during the last decades nonlinear models have become a workhorse for data analysis in many applications . while there is now an extensive literature on data analysis for such models , research on design selectionhas not kept pace , even though there has seen a spike in activity in recent years . identifying optimal designs for nonlinear modelsis indeed much more difficult than the much better studied corresponding problem for linear models . for nonlinear models resultscan typically only be obtained on a case - by - case basis , meaning that each combination of model , optimality criterion and objective of the experiment requires its own proof .another challenge is that for a nonlinear model an optimal design typically depends on the unknown parameters .this leads to the concept of locally optimal designs , which are optimal for a priori chosen values of the parameters .the designs may be poor if the choice of values is far from the true values .where feasible , a multistage approach could help with this . a small initial design is then used to obtain some information about the parameters , and this information is used at the next stage to estimate the true parameter values and to extend the initial design in a locally optimal way to a larger design .the design at this second stage could be the final design , or there could be additional stages at which more design points are selected .the solution presented in this paper is applicable for a one - shot approach for finding a locally optimal design as well as for a multistage approach .the argument that our method can immediately be applied for the multistage approach is exactly as in .  for a broader discussion on the challenges to identify optimal designs for generalized linear models ,many of which apply also for other nonlinear models , we refer the reader to .the work presented here is an extension of , and . the analytic approach in those papers unified and extended many of the results on locally optimal designs that were available through the so - called geometric approach .the extension in the current paper has major consequences for two reasons .first , it enables the application of the basic approach in the three earlier papers to many models for which it could until now not be used . as a result, this paper opens the door to finding locally optimal designs for models where no feasible approach was known so far .second , for a number of models for which answers could be obtained by earlier work , the current extension enables the identification of locally optimal designs with a smaller support .this is important because it simplifies the search for optimal designs , whether by computational or analytical methods .section [ sec4 ] will illustrate the impact of our results .the basic approach in , and , which is also adopted here , is to identify a subclass of designs with a simple format , so that for any given design , there exists a design in that subclass with under the loewner ordering .we will refer to this subclass as a complete class for this problem . here , and are information matrices for a parameter vector under and , respectively .others , such as have called such a class _ essentially _ complete , which is admittedly indeed more accurate , but also more cumbersome .when searching for a locally optimal design , for the common information - based optimality criteria , including - , - , - and -criteria , one can thus restrict consideration to this complete class , both for a one - shot or multistage approach .also , as shown in , this conclusion holds for arbitrary functions of the parameters .ideally , the same complete class results would apply for all a priori values of the parameter vector  .however , it turns out , as we will see in section [ sec4 ] , that there are instances where complete class results hold only for certain a priori values of .=-1   , and identify small complete classes for certain models .they do so by showing that for any design that is not in their complete class , there is a design that is in the complete class such that all elements of are the same as the corresponding elements in  , except that _ one diagonal element _ in is at least as large as that in .this guarantees of course that .the contribution of this paper is that we focus on increasing a _ principal submatrix _ rather than just a single _diagonal element_. this allows us to obtain results for more models than could be addressed by , and , and also facilitates the identification of smaller complete classes for some models considered in these earlier papers .  in section [ sec2 ]we will present the necessary background , while the main results are featured in section [ sec3 ] .the power of the proposed extension is seen through applications in section [ sec4 ] .we conclude with a short discussion in section [ sec5 ] .consider a nonlinear regression model for which a response variable depends on a single regression variable .we assume that the s are independent and follow some exponential distribution with mean , where is the parameter vector , and the values of can be chosen by the experimenter .typically , approximate designs are used to study optimality in this context .an approximate design can be written as , where is the weight for design point and .it is often more convenient to present as ,  ] is called a chebyshev system on  ] , suppose that either \\[-8pt ] & & \mbox{form chebyshev systems for every nonzero vector } q\nonumber\end{aligned}\ ] ] or \\[-8pt ] & & \mbox{form chebyshev systems for every nonzero vector } q.\nonumber\end{aligned}\ ] ]  then the following conclusions hold :  for , if ( [ assumption1 ] ) holds , then for any set } with , there exists a set with , such that  is dominated by .  for ,if ( [ assumption2 ] ) holds , then for any set with , there exists a set with , such that is dominated by .  for ,if ( [ assumption1 ] ) holds , then for any set with , there exists a set with , such that is dominated by .  for ,if ( [ assumption2 ] ) holds , then for any set with , there exists a set with , such that is dominated by  .  since the proof is similar for all parts , we only provide a proof for part ( a ) .let be as in part ( a ) .first consider the special case that . by ( 1a ) of therorem 3.1 in , there exists a set of at most pairs ( with one of the points equal to so that ( [ exist:1 ] ) and ( [ exist:2 ] ) hold for this . by part ( a ) of proposition [ prop0 ] in the , the number of distinct points with must then be exactly .thus we have , and the s and s must alternate by part ( b ) of proposition [ prop0 ] .the result follows now for an arbitrary nonzero by applying proposition [ prop1 ] in the and using ( [ assumption1 ] ) and ( [ exist:2 ] ) .lemma [ lemma2 ] partially extends lemma [ lemma1 ] by observing that larger sets than in lemma [ lemma1 ] are also dominated by sets as in that lemma .  [ lemma2 ] with the same notation and assumptions as in lemma [ lemma1 ] , let , where for cases ( ) , ( ) , and ( ) of lemma [ lemma1 ] , and for case ( ) . then the following conclusions hold :  for , if ( [ assumption1 ] ) holds , then is dominated by a set of size that includes as one of the points .  for ,if ( [ assumption2 ] ) holds , then is dominated by a set of size that includes as one of the points .  for ,if ( [ assumption1 ] ) holds , then is dominated by a set of size that includes both and as points .  for ,if ( [ assumption2 ] ) holds , then is dominated by a set of size .the results follow by application of lemma [ lemma1 ] .for example , for case ( a ) , if , the result follows directly from lemma [ lemma1 ] . if , we start with the points in .using lemma [ lemma1 ] , we obtain points in a set that dominates . using lemma [ lemma1 ] again on the largest points other than in , we move one more point to , obtaining a new set with that dominates .continue until the size of the set is reduced to ; this is the desired set .the first main result is an immediate consequence of lemma [ lemma2 ] .[ main1 ] for a regression model with a single regression variable , suppose that the information matrix can be written as in ( [ infor1 ] ) for  ] .  for ease of notation , in the remainder we will write instead of , and means that for all  ] .