper, 1962; Brodie, 1977).

In order to utilize oxygen stores wisely at depth, diving animals must not only reduce locomotor activity but also exhibit an efficient mode of locomotion. The morphological design of the fin whale is well equipped for efficient, high-speed swimming (Bose and Lien, 1989; Bose et al.,1990). Fin whales are theoretically capable of speeds as high as 13 m s–1 (Bose and Lien,1989), and maximum speeds of up to 10 m s–1 have been reported (Gambell, 1985). Average speeds observed over long-distance tracks, however, are only 0.5–2.0 m s–1(Notarbartolo-di-Sciara et al.,2003). In the present study, sustained speeds during ascent(Table 1) were within the range predicted to be efficient (2–10 m s–1) but were quite lower than those predicted to produce maximum propulsive efficiency (6–8 m s–1) by unsteady hydrofoil theory(Bose and Lien, 1989). Maximum speeds estimated for fin whales on descent (5.7±0.3 m s–1; Table 1)were significantly lower than maximum swim speeds observed in both captive and wild delphinids (Rohr et al.,2002). With respect to body size, the speeds observed in this study by fin whales are very low in comparison with odontocetes.

The kinematics of the body and flukes during lunges depict an exceptionally dynamic event. Body acceleration driven by a bout of fluking is immediately met by a relatively larger deceleration, probably due to the opening of the mouth. Lowering of the mandibles increases the surface area of the body,specifically the buccal cavity, perpendicular to flow. The moving buccal cavity meeting the stationary volume of prey-laden seawater provides the pressure needed to expand the ventral groove blubber in proportion to the square of velocity (Orton and Brodie,1987). A large part of the kinetic energy of the body should be converted into potential energy stored in the stretched ventral groove blubber. The Y-shaped fibrocartilage skeleton that lies within the musculature of the ventral pouch may provide structural rigidity to the region or act as a tendon to distribute forces involved in the feeding process(Pivorunas, 1977). Once the buccal cavity is filled, the elongated, bloated tadpole' profile of the body(Orton and Brodie, 1987) must also increase drag on the body and contribute to the overall deceleration of the body.

Accelerating a large body is energetically demanding. This appears to be the reason why lunge-feeding is so costly and thus limits dive time. Since drag on the body is proportional to the square of its instantaneous velocity,the thrust and energy needed to overcome drag will be high during a lunge. In addition, drag should become dramatically larger when the mouth is agape,dissipating the kinetic energy of the body. Our data support the hypothesis by Acevedo-Gutierrez et al. that the rapid changes in speed associated with lunge-feeding at depth are energetically expensive and limit dive capacity in rorqual whales (Acevedo-Gutierrez et al.,2002). Quite the opposite seems to be the case for bowhead and right whales, which appear to swim at relatively constant speeds(Nowacek et al., 2001) and continuously filter feed via both hydrodynamic and ram hydraulic pressures (Werth, 2004; Lambertsen et al., 2005). According to mechanical principles, this foraging strategy should be energetically more efficient than lunge-feeding since the body maintains a relatively constant speed and thus accelerations of the center of mass will be minimized. Bowhead whale dive behavior is consistent with this hypothesis;they exhibit longer dive durations and shorter recovery times between dives than a larger blue whale diving to the same depth(Dorsey et al., 1989; Würsig and Clark, 1993; Krutzikowsky and Mate, 2000; Croll et al., 2001).

Steady fluking that occurs during the ascent phase of a dive must also come at an energetic cost since the whale is negatively buoyant, but this should not be as costly as lunging given that speed on ascent remained relatively constant (Fig. 6). Therefore,it is not surprising to observe rorquals performing deep dives of limited duration (Panigada et al.,1999), as long as the number of lunges per dive is low. Body acceleration observed over the course of ascent decreases steadily(Fig. 8B), perhaps indicating fatigue. However, it may also indicate a decrease in motivation associated with gradually changing buoyant forces near the end of ascent. From these kinematic data it is unclear whether fin whales were exceeding their aerobic capacity during foraging dives.

The maximum speeds recorded at jaw opening match the predictions made by Orton and Brodie for the hydrodynamic forces needed to expand the ventral groove blubber if feeding was exclusively powered by the locomotor muscles(Orton and Brodie, 1987). Also, the slight downward pitch of the body just prior to the lunge may help to open the mouth by lowering the pressure on the underside of the head via the Bernoulli effect. Our results show that fin whales fluke throughout each lunge (Figs 9, 10), even after jaw opening,supporting the hypothesis that prey-laden water is enveloped by the buccal cavity (Orton and Brodie,1987). The timing of jaw opening is critical for successful prey capture in order to avoid pushing prey away with a bow wave(Brodie, 1977) and is likely facilitated by the tactile sensing of prey via vibrissae on the mandibles (Ogawa and Shida,1950; Gaskin,1982). Contraction of the buccal cavity must occur when the whale is gliding between lunges. If we assume that engulfment is accomplished in the time elapsed during a fluking bout (16.2±3.9 s; Table 2) and that a fin whale engulfs approximately 30 m3 of water and prey during a lunge (see a previous estimate of 70 m3 for a blue whale)(Pivorunas, 1979), water must be filtered at a rate of nearly 1 m3 s–1, since durations between each consecutive lunge were 44.5±19.1 s. However, we do not know the extent to which the buccal cavities were filled during each lunge, as there have been previous accounts of fin and sei whales engulfing prey without the buccal cavity becoming `enormously expanded'(Pivorunas, 1979). In addition, estimates for the volume of engulfed prey and water are entirely anecdotal.

