in a recent series of papers evidence has been presented suggesting a correlation between nuclear decay rates and solar activity .this evidence came from analyses of data from three independent sources .the first was data taken at purdue university during a period which included the solar flare of 2006 december 13 , which exhibited a statistically significant dip in the counting rate of coincident in time with the solar flare .the second was data from a measurement of the half - life ( t ) of at brookhaven national laboratory ( bnl ) , and the third data set was an extended study of detector stability via measurements of the decay rates of , , and at the physikalisch - technische bundesanstalt ( ptb ) in germany .the bnl and ptb data revealed periodic variations in measured decay rates which approximately correlated with earth - sun distance .taken together the data of refs . raise the possibility that nuclear decays are being influenced by the sun through some as yet unknown mechanism , perhaps involving solar neutrinos .the data presented in ref . were a series of successive four - hour counts taken over the course of two months as part of a half - life measurement .the selection of a four - hour count time allowed for a time resolution capable of detecting the solar flare , which lasted approximately 43 minutes .due to the short counting period , the counts were presumably insensitive to typical seasonal environmental effects , such as temperature , pressure , background , etc .however , the same can not be said of the data of refs . since the annually varying decay rates reported in these references could in principle arise from a number of conventional systematic influences on the detectors employed in these experiments rather than from a modulation of the decay rates themselves .the object of the present paper is to analyze the bnl and ptb data in detail in an effort to disentangle environmental influences on the detector systems used in each experiment from a possible contribution originating from the sun , which could affect the intrinsic decay rates .the detector system used in the data set associated with the 2006 december 13 solar flare will also be discussed .although there are many mechanisms through which experimental detectors can be influenced by environmental factors , a useful starting point for the present analysis is the recent paper by semkow , et al . in which the authors attempt to explain both the bnl and ptb data in terms of seasonal temperature variations . in what followswe will analyze these experiments in detail and show explicitly why the mechanisms proposed by semkow et al . to explain the bnl / ptb data are not likely to be correct .this analysis can thus serve as a template for similar analyses of other past and future experiments on half - life determinations to evaluate possible seasonal environmental influences .in the course of experiments at the ptb which were intended to evaluate the stability of detector systems employed in the measurement of half - lives , e.g. ( t=4936.6(20)d ) , the authors reported an oscillation in the decay rate of the long - lived standard , ( t=1600(7)y ) used in the measurements .these fluctuations were attributed by the authors to a `` discharge effect on the charge collecting capacitor , the cables and insulator to the ionization chamber electrode caused by background radioactivity such as radon and daughters which are known to show seasonal concentration changes '' .it was also proposed elsewhere that the variations in the measured activity of the source were caused by variations of the density of the argon working gas within the detector chamber itself , caused by seasonal changes in the ambient temperature .this and other potential systematic effects , such as changes in ambient pressure , humidity , background radiation or electronic fluctuations , are indeed possible explanations for the observed variations and should be examined . an alternative proposal to these explanations , as suggested in ref . , was that the oscillations of the measurements were not caused by local perturbations to the detector system , but were due to the influence of fields or particles such as neutrinos emanating from the sun . inwhat follows , we will examine possible environmental influences on the detector system , and determine which of these , if any , could be responsible for the fluctuations in the measurements .  to evaluate which environmental influences may have caused the fluctuations in the detector , it is helpful to understand how the detector utilized in the ptb experiments works .the measurements were taken with a high - pressure 4 ionization chamber , model ig12 , manufactured by 20th century ( now centronic , ltd . ) , which was filled with argon to 20 atm ( 2 mpa ) by the manufacturer , then welded shut prior to delivery to the user ( trevor alloway , centronic ltd ., private communication , 2009 ) .therefore , the number of atoms of the working gas , argon , within the detector is constant throughout the lifetime of the detector , irrespective of the external temperature , pressure or relative humidity .the ionization chamber for the ptb experiments was operated in a `` current '' mode , as is typical with ionization chambers in this application . here, the output current of the detector is generated by the collection of electron - ion pairs created by the ionization of the argon gas inside .these electron - ion pairs arise from photon interactions within the gas , and from the interaction of photoelectrons and compton electrons generated by photon interactions in the wall of the re - entrant tube , which then enter the working gas also causing ionization .ionization chambers operate at sufficiently high bias voltages to prevent recombination of the electron - ion pairs generated by the interactions of photons , photoelectrons , and compton electrons in the gas . in the case of the ptb measurements , that voltage was 500v .the electrodes ( anode and cathode ) of the detector are contained within the chamber of the detector itself , and these collect the electrons and ions created via the radiation interactions .the current output is then the result of the of electrons collected at the anode , which is ( http://www.centronic.co.uk/ionisation.htm , 2009 ) .one obvious systematic variable that can be quickly examined is variation in the bias voltage . however , ionization chambers are known for their insensitivity to applied bias voltage changes , since the ionization current is essentially independent of the applied voltage in the ion chamber region .this is due to the fact that , as pointed out above , the bias voltage applied in an ionization chamber is chosen to be sufficiently high to prevent appreciable recombination of the generated electron - ion pairs , while at the same time not providing enough energy to cause multiplication of electron - ion pairs within the detector .it follows that the output current from the detector will be based solely on the number of electron - ion pairs generated by the incident radiation and collected at the electrodes within .since it is the ionization of the argon gas within the detector that provides the measure of the energy deposited within by the incident radiation , and thus the activity of the sources being analyzed , it follows that changes in the properties of the gas within the detector , such as the atom density , could affect the measurements .however , since the number of moles ( ) of argon gas within the ptb detector is constant , the density of the gas will also be constant , unless the volume of the detector changes .using the ideal gas law , , where is the gas constant , is the temperature , and is the pressure , it follows that if the volume is also constant ( a non - constant volume will be addressed below ) then the detector system is described by , where and denote the initial and final conditions , respectively .hence , the only effect of a change in room temperature is to change the pressure inside the detector . and, since the ionization potential of argon is known to be a constant over all pressures from 1 to 200 bar , it follows that there should be no change in the response of the detector to changes in the room temperature if the volume of the detector is constant .we next consider the small effects that may arise from a volume change due to expansion or contraction of the steel walls of the ptb detector as the room temperature increases or decreases .the outer case and the bulkheads of the detector chamber are manufactured using mild steel ( trevor alloway , centronic ltd ., private communication , 2009 ) , and typical coefficients of linear expansion , , for steel range from to . for our analysis here , we will use an average value of , and assume that the expansion of the metal is isotropic . based on the stated dimensions of the detector , radius 9.25 cm and length 42.7 cm , and neglecting the small volume of the re - entrant tube ( well ), the initial volume , , of the chamber at the initial temperature can be calculated to be .9 .the change in volume , , arising from a change , , in temperature is \left [ l\left ( 1+\alpha \delta t\right ) \right ] \\ & \cong & v_{o}\left ( 1 + 3\alpha \delta t\right ) , \label{volume}\end{aligned}\ ] ] and hence the new volume of the cylinder will be 11478.3 for a 1 increase in temperature in the room .the change in volume is thus parts in 100,000 , and would then result in a fractional density change of , which is two orders of magnitude smaller than the density change proposed in ref . to explain the ptb data .additionally , it should be noted that this is a _ decrease _ in density , not an increase as claimed in ref . , since the same number of gas atoms now occupy a negligibly larger volume .we can therefore conclude that temperature increases would lead to lower density of the working medium in the detector , and thus to lower counts in the summer , as observed in the ptb data .however , in order to achieve a fractional change in density , which would cause a corresponding fractional change in the ionization and subsequent current , the room temperature would need to change by , which is unrealistic .in contrast to the operation of the centronic ig12 ionization chamber employed in ref . and described above , whose density dependence is characterized we next consider a hypothetical detector in which the pressure , , and the volume , , are held constant , which is the setup studied in ref . . in this case, we find from the ideal gas law that , so that .the change in intensity of -rays passing through a thickness of absorbing material with density is given by where is the absorption coefficient , and this leads to the familiar result  . without loss of generality, we can express the counting rate as , where is an appropriate normalization constant , and hence , . to obtain the temperature dependence of , we then replace by which gives although this reproduces eq . 