[ main2 ] for a regression model with a single regression variable  , let  ] .suppose that either or is positive definite for all  ] . among all , , and ,suppose that of them are negative .let denote the subscripts for these negative terms , and note that must be even .note also that the labels do not depend on the choice of the vector since do not depend on .finally , note that for any with , if we replace by , then the signs of and are switched while all others remain unchanged .we now change some of the s to .this is done for those that satisfy for some value of .denote the new -functions by .notice that . from the last observation in the previous paragraph ,it is easy to check that , , for the functions that correspond to this new set of -functions . by proposition [ prop3 ] in the , and are chebyshev systems on  ] is allowed to be .in this situation , for any given design , we can choose , and the conclusion of the theorem holds .similarly , can be in case ( b ) , and the interval can be unbounded at either side for case ( d ) .as noted at the end of section [ sec2 ] , the results in , , and correspond to .the extension in this paper allows the choice of larger values of where feasible .larger values of lead to designs with smaller support sizes .the reason for this is that the value of in theorems [ main1 ] and [ main2 ] corresponds to the number of equations in ( [ exist:1 ] ) . for a particular model ,this number is smaller for larger  .since the support size of the designs is roughly half the value of  , the support size is smaller for larger values of .we will provide some examples of the application of theorems [ main1 ] and [ main2 ] in the next section , and will offer some further thoughts on the ease of their application in section [ sec5 ] .whether the model is for continuous or discrete data , with homogeneous or heterogeneous errors , theorems [ main1 ] and [ main2 ] can be applied as long as the information matrix can be written as in ( [ infor1 ] ) . as the examples in this section will show , in many cases the result of the theorem facilitates the determination of complete classes with the minimal number of support points .dette , melas and wong ( ) studied exponential regression models , which can be written as where the s are i.i.d . with mean 0 and variance , and ] as the value of the single regression variable , which in this case refers to time . here is the parameter vector , where is the baseline logarithm of the tumor volume , is the final growth rate and is the rate at which killed cells get washed out .the size of the parameter relative to determines whether regrowth is monotonic ( ) or not . recently studied this model and showed that a -optimal design for can be based on four points , including and .we will now show that theorem [ main2 ] extends this conclusion to other optimality criteria and functions of interest .the information matrix for under design can be written in the form of ( [ infor1 ] ) with \\[-8pt ] c(\theta , c)&= & % \pmatrix { 1 & & & \cr e^{c } & e^{2c } & & \cr c & c e^{c } & c^2 & \cr c e^{c } & c e^{2c } & c^2 e^{c } & c^2 e^{2c } } % , \nonumber % \end{aligned}\ ] ] where . with a proper choice of functions, it can be shown that the result in yields a complete class of designs with at most five points , including and .we can again do better with theorem [ main2 ] .define , , , , and this yields , , , , and clearly is a positive definite matrix .therefore , by part ( c ) of theorem [ main2 ] , we reach the following conclusion .  for the linexp model ( [ linexp ] ) , the designs with at most four points , including and , form a complete class .demidenko ( ) , using a two - compartment model , developed a double - exponential regrowth model to describe the dynamics of post - irradiated tumors .the model can be written as + \varepsilon_{ij } , % \ ] ] with independent and  ] .let , and let be coefficients that satisfy the following equations : then we have :   if and one is not zero , then all are nonzero ; moreover all  s for odd must then have the same sign , which is opposite to that of the s for even .  for part ( a ) ,if , we can expand to a set of distinct points , taking for the added points .thus without loss of generality , take .consider the matrix then ( [ prop0:1 ] ) can be written as where . since is a chebyshev system , is nonsingular , so that .  for part ( b ) , if one is 0 , then it follows from part ( a ) that all s are 0 . therefore ,if at least one is nonzero , then all of them must be nonzero . with the notation from the previous paragraph, we can write ( [ prop0:1 ] ) as where .it follows that by the chebyshev system assumption , the denominator in ( [ addthis ] ) is positive , while the numerator is positive for and negative otherwise .the result in ( b ) follows .[ prop1 ] let be a chebyshev system on an interval  ] , we can conclude that if is also a chebyshev system .  with the equations in ( [ prop1:1 ] )can be written as where and are as defined in the proof of proposition [ prop0 ] .further , ( [ prop1:2 ] ) is equivalent to using ( [ prop1:3 ] ) to solve for , and using that , we see that ( [ prop1:4 ] ) is equivalent to from an elementary matrix result [ see , e.g. , theorem 13.3.8 of ] , the left - hand side of ( [ prop1:5 ] ) can be written as where \\[-8pt ] & & \qquad= \pmatrix { \psi_0(c_1 ) & \psi_0(\tilde{c}_1 ) & \cdots & \psi_0(c_n ) & \psi _ 0(\tilde{c}_n ) \cr \psi_1(c_1 ) & \psi_1(\tilde{c}_1 ) & \cdots & \psi_1(c_n ) & \psi _1(\tilde{c}_n ) \cr \vdots & \vdots & \ddots & \vdots & \vdots\cr \psi_{k-1}(c_1 ) & \psi_{k-1}(\tilde{c}_1 ) & \cdots & \psi_{k-1}(c_n ) & \psi_{k-1}(\tilde{c}_n)\cr \psi_{k}(c_1 ) & \psi_{k}(\tilde{c}_1 ) & \cdots & \psi_{k}(c_n ) & \psi _ { k}(\tilde{c}_n ) } .\nonumber\end{aligned}\ ] ] since both and are chebyshev systems and , it follows that ( [ prop1:6 ] ) is negative , which is what had to be shown .[ prop2 ] let be a chebyshev system on an interval  ] , we can conclude that if is also a chebyshev system .[ prop3 ] consider functions on an interval  ] , ( [ prop3:3 ] )is equivalent to recall that the entries in the last rows of this matrix are by definition simply values of , .hence , applying the same arguments used for ( [ prop3:1 ] ) to ( [ prop3:4 ] ) and using that for  ] , which is precisely our assumption .thus the conclusion follows .

Abstract: we extend the approach in [ _ ann . statist . _ * 38 * ( 2010 ) 24992524 ] for identifying locally optimal designs for nonlinear models . conceptually the extension is relatively simple , but the consequences in terms of applications are profound . as we will demonstrate , we can obtain results for locally optimal designs under many optimality criteria and for a larger class of models than has been done hitherto . in many cases the results lead to optimal designs with the minimal number of support points .  .


since first experimental demonstrations in 1991 , atom interferometry has developed into a powerful tool for the ultra precise measurement of accelerations and rotations .it is now used in various laboratories for experiments in the fields of fundamental physics and metrology . in principle , this new technique is also ideally suited for high - accuracy field research such as gravity mapping , geophysics , seismology or navigation and could substantially exceed that of classical gravimeters . however , due to the complexity of these experiments they were so far confined to laboratory environments . only in recent yearsefforts have been undertaken to develop mobile atom interferometers that might in future versions also be used on satellite missions .our transportable high - precision gravimeter gain ( gravimetric atom interferometer ) is designed specifically for geophysical on - site measurements .the working principle of a gravimetric atom interferometer has been described in detail elsewhere . in short : an ensemble of laser - cooled neutral atoms ( in our case rubidium 87 ) is prepared in a 3d magneto - optical trap ( mot ) , further cooled in optical molasses , and launched upwards . during their parabolic flight , the atoms are subjected to three pulses from counterpropagating laser beams , thereby inducing two - photon raman transitions that transfer them between the two hyperfine ground states via a stimulated raman process .the sequence consists of one - , one - , and finally another -pulse , which constitute an atom optic beam splitter , mirror and recombiner , respectively .thereby the atomic wave packet is split into two parts that travel on different trajectories due to momentum transfer from the photons . at the output of the atom interferometer the transition probability from one hyperfine state to the otheris given by , where is the accumulated phase difference between wave packets and the contrast of the measurement .this includes an acceleration contribution of with being the time between two consecutive raman pulses , the sum of the two individual couterpropagating raman laser wavenumbers , and the local gravitational acceleration .  in this paper , we describe in detail the realization of the atomic gravimeter s laser system and characterize its subcomponents .the sensor s physics package ( i.e. main chamber and attached optics and electronics , vibration isolation system , control system ) , first high - precision gravity measurements as well as the instrument s performance in the field will be discussed in future publications .naturally , the laser system for a mobile atom interferometer has to be mobile and compact as well .our system is designed to be `` truckable '' ( i.e. easily transportable by a small truck from one gravity measurement site to the next ) , requiring a rethermalization and readjustment time of less than a day .it will operate at gravimetry reference points where the absolute gravity value is measured in regular intervals .these points are usually selected to be inside of buildings with an environment that is relatively stable , as conventional gravimeters are sensitive to changes in their operating environment ( although harsher conditions can be found at more unusual measurement sites ) .we can thus expect to typically encounter environments with temperature variations of one or two kelvin , low vibrational noise level and no direct sun light .this is , however , still considerably worse than conditions usually found in laboratory cold atom experiments .standard laser systems for these experiments are not only quite complex , but also can rarely endure significant mechanical vibrations , thermal fluctuations of even a few kelvin or electromagnetical noise , without losing laser frequency locks or a significant decrease in optical power output .a first step in solving these problems was the redesign from scratch of almost all optical mounts , as standard laboratory equipment rarely offers sufficient mechanical stability and is in most cases simply too large for our purposes ( figure [ optics ] ) .many of these mount designs are adapted from the quantus drop tower project and have proven their stability even under extreme accelerations of up to 50 .we have mounted all optics in four closed , compact modules with 1 cm thick walls and a beam height of 20 mm for high ruggedness .light is transferred between the modules by means of polarisation maintaining optical fibres . except for the reference laser , which is even smaller, each module s base area is 42 by 42 cm with a height of a few centimeters .the base plate of two of these modules is a custom made thorlabs aluminium honeycomb breadboard with a 1 cm grid of m3 threads which provides stability as well as flexibility for possible future modifications .the other two modules use 25 mm thick aluminum slabs as base plates with the four walls and two additional divider walls mounted in a force - fit such that under mechanical stress the assembled modules behave as if manufactured from one piece .fibre and electrical connections are mounted on the walls of each module , the total height of each module varies between 75 and 105 mm .  