Lunges occurred in two distinct modes, which were distinguished by the degree of body roll at the moment of jaw opening (Figs 9, 10). Lateral lunges involved a 90° roll to the same side on each lunge, while regular lunges involved no significant roll as the body reached maximum velocity. Why Balaenoptera roll during lunges is not known. Rotating about the longitudinal axis may orient the jaws in such a way as to capture prey by anticipating their escape trajectory (F. E. Fish, personal communication). Lateral lunges may also be a way to pin or drive prey against a barrier, such as the sea surface or sea floor. Being negatively buoyant, a 90° roll may help with maneuvers in the plane perpendicular to buoyancy, where they are weight neutral (B. Ahlborn, personal communication). Both behaviors have been previously observed among the rorquals(Andrews, 1909; Tomilin, 1957; Jurasz and Jurasz, 1979; Watkins and Schevill, 1979; Gaskin, 1982; Hain et al., 1995; Corkeron,1999). Gaskin, who reported both lateral and regular lunges for fin whales off the southwestern coast of Nova Scotia(Gaskin, 1982), suggested that regular lunges were generally less effective and mainly directed towards fish rather than euphuasiids.

Cetacean maneuvers are primarily driven by lift derived from the flukes and by the asymmetrical orientation and/or movement of the flippers(Edel and Winn, 1978; Fish, 2002; Fish, 2004; Fish et al., 2006). Our data show that body rotation, particularly roll, occurs during fluking bouts associated with lunges (Fig. 9). However, rolling was sometimes observed during glides,especially in individuals that exhibited regular lunges, suggesting that fin whales employ both powered and non-powered lift-based mechanisms to maneuver. Edel and Winn reported flipper movement and twisting of the caudal stock and fluke during banked turns in the humpback whale(Edel and Winn, 1978). Fin whales performing lateral lunges at the sea surface exhibited strong fluking in coordination with lateral extensions of a pivotal flipper(Gaskin, 1982). We were not able to discern flipper movement during lunges, so the extent to which torque generated by the fluke was enhanced by control surfaces remains undetermined for fin whales.

Although three-dimensional dive behavior has been described for phocid seals (Davis et al., 2001; Davis et al., 2003; Mitani et al., 2003; Mitani et al., 2004),comprehensive data showing maneuvers effected by six kinematic degrees of freedom have not been presented for any marine mammal. Our analysis was only able to resolve two kinematic degrees of freedom with respect to body rotation, so changes in yaw that occurred through the dive cycle were not unknown. As a result, the roll moments recorded could have been associated with maneuvering, as has been observed for beaked whales maneuvering to capture prey (Madsen et al.,2005). Considering the repeatable and modal nature of the behaviors observed (Fig. 8), we suspect that fin whales were spinning about their longitudinal axis, executing lateral lunges similar to what was qualitatively described for humpback whales at the sea surface (Jurasz and Jurasz,1979) and along the sea floor(Hain et al., 1995). However,lateral lunge-feeding behavior can involve a curvilinear trajectory(Gaskin, 1982). During regular lunges, roll moments were often observed just before maximum velocity, perhaps indicating a maneuver or banked turn towards a prey patch. These types of complex maneuvers typically involve temporal coupling of yaw and roll moments in flying animals (Schilstra and Van Hateren, 1999; Horisawa et al., 2003). More tagging efforts are necessary in order to determine body yaw and thus resolve full three-dimensional dive behavior in the context of local bathymetry and prey distribution(Mitani et al., 2003; Mitani et al., 2004; Watanabe et al., 2003).

We especially thank Bill Burgess at Greeneridge Sciences, Inc. for building an excellent, high-resolution Bioacoustic Probe. We also thank Bill Burgess,Frank Fish, Robert Dudley, Jerry Kooyman, Torre Knower, Megan McKenna, Jessica Meir, Paul Murphy, Nick Pyenson, Melissa Soldevilla and Greg Szulgit for providing useful comments on the manuscript. We thank Sean Wiggins for assistance with visualizing local sea floor bathymetry. Funding was provided by the United States Navy (SERDP Robert Holst, CNO-N45 Frank Stone, ONR Bob Gisiner) and NSERC to R.E.S.

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text:
# Quick proof of the fact that the ring of integers of Q(\zeta_n) is Z[\zeta_n]?

I cannot find a good reference for the proof that the ring of integers in a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$. The proof I usually find does an induction on the number of prime factors of $n$, coupled with a lengthy and somewhat computational proof in the case where $n$ is the power of a prime.

Do you know a quicker and possibly more conceptual approach?

-
I have rolled back to the previous version, since LaTeX is rendered in the title too. Is there any reason why it was removed? –  Andrea Ferretti Mar 6 '10 at 17:04
Ok, I have reverted to the version without LaTeX, since that is not rendered on the home page. –  Andrea Ferretti Mar 6 '10 at 17:16

The ring of integers $R_n$ of ${\mathbf Q}(\zeta_n)$ contains ${\mathbf Z}[\zeta_n]$ as a subring with finite index. To show the containment of rings is an equality, it suffices to show the inclusion ${\mathbf Z}[\zeta_n] \rightarrow R_n$ becomes an isomorphism after tensoring with ${\mathbf Z}_p$ for all $p$, and this basically boils down to showing the ring of integers of ${\mathbf Q}_p(\zeta_n)$ is ${\mathbf Z}_p[\zeta_n]$ for all $p$. Now you have to know something about how to compute rings of integers in unramified and totally ramified extensions of local fields. I'm leaving off some details here, admittedly, and since I used the terrible word "compute" maybe this isn't an answer you are looking for.

You didn't tell us whether you were okay with the inductive argument but disliked (apparently) the prime-power case or you were unhappy with both aspects.

