Nicolas Vasselin

François Fages

Mixture Model-CMA-ES 2016 internship report

In the following, we report on our attempt to improve the CMA-ES global optimization algorithm based on two ideas: rst the use of Sobol's quasi-random low discrepancy numbers instead of pseudo-random numbers, second the design of an alternative to sequential restarts to dynamically adapt the population size, using a mixture model extension of CMA-ES (MM-CMA-ES). On the standard Coco benchmark for evaluating global stochastic optimization methods, the use of Sobol numbers shows a quite uniform improvement, as was already shown by [teytaud] last year. On the other hand, MM-CMA-ES does not show speed-up w.r.t. CMA-ES with IPOP restart strategy, even on objective functions with many local minima such as the Rastrigin function. The reason is the overhead in the number of evaluation of the objective functions, introduced by the MM strategy, and the very subtle eect of the adaptive step size strategy of CMA-ES to escape from the covering of several local minima by one (large) normal distribution. A rst approach to the CMA-ES would be Nikolaus Hansen's The CMA Evolution Strategy: A Tutorial.

Sobol's low-discrepancy quasi-random sequence intuitively "lls better" the space.

In the following results (appendices B and C), the implementation of [2] was used, with a broader initialization allowing for sampling in larger dimensions (see appendix A). Its computational cost proved not to be signicant. We see in the results that sobol-CMA-ES slightly outperforms CMA-ES on every benchmarking function, in every dimension, both when used without restarts, or with the IPOP restart strategy as described in [3] (as the rst restarting regime). Similar results were observed in [START_REF] Teytaud | Quasi-random numbers improve the CMA-ES on the BBOB testbed[END_REF]. CMA-ES does not adapt the size of its population in a run. This makes it perform poorly in most testbed functions, especially in higher dimensions and on functions with many local minima. The classical way to adaptively change the population size is by doing sequential, budget-restricted restarts of the CMA-ES and doubling its population size each time. MM-CMA-ES is an attempt at nding a better strategy to adapt the size of the population, by using a Mixture Model to explore the search space in a parallel way. The goal is to avoid restart-related information loss and to outperform IPOP-CMA-ES. Incidentally, it allows for greater parallelism and nding more local optima.

EM-CMA-ES

This strategy uses the Expectation-Maximization algorithm to cluster the selected points into a new number of villages. Here are its zoomed-out steps: The results are not as good as the basic CMA-ES strategy. Figure 2 shows the best evaluation found in y-axis, with respect to the number of target function evaluation in x-axis in log scale, evaluated over a sphere, Rastrigin's, and Rosenbrock's functions from left to right. EM-CMA-ES is signicantly slower over the sphere and Rosenbrock's function, but it is almost as good as CMA-ES on Rastrigin's function. This can be explained as follows: in EM-CMA-ES, when a village is reaching the bottom of a local minimum, few or none of its population will be selected to reach the next step, because the other villages will be in deeper pits of potentials. However, while the population of that village is not selected, it means that more than the half of the population of other villages will be remembered. As CMA-ES performs best when the better half of the population is remembered, the other villages will converge slower, because more of their population will be remembered and pulling them in more directions. The clustering of EM-CMA-ES was expected to cut short to hesitation between two local optima by splitting the distribution and exploring both optima at the same time. However, CMA-ES does not waste much time when covering two local minima, because of the step-size adaptation mechanism. The step-size adaptation uses a vector "path" variable p c , whose update is

p (g+1) c ← (1 -c c )p c + αC ( g) -1 2 m (g+1) -m (g) σ (g)
, where c c and α are some learning coecients. This path variable p c holds the memory of the means' movements, unaected by the step-size nor the covariance matrix. Therefore, its norm tells whether the distribution is going forward or staying in place. If it is going somewhere, i.e the mean is always displaced towards the same direction, then p c will grow. Besides, it means the distribution is following a slope toward a region of greater interest, and so it should move faster. If, on the contrary, the means don't move much between each generation, then p c will decay; besides, it means that the distribution is hovering over a minimum, and so it should converge. The actual step-size update is as follows:

σ (g+1) ← σ (g) exp(β( pc E N (0,I) -1)).
The step-size σ will quickly increase when σ is greater than its expected value, and quickly decrease when it's smaller. That's why CMA-ES doesn't lose much time when hovering over two minima: as the mean will stay roughly immobile between the two pits of potential, the step-size will decrease, reducing the coverage of both pits to focus in the region between them. Then, one better point in one or the other pit will drag the distribution in it. Thus, CMA-ES chooses very quickly and at random between two similar optima. Furthermore, when hovering above many local optima, CMA-ES is more likely (and even more as its population increases) to choose the better one, skipping the coslty exploration of all of them.

poison-CMA-ES

This strategy uses a small, constant per-village population, but a variable number of villages, and so a variable global population. It follows the CMA-ES steps for each villages, with the following dierences:

• After sampling, check in every village for the division criterion:

∀(i, j) ∈ [[1, µ]] 2 di→j d d (i:λ+j:λ) > d t
, where d i→j is the distance between x i and x j , d d is the largest eigenvalue of the covariance matrix times the step-size σ, i : λ and j : λ are the ranks of x i and x j , and d t is an arbitrary threshold.