1 of ref . , we see from the preceding discussion that this result only applies to a detector for which .this is not the case for the ptb detector , which is characterized by as noted above .one can illustrate the practical difference between these two cases by using the values for , , and quoted in ref . : , , and .we find , and hence , . by way of contrast , for the actual ptb detector , the only dependence of on is through the small thermal expansion of the detector chamber , as discussed above . using a coefficient of volume expansion for steel, we find from the preceding analysis .for =300k , we then find for the assumed temperature change in ref . , for the actual ptb detector .this is times smaller than the value claimed in ref . from an inappropriate application of the ideal gas law to the actual ptb detector .it follows from the preceding discussion that the expected effects of temperature on the ig12 ionization chamber used in the ptb measurements are much smaller than the measured fluctuations in the data .we then turn to other possible environmental effects , such as humidity , ambient air pressure , or ambient air density , which are less likely to affect the detector chamber itself since the detector is a closed system .moreover , these effects are not likely to have an effect on the very small air gap between the source and wall of the detector well , since the 4 ionization chamber is measuring the photons ( gammas and bremsstrahlung ) emitted by the sources , the fluxes of which would have little dependence on the marginal air density changes in the small air gap .we next turn to the possible contributions from background radiation of cosmic and terrestrial origins , such as the variations considered by siegert , et al . , and schrader as the cause of the annual variation of the measurements .although concentrations do exhibit seasonal as well as diurnal variations , the actual activities are quite small , on the order of 15 - 90 , which are far below the activities of the sources being measured . specifically , an analysis performed by abbady , et al . in hannover ,germany ( which is km from the ptb in braunschweig ) , measured indoor concentrations of inside the centre for radiation protection and radioecology ( zsr ) at hannover university .the data were collected for the period of one year , with measurements taken every two hours daily , from which monthly averages were calculated for each two hour interval .the data , shown in table 1 of ref . , do exhibit a small annual variation in the concentration of inside the zsr offices ( an institutional building which can be assumed to be somewhat similar to the institutional buildings at ptb ) , with the minimum and maximum concentrations occurring in february and may , respectively .interestingly , the day / night variation shown in ref . appears to be larger than the annual variation , with the minimum and maximum occurring at 1600 and 0800 , respectively .plots of the time - averaged data and the monthly averaged data from ref . are shown in figs .[ fig : hourlyrn ] and [ fig : monthrn ] .the diurnal effect is clearly evident , with the early morning maximum activity concentration possibly happening while the heating , ventilation and air conditioning ( hvac ) systems are off or in a lower functional state , as is typical in large institutional buildings .it should also be noted that concentrations are known to be dependent on air density , which is in turn dependent on temperature , relative humidity and pressure , as well as on precipitation .it follows that since the warmest part of the day is generally late afternoon , and the coolest is early morning , a diurnal effect is an understandable result .the annual variation is less obvious , but it appears that a maximum occurs in summer / autumn . hourly averaged concentration obtained from table 1 of abbady et al .the data were taken at the centre for radiation protection and radioecology ( zsr ) at hannover university from july 2000 to june 2001.,height=257 ]   monthly averaged concentration obtained from table 1 of abbady et al .the data were taken from june 2000 through may 2001 , and are arranged here january through december .see caption to figure [ fig : hourlyrn ] for further details.,height=257 ]  while the preceding information and analyses do not completely rule out an effect caused by , it appears that not only are the resulting activities quite small , but additionally the phase and period of the concentrations do not match those of the measurement variations reported in ref . and later analyzed in ref . .other contributions to background in the ptb measurements would be of terrestrial and cosmic origins , such as muons , or photons from decay .efforts were made by siegert , et al . to reduce the background by use of significant lead shielding around the detector .however , it should be noted that the data provided by schrader ( h. schrader , private communication , 2006 ) for the analysis presented in ref . were already corrected for background , which obviates the need for analyses of these background effects .nevertheless , a discussion of those possible backgrounds is presented below .an analysis of environmental radiation at ground level was conducted by wissmann at the ptb , utilizing data from the cosmic radiation dosimetry site , situated on a lake near the ptb laboratory .the results of the analysis , which corrected for fluctuations in weather and solar effects , exhibited an annual variation of , which is equivalent to a dose variation equal to . given the initial amount of in the ptb source , 300 g , which would have had an initial activity of 11.1 mbq , the activity would be .8 mbq after 40 years due to the ingrowth of the radioactive daughters . utilizing the more conservative activity value of 11.1 mbq, the gamma dose rate from the source can be estimated to be , which is 7 orders of magnitude larger than the variation in the cosmic dose rate reported in ref .we can therefore conclude that seasonal variations in cosmic radiation background would likely have little if any effect on the ptb detector system .  returning briefly to the background radiation from ,it should also be noted since decays to , the decay chain of is identical to the decay chain of . if the daughters are in equilibrium with the relatively constant concentration of bq , the activity of and its daughters at any time will be , which equates to a gamma dose rate of .this value is five orders of magnitude smaller than the dose rate of the source itself .hence , it can be safely assumed that even with a variation of the atmospheric concentration of ( which is significantly larger than the actual variation noted in ref . ) , the even larger activity of would yield a new gamma dose rate of , which is still four orders of magnitude smaller than the dose from the source itself .this effectively rules out a discharge effect on cables , etc . arising from charged particles or photons produced by daughters as the cause of the measured variations reported in ref .the and samples studied by siegert , et al . were measured with three different semi - conductor detectors as part of the evaluation of detector system stability previously discussed . in the course of these measurements ,a separate calibration source was used to measure the changes in efficiency of the ge(li ) detector which was one of the three semi - conductor detectors examined . a plot of measurements of the efficiency of a ge(li ) detector versus time , which is presented in fig . 2 of ref . , exhibits a similar period of 1 year , shifted by half a period ( or about 6 months ) , with an amplitude of about 0.5% .the efficiency of a detector is determined by comparison of the measured counts taken with the detector to the expected output of a known source in a fixed geometry .thus , for the ge(li ) measurement of the point source , the minima of the efficiency curve represent a deficit in counts ( with respect to the expected counts based on the known activity and half - life of the standard ) , while the maxima represent an excess of counts . specifically , the minima of the efficiency plot fall near the beginning of the calendar year for the ge(li ) detector , which is opposite to the signal seen in the ionization chamber measurements of .this appears to call into question the conclusion of jenkins , et al . that decay rates are simply dependent on the distance of the earth from the sun. however , the fact that the period of the data is the same , but shifted by half a period , still suggests a possible link to the earth - sun distance .  this possibility is supported by an examination of the decay modes of which has two branches , k - capture ( 72.10% ) and decay ( 27.90 ) .the 1408 kev photon measured with the ge(li ) detector only arises from the k - capture branch , and is seen in 21.07% of the decays ( there is no photon of that energy in the decay branch) .in contrast to the ge(li ) detector , the ionization chamber is incapable of discriminating among the different photons measured , and hence could not track the 1408 kev photon separately , as it would be lost in the sum of all the photons emitted by the sample .the phase of the data could thus be understood if k - capture modes responded to an external influence oppositely from decays .  