-
I'm fine with the inductive argument; it is a simple reduction step using the fact that $\mathbb{Q}(\zeta_{mn})$ is the composite of $\mathbb{Q}(\zeta_{m})$ and $\mathbb{Q}(\zeta_{n})$ if $m$ and $n$ are relatively prime. But the standard proof for the prime power case is not very illuminating. Your answer and that of Arno make me understand better what's going on. –  Andrea Ferretti Mar 7 '10 at 0:06
Suppose K = Q(a) where a is the root of a poly. in Z[x] that's Eisenstein with respect to p. Using the Eisenstein condition, the natural map Z[a] --> O_K/pO_K has kernel pZ[a], so Z[a] and pO_K meet in pZ[a]. Therefore the group O_K/Z[a] has no element of order p, so [O_K:Z[a]] is not divisible by p. At the same time this index is a factor of disc(f), so if you are in the case that disc(f) is a p-power (up to sign), then [O_K:Z[a]] is both not divisible by p and is a power of p, so the index is 1: O_K = Z[a]. When a = zeta_{p^r} - 1 this happens, so O_K = Z[zeta_{p^r}-1} = Z[zeta_{p^r}]. –  KConrad Mar 7 '10 at 1:21
Very nice. In the course I am teaching on local fields, we just did the structure theory of unramified and totally ramified extensions, so this will make a great homework problem! –  Pete L. Clark Mar 7 '10 at 3:18

In case of a prime power it is not so hard. Denote $R$ the ring of integers of ${\mathbf Q}(\zeta_n)$. First note that $X^n -1$ is separable mod $p$ when $p$ does not divide $n$. This implies $R [1/p] = {\mathbf Z}[\zeta_n, 1/p]$. To check that $R = {\mathbf Z}[\zeta_n]$ it then suffices to show that the local rings of ${\mathbf Z}[\zeta_n]$ at all primes above $p$ are DVRs. But you have the explicit element $\lambda = 1 - \zeta$ that you defined above and its norm equals $p$. This implies that there is only one prime above $p$ and that this ideal is generated by $\lambda$. In particular the local ring at this prime is regular and hence a DVR.

-
after a look in Langs alg number theory, I think the "taking-tensor-products-to-reduce-to-prime-powers" argument is the easiest. It is really quick: take a look on p 68 prop 17, you only need some basic properties of the different of a number field. –  Arno Kret Mar 8 '10 at 9:33

I don't know if this is what you're looking for, but I like the proof in Neukirch's Algebraic Number Theory Section I.10. Conceptually the idea (for $n$ a prime power) is that if the discriminant is a prime power and there is an element $\lambda$ whose norm is that prime, then $\mathbf{Z}[\lambda]$ should be the ring of integers. Here, $\lambda = 1- \zeta$ so $\mathbf{Z}[\lambda] = \mathbf{Z}[\zeta]$.

Then for general $n$ you use the fact that $\mathbf{Q}(\zeta_n)$ is the compositum of $\mathbf{Q}(\zeta_{\ell^r})$, and $\zeta_{\ell^r}$ is a power of $\zeta_n$.

-
I fear this is the "computational" proof I already know. By the way, is your second sentence an actual theorem? This may help streamlining the proof. –  Andrea Ferretti Mar 6 '10 at 16:49
I don't know that it's a significant enough phrase to call a theorem, but it's a good way to find the ring of integers when you have a totally ramified prime in $\mathbf{Z}[\lambda]$. –  stankewicz Mar 6 '10 at 16:58
@stankewicz: Ok, but apart from what we call it, is that true? Namely: if $\lambda$ is an algebraic integer in a number field with norm $p$ and if the discriminant is a power of $p$, then the ring of intgers is $\mathbb{Z}[\lambda]$. –  Andrea Ferretti Mar 6 '10 at 17:07
@Ferretti: If I were to state it as a theorem, it would be this: Let $K$ be an algebraic number field whose discriminant is a power of $p$ a prime number which is totally ramified in $K$. If $\mathbf{Z}[\lambda]$ is an order of $K$ in which $p$ totally ramifies then $\mathbf{Z}[\lambda]$ is the ring of integers. –  stankewicz Mar 6 '10 at 17:23
@Andrea Ferreti: The discriminant of $\mathbf Z[\lambda]$ is the discriminant of the ring of integers $R$ times the $d$th power of the index of $\mathbf Z[\lambda]$ in $R$, where $d$ is the degree of the extension. So if the discriminant of $\mathbf Z[\lambda]$ has no $d$th powers, one must have $\mathbf Z[\lambda]=R$. –  ACL May 27 '10 at 15:27

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# Simple inequalities.

1. Oct 25, 2006

### MathematicalPhysicist

i need to prove the following:
1)(1+1/n)^n<3 for every n>=3.
2) (x^n+y^n)/2>=((x+y)/2)^n for every n natural and every x,y>=0.
3) |a+1/a|>=2 for every a different than 0.

for the first i thought to use induction and to use the fact of increasing sequence (1+1/n)^n or of the decreasing sequence (1+1/n)^n+1, but i didnt get much far, i also tried to use the fact of 1/n<=1/3, but im stuck.

for the second i tried by induction: ((x+y)/2)^n+1<=(x^n+y^n)/2(x+y)/2 but im also stuck.
for the third i tried dissecting it into parts, i.e 0<a<1 a>1 or -1<a<0 a<-1 but i got stuck in a>1.
i mean for a>1 we can write a=h+1 when h>1 and thus h+1+1/(h+1)>2+1/(h+1)>2 but im having a problem with 0<h<1 then 1<h+1<2 1/2<1/(h+1)<1
3/2+h<1+h+1/(h+1) but form here i couldnt conclude that it's bigger than 2.

2. Oct 25, 2006

### StatusX

For the first, start by showing that sequence is bounded above by the other definition of e in terms of the sum of the inverse factorials, and then bound that series by a geometric series.

For the second, you're on the right track. Try to write what you get from that product as a sum of the expression you want ($(x^{n+1}+y^{n+1})/2$), and some other term, and show this extra term is nonpositive, so that the product is less than or equal to the expression you want.

Finally, for the last, focus on a>0, and note that your're trying to show there are no solutions to a+1/a<2. This can be rewritten as a quadratic inequality that you can show to have no real solutions.

3. Oct 25, 2006

### MathematicalPhysicist

but from the prodcut i get [(x^(n+1)+y^(n+1))/2+(x^ny+y^nx)/2]/2 how do i show that it's less than (x^(n+1)+y^(n+1))/2?
i mean obviously x^ny+xy^n is positive cause x,y>=0
for the first question, how would i solve this question without the knowledge of e and its definition?

and about the last question, **** it was that easy. (-:
thanks.