• If the division criterion is met, split the village in two. Determine the distribution of the daughters using the two matching points as means (disregarding all the other points) and using a normal CMA-ES update, without the µ-rank update of the covariance matrix.

• After the distribution updates, check the fusion criterion between every villages:

m i -m j 2 max(d i d ,d j d ) < f t
, where m i is the mean of village i, and d i d is the dened as in the division criterion, and f t is another arbitrary threshold.

• If met, fuse the two villages by taking the arithmetic mean of their means m, covariance matrixes C, covariance paths p C , step-size path p σ , the geometric mean of their step-size σ.

• Do the poison update.

Note that while the population of each village is constant and equal, the global population is variable.

The poison update is done as follows : each village, at each step, leaves behind it a drop of poison, which is formed by the quadruplet (m, C, σ, f bestever), where f bestever is the best target function evaluation done so far by this village (or its ancestors). Then, every village v updates its toxicity level with respect to each poison drop p by adding to it the quantity

1 ( m v -m p * C v -C p ) 2 * |σ v -σ p | , only if f bestever p < f bestever v . If it
brings the toxicity level over the poison threshold p t , the village is paused and stored in a sleeping queue, where the villages are stored according to their f bestever. Finally, when there are no villages left, and if there are still some villages in the sleeping queue, the poison threshold p t is doubled, and every village in the sleeping queue that are now under the new threshold are awakened. This allows to dynamically adapt the poison threshold, preventing an early end of all villages.

The results are observable in E. We can see that on the low dimensions (2,3 and 5), the two algorithm are about as ecient. poison-CMA-ES is signicantly better on the function 5 Linear Slope, which tests the ability of the algorithm to quickly move a long distance by putting the optimum on the border of the domain of the interest. This is because splits allow the distributions to travel long distances signicantly faster than in the standard CMA strategy. We also observe overall better performances over the Gallagher functions (21 and 22) in low dimensions, which hints toward a better management of many local optima by poison-CMA-ES, which was the researched goal. However, this is not generalized to others such functions (Rastrigin). The current set of parameters of poison-CMA-ES is t only for lower dimensions. We can see its performances crumble on dimensions 10 and 20, except on function 5 Linear Slope. Note that poison-CMA-ES under another denition of complexity adapted to its parallel nature, poison-CMA-ES may outperform CMA-ES: under the hypothesis of roughly constant evaluation cost of the target function, a village-parallel implementation of poison-CMA-ES would actually be faster than the sequential IPOP-CMA-ES.

Conclusion

The use of a low-discrepancy quasi-random sequence such as Sobol's is a free increase in performances over all benchmarking function and in all dimensions.

The EM-CMA-ES currently needs too large a starting population and induces a slowdown of the search of interesting regions. However, the clustering algorithm is still roughly used; maybe a ner, more local one could circumvent such issues. Another potential improvement would be a budget management between the villages, giving priority to the most promising villages while keeping exploring new regions. The current set of parameters of poison-CMA-ES were empirically found based on low-dimension experiments. There is probably room for signicant improvement on that regard, especially on higher dimensions. Another way to gain performances may be a more economical way to adapt the poison threshold. This function needs to be called once with a negative value of n in order for it to initialize. Then, it will ll the x vector, starting from index 1 (and not 0 !), with the next vector in sobol's quasi-random sequence.

#

} f s c a n f ( fp , " d s a m_i \n " ) ; f o r ( k=1;k<=MAXDIM; k++){ f s c a n f ( fp ,"% l d %l d %l d ",& i ,&mdeg [ k ] ,& i p [ k ] ) ; f o r ( i =0; i <mdeg [ k ] ; i ++){ f s c a n f ( fp ,"% l d ",& i v [ k+i * MAXDIM] ) ; } f s c a n f ( fp , " \ n " ) ; } i n i t i a l i z e d =1; } i f ( * n < 0){ f o r ( k=1;k<=MAXDIM; k++) i x [ k ]=0; i n =0; i f ( i v [ 1 ] != 1) r e t u r n ; f a c =1.0/(1L << MAXBIT) ; f o r ( j =1,k=0; j<=MAXBIT; j ++,k+=MAXDIM) i u [ j ] = &i v [ k ] ; f o r ( k=1;k<=MAXDIM; k++) { f o r ( j =1;( unsigned ) j<=mdeg [ k ] ; j++) i u [ j ] [ k ] <<= (MAXBIT-j ) ; f o r ( j=mdeg [ k ]+1; j<=MAXBIT; j++) { ipp=i p [ k ] ; i=i u [ j-mdeg [ k ] ] [ k ] ; i ^= ( i >> mdeg [ k ] ) ; f o r ( l=mdeg [ k ] -1; l >=1; l --) { i f ( ipp & 1) i ^= i u [ j-l ] [ k ] ; ipp >>= 1 ; } i u [ j ] [ k]= i ; } } } e l s e { im=i n++; f o r ( j =1; j<=MAXBIT; j++) { i f ( ! ( im & 1 )
) break ; im >>= 1 ; } i f ( j > MAXBIT) FATAL("MAXBIT too s m a l l i n sobseq " , 0 , 0 , 0 ) ; im=(j -1) * MAXDIM; f o r ( k=1;k<=intMin ( * n ,MAXDIM) ; k++ All ,3 ] ] ; P r i n t [ " U n f i n i s h e d v i l l a g e s : " , Length [ a l m o s t R e s u l t s ] ] ; P r i n t [ " Terminated v i l l a g e s : " , t e r m i n a t e d V i l l a g e s ] ; r e s u l t s = Catenate [ { r e s u l t s , a l m o s t R e s u l t s } ] ; { r e s u l t s , c o u n t e v a l s } ] ;