in refs . the conjecture was put forward that a field or particle emanating from the sun , possibly neutrinos , might be enhancing or interfering with the decay of radioactive nuclides .since the quantum mechanical details of the k - capture and decays are quite different , it is in fact possible that the response of the k - capture branch would be different from that of the decay branch .if these were in fact opposite effects , i.e. the decay rate for the decay branch would be increasing when the k - capture rate was decreasing , these competing effects would tend to damp out any fluctuations . that such a possibility could actually occuris supported by data from ( which also decays via k - capture ) acquired during the solar flare of 2006 december 13 , .it was observed that the count decreased from the expected rate during the three - day period encompassing the solar flare .thus , if we assume that the solar neutrino flux increased during the solar flare , then the response of and to increased solar flux would have the same phase in the sense that both decay rates decreased in response to an increase in solar neutrino flux .we next turn to an examination of the environmental sensitivity of the gas proportional counter and sample changing system used by alburger , et al . and harbottle , et al . to measure the half - life of .the detector in question was cylindrical , .5 in .diameter , 2 in .depth , with a 1 in .end window , operating on p-10 gas .the detector pressure was held constant by a device on the gas outlet vented to a barostated enclosure to limit fluctuations in the density of the p-10 gas .the detector itself was mounted on an automatic precision sample changer , which was entirely contained within the aforementioned barostated enclosure , as described in refs .the window material was 0.006 mm kapton ( dupont h - film ) , with gold vacuum - deposited on both sides ( 40 - 50  on the outer surface , 120 - 150  on the inner surface ) .a bias voltage of 2150 v was applied to the detector , which was the voltage closest to the center of the beta counting plateau .gas proportional detectors utilize gas multiplication , where free electrons generated by the ionization of the gas by the incoming particles are accelerated by the strong electric field ( created by the bias voltage ) to sufficient kinetic energy to cause further ionizations .electrons freed in these ionizations can also be accelerated to cause additional ionizations , as long as their energy is greater than the ionization energy of the neutral gas molecules .this gas multiplication process , or cascade , is known as a townsend avalanche , which terminates when all free electrons are collected at the anode . with a well designed and maintained detector ,the number of secondary ionizations is proportional to the number of initial ionizations , but multiplied by a factor of `` many thousands'' .almost all primary ion pairs are formed outside the multiplying region , which is confined to a small volume around the anode , and the electrons from the primary ionizations drift to that region before multiplication takes place , `` therefore , each electron undergoes the same multiplication process regardless of its position of formation , and the multiplication factor will be [ the ] same for all original ion pairs . '' if the charge pulse of all of the electrons collected at the anode per event is larger than the discriminator level setting after amplification , a count is then recorded by the scaler .the two sources used in the bnl experiment , and , are described in detail in alburger , et al ., sec . 2 of ref .the isotope undergoes -decay ( 100 , ) to with a t=157(19 ) y .the radioactive daughter , , ( which is in secular equilibrium with ) undergoes -decay ( 100 , = 1710.5(21 ) kev ) to ( stable ) with a =14.262(14 ) d .the isotope , on the other hand , decays via competing -decay branches ( , 98.1(1) , =708.6(3 ) kev ;  .great care was taken by the bnl group in preparation of the source to ensure that no contaminants would be included that would confuse the data collection and analysis .analyses of the spectra of each of the sources were conducted with a plastic scintillator to test for impurities or other problems , and the spectra were as expected .the betas were not noticeably evident in the spectrum from the plastic scintillator , which was likely due to the effective thickness of the sample ( 17 mg/ ) that served to degrade and smear the low - energy betas , as noted by alburger , et al . . however , they did appear with the expected equal distributions when part of the same material was measured in a liquid scintillator .the spectrum was as expected , as reported by alburger , et al . .the gas proportional detector described above was utilized in a differential counter system described in detail in ref . , which used a precision sample changing system that would alternate the sources during counting runs .the system was designed and built to allow maximum reproducibility , e.g. by utilizing precision micrometer heads which adjusted the source / detector distance to within .001 mm . as stated before ,this system was contained within a `` pressure - regulated box in order to minimize the differential energy - loss effects that would occur with changes in barometric pressure '' .tests of the sensitivity of the counting system to changes in the box pressure , as well as to detector bias voltage , detector gas flow , and discriminator setting were carried out prior to the series of measurements , and the estimated effects on the counts and the si / cl ratio are detailed in table 1 of ref . .notwithstanding the tests described above , alburger , et al .identified an unexpected periodic fluctuation in the si / cl ratio with a period of approximately 1 year .having virtually eliminated other systematic variables , or at least having controlled them to the extent possible , alburger s bnl group was left with temperature and humidity as possible explanations , which are known to change the density of air ( even if the pressure of the box containing the detector system is held constant ) .increases in temperature will decrease the density of air , as will increases in relative humidity , although humidity has a much smaller effect than temperature . to summarize , warmer more humid air , i.e. summer air , is less dense than cooler , drier winter air .this is depicted in fig .[ fig : air - den ] , which plots the temperature and relative humidity dependence of the density of air , based on equations developed and recommended by the comit international des poids et mesures ( cipm ) .plot of air density as a function of temperature and relative humidity .the data were calculated based on the formulae presented in refs . , height=253 ]  the alburger group at bnl did not begin collecting data on temperature and humidity until the last five months of the four year experiment , and during that time saw a range of temperatures of 72.4 - 74.7 , and a relative humidity range of 35 - 76% .however , further investigations into data collected in other laboratories within the same building suggested that the average temperatures remained well within the range of 70 - 76 year round .the bnl group noted that the fluctuations appear to follow the same annual cycle as outdoor temperature .they also noted that if the indoor temperature and relative humidity were to track the outdoor conditions , then the data appeared to indicate that the fluctuations were environmentally based .the cooler , drier , and denser winter air could attenuate the lower energy betas , and would thus affect the ratio si / cl because the lower energy counts would be more weather dependent .however , alburger , el al . concluded that `` in order to produce the variations of standard deviations , the large humidity changes would have to be combined with temperature variations over a range of at least , which is larger than the probable actual range .we therefore conclude that systematic periodic variations are present but that they can not be fully accounted for by our tests or estimates'' .the conclusions of the bnl group can be supported by estimating the effect of temperature and relative humidity on the range of betas over the spectrum of energies that would be seen from both sources .the range of an electron can be estimated by utilizing an analysis by katz and penfold , which provided a range - energy relationship given by : the linear range can be derived by dividing by the density of the propagating medium , air , and expressing the electron energy , in mev ( it should be noted that these linear ranges are not necessarily straight line paths ) .we utilize the variations suggested by alburger , et al . for temperature ( t )( 73 ) and relative humidity ( rh ) ( 35 - 76% ) , and assume that the maximum density would be associated with the minimum t and rh ( 68 , rh=35% ) , and the minimum air density with the maximum t and rh ( 78 , rh=76% ) .the air densities are then found to be 1.185mg/ and 1.155 mg/ , respectively .the energy - dependent linear ranges of energetic electrons for each condition set are shown in fig .[ fig : esd ] , which gives the linear range of betas of various low energies in air .we have neglected the effects of the energy losses due to transport through the detector window or sources , which are assumed to be independent of temperature .it should be noted again , that the source was 4 mm from the detector window , and the source 1 mm away .[ fig : esd ] confirms the conclusions of the bnl group that changes in t and/or rh would have been too small to explain the fluctuations in their data .these conclusions are further supported by an mcnp analysis which we have carried out , the results of which will be described below . before turning to this analysis, we first consider the remaining question of backgrounds which could impact the bnl detector system .  linear range of betas in air as a function of beta energy .the graphs are calculated using the results of katz and penfold in ref .the air densities shown are based on the maximum and minimum values of temperature and humidity quoted by alburger , el al . in ref . , height=253 ]  the bnl group measured an initial background counting rate of 6.6(3)/min .this background measurement was repeated after the conclusion of all runs by counting with the source removed from its holder , and the still in place ( to ensure that there was no cross - talk between the sources ) .the rate was found to be 6.5(3)/min .since the average count rates for the measurements were initially substantially larger , 21,500/min and 14,800/min for the source and source , respectively , the background count rates were considered insignificant , and hence were ignored by the bnl group .additionally , since the sources were alternated during the counting procedure for each run , 30 minutes each for 20 cycles , the background would likely be the same for both of the isotopes .the monte carlo radiation transport software package , mcnpx , was used to determine the effects of changes in ambient air temperature on the proportional detector system used by alburger et al . .the air density between the source and detector window was varied over the temperature range 40-90 by use of the ideal gas law , corresponding to an approximate density range between 1.1350 and 1.2486 for dry air . while humidity also affects total air density ,this effect is presumed to be small ( see fig .