4. Oct 25, 2006

### StatusX

1. Keep going. You know what you want, so just subtract what you found from this, and show what you get is positive (pay attention to factors of 2 and 4).

2. You don't need to assume anything, just show:

$$(1+\frac{1}{n})^n \leq \sum_{m=0}^\infty \frac{1}{m!}$$

expand the LHS and this shouldn't be too hard.

5. Oct 26, 2006

### MathematicalPhysicist

ok, thanks.
btw what's wrong with the latex?


text:
# Math Help - Prove that the sequence is increasing

1. ## Prove that the sequence is increasing

Prove that the sequence $a_n=\frac{n^n}{n!}$ is increasing.

2. ## Re: Prove that the sequence is increasing

Compute $\frac{a_{n+1}}{a_n}$.

3. ## Re: Prove that the sequence is increasing

Originally Posted by alexmahone
Prove that the sequence $a_n=\frac{n^n}{n!}$ is increasing.
Can you show $\frac{{a_{n + 1} }}{{a_n }} = \left( {1 + \frac{1}{n}} \right)^n ~?$

4. ## Re: Prove that the sequence is increasing

$a_{n+1}=\frac{(n+1)^{n+1}}{(n+1)!}=\frac{(n+1)^n}{ n!}>\frac{n^n}{n!}=a_n$

So, $a_n$ is increasing.


text:
# Revision history [back]

### How do I restart a suspended VM after a reboot?

I was experimenting with managing server recovery after a reboot and put several vms in suspend status. To my horror I discovered that I cannot resume them. The problem appears to be described here (https://bugs.launchpad.net/nova/+bug/1052696), but does not offer any suggestion how to recover from this scenario. I have tried 'resume instance' from horizon - errors in nova-api.log: 2013-08-12 21:16:40.069 INFO nova.osapi_compute.wsgi.server [req-58ab0c59-7070-4ae3-af51-e12853cd6c30 3036f84a152d4b9e9a8e456e386c2d7d d31d9dff59e34b0d86dac87abcb05a0f] 172.30.1.24 "GET /v2/d31d9dff59e34b0d86dac87abcb05a0f/flavors/2542733c-e219-4e17-8830-136ba823190d HTTP/1.1" status: 200 len: 703 time: 0.0099411

I have tried to manually reset the state (http://docs.openstack.org/trunk/openstack-compute/admin/content/reset-state.html) nova reset-state <id>

I have tried using virsh shutdown <id>

but so far VM instances are in status of Shutdown or Suspended and Power status is Shutdown - cannot get them to boot.

### How do I restart a suspended VM after a reboot?

My setup is: Ubuntu 13.04 / grizzly with grizzly and grizzly-backports in etc/apt/sources.list.d/grizzly.list

I was experimenting with managing server recovery after a reboot and put several vms in suspend status. To my horror I discovered that I cannot resume them. The problem appears to be described here (https://bugs.launchpad.net/nova/+bug/1052696), but does not offer any suggestion how to recover from this scenario. I have tried 'resume instance' from horizon - errors in nova-api.log: nova-api.log:

2013-08-12 21:16:40.069 INFO nova.osapi_compute.wsgi.server [req-58ab0c59-7070-4ae3-af51-e12853cd6c30 3036f84a152d4b9e9a8e456e386c2d7d d31d9dff59e34b0d86dac87abcb05a0f] 172.30.1.24 "GET /v2/d31d9dff59e34b0d86dac87abcb05a0f/flavors/2542733c-e219-4e17-8830-136ba823190d HTTP/1.1" status: 200 len: 703 time: 0.0099411

I have tried to manually reset the state (http://docs.openstack.org/trunk/openstack-compute/admin/content/reset-state.html) nova reset-state <id>

I have tried using virsh shutdown <id>

but so far VM instances are in status of Shutdown or Suspended and Power status is Shutdown - cannot get them to boot.

 3 No.3 Revision Jobin 492 ●21 ●28 ●45 http://thelittleican.b...

### How do I restart a suspended VM after a reboot?

My setup is: Ubuntu 13.04 / grizzly with grizzly and grizzly-backports in etc/apt/sources.list.d/grizzly.list

etc/apt/sources.list.d/grizzly.list

I was experimenting with managing server recovery after a reboot and put several vms in suspend status. To my horror I discovered that I cannot resume them. The problem appears to be described here (https://bugs.launchpad.net/nova/+bug/1052696), but does not offer any suggestion how to recover from this scenario. I have tried 'resume instance' resume instance'from horizon - errors in nova-api.log:nova-api.log:

2013-08-12 21:16:40.069 INFO nova.osapi_compute.wsgi.server [req-58ab0c59-7070-4ae3-af51-e12853cd6c30 3036f84a152d4b9e9a8e456e386c2d7d d31d9dff59e34b0d86dac87abcb05a0f] 172.30.1.24 "GET /v2/d31d9dff59e34b0d86dac87abcb05a0f/flavors/2542733c-e219-4e17-8830-136ba823190d HTTP/1.1" status: 200 len: 703 time: 0.00994110.0099411

I have tried to manually reset the state (http://docs.openstack.org/trunk/openstack-compute/admin/content/reset-state.html) nova reset-state <id>

I have tried using virsh shutdown <id><ID>

but so far VM instances are in status of Shutdown or Suspended and Power status is Shutdown - cannot get them to boot.


text:
# 4.6: Mössbauer Spectroscopy

In 1957 Rudolf Mössbauer achieved the first experimental observation of the resonant absorption and recoil-free emission of nuclear γ-rays in solids during his graduate work at the Institute for Physics of the Max Planck Institute for Medical Research in Heidelberg Germany. Mössbauer received the 1961 Nobel Prize in Physics for his research in resonant absorption of γ-radiation and the discovery of recoil-free emission a phenomenon that is named after him. The Mössbauer effect is the basis of Mössbauer spectroscopy.

