the cocoa project dates back to 1987 under the lead of l. robbiano : the aim was to create a software laboratory for studying commutative algebra and especially grbner bases , and which is welcoming even to mathematicians who are wary of new - fangled computers ,  since then the realm of applicability of grbner baseshas continually expanded , so the researchers interested in using them now come from a broad palette of subject areas ranging from the theoretical to quite practical topics .so there are still the `` pure '' mathematicians as at the outset , but now also `` programming '' mathematicians , and statisticians , computer scientists , and so on .a factor crucial in making grbner basesrelevant to practical problems is the interim progress in computer hardware and software techniques .  since its beginningthe cocoaproject has evolved and has been rewritten , and now comes in the form of a very flexible software combination cocoa-5/cocoalib , while maintaining its tradition of being _ user - friendly _ so it offers grbner basesfor all levels of interest and programming ability : _ a grbner basisfor everyone_. this means that the `` cocoaexperience '' covers a wide range : from the basic , quick call in the interactive system cocoa-5  , to the problem - specific optimized implementations , to the computer computer communication with the open source c++ software library , cocoalib  , or with the prototype openmath - based server .the importance that grbner baseshave acquired derives from the fact they enable or facilitate so many other computational mathematical results .a natural consequence is that a grbner basisis almost never the final answer that is sought , but just a stepping stone on the way to the goal .it is very rare that anyone really wants to have a grbner basis , let alone actually look at one  usually they are frighteningly large beasts !   cocoa-4 was widely appreciated for its ease of use , and the naturalness of its interactive language .however , it did have limitations , and several `` grey areas '' .we designed the new cocoa-5 language to strike a balance between backward - compatibility ( hoping not to alienate existing cocoa-4 users ) and greater expressibility with a richer and more solid mathematical basis ( eliminating those `` grey areas '' ) .so , what s not new ? superficially the new cocoa-5 language / system closely resembles cocoa-4 because we kept it largely backward compatible ; at the same time it improves the naturalness and ease of use of the old system .we are very aware that a number of cocoa users are mathematicians with only limited programming experience , for whom learning cocoa was a `` big investment '' , and who are reluctant to make another big investment  that is why we wanted to make the passage to cocoa-5as painless as possible .so , if almost nothing has changed , what s new ?the clearly defined semantics of the cocoa-5new language make it both more robust and more flexible ; it provides greater expressibility and a more solid mathematical basis .in particular , it offers full flexibility for the field of coefficients : _e.g. _ fraction fields and algebraic extensions , and even heuristically guaranteed floating point arithmetics with rational reconstruction ( see section [ sec : twinfloats ] ) .however , under the surface , the change is radical since its mathematical core , cocoalib , has been rewritten from scratch , to be faster , cleaner and more powerful than the old system , and also to be used as a c++ library .the glib answer is : _ as little as possible ! _a more honest answer is that that is our aim , but there is more work still to do . in any case , using cocoa-5will not magically make you into a c++ programmer , so some differences will necessarily remain .cocoa-5is an interactive , interpreted environment which makes it far better suited to `` rapid prototyping '' than the rigid , statically typed regime of c++ .to keep it simple to learn , cocoa-5has only a few data types : for instance , a power - product in cocoa-5 is represented as a monic polynomial with a single term , _i.e. _ a ring element ( of a polynomial ring ) ; in contrast , in cocoalibthere is a dedicated class , ` ppmonoidelem ` , which directly represents each power - product , and allows efficient operations on the values ( _ e.g. _ without the overhead of the superfluous coefficients ) .cocoalib  contains ( practically ) all the mathematical knowledge and ability while cocoa-5offers convenient access to cocoalib s capabilities .programming with cocoalibtends to be more onerous than with cocoa-5largely because of c++ s demanding , rigid rules ; the reward is typically faster computation ( sometimes much faster ) .also , of course , those who want to use cocoalib s abilities in their own c++ program have to use cocoalib , see section [ sec : cocoalib ] .this is why everything which can be computed with cocoa-5 should be just as readily computable with cocoalib .some cocoa-5functions are still implemented in cocoa-5 packages ; but they are being steadily translated into c++ .( we do nt say which ones , because the list is constantly shrinking )the simplest context for grbner bases is for ideals in  ] with  a prime . these are also the easiest cases to give to cocoa .  an essential ingredient in the definition of a grbner basisis the term - ordering : a total ordering on the power - products which respects multiplication , and where 1 is the smallest power - product . in cocoathe term - orderingis specified at the same time as the polynomial ring ; all grbner basesof ideals in that polynomial ring will automatically be computed with respect to that ordering  unless otherwise specified , the ordering is `` degree reverse lexicographic '' , which is known to be the `` best general choice '' .cocoahas a fully general ( matrix - based ) implementation of term - orderings , with efficient handling of some common special cases .here is an example of computing a grbner basisin cocoa :  .... /**/use qq[x , y , z ] ; /**/ i : = ideal(x^3 + x*y^2 -2*z , x^2*y^3 -y*z^2 ) ; /**/gbasis(i ) ; [ x^3 + x*y^2 -2*z , x^2*y^3 -y*z^2 , x*y^5 -2*y^3*z + x*y*z^2 , .... ] ....  as mentioned above , the default ordering is ` stddegrevlex ` , which _ generally _ gives the best performance and smallest answer .however , other gradings and orderings may be useful for studying specific problems : for instance , an important family are the _ elimination orderings _ which are used implicitly in section [ sec : elim ] .another well - known ordering is `lex ` , which gives a grbner basiswith a particular _ shape _ useful for solving polynomial systems ( see , for example , the kreuzer robbiano book  , sec.3.7 ) .their practical usefulness is limited by the fact that ` lex ` bases tend to be particularly big and ugly , and are frequently rather costly to compute .  in cocoathe term - orderingis an intrinsic property of a polynomial ring : this means that  ] with ` stddegrevlex ` .here is an example of computing a ` lex ` grbner basis ; note how the term - ordering is specified at the start along with the polynomial ring .  .... /**/use qq[x , y , z ] , lex ; //specify ordering together with ring /**/i : = ideal(x^3 + 3 , y - x^2 , z - x - y ) ; /**/ reducedgbasis(i ) ;  // basis is wrt .lex ordering [ z^3 + 9*z -6 ,  y -(1/4)*z^2 -(3/4)*z -3/2 ,  x + ( 1/4)*z^2 -(1/4)*z + 3/2 ] ....  in the last example above we used the command ` reducedgbasis ` which computes a reduced grbner basiswhich is a `` cleaned up '' grbner basis with only non - redundant , monic , fully reduced elements  it is unique ( up to the order of the elements ) .the easy examples above show the definition of a polynomial rings with rational coefficients , but the choice in cocoais quite wide .for example where may also be quite large , or algebraic extensions , of fraction fields .  ..../**/ use zz/(100000000000000000000000000319)[x ] ; /**/reducedgbasis(ideal(3*x-1 ) ) ; [ x -33333333333333333333333333440 ]  /**/ use r : : = qq[i ] ; /**/qqi : = r / ideal(i^2 + 1 ) ; /**/ use qqi[x ,y , z ] ; /**/ i : = ideal(i*x^3 -z , x^2*y^3 -i*y*z^2 ) ; /**/reducedgbasis(i ) ; [ x^3 + ( i)*z , y^3*z + x*y*z^2 , x^2*y^3 + ( -i)*y*z^2 ]  /**/ use r : : = qq[sqrt2 , sqrt3 ] ; /**/ k : = r / ideal(sqrt2 ^ 2 -2 , sqrt3 ^ 2 -3 ) ; /**/use k[x , y , z ] ; /**/ i : = ideal(sqrt3*x^2 -y , x*y -sqrt2*z ) ; /**/reducedgbasis(i ) ; [ y^2 + ( -sqrt2*sqrt3)*x*z , x*y + ( -sqrt2)*z , x^2 + ( ( -1/3)*sqrt3)*y ]  /**/ use qqab : : = qq[a , b ] ; /**/k : = newfractionfield(qqab ) ; /**/use k[x , y , z ] ; /**/i : = ideal(x^3 -a*z , x^2*y^3 -b*y*z^2 ) ; /**/ reducedgbasis(i ) ; /**/ [ x^3 -a*z , y^3*z + ( -b / a)*x*y*z^2 , x^2*y^3 -b*y*z^2 ] ....please note in this last example is actually the field with no specialization of .so the grbner basisrepresents the _ generic _ case , meaning that an algebraic expression in which is not identically zero is considered to be non - zero .the problem of considering all possible specializations of the parameters is known as _ comprehensive grbner basis _ , and is not ( yet ) implemented in cocoa .another family of computationally interesting rings in cocoais given by ` newringtwinfloat(bitprec ) ` .it will be presented in detail in sections [ sec : gin ] and [ sec : twinfloats ] .there is a notion of _ universal grbner basis _ which is a grbner basisfor every term - ordering .the cocoafunction will compute one such basis ; this function is based on the computation of the grbner fan ( a richer structure , see below ) which gives all possible reduced grbner bases : we can take the union of all of them to produce the universal basis .the following example shows that the maximal minors of a matrix of indeterminates form a universal grbner basisof the ideal they generate :  .... /**/ use r : : = qq[a , b , c , d , e , f , g , h , i , j , k , l ] ; /**/ i : = ideal(minors(mat([[a , b , c , d],[e , f , g , h],[i , j , k , l]]),3 ) ) ; /**/indent(universalgbasis(i ) ) ; [  d*g*j -c*h*j -d*f*k + b*h*k + c*f*l -b*g*l ,  d*g*i -c*h*i -d*e*k + a*h*k + c*e*l -a*g*l ,  d*f*i -b*h*i -d*e*j + a*h*j + b*e*l -a*f*l ,  c*f*i -b*g*i -c*e*j + a*g*j + b*e*k -a*f*k ] /**/eqset(-1*gens(i ) , reducedgbasis(i ) ) ; true ....  the * grbner fan * of an ideal was defined by mora and robbiano in 1988 ( ) : it is a ( finite ) fan of polyhedral cones indexing the reduced grbner basesof the ideal .this has been implemented by jensen in his software _ gfan _( ) which he has recently linked into cocoa ; we note that cocoa s fully general approach to representing term - orderings was essential in making this integration possible .the grbner fan is useful because many well - known theoretical applications of grbner basesrely on the existence of a grbner basisof an ideal with prescribed properties , such as having a certain cardinality , or comprising polynomials of a specified degree , or all squarefree .for example , if an ideal  ] is koszul .the function ` groebnerfanideals(i ) ` returns all reduced grbner basesof the ideal  as a list of ideals : we chose to encode each basis in an ideal because in cocoa an ideal  stores internally various pieces of information , including the term - ordering and the grbner basis , so putting the result in an ideal instead of a simple list of polynomials is more efficient for any subsequent operations .the following ideal , example 3.9 from sturmfels s book , has 360 distinct reduced grbner bases :  .... /**/ use r : : = qq[a , b , c ] ; /**/ i : = ideal(a^5+b^3+c^2 - 1 , a^2+b^2+c-1 , a^6+b^5+c^3 - 1 ) ; /**/l : = groebnerfanideals(i ) ; /**/len(l ) ; 360 ....  