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Two-timescale response of a large Antarctic ice shelf to climate change

Abstract

A potentially irreversible threshold in Antarctic ice shelf melting would be crossed if the ocean cavity beneath the large Filchner–Ronne Ice Shelf were to become flooded with warm water from the deep ocean. Previous studies have identified this possibility, but there is great uncertainty as to how easily it could occur. Here, we show, using a coupled ice sheet-ocean model forced by climate change scenarios, that any increase in ice shelf melting is likely to be preceded by an extended period of reduced melting. Climate change weakens the circulation beneath the ice shelf, leading to colder water and reduced melting. Warm water begins to intrude into the cavity when global mean surface temperatures rise by approximately 7 °C above pre-industrial, which is unlikely to occur this century. However, this result should not be considered evidence that the region is unconditionally stable. Unless global temperatures plateau, increased melting will eventually prevail.

Introduction

In order to constrain long-term projections of sea-level rise, it is crucial to understand the likely response of the Filchner–Ronne Ice Shelf (FRIS) to climate change. FRIS is one of Antarctica’s largest and most dynamically important ice shelves, which accounts for over 10% of the discharge of the entire Antarctic Ice Sheet1 and plays a crucial role in global thermohaline circulation2. Unlike ice shelves in the Amundsen Sea, which are already in contact with relatively warm water flowing from the deep ocean (≈1 °C)3, the cavity beneath FRIS is currently filled with cold water (≈−2 °C)2. This results in relatively low ice shelf basal melt rates with some regions of refreezing. Whether the FRIS cavity remains in this cold state, or transitions to a warm state akin to the Amundsen Sea, will have significant implications for its source glaciers and ultimately global sea level rise4.

Circulation beneath FRIS2 is driven by density gradients between saltier, denser water on the surrounding continental shelf and fresher, lighter water within the cavity. Cold and dense High Salinity Shelf Water (HSSW) flows into the cavity from the strong sea ice formation regions of the Ronne Depression and Berkner Bank (Fig. 1a), where salinity is enhanced by brine rejection. As it travels, the HSSW melts the base of the ice shelf, becoming colder and fresher as meltwater is entrained. The resulting Ice Shelf Water (ISW), supercooled relative to the surface freezing point, exits the cavity through the Filchner Trough.

For decades, it was thought that FRIS melting would decrease with climate change. Observations show that FRIS melt rates are highest when HSSW flow through the cavity is strongest5. Nicholls6 hypothesised that climate warming would provoke a response similar to springtime warming, whereby sea ice formation decreases, the flow of HSSW weakens, and melt rates are reduced.

More recently, Hellmer et al.7 and related studies8,9,10 proposed an alternative hypothesis: that FRIS melting would increase with climate change, triggered by a fundamental change in cavity circulation. Using an ice-ocean model forced with a climate change projection, Warm Deep Water (WDW) from offshore was simulated to enter the cavity through the Filchner Trough, dramatically increasing basal melt rates. A similar circulation change has been triggered in other ocean models by manually varying the cavity salinity or atmospheric forcing11,12. However, it remains unclear how easily this circulation change could occur in reality, and over what timescales.

These two conflicting hypotheses for the future of FRIS melting—whether it will decrease or increase—have never been reconciled. Recent advances in Antarctic modelling warrant a re-examination of this topic. All previous simulations projecting WDW intrusion in the 21st century were forced by the same climate model projection, which is now a decade old13. In the years since, global climate models have reduced their atmospheric biases in the Antarctic region14, which increases confidence in their projections. Regional ice-ocean models, which can downscale these results to ice shelf cavities, have become more sophisticated15 with higher resolution, coupled ice sheet dynamics, and reduced present-day biases. Even so, future projections of ice shelf melting are scarce in the scientific literature.

In this study, we find a two-timescale response of the FRIS cavity to climate warming (Fig. 1b, c). The first stage is an extended period of weakened circulation and reduced basal melting. In the second stage, FRIS melting increases as WDW flows into the cavity via the Filchner Trough. These two stages roughly correspond to the contrasting paradigms of Nicholls6 and Hellmer et al.7, showing they are two sides of the same coin. However, the second stage requires extremely strong greenhouse forcing in our simulations, indicating that FRIS melting is unlikely to increase in the 21st century.

Results

Experimental design

Here we present simulations with the coupled ice sheet-ocean model ÚaMITgcm, which simulates ocean circulation, sea ice processes, ice shelf basal melting, and ice sheet flow in the Weddell Sea and FRIS region (Methods). It is forced in the atmosphere and at the lateral ocean boundaries with output from the global climate model UKESM (United Kingdom Earth System Model)16. Present-day simulations with this combination of models produce basal melt rates and water mass characteristics exhibiting good agreement with observations, with minimal grounding line drift (Supplementary Note 2).

We simulate three idealised scenarios following the CMIP6 protocol (Coupled Model Intercomparison Project phase 6)17: piControl holds CO2 fixed at pre-industrial levels, abrupt-4xCO2 instantaneously quadruples CO2 and then holds it fixed, and 1pctCO2 increases CO2 by 1% per year. All simulations run for 150 years. To investigate longer-term impacts, we then run 50-year extensions of each simulation, where the last decade of UKESM forcing is repeated 5 times. These idealised scenarios were chosen over realistic climate projections, which are shorter with less extreme changes in CO218, and may not induce sufficient warming to trigger a circulation change beneath FRIS.

The computational expense of ÚaMITgcm means that a large ensemble of long simulations would be impractical. However, the signal-to-noise ratio of forced change to internal climate variability is relatively small in scenarios with such extreme CO2 forcing. We therefore use a single ensemble member for each UKESM scenario. UKESM was chosen from CMIP6 due to early data availability, as well as its good representation of the present-day Antarctic region14.

Freshening of continental shelf

Both stages of the response are ultimately driven by freshening of the source waters which feed the FRIS cavity. Average salinity on the continental shelf in front of FRIS (Fig. 2) decreases by approximately 0.7 psu in 1pctCO2 and almost 1 psu in abrupt-4xCO2, relative to the piControl mean. While the rate of freshening is relatively steady throughout 1pctCO2, in abrupt-4xCO2 it largely plateaus after 80 years, in line with global temperature change.

This freshening is caused by a combination of local and remote factors, which we examine using a salt budget analysis for the abrupt-4xCO2 simulation. Integrated salt fluxes over the continental shelf in front of FRIS (Fig. 3a) show that surface processes are the main cause of the freshening. Surface salt flux anomalies are strongly negative during the entire simulation.

Figure 3b further decomposes the surface processes, showing anomalies in surface freshwater fluxes (note the opposite sign to Fig. 3a, which shows salt fluxes). The increased freshwater is primarily the result of increased sea ice melting. A warmer atmosphere means more sea ice is melting locally, rather than being exported from the continental shelf and later melting offshore. Sea ice freezing initially strengthens (more negative freshwater flux) owing to the greater expanse of open water at the start of each freeze season; however, it only partially compensates for the increased melting. Towards the end of the simulation, sea ice freezing weakens (more positive freshwater flux) as the length of the freeze season becomes shorter. There is a very small contribution from increased precipitation over the continental shelf.

While surface processes dominate, Fig. 3a also shows a freshening contribution from advection beginning around year 40, which indicates that fresher water is being advected into the region along the upstream coastal current. This fresher coastal current is partially caused by increased ice shelf melting in the upstream Eastern Weddell sector, within the ÚaMITgcm domain (eastern boundary 30E). It also reflects freshening in UKESM, which then enters the ÚaMITgcm domain at the eastern boundary. The Southern Ocean freshening seen in UKESM is a common response to climate warming among CMIP models19 and is primarily caused by increased precipitation, with a smaller role for surface melting of the Antarctic Ice Sheet. Precipitation over the Antarctic Ice Sheet in UKESM is not immediately routed to the ocean, but rather is handled by a snow melting scheme. Note that the relatively low resolution of UKESM may decrease confidence in its ability to accurately simulate coastal dynamics20. Therefore, some uncertainty exists in the relative role of advection in this salt budget, which could be important for FRIS melt rates21.

The changes in surface fluxes and advection are partially offset by increased diffusion. After year 80, a new equilibrium is reached and the total salt flux anomalies oscillate around 0, indicating the freshening has plateaued (as in Fig. 2).

In summary, the main causes of freshening are as follows:

1. 1.

An increase in local sea ice melting

2. 2.

The advection of fresher water into the region (caused by increased ice shelf melting upstream, and increased precipitation over the Southern Ocean)

3. 3.

A decrease in sea ice formation, but only in the later stages of the simulation

These mechanisms differ from the hypothesis of Nicholls6, which relied entirely on a reduction in local sea-ice formation.

Continental shelf freshening has implications for the density gradients which drive circulation beneath FRIS. There are two density gradients to consider:

1. 1.

between the dense source HSSW (in Ronne Depression and over Berkner Bank) and the less dense FRIS cavity; and

2. 2.

between the dense outflowing ISW (in Filchner Trough) and the less dense offshore WDW.

A reversal of the first density gradient characterises Stage 1 of the response, when HSSW inflow slows and ice shelf melting decreases. A reversal of the second density gradient characterises Stage 2, when WDW flows into the cavity and ice shelf melting increases.

Figure 4 illustrates how these density gradients change as the climate warms and the continental shelf freshens. The complexity of ocean circulation means that individual density transects are not simple determinants of the direction of flow, which also depends on topography, wind stress, and many other factors. However, changes in density gradients do influence, and can reasonably explain, circulation changes seen in the simulation (see Supplementary Movie 1).

In the Ronne Depression (Fig. 4a), bottom density is initially relatively constant along the entire length of the depression. HSSW flows freely downslope, southward from the continental shelf into the western Ronne cavity, and then into the rest of the FRIS cavity which is less dense. Beginning in Stage 1, the continental shelf freshens more quickly than the cavity. This creates a density barrier by which the continental shelf (point 3) is less dense than the cavity (point 2) and especially the deep cavity (point 1), impeding the flow of HSSW beneath the ice shelf.

In the Filchner Trough (Fig. 4b), the cavity (point 4) and continental shelf are initially more dense than the offshore WDW (point 5), creating a barrier to onshore flow. In Stage 1, the continental shelf freshens, which allows small incursions of WDW to cross the continental slope. However, the Filchner cavity is still sufficiently dense that the WDW incursions do not enter the cavity; rather, they are overpowered by ISW flowing in the opposite direction. In Stage 2, the cavity has freshened so much that WDW can flow along the entire length of the Filchner Trough and access the ice shelf.

Stage 1: ice shelf melting decreases

A reversed density gradient between the inflowing HSSW and the FRIS cavity has profound impacts on the ice shelf. Weaker circulation (Fig. 5a) means that HSSW, the main source of heat to the cavity, is not efficiently renewed. The water within the cavity becomes relatively stagnant, and cools as its heat is extracted by the ice shelf (Fig. 5b). The reduction in thermal driving, particularly near the deep grounding lines, leads to reduced basal melting (Fig. 5c). Compared to the piControl mean, basal mass loss is 20% lower averaged over Stage 1 of 1pctCO2, and 27% lower for abrupt-4xCO2.

Although basal melt rates are lower, the longer residence time induced by slower circulation means that more heat is ultimately extracted from the source HSSW through ice shelf melting. The resulting supercooled ISW, which flows out into Filchner Trough, becomes colder throughout Stage 1 (blue regions in Fig. 6a).

Stage 2: ice shelf melting increases

The freshening signal gradually propagates through the FRIS cavity and into the ISW outflow (Fig. 6b). Eventually, the direction of flow through Filchner Trough reverses, so that pulses of WDW enter the cavity (Fig. 1c). This response is in line with previous analyses9,11. Modified WDW also enters the cavity from its original region of inflow, the Ronne Depression (Supplementary Note 3 and Supplementary Movie 1). It follows the existing pathways of HSSW inflow, but the temperature of the water column no longer drops as low as the surface freezing point.

In 1pctCO2, WDW intrusion begins in the last 5 years of the main simulation (model year 145; Fig. 5b). This pulse is repeated each decade of the 50-year extension, indicating that interannual variability in the forcing modulates the initial WDW intrusion. The cavity slowly warms following these regular pulses, and basal mass loss increases (Fig. 5c), but does not yet reach levels substantially higher than pre-industrial.