as laser sources we have built compact external cavity diode lasers ( ecdl ) that have been adapted from a design developed at syrte using sharp gh0781ja2c laser diodes .they include an interference filter as a wavelength selector that is one order of magnitude less sensitive to angular misadjustments than gratings found in conventional ecdl configurations which makes it suitable for our application .the output power is up to 50 mw and the intrinsic linewidth less than 10 khz .the cavity length is 80 mm which gives a free spectral range ( fsr ) of approximately 1.9 ghz .control of the laser frequency is achieved by tuning of either the laser diode current or the cavity length ( via a piezo ) . applied separately , the mode - hop - free tuning range is limited to 200 mhz or the fsr of 1.9 ghz , respectively .however , by controlling both parameters simultaneously at a fixed gain ratio , a mode - hop free tuning range of about 9 ghz and thereby over more than the complete d2 line was achieved ( figure [ ecdlspectroscopy ] ) .  to provide the various optical frequencies required to operate an atomic fountain interferometer and yet still maintain sufficient flexibility for future enhancements like quasi - continuous operation ( many clouds of atoms in flight at the same time ) , a total of five ecdls is employed .they are organized in four distinct modules : one reference laser module , two cooling laser modules and one raman laser module ( figure [ schematic ] ) .the modules are mounted in a standard 19 " electronics rack that has been fitted with inflatable air bags between its main body and its base plate which serve as a passive vibration isolation and also as a shock absorber for transport over rough terrain .a test transport over a snow - covered cobblestone surface has resulted in a decrease of fibre coupling efficiency of less than 30 percent .the rack is dimensioned such that it can fit through standard doors and is thereby easily transportable to different locations ( figure [ photo1 ] ) . for easy access ,the laser modules are mounted on telescopic rails .the complete laser system and its control electronics could theoretically be mounted in just one rack we chose , however , to also include other gravimeter electronics such as computer control , a backup power system , power supplies and diagnostic equipment which made a second rack necessary and enabled us to move mechanically noisy equipment ( i.e. anything that includes cooling fans ) away from the optics .the dimensions of the complete system are  ( computer and control electronics rack ) plus  ( laser rack ) , adding up to a total volume of m .power consumption is less than 1 kw .the reference laser module houses an ecdl that is stabilized 40 mhz below the f=3 f=4 transition by modulation transfer spectroscopy .the control signal is used to stabilize the laser wavelength via the the laser diode s current controller . however , since this parameter is limited to a mode - hop free tuning range of 200 mhz , a second control path is employed : low - frequency components ( below 100 hz ) of the control signal are applied to both the laser diode current as well as the piezo voltage at the fixed gain ratio determined to give us a mode - hop free tuning range of 9 ghz ( see above ) . this way , slow drifts due to temperature variations or mechanical stress are compensated and the laser stays locked over days at its required frequency . in a laboratory test, the reference laser stayed locked even when slowly changing the module s temperature by 10 kelvin over two hours .the reference laser is also impervious to moderately strong hits of a metallic wrench to its base plate . comparing the system s open - loop and closed - loop frequency responses using a network analyzer , we measured a locking bandwidth of approximately 300 khz ( at 3db below unity gain ) .this bandwidth is limited by electronics and cable lengths and not by the optical spectroscopy itself , as we reached a regime in which the atoms are pumped quickly into the desired internal states by using a pumping beam power of half a milliwatt with a beam diameter of one millimeter .an atomic fountain setup requires a variety of optical frequencies in order to trap , cool , launch , select and detect the atoms , as can be seen in figure [ levels ] . in our vacuum chamber , we capture the atoms in a magneto - optical trap ( mot ) .since the atom interferometer s sensitivity scales with the square root of both the repetition rate and the number of atoms , we aim to trap as many atoms as possible as quickly as possible . to achieve this, we use a large mot volume ( beam diameter 35 mm ) at high laser power ( mw ) . in order to launch the atoms , both upper and lower mot beams have to be detuned with respect to each other to achieve a moving molasses configuration .additionally , a repumper is required , as well as blow - away and detection light in both f=1 and f=2 frequency classes .a total of 11 separate fibre output ports at different laser powers and frequencies is required for full functionality .  due to the large hyperfine ground splitting of 6835 mhz , light from the f=1 frequency class can not easily ( or with high efficiency ) be shifted to f=2 by means of acousto - optical modulators .hence , we employ two ecdls mounted in the first cooling laser module , one for each frequency class . for frequency stabilization ,light from each ecdl is overlapped with light from our reference laser on a fast photodiode ( hamamatsu g4176 - 03 ) resulting in beat frequencies of about 5080 mhz and 1000 mhz , respectively .the signals frequencies are subsequently divided down to about 19.8 mhz ( factor 256 ) and 100 mhz ( factor 10 ) , respectively . comparing the resulting signal s zero - crossings to those of a stable direct digital synthesizer ( dds ) reference frequency in a hittite hmc440qs16 g digital phase - frequency detector ( pfd ) gives an error signal that we use to phase - lock the beat signal onto the dds reference at a bandwidth of 200 khz .this setup enables us to reach any desired laser frequency simply by changing the dds frequency .the f=2 laser light is split into two halves of 12 mw each that are used to seed two eagleyard tapered amplifiers ( tpa-0780 - 01000 ) , thereby amplifying laser power to two times one watt . both of these high - power beams mode - cleaned by an optical fiber at a coupling efficiency of only between 50 and 60 percent due to imperfections in the tapered amplifier s output laser beam profile .a third fibre is employed for the f=1 light ( 17 mw in - fibre ) .  in the second cooling laser modulethe light is frequency shifted and switched using acousto - optical modulators ( aoms ) , one for each of these three beams .this enables separate frequency and switching control of upper mot , lower mot and repumper beams .not all eleven output ports will have to be used simultaneously , as for instance blow - away beams and mot light are not required at the same time . by changing the aoms frequencies , the first order diffraction beam angle varies slightly .we make use of this effect and hit different mirrors at different aom frequencies , one of which directs the light into one fibre , the other one reflects the light back into the aom for a double pass configuration .effectively , this enables us to switch between different fibre outputs at different laser frequencies within less than a microsecond without losing any light power at ports that are not in use at any given moment .since the most light power is needed in the mot phase of our experiment , the aoms have been selected and adjusted so that they work at their center frequency and therefore peak efficiency of about 80 percent ( single - pass ) in that configuration . for producing blow - away anddetection beams , less light power is sufficient , so we chose this configuration for driving the aoms far detuned from their center frequency which resulted in single - pass efficiencies of 50 percent ( driving an 80 mhz aom at 100 mhz , crystal technology model 3080 - 125 ) and 60 percent ( driving a 200 mhz aom at 171 mhz , crystal technology model 3200 - 121 ) .  