) { i x [ k ] ^= i v [ im+k ] ; x [ k]= i x [ k ] * f a c ; } } } B sobol-CMA-ES vs CMA-ES
] := Module [ { \ [Mu] , wPrime , \ [Mu] e f f , w, c \ [ Sigma ] , d \ [ Sigma ] , cc , c1 , c \ [Mu] } , ( * \ [Mu]= Floor [ \ [ Lambda ] / 2 ] ; * ) I f [ \ [ Lambda ] == 1 , P r i n t [ " Warnin : \ [ Lambda ] = = 1 " ] ] ; \ [Mu] = \ [ Lambda ] ; wPrime = N[ Table [ Log [ ( \ [ Lambda ] + 1 ) / 2 ] -Log [ i ] , { i , \ [Mu ] } ] ] ; \ [Mu] e f f = (Sum [ wPrime [ [ i ] ] , { i , \ [Mu] } ] ^2 ) / Sum [ wPrime [ [ i ] ] ^2 , { i , \ [Mu ] } ] ; w = Table [ 0 , { i , \ [ Lambda ] } ] ; For [ i = 1 , i <= \ [Mu] , i ++, w [ [ i ] ] = wPrime [ [ i ] ] / Sum [ wPrime [ [ j ] ] , { j , \ [Mu ] } ] ] ; c \ [ Sigma ] = ( \ [Mu] e f f + 2 ) / ( n + \ [Mu] e f f + 5 ) ; d \ [ Sigma ] = 1 + 2 Max[ 0 , Sqrt [ Max[ 0 , \ [Mu] e f f -1 ] / ( n + 1 ) ] -1 ] + c \ [ Sigma ] ; cc = (4 + \ [Mu] e f f /n ) / ( n + 4 + 2 \ [Mu] e f f /n ) ; c1 = \ [ Alpha ] cov / ( ( n + 1 . 3 ) ^2 + \ [Mu] e f f ) ; c \ [Mu] = Min [ 1 -c1 , \ [ Alpha ] cov ( \ [Mu] e f f -2 + 1 / \ [Mu] e f f ) / ( ( n + 2)^2 + \ [ Alpha ] cov \ [Mu] e f f / 2 ) ] ; { \ [ Lambda ] , \ [Mu] , w, c \ [ Sigma ] , d \ [ Sigma ] , cc , c1 , c \ [Mu] , \ [Mu] e f f } ] ; ( * i n i t P a
; I f [N[ Norm [ sqC . sqC -C [ [ 1 ] ] ] ] > 10^-10, None , sqrMatWarnings + + ] ; ] ] ] ; {C [ [ 1 ] ] , B, D, sqC , invSqC , newTerminate } ] ; ( * updateEigensystem * ) samplePop [ { param_ , C_, M_, pC_, p \ [ Sigma ]_, \ [ Sigma ]_, terminate_ } ] := Module [ { z , y , x , f x } , Table [ z = RandomVariate [ norm , n ] ; y = C [ [ 4 ] ] . z ; x = M + \ [ Sigma ] y ; c o u n t e v
] ; { \ [ Lambda ] , \ [Mu] , w, c \ [ Sigma ] , d \ [ Sigma ] , cc , c1 , c \ [Mu] , \ [Mu] e f f } = param ; newC = Sum [w [ [ i ] ] c l u s t e r [ [ i , 5 , 2 , 1 ] ] , { i , \ [Mu] } ] ; ( * to be updated with updateEigenSystem * ) newM = Sum [w [ [ i ] ] c l u s t e r [ [ i , 5 , 3 ] ] , { i , \ [Mu ] } ] ; newPc = Sum [w [ [ i ] ] c l u s t e r [ [ i , 5 , 4 ] ] , { i , \ [Mu ] } ] ; newP \ [ Sigma ] = Sum [w [ [ i ] ] c l u s t e r [ [ i , 5 , 5 ] ] , { i , \ [Mu ] } ] ; new \ [ Sigma ] = GeometricMean
[ Sigma ] = (1 -c \ [ Sigma ] ) newP \ [ Sigma ] + Sqrt [ c \ [ Sigma ] (2 -c \ [ Sigma ] ) \ [Mu] e f f ] newC [ [ 5 ] ] . yw ; new \ [ Sigma ] = new \ [ Sigma ] Exp [ c \ [ Sigma ] / d \ [ Sigma ] (Norm [ newP \ [ Sigma ] ] / \ [ Chi ] n -1 ) ] ; newPc = (1 -cc ) newPc + Sqrt [ cc (2 -cc ) \ [Mu] e f f ] yw ; newC = (1 -c1 -c \ [Mu] Total [w ] ) newC [ [ 1 ] ] + c1 Transpose [ { newPc } ] . { newPc} + c \ [Mu] Sum [ w [ [ i ] ]

Figure 1 :

 1 Figure 1: Uniform random sampling (left) vs Sobol's sequence (right)

1 .