[ fig : air - den ] ) relative to the effects of temperature change , and hence was ignored in this study .additionally , the 40 - 90 range that was analyzed encompasses both the minimum and maximum air densities expected for a wide range of both temperature and humidity in the laboratory .the physical parameters of the detector system ( including the proportional detector , source , and source ) were obtained from the literature , the bnl group s experimental notes ( private communication , d. alburger and g. harbottle , 2005 ) , and measurements of both the actual sources used in the experiment and a proportional detector of similar design .  the cylindrical end - window proportional detector was modeled as a stainless steel ( type 304 ) cylindrical shell with inner diameter 3.96875 cm , length 5.08 cm , and 3 mm wall thickness .the detector window was modeled following harbottle et al . as 0.006 mm kapton plated with vacuum - deposited gold on both the inner ( 125 ) and outer ( 45 ) surfaces .the detector volume was filled with p-10 gas ( 90% argon , 10% methane by volume ) with a calculated density of 1.6773 . this gas density within the detectorwas held constant .  due to the possibility that the lower energy betas from  were more easily affected by changes in air density than the higher energy - betas , each source was modeled separately , which also allowed us to account for the difference in proximity of each source to the detector .the source , which was modeled in detail as described by alburger et al . , consisted of a brass base 3.18 cm in diameter and approximately 0.3175 cm ( 1/8 in . )thick with a 1.91-cm - diameter , 0.8-mm - deep recess .the recess contained 47.7 mg of sio covered by 9.263 cm of aluminum foil .the mcnpx electron source volume was defined uniformly over the volume occupied by the sio . according to alburger, the source was created approximately 13 years prior to the start of the half - life experiment , and hence was certainly in secular equilibrium with its radioactive daughter , , during the experimental time frame . the source energy spectrum used in these mcnpx models consisted of equal contributions from the ( e = 225 kev ) and ( e = 1710.5 kev ) beta energy spectra calculated according to fermi theory , which allowed us to incorporate the loss of the much lower energy betas from .  as was the case for the source ,the source was placed on a brass base 3.18 cm in diameter and 0.3175 cm thick , but without a recess machined into the top .the was spread over a diameter of 1.91 cm in the center of the top surface of the base .thus , the mcnpx electron source was defined as a flat disk 1.91-cm in diameter on the surface of the brass base with an energy spectrum equal to that of ( e = 709.2 kev ) calculated according to fermi theory .  according to the bnl experimental notes , the distance of each source from the detector windowwas adjusted in order to obtain similar count rates from each , and this was achieved by having the source 1.000 mm from the detector window , while the was placed at a distance of 4.000 mm .the source position in the mcnpx models for each source reflected these distances .  as stated before , alburger , et al .reported that the temperature variation was well within the range 70 - 76 .we found that tally values varied so little over this range that they were statistically constant to within the minimum achievable relative error in mcnpx .thus , a large temperature range of 40 to 90 was chosen to improve the statistical determination of the function relating each tally to air temperature ( and associated density ) .the results presented here were obtained using ( for ) or ( for - ) source electrons for each density variation , and this provided relative errors of for the particle / energy current tallies , and for the energy deposition tallies .a number of mcnpx tallies were used to test the assertion that variations in air density with temperature would result in an associated increase or decrease in the probability that an electron with a given energy will traverse the air gap between the source and detector window .a linear least - squares fit was applied to the results of each tally as a function of air temperature , and these results are summarized in table 1 .[ table1 ]  [ cols="<,<,^,^,^,^ " , ]   of particular interest are the temperature dependent variations in the energy deposition within the detector ( fig .[ fig : siclratedep ] ) , the electron current at the detector window ( fig .[ fig : ratwineleccurr ] ) , and the energy current at the detector window ( fig .[ fig : ratwinencurr ] ) , all of which can be used as surrogates for detector count rate . for each of these tallies , ratios of the value per source electron for to that of were calculated .energy deposition , electron current , and energy current ratios were found to vary by , , and , respectively .these results are echoed by the tally variations as a function of temperature for - and individually in table 1 .as expected , all individual tally values increase with increasing temperature due to the associated decrease in air density between the source and detector . in all cases ,the rate of change is greater for due to its lower energy spectrum ( relative to ) and greater source - detector distance . in general ,detector events arising from the source are due almost entirely to betas because the low - energy betas are largely attenuated by the source itself .this explains the negative slope associated with the si / cl ratios as temperature increases .the ratio of to energy deposition in the proportional detector . dashed lines indicate the 95% confidence interval for the calculated linear least - squares fit of the data.,height=264 ]  the ratio of to electron current ( number of electrons ) across the proportional detector window ( into the detector ) . dashed linesindicate the 95% confidence interval for the calculated linear least - squares fit of the data.,height=264 ]  the ratio of to energy current ( mev ) across the proportional detector window ( into the detector ) .dashed lines indicate the 95% confidence interval for the calculated linear least - squares fit of the data.,height=264 ]  the ratios of electron current and energy current across a plane just above the source , and above the detector window were also examined ( see figures [ fig : sourcedetwineleccurr ] and [ fig : sourcedetwinenergycurr ] ) .these ratios directly address the assertion that changes in air density result in count rate variations due to increased or decreased absorption of electrons in the air gap between source and detector . the electron and energy current ratio dependencewas found to be and , respectively , and for , and , respectively .these temperature dependencies are all approximately 2 orders of magnitude smaller than the periodic variations observed by alburger et al.  the ratios of electron current exiting the source toward the detector and electron current across the proportional detector window ( into the detector ) for and . data have been shifted up by 0.2465 to facilitate visualization of both curves . dashed and dotted lines indicate the 95 confidence interval for the calculated linear least - squares fit of the data for and , respectively.,height=264 ]  the ratios of energy current exiting the source toward the detector and energy current across the proportional detector window ( into the detector ) for and .the data have been shifted up by 0.2390 to facilitate visualization of both curves . dashed anddotted lines indicate the 95% confidence interval for the calculated linear least - squares fit of the data for and , respectively.,height=264 ]  as we noted previously , following a number of experimental tests , alburger et al . concluded that it was possible that part of the 0.3% annual variation in observed count rates could be attributed to changes in ambient temperature ( between 70 and 76 ) , and relative humidity ( 35 to 76% ) .however , the results obtained in this monte carlo analysis indicate that a temperature variation much larger than 6 would be required to generate the observed count rate oscillations .given that the ratio of energy deposition in the detector varies by approximately , the temperature of the air separating the source from the detector would have had to vary by well over 100 to account for the observed data .we therefore conclude that changes in temperature and relative humidity alone could not have caused the observed periodic count rate variations .in fig . [ fig : siclratio ] , we exhibit the count rate ratio as a function of time , along with the outdoor temperature variation for the area surrounding the bnl and .we see that si / cl approximately correlates with and anti - correlates with temperature , in both cases with observable phase shifts . since at this stage we have no theoretical reason to suppose that si / cl should correlate ( rather than anti - correlate ) with , both of the correlations in fig .[ fig : siclratio ] are significant .however , we have already ruled out temperature as the primary explanation of the si and cl individual fluctuations , and hence we wish to understand whether influences emanating from the sun , such as neutrinos , could be compatible with these data .  the counting rate ratio for the bnl data as a function of time , along with and the outdoor temperature measured by noaa near bnl .the correlation with and the anti - correlation with temperature are both evident .see text for further discussion.,height=253 ]  in such a picture , the effect of analyzing the ratio si / cl is to largely cancel out small fluctuations in voltage , currents , temperature , background , etc . , which are common to both and , while leaving intact the presumably larger effects due to the variation of . from this perspective , the fact that the si / cl ratio does not exactly track is understandable in terms of the differing sensitivities of si and cl to external perturbations : the same nuclear structure mechanisms which endow si and cl with very different lifetimes ( 172y and 301,000y , respectively ) could presumably result in different responses of these nuclei to small perturbations .evidence of this effect was first pointed out by ellis , who saw an annual periodic signal in , but not in when both nuclides were measured for several years on the same detector system .therefore , a long - term periodic signal due to the variation of would survive in the ratio of si / cl .this can be understood quantitatively by writing }{a\left ( \rm{cl } \right)\left[1+\epsilon\left(\rm{cl}\right)\cos{\omega t}\right ] } \\ & \cong & \dfrac{a\left(\rm{si}\right)}{a\left(\rm{cl}\right ) } \left\lbrace  1 + \left[\epsilon\left(\rm{si}\right)- \epsilon\left(\rm{cl}\right ) \right ] \cos{\omega t}\right\rbrace .