to avoid unwanted scattered light , additional mechanical shutters ( sunex sht934 , switching time 1 - 2 ms )are used at each of the eleven output ports .we have not observed any influence of the shutters mechanical noise on the fibre coupling efficiency .a schematic of this setup is shown in figure [ coolsetup ] , the total output power of this system in mot configuration is six times 60 mw cooling light plus 5 mw repumper light ( in - fibre , coupling efficiency larger than 80 percent ) .in order to induce an optical raman transition between the hyperfine ground states of the atoms , a pair of two lasers with a fixed phase relation and a frequency difference of 6.835 ghz ( the ground state hyperfine splitting ) is required to drive the two - photon transition via an intermediate level , as seen in figure [ levels ] . in this section, we will describe our raman laser system and its performance .we employ two ecdls ( figure [ ramansetup ] ) which emit light that is amplified to 1 watt using tapered amplifiers and then overlapped .rubidium vapour cells are employed to suppress unwanted amplified spontaneous emissions on atomic resonances .an aom is used for fast switching and pulse - shaping of the raman pulses .identical pulse - shaping on both laser beams is ensured by an intra - module fibre common to both lasers where the light is mode - cleaned before entering the aom .the raman master laser ecdl is phase - locked to the reference laser using a setup similar to the one employed in the cooling laser system , however the pfd we use here is an analog devices adf4108 .the raman slave laser ecdl is stabilised in frequency and phase in respect to the raman master laser .any phase noise between the two raman lasers will be imprinted onto the atoms and will therefore directly limit the gravimeter s sensitivity .this is discussed in detail in section [ sec : limit ] .accordingly , the requirements in noise and locking bandwidth are much higher here than for the locks of the raman master or the two cooling lasers . for this phase lock between the two ecdls , light from both lasersis overlapped on a fast photodiode that is placed behind the intra - module fibre so that all noise sources that are not common to both beams ( i.e. anything before overlapping and mode - cleaning done by the fibre ) can be compensated for by the phase lock .the resulting beat signal of 6835 mhz is mixed down using a stable 6735 mhz reference .the resulting 100 mhz signal is phase - locked onto a dds reference frequency using a motorola mc100ep140 pfd . to overcome bandwidth limitations and thereby residual phase noise imposed by long cables and the laser diode current controller s response time , an additional high frequency control path is employed that modulates the ecdl laser diode s current directly via a small signal n - channel fet as a voltage to current converter and a lag - lead compensation network .total cable lengths for this fast path add up to less than a meter .the 6735 mhz mixing - down frequency is generated by a frequency chain that uses a spectra dynamics dlr-100 system as a frequency reference .the dlr-100 includes an ultra - low noise 100 mhz quartz that is locked to the 10th harmonic of a frequency - doubled 5 mhz quartz for even lower phase noise at low frequencies .the 100 mhz signal is multiplied to 6800 mhz and is then used to lock a dielectric resonator oscillator ( dro ) to 6735 mhz .this frequency chain is basically identical to the one described in .  by comparing the downconverted signal with the 100 mhz dds reference using an independent mixer , we measured the phase noise spectral density of this optical phase lock loop ( opll ) , see figure [ psd ] . between 100 hz and 60 khz , where our interferometer is most sensitive to noise ( as shown in section [ sec : limit ] ), it largely stays below a level of -120 dbrad/hz ( 1 / hz ) .the achieved locking bandwidth is slightly above 4 mhz .however , since in this measurement any noise in either the frequency chain or the reference quartz is cancelled out , both of these noise sources have to be taken into account additionally in order to evaluate the performance of the complete system : thus , figure [ psd ] also shows the phase noise spectral density of the frequency chain as described and characterized in and that of the reference quartz system .in this section , we will calculate the limit to our gravimeter s sensitivity due to phase noise from the raman laser system . of course , these are not the only noise sources in our gravimeter , as we are amongst others sensitive to mechanical vibrations and detection noise .these noise sources will be evaluated in future publications . in a setup like ours , the precision with which we are able to measure local in a single measurementis limited by the phase uncertainty as follows :    the contribution of the raman laser s power spectral density affects via a transfer , or weighting function , , i.e. can be evaluated as    in our setup we employ three raman pulses : one -pulse , one -pulse , and finally another -pulse . assuming square raman pulses of duration , separated by time , the explicit form for as derived in is  \right|^2\end{array}\ ] ]  with being the rabi oscillation frequency of the raman transition . due to a highly oscillatory behavior of , however ,after the 16th oscillation only the average value is calculated ( figure [ sens_function ] ) in order to avoid aliasing effects due to our limited data point spacing at higher frequencies . also of noteis the band pass filter behavior of whose effective lower cutoff frequency scales with , whereas the upper cutoff frequency scales with .  to evaluate the limit that our laser system imposes on gravimeter sensitivity , we calculate the root - mean - square of the spectra of the three contributing sources of phase noise in our system ( opll , frequency chain , quartz ) and multiply it with . integrating over the complete frequency spectrum, we obtain our limits for and consequently for .assuming standard operating parameters for our gravimeter of =150 ms and =100 , our single - shot sensitivity will be limited to by raman laser phase noise .  for different and , the band pass behavior of the weighting function changes and so does our sensitivity to certain phase noise frequencies .the resulting sensitivity limits are shown in figure [ tauvst ] . to illustrate which frequencies are the dominant contributors to these limit calculations , we have plotted an accumulation integral for different pulse lengths ( figure [ acc_tau ] ) . in this diagram , is plotted against in . as can be seen , our 4 mhz servo bump ( see figure [ psd ] ) does not significantly contribute to the overall gravimeter sensitivity at long pulse lengths , at shorter pulse lengths , however , the noise spectrum in the mhz range becomes a dominating factor .in contrast , a variation of pulse spacing instead of increases our sensitivity to low - frequency phase noise which is dominated by our quartz reference .this contribution is , however , not nearly as significant as the fact that the sensitivity to scales with , so we want to keep as long as possible in our apparatus ( see figure [ tauvst ] ) .we have designed and built a laser system for atom interferometry applications that is mobile and robust , yet still offers improvements over many conventional laboratory - based systems .this system enables us to operate a highly precise atom interferometer outside of standard laboratory conditions and thereby opens up new possibilities for geophysical gravity measurements .this work is supported by the european commission ( finaqs , contr .012986 - 2 nest ) , by esa ( sai , contr .20578/07/nl / vj ) and by esf / dfg ( euroquasar - iqs , dfg grant pe 904/2 - 1 ) .we further thank lne - syrte for design and construction of the frequency chain and the quantus team for a fruitful collaboration .g. stern , b. battelier , r. geiger , g. varoquaux , a. villing , f. moron , o. carraz , n. zahzam , y. bidel , w. chaibi , f. p. dos santos , a. bresson , a. landragin and p. bouyer , _ eur .j. d _ * 53 * , 353 ( 2009 ) .g. m. tino , l. cacciapuoti , k. bongs , c. j. borde , p. bouyer , h. dittus , w. ertmer , a. goerlitz , m. inguscio , a. landragin , p. lemonde , c. lammerzahl , a. peters , e. rasel , j. reichel , c. salomon , s. schiller , w. schleich , k. sengstock , u. sterr and m. wilkens , _ nucl .b. _ * 166 * , 159 ( 2007 ) .s. g. turyshev , u. e. israelsson , m. shao , n. yu , a. kusenko , e. l. wright , c. w. f. everitt , m. kasevich , j. a. lipa , j. c. mester , r. d. reasenberg , r. l. walsworth , n. ashby , h. gould and h. j. paik , _ int . j. mod .* 16 * , 1879 ( 2007 ) .t. van zoest , n. gaaloul , y. singh , h. ahlers , w. herr , s. t. seidel , w. ertmer , e. rasel , m. eckart , e. kajari , s. arnold , g. nandi , w. p. schleich , r. walser , a. vogel , k. sengstock , k. bongs , w. lewoczko - adamczyk , m. schiemangk , t. schuldt , a. peters , t. koenemann , h. muentinga , c. laemmerzahl , h. dittus , t. steinmetz , t. w. haensch and j. reichel , _ science _ * 328 * , 1540 ( 2010 ) .

Abstract: we present a modular rack - mounted laser system for the cooling and manipulation of neutral rubidium atoms which has been developed for a portable gravimeter based on atom interferometry that will be capable of performing high precision gravity measurements directly at sites of geophysical interest . this laser system is constructed in a compact and mobile design so that it can be transported to different locations , yet it still offers improvements over many conventional laboratory - based laser systems . our system is contained in a standard 19 " rack and emits light at five different frequencies simultaneously on up to 12 fibre ports at a total output power of 800 mw . these frequencies can be changed and switched between ports in less than a microsecond . the setup includes two phase - locked diode lasers with a phase noise spectral density of less than 1 / hz in the frequency range in which our gravimeter is most sensitive to noise . we characterize this laser system and evaluate the performance limits it imposes on an interferometer .