In abrupt-4xCO2, the first major WDW pulse occurs during years 80–90, causing a warming of ≈0.2 °C in the cavity and a doubling of melt rates compared to piControl (Fig. 5b, c; Supplementary Movie 1). After this initial pulse, the WDW recedes and melt rates approach mean piControl values, until year 140 when a much larger WDW pulse begins. By the end of the 50-year extension, melt rates are a factor of 21 higher than the piControl mean, and cavity temperatures 2.7 °C warmer. These changes are similar in magnitude to previous studies7, but the greenhouse forcing is much more extreme.

Impact on the ice sheet

The simulated changes in basal melt rates affect ice shelf thickness and the velocity of upstream glaciers. Compared to piControl, during Stage 1 the ice shelf is generally thicker, especially near the deep grounding lines (Fig. 7a). Since weakening circulation beneath FRIS reduces refreezing as well as basal melting, the ice shelf is thinner in regions of net refreezing (central Ronne Ice Shelf and east of Berkner Island; see Fig. 1). The flow of ice streams is generally slower (Fig. 7c), particularly the Slessor Glacier and Foundation Ice Stream, in response to increased buttressing at the grounding line.

These changes are reversed during Stage 2, and by the end of the 50-year extension there is significant and widespread thinning across the ice shelf (Fig. 7b). All of the upstream glaciers accelerate, particularly the Institute Ice Stream and the Slessor Glacier (Fig. 7d). The grounding line retreats relative to piControl, with the most significant retreat near the Institute Ice Stream. The final contribution to global sea level rise is only 1 cm (Methods). However, this rise occurs entirely over the last 40 years of the extension, and is rapidly accelerating. The full response of the ice sheet to increased basal melting of FRIS would occur over timescales much longer than can be simulated with a coupled ice-ocean model.

Timescales of change

Stage 1 is a gradual trend, which is initially obscured by decadal variability. It becomes detectable (Methods) after 69 years in 1pctCO2 and 14 years in abrupt-4xCO2, which corresponds to ≈3 °C and ≈5 °C of warming respectively. Observed global mean near-surface air temperature is now ≈1 °C above pre-industrial, which corresponds to ≈30 years of the 1pctCO2 experiment (Fig. 8). Our simulations therefore suggest that Stage 1 is unlikely to be discernible in observations until warming proceeds further.

Stage 2 begins when the Filchner Ice Shelf front transitions from a net exporter of supercooled ISW to a net importer of WDW above the surface freezing point (Methods). This transition occurs after 79 years in abrupt-4xCO2 and 147 years in 1pctCO2. In both cases, this corresponds to ≈7 °C of average global near-surface warming relative to pre-industrial.

UKESM climate projections suggest that 7 °C of warming could only be reached in the 21st century under the highest emissions scenarios. Figure 8 compares global mean near-surface air temperature anomaly in the idealised climate scenarios used in this study (1pctCO2 and abrupt-4xCO2) with more realistic climate change scenarios which could occur over the 21st century (the ScenarioMIP experiments)18. It is only the most extreme scenario (SSP5-8.5) which exceeds 7 °C of warming by the end of the century. The trajectory of SSP3-7.0 suggests it would also exceed this threshold in the 22nd century, but this is not obvious for the cooler scenarios (SSP2-4.5 and SSP1-2.6). With regards to the 3–5 °C warming required for detectability of Stage 1, only the most moderate scenario (SSP1-2.6) stays clearly below this range. In all other scenarios, we would expect Stage 1 to become detectable at some point in the second half of the 21st century.

This comparison should be interpreted with caution, as UKESM is a single model with a higher climate sensitivity than the CMIP6 ensemble mean22. The true timescales could therefore be even longer. Other forcing agents in the ScenarioMIP experiments, such as changes in stratospheric ozone, could also have a regional effect on the Antarctic climate23 which may confound the comparison.

Discussion

Our simulations are the first to identify a two-timescale response of FRIS melting to a warming climate. This result resolves previous disagreement as to whether FRIS melting would increase or decrease in response to climate change. Stage 1 was previously hypothesised based on one year of observations6, but no model projections were provided to support this hypothesis. Conversely, previous modelling studies using two closely-related ice-ocean models7,8,9 advanced directly to Stage 2 and did not appear to simulate a discernible Stage 1. We hypothesise that these studies may have been overly sensitive to WDW inflow, as they were all forced with the same climate model projection, now two generations old, with no bias corrections. The idealised study of Hazel and Stewart11 is not comparable to our results, as it applied large step-changes in atmospheric variables which triggered an immediate Stage 2 response, with little opportunity for transient changes.

The challenge now is to bound the timescales of Stage 1 and Stage 2. Our results call for a larger model intercomparison project of ice shelf melting projections in the Weddell Sea, to quantify the model structural uncertainty in these estimates. Internal climate variability may control the precise timing of WDW pulses, so a range of CMIP models and ensemble members should be used for forcing, as well as a range of resolutions in the ocean and atmosphere. Fully coupled models, which simulate the atmosphere, ocean, and ice sheet simultaneously, should be included as they are developed. These models would simulate atmosphere-ocean feedbacks, which may be important to FRIS, but which cannot be captured by ice-ocean models such as ÚaMITgcm.

At the same time, observational programmes should monitor the FRIS cavity and continental shelf for signs of Stage 1 beginning. It is not possible to make judgements based on existing observations, which have been sparse until recently. Satellite estimates of basal melt rates24,25 have not existed for long enough to detect a climatological trend in FRIS melting. The few in-situ measurements which exist beneath the ice shelf show large variability related to the strength of circulation2,5, which masks any trend in cavity water mass properties. Continued observations would allow for better detection of trends in the region, which could then be used to constrain models. In particular, long-term monitoring should continue in the Filchner Trough26, as it could detect cooling and freshening of ISW.

Finally, our results suggest that the FRIS region may be less sensitive to climate change than previously thought. In order for melt rates to increase, an extraordinary amount of global warming (7 °C in the UKESM) is required. This far exceeds the pledges of the Paris Agreement, which aims to keep warming below 2 °C. Unless these global mitigation efforts fail dramatically, there is a good chance that the FRIS sector of the Antarctic Ice Sheet could be preserved. However, if global temperatures continue to climb, decreased melting from FRIS should not be considered evidence that the region is unconditionally stable. Our study finds that if continental shelf freshening proceeds far enough, reduced melting will eventually give way to increased melting as warm water floods the cavity. A reduction in FRIS melting could paradoxically be an early warning sign for a later destabilisation of the region.

Methods

Simulations are performed with a regional Weddell Sea configuration of the ÚaMITgcm model. This coupled framework consists of three parts:

1. 1.

the MITgcm ocean model (Massachusetts Institute of Technology general circulation model)27,28, which includes ice shelf thermodynamics and sea ice

2. 2.

the Úa ice flow model29, which governs the movement and geometry of the ice shelves and ice sheet

3. 3.

a custom-built coupler that handles the exchange of information between the two models

MITgcm ocean

The ocean component is based on the configuration from Naughten et al.30, which uses a Weddell Gyre domain with open boundaries at 30E and 61S. Horizontal resolution is 0.25 degrees scaled by cosine of latitude, and there are 120 vertical levels with 25 m resolution throughout the FRIS cavity. Bathymetry follows Bedmap231 with updates in the Filchner Ice Shelf cavity from Rosier et al.32 and the grounded iceberg A23-A as defined by Schaffer et al.33. The FRIS ice shelf draft is given by Úa as described below; outside the Úa domain (e.g. for the Larsen and Fimbul ice shelves) the ice shelf draft follows Bedmap2. In regions with thin water columns and steeply sloping ice shelves, the bathymetry is deepened by the minimum amount to prevent the formation of spurious subglacial lakes34, a process hereafter referred to as digging.

To ensure numerical stability on the long timescales used in this study, we do not use the z* coordinate system as in Naughten et al.30, but instead employ MITgcm’s linear free surface scheme. We use an ice shelf drag coefficient of 0.0025 following new observational evidence35. The MITgcm code has also been updated to version 67k. All other parameterisations, including the GM-Redi parameterisation of geostrophic eddies, the KPP vertical mixing scheme, and sea ice elastic-viscous-plastic rheology, follow Naughten et al.30.

Úa ice sheet

Úa is a finite-element ice flow model based on the shallow ice stream approximation, with spatially varying resolution, automatic mesh refinement, and adaptive time-stepping29,36. Further numerical details are provided by Gudmundsson37. The domain used here consists of the FRIS catchment region38 with zero inflow conditions on the boundaries. Resolution ranges from 300 m to 50 km, with the finest resolution near grounding lines and ice shear margins. A minimum ice sheet thickness of 10 m is imposed. The ice front is constant in time, and any ice which flows past this boundary is assumed to have calved. The freshwater fluxes resulting from calving are not passed to the ocean, as iceberg freshwater flux is already accounted for in MITgcm (see UKESM Forcing below).

This configuration assumes a meteoric ice density of 917 kg/m3 with firn depths provided by the RACMO2.1 firn densification model39. The surface mass balance is a climatological record also obtained from RACMO2.140 and the basal mass balance is given by MITgcm, as described below. Úa is initialised with the same bathymetry as MITgcm, and ice surface topography from Bedmap231. We invert for the basal slipperiness in the Weertman sliding law and the rate factor in Glen’s Flow Law, using surface ice velocity from the MEaSUREs 2 dataset41 which covers the years 1996–2016. The inversion is based on an iterative adjoint method with Thikonov regularisation, and minimises the cost function between observations and model output of surface velocities. There is an additional term in the cost function to penalise large changes in ice thickness, which reduces drift in the subsequent simulations.

Coupling

The coupling algorithm originated with De Rydt and Gudmundsson42 but has been rewritten and extended so it is suitable for a realistic ocean with time-varying forcing and sea ice. For each coupling segment, Úa and MITgcm run independently for 1 year, then stop, exchange data, and restart. This design is known as offline coupling, and is standard practice for most ice sheet-ocean models in the Marine Ice Sheet Ocean Model Intercomparison Project (MISOMIP)43.

At each coupling step, MITgcm sends Úa the ice shelf basal melt rates, which Úa then linearly interpolates to all ice shelf nodes. No basal melting is applied in partially grounded elements, to avoid unphysical melting at connected nodes upstream of the grounding line. Úa sends MITgcm the ice shelf geometry and floating mask, linearly interpolated to the MITgcm grid. Each MITgcm cell containing at least one Úa ice shelf node is considered to be floating.

The MITgcm restart file is then edited to suit the new geometry. As much information as possible is preserved from the previous ocean state, unlike the re-initialisation approach of De Rydt and Gudmundsson42. Newly opened cells are filled with the properties (temperature, salinity, and free surface) of nearest neighbours, preferencing vertical neighbours. This ensures that newly opened cells at the ice base are filled with conditions from the ice-ocean boundary layer wherever possible. Newly closed cells become part of the land mask with no changes to the properties of surrounding cells.

The velocity in newly opened cells is first set to zero. If a given water column is newly opened, its velocity remains at zero. Otherwise, the velocity is corrected to preserve the barotropic transport field, by adding or subtracting a small value from the entire water column. This correction is necessary to ensure numerical stability. The barotropic transport field is not preserved in the case where an entire water column closes.

The pressure load anomaly of the ice shelf is recalculated at each coupling step, assuming the water displaced by the ice shelf has the temperature and salinity of the uppermost ocean layer. Digging of the MITgcm bathymetry is recalculated each coupling step as the ice shelf draft changes, and previous digging is reversed if it is no longer necessary. Úa does not see these changes in the bathymetry.

UKESM forcing

MITgcm is forced with output from the UKESM1-0-LL submission to CMIP644,45,46. This forcing consists of daily fields for atmospheric variables (near-surface temperature, specific humidity, and winds; surface downwelling shortwave and longwave radiation; sea level pressure; and precipitation) and monthly fields for ocean and sea ice variables on the lateral boundaries (ocean temperature, salinity, and horizontal velocity; sea ice area, thickness, velocity, and snow thickness). The atmospheric fields are used to calculate bulk fluxes within the MITgcm code, which then force the ocean and sea-ice models. We use the first ensemble member of UKESM (r1i1p1f2).