storing all the possible different ( reduced ) grbner basesis practicable only for a small example ; larger ideals may have thousands or even millions of different grbner bases .typically we are interested only in those bases satisfying a certain property .so in cocoathere is the function ` callongroebnerfanideals ` which calls a given function on each grbner basissuccessively without having to store them all in a big list ( if the computer s memory can handle it ! ) . using this cocoafunctionneeds a little technical ability , but might make the difference between getting an answer or not because the computer s memory filled up :  .... define gboflen3(i )  if len(gbasis(i))=3 then   println ; println ordmat(ringof(i ) ) ;   println gbasis(i ) ;  endif ; enddefine ;  /**/ use r : : = qq[a , b , c ] ; /**/i : = ideal(a^5+b^3+c^2 - 1 , b^2+a^2+c-1 , c^3+a^6+b^5 - 1 ) ; /**/callongroebnerfanideals(i , gboflen3 ) ; matrix(zz ,  [ [ 3 , 7 , 7 ] ,  [ 3 , 6 , 8 ] ,  [ 0 , 0 , -1 ] ] ) [ b^2+c+a^2 - 1 ,  a^5+c^2-b*c - a^2*b+b-1 ,  c^3+b*c^2 + 2*a^2*b*c+a^4*b - a*c^2+a*b*c+a^3*b-2*b*c-2*a^2*b - a*b+b+a-1 ] matrix(zz ,  [ [ 6 , 7 , 14 ] ,  [ 6 , 5 , 15 ] ,  [ 0 , 0 , -1 ] ] ) [ c+b^2+a^2 - 1 ,  -b^6 - 3*a^2*b^4 - 3*a^4*b^2+b^5 + 3*b^4 + 6*a^2*b^2 + 3*a^4 - 3*b^2 - 3*a^2 ,  a^5+b^4 + 2*a^2*b^2+a^4+b^3 - 2*b^2 - 2*a^2 ] ....  here we see explicitly that cocoarepresents some term - orderings via matrices of integers .see section [ sec : elim ] for an example of how to ask cocoato compute a grbner basiswith a term - ordering given by a matrix .let  ] ) then . and here we have to admit that the acronym _ gin _ sounds nicer than !the definition of _ gin _ suggests an obvious algorithm for computing it ( see section [ sec : morecoefficients ] for an example with _coefficients ) .however , it quickly becomes apparent that the coefficients in become unwieldy except for the very simplest cases ; so the obvious approach is utterly hopeless .instead we can pick an explicit , random change of coordinates , and then compute ; the coordinate changes for which form a zariski - open set .this approach can be used when  is infinite ; if the random change of coordinates is chosen from a large set then will indeed be with high probability .this is what cocoadoes .while choosing random changes of coordinates with large coefficients increases the probability of getting the correct result , it also tends to produce large coefficients in the transformed polynomials . in this examplethe original polynomials have very small coefficients , but there is a coefficient with almost 50 digits in the tranformed polynomials :  .... /**/ use p : : = qq[x , y , z ] ; /**/i : = ideal(y^20 -x^5*z^6 ,  x^2*z^3 -y*z^2 ) ; /**/l : = [ sum ( [ random(-1000,1000)*indet(p , j ) | j in 1 .. 3 ] )                           | i in 1 .. 3 ] ; /**/l ; [ 941*x -70*y -981*z , 974*x + 342*y -789*z , 398*x + 112*y -942*z ] /**/g : = polyalgebrahom(p , p , l ) ; /**/gi : = ideal(apply(g , gens(i ) ) ) ;  gi ; ideal(101148714265738572816020149152880657871065317376*x^20 + ... ,     ....... ) ....  with coefficients like that , computing the grbner basisof the transformed ideal over the rationals would be quite expensive ! thus , when choosing random coefficients , one needs to strike a balance between a wide range , so the transformation is `` generic enough '' , but not too wide , to limit coefficient growth during the computation .the implementation for computing in cocoauses a special approximate floating - point representation for rational coefficients , namely _ twin - floats _ ( see section [ sec : twinfloats ] ) .the grbner basisof the twin - float transformed ideal will only have approximate twin - float coefficients , but this does not matter because we need only the leading power - products of the polynomials in the basis .twin - float numbers have fixed - precision ( so do not grow in size the way rational numbers do ) , and employ heuristics to guarantee the correctness of results .this allows the implementation to make random coefficient choices from a wide range ( in fact , integers between and ) without paying the price for calculating with transformed polynomials having complicated rational coefficients .if the chosen precision for the twin - floats is too low , this will be signalled ; and the computation can be restarted choosing a higher precision .cocoa s function ` gin ` does this all automatically . moreover , it tries a second random change of coordinates , just to make sure it gets the same leading term ideal .the internal workings can be seen via printed messages when using the option ` verbose ` .  .... /**/gin(i , " verbose " ) ; [  15915*x ,  872152*x -383743*y ,  412211*x -406393*y -383480*z ] -- trying with floatprecision 64 [  -925894*x ,  327379*x -729412*y ,  -945709*x + 550455*y + 499099*z ] -- trying with floatprecision 64 ideal(x^5 , x^4*y^16 , x^3*y^18 , x^2*y^20 , x*y^22 , y^24 ) ....elimination is a central topic in computational commutative algebra ( see for example the text book by kreuzer and robbiano  , sec . 3.4 ) and its applications are countless .elimination means  blue :  given an ideal  ] where the indeterminates have been `` eliminated '' .given its usefulness , elimination is an operation offered in almost all computer algebra systems .we can be sure that all such elimination functions internally compute a grbner basiswith respect to an * elimination ordering * for the subset of indeterminates to be eliminated : with such an ordering the subset of polynomials in the grbner basiswhose leading terms are not divisible by any of the are exactly the generators we seek for the ideal  ] .more precisely , let be elements in  ] and _ eliminating _ all the parameters , as we saw in the example .unfortunately this extraordinarily elegant tool often turns out to be quite inefficent , resulting in long and costly computations . knowing how to exploit special properties of a given class of examplesmight make a huge difference .if the algebra we want to present is generated by power - products then the elimination can be computed by the cocoafunction ; toric ideals are prime and generated by binomials .again we consider  ] :  .... /**/use qq[x , y , z ] ; /**/ toric(rowmat([1,2,3 ] ) ) ; // just the list of exponents ideal(-x^2 + y , x^3 -z )  /**/use qq[x , y , z ] ; /**/ toric(rowmat([3,4,5 ] ) ) ; ideal(-y^2 + x*z , x^3 -y*z , -x^2*y + z^2 ) ....  with a very slightly more challenging example we can clearly measure the advantage in using the specialized function over the general function :  .... /**/ use r : : = zz/(2)[x[1 .. 6 ] , s , t , u , v ] ; /**/l : = [ s*u^20 , s*u^30 , s*t^20*v , t*v^20 , s*t*u*v , s*t^2*u ] ; /**/expl : = [ [ 1 ,  1 ,  1 ,  0 , 1 , 1 ] ,        [ 0 ,  0 , 20 ,  1 , 1 , 2 ] ,        [ 20 , 30 ,  0 ,  0 , 1 , 1 ] ,        [ 0 ,  0 ,  1 , 20 , 1 , 0 ] ] ;  /**/i : = ideal([x[i ] - l[i ] | i in 1 .. 6 ] ) ; /**/ t0 : = cputime ( ) ; ie : = elim([s , t , u , v ] , i ) ; timefrom(t0 ) ; 9.274 /**/ t0 : = cputime ( ) ; it : = toric(expl ) ; timefrom(t0 ) ; 0.032 ....  the cocoafunction employs a non - deterministic algorithm : so the actual set of ideal generators produced might vary .  for further details on the algorithms implemented in cocoasee bigatti , la scala , robbiano .the article describes three different algorithms ; the default one in cocoais eati ( _ elimination algorithm for toric ideals _ ) .  for more details on the specific function type ` ?toric ` into cocoa(or read the pdf manual , or the html manual in the web - site ) .   as mentionedearlier elimination provides a general solution to the implicitization problem , but this solution is more elegant than practical .we can do rather better in the special case of implicitization of a hypersurface .one immediate feature is that the result is really just a single polynomial since the eliminated ideal must be principal .it is well - known that buchberger s algorithm usually works better with homogeneous ideals ( though there are sporadic exceptions ) . yet the very construction of the eliminating ideal looks intrinsically non - homogeneous . but with a little well - guided effort we can transform the problem into the calculation of a grbner basisof a homogeneous ideal .  if the are all polynomials then we take a new indeterminate ( say  ) and use it to homogenize each to produce .now we do have a homogeneous ideal provided we give weights to the indeterminates by setting for each  .since we are in the special case of a hypersurface , it can be shown that the ( non - zero ) polynomial of lowest degree in is unique up to scalar multiples ; its dehomogenization is then the polynomial we seek !we get two advantages from the homogeneous ideal : we gain efficiency by using buchberger s algorithm degree - by - degree , and we can stop as soon as the first basis polynomial is found  most probably there will still be many pairs to process .see abbott , bigatti , robbiano  for all details and proofs , and also how to `` correctly homogenize '' parametrizations by rational functions ..... /**/ use p : : = qq[s , t , x , y , z ] ; /**/elim([s , t ] ,  ideal(x - s^2 , y - s*t , z - t^2 ) ) ; ideal(y^2 -x*z )  /**/ use r : : = qq[s , t ] ; /**/p : : = qq[x , y , z ] ; /**/implicithypersurface(p , [ s^2 , s*t , t^2 ] , " elimth " ) ; ideal(y^2 -x*z ) ....  in the same paper we describe another algorithm which uses a completely different technique , a variant of the buchberger - mller algorithm ( see section [ sec : bm ] ) , based on linear algebra .it is well - suited to low degree hypersurfaces .  .... /**/implicithypersurface(p , [ s^2 , s*t , t^2 ] , " direct " ) ; ideal(y^2 -x*z ) ....  both algorithms , in the case of rational coefficients , use modular methods computing the result for some primes , combining them using chinese remaindering , and reconstructing the rational coefficients of  using the fault - tolerant rational reconstruction described in section [ sec : faulttolerant ] .let be a non - empty , finite set of points in , then the set of all polynomials in  ] . if contains several points we could just intersect the ideals for each single point , and these intersections may be determined via grbner basiscomputations ; while fully effective and mathematically elegant this approach is computationally disappointing .a more efficient method is the buchberger - mller algorithm  .somewhat astonishingly this uses just simple linear algebra to determine the grbner basis .the original algorithm was closely analysed in  , then later generalized to zero - dimensional schemes  , where it turned out that it also incorporates the well - known fglm algorithm for `` changing term - ordering '' of a grbner basis .the bm algorithm is also amenable to a modular approach , a technique for achieving particularly efficient computation with rational numbers .the cocoaimplementation uses this approach .  .... /**/p : : = qq[x , y ] ; /**/ points : = mat([[10 , 0 ] , [ -10 , 0 ] , [ 0 , 10 ] , [ 0 ,-10 ] ,            [ 7 , 7 ] , [ -7 , -7 ] , [ 7 , -7 ] , [ -7 , 7 ] ] ) ; /**/indent(idealofpoints(p , points ) ) ; ideal (  x^2*y + ( 49/51)*y^3 + ( -4900/51)*y ,  x^3 + ( 51/49)*x*y^2 -100*x ,  y^4 + ( -2499/2)*x^2 + ( -2699/2)*y^2 + 124950 ,  x*y^3 -49*x*y  ) ....  the simple use of linear algebra in the bm algorithm makes it a good candidate for identifying `` almost - vanishing '' polynomials for sets of _ approximate _ points : for instance , the points in the example above `` almost lie on '' a circle of radius centred on the origin . the notion of grbner basisdoesnot generalize well to an `` approximate context '' because the algebraic structure of a grbner basisis determined by zariski - closed conditions ( _ i.e. _ the structure is valid when certain polynomials vanish ) ; instead the notion of a _ border basis _ is better suited since its structure is valid dependent on a zariski - open condition ( _ i.e. _ provided a certain polynomial _ does not _ vanish ) .so long as the approximate points are not too few nor too imprecise the bm algorithm can compute at least a partial border basis , and this should identify any `` approximate polynomial conditions '' which the points the almost satisfy ( see abbott , fassino , torrente ) .we can ask cocoato allow a certain approximation on the coordinates of the points :  .... /**/ epsilon : = colmat([0.1 , 0.1 ] ) ; //coord approximation 0.1 /**/av : = tmpnbm(p , points , epsilon).almostvanishing ; indent(av ) ; [  x^2 + ( 4999/5001)*y^2 -165000/1667 , // almost a circle  x*y^3 -49*x*y ,  y^5 -149*y^3 + 4900*y ]  /**/ epsilon : = colmat([0.01 , 0.01 ] ) ; //coord approximation 0.01 /**/av : = tmpnbm(p , points , epsilon).almostvanishing ; /**/indent(av ) ;  //  with this epsilon , not near a conic [  x^2*y + ( 49/51)*y^3 + ( -4900/51)*y ,  x^3 + ( 51/49)*x*y^2 -100*x ,  y^4 + ( -2499/2)*x^2 + ( -2699/2)*y^2 + 124950 ,  x*y^3 -49*x*y ] ....it is well known that computations with coefficients in can often be very costly in terms of both time and space . for grbnerbasesover we are free to multiply the polynomials by any non - zero rational ; so we can clear denominators and remove integer content .this does yield some benefit , but is not wholly satisfactory .sometimes the grbner basishas complicated coefficients ( _ i.e. _ we mean _ big numerators and denominators _ ) , but more often the coefficients in the answer are reasonably sized , while the computation to obtain them involved far more complicated coefficients : this problem is known _ intermediate coefficient swell_.  the phenomenon of coefficient swell is endemic in computer algebra , and many techniques have been investigated to tackle this problem .we illustrate what cocoaoffers .cocoaoffers floating - point arithmetic with a heuristic guarantee of correctness : the aim is to combine the speed of floating - point computation with the reliability of exact rational arithmetic  for a fuller description see the article  .normally a twin - float computation will produce either a good approximation to the correct result or an indication of failure ; strictly , there is a very small chance of getting a wrong result , but this never happens in practice .  to perform a computation with twin - floatsthe user must first specify the required precision ; cocoawill then check heuristically that the result of every computation has at least that precision .if the check fails then cocoasignals an `` insufficient precision '' error ; the user may then restart the computation specifying a higher precision . although twin - float values are , by definition , approximate , all input values are assumed to be exact ( so they can be converted to a twin - float of any precision ) .it is also possible to reconvert a twin - float value to an exact rational number .like all other twin - float operations , this conversion may fail ( because of `` insufficient precision '' ) .printing out a twin - float automatically checks if the value can be converted to a rational as rationals are easier to read and comprehend .  .... /**/rr16 : = newringtwinfloat(16 ) ; /**/ use rr16_x : : = rr16[x , y , z ] ; /**/f : = 12345678*x+1/456789 ; /**/f ;      // both coeffs are printed as rationals 12345678*x + 1/456789 /**/f * 10 ^ 3 ;  //first coeff is printed in " floating - point " 0.12345678 * 10 ^ 11*x + 1000/456789 /**/f * 10 ^ 5 ;  // both coeffs are printed in " floating - point " 0.12345678 * 10 ^ 13*x + 0.2189194573 ....  twin - floats include a ( heuristically guaranteed ) test for zero ; this means it is possible to compute grbner baseswith twin - float coefficients .one reason for wanting to do this is that often the grbner basisover the rationals involves `` complicated fractions '' ( _ i.e. _ whose numerator and denominator have many digits ) , and arithmetic with such complicated values can quickly become very costly .in contrast , with twin - floats the arithmetic has fixed cost ( dependent on the precision chosen , of course ) .these characteristics are exploited in cocoafor the computation of described in section [ sec : gin ] .a widely used technique for avoiding intermediate coefficient swell is to perform the computation modulo one or more prime numbers , and then lift / reconstruct the final result over .we call this the * modular approach*. there are two general classes of method : * hensel lifting * and * chinese remaindering * , the first is not universally applicable but does work well for polynomial gcd and factorization , while the second is widely applicable and works well in most other contexts .the modular approach has been successfully used in numerous contexts , here are a few examples : polynomial factorization  , determinant of integer matrices  , ideals of points ( see section [ sec : bm ] ) , and implicitization ( see section [ sec : implicit ] ) .  in any specific applicationthere are two important aspects which must be addressed before a modular approach can be adopted , and there is no universal technique for addressing these issues .  *knowing how many different primes to consider to guarantee the result ( _ i.e. _ find a realistic bound for the size of coefficients in the answer ) ; * handling _ bad primes _ : namely those whose related computation follows a different route , yielding an answer with the wrong `` shape '' ( _ i.e. _ which is not simply the modular reduction of the correct non - modular result ) .  in the context of grbner baseswedo not have good , general solutions to either of these issues .one of the first successes in applying modular techniques to grbner basiscomputation appeared in  . finding good ways to employ a modular approach for grbner basesis still an active area .cocoadoes not currently use a modular approach for general grbner basiscomputations .a vital complement to the modular computation is the reconstruction of the final , rational answer from the modular images .cocoaoffers functions for combining to two residue - modulus pairs into a single `` combined '' residue - modulus pair ( _ i.e. _ using the chinese remainder theorem ) .it also offers functions for determining a `` simple '' rational number coresponding to a residue - modulus pair ; this is called * rational reconstruction*. correct reconstruction can still be achieved even in the presence of a few `` faulty residues '' ( see  ) ; this fault - tolerance was exploited in the functions for hypersurface implicitization ( see section [ sec : implicit ] ) .here we see how two modular images can be combined in cocoa(using in this case ) , and then the correct rational result is reconstructed from the combined residue - modulus pair .  .... /**/p1 : : = zz/(12347)[x ] ; /**/p2 : : = zz/(23459)[x ] ; /**/f1 : = readexpr(p1 , " x/1234 - 1/5 " ) ;  f1 ;  // in p1 -5293*x -4939 /**/f2 : = readexpr(p2 , " x/1234 - 1/5 " ) ;  f2 ;  // in p2 -1806*x -4692  /**/ use p  : : = qq[x ] ; /**/combined:=crtpoly(-5293*x-4939 , 12347 ,  -1806*x-4692 , 23459 ) ; /**/ combined ;  // in p record[modulus : = 289648273 , residue : = 79571122*x + 115859309 ] /**/ratreconstructpoly(combined.residue , combined.modulus ) ; ( 1/1234)*x -1/5 ....  this reconstruction is exploited in cocoafor the computation of the implicitization of hypersurfaces described in section [ sec : implicit ] .as mentioned in section [ sec : cocoalib - vs - cocoa5 ] , our aim is to make computation using cocoalib as easy as using cocoa-5 . to illustrate this , hereis the first example from section [ sec : gbwithease ] but in c++ :  .... ring p = newpolyring(ringqq ( ) , symbols("x , y , z " ) ) ; ideal i = ideal(readexpr(p , " x^3 + x*y^2 - 2*z " ) ,         readexpr(p , " x^2*y^3 -y*z^2 " )  ) ; cout < < gbasis(i ) ; ....  in comparison to cocoa-5 , this c++ code is more cumbersome and involved , though we maintain that it is still reasonably comprehensible ( once you know that ` cout < < ` is the c++ command for printing ) .we have designed cocoa-5and cocoalibtogether with the aim of making it easy to develop a prototype implementation in cocoa-5 , and then convert the code into c++ .to facilitate this conversion we have , whenever possible , used the same function names in both cocoa-5and cocoalib , and we have preferred traditional `` functional '' syntax in cocoalib over object oriented `` method dispatch '' syntax ( _ e.g. _ ` gbasis(i ) ` rather than ` i.gbasis ( ) ` ) .this means that most of the cocoa-5examples given here require only minor changes to become equivalent c++ code for use with cocoalib .  naturally , a grbner basisis rare computed in isolation ; it is usually just one step in a higher - level computation .we have designed cocoalibto be easy to use , _ e.g. _ for preparing the generators of the ideals whose grbner basesare to be computed , and also for further processing with the grbner basesafter they have been computed . to maintain the `` friendly '' tradition of cocoasoftware ,our design of cocoalibfollows these aims :  * designed to be * easy and natural to use * * execution speed is * good * * well - documented , including * many example programs * * * free and open source * c++ code ( gpl3 licence ) * source code is * clean and portable * ( currently c++03 ) * design respects the underlying * mathematical * structures + ( inheritance , no templates ) * * robust * ( motto : `` no nasty surprises '' ) , exception - safe , thread - safethe cocoasoftware aims to make it easy for everyone to use grbner bases , whether directly or indirectly through some other function .the cocoa-5system is designed to be welcoming to those with little computer programming experience , while the cocoaliblibrary aims to make it easy for experienced programmers to use grbner basesin their own programs .b. buchberger , h.m .mller , _ the construction of multivariate polynomials with preassigned zeros _ , proceedings of the european computer algebra conference ( eurocam 82 ) , * 144 * , lecture notes in comp.sci . , ( 1982 ) , pp . 2431 .