el estudio de las ecuaciones de los fluidos incompresibles tiene cada vez un mayor inters , tanto desde el punto de vista ms terico ( integrales singulares ... ) como desde el enfoque ms aplicado ( simulaciones numricas ... ) .las ecuaciones que aparecen modelizando problemas de mecnica de fluidos son variadas , pero las ms importantes son las de euler y navier - stokes. de hecho por demostrar ( o refutar ) la existencia global de solucin clsica para navier - stokes el instituto clay otorga un premio de un milln de dlares .consideramos como dominio espacial el plano o el espacio enteros , con y exigimos que la velocidad del fluido sea cero en el infinito .sea adems el campo de velocidades inicial .entonces las ecuaciones de euler ( 1707 - 1783 ) para la velocidad de un fluido incompresible son con y .este sistema de ecuaciones es la segunda ley de newton en el caso de un continuo de partculas .las ecuaciones de navier ( 1785 - 1836 ) y stokes ( 1819 - 1903 ) para un fluido incompresible con densidad son donde .  en las ecuaciones de navier - stokes se ha a~nadido el rozamiento entre partculas del fluido modelizndolo con un laplaciano .observamos que es un sistema de evolucin _extra~ no _ en el sentido de que la derivada temporal de la presin no aparece .eso nos indica que la presin se puede obtener de la velocidad .para ver esto basta tomar la divergencia de la ecuacin de conservacin del momento , ahora podemos utilizar la funcin de green para el laplaciano ( que para es conocida ) y obtener .tomamos el gradiente en esta expresin y obtenemos un sistema de ecuaciones no - locales cerrado para ( donde ahora nos restringimos a velocidades incompresibles ) .adems hemos obtenido que la presin acta como un multiplicador de lagrange para la restriccin . los resultados de los que se dispone cierran la teora en .en el plano se conoce la existencia global ( en tiempo ) de soluciones clsicas .para el problema completo se tiene un teorema de existencia local ( en tiempo ) de solucin clsica ( ver ) .esta diferencia es porque las ecuaciones en los casos bidimensional y tridimensional son sensiblemente distintas .para verlo necesitamos definir una cantidad que nos de informacin sobre cunto _ gira _ el fluido .esta cantidad que denotaremos se llama vorticidad y es el rotacional de la velocidad ,  con el campo de velocidades del fluido .en concreto se tiene que las ecuaciones para la vorticidad son distintas : para un fluido no viscoso e incompresible , es decir , que sigue las ecuaciones de euler , en el caso se tiene que es una ecuacin de transporte , mientras que en el caso se tienen las siguientes ecuaciones en el caso de considerar un fluido viscoso se ha de a~ nadir un trmino .la diferencia entre ambos casos aparece tambin en las longitudes de onda que estn ntimamente relacionadas con el fenmeno de la _ turbulencia _ ( ver ) .la turbulencia tiene como efecto principal facilitar que dos fluidos se mezclen , por lo tanto , llegados a este punto podemos _experimentar _ un teorema .para este peque~ no juego necesitamos dos vasos peque~ nos llenos hasta arriba uno de ellos de agua y el otro de vino . la cuestin es : cmo conseguimos cambiar los lquidos de vaso sin usar un tercer recipiente y sin que se mezclen ?para responder a esta pregunta hemos de conocer cmo es la turbulencia en tres dimensiones y qu diferencia hay con dos dimensiones .as , si conseguimos una manera de _ reducir el problema tridimensional a uno bidimensional _hemos acabado , porque en dos dimensiones _no hay turbulencia _ y sta es la culpable de que los lquidos se mezclen .para conseguir esta reduccin en las dimensiones lo que hacemos es tapar el vaso de agua con un carn y colocarlo con cuidado encima del vaso de vino .si lo hemos hecho bien no se ha salido ni una gota .ahora abrimos una rendija minscula entre los vasos y el carn .el agua es ms densa , por lo tanto comenzar a bajar mientras que el vino subir ... y todo esto sin mezclarse ! ( ver figura [ figura1 ] )     tras esta excursin por las ciencias experimentales volvamos a las matemticas .la vorticidad es una cantidad que aparece en el conocido criterio de existencia global de beale - kato - majda ( ver ) .sean y dos constantes tales que entonces una solucin clsica de las ecuaciones de euler existe al menos hasta tiempo .adems , si es el tiempo mximo de existencia ( es decir , aparece una singularidad ) , entonces [ teo1 ]  este teorema nos dice que si controlamos _ lo que gira _ el fluido entonces no hay singularidades , por lo tanto conocer cmo se comporta la vorticidad es crucial para intentar entender qu hace el fluido . en efecto , tambin es interesante porque si la conocemos podemos recuperar la velocidad gracias a la frmula de biot - savart ( ver ) . en el caso bidimensional la ley de biot - savart es con ahora bien , con lo que acabamos de mencionar podemos ver que el problema de las ecuaciones de euler en el caso bidimensional podemos formularlo de manera equivalente como un _escalar activo _ , es decir , un escalar que es transportado por el fluido de manera que adems podemos recuperar la velocidad del fluido si conocemos el escalar .en efecto : recordemos que tenamos la ecuacin para la vorticidad ( que en dos dimensiones es un escalar ) , esto unido a cierra el problema para la vorticidad .adems si asumimos que nuestro dato inicial est acotado , usando el teorema [ teo1 ] , tenemos que la solucin clsica del sistema existe globalmente .el estudio matemtico de los escalares activos tiene una gran relevancia en cuanto que son sistemas _que conservan el carcter no local de un fluido incompresible .adems de que hay multitud de aplicaciones donde el problema fsico se puede modelizar con un escalar activo .por ejemplo tenemos el caso de la ecuacin _ quasigeostrfica _ ( ver , , , y las referencias all expuestas ) o de la _ ley de darcy _ ( ver , , , , , ) .la ecuacin quasigeostrfica modeliza la evolucin de la temperatura de grandes masas de aire en grandes escalas espaciales y es muy estudiada como modelo de la _ frontognesis _ ( la formacin de frentes de aire a distinta temperatura ) .este problema es de inters en meteorologa , porque ya se sabe que  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ la falta de acierto de quienes predicen el tiempo se ha hecho ya proverbial , y sin embargo no hay ningn meteorlogo competente que no opine que los procesos atmosfricos estn causalmente determinados .max planck ( extrado de ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  la ecuacin quasigeostrfica en dos dimensiones espaciales es donde es la transformada de riesz ( ver ) y es la temperatura del aire .la ley de darcy modeliza un fluido incompresible que se mueve a bajas velocidades por un medio poroso .as si es la densidad del fluido se tiene el problema este problema puede reducirse a un escalar activo tomando el rotacional dos veces en la ecuacin , obteniendo podemos modelizar la transferencia de calor interno del fluido en un medio poroso con la misma ecuacin si a le damos el sentido de una temperatura ( ver ) .por lo tanto , tanto en el caso de la ecuacin y de puede interesarnos a~ nadir un trmino de difusin del calor .sin embargo estos trminos de difusin no tienen por qu ser el tpico laplaciano , puede ser necesario a~ nadir una _potencia fraccionaria del laplaciano_. el laplaciano en el espacio de fourier ( con variables ) tiene una expresin sencilla , es un multiplicador : podemos definir el operador de la siguiente manera y equivalentemente notemos que tambin podemos escribir el resultado de aplicar el laplaciano fraccionario como la siguiente convolucin : donde es una constante de normalizacin .los operadores y ( o su versin ) son lo que nosotros entendemos por _ potencias fraccionarias del laplaciano_. as las ecuaciones y con difusin no local son y   para irnos aproximando a estos problemas podemos plantearnos otros ms sencillos o simplificados .por ejemplo , consideremos un escalar activo donde es un operador integral singular que adems nos garantiza que .queremos simplificar este problema de manera que sea fcilmente abordable , pero no tiene que estar tan simplificado que no nos de ninguna informacin . lo primero que hacemos es reducir el nmero de variables espaciales a una , por lo tanto .adems podemos simplificar tomndolo igual a la identidad , es decir , pierde su carcter integral singular .la ecuacin resultante de estas simplificaciones es la _ ecuacin de burgers viscosa _si y la _ ecuacin de burgers no viscosa _ si ( ver y ) : como la condicin en una dimensin no tiene sentido fsico ( en una dimensin no hay choques ) con esta ecuacin tenemos un modelo unidimensional de las ecuaciones y .as nuestra ecuacin es el modelo ms sencillo que nos da informacin tanto sobre las ecuaciones , como .este texto est organizado de la siguiente manera : en la seccin 2 probaremos la existencia local de solucin clsica para toda la familia de ecuaciones . en la seccin 3 obtendremos una ley de conservacin y unos principios del mximo que se tienen para las soluciones clsicas de la ecuacin . en la seccin 4 daremos un criterio de existencia de solucin clsica anlogo al de beale - kato - majda . en la seccin 5 veremos que hay _blow up _ para en y en la seccin 6 haremos simulaciones numricas de la solucin de .el problema que estudiaremos en esta seccin y las siguientes es la ecuacin ( donde cambiamos la notacin por la ms corriente ) : \times{\mathbb r}\\ u(0,x)=f(x),\\ \end{array}\right . \label{burg}\ ] ] donde es el tiempo de existencia , y .sobre el dato inicial impondremos las condiciones necesarias cuando veamos el teorema de existencia local , pero por el momento podemos suponer que .nos restringiremos a las _ soluciones clsicas _ o _ soluciones fuertes_. este tipo de soluciones es tan suave como sea necesario para dar el sentido usual a las derivadas parciales que aparecen en la ecuacin .as para la ecuacin de burgers queremos que tenga al menos una derivada en y dos en .es decir , queremos que .en esta seccin probaremos la existencia local de solucin clsica para la ecuacin con y  ] .adems , por las propiedades de los ncleos , se puede demostrar que   tenemos as el siguiente resultado :  dado una dato inicial con el problema tiene una nica solucin .   