The Mössbauer effect can be described very simply by looking at the energy involved in the absorption or emission of a γ-ray from a nucleus. When a free nucleus absorbs or emits a γ-ray to conserve momentum the nucleus must recoil, so in terms of energy:

$E_{ \gamma - ray} \ = \ E_{\text{nuclear transition}}\ -\ E_{\text{recoil}} \label{1}$

When in a solid matrix the recoil energy goes to zero because the effective mass of the nucleus is very large and momentum can be conserved with negligible movement of the nucleus. So, for nuclei in a solid matrix:

$E_{\gamma - ray} \ =\ E_{\text{nuclear transition}} \label{2}$

This is the Mössbauer effect which results in the resonant absorption/emission of γ-rays and gives us a means to probe the hyperfine interactions of an atoms nucleus and its surroundings.

A Mössbauer spectrometer system consists of a γ-ray source that is oscillated toward and away from the sample by a “Mössbauer drive”, a collimator to filter the γ-rays, the sample, and a detector.

Figure $$\PageIndex{2}$$ hows the two basic set ups for a Mössbauer spectrometer. The Mössbauer drive oscillates the source so that the incident γ-rays hitting the absorber have a range of energies due to the doppler effect. The energy scale for Mössbauer spectra (x-axis) is generally in terms of the velocity of the source in mm/s. The source shown (57Co) is used to probe 57Fe in iron containing samples because 57Co decays to 57Fe emitting a γ-ray of the right energy to be absorbed by 57Fe. To analyze other Mössbauer isotopes other suitable sources are used. Fe is the most common element examined with Mössbauer spectroscopy because its 57Fe isotope is abundant enough (2.2), has a low energy γ-ray, and a long lived excited nuclear state which are the requirements for observable Mössbauer spectrum. Other elements that have isotopes with the required parameters for Mössbauer probing are seen in Table $$\PageIndex{1}$$.

 Most commonly examined elements Fe, Ru, W, Ir, Au, Sn, Sb, Te, I, W, Ir, Eu, Gd, Dy, Er, Yb, Np Elements that exhibit Mössbauer effect K, Ni, Zn, Ge, Kr, Tc, Ag, Xe, Cs, Ba, La, Hf, Ta, Re, Os, Pt, Hg, Ce, Pr, Nd, Sm, Tb, Ho, Tm, Lu, Th, Pa, U, Pu, Am

### Mössbauer Spectra

The primary characteristics looked at in Mössbauer spectra are isomer shift (IS), quadrupole splitting (QS), and magnetic splitting (MS or hyperfine splitting). These characteristics are effects caused by interactions of the absorbing nucleus with its environment.

Isomer shift is due to slightly different nuclear energy levels in the source and absorber due to differences in the s-electron environment of the source and absorber. The oxidation state of an absorber nucleus is one characteristic that can be determined by the IS of a spectra. For example due to greater d electron screening Fe2+ has less s-electron density than Fe3+ at its nucleus which results in a greater positive IS for Fe2+.

For absorbers with nuclear angular momentum quantum number I > ½ the non-spherical charge distribution results in quadrupole splitting of the energy states. For example Fe with a transition from I=1/2 to 3/2 will exhibit doublets of individual peaks in the Mössbauer spectra due to quadrupole splitting of the nuclear states as shown in red in Figure $$\PageIndex{2}$$.

In the presence of a magnetic field the interaction between the nuclear spin moments with the magnetic field removes all the degeneracy of the energy levels resulting in the splitting of energy levels with nuclear spin I into 2I + 1 sublevels. Using Fe for an example again, magnetic splitting will result in a sextet as shown in green in Figure $$\PageIndex{2}$$. Notice that there are 8 possible transitions shown, but only 6 occur. Due to the selection rule ІΔmIІ = 0, 1, the transitions represented as black arrows do not occur.

## Synthesis of Magnetite Nanoparticles

Numerous schemes have been devised to synthesize magnetite nanoparticles (nMag). The different methods of nMag synthesis can be generally grouped as aqueous or non-aqueous according to the solvents used. Two of the most widely used and explored methods for nMag synthesis are the aqueous co-precipitation method and the non-aqueous thermal decomposition method.

The co-precipitation method of nMag synthesis consists of precipitation of Fe3O4 (nMag) by addition of a strong base to a solution of Fe2+ and Fe3+ salts in water. This method is very simple, inexpensive and produces highly crystalline nMag. The general size of nMag produced by co-precipitation is in the 15 to 50 nm range and can be controlled by reaction conditions, however a large size distribution of nanoparticles is produced by this method. Aggregation of particles is also observed with aqueous methods.

The thermal decomposition method consists of the high temperature thermal decomposition of an iron-oleate complex derived from an iron precursor in the presence of surfactant in a high boiling point organic solvent under an inert atmosphere. For the many variations of this synthetic method many different solvents and surfactants are used. However, in most every method nMag is formed through the thermal decomposition of an iron-oleate complex to form highly crystalline nMag in the 5 to 40 nm range with a very small size distribution. The size of nMag produced is a function of reaction temperature, the iron to surfactant ratio, and the reaction time, and various methods are used that achieve good size control by manipulation of these parameters. The nMag synthesized by organic methods is soluble in organic solvents because the nMag is stabilized by a surfactant surface coating with the polar head group of the surfactant attached to and the hydrophobic tail extending away from the nMag (Figure $$\PageIndex{3}$$). An example of a thermal decomposition method is shown in Figure $$\PageIndex{3}$$.

## Mössbauer Analysis of Iron Oxide Nanoparticles

### Spectra and Formula Calculations

Due to the potential applications of magnetite nanoparticles (Fe3O4, nMag) many methods have been devised for its synthesis. However, stoichiometric Fe3O4 is not always achieved by different synthetic methods. B-site vacancies introduced into the cubic inverse spinel crystal structure of nMag result in nonstoichiometric iron oxide of the formula (Fe3+)A(Fe(1-3x)2+ Fe(1+2X)3+Øx)BO4 where Ø represents B-site vacancy. The magnetic susceptibility which is key to most nMag applications decreases with increased B-site vacancy hence the extent of B-site vacancy is important. The very high sensitivity of the Mössbauer spectrum to the oxidation state and site occupancy of Fe3+ in cubic inverse spinel iron oxides makes Mössbauer spectroscopy valuable for addressing the issues of whether or not the product of a synthetic method is actually nMag and the extent of B-site vacancy.