Abstract: we present a survey on the developments on grbner basesshowing explicit examples in cocoa .   the cocoa project dates back to 1987 : its aim was to create a `` mathematician''-friendly laboratory for studying commutative algebra , most especially grbner bases . since then , always maintaining this `` friendly '' tradition , it has evolved and has been completely rewritten .   cocoaoffers grbner bases for all levels of interest : from the basic quick call in the interactive system cocoa-5  , to problem - specific optimized implementations , to the computer computer communication with the open source c++ software library , cocoalib  , or the prototype openmath - based server .   the openness and clean design of cocoalib and cocoa-5 are intended to offer different levels of usage , and to encourage external contributions .


a genuine connection between the laws of black holes mechanics and those of thermodynamics is widely believed today .it started out with a mere analogy connecting these two sets of laws which was first pointed out by bardeen , carter and hawking . only after the discovery of hawking radiation , it has been realized that these thermodynamic relations describe the thermal properties of a black hole . in this workhawking was able to show that a black hole behaves quantum mechanically as a black body radiator , with a temperature proportional to its surface gravity and entropy proportional to its horizon area .based on the above connection between entropy and horizon area , and that of temperature and surface gravity , jacobson found that einstein field equations can follow exactly from the fundamental thermodynamic relation between heat , entropy , and temperature , i.e. , clausius relation .the argument is based on demanding that clausius relation holds for all the local rindler causal horizons through each spacetime point .jacobson interpreted as the energy flux and as unruh temperature seen by an accelerated observer just inside the horizon .jacobson s work can be stated as follows ; from a thermodynamic point of view , einstein field equations are nothing but an equation of state for the spacetime under consideration . inspired by jacobson approach , cai et.al derived friedman equations of -dimensional friedman - robertson - walker ( frw ) universe , from the clausius relation with the apparent horizon of frw universe , assuming that the entropy is proportional to the area of apparent horizon .the derivation of cai et al . was based on assuming that the apparent horizon has an entropy , and a temperature ;  s&= & , [ entropy ] + t&= &  [ hawking ] notice here that the entropy is proportional to the area , but the temperature is not proportional to the surface gravity , since = - ( 1- ) . as a result , the first law in this context is an approximation , where , has to be satisfied .this has been noticed and discussed by the authors in .the drawback of this approximation is that it constrains possible equations of state ( eos ) to be , i.e. , eos has to be close enough to that of the vacuum energy or de sitter spacetime .this can be shown clearly through the cosmological equation for a non - vanishing . to improve this thermodynamic description of frw cosmology , akbar and cai used the misner - sharp energy relation inside a sphere of radius of an apparent horizon and rewrite the dynamical friedman equation in the form of the first law of thermodynamics with a work term .de= t ds+ w dv , where the work density is given in terms of the energy density and the pressure of matter in the universe as follows :  w=(-p ) .in addition , they considered the misner - sharp energy to be the total energy of the matter existing inside the apparent horizon which is given by , where is the volume of the apparent horizon .  motivated again , by black hole physics , there is an another interesting connection between the hawking radiation and the uncertainty principle where the black hole can be modeled as -dimensional sphere of size equal to twice of schwarzschild radius .since the hawking radiation is a quantum process , one could assume that the emitted particles should obey the heisenberg uncertainty relation .from this , one can derive exactly the hawking temperature and the thermodynamic properties of the black hole . from the black hole thermodynamics, it has been realized that ; as the black hole approaches zero mass , its temperature approaches infinity with infinite radiation rate which is considered a _catastrophic _ evaporation of the black hole .  a generalized uncertainty principle(gup ) was proposed by different approaches to quantum gravity such as string theory and black hole physics in which planck length plays an important role .the existence of planck length as a minimal observable length is a universal feature among all approaches of qg .this minimal length works as a natural cutoff , which is expected to have a crucial role in resolving curvature singularities in general relativity . by employing this gup, it was found that the black hole thermodynamics quantities , such as temperature , and entropy are completely modified such that the end - point of hawking radiation is not catastrophic anymore .this is because the gup implies the existence of black hole _remnants _ at which the specific heat vanishes and , therefore , the black hole can not exchange heat with the surrounding space .this means that gup prevents the black hole from evaporating completely , just like the standard uncertainty principle prevents the hydrogen atom from collapsing .it is worth mentioning that the gup modifies , significantly , the entropy - area law as has been pointed out by several authors ( see for example ref.s ) .  in this work we study exact modifications of the entropy - area law due to gup , which is used to modifying friedmann equations derived by akbar and cai in at very high densities .it is worth mentioning that gup has been studied with friedmann equations through different frameworks in . in this paper , we first derive the modified friedmann equations for a general form of the entropy as a function of area using the first law of thermodynamics .this generalization can host all possible correction to the entropy - area law like the logarithmic correction and power law corrections which are motivated by different approaches to quantum gravity such as string theory and loop quantum gravity .we then investigate an important entropy - area law which is motivated by the gup and study the modified friedmann equations derived through the first law approach introduced by akbar and cai in .studying the resulted friedmann equations reveals the existence of a maximum energy density closed to planck density . assuming a general continuous pressure leads to bounded curvature invariants and a general nonsingular evolution . as a result, the maximum energy density is reached in a finite time and there is no cosmological evolution beyond this point from a spacetime prospective .the existence of maximum energy density and a general nonsingular evolution is independent of the equation of state and the spacial curvature . as an example we study the evolution of the equation of state , using a phase - space diagram , to show the existence of a maximum energy density and a finite time to reach it .our results reveal that the big bang singularity is not accessible in this description since the spacetime itself can not be extended beyond planck density as a result of the gup and the thermodynamic approach to gravity which modifies friedmann equations .the paper is organized as follows , in sec .[ ah ] , we review the derivation of friedmann equations from the first law of thermodynamics due to akbar and cai , assuming that the entropy is proportional to the area of apparent horizon . in sec .[ general ] , we extend this procedure for an arbitrary entropy , which could host various possible correction to entropy - area law , to obtain a set of modified friedmann equations . in sec .[ gup ] , we review the generalized uncertainty principle and calculate the exact entropy - area law from the first law of thermodynamics . in sec .[ frw - gup ] , we calculate the modified friedmann equations due to the exact entropy - area law obtained from gup using the first law of thermodynamics . in sec .[ raychud ] , we discuss direct implications of the modified friedmann equations arguing for the existence of a maximum energy density closed to planck density . assuming a continuous pressure we show that all curvature invariants are finite and the previous two features are independent of the equation of state and the spacial curvature . as an example we study the evolution of the equation of state , using a phase - space diagram , to show the existence of a maximum energy density and a finite time to reach it . at the endwe conclude by showing the general implications of the modified friedmann equations that we found due to the exact entropy - area law obtained from the gup .in this section , we review the derivation of friedmann equations from the first law of thermodynamics relation with the apparent horizon of frw universe with assuming that the entropy is proportional to the area of the apparent horizon .the -dimensional friedmann - robertson - walker ( frw ) universe is described by the following metric : ds^2=h_a bdx^a dx^b+^2 d^2_n-1 , [ metric ] where , is the metric of -dimensional sphere , and the spatial curvature constant takes the values for a flat , closed and open universe , respectively .the dynamical apparent horizon is determined by the relation , which would give the radius of the apparent horizon to be :  [ app - radius ] _ a = a r= , where is the hubble parameter . by assuming that the matter which occupy the frw universe forms a perfect fluid , so the energy - momentum tensor would be : t_=(+p)u_u_+pg_. where is the four velocity of the fluid .the energy conservation law ( ) leads to the continuity equation  + n h(+p)=0.[continuity ] based on the arguments of , one can define work density as follows  w=- t^ab h_ab , where is the projection of the energy - momentum tensor in the normal direction . for frw universe with perfect fluid , the work density will be  w&= & ( -p ) .now , we give a brief review for the procedure that has been followed by akbar - cai in .considering the first law of thermodynamics as follows :  de= t ds+ w dv .[ 1st ] let us calculate term by term in eq .( [ 1st ] ) .we start with .the term represents the infinitesimal change in the total energy during small interval of time .since introduces the total energy of matter inside the apparent horizon ( misner - sharp energy ) , so it can take the following form e=  v , where is the volume of n - dimensional sphere with radius and .hence , the differential element of would be de&= & dv+ v  d + & = & n_n_a^n-1 d_a+_n_a^n d.[denergy ] by using the continuity equation of eq .( [ continuity ] ) in eq .( [ denergy ] ) , we get ; de= n_n_a^n-1 d_a- n _ n _ a^n ( + p ) h dt [ denergy1 ] turning to the other term , it can be written as follows : w dv= n _n _ a^n-1 ( -p)d_a . [ dwork ] for the term , we should use definition of hawking temperature of eq .( [ hawking ] ) as well as the entropy - area law .we start with hawking temperature which is defined in terms of surface gravity as follows :  t= where introduces the surface gravity for the metric of eq .