While this version of UKESM does not include ice shelf cavities, it prescribes a steady three-dimensional coastal freshwater flux representing ice shelf basal melting47,48. The spatial distribution of this flux follows the observations of Rignot et al.24. UKESM also considers steady iceberg calving through a statistical iceberg model49. To conserve mass, the sum of the meltwater flux and the calving flux is equal to the modelled pre-industrial surface mass balance of Antarctica (not including surface runoff, which is a separate flux to the ocean). This freshwater flux implicitly enters the MITgcm domain via the lateral boundary conditions. Therefore, the meltwater from ice shelves and icebergs is accounted for both within the ÚaMITgcm domain and in the external forcing from UKESM.

The initial conditions in MITgcm are a combination of UKESM output for the first month of simulation, and the Southern Ocean State Estimate (SOSE)50 January climatology. SOSE conditions for temperature and salinity are applied on the continental shelf (defined by the 2500 m isobath on the continental slope) and extrapolated into ice shelf cavities. UKESM temperature and salinity are used elsewhere in the domain, and sea ice initial conditions (sea ice area, thickness, and snow thickness) come exclusively from UKESM. This merged approach to initial conditions was chosen due to a poor representation of the continental shelf in the low-resolution UKESM ocean.

The MITgcm configuration has been shown to produce realistic results when forced with atmospheric reanalyses30. In this domain, the only significant bias of UKESM’s historical simulation compared to the ERA5 reanalysis51 is the coastal winds around Antarctica. These winds are generally too weak and diffuse, which is expected given the low resolution of UKESM. Coastal winds, including katabatic winds and barrier winds, are directed by small-scale features of the ice sheet surface topography which can only be properly captured at high atmospheric resolution52. We counteract this bias using a coastal wind correction, which is applied to the UKESM output when it is passed to MITgcm. This method extends the work of Mathiot et al.52 by applying a spatially varying, time-constant correction to winds near the coastline. The winds are converted to local polar coordinates, and are scaled and rotated such that their time-averaged magnitude and angle agrees with the ERA5 reanalysis over the observational period. Further details of the correction are provided in Supplementary Note 1. With this correction, present-day conditions simulated by ÚaMITgcm in the FRIS region largely agree with observations, as shown in Supplementary Note 2.

To prevent sea level drift in MITgcm, the ocean velocity boundary conditions are modified each year. This is necessary due to differences in the UKESM and MITgcm freshwater budgets which would otherwise lead to a net convergence or divergence of volume within the domain. The correction adds or subtracts a small value from the normal velocities at the open boundaries, to produce a volume flux, which relaxes the area-averaged free surface to zero over a 1-year timescale. Interannual variability in the freshwater budget means that this value needs to be updated every year, as a rolling correction.

Freshwater fluxes from iceberg melt are given by the monthly climatology of Storkey et al.53 and do not change over the simulations. Similarly, the surface mass balance used to force Úa remains constant in time.

Spinup

ÚaMITgcm was spun up in three stages to ensure numerical stability and consistency between the two component models:

1. 1.

An initial standalone ice sheet simulation of 10 years was performed with Úa, forced with basal melt rates from the piControl average of a standalone MITgcm simulation performed previously. The purpose of this Úa simulation was to overcome the initial adjustment period, which can trigger brief but rapid changes in ice geometry. The final geometry from this simulation was used to redefine the MITgcm domain.

2. 2.

An ocean-only spinup of 10 years was then performed, forced by years 2880–2889 of the piControl simulation in UKESM. These years were chosen so that the transient simulations would later branch from the correct point of piControl.

3. 3.

The coupled ice sheet was activated for a further 20 years of spinup under piControl forcing (years 2890–2909).

Three simulations branched from this point, each running for 150 years: the continuation of the piControl case (years 2910–3059), abrupt-4xCO2, and 1pctCO2. The final decade of each forcing scenario was then repeated 5 times for a 50-year extension.

Calculation of Stage 1 and Stage 2 thresholds

To determine the threshold of detectability for Stage 1, we compare two annually averaged timeseries (FRIS cavity temperature and basal mass loss, as in Fig. 5b, c) between each transient simulation and piControl. For each variable v and each year t, we calculate the mean μ and standard deviation σ of the 20-year period around t. The variable v is deemed to be outside the bounds of pre-industrial variability when the range (μσ, μ+σ) no longer overlaps between the two simulations. More specifically, when

$$\mu ({v}_{(t-10:t+9)})+\sigma ({v}_{(t-10:t+9)}){\,}< {\,}{\mu }_{PI}({v}_{(t-10:t+9)})-{\sigma }_{PI}({v}_{(t-10:t+9)})$$
(1)

where μ and σ refer to the transient simulation (abrupt-4xCO2 or 1pctCO2), and μPI and σPI to piControl. Stage 1 is considered to be detectable for the first value of t where this holds for both variables, i.e. cooler temperature and lower basal mass loss.

To determine the threshold at which Stage 2 begins, we calculate the temperature difference from the surface freezing point (TTf), which varies with the local salinity but not with depth. This quantity is averaged over the ocean cells at the Filchner Ice Shelf front at all depths between the ice shelf base and the seafloor. If the integrated result is negative, this transect has net supercooling with respect to the surface freezing point, indicating a dominance of Ice Shelf Water. This is the case for every year of piControl. Stage 2 begins when the annually averaged value first becomes positive, indicating a new regime of warm water transport into the ice shelf cavity.

Calculation of sea level rise contribution

We follow the method of Goelzer et al.54 to convert changes in total ice volume above flotation (VAF) to a global sea level contribution (see their Eq. 2). To account for drift in the ice sheet model, we use the difference in VAF between the transient simulation (here abrupt-4xCO2) and the piControl simulation at the same point in time.

Data availability

The UKESM output used to force the model is freely available in the CMIP6 archive44,45,46. The output from the ÚaMITgcm simulations can be obtained from the corresponding author upon request.

Code availability

The latest version of the ÚaMITgcm source code is publicly accessible at https://github.com/knaughten/UaMITgcm. These simulations use the WSFRIS configuration, and the specific version of ÚaMITgcm used here has been archived55 with doi:10.5281/zenodo.3876453. All pre- and post-processing code are publicly accessible at https://github.com/knaughten/mitgcm_python.

References

1. 1.

Gardner, A. S. et al. Increased West Antarctic and unchanged East Antarctic ice discharge over the last 7 years. The Cryosphere 12, 521–547 (2018).

2. 2.

Nicholls, K. W., Østerhus, S., Makinson, K., Gammelsrød, T. & Fahrbach, E. Ice-ocean processes over the continental shelf of the southern Weddell Sea, Antarctica: a review. Rev. Geophys. 47, 1–23 (2009).

3. 3.

Jacobs, S. et al. The Amundsen Sea and the Antarctic ice sheet. Oceanography 25, 154–163 (2012).

4. 4.

Reese, R., Gudmundsson, G. H., Levermann, A. & Winkelmann, R. The far reach of ice-shelf thinning in Antarctica. Nature Climate Change 8, 53–57 (2018).

5. 5.

Nicholls, K. W. & Østerhus, S. Interannual variability and ventilation timescales in the ocean cavity beneath Filchner-Ronne Ice Shelf, Antarctica. J. Geophys. Res. 109, C04014 (2004).

6. 6.

Nicholls, K. W. Predicted reduction in basal melt rates of an Antarctic ice shelf in a warmer climate. Nature 388, 460–462 (1997).

7. 7.

Hellmer, H. H., Kauker, F., Timmermann, R., Determann, J. & Rae, J. G. L. Twenty-first-century warming of a large Antarctic ice-shelf cavity by a redirected coastal current. Nature 485, 225–228 (2012).

8. 8.

Timmermann, R. & Hellmer, H. H. Southern Ocean warming and increased ice shelf basal melting in the twenty-first and twenty-second centuries based on coupled ice-ocean finite-element modelling. Ocean Dyn. 63, 1011–1026 (2013).

9. 9.

Hellmer, H. H., Kauker, F., Timmermann, R. & Hattermann, T. The fate of the Southern Weddell Sea Continental Shelf in a warming climate. J. Climate 30, 4337–4350 (2017).

10. 10.

Timmermann, R. & Goeller, S. Response to Filchner-Ronne Ice Shelf cavity warming in a coupled ocean-ice sheet model. Part I: the ocean perspective. Ocean Sci. 13, 765–776 (2017).

11. 11.

Hazel, J. E. & Stewart, A. L. Bistability of the Filchner-Ronne Ice Shelf Cavity Circulation and Basal Melt. J. Geophys. Res.: Oceans 125, 1–21 (2020).

12. 12.

Daae, K. et al. Necessary conditions for Warm inflow toward the Filchner Ice Shelf, Weddell Sea. Geophys. Res. Lett. 47, e2020GL089237 (2020).

13. 13.

Collins, M. et al. Climate model errors, feedbacks and forcings: a comparison of perturbed physics and multi-model ensembles. Climate Dyn. 36, 1737–1766 (2011).

14. 14.

Bracegirdle, T. J. et al. Improvements in circumpolar southern hemisphere extratropical atmospheric circulation in CMIP6 Compared to CMIP5. Earth Space Sci. 7, 1–12 (2020).

15. 15.

Asay-Davis, X. S., Jourdain, N. C. & Nakayama, Y. Developments in simulating and parameterizing interactions between the Southern Ocean and the Antarctic Ice Sheet. Curr. Climate Change Rep. 3, 316–329 (2017).

16. 16.

Sellar, A. A. et al. UKESM1: description and evaluation of the U.K. Earth System Model. J. Adv. Modeling Earth Syst. 11, 4513–4558 (2019).

17. 17.

Eyring, V. et al. Overview of the Coupled Model Intercomparison Project Phase 6 (CMIP6) experimental design and organization. Geosci Model Dev. 9, 1937–1958 (2016).

18. 18.

O’Neill, B. C. et al. The Scenario Model Intercomparison Project (ScenarioMIP) for CMIP6. Geosci. Model Dev. 9, 3461–3482 (2016).

19. 19.

de Lavergne, C., Palter, J. B., Galbraith, E. D., Bernardello, R. & Marinov, I. Cessation of deep convection in the open Southern Ocean under anthropogenic climate change. Nat. Climate Change 4, 278–282 (2014).

20. 20.

Newsom, E. R., Bitz, C. M., Bryan, F. O., Abernathey, R. & Gent, P. R. Southern Ocean deep circulation and heat uptake in a high-resolution climate model. J. Climate 29, 2597–2619 (2016).

21. 21.

Bull, C. Y. S. et al. Remote control of Filchner-Ronne Ice Shelf melt rates by the Antarctic Slope Current. J. Geophys. Res.: Oceans 126, e2020JC016550 (2020).

22. 22.

Zelinka, M. D. et al. Causes of higher climate sensitivity in CMIP6 models. Geophys. Res. Lett. 47, 1–12 (2020).

23. 23.

Bracegirdle, T. J. et al. Twenty first century changes in Antarctic and Southern Ocean surface climate in CMIP6. Atmos. Sci. Lett. 21, e984 (2020).

24. 24.

Rignot, E. J., Jacobs, S. S., Mouginot, J. & Scheuchl, B. Ice-Shelf Melting Around Antarctica. Science 341, 266–270 (2013).

25. 25.

Moholdt, G., Padman, L. & Fricker, H. A. Basal mass budget of Ross and Filchner-Ronne ice shelves, Antarctica, derived from Lagrangian analysis of ICESat altimetry. J. Geophys. Res.: Earth Surf. 119, 2361–2380 (2014).

26. 26.

Ryan, S. et al. Exceptionally warm and prolonged flow of warm deep water toward the Filchner-Ronne Ice Shelf in 2017. Geophys. Res. Lett. 47, e2020GL088119 (2020).

27. 27.

Marshall, J., Hill, C., Perelman, L. & Adcroft, A. Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res. 102, 5733–5752 (1997).

28. 28.

Losch, M. Modeling ice shelf cavities in a z coordinate ocean general circulation model. J. Geophys. Res. 113, C08043 (2008).

29. 29.

Gudmundsson, G. H., Krug, J., Durand, G., Favier, L. & Gagliardini, O. The stability of grounding lines on retrograde slopes. Cryosphere 6, 1497–1505 (2012).

30. 30.

Naughten, K. A. et al. Modelling the influence of the Weddell Polynya on the Filchner-Ronne Ice Shelf cavity. J. Climate 32, 5289–5303 (2019).