 1 Sampling according to each village's law and population 2. Ranking all the population, keep only the better half 3. Clustering of what's left using EM 4. Villages composition: weighted mean of previous villages distributions 5. Villages updates: cf CMA-ES The Mathematica code is provided in D.

Figure 2 :

 2 Figure 2: EM-CMA-ES (yellow) vs CMA-ES (blue)

Appendices [ 2 ]

 2 William H. Press; Saul A. Teukolsky; William T. Vetterling; Brian P. Flannery. Numerical recipes in C (2nd ed.): the art of scientic computing. Cambridge University Press New York, NY, USA c 1992,.SBN:0-521-43108-5 [3] Nikolaus Hansen. Benchmarking a BI-Population CMA-ES on the BBOB-2009 Function Testbed. ACM-GECCO Genetic and Evolutionary Computation Conference, Jul 2009, Mon-treal, Canada. 2009. <inria-00382093> A Sobol's quasirandom sequence C implementation

Figure 3 :

 3 Figure3: For each graph, corresponding to functions f1 to f24 in the Bbob/Coco framework, the x-axis is the run length for the CMA-ES in log-10 scale, whereas the y-axis is the run length in log-10 scale for the sobol-CMA-ES. When the points are above the diagonal, sobol-CMA-ES outperforms CMA-ES. 8

Figure 4 :

 4 Figure4: For each graph, corresponding to functions f1 to f24 in the Bbob/Coco framework, the x-axis is the run length for the IPOP-CMA-ES in log-10 scale, whereas the y-axis is the run length in log-10 scale for the sobol-IPOP-CMA-ES. When the points are above the diagonal, sobol-IPOP-CMA-ES outperforms IPOP-CMA-ES.

  r a m e t e r s * ) updateEigensystem [ { param_ , C_, M_, pC_, p \ [ Sigma ]_, \ [ Sigma ]_, terminate_ } ] := Module [ {B, D, sqC , invSqC , newTerminate } , newTerminate = F a l s e ; I f [ ! P o s i t i v e D e f i n i t e M a t r i x Q [C [ [ 1 ] ] ] , P r i n t [ " P o s i t i v e D e f i n i t e Matrix warning " ] ; newTerminate = True ; , {D, B} = N[ Eigensystem [C [ [ 1 ] ] ] ] ; I f [ Length [ Pick [D, U n i t i z e [ Im [D] ] , 1 ] ] > 0 , P r i n t [ " Complex e i g e n v a l u e s : " , D ] ; newTerminate = True ; , I f [ Length [ Pick [D, P o s i t i v e [D ] ] ] < n , P r i n t [ " Negative e i g e n v a l u e : " , D ] ; newTerminate = True ; , B = Transpose [B ] ; invSqC = B. DiagonalMatrix [ 1 / Sqrt [D ] ] . Transpose [B ] ; D = DiagonalMatrix [ Sqrt [D ] ] ; sqC = B.D

  a l s ++; f b e s t e v e r = Min [ f b e s t e v e r , f [ x ] ] ; Sow [ 1 -f b e s t e v e r / f I n i t ] ; { f [ x ] , z , y , x , {param , C, M, pC , p \ [ Sigma ] , \ [ Sigma ] , t e r m i n a t e }} , 2 param [ [ 1 ] ] ] ] ; ( * samplePop * ) u p d a t e D i s t r i b u t i o n [ c l u s t e r _ ] := Module [ { \ [ Lambda ] , \ [Mu] , w, c \ [ Sigma ] , d \ [ Sigma ] , cc , c1 , c \ [Mu] , \ [Mu] e f f , param , yw , newC , newM, newPc , newP \ [ Sigma ] , new \ [ Sigma ] , newTerminate } , ( * {param_ ,C_,M_,pC_, p \ [ Sigma ]_, \ [ Sigma ]_, terminate_ } * ) \ [ Lambda ] = Length [ c l u s t e r ] ; param = i n i t P a r a m e t e r s [ \ [ Lambda ]

  [ c l u s t e r [ [ All , 5 , 6 ] ] ] ; newC = updateEigensystem [ { param , {newC} , newM, newPc , newP \ [ Sigma ] , new \ [ Sigma ] , F a l s e } ] ; I f [ newC [ [ 6 ] ] , newTerminate = True ; , ( * begin CMAES-l i k e updates * ) yw = Sum [w [ [ i ] ] c l u s t e r [ [ i , 3 ] ] , { i , \ [Mu ] } ] ; newM += new \ [ Sigma ] yw ; newP \