( [ metric ] ) and is defined as  & = & _ a(h^ab_b ) + & = & - ( 1- ) besides , we use the entropy - area law of eq .( [ entropy ] ) and the expression of area for n - dimensional sphere .this yields at the end the following expression  t ds&= & d ( ) + & = & - ( _ a^n-2 ) [ tds ] by substituting eqs . ( [ denergy1 ] , [ dwork ] and [ tds ] ) into the first law of thermodynamics of eq .( [ 1st ] ) , we get = ( + p ) h dt .[ 1frw ] using eq .( [ app - radius ] ) which yields  , one simply finds that eq .( [ 1frw ] ) introduces the dynamical friedman equation .-=-(+p ) [ doth ] using the continuity equation ( [ continuity ] ) and integrating eq .( [ doth ] ) , one simply gets h^2+= , which is the friedmann equation for -dimensional frw universe .in this section , we derive the general modified friedmann equation for a general expression of the entropy as a function of area through the apparent horizon approach and first law of thermodynamics .suppose that the general expression for the entropy - area relation takes the following form : s&= & , + & = & , where .using the first law of thermodynamics , and following the same procedure that we reviewed in sec .[ ah ] , we get  f^(a ) = ( + p ) h dt [ dr ] again , using eq.([app - radius ] ) which yields , d_a= -h _a^3(-k / a^2 ) dt one simply finds that eq .( [ 1frw ] ) introduces the dynamical friedman equation . ( - ) f^(a)= - ( -p ) [ fr1 ] using the continuity equation of eq .( [ continuity ] ) , and with few calculations , we can rearrange eq .( [ dr ] ) as follows f^(a ) = - dthe last equation can be integrated to give the general modified friedmann equation for -dimensional frw universe due to a general form of the entropy using the first law of thermodynamics .- = f^(a ) [ fr2 ] again , if we set , the first friedmann equation will be satisfied .these modified equations was gotten first in using the clausius relation .we are deriving them here with considering the contribution of the work in the first law of thermodynamics .we first review briefly the generalized uncertainty principle ( gup ) and secondly we review its effect on the area - entropy law .the existence of a minimum measurable length originates as an intriguing prediction of various frameworks of quantum gravity such as string theory and black hole physics .this implies a direct modification of the standard uncertainty principle :  x , [ gup ] where is the planck length and is a dimensionless constant which depends on the quantum gravity theory . the new correction term in eq .( [ gup ] ) becomes effective when the momentum and length scales are of order the planck mass and of the planck length , respectively .it was straightforward to find that eq.([gup ] ) implies the existence of minimal measurable length scale as follows :  x x_min = 2  _p by some manipulations , gup can be represented by another form as follows : there has been investigations devoted to study the impact of gup on the black hole thermodynamics and to the bekenstein hawking ( black hole ) entropy ( e.g. , ) .these studies are based on the argument that hawking radiation is a quantum process and it should respect the uncertainty principle .  herewe review the analysis of and use the arguments in which say that the uncertainty principle can be represented by the lower bound , so one can get for the gup case :  e .[ energy0 ] for any black hole absorbing or emitting a quantum particle whose energy and size , the area of the black hole would change by an amount .a 8 _ p^2 e r , the quantum particle itself implies the existence of finite bound given by a_min 8 _ p^2 e x. using gup by substituting ( [ energy0 ] ) into ( [ darea ] ) , we get a_min .[ area ] the value of in this analysis is set to be the inverse of surface gravity where is the schwarzschild radius , where this is probably the most sensible choice of length scale in the context of near - horizon geometry . this implies the following identity : x^2 = .[ dx ] substituting eq .( [ dx ] ) into eq .( [ area ] ) , we get a_min .the area change is then determined as : a_min , where the parameter will be fixed from the bekenstein - hawking entropy formula . according to ,the black hole s entropy is conjectured to depend on the horizon s area . from the information theory , it has been stated that the minimal increase of entropy should be independent of the area .it is just one bit of information which is .= = . according to , the bekenstein - hawking entropy formula has been used to calibrate the the constants , so the we have = = [ dentropy ] to simplify the expression of eq .( [ dentropy ] ) , we set , so we get = .[ exact ]  in this paper , we are interested in the _ exact _ form of the entropy instead of the approximated one that was used in , so we will not make any approximation for the expression of .the exact form of eq .( [ exact ] ) would enable us to study the behavior of a general solution of the modified friedmann equations ( using its fixed points ) due to gup through the first law of thermodynamics . to get the exact expression of the entropy, we integrate eq .( [ exact ] ) to yield :  s= & & + s_0 [ exactentropy ] where is an integration constant .we find that eq .( [ exactentropy ] ) modifies bekenstein - hawking entropy as a result of a minimum length scale or the gup considered above . in the next section, we will calculate the modified friedmann equations due to the modified entropy , eq .( [ exactentropy ] ) , of the apparent horizon of frw universe .in this section , we implement the modified entropy of eq.([exactentropy ] ) in the first law of thermodynamics relation , and derive the modified friedmann equations with the apparent horizon approach .now , we consider our current case of modified entropy due to gup in eq .( [ exact ] ) and eq .( [ exactentropy ] ) . by substituting eq .( [ exact ] ) into the general modified friedmann equations eq .( [ fr1 ] ) and eq .( [ fr2 ] ) with , we get the following expression :  ( - ) = -4 g ( + p ) , [ mfr1 ] & = & - 16 _ p^2 da , + & = & 2 ( -)+c,[mfr2 ] where is a constant of integration and it can be fixed from the initial conditions in eq .( [ mfr2 ] ) .as the universe expands , the area of apparent horizon is supposed to go to infinity with the density having vacuum energy density , where is the cosmological constant . from this argument, takes the following value : c= + = ( + ) .the area of the apparent horizon is given in sec .( [ ah ] ) as :   a= 4accordingly , the modified friedmann equations ( [ mfr1 ] ) and ( [ mfr2 ] ) due to gup will be  ( - ) & = & ( h^2+)+43 , [ mfra ] + - 4 g(+p)&=&(- ) .[mfrb ] we find that eq .( [ mfra ] ) and eq .( [ mfrb ] ) give the exact modified friedmann equations due to gup using akbar - cai approach . in the next section we are going to discuss consequences of eq .( [ mfra ] ) and eq .( [ mfrb ] ) on the behavior of the frw cosmology .it is interesting to notice that eq .( [ mfra ] ) leads to a bounded energy density since the inequality must be satisfied , otherwise , the density is complex .let us analyze this more closely for cases with , and . in order to keep the densityreal the previous inequality requires `` a '' to have a minimum and `` h '' to have a maximum .this leads to a maximum energy density ( since is tiny ) , where is planck s density .having a maximum value for ( or since they are related ) does not necessarily means a finite curvature or nonsingular behavior .the reason is that curvature invariants such as ricci scalar , riemann tensor squared , etc . , depend not only on but also on . but the behavior of is controlled by both and as it is clear from eq .( [ mfrb ] ) . to check whether has a diverging behavior or not we do nt have to know the equation of state, it is enough to know some generally properties of pressure . herewe will follow the discussion in ref . on general properties of pressure which leaves the mathematical problem addressed by eq .( [ mfra ] ) well defined . herewe assume that the pressure , or is a continues of function of its arguments , in order to have a well defined mathematical evolution for or .this ensures existence and uniqueness of the general solution of the continuity equation or eq .( [ mfrb ] ) .in addition , having a discontinues pressure in or could imply a noncausal evolution since it leads to a divergent , or an unbounded speed of sound !therefore , it is natural to assume a continues or . as a result, one observes that if has a minimum value and is bounded , then must be bounded given eq .( [ mfrb ] ) .therefore , all curvature invariants of the friedmann - robertson - walker metric are finite since they can be expressed as functions of and .but what about the case , which potentially could still leads to a singular solution . herewe need to satisfy the inequality , for this case too . for a generalcontinues pressure , such that as , the previous inequality leads to a minimum value for and maximum value for .but for the specific case where as the scale factor can go to zero and the hubble rate to infinity without violating the inequality . in this case, it is interesting to notice that both density and pressure are finite .a finite pressure can be shown through taking the derivative of with respect to time using and calculating using eq .( [ mfrb ] ) . analyzing this case we found that it leads to a specific eos , namely , which gives a de sitter evolution as .our conclusion is that the above modified friedmann equation leads to a bounded energy density with a maximum value given by the planck s density for any equation of state and all values .in addition , by assuming a continues pressure one can show that curvatures invariants are finite as a result of maximum density and minimum scale factor , therefore , the solution is nonsingular .our results reveals the limitation or the breaking down of the spacetime description of gravity near planck scale energies which was not evident from general relativity . as an example we will discuss next the eos and its raychaudhuri equation which describes a nonsingular universe which starts from a finite time with a planck densitynow to show some of the features discussed above let us consider the usual equation of state , in the light of this modified friedmann equation and check the behavior of the universe in early times . to dothat one can study raychaudhuri equation or eq .( [ mfrb ] ) on the following form  = f(h ) [ raych ]  using the modified friedmann equations of eq .( [ mfra ] ) and eq .( [ mfrb ] ) , and the above equation of state , we get the following form of raychaudhuri equation .( -)&&=- ( 1 + ) + & & ( ) .  for the flat case with the raychaudhuri equation takes the following form  = & & - ( 1 + ) + & & [ mf ]  using the general analysis of the eq . ( [ raych ] ) in for frw cosmology .the above first - order system is well studied in dynamical system in cosmological context , see e.g. , , or see for more general applications . knowing fixed points ( i.e. , its zeros .let us call them ) and its asymptotic behavior enables one to qualitatively describe the behavior of the solution without actually solving the system . drawing the phase - space diagram for for eq .( [ mf ] ) for ( i.e. , plotting versus ) we get the following behavior in fig.([solution ] )   versus ,   . has a maximum value .