veremos que ,h^k) ] , donde ( ver proposicin [ estimeps ] ) .para ello tenemos que ver que en es cauchy y entonces utilizando las estimaciones de la proposicin [ estimeps ] concluiremos que es una sucesin de cauchy en con .el caso extremo de lo trataremos al final . es una sucesin de cauchy en ,h^s) ] .sean dos nmeros positivos .entonces se tiene veamos el caso de los operadores difusivos : integrando por partes el segundo sumando obtenemos que y por lo tanto , usando las propiedades de las suavizaciones ( ver ) recordemos que , fijo , tenemos la cota uniforme y por lo tanto tenemos que para la integral se hace igual .hay que sumar y restar los trminos y aplicar las mismas desigualdades ( sobolev , cauchy - schwartz ... ) .una vez que hemos probado que es una sucesin de cauchy en ,l^2) ]como el lmite de en la topologa usual de ese espacio .adems podemos mejorar el resultado para cubrir el caso extremo .as hemos de ver que est en el espacio .,h^s) ] .para probarlo hemos de razonar con anlisis funcional .como est acotada en ( proposicin [ estimeps ] ) tiene una subsucesin que converge dbilmente en este espacio .es decir en principio no podemos asegurar que .para concluir la igualdad utilizamos las convergencias dbiles , pues la convergencia fuerte implica la convergencia dbil , en los espacios con adems del hecho de que y obtenemos que si tenemos una contradiccin .gracias a los lemas anteriores podemos demostrar el siguiente resultado :  sea , .entonces existe una nica ,h^k)\cap c^1([0,t],h^{k-2}) ] como lmite uniforme en compactos de .para obtener la regularidad temporal necesaria para observamos que tiende como distribucin a la derivada dbil de ( que denotamos por ) .en efecto , para toda funcin test , se tiene que adems en , c({\mathbb r})) ] con se concluye que es solucin clsica de la ecuacin de burgers .para concluir que ,h^{k-2}) ] , entonces se tienen los siguientes resultados  * si , * si ,   observamos que la funcin es lipschitz .en efecto : de donde para concluir que la funcin es lipschitz observamos que debemos restringirnos a un intervalo temporal  ] y obtenemos }\max_x({\partial_t}u ( s , x))(t_1-t_2)=l(t_1-t_2).\ ] ] usando el teorema de radamacher tenemos que es derivable en casi todo punto , de manera que , si , donde en la ltima igualdad hemos usado que es mximo .en el caso en el que tenemos que dar signo al trmino .usaremos la expresin como una convolucin .se tiene que en el caso en el que el resultado se reduce a conocer qu signo tiene la segunda derivada en un punto de mximo .para el mnimo se hace de manera anloga .el lector atento se habr percatado de que la ecuacin de burgers tiene una especie de _ teorema de beale - kato - majda_. recordemos que en una dimensin espacial lo ms parecido a es la primera derivada espacial .entonces tenemos el siguiente resultado :  sea .entonces , si existe solucin clsica de al menos hasta tiempo .  en la prueba de la proposicin de las estimaciones _ a priori _ ( proposicin [ estim ] ) logramos la cota de esta expresin obtenemos y aplicando la desigualdad de gronwall ( es decir , integramos la edo ) conseguimos el resultado .consideramos la ecuacin con .esta ecuacin es una ecuacin de transporte unidimensional no - lineal , por lo tanto es susceptible de aplicarle el mtodo de las caractersticas .esa tcnica aplicada a esta ecuacin se puede encontrar en casi todos los manuales de ecuaciones diferenciales .nosotros utilizaremos una tcnica similar a la que utilizamos para probar el principio del mximo en la seccin 2 .queremos encontrar una singularidad para .por la proposicin [ prop2 ] tenemos que no _ explota _ , por lo tanto si hay una singularidad debe estar en alguna de las derivadas espaciales de .como estamos en el caso no viscoso solamente tenemos una derivada espacial , as que es nuestra candidata a cantidad que explota.  as sea una solucin clsica de .por la inmersin de sobolev se tiene que .ahora la idea es aplicar el teorema de rademacher a .de manera anloga a la de la demostracin de la proposicin [ prop2 ] se demuestra que si se tiene que como es el punto de mximo de tenemos que , por lo que la ecuacin para la evolucin del mximo es si ahora resolvemos la edo obtenemos que hemos obtenido el siguiente resultado  sea tal que .entonces solucin clsica de con desarrolla una singularidad en tiempo finito .[ singula ]   el tipo de singularidad que se produce se conoce como __ choque _ _ y es una discontinuidad en , o lo que es lo mismo .si pensamos en como la altura de la superficie de una ola unidimensional lo que ocurre es que las partculas del fluido que integran la ola se mueven ms rpido cuanto ms altas estn , por lo que las ms altas van a acercarse a las ms bajas .entonces cuando las alcancen se produce este _choque_. esta discontinuidad previene que nuestra interfase deje de ser un grafo , porque si pensamos en la fsica de la ola tenemos que las partculas ms rpidas sobrepasan a las ms lentas ( ver figura [ figura2 ] ) .hemos mencionado que la ecuacin de burgers poda ser entendida como una ola , lo que no es completamente descabellado .si es la superficie del agua ( entonces la ola viene dada por la curva ) , y es el campo de velocidades del agua , entonces la ecuacin de la interfase entre el agua y el aire es ahora observamos que en las olas la parte ms alta se mueve a mayor velocidad que la parte ms baja .por lo tanto parece natural ( al menos como primera aproximacin ) hacer la hiptesis ( la constante podemos tomarla igual a 1 si hacemos un cambio de variables ) .as si despreciamos la velocidad vertical , , tenemos la ecuacin de burgers no viscosa como modelo del perfil de una ola .podemos razonar tambin que la curva cambia segn sea su curvatura , de manera que  ] la curvatura de la curva .adems si suponemos curvas suaves de amplitud peque~ na podemos linealizar la curvatura obteniendo el operador .as , con hiptesis razonables al menos en primera aproximacin , obtenemos la ecuacin de burgers viscosa como modelo simplificado del perfil de una ola .una ola unidimensional con unas ciertas caractersticas ( longitud de onda , amplitud ... ) puede modelizarse en un mejor nivel de aproximacin con la ecuacin kdv siguiente : veamos de manera muy resumida cmo puede derivarse formalmente esta ecuacin .para describir una ola consideramos que el agua bajo la superfie tiene un flujo irrotacional ,  para cierta funcin escalar .si suponemos vlidas las ecuaciones de euler para el agua bajo la superficie tenemos que sigue la ley adems , por la incompresibilidad se tiene .( 0,-1.545)(6.24,1.565 ) ( 0.0,0.735)(0.0,-0.065)(0.92163783,0.97220963)(1.92,0.915)(2.9183621,0.85779035)(3.1040077,0.5444405)(4.1,0.455)(5.095992,0.3655595)(6.176627,-0.48499432)(6.18,0.515 ) ( 0.06,-1.485)(0.06,-1.505 ) ( 0.08,-1.525)(6.22,-1.505 ) ( 0.08,0.435)(6.1,0.435 ) ( 1.24,-1.485)(1.24,0.355 ) ( 1.4478126,-0.9)h ( 1.86,0.535)(1.86,0.855 ) ( 2.013125,0.66)a ( 0.16,1.095)(5.8,1.115 )( 3.3092186,1.38)  la coordenada se distingue de la en que acta la gravedad , por lo tanto parece natural hacer un desarrollo de en potencias de , .si consideramos olas peque~ nas en amplitud con respecto a la longitud de onda , entonces tenemos que despreciar los trminos de orden grande en en nuestra expresin para .si adems suponemos que podemos concluir la ecuacin .esta hiptesis se motiva por los desarrollos en serie de potencias anteriores .observamos que la hiptesis para obtener la ecuacin de korteveg - de vries es menos restrictiva que para obtener la ecuacin de burgers , pues exclusivamente suponemos que no que .consideramos por lo tanto discrepancias en los ordenes mayores .en la siguiente seccin nos proponemos dar una aproximacin numrica al problema propuesto en .la forma natural de abordar el problema es usando la transformada de fourier discreta ( dft ) ( ver y ) .sea una discretizacin de , con un nmero par de puntos , dada por , la idea es sustituir el valor de en los nodos , , por su interpolante trigonomtrico , es decir , dado un vector con los valores nodales de ,  ] .estos coeficientes vienen dados por , y puede demostrarse que estn relacionados con los coeficientes de la serie de fourier de mediante , sabiendo como aproximar , es importante saber como aproximar .para ello derivamos el interpolante trigonomtrico anterior obteniendo , nuestra aproximacin de en vendr dada por lo que nosotros llamaremos , de la misma forma , parece apropiado aproximar el operador mediante ,      sea una discretizacin del dominio temporal  ] es la aproximacin de la solucin a nuestro problema en tiempo se tiene que verifica la ecuacin con .

Abstract: en este artculo se presentan las ecuaciones de euler y de navier - stokes , las ms bsicas de las ecuaciones de los fluidos incompresibles , as como unos modelos simplificados de dichos problemas como pueden ser la ecuacin quasigeostrfica u otros escalares activos . las tcnicas matemticas para obtener que estas ecuaciones estn bien puestas en el sentido de hadamard y otras propiedades cualitativas ( principios del mximo , formacin de singularidades ... ) se ilustran en el caso sensiblemente ms sencillo de la ecuacin de burgers con disipacin no - local . se adjunta una seccin dedicada a los mtodos numricos usados para aproximar soluciones a estos problemas . este trabajo tiene su origen en una serie de clases que impartimos en la escuela jae - intro del csic durante el verano del curso 2010 - 2011 y por lo tanto se centra en la popularizacin y divulgacin de la fsica involucrada y en la explicacin detallada de las ideas ms abstractas en los argumentos puramente matemticos .   * palabras clave * : ecuaciones de euler , ecuaciones de navier - stokes , ecuacin quasi - geostrfica , medios porosos , ley de darcy , interfase .   * agradecimientos * : r.granero est financiado por el proyecto mtm2008 - 03754 del ministerio de ciencia e innovacin ( micinn ) .