  Transpose [ { c l u s t e r [ [ i , 3 ] ] } ] . { c l u s t e r [ [ i , 3 ] ] } , { i , \ [Mu ] } ] ; newTerminate = ( new \ [ Sigma ] < \ [ Sigma ] Min ) | | (Max [ c l u s t e r [ [ All , 1 ] ] ] -Min [ c l u s t e r [ [ All , 1 ] ] ] < stopTolFun ) ; ] ; I f [ newTerminate , t e r m i n a t e d V i l l a g e s ++]; {param , {newC} , newM, newPc , newP \ [ Sigma ] , new \ [ Sigma ] , newTerminate } ] ; ( * u p d a t e D i s t r i b u t i o n * ) run [ v i l l a g e s _ ] := Module [ { pop , c l u s t e r s , temp1 , temp2 , n e w V i l l a g e s } , ( * parameters , pop , {C, B,D, sqC , invSqC } ,M, pC , p \ [ Sigma ] , \ [ Sigma ] , t e r m i n a t e * ) ( * g e n e r a t e p o p u l a t i o n * ) pop = Sort [ F l a t t e n [ Map [ samplePop , v i l l a g e s ] , 1 ] ] ; pop = Take [ pop , Floor [ Length [ pop ] / 2 ] ] ; I f [ Depth [ pop [ [ All , 4 ] ] ] != 3 , P r i n t [ " I n v a l i d pop depth : " , Depth [ pop [ [ All , 4 ] ] ] ] ] ; temp1 = F i n d C l u s t e r s [ pop [ [ All , 4 ] ] , Method -> " GaussianMixture " ] ; temp1 = S e l e c t [ temp1 , Length [#] > 1 & ]; maxVillages = Max [ Length [ temp1 ] , maxVillages ] ; ( * rematch every p o i n t with i t s v i l l a g e i n f o * ) c l u s t e r s = { } ; For [ i = 1 , i <= Length [ temp1 ] , i ++, temp2 = { } ; For [ j = 1 , j <= Length [ temp1 [ [ i ] ] ] , j ++, temp2 = Append [ temp2 , F i r s t [ S e l e c t [ pop , # [ [ 4 ] ] == temp1 [ [ i , j ] ] & ] ] ] ; ] ; c l u s t e r s = Append [ c l u s t e r s , Sort [ temp2 ] ] ; ] ; n e w V i l l a g e s = Map [ u p d a t e D i s t r i b u t i o n , c l u s t e r s ] ; ( * take out the terminated v i l l a g e s * ) r e s u l t s = Catenate [ { r e s u l t s , S e l e c t [ newVillages , Last [#] & ] [ [ All , 3 ] ] } ] ; n e w V i l l a g e s = S e l e c t [ newVillages , ! Last [#] &] ; n e w V i l l a g e s = Map [ { # [ [ 1 ] ] , updateEigensystem [ # ] , # [ [ 3 ] ] , # [ [ 4 ] ] , # [ [ 5 ] ] , # [ [ 6 ] ] , #[[ 7 ] ] } &, n e w V i l l a g e s ] ; n e w V i l l a g e s = S e l e c t [ newVillages , ! # [ [ 2 , 6 ] ] & ] ; n e w V i l l a g e s ] ; ( * run * ) v i l l a g e I n i t = {{} , { I d e n t i t y M a t r i x [ n ] } , y I n i t ,

Figure 5 :

 5 Figure5: For each graph, corresponding to functions f1 to f24 in the Bbob/Coco framework, the x-axis is the run length for the sobol-IPOP-CMA-ES in log-10 scale, whereas the y-axis is the run length in log-10 scale for the poison-CMA-ES (which uses Sobol's quasi-random sequence too).When the points are above the diagonal, poison-CMA-ES outperforms sobol-IPOP-CMA-ES.

Table [

 [

	0 . , n ] ,
	Table [ 0 . , n ] , \ [ Sigma ] I n i t , F a l s e } ;
	v i l l a g e I n i t [ [ 1 ] ] = i n i t P a r a m e t e r s [ Floor [ \ [ Lambda ] I n i t / 2 ] ] ;
	v i l l a g e I n i t [ [ 2 ] ] = updateEigensystem [ v i l l a g e I n i t ] ;
	a l m o s t R e s u l t s =
	NestWhile [
	run , { v i l l a g e I n i t } , (# != {}) && ( c o u n t e v a l s < maxfeval ) & ] [ [