As with most analysis using multiple instrumental methods in conjunction is often helpful. This is exemplified by the use of XRD along with Mössbauer spectroscopy in the following analysis. Figure $$\PageIndex{4}$$ shows the XRD results and Mössbauer spectra “magnetite” samples prepared by a Fe2+/Fe3+ co-precipitation (Mt025), hematite reduction by hydrogen (MtH2) and hematite reduction with coal(MtC). The XRD analysis shows MtH2 and MT025 exhibiting only magnetite peaks while MtC shows the presence of magnetite, maghemite, and hematite. This information becomes very useful when fitting peaks to the Mössbauer spectra because it gives a chemical basis for peak fitting parameters and helps to fit the peaks correctly.

Being that the iron occupies two local environments, the A-site and B site, and two species (Fe2+ and Fe3+) occupy the B-site one might expect the spectrum to be a combination of 3 spectra, however delocalization of electrons or electron hopping between Fe2+ and Fe3+ in the B site causes the nuclei to sense an average valence in the B site thus the spectrum are fitted with two curves accordingly. This is most easily seen in the Mt025 spectrum. The two fitted curves correspond to Fe3+ in the A-site and mixed valance Fe2.5+ in the B-site. The isomer shift of the fitted curves can be used to determined which curve corresponds to which valence. The isomer shift relative to the top fitted curve is reported to be 0.661 and the bottom fitted curve is 0.274 relative to αFe thus the top fitted curve corresponds to less s-electron dense Fe2.5+. The magnetic splitting is quite apparent. In each of the spectra, six peaks are present due to magnetic splitting of the nuclear energy states as explained previously. Quadrupole splitting is not so apparent, but actually is present in the spectra. The three peaks to the left of the center of a spectrum should be spaced the same as those to the right due to magnetic splitting alone since the energy level spacing between sublevels is equal. This is not the case in the above spectra, because the higher energy I = 3/2 sublevels are split unevenly due to magnetic and quadrupole splitting interactions.

Once the peaks have been fitted appropriately, determination of the extent of B-site vacancy in (Fe3+)A(Fe(1-3x)2+ Fe(1+2X)3+Øx)BO4 is a relatively simple matter. All one has to due to determine the number of vacancies (x) is solve the equation:

$\frac{RA_{B}}{RA_{A}} = \frac{2-6x}{1-5x} \label{3}$

where RAB or A = relative area

$\frac{Area\ A\ or\ B\ site\ curve}{Area\ of\ both\ curves} \label{4}$

of the curve for the B or A site respectively

The reasoning for this equation is as follows. Taking into account that the mixed valance Fe2.5+ curve is a result of paired interaction between Fe2+ and Fe3+ the nonstochiometric chemical formula is (Fe3+)A(Fe(1-3x)2+Fe(1+2X)3+Øx)BO4. The relative intensity (or relative area) of the Fe-A and Fe-B curves is very sensitive to stoichiometry because vacancies in the B-site reduce the Fe-A curve and increase Fe-B curve intensities. This is due to the unpaired Fe5x3+ adding to the intensity of the Fe-A curve rather than the Fe-B curve. Since the relative area is directly proportional to the number of Fe contributing to the spectrum the ratio of the relative areas is equal to stoichiometric ratio of Fe2.5+ to Fe3+, which yields the above formula.

Example Calculation:

For MtH2 RAA/RAB = 1.89

Plugging x into the nonstoichiometric iron oxide formula yeilds:

$\frac{RA_{B}}{RA_{A}} = \frac{2-6x}{1-5x} \label{5}$

solving for x yields

$x=\frac{2-\frac{RA_{A}}{RA_{B}}}{5 \frac{RA_{A}}{RA_{B}}\ +\ 6} \label{6}$

(Fe3+)A(Fe 1.95722+ Fe0.03563+)BO4 (very close to stoichiometric)

Figure $$\PageIndex{2}$$: Parameters and nonstoichiometric formulas for MtC, Mt025, and MtH2
Sample RAB/RAA X Chemical Formula
MtH2 1.89 0.007 (Fe3+)A(Fe0.9792+Fe1.0143+)BO4
MtC 1.66 0.024 (Fe3+)A(Fe0.9292+Fe1.0483+)BO4
Mt 025 1.60 0.029 (Fe3+)A(Fe0.9142+Fe1.0573+)BO4

## Chemical Formulas of Nonstoichiometric Iron Oxide Nanoparticles from Mössbauer Spectroscopy

### Chemical Formula Determination

Magnetite (Fe3O4) nanoparticles (n-Mag) are nanometer sized, superparamagnetic, have high saturation magnetization, high magnetic susceptibility, and low toxicity. These properties could be utilized for many possible applications; hence, n-Mag has attracted much attention in the scientific community. Some of the potential applications include drug delivery, hyperthermia agents, MRI contrast agents, cell labeling, and cell separation to name a few.

The crystal structure of n-Mag is cubic inverse spinel with Fe3+ cations occupying the interstitial tetrahedral sites(A) and Fe3+ along with Fe2+ occupying the interstitial octahedral sites(B) of an FCC latticed of O2-. Including the site occupation and charge of Fe, the n-Mag chemical formula can be written (Fe3+)A(Fe2+Fe3+)BO4. Non-stoichiometric iron oxide results from B-site vacancies in the crystal structure. To maintain balanced charge and take into account the degree of B-site vacancies the iron oxide formula is written (Fe3+)A(Fe(1-3x)2+ Fe(1+2X)3+Øx)BO4 where Ø represents B-site vacancy. The extent of B-site vacancy has a significant effect on the magnetic properties of iron oxide and in the synthesis of n-Mag stoichiometric iron oxide is not guaranteed; therefore, B-site vacancy warrants attention in iron oxide characterization, and can be addressed using Mössbauer spectroscopy.