[ solution ]  as one can notice we have a future fixed point at , which we can provide the asymptotic behavior of our universe at late times . in additionthis fixed point is reached in an infinite time since by integrating eq .( [ mf ] ) we get t= _dh f(h ) = , where is the initial value of the hubble rate and f(h ) is f(h)=- ( 1-(1- h^2 ) ^)the interesting observation in this diagram is the existence of a maximum value for the hubble rate , beyond which there is no evolution allowed .we have calculated the time to reach the maximum hubble rate t_m= -_h_0^h_max dh f(h)= finite , since all curvature invariants of the frw metric are functions of the hubble rate and it first - time - derivative , it is straight forward to show that they are all finite as a result of a maximum density and the eos given the above modified friedmann equations .it is interesting to observe that the big bang singularity is not accessible in this description since the spacetime itself can not be extended beyond planck density as a result of the minimum length or the gup and the thermodynamic approach to gravity provided which modifies friedmann equations .we generalize akbar cai derivation of friedmann equations from the first law of thermodynamics , to include an arbitrary entropy - area law which could include possible corrections arise from different approaches to quantum gravity . studyingthe resulted friedmann equations for an entropy - area law motivated by the generalized uncertainty principle ( gup ) revealed the existence of a maximum energy density with a value around planck density . allowing for a general continuous pressure to bounded curvature invariants and a general nonsingular evolution . in this case, the maximum energy density is reached in a finite time and there is no cosmological evolution beyond this point from a spacetime prospective .the existence of maximum energy density and a general nonsingular evolution is independent of the equation of state and the spacial curvature . as an example we study the evolution of the equation of state , using a phase - space diagram , to show the existence of a maximum energy density and a finite time to reach it .our results reveal that the big bang singularity is not accessible in this description since the spacetime itself can not be extended beyond planck density as a result of the minimum length or the gup and the thermodynamic approach to gravity which modifies friedmann equations .b. majumder , phys .b * 703 * , 402 ( 2011 ) .[ arxiv:1106.0715 [ gr - qc ] ] ; a. f. ali , jhep * 1209 * , 067 ( 2012 ) [ arxiv:1208.6584 [ hep - th ] ] ; a. f. ali , h. nafie and m. shalaby , europhys .lett .  * 100 * , 20004 ( 2012 ) .m. maggiore , m. maggiore , phys .b * 304 * ( 1993 ) 65 [ arxiv : hep - th/9301067 ] ; m. maggiore , phys .d * 49 * ( 1994 ) 5182 [ arxiv : hep - th/9305163 ] ; m. maggiore , phys .b * 319 * ( 1993 ) 83 [ arxiv : hep - th/9309034 ] ; l. j. garay , int . j. mod .phys .  a * 10 * ( 1995 ) 145 [ arxiv : gr - qc/9403008 ] ; s. hossenfelder , m. bleicher , s. hofmann , j. ruppert , s. scherer and h. stoecker , phys . lett .b * 575 * ( 2003 ) 85 [ arxiv : hep - th/0305262 ] ; c. bambi and f. r. urban , class . quant .* 25 * ( 2008 ) 095006 [ arxiv:0709.1965 [ gr - qc ] ] ; a. f. ali , s. das and e. c. vagenas , phys .b * 678 * , 497 ( 2009 ) [ arxiv:0906.5396 [ hep - th ] ] .f. scardigli , phys .b * 452 * ( 1999 ) 39 [ arxiv : hep - th/9904025 ] .b. majumder , astrophys .space sci .  *336 * , 331 ( 2011 ) [ arxiv:1105.2425 [ gr - qc ] ] .j. e. lidsey , phys .d * 88 * , 103519 ( 2013 ) [ arxiv:0911.3286 [ hep - th ] ] .r. b. mann and s. n. solodukhin , phys .d * 55 * , 3622 ( 1997 ) [ hep - th/9609085 ] ; r. k. kaul and p. majumdar , phys .* 84 * , 5255 ( 2000 ) [ gr - qc/0002040 ] ; s. das , p. majumdar and r. k. bhaduri , class .* 19 * , 2355 ( 2002 ) [ hep - th/0111001 ].   g. amelino - camelia , m. arzano and a. procaccini , phys .d * 70 * , 107501 ( 2004 ) [ gr - qc/0405084 ] ; e.m .lifshitz , l.p . pitaevskii and v.b .berestetskii , landau - lifshitz course of theoretical physics , volume 4 : quantum electrodynamics , ( reed educational and professional publishing , 1982 ) .d. christodoulou , phys .* 25 * , 1596 ( 1970 ) ; d. christodoulou and r. ruffini , phys .d * 4 * , 3352 ( 1971 ) .

Abstract: inspired by jacobson s thermodynamic approach , cai et al have shown the emergence of friedmann equations from the first law of thermodynamics . we extend akbar cai derivation of friedmann equations to accommodate a general entropy - area law . studying the resulted friedmann equations using a specific entropy - area law , which is motivated by the generalized uncertainty principle ( gup ) , reveals the existence of a maximum energy density closed to planck density . allowing for a general continuous pressure leads to bounded curvature invariants and a general nonsingular evolution . in this case , the maximum energy density is reached in a finite time and there is no cosmological evolution beyond this point which leaves the big bang singularity inaccessible from a spacetime prospective . the existence of maximum energy density and a general nonsingular evolution is independent of the equation of state and the spacial curvature . as an example we study the evolution of the equation of state through its phase - space diagram to show the existence of a maximum energy which is reachable in a finite time .  * * adel awad[multiblock footnote omitted ] and ahmed farag ali +  +  +  +  +  +


plants are complex biological systems , which feed , develop and function thanks to a variety of elementary and cooperative processes that ensure transport of energy and matter in the form of mineral elements , carbohydrates , and hormones in watery solutions called sap. two categories of sap are transported in conductive tissues : xylem sap transported in xylem , a continuum of dead cells forming a pipe that opposes little resistance to the transport of water and mineral elements from roots to leaves ; and phloem sap , containing water and sugar , formed in the leaves by photosynthesis , and transported in phloem ( sieve tube elements ) from the leaves to the roots .explanations for xylem sap transport includes the cohesion - tension theory and multiforce theories ; and the pressure flow hypothesis proposes a mechanism for phloem sap transport .notwithstanding their relative merits , it is of interest to note that as saps contain ions , a net voltage may be recorded within plants .furthermore , as a living plant is naturally submitted to a temperature gradient between its roots and its leaves , heat may also flow from its hotter to its colder regions .these two facts lead to the questions of the existence and strength of a coupling between electrical current and heat flux in plants , and whether this coupling plays a significant role in the plants lives .answers to these questions are of importance to gain further insight into plant homeostasis , closely related to the le chatelier - braun principle in chemical thermodynamics , and to develop further the field of plant electrophysiology by analysing the impact of thermoelectric coupling on photosynthesis , plant respiration , and any other processes involving bioelectrochemical phenomena .biophysical processes , which include energy and matter transport in dedicated structures , are essentially nonequilibrium in nature .their description involves the notions of thermodynamic forces and their conjugate fluxes which , in a linear response description , are proportional to each other . in a general manner ,transport phenomena are nonequilibrium irreversible processes : fluxes within a system result from applied external constraints , and they are accompanied by energy dissipation and entropy production . as the forces applied to a systemderive from potentials , the description of the system s nonequilibrium properties relies on the definition of these potentials and their degree of coupling .thermoelectricity is sometimes mistakenly viewed as a phenomenon pertaining to solid - state systems , but this effect has also been reported in liquids and gels .now , considering more specifically electrical phenomena in plants , one should also account for their dependence on temperature and the local variations of this latter . as a matter of fact, one learns from the equilibrium thermodynamics of charged fluids that heat and electricity are fundamentally related through the gibbs - duhem relation ; so , since plants , as any other system , are subjected to the laws of thermodynamics , one may anticipate that a proper understanding of the coupled transport of heat and electricity in plants will pave the way towards the potentially rich new field of thermoelectricity - based plant energy conversion , and harvesting metabolic energy from plants .thermoelectricity as part of non - isothermal processes in living organisms , has already been mentioned ; but except for the case of insects like hornets and bees , for which genuine thermoeletric effects have been analysed , the other studies simply reported the record of the temperature rise of colocasia odora leaves using a thermocouple , or analysed pyroelectricity , mistakenly taken as a thermoelectricity , amongst the nonequilibrium processes at the origin of life .we propose here to consider bean sprouts as an illustrative case of a living plant system whose physiological properties also entail thermoelectric effects , with a particular focus on the thermovoltage response of the plant .while a voltage difference along the stem of a plant has already been measured and analysed , its interpretation does not account for a likely the thermoelectric origin .nevertheless , it is clear that the source of any voltage difference is due to concentration gradients of charges inside the plant , regardless of the cause of the gradients .the underlying processes may thus originate in various electrochemical processes , including acid - base and redox effects . in the present work we focus on the voltage response due to a temperature difference .the system is presented in fig .the stem and leaves are kept at constant temperature while the temperature of the roots can be modified ; the thermoelectric response is recorded on both the plant ( a ) and a control wire ( b ) .thermoelectricity differs from electrochemistry as the latter considers isothermal systems only .thermoelectric effects , on the contrary , manifest themselves in non - isothermal conditions and their thermodynamic description is best done with the coupled variables , where is the electrochemical potential and the temperature of the considered system . in a force - flux approach , as a temperature bias is applied to a conductor , both heat and charges are transported in coupled flows since each electron carries an electric charge , , and energy ; the thermoelectric coupling parameter or seebeck coefficient is thus defined as :    where denotes the spatial gradient .equation ( [ seebeck ] ) may be viewed as a generalization of the gibbs - duhem relation to the dynamical response of the coupled intensive variables by defining the thermodynamic forces acting on the system as the gradients of the thermodynamic potentials .each force is conjugated to a flux which is proportional to the time derivative of the corresponding extensive parameter , and it follows that forces and fluxes are linearly combined as proposed by onsager .note that the coefficient , which in eq .