Sobol-CMA-ES

C sobol-IPOP-CMA-ES vs IPOP-CMA-ES


S. ATTARDO (ED.), SAGE ENCYCLOPEDIA OF HUMOR STUDIES (2014)

PASTICHE BY YEN-MAI TRAN-GERVAT, UNIVERSITÉ SORBONNE NOUVELLE -PARIS 3, CERC-EA 172

Pastiche is a French word which began to be used regularly in English in the 1880s and 1890s. In 18 th century France, it meant "a work of art imitating another author's style" (e.g., Dictionnaire de l'Académie, 5 th edition, 1798 : "En Peinture, on appelle Pastiches, Des tableaux où un Peintre a mêlé la manière d'un autre à la sienne, a emprunté son goût, son coloris, ses formes favorites"). The French word was itself derived from the Italian pasticcio, meaning (from late Latin antiquity) a kind of mixed pastry, then indicating a musical form made up of various parts imitated from other composers (in the same dictionary : "Mot emprunté des Italiens, et signifiant, dans la langue des Arts, Mélange, Composition mélée", then : "En Musique, on appelle Pastiche, un opéra composé de morceaux de différens Maîtres"). The English "pastiche" still shows these two influences: the "musical medley" meaning and more generally the idea of a "hodgepodge" or incongruous mixture derived directly from the Italian term; whereas the Oxford English Dictionary definition of pastiche as "an artistic work in a style that imitates that of another work, artist, or period" comes from French. It must be noted right away that pastiche is not necessarily humorous: for example, pastiche as a way of learning good writing by imitating great authors, as teachers in 19 th century French schools made their pupils practice doing, was a very serious matter. The same is true of some artistic instances of pastiche, depending on the conditions of their reception. For example, the cover illustration on Richard Dyer's Pastiche prompts the reader to interpret the 2002 film Far From Heaven, starring Julian Moore in a story that takes place in the 1950s, as a pastiche. In fact the film itself makes a very touching drama.

However, discussing Far From Heaven in 2003, Peter Bradshaw, critic for The Guardian newspaper, described how everybody had been enjoying the first five minutes of the film, which struck Bradshaw as a rather detailed "Hi-honey-I'm-home" 1950s skit, at the previous year's Venice Film Festival, before recounting how the giggling disappeared to be replaced by an absolute approval of every aspect of its elaboration. The "first giggling" effect and its link with the comic dimension of pastiche will be discussed below.

Elements of Definition

It is complex to define what pastiche is and how to distinguish it from other forms usually associated with it. Attempting to distinguish parody from related forms such as pastiche, Margaret Rose reviews different definitions of "pastiche" in English, underlining their negative critical connotations, due to the derivative aspect in art of imitation as opposed to originality. An example is Russell Sturgis' definition, which is general enough to extend to other art forms: "A work of art produced in deliberate imitation of another or several others, as of the works of a master taken together" (Sturgis, 1902, vol. 3, p. 73). This is close to Dyer's "pastiche is a kind of imitation that you are meant to know is an imitation" (Dyer, 2007, p. 1 ). However, the latter insists on the spectator's or reader's point of view, whereas the former focuses on the author's intention. Accordingly, the definition of humorous pastiche proposed here can refer either to an entire work or to parts of that work: a deliberate imitation of a style or genre, with an amusing effect.

Differentiating Pastiche from Parody

Pastiche is very close to parody and is regularly confused with it in the public mind, due to the complex and mixed nature of all parodic works, which are humorous remakes of linguistic or artistic performances, according to Margaret Rose. In most practical cases, the two words are used interchangeably about any work that humorously rewrites a recognizable source or style.

The French novelist Marcel Proust spoke alternatively of "pastiche" and parodie about his own imitations of 19th century novelists; in Italian, Umberto Eco (b. 1932) does the same about his Diario minimo (Misreadings). Similarly in Mikhail Bakhtin's polyphonic theory of enunciation, pastiche and parody are effectively the same thing: the presence or imitation of another discourse in one's own speech.

Clear landmarks for differentiating pastiche from parody are nevertheless given by French critic Gérard Genette, who attempts to distinguish various categories of "hypertextuality," the word he proposed for "any relationship uniting a text B (which I shall call hypertext) to an earlier text A (I shall, of course, call it the hypotext), upon which it is grafted in a manner that is not that of commentary." (Genette, 1997, p. 5) This structuralist approach is not devoid of humor itself, and Genette was aware that it could be unsatisfactory for the analysis of "general parody"; but it is a useful basis for defining pastiche. Setting aside the subtleties of Genette's six hypertextual categories (he himself admits they are bound to dissolve when applied to any complex work), two kinds of "relations" between texts emerge, transformation and imitation. In fact these are the principal differences between parody and pastiche. Parody is predominantly a transformative technique (Linda Hutcheon terms it "repetition with critical distance, which marks difference rather than similarity" [Hutcheon, 1985: 6]). Pastiche however is an imitative technique, characterized by marking similarities rather than differences.

A good illustration of this distinction is provided by Martin Rowson's graphic novel (cartoon) version of Laurence Sterne's early novel, Tristram Shandy (1759-67; 1996). Adapting Sterne's own playful rewriting of various books, the cartoonist uses parody and pastiche as two clearly different forms. His "plates" illustrating the story of Slawkenbergius (adapting Book IV of Tristram Shandy, a tale about a long-nosed traveler) are respectful parodies of well-known pictures-successively Albrecht Dürer's Knight, Death and the Devil (1513), William Hogarth's Southwark Fair (1733/4) and Aubrey Beardsley (1876-98)'s The Eyes of Herod-whose comic meaning relies on slight changes of content arising from the story being told. This is parody.