4.6: Mössbauer Spectroscopy is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Pavan M. V. Raja & Andrew R. Barron via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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# Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is widely used in the fields of biochemistry and chemistry in order to estimate the pH value of a given buffer solution when the concentration of the acid and its conjugate base (or the base and the corresponding conjugate acid) are known.

An equation that could calculate the pH value of a given buffer solution was first derived by the American chemist Lawrence Joseph Henderson. This equation was then re-expressed in logarithmic terms by the Danish chemist Karl Albert Hasselbalch. The resulting equation was named the Henderson-Hasselbalch Equation.

## Henderson-Hasselbalch Equation Derivation

The ionization constants of strong acids and strong bases can be easily calculated with the help of the direct methods. However, when it comes to weak bases and weak acids, the same methods cannot be used since the extent of ionization of these acids and bases are very low, i.e. weak acids and bases hardly ionize.

Thus, in order to approximate the pH of these types of solutions, the Henderson-Hasselbalch Equation is used.

Let us take an example of ionization of weak acid HA:

$HA~ +~ H_{2}O~ ⇋ ~H^{+}~ +~ A^{−}$

Acid dissociation constant, K a can be given as:

K a = $\frac{[H^+][A^-]}{[HA]}$

Taking, negative log of RHS and LHS:

$-log~K_a$ = $-log~\frac{[H^+][A^-]}{[HA]}$

$\Rightarrow -log~K_a$ = $-log~[H^+] ~ – ~log~\frac{[A^-]}{[HA]}$

As we know, $-log~[H^+]$ = $pH$ and $-log~K_a$ = $pKa$,

Above equation can be written as,

$pK_a$ = $pH~ -~ log~\frac{[A^-]}{[HA]}$

Rearranging the equation,

$\Rightarrow pH$ = $pK_a ~+ ~log\frac{[A^-]}{[HA]}$

The above equation is known as Henderson-Hasselbalch equation, popularly known as Henderson equation. It is very useful for estimating the pH of a buffer solution and finding the equilibrium pH in acid-base reactions. From the equation we can infer when $pH$ = $pK_a$

$log~\frac{[A^-]}{[HA]}$ = $0$

$[A^-]$ = $[HA]$

That is, when $pH$ = $pK_a$, concentration of both the species are same or in other words, acid will be half dissociated.

Similarly, for a weak base B:

$B~ +~ H_2O~ ⇋ ~ OH^− ~+~ HB^+$

Base dissociation constant, Kb, of the base can be given as,

$K_b$ = $\frac{[BH^+][OH^-]}{[B]}$

Taking negative log of RHS and LHS

$-logK_b$ = $-log \frac{[BH^+][OH^-]}{[B]}$

$\Rightarrow -logK_b$ = $-log~[OH^-]-log\frac{[BH^+]}{[B]}$

As we know, $-log~[OH^-]$ = $pOH$ and $-logK_b$ = $pKb$,

Above equation can be written as,

$pK_b$ = $pOH ~- ~log\frac{[BH^+]}{[B]}$

Rearranging the equation,

$\Rightarrow pOH$ = $pK_b + log~\frac{[BH^+]}{[B]}$

### Limitations of Henderson-Hasselbalch equation:

The Henderson-Hasselbalch equation makes correct predictions for weak acids and bases but fails to predict accurate values for the strong acids and strong bases as it assumes that the concentration of the acid and its conjugate base at chemical equilibrium will remain the same as the formal concentration.

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#### Practise This Question

Which of the following are the acceptable values for molecularity of a reaction?


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What Do You Expect the Nature of Hydrides Is, If Formed by Elements of Atomic Numbers 15, 19, 23 and 44 with Dihydrogen? Compare Their Behaviour Towards Water. - Chemistry

ConceptHydrides Ionic Or Saline Hydrides

Question

What do you expect the nature of hydrides is, if formed by elements of atomic numbers 15, 19, 23 and 44 with dihydrogen? Compare their behaviour towards the water.

Solution 1

The elements of atomic numbers 15, 19, 23, and 44 are phosphorus, potassium, vanadium, and ruthenium respectively.

1) Hydride of phosphorus

Hydride of nitrogen (PH3) is a covalent molecule. It is an electron-rich hydride owing to the presence of excess electrons as a lone pair on phosphorus.

2) Hydride of potassium

Dihydrogen forms an ionic hydride with potassium owing to the high electropositive nature of potassium. It is crystalline and non-volatile in nature.

3) Hydrides of Vanadium and Ruthenium

Both vanadium and ruthenium belong to the d–block of the periodic table. The metals of d–block form metallic or non–stoichiometric hydrides. Hydrides of vanadium and ruthenium are therefore, metallic in nature having a deficiency of hydrogen.

4) Behaviour of hydrides towards water

Potassium hydride reacts violently with water as: KH_((s)) + H_2O_((aq)) -> KOH_((aq)) + H_(2(g))

Phosphorus (PH3)  is covalent hydride and slightly soluble in water.

Hydrides of vanadium and Ruthenium do not react with water. Hence, the increasing order of reactivity of the hydrides is (V, Ru) H < NH< KH.

Solution 2

Atomic No. 15 is of phosphorus. The hydride is PH3 and its nature is covalent. Atomic No. (Z = 19) is of potassium. The hydride is KH and it is ionic in nature. Atomic No. (Z = 23) is of vanadium. The hydride is VH. It is interstitial or metallic. Atomic No. 44 is of ruthenium, its hydride is interstitial or metallic.

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# Volume under the elliptic paraboloid

1. Nov 6, 2008

Greetings again,

this is another I'm stuck on...

Calculate the volume under the elliptic paraboloid z = x^{2} + 4y^{2} and over the rectangle R = [-1, 1] \times [-4, 4].