( [ seebeck ] ) is defined locally , is also often expressed as , where denotes the global difference between two distant points of the system .these general considerations apply for any system containing free charge carriers . as a consequence , for a given temperature difference, the measurement of a thermovoltage directly gives an estimation of the average ratio of charge concentrations in separate parts of the system .therefore , any system containing free mobile carriers may be exhibit a thermoelectric signature .one should note that depending on the specific properties of the considered system , this signature may intertwine or not with those of other temperature - dependent processes ; hence the need for specific system - dependent approaches to extract the thermoelectric signal . in the case of liquids and gels ,the studies usually focus on the thermoelectrophoresis parameters . for solid state - systems ,the conduction is ensured by mobile electrons and holes , leading to a possible steady - state electrical current . for ionic carriers , in liquids and gels ,such a steady - state current may only take place if a redox process occurs at the ends of the system .the experiment aims to identify and characterize the thermoelectric response of a living plant ( a bean sprout in the present work ) . as the response of the plant necessarily entails physiologic effects ,the measured data is compared to the purely thermoelectric response of a control wire . as depicted on fig .[ fig : figure1 ] , the temperature of the plant s roots is imposed by a thermostatic bath ; two thermocouples are used to measure the temperatures of the roots and the leaves , and two electrodes are placed respectively in contact with the roots and the leaves of the living plant .the roots are bathed in an aqueous dilute kcl solution .the electrodes connected to the leaves are made of ag / agcl wires .all the temperatures and voltages are recorded using a keithley k2700 scanning multimeter and a k182 nanovoltmeter .the thermoelectric control wire ( cw ) is made of two chromium - nickel - steel junctions , with a seebeck coefficient / k ; it is placed in the same configuration as that of the plant .the recorded voltage at the ends of the control wire describes a _ pure _ thermoelectric response , exempt from any influence of the physiologic process .( upper left panel ) and of the plant ( upper right panel ) , both submitted to a time - varying temperature difference , reported on the lower panel .the magnitude of measured voltage has been multiplied by a factor 2000 in order to report it on the same scale as that of .the red - dotted lines indicate the average slope of the collected data . in the lower panel ,the scale is that of the temperature difference ; the voltages are also depicted in this lower panel to see how close their time - dependent behaviour is to that of the measured temperature difference.,scaledwidth=50.0% ]  the voltage response of the living plant and that of the control wire to the temperature difference are both shown on fig .[ fig : fig2 ] , where the time evolution of is also displayed , illustrating the dynamics of the plant and that of the control wire .these data were recorded over a period of four hours for various temperature differences and variation rates .the two upper panels show separately the parametric plots characterising the thermoelectric response of the control wire and that of the plant .notice that the response of the control wire is strictly proportional to the temperature difference , as expected for a thermoelectric material , with a _ passive _ seebeck coefficient .conversely , the plant exhibits a complex response , which includes the effects of its physiological processes thus giving rise to an _ active _ seebeck response .the curves represented on the lower panel are displayed together on the same scale to qualitatively show how the responses ( voltages and ) follow the applied constraint ( temperature difference ) .the figures on the -axis correspond to in .the values taken by the voltages and may be determined from the curves using a scaling factor that corresponds to the relevant seebeck coefficient .the control wire shows a faithful homothetic response to the temperature difference , with a linear factor of scaling of / k as expected for the seebeck coefficient of the corresponding chromium - nickel - steel thermocouple .the response of the plant gives an average seebeck coefficient of 2.5 mv / k .such a large value in a solid - state material would correspond to a system with low electrical conductivity . here , this results from ion movements through xylem and phloem saps circulating in opposite directions .  to analyze precisely these data , it is convenient to split each panel of fig .[ fig : fig2 ] into three parts , which correspond to three distinct time domains .the first and third ones , figs.[fig : fig3 ] and [ fig : fig5 ] , respectively , describes the plant s response when its roots are heated ; while the second , fig.[fig : fig4 ] , reports the response over the cooling sequence ..,scaledwidth=50.0% ]      depending on the precise thermal condition supported by the plant , we can identify different thermoelectric responses . consider the first warming sequence of fig .[ fig : fig3 ] .one may identify five events ( labelled from a to e ) . during the first period ( fig .[ fig : fig3]a ) , the temperature difference is roughly constant .the plant presents a large fluctuating thermoelectric response while the control wire is characterized by a constant seebeck coefficient . during the second period , shown on fig .[ fig : fig3]b , the plant s reaction is essentially sensitive to the temperature variation rate .this sensitivity has already been observed in studies devoted to plant sensing .the processes taking place within the zones a and b can be also observed in the c and d zones ; and when the rate is reduced , we observe a linear response for the plant , as shown in the zone e. it thus appears clearly that the passive response , probably mostly due to the xylem properties , can be strongly modified by active physiological processes .this latter appears to be particularly sensitive to the time derivative of the stress , but not so much on the intensity of the stress itself ; this observation is perfectly consistent with previous findings .we now turn to the cooling sequence , fig .[ fig : fig4 ] .the cooling rate is approximately constant , except for the two events when it is clearly larger than the average as indicated with the labels a and b , on the upper right panel .these events relate to time delays , also apparent in the measured data for the control wire on the upper left panel : these are signatures of the change of rate .note that although these may serve as a probe , time delays must be considered with care , since in the case of very large time response of a system , they may lead to erroneous interpretations .the shape of the plant response for events taking place in the zones a and b is very similar to that of figs .[ fig : fig3]b and [ fig : fig3]d , which confirms the acute sensitivity of the plant to the rate .this physiological response is the fastest response to a stress that a plant can use , as it has no means to avoid it .the last sequence depicted on fig .[ fig : fig5 ] concerns the plant s response to warming at a reduced rate .it is essentially linear , thus confirming the presence of a threshold rate process for the plant s response to a given stress .it also shows unambiguously that thermoelectricity naturally partakes in the complex processes that contribute to a plant s life .all the effects discussed in the present work are related to the internal modifications of the ionic concentrations inside the plant , including the regulation of cellular ion transporters allowing ion release into the xylem sap toward the shoots .it is clear that while the response is mainly driven by a classical thermoelectric process with a giant seebeck coefficient , the physiology of the plant interferes in a complex way with it .a complete numerical model , including series and parallel assemblies of living ( phloem sieve tube elements , root and endodermis cells ) and dead cells ( xylem tracheids and vessels ) , will be considered for further investigation of the interplay between physiologic and thermoelectric response to a thermal stress .further , since there is no limitation in the number of thermodynamic potentials which may be coupled , we may account for pressure and derive a complete response of a system to the direct and the cross- effects of the three thermodynamic potentials , .then the lowering of a local pressure gradient , by the coupled effect of the temperature and the electrochemical potential becomes possible .this latter process may contribute to the motion of saps in plant , and especially in trees .the reported measurements are fully scalable from the unique cell size to a complete plant or tree . in addition tothe stationary response observed here , the voltage fluctuations are also of great interest since they permit the study of the threshold of the physiological response through its noisy signals .as observed in other systems , the noise response is a very sensitive probe of the emergence of a macroscopic response .taking each cell as a fluctuator , the convolution of the individual responses may lead to different signatures depending on the considered scale of the sample .seeds of bush bean ( phaseolus vulgaris l. cv .contender ) were sown into pots containing vermiculite as soil .the bean seedlings were grown in a growth chamber at with a cycle of 12 hours of light ( 40 photon m s ) , 12 hours of darkness .the plants were watered at the bottom of the pots every three days .the 21 days old plants were removed from the vermiculite .roots were carefully rinsed with water to remove vermiculite particles and then transferred in dilute kcl solution ( 120 mm ) for experiments .two electrodes were placed respectively at the surface of the root and the stem of the living plant .the electrodes connected to the stems are made of ag / agcl wires moistened with 200 mm kcl and wrapped in cellophane to provide appropriate contact with the plant surface .

Abstract: plants are sensitive to thermal and electrical effects ; yet the coupling of both , known as thermoelectricity , and its quantitative measurement in vegetal systems never were reported . we recorded the thermoelectric response of bean sprouts under various thermal conditions and stress . the obtained experimental data unambiguously demonstrate that a temperature difference between the roots and the leaves of a bean sprout induces a thermoelectric voltage between these two points . basing our analysis of the data on the force - flux formalism of linear response theory , we found that the strength of the vegetal equivalent to the thermoelectric coupling is one order of magnitude larger than that in the best thermoelectric materials . experimental data also show the importance of the thermal stress variation rate in the plant s electrophysiological response . therefore , thermoelectric effects are sufficiently important to partake in the complex and intertwined processes of energy and matter transport within plants .