\label{eqn : siclperiod } \end{aligned}\ ] ] here a , b , and are constants , with , and has been assumed . in summary ,[ fig : siclratio ] is compatible with a picture in which any differences between si / cl and are due to the varying sensitivities of si and cl to an external perturbation coming from the sun , as seen in ellis data .although a differential temperature effect could in principle also explain fig .[ fig : siclratio ] , this would require temperature - dependent effects which are much larger than allowed by the preceding analysis .the preceding discussion leads naturally to the question of whether the observed effects in the bnl and ptb data could arise from a change in the gravitational potential of the sun at the respective detectors as the earth passes from perihelion to aphelion and back . in the framework of conventional general relativity ( gr ) all clocks at a given space - time point run at the same rate , irrespective of their internal mechanism . in terms of eq .[ eqn : siclperiod ] this would correspond to setting , in which case there would be no fluctuation in the ratio si / cl . since this is in obvious disagreement with the bnl data , conventional gr can be ruled out as an explanation of the observed effects .however , this conclusion does not necessarily hold for various alternatives to gr .will has shown that in non - metric theories , clocks of different construction can in fact behave differently in a gravitational field .although such a mechanism is possible in principle , it fails in practice on quantitative grounds .the sun s gravitational potential at the earth is , and the fractional change between perihelion and aphelion is .an effect this small would be undetectable given the bnl and ptb statistics , even if the coefficients multiplying in a non - metric theory were relatively large .additionally , constraints on parameters which measure deviations from gr are sufficiently restrictive to preclude the possibility of explaining the bnl and ptb data in terms of any known alternative to gr .a class of gravity - related theories which can not be excluded at present are those in which scalar fields are introduced to induce a time variation in fundamental constants such as the fine structure constant .we have discussed elsewhere the possibility that although the simplest such theories can not account for the bnl or ptb data , theories in which two or more scalar fields are introduced which influence both and the electron / proton mass ratio ( and possibly other quantities ) might work .we turn in this section to a discussion of possible environmental influences on the nai detector system used in during the 2006 december 13 solar flare .this is motivated by the potential connections between the decay data collected in the time period surrounding the flare , and the bnl and ptb decay data which are the primary focus of the present paper . as we have already noted in the introduction ,the short duration of the solar flare allows us to rule out seasonal temperature , humidity and pressure influences on the detector system as potential explanations of the observed decrease in the counting rate .hence the focus of this section will be on other possible `` environmental '' influences on the detector system during the solar flare , such as neutrons , muons , etc .although cosmic ray particles are known to be capable of influencing some detectors , there is a generic argument against any mechanism which attributes the drop in the count rate to cosmic rays .this is based on the observation that the signal for a decay in our system is the detection of the characteristic 834.8 kev photon resulting from the electron - capture process ,    hence the 834.8 kev photon which signals the electron - capture process is uniquely associated with the specific transition of the excited from the excited state to the ground state ( g.s . ) .the data points utilized in ref . the integral count within the region of interest ( roi ) set on the 834.8 kev photon for a four hour live - time .even if there had been a large change in the cosmic ray flux of neutrons or muons during the flare , the chance that the nuclear or electromagnetic interactions induced by these particles would have accidentally produced a photon at 834.8 kev , or deposited the energy equivalent of 834.8 kev within the nai crystal , is extremely small . moreover , the fact that a deficit of counts appeared in the roi , and not an increase , indicates that this already very remote possibility is highly unlikely .it is also unlikely that counts were selectively removed from the roi as a result of changes to the cosmic ray flux .an alternative explanation might be that an increase in the cosmic ray flux would generate a large distribution of energy depositions across the energy spectrum , and thus change the system dead time , leading to a loss of counts within the roi .dead time was tracked as a variable , however , and did not show changes over the course of the flare when compared to the counts preceding or following the flare period .additionally , inspection of the spectra files also show no significant changes in the counts of channels above or below the 834.8 kev peak .the above argument is bolstered by observations made by other groups during the 2006 december 13 flare .although an analysis of data from world - wide neutron monitors by btikofer , et al . and plainaki , et al . exhibited a % increase in the 1-minute data at oulu and apatity , the overall neutron flux was still so small as to have a negligible effect on the detector system .this is particularly true in light of the fact that the nai detector is insensitive to neutrons , and the flux of neutrons was too low to affect the sample directly .similar arguments allow us to exclude a change in the flux of cosmic ray muons resulting from a forbush decrease as a possible explanation of the flare data .a forbush decrease is a rapid change in the flux of cosmic rays resulting from plasma clouds emitted by the sun during a solar storm .evidence for a forbush decrease during the 2006 december 13 flare obtained from the muon telescopes at the basic environmental observatory ( beo ) in bulgaria and from nagoya , japan are shown in figs . 5 and 6 of ref . , respectively .the beo data exhibited a sharp forbush decrease beginning at :30 - 16:30 ut on 2006 december 14 , and reached a maximum at 05:00 ut on december 15 .the beginning of the forbush decrease thus occurred 37 hours _after _ the dip in the counting rate reported in ref . , which coincided in time with the solar flare at :40 ut on december 13 . additionally , the minimum in the muon count rate associated with the forbush decrease was observed more than 2 days later .the data from the beo are supported by data obtained at nagoya , which exhibited a similar time - dependence . given the timing of the forbush decrease relative to the dip in the counting rate we can thus reasonably conclude that the dip was not likely the result of the response of the detector system to the forbush decrease .the preceding arguments can be further strengthened by noting from ref . that the count rate started to decrease .7 days _ before _ the flare .it follows that the beginning of the decrease in the count rate preceded the beginning of the forbush decrease by more than 70 hours , and hence was not caused by it .  in summary , we can rule out environmental explanations of the correlation between the solar flare of 2006 december 13 and the observed dip in our data on several grounds .these include , ( a ) the observation that the solar flare lasted approximately 43 minutes , which is too short a time for seasonal environmental influence to have affected the detector system .( b ) the failure of these mechanisms to produce the characteristic 834.8 kev photon arising from electron capture , or to account for the .7 day precursor signal observed in the decay data ; ( c ) the timing of the forbush decrease which followed , rather than preceded , the dip in the data .additionally , the observed changes in cosmic ray fluxes would have been too small quantitatively to account for the data .as we noted in the introduction , there are at present two competing general explanations for the apparent fluctuations observed in the bnl and ptb data : ( a ) they arise from the responses of the respective detector systems to seasonal variations in temperature , pressure and humidity , and possibly other factors such as radon buildup .( b ) the fluctuations arise from the decay process itself , due to some as yet unknown influence possibly originating from the sun . by modeling the respective detector systems in detail , we have shown here that alternative ( a ) is an unlikely explanation for the observed bnl and ptb data , and hence by implication alternative ( b ) is more likely to be correct .the resulting inference , that nuclear decay rates are being directly influenced by solar activity is further supported by our analysis of decay data acquired during the solar flare of 2006 december 13 , which we have also analyzed .our conclusion that the dip in the data , which was coincident in time with the flare , was not likely attributable to changes in the cosmic ray flux during the flare , further strengthens the case that nuclear decays are being directly affected by solar activity .this inference can evidently be checked in a variety of experiments such as those described on ref . , some of which are already in progress .the authors wish to thank d. alburger and g. harbottle for making their raw bnl data , apparatus , and samples available to us , and for many helpful conversations .we are also deeply grateful to h. schrader for the raw ptb decay data , as well as for invaluable discussions on his experiment during our visit to the ptb .we are indebted to our colleagues who have helped us in this work , including v. barnes , j. buncher , t. downar , d. elmore , a. fentiman , j. heim , m. jones , a. karam , d. krause , a. longman , j. mattes , e. merritt , h. miser , t. mohsinally , s. revankar , b. revis , a. treacher , j. schweitzer , and f. wissman .the work of ef was supported in part my the u.s .department of energy under contract no .de - ac02 - 76er071428 .j. h. jenkins , e. fischbach , j. b. buncher , j. j. mattes , d. e. krause , j. t. gruenwald , evidence of correlations between nuclear decay rates and earth - sun distance , astropart .phys . 32 ( 1 ) ( 2009 )