Later on, to access the transposed version of Book V, when "some dangerous nerd" has downloaded the entire text of Tristram Shandy, the "reader" inside Rowson's graphic novel has to hack into this digitalized text in order to "find out what happens next." But this operation sets off a huge "bug" which generates alternative versions of Sterne's story, each in the style of a 20 th century novelist or movement and each illustrated with a caricature of the author and a pastiched speech bubble. Martin Amis, Raymond Chandler's Philip Marlowe, South American magical realism, D. H. Lawrence, and T. S. Eliot are all imitated. Although each stylized story-bit still contains elements linking it with Tristram Shandy (e.g. characters' names), its comic effect derives from pastiche since it relies on its resemblance to its "hypotext" rather than on transformation of content or context.

Pastiche and Caricature in the Past

In the 17 th and 18 th centuries, pastiche of the mock-heroic variety was more highly regarded than caricature (also called burlesque travesty), because its principle was to preserve the dignity of the noble genre and high style being imitated, even though applied to a trifling incident. The canonical example of such a heroicomical poem is Alexander Pope's The Rape of the Lock (1712).

Another noted exponent was novelist Henry Fielding, whose writing included what he called "burlesque in diction" (preface to Joseph Andrews, 1742) purely for entertainment, for example in heroic descriptions of battles involving dogs or drunkards. Nowadays this would be called "mock heroic pastiche" and its mainly imitative principle bears out Fielding's conception of the novel as a "comic epic in prose." However Fielding condemned caricatura and its exaggerated satiric deformation as contradicting his aim of drawing comic characters from nature.

During periods of intense renewal of forms and genres such as occurred during the Romantic period (late 18 th and early 19 th centuries), pastiche was a very useful, hence widespread form of writing. Paul Aaron demonstrates how the French Romantics used pastiche to explore all kinds of ironic or eccentric forms to condemn ancient literary forms and compromise new ones. For the authors of the revolutionary generation that discovered the notion of personal originality in art, criticizing outworn classical forms by imitating them was a way of experimenting with what a century later was theorized by the Russian formalists as the "laying bare" of mechanical devices.

Pastiche and Satire

Besides parody, satire is commonly associated with pastiche. Although the imitation in pastiche must be recognizable (or else it could not be identified as pastiche), satirical exaggeration of the model's typical traits may tend toward caricature. Then, satire is aimed at the hypotext itself, whose "saturation" (Gérard Genette's term) with these traits is mocked by copying and exhibiting them to the reader. The difference between pastiche and caricature here is a matter of degree: pastiche is more neutral and less satirical in purpose. In the Rowson example above, Martin Amis is pastiched: his stylistic habits are playfully summed up in one speech bubble. But on the next page, T. S. Eliot is the object of a caricature: his style is ridiculed into being nothing more than meaningless mumbles. Martin Rowson publicly declared his irritation with Eliot and also adapted The Waste Land as a satirical graphic transposition. In Far From Heaven, the film mentioned earlier, the critic's first five minutes of "giggles" result from the same mechanism: at first sight, Todd Haynes' reconstruction of a 1950s aesthetic passes for a caricature of past narrative forms saturated with easily recognizable traits. But then, Dyer points out, the story reveals the re-creation to be meaningful and touching-a serious pastiche.

Is Pastiche "Blank Parody"?

The relative neutrality of pastiche, compared to both parody (in the transforming relationship) and caricature (in the imitative relationship), led postmodernist scholar Frederic Jameson to regard pastiche as "blank parody." Showing how the concept of a norm in postmodern art and literature is replaced by "private styles and mannerisms," Jameson pessimistically regards pastiche as similarly replacing value-laden parody. He calls it "blank parody, a statue with blind eyeballs" (1991:17). It may, he concedes, involve some humor, being "at the least compatible with addiction-with a whole historically original consumers' appetite for a world transformed into sheer images of itself and for pseudoevents." In other words, post-modern pastiche is an empty form because it is the perpetual reflection of a self-centered consumer society which has lost its sense of history.

Nevertheless, today pastiche often remains one of the conditions for a satiric parody to be fully effective, serving as the background against which the parody takes place. For instance, when the Reduced Shakespeare Company (the name parodies the Royal Shakespeare Company) gave an abridged version of Titus Andronicus in the form of a culinary TV program, that type of show was pastiched without any evident critical intent, while Shakespeare's play was very funnily parodied. Satire here was not aimed at cooking programs but at the accumulation of violent deaths in the Bard's first tragedy. The parodic RSC imitated a popular form to remind their public joyfully about Shakespearian (or Jacobean) aesthetics of cruelty. In this case, pastiche actually helps maintain historical and cultural heritage.

Playful rather than satirical tone underlying humorous pastiche is also exemplified by American Bill Watterson's comic strip Calvin andHobbes (1985-1995). Here the reader is invited to share the viewpoint of a little boy named Calvin (after John Calvin, 16th century French Reformation theologian and paired with a stuffed tiger named after Thomas Hobbes, 17th century English political philosopher). When Calvin's mother tries to make him eat his dinner, for example, what we see through the boy's eyes is a monstrous alien from science fiction comics or TV shows. This is graphical pastiche and its humorous effect relies on reusing visual codes associated with another genre (sci-fi) to depict a small boy's fantasies. Through pastiched images that are part of each generation's common cultural background, the adult reader is put back in touch with his or her own former childlike imaginings. The young reader may also recognize his or her own fantasies and perhaps, by identifying the imitation as such, may apprehend them differently with humorous distance (distancing) and a knowing amusement.