a quantum state of a qubit can be remotely prepared with the help of a preshared nonclassical bipartite resource state and a classical communication channel .this is called remote state preparation ( rsp )  .the basic idea lies as follows  : alice and bob share a bipartite nonclassical state , for example , a maximally entangled qubit ( ebit ) in the ideal case . after a local measurement along certain direction at alice s side , alice sends bob one classical bit ( cbit ) , which tells bob whether or not to flip his qubit , in order to obtain the state as desired .several experiments for this scheme have been carried out , e.g. , by nuclear magnetic resonance ( nmr ) techniques  .  in the ideal case ,when the resource state is a maximally entangled state , the desired target state can be prepared perfectly at bob s side , i.e. , with fidelity equal to one . however , in practice , the resource state for rsp is usually a mixed one , thus we can not always complete a perfect preparation .the efficiency of successfully doing this task can be measured by rsp - fidelity  . since rsp can not be carried out without a nonclassical resource state , rsp - fidelity to some extentdescribes the quantumness " of the bipartite resource state , which is similar with quantum discord  .recently , a surprising discovery has been made , which connects the rsp - fidelity with the geometric measure of quantum discord ( gmqd )  .they find that the necessary condition for a resource state to afford non - zero fidelity of rsp is that it must have nonzero quantum discord , rather than nonzero entanglement .furthermore , they explicitly show that for a broad class of states , the rsp - fidelity is equal to the gmqd .hence their work links gmqd to an operational meaning .some works point out that quantum discord can in- crease via local operations  .for instance , a state with zero discord can be transformed into a state with non - zero discord by a local operation acting on the first qubit such that , , with .it can be verified that is a trace - preserving and completely positive map .this fact makes us conjecture that rsp - fidelity , which is accessible in experiment , may share the similar property of local increase .  in this paper, we study the influence of local operations on the rsp - fidelity .we prove that the rsp - fidelity does not increase for unital local operation .however , for nonunital local operation , such as local amplitude damping channel , we find that some resource states can be enhanced to increase the rsp - fidelity .we give the optimal parameter of symmetric local amplitude damping channel for enhancing bell - diagonal resource states .meanwhile , we observe some interesting dynamic behaviors of rsp - fidelity .rsp - fidelity under local decoherence has the effect of sudden change and can vanish at instant .particularly , the behavior of vanishing at instant indicates rsp - fidelity is more fragile than discord against decoherence , and therefore should take more enhancement .we organize the remains of this paper as follows . in sec .[ secii ] , we briefly review the rsp protocol and the connection between rsp - fidelity and gmqd . in sec .[ seciii ] , we study the influence of two types of local operations , the unital and the nonunital ones , on the rsp - fidelity. a necessary condition for local increase of rsp - fidelity is obtained .we also give a criterion of enhancibility for bell - diagonal states under a symmetric local amplitude damping channel , as well as the optimal parameter of the channel .finally , in sec .[ seciv ] , we summarize our results .in this section , we first introduce rsp based on the protocol in ref .. then we make a quantitative comparison between rsp - fidelity and gmqd .   the concrete rsp protocol that our paper deals with comes from pati  , where alice intends to remotely prepare a pure qubit state from an ensemble represented by a great circle on the bloch sphere using only one cbit and one preshared nonclassical bipartite resource state .the basic idea ( also described by daki _et al . _ ) is as follows : the target qubit , which is denoted as or a bloch vector , is chosen from the great circle of the bloch ball orthogonal to a unit vector .a nonclassical state is preshared as resource state by both parties . since in rspalice knows exactly what the target state is , she can choose such a unit vector to perform a local measurement along that would help to prepare the target state at bob s side with the highest achievable fidelity  .for instance , in the case when alice and bob have an singlet state as resource state , alice can perform a measurement along the direction and then send bob the measurement outcome as one cbit information . for ,bob applies a rotation about , and for , bob does nothing .after this , the resulting state on bob s side will be nothing but the target state .there is nothing special about sharing a singlet state as resource state .one can use any other maximally entangled states to achieve the same goal up to a change of measurement direction at alice s side . in general , when the resource state is not maximally entangled , the resulting state may differ from the target state . in order to evaluate the efficiency of rsp , daki _ et al . _ introduce the notion of payoff - function . in each run , by deliberately choosing the measurement direction , alice can optimize the payoff - function .we denote the optimized payoff - function by .the rsp - fidelity averages the optimized payoff - function and is minimized over all on bloch sphere , i.e. , the rsp - fidelity captures two most natural aspects , ( ) , for the resource state maximally entangled , while ( ) for the maximally mixed .  by representing a two - qubit resource state in terms of pauli matrices  , \label{eqbloch}\ ] ] where , are local bloch vectors , the rsp - fidelity can be written as here the coefficients form a real matrix denoted by , and the eigenvalues of . for simplicity, we will skip the derivation of this expression , one can consult ref . for details .for rsp with only classical resource , where , non - zero rsp - fidelity is impossible . since rsp - fidelity is a function of state , we can compare it with gmqd and this is what we are going to do in the next subsection .gmqd is introduced by the motivation for capturing total quantum correlation  .the original paper defines gmqd as the square of the hilbert - schmidt distance from the given state to the set of classical - quantum states , i.e. , where ^{1/2} ] .the parameter is responsible for a wide range of physical phenomena .there exist some time - dependent decoherence models , where is replaced by a time - varying function with a constant characterizing the speed of the processes .  for simplicity, we only consider the symmetric situation in which the damping rates on both sides are equal .in addition , we restrict the resource states for rsp to bell - diagonal states , which are diagonal in the bell basis ( , ) .a bell - diagonal state can also be expressed by pauli matrices as the condition for being a proper density matrix imposes that the point made of coordinates must lie inside the tetrahedron with the four bell states being the corners ( see fig .[ fig2](a ) )  .the bell - diagonal states may arise in a wide variety of physical situations and play an important role in quantum information processings  .moreover , there always exists a local transformation that can transform a given mixed state to the corresponding bell - diagonal form  .given an initial two - qubit state , its evolution under a amplitude damping channel can be modeled in the kraus form we can calculate the output of an arbitrary bell - diagonal state with the form of eq .( [ eqbelldiag ] ) under a symmetric amplitude damping channel with parameter , and arrive at .\end{split}\end{aligned}\ ] ] calculate both the rsp - fidelity and the gmqd for , we have , \end{split } \label{eqfunderch}\end{aligned}\ ] ] .\end{split } \label{eqdgunderch}\end{aligned}\ ] ]  .the initial state used here is given by eq .( [ eqbelldiag ] ) with , and .,scaledwidth=47.0% ]  in fig .[ fig1 ] , we plot both the rsp - fidelity and the gmqd of as functions of scaled time .the initial resource state used is given by eq .( [ eqbelldiag ] ) with , and . in this case , although the resource state is initially bell - diagonal , but it can not maintain the bell - diagonal form as long as it s evolution begins . this is why the curves of and do not always match .we can observe that is always satisfied , which coincides with the result in sec .[ seciib ] . andjust as we expected , we find both rsp - fidelity and gmqd can increase during the time evolution under certain conditions . also , due to the maximization procedures in eqs .( [ eqfunderch ] ) and ( [ eqdgunderch ] ) , both rsp - fidelity and gmqd have a _ sudden change _effect during the time evolution .the phenomena of sudden change in the dynamics of both gmqd and quantum discord have been discussed in detail in ref .yet most interestingly , rsp - fidelity vanishes at instant .this is quite different from discord and may show one different facet of rsp - fidelity from gmqd . as we know, when two qubits subject to independent markovian decoherence , the evolution of discord decays only in asymptotic time  . in this sense ,compared with quantum algorithms based only on quantum discord , the enhancement of rsp - fidelity is more worthy of consideration .but it should be pointed out , in the situation of non - markovian , quantum discord can vanishes , see  .   in the above subsection ,we have considered the symmetric local amplitude damping channel as decoherence noise , which lands us in a passive position .however since the amplitude damping channel is the quantum operation which we can experimentally control , we can impose it on the resource state for enhancing as our aim . in this subsection, we will give the criterion for testing the enhancibility of a bell - diagonal state under symmetric local amplitude damping channel , as well as the optimal parameter of such channel .for this purpose , we need to thoroughly discuss the expression of . since the maximization procedure in eq .( [ eqfunderch ] ) complicates the calculation , we divide the discussion into two cases , depending on the sign of .we set .to get rid of the maximum operator in eq .( [ eqfunderch ] ) , we need to note the sign of .when , by observing the spatial relation between the straight line segment ,  ] , we can easily express from eq .( [ eqfunderch ] ) in piecewise form , & q_1\leq q\leq 1 , \\\frac{1}{2}q ^2\left(c_1 ^ 2+c_2 ^ 2\right ) & 0\leq q < q_1 . \end{cases}\end{split } \label{eqfpiecewise}\ ] ] here is the the smaller root of the equation .subsequently , we treat , instead of , as the independent variable of . for  ] .while for , through an analysis of eq .( [ eqdf ] ) , we observe that is the only possible local maximum point of . by inserting into eq .( [ eqfpiecewise ] ) , the existence of a symmetric local amplitude damping channel of parametrer for which can be checked by   in the above discussion , we considered the situation of .when , holds for all value of .   when , the straight line segment ,  ] intersect only once at . thus by eq .( [ eqfunderch ] ) , the expressions of and in this case are the same with case 1 . and for $ ] , a short analysis of eq .( [ eqdf ] ) shows that is the only possible local maximum point of .hence we arrived at the same fomula as eq .( [ eqtest ] ) in case 1 for checking the existence of a symmetric local amplitude damping channel for enhancing ,   then , when , holds for all value of .   in the above, we have discussed the enhancibility of a bell - diagonal state for rsp in two different cases . as we can see , for both cases , we come to the same conclusion that if the condition eq .( [ eqtest ] ) holds , there exists a optimal symmetric local amplitude damping channel with parametric use which we can enhance the corresponding bell - diagonal state .  in fig .[ fig2](a ) , we have plotted the region that represents the bell - diagoal states satisfying the condition of eq .( [ eqtest ] ) . in fig .[ fig2](b ) , we depict the increment of rsp - fidelity of a one - dimensional class of bell - diagonal states before and after the action of the optimal symmetric local amplitude damping channel .as we can see , even the classical state with coordinates can be used for rsp ( with non - zero rsp - fidelity ) after a proper local operation .in this paper , we study the influence of local operations on the rsp - fidelity .we find that rsp - fidelity does not increase for any unital local operation , while the resource sates may be enhanced by suitable nonunital local operations .we use the symmetric local amplitude damping channel acting on bell - diagonal state to exemplify that nonunital channel under specific condition can enhance the fidelity of rsp .we also use the same example to show that rsp - fidelity can vanish at instant under local markovian decoherence , which is different from gmqd . at last , we give the optimal parameter of symmetric local amplitude damping channel for enhancing bell - diagonal resource states [ eq .( [ eqpopt ] ) ] , and the criterion for the enhancibility of bell - diagonal states [ eq .( [ eqtest ] ) ] .we hope that our research will help to promote the future investigations on the power of local operation for enhancing quantum information protocols .also we look forward to seeing more works of revealing the essence of increasing quantum correlation by local operation .this research is supported by the nnsf of china , grants no .

Abstract: remote state preparation ( rsp ) is a quantum information protocol which allows preparing a quantum state at a distant location with the help of a preshared nonclassical resource state and a classical channel . the efficiency of successfully doing this task can be represented by the rsp - fidelity of the resource state . in this paper , we study the influence on the rsp - fidelity by applying certain local operations on the resource state . we prove that rsp - fidelity does not increase for any unital local operation . however , for nonunital local operation , such as local amplitude damping channel , we find that some resource states can be enhanced to increase the rsp - fidelity . we give the optimal parameter of symmetric local amplitude damping channel for enhancing bell - diagonal resource states . in addition , we show rsp - fidelity can suddenly change or even vanish at instant under local decoherence .