Abstract: in a recent series of papers evidence has been presented for correlations between solar activity and nuclear decay rates . this includes an apparent correlation between earth - sun distance and data taken at brookhaven national laboratory ( bnl ) , and at the physikalisch - technische bundesanstalt ( ptb ) . although these correlations could arise from a direct interaction between the decaying nuclei and some particles or fields emanating from the sun , they could also represent an `` environmental '' effect arising from a seasonal variation of the sensitivities of the bnl and ptb detectors due to changes in temperature , relative humidity , background radiation , etc . in this paper , we present a detailed analysis of the responses of the detectors actually used in the bnl and ptb experiments , and show that sensitivities to seasonal variations in the respective detectors are likely too small to produce the observed fluctuations .   ionization chambers , proportional counters , beta decay , gamma decay , energy loss 23.40.-s , 29.40.cs , 23.40.-s , 29.40.-n


in many engineering applications , communications delays between subsystems can not be neglected .such an example is bilateral teleoperation , where signals can experience significant transmission delays between local and remote sites . throughout the last decade ,both internet and wireless technologies have vastly extended practical communication distances .information exchange and cooperation can now occur in very widely distributed systems , making the effect of time delays even more central .  in the context of telerobotics, proposed a control law for force - reflecting teleoperators which preserves passivity , and thus overcomes the instability caused by time delays .the idea was reformulated in  in terms of scattering or `` wave '' variables .transmission of wave variables across communication channels ensures stability without knowledge of the time delay .further extensions to internet - based applications were developed  , in which communication delays are variable .recently , extended the application of wave variables to a more general context by performing a nonlinear contraction analysis  of the effect of time - delayed communications between contracting systems .this paper modifies the design of the wave variables proposed in  .specifically , a simplified form provides an effective analysis tool for interacting nonlinear systems with time - delayed feedback communications . for appropriate coupling terms , contraction as a generalized stability property is preserved regardless of the delay values .the result also sheds a new light on the well - known fact in bilateral teleoperation , that even small time - delays in feedback pd controllers may create stability problems for simple coupled second - order systems , which motivated approaches based on passivity and wave variables  .the approach is then applied to study the group cooperation problem with delayed communications .we show that synchronization with linear protocols  is robust to time delays and network connectivity , without requiring the delays to be known or equal in all links . in a leaderless network ,all the coupled elements tend to reach a common state which varies according to the initial conditions and the time delays , while in a leader - followers network the group agreement point is fixed by the leader .the approach is suitable to study both continuous and discrete - time models .after brief reviews of nonlinear contraction and wave variables , section uses simplified wave variable forms to analyze time - delayed feedback communications .the same approach is then applied in section to study group cooperation .concluding remarks are offered in section .inspired by the use of passivity  and wave variables  in force - reflecting teleoperation , performed a contraction analysis of the effect of time - delayed communications .as we will discuss in this section , the form of the transmitted variables in  can be simplified and applied to analyze time - delayed feedback communications .we first summarize briefly some basic definitions and main results of nonlinear contraction theory  .consider a nonlinear system with and continuously differentiable .for any virtual displacement as a smooth function of the initial condition and of time , , then . ] between two neighboring trajectories , we have where is the largest eigenvalue of the symmetric part of the jacobian .hence , if is uniformly strictly negative , any infinitesimal length converges exponentially to zero . by path integration at fixed time , this implies that all the solutions converge exponentially to a single trajectory , independently of the initial conditions .  more generally , considering a coordinate transformation with square matrix uniformly invertible ,we have so that exponential convergence of to zero is guaranteed if the _ generalized jacobian matrix _ is uniformly negative definite .again , this implies in turn that all the solutions of the original system ( [ eq : nonlinear_form ] ) converge exponentially to a single trajectory , independently of the initial conditions .such a system is called _ contracting _ , with _ .  in the sequel, we will also use _ asymptotical contraction _ to refer to asymptotic convergence of any to zero , which implies global asymptotic convergence to a single trajectory .wave variables  are used in bilateral teleoperation systems to guarantee the passivity  of time - delayed transmissions .the idea was generalized in  by conducting a contraction analysis on time - delayed transmission channels .as illustrated in figure [ fig : teleoperation ] , considers two interacting systems of possibly different dimensions , where , are constant matrices and , have the same dimension .communication between the two systems occurs by transmitting intermediate `` wave '' variables , defined as because of time - delays , one has where and are two positive constants .it can be proved that , if both and are contracting , the overall system is asymptotically contracting independently of the time delays .  note that subscripts containing two numbers indicate the communication direction , e.g. , subscript `` '' refers to communication from node to .this notation will be helpful in section [ sec : group - delays ] , where results will be extended to groups of interacting subsystems .let us now consider modified choices of the transmitted variables above .specifically , for interacting systems ( [ eq : two - delay - systems ] ) , define now the transmitted variables as where and are two strictly positive constants .consider , similarly to , the differential length where note that this yields if and are both contracting with identity metrics ( i.e. , if and are both uniformly negative definite ) , then , and is bounded and tends to a limit .applying barbalat s lemma  in turn shows that , if is bounded , then tends to zero asymptotically , which implies that , , and all tend to zero .regardless of the values of the delays , all solutions of system ( [ eq : two - delay - systems ] ) converge to a single trajectory , independently of the initial conditions . in the sequelwe shall assume that can indeed be bounded as a consequence of the boundedness of .this result has a useful interpretation . expanding system dynamics ( [ eq : two - delay - systems ] ) using ( [ eq : feedback - wave - variable ] ) yields so that the above communications in fact correspond to simple feedback interactions .if we assume further that and have the same dimension , and choose , the whole system is actually equivalent to two diffusively coupled subsystems note that constants and and thus coupling gains along different diffusion directions can be different. we can thus state  [ th : two - delayed - diffusion ] consider two subsystems , both contracting with identity metrics , and interacting through time - delayed diffusion - like couplings ( [ eq : two - delayed - diffusion ] ) .then the overall system is asymptotically contracting ._ remarks _  * theorem [ th : two - delayed - diffusion ] does not contradict the well - known fact in teleoperation , that even small time - delays in bilateral pd controllers may create stability problems for coupled second - order systems  , which motivates approaches based on passivity and wave variables .in fact , a key condition for contraction to be preserved is that the diffusion - like coupling gains be symmetric positive semi - definite _ in the same metric _ as the subsystems , as we shall illustrate later in this section .* if there are no delays , i.e. , , then .thus , the analysis of the differential length yields that where this implies that and tend to zero exponentially for contracting dynamics and , i.e. , the overall system is exponentially contracting . *if and are both contracting and time - invariant , all solutions of system ( [ eq : two - delayed - diffusion ] ) converge to a unique equilibrium point , whose value is independent of the time delays and the initial conditions . indeed , in a globally contracting autonomous system , all trajectories converge to a unique equilibrium point  , which implies that if and are both contracting and time - invariant , an equilibrium point must exist for system ( [ eq : two - delayed - diffusion ] ) when . in turn , this point remains an equilibrium point for arbitrary non - zero and .since the delayed system ( [ eq : two - delayed - diffusion ] ) also preserves contraction , all solutions will converge to this point independently of the initial conditions and the explicit values of the delays .* theorem [ th : two - delayed - diffusion ] can be extended directly to study more general connections between groups , such as bidirectional meshes or webs of arbitrary size , and parallel unidirectional rings of arbitrary length , both of which will be illustrated through section [ sec : group - delays ] .inputs to the overall system can be provided through any of the subgroup dynamics .