quantum game theory , an interdisciplinary field that combines quantum theory and game theory , has been investigated for fifteen years . the first attempt to describe a game in the quantum domain applied to finite noncooperative games in the normal form , , .the general idea ( in the case of bimatrix games ) was based on identifying the possible results of the game with the basis states .soon after quantum theory has also found an application in duopoly problems , .it has been a challenging task as the players strategy sets in the duopoly examples are ( real ) intervals and therefore there are continuum of possible game results .the scheme presented in adapts the marinatto - weber quantum game scheme to the stackelberg duopoly example .the model relates actions in the duopoly with the probabilities of applying the bit - flip operators . on the other hand ,the model introduced in is a new framework compared with the quantum game schemes .it was defined to consider the cournot duopoly problem where each strategy in the classical game corresponds to a specific unitary operator .the model entangles the players quantities and the relation depends on a degree of the entanglement .the iqbal - toor and li - du - massar schemes undoubtedly brought new ideas to the field of quantum games .these remarkable schemes have found an application in many duopoly problems .the former scheme was further investigated , for example , in , , , , , the latter one in ,  , , , , , , , , , .the aim of the paper is to pay attention to some properties of the quantum duopoly schemes that might be thought unsuitable ( in the case of the iqbal - toor scheme ) and specify the nash equilibrium analysis ( in the case of the li - du - massar scheme ) .we recall the version of the cournot model with two firms that have been the subject of research in the quantum domain . based on , firm 1 and firm 2 offer quantities and , respectively , of a homogeneous product .the price of the product depends on the total quantity .the higher the quantity is , the lower the price of the product . more formally , the cournot s duopoly model defines a strategic form game , where  1 . is a set of players , 2 . is a player s strategy set , 3 . is a player s payoff function given by formula here represents the market price of the product , and is a marginal cost with   a nash equilibrium is the most commonly used solution concept to study duopoly examples .it is defined as a profile of strategies of all players in which each strategy is a best response to the other strategies . in view of the cournot duopoly itis a strategy profile that satisfies the following system of inequalities : it implies that the game has a unique nash equilibrium with the equilibrium payoff for each player .it is appropriate at this point to note that in literature one can find ( [ classicprice ] ) with requirement and a statement that the cournot duopoly has the unique nash equilibrium .in fact , is crucial to the uniqueness of the equilibrium .if and one of the players , say player 1 , chooses then the player 2 s set of best replies is . by completely symmetric arguments, is player 1 s best reply to .thus there would be continuum many equilibria such that with payoff 0 for both players .in this section we discuss the two main quantum approaches to the problem of duopoly .both schemes show how to define game with uncountable sets of strategies .   in paper  the authors adapted the marinatto - weber quantum scheme for games to the problem of the stackelberg s duopoly . we restrict ourselves to the cournot s duopoly , where the players choose their actions simultaneously instead of a sequential order. it does not affect the framework of the quantum scheme but simplifies the analysis .the key idea relies on identifying players actions in the classical duopoly with probabilities that determine the final state in the quantum model of games .then , by appropriately defined measurement operators , the scheme , in particular case , is supposed to output the classical game .formally , the final state associated with the iqbal - toor scheme has the form where and  are the probabilities of choosing by player 1 ( player 2 ) the identity operator and the pauli operator , respectively . in order to associate player s actions for with final state ( [ finalstate ] ) the authors defined the following probability relations : as a result , if and the probabilities and are given by equation ( [ probabilities ] ) , the final state ( [ finalstate ] ) can be written as .\ ] ] given the payoff operator , ,\ ] ] player s payoff is of the form   let us consider scheme ( [ finalstate])-([payoff ] ) in view of the cournot s duopoly .the first problem we can see is that model ( [ finalstate])-([payoff ] ) in fact does not reproduce the game defined by ( [ classicpayoff])-([classicprice ] ) .the payoff function ( [ payoff ] ) coincides with formula ( [ classicpayoff ] ) for but it does not take into account the market price equal to zero in the case . as a result , if , for example , , player 1 s payoff function in the classical case comes down to and it is different in general from given by ( [ payoff ] ) .  the scheme ( [ finalstate])-([payoff ] ) will generalize the classical cournot sduopoly if we modify operator ( [ operatorm ] ) to include the case .let us define & if~~q_{1 } + q_{2 } \leqslant a \\ ( 1 + q_{1})(1+q_{2})q_{i}(-c|00\rangle \langle 00| ) & if~~q_{1 } + q_{2 } > a. \end{cases}\ ] ] then , where is given by ( [ finmw ] ) , determines the complete payoff function ( [ classicpayoff ] ) .it is worth noting that there are many ways to define the measurement operator that determines the same outcome for .let operators ( [ operatorm ] ) for be replaced by \\m''_{2 } = ( 1+q_{1})(1 + q_{2})[(a - c)q_{2}|00\rangle \langle 00| - q_{2}|01\rangle \langle 01| - |11\rangle \langle 11| ] .\end{array}\ ] ] then for .simultaneously , the payoff function defined by ( [ m2prim ] ) is different from one determined by ( [ operatorm ] ) for other initial states . for example , in the case of the initial state , the final state has the form .\ ] ] then , the payoff functions corresponding to measurements and are given by formulae , \quad \mathrm{tr}(\rho_{\mathrm{fin}}m''_{1 } ) = q_{1}\left[(a - c)q_{1}q_{2 } - q_{2}\right ] - 1.\ ] ]  equations ( [ roznefunkcje ] ) suggest that we can modify model ( [ finalstate])-([payoff ] ) in various ways , each time obtain new results , for example , with respect to nash equilibria . however , the scheme ( [ finalstate])-([payoff ] ) has a feature that may question the correctness of the model .namely , it outputs non - classical results already for separable initial states . in order to see that ,let us use results ( [ fin11 ] ) and ( [ roznefunkcje ] ) obtained by letting .then for arbitrary and suitably large expression as a function of variable increases monotonically and . in the same manner we can see that if is suitably large , the higher quantity is , the more player 2 gains . therefore , scheme ( [ finalstate])-([payoff ] ) with the initial state the players to obtain arbitrary large payoffs .by contrast , each player s set of available payoffs in the classical cournot duopoly is bounded from above .the upper bound is .it corresponds to the monopoly quantity given by solving .an easy computation shows that there is no another form of to obtain the scheme ( [ finalstate])-([payoff ] ) that outputs the classical game for , and is a nontrivial generalization of ( [ classicpayoff ] ) .indeed , let us consider the general form of a payoff operator for , where .suppose operator , satisfies the equation if for , the final state described by ( [ finalstate ] ) assumes the form where means addition modulo 2. then condition ( [ condition ] ) determines a system of four equations that has the unique solution as a result , operator ( [ ogolnym ] ) comes down to the trivial payoff operator that implies the outcome for each density operator defined on . clearly , in the game determined by ( [ finalstate])-([payoff ] ) and there is no profitable nash equilibrium . for the fixed profile ,player would benefit by choosing .thus , one may question , if the players are able to obtain superior payoffs compared to the classical case .however , as it has been mentioned , that model does not take into account the proper price function ( [ classicprice ] ) .it can not then be considered in terms of a generalization of the cournot model .let us now replace ( [ operatorm ] ) with ( [ poprawionym ] ) .this gives & \mbox{if}~q_{1 } + q_{2 } \leqslant a , \\ -cq_{1}q_{2 } & \mbox{if}~q_{1 } + q_{2 } > a. \end{cases}\ ] ] let us assume now that and consider a strategy profile . if then \nonumber\\ & \geqslant \left(\frac{a}{2 } - q_{1}\right)\left(\frac{a}{2 } + q_{1 } - \frac{a}{2}\right ) = \left(\frac{a}{2 } - q_{1}\right)q_{1 } \geqslant 0,\end{aligned}\ ] ] where the second to last inequality follows from the fact that the assumption implies in the case , we have thus proved that is a player 1 s best response to . in an exactly similar way we can show that is a best response to .this means that is a nash equilibrium .we now conclude from comparing the equilibrium payoffs in the classical cournot duopoly and defined by function ( [ function11 ] ) that the latter payoff can be much greater .   and for initial numbers , where .