31. 31.

Fretwell, P. et al. Bedmap2: Improved ice bed, surface and thickness datasets for Antarctica. Cryosphere 7, 375–393 (2013).

32. 32.

Rosier, S. H. R. et al. A New Bathymetry for the Southeastern Filchner-Ronne Ice Shelf: implications for modern oceanographic processes and glacial history. J. Geophys. Res.: Oceans 123, 4610–4623 (2018).

33. 33.

Schaffer, J., Timmermann, R., Arndt, J. E., Kristensen, S. S. & Mayer, C. A global, high-resolution data set of ice sheet topography, cavity geometry, and ocean bathymetry. Earth Syst. Sci. Data 8, 543–557 (2016).

34. 34.

Jordan, J. R. et al. Ocean-forced ice-shelf thinning in a synchronously coupled ice-ocean model. J. Geophys. Res.: Oceans 123, 864–882 (2018).

35. 35.

Davis, P. E. & Nicholls, K. W. Turbulence observations Beneath Larsen C Ice Shelf, Antarctica. J. Geophys. Res.: Oceans 124, 5529–5550 (2019).

36. 36.

Gudmundsson, G. H., GHilmarG/UaSource: Ua2019b (Version v2019b), https://doi.org/10.5281/zenodo.3706624 (2020).

37. 37.

Gudmundsson, G. H., Ua Compendium, University of Northumbria, https://github.com/GHilmarG/UaSource/blob/master/UaCompendium.pdf (2020).

38. 38.

Rignot, E. J., Mouginot, J. & Scheuchl, B. Ice flow of the Antarctic ice sheet. Science 333, 1427–1431 (2011).

39. 39.

Ligtenberg, S. R., Helsen, M. M. & Van Den Broeke, M. R. An improved semi-empirical model for the densification of Antarctic firn. Cryosphere 5, 809–819 (2011).

40. 40.

Lenaerts, J. T. M., van den Broeke, M. R., van de Berg, W. J., van Meijgaard, E. & Munneke, P. K. A new, high-resolution surface mass balance map of Antarctica (1979-2010) based on regional atmospheric climate modeling. Geophys. Res. Lett. 39, L04501 (2012).

41. 41.

Rignot, E. J., Mouginot, J. & Scheuchl, B. MEaSUREs InSAR-Based Antarctica Ice Velocity Map, Version 2, https://doi.org/10.5067/D7GK8F5J8M8R (2017).

42. 42.

De Rydt, J. & Gudmundsson, G. H. Coupled ice shelf-ocean modeling and complex grounding line retreat from a seabed ridge. J. Geophys. Res.: Earth Surf. 121, 865–880 (2016).

43. 43.

Asay-Davis, X. S. et al. Experimental design for three interrelated marine ice sheet and ocean model intercomparison projects: MISMIP v. 3 (MISMIP +), ISOMIP v. 2 (ISOMIP +) and MISOMIP v. 1 (MISOMIP1). Geosci. Model Dev. 9, 2471–2497 (2016).

44. 44.

Met Office Hadley Centre, WCRP CMIP6: Met Office Hadley Centre (MOHC) UKESM1-0-LL model output for the “1pctCO2” experiment, https://catalogue.ceda.ac.uk/uuid/1413ada259844d749b462b6b490d58ff (2019).

45. 45.

Met Office Hadley Centre, WCRP CMIP6: Met Office Hadley Centre (MOHC) UKESM1-0-LL model output for the “abrupt-4xCO2” experiment, https://catalogue.ceda.ac.uk/uuid/3663b0d9d47443a786eca5ccdb32c18e (2019).

46. 46.

Met Office Hadley Centre, Met Office Hadley Centre (MOHC) UKESM1-0-LL model output for the “piControl” experiment, https://catalogue.ceda.ac.uk/uuid/208e35d6d82444b18b1f3b938cdc080f (2019).

47. 47.

Sellar, A. A. et al. Implementation of U.K. Earth System Models for CMIP6. J. Adv. Modeling Earth Syst. 12, e2019MS001946 (2020).

48. 48.

Mathiot, P., Jenkins, A., Harris, C. & Madec, G. Explicit representation and parametrised impacts of under ice shelf seas in the z* coordinate ocean model NEMO 3.6. Geosci. Model Dev. 10, 2849–2874 (2017).

49. 49.

Marsh, R. et al. NEMO-ICB (v1.0): Interactive icebergs in the NEMO ocean model globally configured at eddy-permitting resolution. Geosci. Model Dev. 8, 1547–1562 (2015).

50. 50.

Mazloff, M. R., Heimbach, P. & Wunsch, C. An Eddy-Permitting Southern Ocean State Estimate. J. Phys. Oceanogr. 40, 880–899 (2010).

51. 51.

Copernicus Climate Change Service, ERA5: Fifth generation of ECMWF atmospheric reanalyses of the global climate, https://www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era5 (2017).

52. 52.

Mathiot, P. et al. Introducing katabatic winds in global ERA40 fields to simulate their impacts on the Southern Ocean and sea-ice. Ocean Modelling 35, 146–160 (2010).

53. 53.

Storkey, D. et al. UK Global Ocean GO6 and GO7: a traceable hierarchy of model resolutions. Geosci. Model Dev. 11, 3187–3213 (2018).

54. 54.

Goelzer, H., Coulon, V., Pattyn, F., De Boer, B. & Van De Wal, R. On calculating the sea-level contribution in marine ice-sheet models. Cryosphere 14, 833–840 (2020).

55. 55.

Naughten, K. A. & De Rydt, J. knaughten/UaMITgcm: UaMITgcm WSFRIS configuration (Version v1.0), https://doi.org/10.5281/zenodo.3876453 (2020).

56. 56.

Morice, C. P., Kennedy, J. J., Rayner, N. A. & Jones, P. D. Quantifying uncertainties in global and regional temperature change using an ensemble of observational estimates: The HadCRUT4 data set. J. Geophys. Res. Atmos. 117, D08101 (2012).

Acknowledgements

The authors are grateful to Hilmar Gudmundsson for developing and maintaining the Úa source code, and to Keith Nicholls, Keith Makinson, and John King for helpful discussions while writing this manuscript. David Munday provided advice regarding the MITgcm salt budget analysis, and Markus Janout provided the PS111 observational data for model evaluation. Computational resources were provided by the ARCHER UK National Supercomputing Service as well as the JASMIN data analysis facility. This research is part of the Filchner Ice Shelf System (FISS) project NE/L013770/1.

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Authors

Contributions

K.A.N. devised and carried out the experiments, analysed the output, and wrote the manuscript. K.A.N., J.D.R., and S.H.R.R. developed the ÚaMITgcm model. A.J. and P.R.H. assisted with experimental design and model troubleshooting. J.K.R. provided the UKESM forcing and additional analysis. All authors contributed to the development of ideas and provided feedback on the manuscript.

Corresponding author

Correspondence to Kaitlin A. Naughten.

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Peer review informationNature Communications thanks Xylar Asay-Davis and other, anonymous, reviewers for their contributions to the peer review of this work. Peer review reports are available.

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Naughten, K.A., De Rydt, J., Rosier, S.H.R. et al. Two-timescale response of a large Antarctic ice shelf to climate change. Nat Commun 12, 1991 (2021). https://doi.org/10.1038/s41467-021-22259-0

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SMS scnews item created by Kevin Coulembier at Mon 14 Mar 2016 0932
Type: Seminar
Distribution: World
Expiry: 1 Apr 2016
Calendar1: 18 Mar 2016 1200-1300
CalLoc1: Carslaw 375
CalTitle1: Essential dimension of generic symbols
Auth: kevinc@pkevinc.pc (assumed)

# Algebra Seminar: McKinnie -- Essential dimension of generic symbols

Kelly McKinnie (University of Montana)

Friday 18 March, 12-1pm, Place: Carslaw 375

Essential dimension of generic symbols

The essential dimension of an algebraic object is loosely defined as
the minimal number of independent parameters needed to define the
object over a base field. For example the essential dimension of the
tensor product of  n generic symbol algebras (x_i,y_i) over C is 2n as
expected. In this talk I will discuss generic symbols in both
characteristics 0 and p and their essential dimensions.

If you are registered you may mark the scnews item as read.
School members may try to .

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# Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global sections ect) but I am missing the bigger picture.

Why should I care if a functor has a left adjoint? What does it tell me about the functor?

The answer to this question is the same as the answer to every question of this genre (“why should I care about groups,” “why should I care about rings”): they show up everywhere and are an extremely useful organizing principle.

There is a meta-principle that any time you’re trying to understand something about categories, it’s a good idea to restrict to the special case of posets first, regarded as categories where $a \le b$ means there is a single arrow $a \to b$. For example:

• Products are the same thing as infima. In particular, the terminal object is the empty infimum, which is the maximum element. (This may seem backwards, but it really is what you get out of the definitions.)
• Coproducts are the same thing as suprema. In particular, the initial object is the empty supremum, which is the minimum element.

So what’s a pair of adjoint functors in this context? Well, if $P, Q$ are two posets, then a functor $f : P \to Q$ is just an order-preserving function. So a pair of adjoint functors is first of all a pair $f : P \to Q, g : Q \to P$ of order-preserving functions. The definition I think is most convenient when studying posets is that $f, g$ must satisfy

for all $a \in P, b \in Q$. But this is the same thing as requiring that

This relationship is called a Galois connection. Important examples of Galois connections include:

• $K \to L$ is a finite Galois extension, $P$ is the poset of subgroups of $\text{Gal}(L/K)$, $Q$ is the poset of subfields $K \to M \to L$, $f$ sends a subgroup to its fixed field, $g$ sends a subfield to $\text{Gal}(L/M)$. (One has to reverse the order of one of these posets for this to work.)
• $P$ is the poset of ideals of $\mathbb{C}[x_1, ... x_n]$, $Q$ is the poset of subsets of $\mathbb{C}^n$, $f$ sends an ideal to the set of points defined by its elements vanishing, $g$ sends a set of points to the ideal of functions vanishing on it. (Again, one has to reverse the order of one of these posets.)

Galois connections exist in extreme generality and are, by themselves, already an important organizing principle in mathematics. So adjoint functors are even more important than that!

Edit: It’s probably worth explaining what’s going on in the above examples abstractly. Let $A, B$ be two sets, and let $r : A \times B \to \{ 0, 1 \}$ be a relation. Then $r$ induces an order-reversing Galois connection (a pair of contravariant adjoint functors) between the poset $\mathcal{P}(A)$ of subsets of $A$ and the poset $\mathcal{P}(B)$ of subsets of $B$ as follows: if $S \subset A$ then $f(S) = \{ b \in B : r(a, b) = 1 \forall a \in S \}$ and if $S \subset B$ then $g(S) = \{ a \in A : r(a, b) = 1 \forall b \in S \}$. I’ll leave it as an exercise to figure out what $A, B, r$ are in the above examples.

Note also that the fact that left adjoints preserve colimits and right adjoints preserve limits continues to hold for Galois connections, and shows that some of the properties of the Galois connections above are purely formal (in the sense that they follow from this “abstract nonsense”). Unfortunately it is generally not emphasized which properties those are.

The Wikipedia article does a nice job of explaining some broad general motivation (and has a lot of good discussion on this question besides): very roughly, an adjoint is the best substitute for an inverse that exists in a lot of cases that we care about. You can sort of see how this works in the above examples.

An important property of adjoint pairs is that they restrict to equivalences on subcategories, and this is what we get in the Galois theory and algebraic geometry examples above: the first adjoint pair is an equivalence by the fundamental theorem of Galois theory, and the second adjoint pair restricts to an equivalence between reduced ideals and varieties by the Nullstellensatz.

Since your question is tagged [algebraic-geometry], here is an important non-example related to the second half of Arturo’s answer. There is a functor $F : \text{Aff} \to \text{Set}$ sending an affine scheme to its set of points (the prime ideals of the corresponding ring), and it does not have a left adjoint: there is no “free affine scheme” on a set. The reason is that $F$ does not preserve limits. (Note that a functor has a left adjoint if and only if it is a right adjoint.) In fact, it does not even preserve products. The product of two affine schemes $\text{Spec } R, \text{Spec } S$ is $\text{Spec } R \otimes_{\mathbb{Z}} S$, and it is a basic property of schemes that this is not the set-theoretic product.