* :* consider two identical second - order systems coupled through time - delayed feedback pd controllers with , .if , partial contraction analysis  shows that and converge together exponentially regardless of initial conditions , which makes the origin a stable equilibrium point . if , a simple coordinate transformation yields & = & \left [ \begin{array}{c } \omega y_1 - b x_1-\omega x_1  \end{array } \right ] + { \bf k } ( \left [ \begin{array}{l } x_2(t - t_{21 } ) \\ y_2(t - t_{21 } ) \end{array } \right ] -  \left [ \begin{array}{l } x_1(t ) \\y_1(t ) \end{array } \right ] ) \\ \left [ \begin{array}{l } \dot{x}_2 \\ \dot{y}_2 \end{array } \right ] & = & \left [ \begin{array}{c } \omega y_2 - b x_2  \\             -\omega x_2  \end{array }\right ] + { \bf k } ( \left [ \begin{array}{l } x_1(t - t_{12 } ) \\ y_1(t - t_{12 } ) \end{array } \right ] -  \left [ \begin{array}{l } x_2(t ) \\y_2(t ) \end{array } \right])\end{aligned}\ ] ] where \ \ \ \ \ \ \ \ \rm{and}\ \ \ \ \ \ \ { \bf f}_2=\left [ \begin{array}{c } \omega y_2 - b x_2  \\                  -\omega x_2  \end{array } \right]\ ] ] are both contracting with identity metric  . however , the transformed coupling gain \ ] ] is neither symmetric nor positive semi - definite for any .contraction can not be preserved in this case , and the coupled systems turn out to be unstable for large enough delays as the simulation result in figure [ fig : mass - spring - damping](a ) illustrates . in figure[ fig : mass - spring - damping](b ) we set so that the overall system is contracting according to theorem [ th : two - delayed - diffusion ] .[ ex : two - delayed - mechanical - systems ]  the instability mechanism in the above example is actually very similar to that of the classical smale model  of spontaneous oscillation , in which two or more identical biological cells , inert by themselves , tend to self - excited oscillations through diffusion interactions . in both cases ,the instability is caused by a non - identity metric , which makes the transformed coupling gains lose positive semi - definiteness .note that the relative simplicity with which both phenomena can be analyzed makes fundamental use of the notion of a metric , central to contraction theory .different simplifications of the original wave - variable design can be made based on the same choice of , yielding different qualitative properties .for instance , the transmitted signals can be defined as which leads to and thus also preserves contraction through time - delayed communications . similarly to the previous section , if both and are contracting and time - invariant , the whole system tends towards a unique equilibrium point , regardless of the delay values . at this steady state , and , which immediately implies that and , if with of full rank , that thus , contrary to the case ( [ eq : feedback - wave - variable ] ) of diffusion - like couplings , the remote tracking ability of wave variables is preserved .* :* consider the example of two second - order systems where is an external force , and , are internal forces undergoing time delays . performing a coordinate transformation , we get new equations & = & \left [ \begin{array}{c } \omega_1 y_1 - b_1 x_1  \\           -\omega_1 x_1 + \frac{f_e}{\omega_1 }  \end{array } \right]\ +\  \left [ \begin{array}{l } 0 \\ \frac{f_{21}}{\omega_1 } \end{array } \right ] \\ \left [ \begin{array}{l } \dot{x}_2 \\ \dot{y}_2 \end{array } \right ] & = & \left [ \begin{array}{c } \omega_2 y_2 - b_2 x_2  \\-\omega_2 x_2  \end{array } \right]\ + \  \left [ \begin{array}{l } 0 \\ \frac{f_{12}}{\omega_2 } \end{array } \right ] \end{aligned}\ ] ] the signals being transmitted are defined as simplified wave variables note that here  ] , so that since is lower bounded it converges to a finite limit as , which implies that , tends to zero , which in turn implies that , tends to zero .the rest of the proof is then similar to that of theorem [ th : linear_schooling_delayed ] .niemeyer , g. , and slotine , j.j.e .( 2001 ) towards bilateral internet teleoperation , in _ beyond webcams : an introduction to internet telerobotics _ , eds . k. goldberg and r. siegwart , the mit press , cambridge , ma .olfati - saber , r. , and murray , r.m .( 2004 ) consensus problems in networks of agents with switching topology and time - delays , to appear in the special issue of the _ ieee transactions on automatic control on networked control systems _slotine , j.j.e ., wang , w. , and el - rifai , k. ( 2004 ) contraction analysis of synchronization in networks of nonlinearly coupled oscillators , sixteenth international symposium on mathematical theory of networks and systems , belgium        xie g. , wang l. ( 2004 ) stability and stabilization of switched linear systems with state delay : continuous - time case , _ the international symposium on mathematical theory of networks and systems , belgium _  xie g. , wang l. ( 2004 ) stability and stabilization of switched linear systems with state delay : discrete - time case , _ the international symposium on mathematical theory of networks and systems , belgium _

Abstract: we study stability of interacting nonlinear systems with time - delayed communications , using contraction theory and a simplified wave variable design inspired by robotic teleoperation . we show that contraction is preserved through specific time - delayed feedback communications , and that this property is independent of the values of the delays . the approach is then applied to group cooperation with linear protocols , where it shown that synchronization can be made robust to arbitrary time delays .


the fact that the correlation function in redshift space is distorted from the real space correlations is well known .davis and peebles ( 1983 ) have demonstrated quantitatively that the correlation function , expressed as a function of the line - of - sight and perpendicular separations , has a significant anisotropy . by measuring the elongation of this function along the axis , a value of the one - dimensional velocity dispersion can be inferred .they found that the velocity distribution was well described by an exponential distribution and that the value of the dispersion was relatively independent of scale , at around 350 km / s .peebles ( 1980 ) gives a nice summary why in linear theory the typical velocities of the galaxies scale with . with the advent of biased galaxy formation ( bardeen 1986 ), the velocities have been rescaled by a bias factor , leading to velocities dependent upon the combination .  in kaiser s ( 1987 )pioneering work , the effect of the infall due to linear theory was identified and it was shown that in the limit that the lines of sight to the two galaxies are approximately parallel to each other , the power spectrum is enhanced as a function of the directional cosine between the wave vector and the line - of - sight as   in related work , lilje and efstathiou(1989 ) calculated the angular average of the redshift correlation function directly related to this expression .hamilton(1992 ) expanded the redshift space correlation function into components , multiplied with the angular multipoles .he showed that only the quadrupole and hexadecapole terms arise in linear theory , in agreement with kaiser s expression .other promising approaches have involved an expansion into orthogonal eigenfunctions ( heavens and taylor 1994 ) , or restricting the surveys to small opening angles so that the plane - parallel approximation still holds ( cole , fisher and weinberg 1994,1995 ) .these papers attempted to measure the parameter from the quadrupole to monopole ratio .zaroubi and hoffmann ( 1996 ) outlined how to compute a linear expansion of the redshift space correlation function for a general geometrical configuration and provided numerical estimates of the redshift distortions .additionally , hamilton and culhane ( 1996 ) , hereafter hc96 , have introduced a novel integral transform , rotationally invariant and commuting with the redshift distortion operator , which makes the transformation very elegant and simple .  herewe present simple and intuitive expressions to calculate the redshift distortions which exploit the fact that the inherent geometry of the problem is two dimensional .indeed , by rotating all pairs in a redshift survey to a common plane and analyzing the redshift distortions in that plane , we obtain particularly simple analytic results .these results smoothly approach the kaiser / hamilton plane - parallel limit in which angle between the galaxy pair in a survey is small . additionally , these expressions allow for a much more productive use of redshift surveys since all galaxy pairs can be utilized in a given wide angle survey .it can also be used in adjunct with more sophisticated methods of analysis such as construction of eigenvectors for a kl analysis ( vogeley and szalay 1996 ) .the symmetries in the geometric configuration of redshift surveys have been elegantly described in hc96 , who recognized that the correlation function in redshift space is tied to a triangle formed by the observer and the two galaxies which lie upon two arbitrary lines of sight .the correlation function should be a function of this triangle , but otherwise invariant under rotations about the observer .such invariant triangles can be characterized by one size and two shape parameters .we will use two angles to describe the shape of the triangle , and express the scale dependence of the correlation function as a function of , the distance between the two points .  