I'm not sure how to find the limits of z for the triple integral. Can I somehow integrate the function z, and only do a double integral? Perhaps going from -1, 0 and -4, 0 and then double it?

2. Nov 6, 2008


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# How is a MacNeille completion “universal” like a beta-compactification is “universal”?

The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping from its beta-Cpt to the range space (Engelking, Outline, Chapter 5.3). For these reasons, it has performed well as the "universal" space in many applications, notably in Ellis' theory of topological dynamics. What "like" theories are known for the MacNeille completion of a partially ordered set? For example, is the MacNeille completion the "biggest" completion in the same spirit as the Stone-Cech compactification? Can every monotone map from the original POSET to another POSET be extended up through the MacNeille completion? Please accept my apologies for not knowing the partial order theory as well as I know the topological theory. Thank you.

• – Qiaochu Yuan Feb 17 '15 at 20:09
• I also found answers to my questions in the work of Gehrke & colleagues. Gehrke & Priestley published a paper titled "Canonical extensions and completions of posets and lattices" which answered parts of my questions, in particular, the one I so poorly described as the beta-compactification-like question. Proposition 2.1 and their discussion presents compact completions with the universal mapping property, and they show where the MacNeille completion fits in. The G&P paper is strongly related to a 2nd paper by Gehrke, Jansana, and Palmigiano in which these ideas are also illustrated in detail. – James Brewer Mar 4 '15 at 21:35

A subset $A$ of a complete lattice $L$ is said to be join-dense if $L=\{\bigvee^{L} R|R\subseteq A\}$ and $A$ is said to be meet dense in $L$ if $L=\{\bigwedge^{L}R|R\subseteq A\}$. It turns out that the Dedekind-MacNeille completion of a poset $P$ is up-to-an isomorphism preserving $P$ the only complete lattice $L$ with $P\subseteq L$ and where $P$ is both join dense and meet dense in $L$.

As it was mentioned by Matthias Wendt, the Dedekind-MacNeille completion is the smallest completion of $P$. Let me formalize what I mean by smallest.

$\mathbf{Proposition}$ Suppose that $P$ is a poset and $L$ is a complete lattice such that $P\subseteq L$ and $P$ is join-dense in $L$. Then $L$ is up to an isomorphism preserving $P$ the Dedekind-MacNeille completion of $P$ if and only if whenever $M$ is a complete lattice where $P\subseteq M$ and $P$ is join-dense in $M$, then there is a $j:M\rightarrow L$ such that $j(p)=p$ and $j(\bigvee^{M}R)=\bigvee^{L}j[R]$ whenever $R\subseteq M$ (from these properties, one can immediately deduce that the mapping $j$ is always surjective).

In other words, the Dedekind-MacNeille completion of a poset $P$ is the smallest complete lattice in the lattice of completions $L$ of $P$ so that $P$ is join-dense in $L$. And yes, the completions $L$ of $P$ so that $P$ is join-dense in $L$ do form a complete lattice.

There is also another sense in which the Dedekind-MacNeille completion of a poset is the smallest completion.

We say that a poset $(Y,\leq')$ is a minimal completion of a poset $(X,\leq)$ if

1. $X\subseteq Y$

2. $\leq=X^{2}\cap\leq'$ and

3. Whenever $X\subseteq Z\subseteq Y$ and $(Z,\leq'\cap Z^{2})$ is a complete lattice, then $X=Z$.

I showed in this answer that the Dedekind-MacNeille completion of a poset is the unique minimal completion of a poset.

• What? I can think of at least two other reasonable completions, namely the free cocompletion $P^{op} \Rightarrow 2$, and the free completion $(P \Rightarrow 2)^{op}$. These correspond to looking at downward-closed and upward-closed sets respectively. – Qiaochu Yuan Feb 18 '15 at 18:02
• Qiaochu Yuan. I was not being clear about what I meant saying that the Dedekind-MacNeille completion is the only reasonable completion of a poset. I apologize for that. I only meant to say that the Dedekind-MacNeille completion is only completion in which the original poset is both meet dense and join dense and if one wants the original poset to be meet-dense and join-dense in the complete lattice, then the completion is the Dedekind-MacNielle completion. This is the sense in which the Dedekind-MacNielle completion is the only completion of a poset. – Joseph Van Name Feb 18 '15 at 23:19
• Qiaochu Yuan. I agree that for different purposes there are other reasonable notions of a completion of a poset. For instance, anyone interested in forcing would be interested in the Boolean completion of a partially ordered set where the Boolean completion of a separative poset is the unique complete Boolean algebra that contains the poset as a join-dense subset. – Joseph Van Name Feb 19 '15 at 0:39

There are some differences on the categorical level.

The compact Hausdorff spaces are a reflective subcategory of topological spaces, and the Stone-Cech compactification is left adjoint to the inclusion of compact Hausdorff spaces into topological spaces. This basically encodes the universality and is enough reasons for ubiquitous appearance.

For the Dedekind-MacNeille completion, things are a bit different. As a first aside, the Dedekind-MacNeille completion is not the "biggest" something, it is the smallest complete lattice that contains the given partially ordered set. The category of complete lattices is not reflective inside the category of partially ordered sets with monotone maps. What is true is that the complete lattices are injective objects for order-embeddings, and the Dedekind-MacNeille completion is the injective hull of a poset, see the Wikipedia article. If you want the Dedekind-MacNeille completion as a reflector resp. adjoint functor, you have to consider so-called cut-stable maps, see

• M. Erné. The Dedekind-MacNeille completion as a reflector. Order 8 (1991), 159-173.
• Matthias, I was reading research by Twuenissen and Venema (staff.science.uva.nl/~yde/papers/mcnle.pdf). I had not found research describing "sizes" of completions,so I look forward to reading Erné and understanding what you mean by invectives objects and that the D-M is an injective hull of a poset. TY. – James Brewer Feb 17 '15 at 20:50