It follows that the functor $F : \text{Aff} \to \text{Top}$ sending an affine scheme to its set of points in the Zariski topology also does not have a left adjoint. If you’ve ever wondered why the Zariski topology on $\mathbb{A}^2$ isn’t the product topology on $\mathbb{A}^1 \times \mathbb{A}^1$, now you know.


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Research

# Half-linear eigenvalue problem: Limit behavior of the first eigenvalue for shrinking interval

Gabriella Bognár1 and Ondřej Došlý2*

Author Affiliations

1 University of Miskolc, Egyetemváros, Miskolc, 3515, Hungary

2 Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, Brno, 611 37, Czech Republic

For all author emails, please log on.

Boundary Value Problems 2013, 2013:221  doi:10.1186/1687-2770-2013-221

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/221

 Received: 27 June 2013 Accepted: 4 September 2013 Published: 7 November 2013

© 2013 Bognár and Do¿lý; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We investigate the limit behavior of the first eigenvalue of the half-linear eigenvalue problem when the length of the interval tends to zero. We show that the important role is played by the limit behavior of ratios of primitive functions of coefficients in the investigated half-linear equation.

MSC: 34C10.

### 1 Introduction

We consider the eigenvalue problem associated with the half-linear second order differential equation

( r ( t ) Φ ( x ) ) + c ( t ) Φ ( x ) = λ w ( t ) Φ ( x ) , Φ ( x ) : = | x | p 2 x , p > 1 , (1)

with t ( a , b ) , < a < b < , r 1 q , c , w L 1 ( a , b ) , q = p p 1 being the conjugate exponent of p, and r ( t ) > 0 , w ( t ) > 0 . Equation (1) can be written as the first order system

x = r 1 q ( t ) Φ 1 ( u ) , u = ( c ( t ) λ w ( t ) ) Φ ( x ) , (2)

Φ 1 ( u ) = | u | q 2 u being the inverse function of Φ, and the integrability assumption on the functions r 1 q , c, w implies the unique solvability of this system. The original paper of Elbert [1], where the existence and uniqueness results are proved via the half-linear version of the Prüfer transformation, deals with continuous functions in (2), but the idea of the proof applies without change to integrable coefficients when, as a solution x, u, absolutely continuous functions are considered (which satisfy (2) a.e. in ( a , b ) ).

Along with (1), we consider the separated boundary conditions

x ( a ) cos p α r q 1 ( a ) x ( a ) sin p α = 0 , x ( b ) cos p β r q 1 ( b ) x ( b ) sin p β = 0 , (3)

where sin p , cos p are the half-linear goniometric functions, which will be recalled in the next section. Motivated by the paper [2], where the linear Sturm-Liouville differential equation (which is the special case of (1)) is considered, we investigate the limit behavior (as b a + ) of the first eigenvalue of (1), (3) in dependence on α, β. We show that this limit behavior is, in a certain sense, the same as for an eigenvalue problem when boundary conditions (3) are associated with an equation with constant coefficients.

The investigation of half-linear eigenvalue problems is motivated, among others, by the fact that the partial differential equation with the p-Laplacian (which models, e.g., the flow of non-Newtonian fluids, while the linear case p = 2 corresponds to the Newtonian fluid)

Δ p u + c ( x ) Φ ( u ) = λ Φ ( u ) , Δ p ( u ) : = div ( u ( x ) p 2 u ( x ) ) , x R n ,

and the spherically symmetric potential c, can be reduced to an equation of the form (1). For this reason, motivated also by the linear case p = 2 , the problem of dependence of eigenvalues of (1), (3) on the functions r, c, w and the boundary data a, b, α, β was a subject of the investigation in several recent papers. We refer to [3-6] and the references therein.

The paper is organized as follows. In the next section, we recall essentials of the qualitative theory of half-linear differential equations. Section 3 deals with the eigenvalue problem for an equation with constant coefficients. The main results of the paper, limit formulas for the first eigenvalue of (1), (3), are given in Section 4.

### 2 Preliminaries

First, we recall the concepts of half-linear goniometric functions. These functions, in the form presented here, appeared for the first time in [1]. In a modified form, they can also be found in other papers, e.g., in [7].

The half-linear sine function sin p is defined as the solution of the differential equation

( Φ ( x ) ) + ( p 1 ) Φ ( x ) = 0 (4)

given by the initial condition x ( 0 ) = 0 , x ( 0 ) = 1 . The function sin p is π p anti-periodic (and hence 2 π p periodic) with π p : = 2 π p sin ( π / p ) . The derivative ( sin p t ) = : cos p t defines the half-linear cosine function. These functions satisfy the half-linear Pythagorean identity

| sin p t | p + | cos p t | p = 1 . (5)

The half-linear functions tan p and cot p are defined in a natural way as

tan p t = sin p t cos p t , cot p t = cos p t sin p t .

The inverse functions to these functions on ( π p / 2 , π p / 2 ) resp. ( 0 , π p ) are denoted by arctan p and arccot p . By a direct computation, using (5) and the fact that (4) can be written as

x + | x | 2 p Φ ( x ) = 0 ,

one can verify the formulas

[ Φ ( cot p t ) ] = 1 | sin p t | p , ( tan p t ) = 1 | cos p t | p , ( arctan p t ) = 1 1 + | t | p .

The original proof of the unique solvability of (1) (and hence of (2), see [1]) is based on the half-linear version of the Prüfer transformation. Let x be a nontrivial solution of (1), put

x ( t ) = ρ ( t ) sin p φ ( t ) , r q 1 ( t ) x ( t ) = ρ ( t ) cos p φ ( t ) . (6)

Then the Prüfer angle φ solves the first order differential equation

φ = r 1 q ( t ) | cos p φ | p + λ w ( t ) c ( t ) p 1 | sin p φ | p . (7)

The right-hand side of (7) is a Lipschitzian function with respect to φ, hence the standard existence, uniqueness, and continuous dependence on the initial data theory applies to this equation, and these results carry over via (6) to (2) and (1). Observe at this place that the right-hand side of (2) is not Lipschitzian, so this theory cannot be directly applied to (2).

The Prüfer transformation is closely associated with the Riccati-type differential equation

v + λ w ( t ) c ( t ) + ( p 1 ) r 1 q ( t ) | v | q = 0 , (8)

which is related to (1) by the Riccati substitution v = r Φ ( x / x ) . The fact that we have in disposal a Riccati-type differential equation and the generalized Prüfer transformation implies that the linear oscillation theory extends almost verbatim to (1). In particular, similarly to the linear case, the eigenvalues of (1), (3) form an increasing sequence λ n , and the nth eigenfunction has exactly n 1 zeros in ( a , b ) , see [8] and also [[9], Section 5.7]. For some recent references in this area, we refer to [3] and the references therein.

### 3 Equation with constant coefficients

In this section, as a motivation, we consider the equation

( Φ ( x ) ) = λ ( p 1 ) Φ ( x ) , (9)

i.e., r 1 , c 0 , w p 1 in (1). The Riccati equation associated with (9) is

v + ( p 1 ) [ λ + | v | q ] = 0 . (10)

First, consider the case α = β ( 0 , π p ) in (3). Then the first eigenfunction of (9), (3) corresponds to the solution of (10) satisfying

v ( a ) = Φ ( cot p α ) = v ( b ) . (11)

To underline the dependence of eigenvalues of our eigenvalue problem on b, α, β, we denote the first eigenvalue by λ 1 ( b , α , β ) . Also, when α = β , we use the notation λ 1 ( b , α , α ) = : λ 1 ( b , α ) .

Obviously, the solution of (10) satisfying (11) is a constant solution when

λ = λ 1 ( b , α ) = | Φ ( cot p α ) | q = | cot p α | p .

If 0 < α < β < π p , i.e., v ( a ) = Φ ( cot p α ) > Φ ( cot p β ) = v ( b ) , we need v to be decreasing. When the length of the interval ( a , b ) tends to zero, v ( t ) cannot be bounded from below in ( a , b ) , and hence

λ 1 ( b , α , β ) as  b a + .

In the opposite case, when 0 < β < α < π p , i.e., v ( a ) = Φ ( cot p α ) < Φ ( cot p β ) = v ( b ) , we need v to be increasing, and using the same argument as before, we have lim b a + λ 1 ( b , α , β ) = . Finally, if α = 0 and β ( 0 , π p ) , we have v ( a + ) = , and hence lim b a + λ 1 ( b , 0 , β ) = , similarly, if β = π p and α ( 0 , π p ) , we have v ( b ) = , and hence also lim b a + λ 1 ( b , α , π p ) = .

The previous simple considerations are summarized in the next theorem.

Theorem 1Let λ 1 ( b , α , β ) denote the first eigenvalue of (9), (3) with α [ 0 , π p ) , β ( 0 , π p ] . Then

lim b a + λ 1 ( b , α , β ) = { if  α < β , | cot p α | p if  α = β , if  α > β .

Remark 1 In this section, for the sake of simplicity, we have considered a constant coefficients equation in the form (9), i.e., with c = 0 , r = 1 , and w = p 1 . If the functions r, c, w in (1) are constants equal to r , c , w R , a slight modification of the previous considerations shows that for α = β we have

λ 1 ( b , α ) = c w ( p 1 ) r 1 q w | cot p α | p

(actually, this least eigenvalue does not depend on the endpoints a, b).

### 4 Limit behavior of the first eigenvalue

In this section, we consider general half-linear eigenvalue problem (1), (3). We denote

R ( t ) = a t r 1 q ( s ) d s , C ( t ) = a t c ( s ) d s , W ( t ) = a t w ( s ) d s .

In the next theorems, we discuss various asymptotic behavior of ratios of the functions C, R, W for t a + , which implies various limit behavior of the first eigenvalue. The behavior of the higher eigenvalues is described at the end of this section.

In the proofs of the next theorems, given Λ R , φ ( t ) = φ ( t , Λ ) , denotes the Prüfer angle of a solution x of (1) with λ = Λ satisfying (3) at a, i.e., φ ( a , Λ ) = α .

We start with the most interesting case α = β in (3).

Theorem 2Suppose that lim t a + C ( t ) W ( t ) = . Then for any α ( 0 , π p ) , we have

lim b a + λ 1 ( b , α ) = . (12)

Proof We will show that λ 1 ( b , α ) < Λ for any Λ R if b > a is sufficiently close to a. Since the eigenfunction corresponding to the first eigenvalue λ 1 has no zero on ( a , b ) (see, e.g., [10]), we can use the Riccati equation (8) for v = Φ ( cot p φ ) instead of equation (7) for φ. Using the mean-value theorem for Lebesgue integrals in computing the integral a b r 1 q | cot p φ | p , integration of (8) gives

v ( b ) v ( a ) = Φ ( cot p φ ( b ) ) Φ ( cot p α ) = [ C ( b ) W ( b ) Λ ( p 1 ) R ( b ) W ( b ) | cot p φ ( t b ) | p ] W ( b ) ,

where t b ( a , b ) . Hence, for b sufficiently close to a,

Φ ( cot p φ ( b ) ) Φ ( cot p α ) ( C ( b ) W ( b ) Λ ) W ( b ) < 0 ,

i.e., φ ( b ) > α = β . Thus, since φ was the Prüfer angle corresponding to a solution of (1) with λ = Λ satisfying (3) at t = a , we have λ 1 ( b , α ) < Λ for the first eigenvalue when b is in a sufficiently small right neighborhood of a (since we need φ ( b ) = α for λ = λ 1 ( b , α ) ). Therefore, (12) holds. □

Theorem 3Suppose that

lim sup t a + C ( t ) R ( t ) < , lim t a + R ( t ) W ( t ) = ,

and let

α = arccot p ( max { 0 , 1 p 1 lim sup t a + C ( t ) R ( t ) } ) 1 / p ( 0 , π p 2 ] .

Then we have λ 1 ( b , α ) as b a + for any α ( 0 , π p ) [ α , π p α ] .