in this coordinate system , which is illustrated in fig .1 , the two points are located at points and . the observer is at the origin , at rest in comoving frame .the angle between the two normal vectors is , that is .the separation between the two points is , with a normal vector .the angles between and are and respectively .the symmetry axis for and is found by halving the angle .the angle between this symmetry axis and is .we assume that , thus .the angles are related to each other as , and .this introduction implies our choice of the two angles , and , which together with completely describe the shape of the triangle .these angles are particularly convenient since is given by our geometry and smoothly approaches the angle familiar in the plane - parallel limit as tends to 0 .there is another geometric relation between the two angles : if is not 0 , then the smallest value can take is , and its largest value is .this is not due to the choice of coordinates but rather to the geometry of the problem .we will provide an expression for the distortions of the correlation function as a series expansion in terms of the angle , where the coefficients have a simple and dependence , together with a dependence on the parameter that we wish to measure .so far our approach has been quite similar to most earlier work except for the introduction of a convenient set of angles which will simplify the results considerably .it has been a persistent problem to get a simple power spectrum or correlation function beyond the kaiser ( 1987 ) plane - parallel limit .the difficulty lies in that the fourier space transfer function , the ratio of the distorted and undistorted power spectrum contains a strong mode - coupling ( zaroubi and hoffmann 1996 ) .thus , the transfer function is not multiplicative but contains an integral over a non - local kernel with a strong -dependence .this dependence strongly affects the multipole coefficients which would otherwise make a precise determination of quite straightforward .one alternative has been to use only galaxy pairs at small angular separations ( e.g. cole 1994 , 1995 ) .it is interesting to consider what the reasons are for this non - locality . with a finite , the original symmetry of the problem is lost , and the geometry is inherently two - dimensional .this effect is thus arising from ` aliasing ' ( kaiser and peacock 1991 , szalay 1993 , landy 1996 ) , i.e. projections of the spherically isotropic power to the lower dimensions of the survey geometry .interestingly enough , even though the power spectrum is definitely non - local , the redshift space correlation function can still be computed in spherical coordinates , and it can be expressed in a closed form , as will be shown below .  in the following section 2.2 , we will first derive a representation of the redshift distortion problem in terms of spherical tensors .this development , being a representation independent expansion , is independent of our choice of coordinate system described above . in section 2.3, we will reintroduce our coordinates and show how the coefficients of this expansion can be most economically expressed using these coordinates .we can assume without a loss of generality , that the observer is at rest with respect to the cmb frame .as shown by hc96 , this effect can be trivially included .the linear expansion of the overdensity at a redshift space coordinate relates to the real - space overdensity at , with , as a function of the radial velocity and its line - of - sight derivative , as first introduced by kaiser ( 1987 ) , here , where is the selection function which is a slowly varying function of , the radial distance from the observer . due to the fact that the velocity scale is much smaller than the typical depth of todays redshift surveys ,this term is very small and is generally ignored .we will include this term for completeness in our results but would like to make one other point why this term is additionally small . if the redshift survey is defined by a boundary on the sky , a given selection function , and a range of the radial coordinate , then will be averaged , weighted by the selection function . integrating by partsone can show that this term is small and becomes zero as the lower limit becomes 0 and the upper limit , the case of full surveys .the peculiar velocity can be written concisely as a fourier integral , with the usual , as in order to simplify subsequent calculations , we introduce here the spherical tensor as one can then reexpress the redshift - space overdensity at the point in terms of as this use of spherical coordinates makes it easy to express the correlation of the different components , at positions and as where is the legendre polynomial , and is the power spectrum with the definition .the redshift - space two - point correlation function of finite - angle is given in terms of , where we will use . +    \frac{2}{3 } \beta^2\left [ \alpha_1 s_{12}^1 +    \alpha_2 s_{21}^1 \right ]    -\,\beta^2 \alpha_1\alpha_2 s_{11}^2  \label{eq : a8}\end{aligned}\ ] ] both the legendre polynomials and the plane wave can be expanded in terms of spherical harmonics as where and are spherical bessel functions and spherical harmonics , respectively .we can express the integral of three s over with the wigner -symbols .we also introduce the bipolar spherical harmonics which transform as a spherical harmonic with with respect to rotations ( varshalovich 1988 ) . with these can form a rotationally invariant scalar , dependent on , where indicates the shape of the triangle formed by the three unit vectors .any quantity that is a scalar function of is independent of what angles or parameters we may choose to describe the triangle shape .this invariant scalar is symmetric in . with this notation , equation ( [ eq : a6 ] ) reduces to where    the above expression shows the correlation function can be written in terms of a series , factorized into size and shape dependent terms , providing a representation independent expansion .equations ( [ eq : a8 ] ) and ( [ eq : a12 ] ) give an expression of the redshift - space correlation function .re - expanding in terms of , the dependence is contained in the coefficients : it is at this point that the use of any particular coordinate system becomes paramount .since the coefficients functions of the shape of the triangle under consideration , a judicious use of coordinates can simplify the problem considerably , as will be shown below .the terms proportional to have been saparated , since these disappear in the plane - parallel limit . after a somewhat tedious reduction procedure , most of it carried out in mathematica, we obtain the following results : and \nonumber\\  & \,\,\ , + &  \frac15 ( \alpha_1 + \alpha_2 ) \beta^2  \sin\theta  [ \cos^2 \theta p_1(\sin\gamma ) - 2 p_3(\sin\gamma ) ] \label{eq : a44}\\  c_{20 } & = \,\,\ , &  \frac19 \alpha_1 \alpha_2 \beta^2 ( 4 \cos\theta - 1 ) , \label{eq : a45}\\  c_{22 } & = \,\,\ , &  \frac13 \alpha_1 \alpha_2 \beta^2 \sin^2\theta -  \frac23 \alpha_1 \alpha_2 \beta^2 p_2(\cos\gamma ) .\label{eq : a46}\end{aligned}\ ] ] the first set of coefficients , which do not contain , capture most of the relevant physics . for the sake of completeness, the second set gives the -dependent terms , , and which must be included for calculations in which and can not be neglected.  from these expressions , it is evident that a good choice of coordinate system matters a great deal .using the system discussed above ( angles and , separation ) , leads to remarkably simple expressions for the coefficients .other systems we explored generally contained a lot of associated legendre polynomials . it is easily seen in the above expression that taking the limit reproduces kaiser - hamilton s plane - parallel result .we have presented a derivation of the redshift space correlation function between galaxies at two infinitesimal volume elements separated by an arbitrary angle , and have obtained very simple closed form analytic expressions for the correlations .these equations show that the effects of a finite angle are quite important .these are illustrated graphically in figure 2 which indicates how the redshift distortions change as the angle between the two lines of sight is increased from 0 to 120 degrees .since most of the next generation redshift surveys are wide angle with the majority of pairs separated at angles greater than 10 degrees , the use of these expressions will dramatically increase the amount of information which can be extracted from the surveys resulting in more robust measurements and smaller shot noise contamination .also , if the survey is not contiguous but consists of several slices like the las campanas survey ( landy etal 1996 ) , or the 2df survey which will be formed from hundreds of pencilbeams , this approach can use all the data together to estimate the infall distortions .another promising approach to study the large scale behavior of the power spectrum is with building up a karhunen - loeve basis ( vogeley and szalay 1996 ) .the first step in that method is to subdivide the survey into small cells ( in redshift space ) and then construct their correlation matrix .most pairs of cells will have a large relative angle .thus , with the expressions presented here , computing the correlation matrix is a trivial exercise .once the appropriate basis has been created , one can use the fisher matrix to select the set of eigenmodes most sensitive to , yielding an optimal estimation of the value of .  in conclusion , we believe that our results allow a much more elegant treatment of the redshift distortions in a general geometry than before with the added benefit that all galaxy pairs may be utilized to construct a signal .bardeen , j. , bond , j.r . , kaiser , n. , szalay , a.s .1986 , , 304,15 cole , s. , fisher , k.b . , weinberg , d.h .1994 , , 267 , 785 cole , s. , fisher , k.b . , weinberg , d.h .1995 , , 275 , 515 davis , m. , peebles , p.j.e . 1983 , , 267 , 465 hamilton , a.j.s .1992 , , 385 , l5 hamilton , a.j.s . , culhane , m. 1996 , , 278 , 73 heavens , a.f . ,taylor , a.n .1995 , , 275 , 483 landy , s.a . ,schectman , s.a . ,lin , h . ,kirshner , r .oemler , a.a . , tucker , d .1996 , , 456 , 1 lilje , p. , efstathiou , g.p .1989 , , 236 , 851 kaiser , n. 1987 , , 227 , 1 kaiser , n. and peacock , j.a .1991 , , 379 , 482 peebles , p.j.e .1980 , the large scale structure of the universe , ( princeton university press : princeton ) szalay , a.s . ,broadhurst , t.j . ,ellman , n. , koo , d.c . , ellis , r.s .1993 , proc ., 90 , 4853 varshalovich , d.a . , moskalev , a.n . ,khershonski , v.k .1988 , quantum theory of angular momentum ( world scientific ) vogeley , m.s .szalay , a.s .1996 , , 465 , 34 zaroubi , s. , hoffmann , y. 1996 , , 462 , 25

Abstract: using a novel two - dimensional coordinate system , we have derived a particularly simple way to express the redshift distortions in galaxy redshift surveys with arbitrary geometry in closed form . this method provides an almost ideal way to measure the value of in wide area surveys , since all pairs in the survey can be used for the analysis . in the limit of small angles , this result straightforwardly reduces to the plane - parallel approximation . this expansion can also be used together with more sophisticated methods such as for the calculation of karhunen - loeve eigenvectors in redshift space for an arbitrary survey geometry . therefore , these results should provide for more precise methods in which to measure the large scale power spectrum and the value of simultaneously .   # 12 # 1 - 0.2em+-0.2em1