[ figure0 ] ]  for example ( see also the graphs in fig [ figure0 ] ) , if and , the classical equilibrium outcome is 81 whereas .it is worth noting that the problem of nonclassical results determined by the basis states spreads to other duopoly examples based on scheme ( [ finalstate])-([payoff ] ) .let us recall the problem of bertrand duopoly studied in  .in this case two firms ( players ) compete in price .let and be the prices chosen by player 1 and 2 , respectively. then the product quantities are given by and the payoff functions are the quantum scheme for the problem ( [ b1])-([b2 ] ) proceeds similarly to ( [ finalstate])-([payoff ] ) where each in ( [ probabilities ] ) is replaced with ( for the detailed description we refer the reader to ) .it was designed to consider the initial state of the form and depend on a triple and are equal to , \nonumber \\ & & u^{q}_{2}(p_{1 } , p_{2 } ) = ( a - p_{2 } + bp_{1})\left[(p_{2 } - c)\cos^2{\gamma } + \left(p_{1 } + p_{2}\left(-1 -cp_{1 } + p^2_{1}\right)\right)\sin^{2}{\gamma}\right ] . \nonumber\end{aligned}\ ] ] thus , in particular , if and we consider strategy profile in the form of , the formulae comes down to now , it is easily seen that the players payoffs approach infinity as approaches infinity .this is clearly a nonclassical result as in the classical case ( [ b2 ] ) ^ 2}{2(1-b ) } < \infty.\ ] ]the question now arises : given the results above , can the quantum scheme ( [ finalstate])-([poprawionym ] ) be considered reasonable ?the requirement taken from quantum schemes for finite strategic games is satisfied : the duopoly example defined by ( [ classicpayoff])-([classicprice ] ) can be obtained from ( [ finalstate])-([poprawionym ] ) .thus , in this sense , the quantum duopoly based on the marinatto - weber scheme appears to be well - defined . however , there is also some implicit condition embeded in the framework for quantum playing finite strategic games .namely , each payoff profile generated by the quantum scheme lies in the convex hull of set of payoff profiles determined by the classical game .for example , in the marinatto - weber and the eisert - wilkens - lewenstein schemes for games the payoff operator has the form , where are the four payoff pairs that define the game .then for any density operator on determined by the players , the resulting payoff pair is a convex combination of points .if we assumed that equal convex hulls of payoff profiles generated by both the classical and quantum game were a necessary condition for the quantum scheme to be a correct one , then the protocol ( [ finalstate])-([poprawionym ] ) would not be valid .   the quantum protocol introduced in is another way to define the problem of duopoly in the quantum domain .let us recall the formal description of this scheme .let be the initial state and , , where  is the creation ( annihilation ) operator of player i s electromagnetic field .the player s strategies depend on and they are given by formula a quantum measurement on state , described by the observables , gives the quantities equations ( [ quantumquantities ] ) are obtained by using the formula ( see also ) }{\sqrt{b}}\sinh{\lambda\sqrt{\beta } } , \nonumber\end{aligned}\ ] ] where the operators and satisfy the relation = \beta b ] . to show that is the unique best reply in this caselet us note that for .thus we have this means that the player 1 s payoff given by ( [ czescfunkcji ] ) is nonnegative . as a result ,player 1 would make a loss by choosing such that . for function ( [ czescfunkcji ] ) ofvariable is strictly decreasing on .hence , it is optimal for player 1 to take and obtain .note also that player 1 would not be willing to take such that . indeed , since and , it is true that for each   if , problem ( [ argmax ] ) is equivalent to . in this caseit is optimal player 1 to choose . summarizing , player 1 s best reply function is as follows similar arguments to those aboveshow that the player 2 s best reply function is given by formula we recall that a nash equilibrium is a strategy profile at which each player chooses a best response to the other players strategies . as a result , the nash equilibria in the li - du - massar scheme with ( [ poprawionypayoff ] ) can be identified with the points of intersection of graphs determined by the best reply functions and .the graphs are drawn in fig [ figure1 ] .   and given by ( [ bestreply1 ] ) and ( [ bestreply2 ] ) .the intersection represents the unique nash equilibrium in the game .[ figure1 ] ]  in this way , there is the unique nash equilibrium .it is obtained by solving the system of linear equations thus , the solution coincides with the result provided in .namely ,   it is worth noting that the quantum extension of the cournot duopoly introduced in  affects the quantities and and makes use of the payoff function of the classically played duopoly .this implies that the set of payoff profiles generated by the li - du - massar scheme with ( [ poprawionypayoff ] ) is equal to one determined by function ( [ classicpayoff ] ) . in particular , by solving the maximization problem we obtain the set .hence , for each , a symmetric pareto optimal outcome is equal to in both the classical and quantum case .the li - du - massar scheme with the refined payoff function ( [ poprawionypayoff ] ) is defined in accordance with the quantum protocols for finite games .the scheme generalizes the classicaly played cournot duopoly and it keeps the set of feasible payoff profiles unchanged .we saw in subsection [ sectionmw ] that the scheme based on the marinatto - weber approach for bimatrix games does not satisfy the latter condition .the question now is whether these two requirements on a quantum scheme imply the unique quantum model for the cournot duopoly problem .the two well - known and quite different quantum schemes for bimatrix games , introduced in and suggest that the answer ought to be negative .in fact , this is the case .we can define another scheme that is consistent with the requirements above . in what follows ,we give an example of a scheme that is similar in concept to the li - du - massar scheme .the idea is based on ` entangling ' the players quantities and in order to obtain and for some real numbers and .let be the initial state and be an entangling operator , ,\ ] ] where and are the identity and pauli operator , respectively , defined on .the resulting state is then given by let us identify the player s strategies with for .the values and determine two positive operators given by formula the measurement defined by determines the quantities and in the following way : where and are the reduced density operators and of the first and second qubit , respectively . as a result referring to function ( [ classicpayoff ] ) we obtain the following players payoff functions : certainly , scheme ( [ mojscheme1])-([mojscheme2 ] ) is a generalization of the classically played cournot duopoly . if then payoff function ( [ mojscheme2 ] ) boils down to function ( [ classicpayoff ] ) .since the scheme uses the classical payoff function , the corresponding set of feasible payoffs remains the same as in the classical case .if the marginal cost is positive , the set of nash equilibria in the game defined by scheme ( [ mojscheme1])-([mojscheme2 ] ) is as follows : \right\ } & \mbox{if}~\gamma = \pi/4 . \end{cases}\ ] ]  the proof is similar in spirit to that of proposition [ prop1 ] .let satisfy . for solution of is /(2\cos^2{\gamma})\scriptstyle x_{1 } \in [ 0,\infty ) , x_{1 } + x_{2 } \leqslant a\scriptstyle x_{1 } \in [ 0,\infty ) , x_{1 } + x_{2 } \leqslant a\scriptstyle x_{1 } \in [ 0,\infty) ] . if , there are infinitely many solutions such that .this completes the proof .  )depending on the entanglement parameter .[ figure3 ] ]  the payoff corresponding to the nash equilibrium depends on and is given by .\ ] ] in particular , if the final state is maximally entangled ( ) , the resulting equilibrium payoff is pareto optimal and equal to .the theory of quantum games has no rigorous mathematical structure .there is no formal axioms , definitions that would give clear directions of how a quantum game ought to look like .in fact , only one condition is taken into consideration .it says that a quantum game ought to include the classical way of playing the game . as a result , this allows one to define a quantum game scheme in many different ways .the schemes we have studied in section [ section3 ] are definitely ingenious .they make a significant contribution to quantum game theory . our work has shown which of the two schemes ( the iqbal - toor scheme or the li - du - masasr scheme ) might be considered more reasonable . the payoffs in the game determined by the iqbal - toor scheme can go beyond the classical set of feasible payoffs compared with the li - du - masasr scheme .therefore , one may question whether the former scheme outputs the game played in a quantum manner or just another classical game. it might not mean that the latter scheme with ( [ poprawionypayoff ] ) gives the definitive form of the quantum cournot duopoly . at the end of the paper we defined another scheme in terms of quantum theory .we can conclude that the question of quantum duopoly is still an open problem even in the simplest case of the cournot duopoly .it is definitely worth investigating as other new schemes may bring us closer to specify a strict definition of a quantum game .100 meyer d a , quantum strategies , _ phys .lett . _ * 82 * 105255 ( 1999 ) eisert j wilkens m and lewenstein m , quantum games and quantum strategies _ phys .lett . _ * 83 * 3077 - 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Abstract: the aim of this paper is to discuss in some detail the two different quantum schemes for duopoly problems . we investigate under what conditions one of the schemes is more reasonable that the other one . using the cournot s duopoly example we show that the current quantum schemes require a slight refinement so that they output the classical game in a particular case . then we show how the amendment changes the way of studying the quantum games with respect to nash equilibria . finally , we define another scheme for the cournot s duopoly in terms of quantum computation .