Proof Let α ( 0 , α ) ( π p α , π p ) be fixed, and take δ > 0 so small that

( p 1 ) | cot p α | p > lim sup t a + C ( t ) R ( t ) + 2 δ . (13)

Such a positive δ exists according to the definition of the number α . Formula (13) implies that for τ sufficiently close to α, we have the inequality

( p 1 ) | cot p τ | p > C ( t ) R ( t ) + δ

when t is sufficiently close to a. Again, integration of (8), together with the mean value theorem applied to the last integral on the right-hand side of (8), gives

Φ ( cot p φ ( b ) ) Φ ( cot p α ) = [ R ( b ) W ( b ) ( C ( b ) R ( b ) ( p 1 ) | cot p φ ( t b ) | p ) Λ ] W ( b ) < 0 .

Hence φ ( b ) > α . This implies that λ 1 ( b , α ) < Λ for every Λ R , and thus, λ 1 ( b , α ) as b a + . □

Theorem 4Assume that

lim inf t a + C ( t ) R ( t ) > 0 and lim t a + C ( t ) W ( t ) = . (14)

Let

α = arccot p ( 1 p 1 lim inf t a + C ( t ) R ( t ) ) 1 / p [ 0 , π p 2 ) . (15)

Then we have λ 1 ( b , α ) as b a + for any α ( α , π p α ) .

Proof The first formula in (14) implies that C ( t ) > 0 for t in a right neighborhood of a. Take α ( α , π p α ) , and let δ > 0 be so small that

( 1 2 δ ) lim inf t a + C ( t ) R ( t ) > ( p 1 ) | cot p α | p . (16)

Again, such δ > 0 exists according to the definition of α . Hence, for τ sufficiently close to α, from (16), we have for t close to a that

( 1 δ ) C ( t ) R ( t ) > ( p 1 ) | cot p τ | p .

Let Λ R be arbitrary. Similarly as in the proof of the previous theorem,

Φ ( cot p φ ( b ) ) Φ ( cot p α ) = [ C ( b ) W ( b ) ( 1 ( p 1 ) R ( b ) C ( b ) | cot p φ ( t b ) | p ) Λ ] W ( b ) > 0

if b is sufficiently close to a, i.e., φ ( b ) < α , and hence λ 1 ( b , α ) > Λ . Therefore, lim b a + λ 1 ( b , α ) = . □

Theorem 5Assume that

lim sup t a + R ( t ) W ( t ) < .

If

lim t a + ( C ( t ) W ( t ) ( p 1 ) R ( t ) W ( t ) | cot p φ | p ) = L R { ± } ,

then

lim b a + λ 1 ( b , α ) = L .

Proof Let Λ R be arbitrary. Similarly as in the previous theorems,

Φ ( cot p φ ( b ) ) Φ ( cot p α ) = [ C ( b ) W ( b ) ( p 1 ) R ( b ) W ( b ) | cot p α | p (17)

Λ + ( p 1 ) R ( b ) W ( b ) ( | cot p α | p | cot p φ ( t b ) | p ) ] W ( b ) . (18)

If L = , the expression in line (17) tends to −∞ as b a + while remaining terms on the right-hand side of the previous formula are bounded. Hence the expression on the right-hand side is negative for b close to a, which means that φ ( b ) > α for these b. However, this implies that λ 1 ( b , α ) < Λ in a right neighborhood of a, and since Λ was arbitrary, we have lim b a λ 1 ( b , α ) = . The same arguments imply that lim b a + λ 1 ( b , α ) = if L = + .

Finally, if L R , take first Λ = L + ε . Since the last term in (18) tends to zero as b a + , we obtain using the same argument as in the previous part of the proof that λ 1 ( b , α ) < L + ε for b sufficiently close to a. Taking Λ = L ε , we obtain λ 1 ( b , α ) > L ε for b in a right neighborhood of a, and this completes the proof. □

Theorem 6Suppose that

lim t a + R ( t ) W ( t ) < (19)

and α ( 0 , π p ) . If

lim b a + λ 1 ( b , α ) = L exists finite ,

then

lim t a + ( C ( t ) W ( t ) ( p 1 ) R ( t ) W ( t ) | cot p α | p ) = L . (20)

Proof Denote λ ( b ) : = λ 1 ( b , α ) , and let φ ( t , λ ( b ) ) be the Prüfer angle of the solution x of (1) with λ = λ ( b ) = λ 1 ( b , α ) satisfying (3) with α = β . Then

0 = Φ ( cot p β ) Φ ( cot p α ) = Φ ( cot p φ ( b , λ ( b ) ) ) Φ ( cot p φ ( a , λ ( b ) ) ) = C ( b ) λ ( b ) W ( b ) ( p 1 ) R ( b ) | cot p α | p + ( p 1 ) R ( b ) ( | cot p α | p | cot p φ ( t b , λ ( b ) ) | p ) = [ C ( b ) W ( b ) ( p 1 ) R ( b ) W ( b ) | cot p α | p (21)

λ ( b ) + ( p 1 ) R ( b ) W ( b ) ( | cot p α | p | cot p φ ( t b , λ ( b ) ) | p ) ] W ( b ) = 0 (22)

for some t b ( a , b ) . Therefore, from (21), (22), we obtain

C ( b ) W ( b ) ( p 1 ) R ( b ) W ( b ) | cot p α | p = λ ( b ) ( p 1 ) R ( b ) W ( b ) ( | cot p α | p | cot p φ ( t b , λ ( b ) ) | p ) .

Letting b a + and using that

lim t a + λ ( b ) = lim b a + λ 1 ( b , α ) = L R

(which means that λ ( b ) is bounded in a neighborhood of a), we see that

lim b a + φ ( t b , λ ( b ) ) = α ,

and (20) is proved. □

Theorem 7If lim b a + λ 1 ( b , α ) has a finite value for two different values of α 1 and α 2 with α 1 , α 2 ( 0 , π p ) and α 1 π p α 2 , then

lim t a + R ( t ) W ( t ) and lim t a + C ( t ) W ( t )

exist finite, and for each α ( 0 , π p ) , we have

lim b a + λ 1 ( b , α ) = lim t a + C ( t ) W ( t ) ( p 1 ) | cot p α | p lim t a + R ( t ) W ( t ) . (23)

Proof First of all, we have

| cot p α 1 | p | cot p α 2 | p . (24)

Denote λ 1 ( b , α i ) = : μ i ( b ) for i = 1 , 2 , and let φ ( t , μ i ( b ) ) be the Prüfer angle of the solution of (1), (3) with λ = μ i ( b ) and α = β . Since φ ( b , μ i ( b ) ) = φ ( a , μ i ( b ) ) , we have from (8)

C ( b ) μ i ( b ) W ( b ) ( p 1 ) R ( b ) | cot p φ ( t i , b , μ i ( b ) ) | p = 0 , i = 1 , 2 ,

where t i , b ( a , b ) . Hence, one gets

C ( b ) W ( b ) ( p 1 ) R ( b ) W ( b ) | cot p φ ( t i , b , μ i ( b ) ) | p = μ i ( b ) , i = 1 , 2 . (25)

Subtracting these two equations and using the fact that

| cot p φ ( t 1 , b , μ 1 ( b ) ) | | cot p φ ( t 2 , b , μ 2 ( b ) ) |

for b sufficiently close to a (this follows from (24)), we have

R ( b ) W ( b ) = μ 1 ( b ) μ 2 ( b ) | cot p φ ( t 1 , b , μ 1 ( b ) ) | p | cot p φ ( t 2 , b , μ 2 ( b ) ) | p λ 1 ( a + , α 1 ) λ 1 ( a + , α 2 ) | cot p α 1 | p | cot p α 2 | p

as b a + , so lim b a + R ( b ) W ( b ) exists finite, and hence from (25), the same holds for lim b a + C ( b ) W ( b ) , and the conclusion follows from Theorem 5. □

We finish the paper with a brief treatment of the case α β in (3). We show that the situation is similar as in the case of a constant coefficients equation treated in Section 3.

Theorem 8Let α [ 0 , π p ) , β ( 0 , π p ] .

(i) If α < β , then

lim b a + λ 1 ( b , α , β ) = . (26)

(ii) If α > β , then

lim b a + λ 1 ( b , α , β ) = .

Proof We will prove the part (i) only, the proof of (ii) is analogical. Let Λ R be arbitrary, and let φ ( t , Λ ) be the Prüfer angle of the solution x of (1) with λ = Λ satisfying (3) at t = a . Since φ is a continuous function of t, and φ ( a , Λ ) = α < β , we have φ ( b , Λ ) < β if b is sufficiently close to a. Hence, using the same argument as before, we have λ 1 ( b , α , β ) > Λ , which implies (26). □

Remark 2 Until now, we have considered the first eigenvalue λ 1 ( b , α , β ) only. Concerning the asymptotic behavior of higher eigenvalues λ n ( b , α , β ) , we have

lim b a + λ n ( b , α , β ) = (27)

for any α [ 0 , π p ) , β ( 0 , π p ] . This formula follows in the case α < β from the general theory of half-linear eigenvalues problem (see [8,10]), which says that the eigenvalues form an increasing sequence tending to ∞, and from the fact that lim b a + λ 1 ( b , α , β ) = . If α > β , then cot p α < cot p β , but for n N , we have α < β + n π p and higher eigenvalues correspond to the situation when the Prüfer angle of a solution of (1) satisfying the first condition in (3) satisfies φ ( b ) = β + n π p . Hence, the growth of φ must be unbounded when b a + , and hence (27) holds also for α > β .

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally and read and approved the final version of the manuscript.

### Acknowledgements

The research of the first author was carried out as a part of the TAMOP-4.2.2/B-10/1-2010-0008 project in the framework of the New Hungarian Development Plan. The realization of this project is supported by the European Union and co-financed by the European Social Fund. The second author was supported by the Grant GAP 201/11/0768 of the Czech Grant Agency.

### References

1. Elbert, Á: A half-linear second order differential equation. Qualitative Theory of Differential Equations, Vol. I, II, pp. 153–180. North-Holland, Amsterdam (1981)

2. Kong, Q, Wu, H, Zettl, A: Limit of Sturm-Liouville eigenvalues when the interval shrinks to an end point. Proc. R. Soc. Edinb., Sect. A. 138, 323–338 (2008)

3. Benedikt, J, Drábek, P: Estimates of the principal eigenvalue of the p-Laplacian. J. Math. Anal. Appl.. 393, 311–315 (2012). Publisher Full Text

4. Binding, PA, Drábek, P: Sturm-Liouville theory for the p-Laplacian. Studia Sci. Math. Hung.. 40, 375–396 (2003)

5. Kong, L, Kong, Q: Right-definite half-linear Sturm-Liouville problems. Proc. R. Soc. Edinb. A. 137, 77–92 (2007)

6. Kusano, T, Naito, M, Tanigawa, T: Second-order half-linear eigenvalue problems. Fukuoka Univ. Sci. Rep.. 27, 1–7 (1997)

7. Lindqvist, P: Some remarkable sine and cosine functions. Ric. Mat.. 44, 260–290 (1995)

8. Kusano, T, Naito, M: Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations. Rocky Mt. J. Math.. 31, 1039–1054 (2001). Publisher Full Text

9. Došlý, O, Řehák, P: Half-Linear Differential Equations, Elsevier, Amsterdam (2005)

10. Eberhard, W, Elbert, Á: On the eigenvalues of half-linear boundary value problems. Math. Nachr.. 183, 55–72 (1997). Publisher Full Text


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# Associations to the word «Condition»

## Wiktionary

CONDITION, noun. A logical clause or phrase that a conditional statement uses. The phrase can either be true or false.
CONDITION, noun. A requirement, term, or requisite.
CONDITION, noun. (legal) A clause in a contract or agreement indicating that a certain contingency may modify the principal obligation in some way.
CONDITION, noun. The health status of a medical patient.
CONDITION, noun. The state or quality.
CONDITION, noun. A particular state of being.
CONDITION, noun. (obsolete) The situation of a person or persons, particularly their social and/or economic class, rank.
CONDITION, verb. To subject to the process of acclimation.
CONDITION, verb. To subject to different conditions, especially as an exercise.
CONDITION, verb. (transitive) To place conditions or limitations upon.
CONDITION, verb. To shape the behaviour of someone to do something.
CONDITION, verb. (transitive) To treat (the hair) with hair conditioner.
CONDITION, verb. (transitive) To contract; to stipulate; to agree.
CONDITION, verb. (transitive) To test or assay, as silk (to ascertain the proportion of moisture it contains).
CONDITION, verb. (US) (colleges) (transitive) To put under conditions; to require to pass a new examination or to make up a specified study, as a condition of remaining in one's class or in college.
CONDITION, verb. To impose upon an object those relations or conditions without which knowledge and thought are alleged to be impossible.
CONDITION PRECEDENT, noun. (legal) a condition imposed on an agreement that must be satisfied before property is transferred between parties.
CONDITION PRECEDENT, noun. (legal) a condition depending upon an uncertain event which must be fulfilled before an obligation arises or a legal transaction takes effect.
CONDITION SUBSEQUENT, noun. (legal) A condition depending upon an uncertain event which, if fulfilled, brings an end to an obligation or legal transaction.

## Dictionary definition

CONDITION, noun. A state at a particular time; "a condition (or state) of disrepair"; "the current status of the arms negotiations".
CONDITION, noun. An assumption on which rests the validity or effect of something else.
CONDITION, noun. A mode of being or form of existence of a person or thing; "the human condition".
CONDITION, noun. Information that should be kept in mind when making a decision; "another consideration is the time it would take".
CONDITION, noun. The state of (good) health (especially in the phrases in condition' or in shape' or out of condition' or out of shape').
CONDITION, noun. An illness, disease, or other medical problem; "a heart condition"; "a skin condition".
CONDITION, noun. (usually plural) a statement of what is required as part of an agreement; "the contract set out the conditions of the lease"; "the terms of the treaty were generous".
CONDITION, noun. The procedure that is varied in order to estimate a variable's effect by comparison with a control condition.
CONDITION, verb. Establish a conditioned response.
CONDITION, verb. Develop (children's) behavior by instruction and practice; especially to teach self-control; "Parents must discipline their children"; "Is this dog trained?".
CONDITION, verb. Specify as a condition or requirement in a contract or agreement; make an express demand or provision in an agreement; "The will stipulates that she can live in the house for the rest of her life"; "The contract stipulates the dates of the payments".
CONDITION, verb. Put into a better state; "he conditions old cars".
CONDITION, verb. Apply conditioner to in order to make smooth and shiny; "I condition my hair after washing it".

## Wise words

You can change your world by changing your words... Remember, death and life are in the power of the tongue.
Joel Osteen


text:
Go to the content (press return)

# FUTUR. Website for the scientific production of UPC researchers

## Eigenvalue multiplicities for second order elliptic operators on networks

Author
Von Below, J.; Lubary, J.
Type of activity
Journal article
Journal
Operators and Matrices
Date of publication
2013-09
Volume
7
Number
3
First page
587
Last page
602
DOI
https://doi.org/10.7153/oam-07-32
Project funding
ECUACIONES EN DERIVADAS PARCIALES; ANÁLISIS Y APLICACIONES
Abstract
We present some general bounds for the algebraic and geometric multiplicity of eigenvalues of second order elliptic operators on finite networks under continuity and weighted Kirchhoff flow conditions at the vertices. In particular the algebraic multiplicity of an eigenvalue is shown to be strictly bounded from above by the number of vertices if there are no eigenfunctions vanishing in all nodes, and to be bounded from above by the number of edges if there are such eigenfunctions.
Keywords
Elliptic operators on networks, adjacency and transition operators, eigenvalue multiplicities
Group of research
EDP - Partial Differential Equations and Applications


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# Using Gauss's law to determine elctric fields (Calculus based problem)

1. Mar 22, 2010

### camrylx

1. The problem statement, all variables and given/known data
A sphere of radius R has a total charge +Q, uniformly distributed throughout its volume. It is surrounded by a thick spherical shell of inner radius R and outer radius 2R carrying a total charge -Q, also uniformly distributed throughout its volume. Using Gauss's law, determine the electric field as a function of r, the distance from the center of the sphere for the regions:
1. 0<=r<=R
2. R<=r<=2R
3. r>=2R

2. Relevant equations
Gauss's Law but from there no idea

2. Mar 23, 2010

### kuruman

How do you use Gauss's Law? What is the first step? Look in your textbook to get an idea if you have none.

3. Mar 23, 2010

### camrylx

Ok like I know the formula I have a good picture draw but not sure how to tackle this problem.... And by the way the idea of looking in the textbook to get started would be great if there was a textbook... Feel free to recommend one because I'm on my own with that

4. Mar 23, 2010

### kuruman

5. Mar 23, 2010

### camrylx

Ok the Gaussian surface would be a small sphere inside a different sphere. I know where +Q and -Q and i know how to find the $$\sigma$$ and all of that stuff. I know to start with q'=(r^3/R^3)*q and integrate that from 0 to 2pi and i think the 2 radius or something from there. I know coulombs law will come into and will be working this a spherical charge distribution and do some other stuff from there. I've seen a similar problem but im not sure how to fill all of the details in and I struggle with deriving some problems.

6. Mar 23, 2010

### kuruman

OK, Let's start with region 1, r < R. Can you describe the Gaussian surface you will need to find the E field? Fill in the blanks:

It is a sphere of radius _____ centered at _____ .

Then can you find the total electric flux through this sphere?

Finally, can you find the total charge enclosed by this sphere?


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### Let it be known that…

6th February, 2012

…I have just signed the Cost of Knowledge petition.

I don’t think I need say any more, since the issues have been thoroughly discussed elsewhere.


text:
• 0 Vote(s) - 0 Average
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 some questions about sexp bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 07/01/2010, 09:39 AM (06/28/2010, 11:18 PM)tommy1729 Wrote: (question 1) does the following hold : d f^n / d x^n sexp(slog(x) + k) > 0 for all positive integer n and all positive real k ? Though tempting for uniqueness this seems not to be the case, the half iterate of exp(x)-1 (which is conjugate to $e^{x/e}$) has the following powerseries coefficients at 0: Code:`0, 1, 1/4, 1/48, 0, 1/3840, -7/92160, 1/645120,...` « Next Oldest | Next Newest »


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# Data embedding using IBM quantum composer

How do you use the IBM quantum composer to encode some data $$(a,b,c,d)$$ represented by a vector ket in which $$a,b,c,d$$ have been normalized to one? $$a|00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$$

One way to do it is to use the initialize function then decomposed the circuit and print out the qasm code and pull it over to the composer...

There was a similar question to your question here

But here is an example that might help:

from qiskit import QuantumCircuit, IBMQ
import numpy as np
num_qubits = 2
vector = [1,2,3,4] #[a,b,c,d] not normalize
initial_state = vector/np.linalg.norm(vector) #normalize the state
circuit = QuantumCircuit(num_qubits,num_qubits)
circuit.initialize(initial_state, [0,1])
print(circuit)

┌─────────────────────────────────────────────┐
q_0: ┤0                                            ├
│  initialize(0.18257,0.36515,0.54772,0.7303) │
q_1: ┤1                                            ├
└─────────────────────────────────────────────┘
c: 2/═══════════════════════════════════════════════

And now you can use the decomposed function to decomposed this circuit into the gates that is available within the composer:

decomposed_circuit = transpile(circuit, basis_gates=['h', 'x', 'cx', 'ccx', 't', 'tdg', 's', 'sdg', 'p' , 'rz' , 'sx' ,'sxdg','rx', 'ry','rxx' , 'rzz',] )

┌────────────┐┌───┐┌─────────────┐┌───┐
q_0: ┤ RY(2.0344) ├┤ X ├┤ RY(0.17985) ├┤ X ├
├────────────┤└─┬─┘└─────────────┘└─┬─┘
q_1: ┤ RY(2.3005) ├──■───────────────────■──
└────────────┘
c: 2/═══════════════════════════════════════

Extracting the 'qasm' code:

print(decomposed_circuit.qasm())

print(decomposed_circuit.qasm())
OPENQASM 2.0;
include "qelib1.inc";
qreg q[2];
creg c[2];
ry(2.0344439) q[0];
ry(2.300524) q[1];
cx q[1],q[0];
ry(0.1798535) q[0];
cx q[1],q[0];

Now take that and paste it in the QASM code block in the Circuit Composer

Now you have the initialize state that you wanted in the composer.

You can implement a method proposed in this article Transformation of quantum states using uniformly controlled rotations. The authors present a method for transforming any quantum state to any other one. Of course, this allow to change initial state $$|0\rangle^n$$ in the composer to arbitrary $$n$$-qubit. The method works with $$CNOT$$, $$Ry$$ and $$Rz$$ gates only. All these are available in the composer.

According to my knowledge, this method is implemented in Qiskit init function which proposed KAJ226 in the other answer.


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# Confusion regarding continuity equation in electrodynamics

1. Sep 9, 2012

### ppoonamk

Suppose I have two charged particles with charge densities ρ1(r,t) and ρ2 (r,t) with corresponding velocity fields V1(r,t) and V2(r,t). Can I write continuity equation for the combined system? Wouldn't charges moving with different velocities would contribute differently to the current which will violate the continuity equation?

2. Sep 9, 2012

### Ken G

The continuity equation is linear in the charge density-- that means the sum of any two solutions is also a solution.

3. Sep 10, 2012

### Ken G

Perhaps the key point to stress is that if two different charge distributions are associated with two different velocity distributions, you can add the charge densities to get the local charge density, but you don't add the velocities-- there is no local velocity, you'd have to define a charge-weighted velocity to be able to use it like an independent local quantity in the continuity equation. You can think in terms of the charge flux density vector, which is the product of the charge density and the velocity, and then that quantity can be superimposed if you have two separate solutions, but the velocities of the two separate components cannot be meaningfully added the way the charge densities can.


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# Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$

How can I prove that
$$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$
Can anyone help me please?

#### Solutions Collecting From Web of "Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$"

Let’s start with the product of $\;-\ln(1-x)\,$ and $\dfrac 1{1-x}$ to get the product generating function
(for $|x|<1$) :
$$\tag{1}f(x):=-\frac {\ln(1-x)}{1-x}=\sum_{n=1}^\infty H_n\, x^n$$
Dividing by $x$ and integrating we get :
\begin{align}
\sum_{n=1}^\infty \frac{H_n}n\, x^n&=\int \frac{f(x)}xdx\\
&=-\int \frac{\ln(1-x)}{1-x}dx-\int\frac{\ln(1-x)}xdx\\
\tag{2}&=C+\frac 12\ln(1-x)^2+\operatorname{Li}_2(x)\\
\end{align}
(with $C=0$ from $x=0$)
The first integral was obtained by integration by parts, the second from the integral definition of the dilogarithm or the recurrence for the polylogarihm (with $\;\operatorname{Li}_1(x)=-\ln(1-x)$) : $$\tag{3}\operatorname{Li}_{s+1}(x)=\int\frac {\operatorname{Li}_{s}(x)}x dx$$

Dividing $(2)$ by $x$ and integrating again returns (using $(3)$ again) :
\begin{align}
\sum_{n=1}^\infty \frac{H_n}{n^2}\, x^n&=\int \frac {\ln(1-x)^2}{2\,x}dx+\int \frac{\operatorname{Li}_2(x)}x dx\\
&=C+I(x)+\operatorname{Li}_3(x)\\
\end{align}
with $I(x)$ obtained by integration by parts (since $\frac d{dx}\operatorname{Li}_2(1-x)=\dfrac {\ln(x)}{1-x}$) :
\begin{align}
I(x)&:=\int \frac {\ln(1-x)^2}{2\,x}dx\\
&=\left.\frac{\ln(1-x)^2\ln(x)}{2}\right|+\int \ln(1-x)\frac {\ln(x)}{1-x}dx\\
&=\left.\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)\right|+\int \frac{\operatorname{Li}_2(1-x)}{1-x}dx\\
&=\left.\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)-\operatorname{Li}_3(1-x)\right|\\
\end{align}
getting the general relation :
$$\tag{4}\sum_{n=1}^\infty \frac{H_n}{n^2}\, x^n=C+\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)+\operatorname{Li}_3(x)-\operatorname{Li}_3(1-x)$$
(with $C=\operatorname{Li}_3(1)=\zeta(3)$ here)
applied to $x=\dfrac 12$ with $\operatorname{Li}_2\left(\frac 12\right)=\dfrac{\zeta(2)-\ln(2)^2}2$ from the link returns the wished :
\begin{align}
\sum_{n=1}^\infty \frac{H_n}{n^2\;2^n}&=\zeta(3)-\frac{\ln(2)^3}2-\ln(2)\frac{\zeta(2)-\ln(2)^2}2\\
\tag{5}\sum_{n=1}^\infty \frac{H_n}{n^2\;2^n}&=\zeta(3)-\ln(2)\frac{\zeta(2)}2
\end{align}

We will make frequent use of
$$\binom{n+1}{k+1}=\binom{n}{k}\frac{n+1}{k+1}\tag{1}$$
The Generalized Harmonic Numbers of the second order are defined as
$$H_n^{(2)}=\sum_{k=1}^n\frac1{k^2}\tag{2}$$
The factor of $2^{-n}$ in each term reminded me of the Euler Series Transformation. Reversing the series acceleration (series deceleration?), it can be seen that we should look at
\begin{align} \sum_{k=0}^n\binom{n}{k}(-1)^k\frac{H_{k+1}^{(2)}}{k+1} &=\sum_{k=0}^n\binom{n}{k}\frac{(-1)^k}{k+1}\sum_{j=0}^k\frac1{(j+1)^2}\tag{3}\\ &=\frac1{n+1}\sum_{k=0}^n(-1)^k\binom{n+1}{k+1}\sum_{j=0}^k\frac1{(j+1)^2}\tag{4}\\ &=\frac1{n+1}\sum_{j=0}^n\sum_{k=j}^n(-1)^k\binom{n+1}{k+1}\frac1{(j+1)^2}\tag{5}\\ &=\frac1{n+1}\sum_{j=0}^n(-1)^j\binom{n}{j}\frac1{(j+1)^2}\tag{6}\\ &=\frac1{(n+1)^2}\sum_{j=0}^n(-1)^j\binom{n+1}{j+1}\frac1{j+1}\tag{7}\\ &=\frac1{(n+1)^2}\sum_{j=0}^n(-1)^j\sum_{k=0}^n\binom{k}{j}\frac1{j+1}\tag{8}\\ &=\frac1{(n+1)^2}\sum_{k=0}^n\frac1{k+1}\sum_{j=0}^n(-1)^j\binom{k+1}{j+1}\tag{9}\\ &=\frac1{(n+1)^2}\sum_{k=0}^n\frac1{k+1}\tag{10}\\[3pt] &=\frac{H_{n+1}}{(n+1)^2}\tag{11} \end{align}
Explanation:
$\ \;(3)$: use $(2)$
$\ \;(4)$: apply $(1)$
$\ \;(5)$: change order of summation
$\ \;(6)$: $\sum\limits_{j=k}^n(-1)^j\binom{n+1}{j+1}=(-1)^k\binom{n}{k}$
$\ \;(7)$: apply $(1)$
$\ \;(8)$: $\sum\limits_{j=0}^n\binom{j}{k}=\binom{n+1}{k+1}$
$\ \;(9)$: $\sum\limits_{j=0}^k(-1)^j\binom{k+1}{j+1}=1$
$(10)$: $H_{n+1}=\sum\limits_{k=0}^n\frac1{k+1}$

Using $(11)$, the Euler Series Transformation says that
\begin{align} \sum_{k=0}^\infty(-1)^k\frac{H_{k+1}^{(2)}}{k+1} &=\sum_{n=0}^\infty2^{-n-1}\sum_{k=0}^n\binom{n}{k}(-1)^k\frac{H_{k+1}^{(2)}}{k+1}\\ &=\sum_{n=0}^\infty2^{-n-1}\frac{H_{n+1}}{(n+1)^2}\tag{12} \end{align}
Therefore, with a change of indexing, we get
\begin{align} \sum_{n=1}^\infty2^{-n}\frac{H_n}{n^2} &=\sum_{n=1}^\infty(-1)^{n-1}\frac{H_n^{(2)}}{n}\tag{13}\\ &=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^3}+\sum_{n=1}^\infty(-1)^{n-1}\frac{H_{n-1}^{(2)}}{n}\tag{14}\\ &=\frac34\zeta(3)+\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}\sum_{k=1}^{n-1}\frac1{k^2}\tag{15}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=k+1}^\infty\frac{(-1)^{n-1}}{nk^2}\tag{16}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{(-1)^{k+n-1}}{(k+n)k^2}\tag{17}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=1}^\infty(-1)^{k+n-1}\left(\frac1{k^2n}-\frac1{kn(k+n)}\right)\tag{18}\\[6pt] &=\frac34\zeta(3)-\frac12\zeta(2)\log(2)+\frac14\zeta(3)\tag{19}\\[9pt] &=\zeta(3)-\frac12\zeta(2)\log(2)\tag{20} \end{align}
Explanation:
$(13)$: reindex $(12)$
$(14)$: $H_n^{(2)}=\frac1{n^3}+H_{n-1}^{(2)}$
$(15)$: apply $(2)$
$(16)$: change order of summation
$(17)$: reindex $n\mapsto k+n$
$(18)$: $\frac1{(k+n)k^2}=\frac1{k^2n}-\frac1{kn(k+n)}$
$(19)$: $\sum\limits_{k=1}^\infty\sum\limits_{n=1}^\infty\frac{(-1)^{k+n}}{kn(k+n)}=\frac14\zeta(3)$ from $(5)$ and $(7)$ of this answer
$(20)$: addition

\begin{eqnarray}
\sum\limits_{n=1}^\infty \frac{H_n}{n^2 2^n} = \sum\limits_{m=1}^\infty \frac{1}{m} \sum\limits_{n=m}^\infty \frac{1}{n^2 2^n} = \sum\limits_{m=1}^\infty \frac{1}{m} \int\limits_{-\infty}^0(-\xi) \frac{(1/2 \exp(\xi))^m}{1-1/2 \exp(\xi)} d\xi = \\
\int\limits_{-\infty}^0 \xi \frac{\log(1 – 1/2 \exp(\xi))}{1-1/2 \exp(\xi)} d\xi = \\
\int\limits_{1/2}^1 \left(\frac{1}{u} + \frac{1}{1-u}\right) \log(u) \left[\log(2) + \log(1-u)\right] du = \\
\zeta(3) – \frac{1}{12} \pi^2 \log(2)
\end{eqnarray}

I think that all the steps are clear except for the last two ones.In the second last step I substituted for 1 – 1/2 exp(xi).The only non-trivial integrals in here are $\int \log(u)/(1-u) du$ and $\int \log(u) \log(1-u)/(1-u) du$. I compute them now.
The first integral is done by expanding the denominator in a series and integrating term by term.

\int\limits_{1/2}^1 \frac{\log(u)}{1-u} du = \sum\limits_{p=0}^\infty \int\limits_{1/2}^1 u^p \log u du = \sum\limits_{p=0}^\infty \frac{-1+2^{-1-p}}{(p+1)^2} + \log(2) \sum\limits_{p=0}^\infty \frac{2^{-1-p}}{p+1} = -\zeta(2) + Li_2(1/2) + \log(2) Li_1(1/2)

The second integral is done by integrating by parts and using the definition of the polylogarithmic function.

\int\limits_{1/2}^1 \log(u) \frac{\log(1-u)}{1-u} du = \int\limits_{1/2}^1 \log(1-u) Li_2^{‘}(1-u) du = \left.\log(1-u) Li_2(1-u)\right|_{1/2}^1 + Li_3(1/2) = \log(2) Li_2(1/2) + Li_3(1/2)

Now, the only thing that remains is to bring the results together. I am sorry but due to time constraints I am not able to do it right now. I have verified with Mathematica that all the partial results are correct.

Final Note: We can clearly see that the result is expressed though elementary functions and through polylogarithms of order not bigger than three, evaluated at 1/2.From the Wikipedia page on Polylogarithms we learn those polylogarithms at 1/2 are expressed in closed form through $\pi$, $\log(2)$ and the $\zeta$ functions. Having said that we can say that this completes the proof.


text:
### The number of elements in vector spaces over finite fields.

Let $V$ be a vector space over a field $F$. If $n=\dim(V)$ is finite, then any choice of a basis $\beta=\{v_1,\ldots,v_n\}$ gives an isomorphism $F^n\cong V$, by the map $$F^n\to V,\ \ (a_1,\ldots,a_n)\mapsto a_1v_1+\ldots +a_n v_n.$$ Indeed, this map is bijective, due to the result that every $v\in V$ is uniquely a linear combination of elements of the basis. It also preserves addition and scalar multiplication. Suppose now that $F$ is a finite field. Just from the fact that the map above is a bijection, we get that $$\# V=\# (F^n)=(\# F)^n.$$

### The number of elements of finite fields.

An important special case: Recall that for every finite field, the unique smallest natural number $p$ such that $$p\cdot 1=1+\ldots+1=0$$ is a prime number. We obtain an inclusion of $\mathbb{Z}_p$ as a subfield of $F$, consisting of elements $0,1,1+1,\ldots$. But then we may regard $F$ as a vector space over the field $\mathbb{Z}_p$. Using the formula above (with $F$ playing the role of $V$ and $\mathbb{Z}_p$ playing the role of $F$), since $\# \mathbb{Z}_p=p$, we find that $$\# F=p^n.$$ So, we proved that the number of elements in a finite field is always some power of a prime number! Note that the statement itself doesn't involve vector spaces and bases, even though the proof does. (It also true that every power of a prime number really occurs as number of elements of a finite field, in fact a unique one, but this fact is harder.)


text:
The Riemann zeta function as an equivariant Dirac index

Risultato della ricerca: Contributo in rivistaArticolo in rivistapeer review

1 Citazioni (Scopus)

Abstract

Abstract. In this note an interpretation of Riemann’s zeta function is provided in terms of an R-equivariant L^2 -index of a Dirac-Ramond type operator, akin to the one on (mean zero) loops in flat space constructed by the present author and T. Wurzbacher. We build on the formal similarity between Euler’s partitio numerorum function (the S^1 -equivariant L^2 -index of the loop space Dirac-Ramond operator) and Riemann's zeta function. Also, a Lefschetz-Atiyah-Bott interpretation of the result together with a generalisation to M. Lapidus’ fractal membranes are also discussed. A fermionic Bost-Connes type statistical mechanical model is presented as well, exhibiting a “phase transition at (inverse) temperature β = 1”, which also holds for some “well-behaved” g-prime systems in the sense of Hilberdink-Lapidus.
Lingua originale English N/A-N/A 19 International Journal of Geometric Methods in Modern Physics 9 https://doi.org/10.1142/S0219887812500715 Pubblicato - 2012

Keywords

• zeta function, Dirac operators, fractal membranes


text:
Find the product of… - QuestionCove
OpenStudy (anonymous):

Find the product of the real values 'a' such that the equation $ax^{2} + 2y^{2} - 4y + 2(1 - a) = 0$represents an ellipse with latus rectum of length 1.

5 years ago
OpenStudy (anonymous):

Write the equation in standard form

5 years ago
OpenStudy (anonymous):

$ax^2+2(y^2-2y+1)-2a=0$

5 years ago
OpenStudy (anonymous):

$\frac{x^2}{2}+\frac{(y-1)^2}{a}=1$

5 years ago
OpenStudy (anonymous):

Alright, I get how we got to standard form, but how do we find out the values of a? I got this far yesterday, but I don't know what to do from here.

5 years ago
OpenStudy (anonymous):

do u know what is the latus rectum formula for standard form of ellipse

5 years ago
OpenStudy (anonymous):

Isn't it b^2/a? I think that's what someone said when I asked it yesterday.

5 years ago
OpenStudy (anonymous):

2b^2/a

5 years ago
OpenStudy (anonymous):

values of a and b for this equation?

5 years ago
OpenStudy (anonymous):

Well, I don't know because isn't a the bigger one? Would it be: 4/√a = 1 a = 16 OR 2a/√2 = 1 2a = √2 a = √2/2 Then the product: 16√2/2 = 8√2 Is that right?

5 years ago
OpenStudy (anonymous):

if a>b latus rectum=2b^2/a if a<b latus rectum=2a^2/b

5 years ago
OpenStudy (anonymous):

$a=\sqrt{2} \ \ and \ \ b=\sqrt{a}$

5 years ago
OpenStudy (anonymous):

Ok, I figured all that stuff out before I solved for a, so did I get it right 8√2?

5 years ago
OpenStudy (anonymous):

thats right

5 years ago
OpenStudy (anonymous):

Alright. Thank you!

5 years ago