Humorous Pastiche as Imitation

Since pastiche is based on the imitative principle, a final question is whether imitation itself provokes laughter or smiling, independent of the transformations common to pastiche and parody. Gérard Genette believed it did and French philosopher Henri Bergson saw laughter as generally arising from "something mechanical encrusted upon the living" He cited imitation as one instance of this "law" of the comic: "gestures, at which we never dreamt of laughing, become laughable when imitated by another individual. [ . . . ] To imitate any one is to bring out the element of automatism he has allowed to creep into his person." (Bergson [1905(Bergson [ ] 2008 p. 22) p. 22) Moreover, for Bergson, this was "the very essence of the ludicrous": a "deflection of life towards the mechanical" that he believed was the real cause of laughter.

See also

  Bergson's Theory of Humor; Burlesque; Caricature; Cartoons; Comic Books; Comic Strips; Exaggeration; Genres and Styles of Comedy; History of Humor, Early Modern Europe; History of Humor, Modern and Contemporary Europe; History Of Humor, Nineteenth Century Europe; Lampoon; Mock Epic; Parody; Satire; Spoofing; Travesty FURTHER READINGS Aaron, P. (2008). Histoire du pastiche. Le Pastiche littéraire français de la Renaissance à nos jours [History of Pastiche. Literary French pastiche from Renaissance to our times]. Paris: Presses Universitaires de France. Bergson, H. [1905] (2008). Laughter: An Essay on the Meaning of the Comic. Transl. Cloudesley Brereton and Fred Roswell. Orig. ed., Le Rire. Essai sur la signification du comique, Paris: Felix Alcan, 1900. Rockville MD: Arc Manor.


Stefan Mehedinteanu
email: s.mehedinteanu@gmail.com

The Timeline of Universe Based On the Gravitons Release Due of the Merging of the Primordial Micro-black Holes

Keywords: LGO, Inflation, reheating, gravitational charges, gravitonmass, quarks, color gluons, micro black holes (  )BHs merging, epochs, Schwinger effect, BH nuclei of galaxies, SM

Based on the recently LGO discovery, respectively the merging of two black-holes as the base of the space-time deformation, I will advance another scenario for Universe creation and its evolution, respectively, when are created a huge number 62 10  of Microblack-holes (micro virtual black holes BHs  ) as generated due of Quantum fluctuations at Reheating (at 10 11 GeV ;10 -29 s) , and which further, during theirs merging its release gravitons which deform the space-time. Thus, at Confinement the BHs  become, firstly, via Quarks Gluons Plasma (QGP), the quarks of nucleons, and later, another's fractions becomes electrons (also via QGP) and others leptons and bosons (

), and very later Black Holes (BH) as the nuclei of galaxies. Also, is proposed a new test for the finally proof of the model as the deduction of the lifetime value of the   -decay of the free neutron. A correspondence it was established with the recent Standard Model Axion Seesaw Higgs portal inflation (SM*A*S*H) which offers a self-contained description of particle physics from the electroweak scale to the Planck scale and of cosmology from inflation until today.

Introduction

The Universe beginning is when are created a huge number 62 [START_REF] Guth | [END_REF]



of Micro-black-holes (micro-virtual black holes-BH  ) as been produced due of Quantum fluctuations [1a] at Electroweak epoch at 10 11 GeV ;10 -29 s, and which during theirs merging release gravitons which deform the space-time. At Confinement epoch ( V=2GeV) these ex-Micro-black-holes as becoming via Quarks Gluons Plasma (QGP)

  e e -pairs (quarks) which gives an external electrical field ( E ) [1a], that produces later at the interface between normal phase and the superconducting phase a superconductor magnetic field ( B ) generated by the free photons-color gluons (also radiated by BHs  ) condensate a Meissner effect. Scientists first caught wind of this strange phenomenon when they pointed the European Southern Observatory's Very Large Telescope (VLT) at the neutron star RX J1856.5-3754, which has a magnetic field billions of times stronger than our sun's, reports Popular Mechanics. They noticed that there was a high linear polarization of light waves in the empty space around the star. This is unusual because, according to conventional relativity at least, light should pass freely through a vacuum without being altered. But in this instance it wasn't passing freely at all. The virtual particles that result from this interaction are not really particles at all, but rather fluctuations in quantum fields. Though they're merely "virtual," the fluctuations nevertheless act like particles in that they can affect light, that it could be our proposal as BH  pairs. Virtual particles are often popularly described as coming in pairs, a particle and antiparticle, which can be of any kind. These pairs exist for an extremely short time, and then mutually annihilate. In some cases, however, it is possible to boost the pair (like  W bosons ) apart using external energy so that they avoid annihilation and become actual particles. In other words the Quantum is a real world. After the discovery of quark ) (q jets in 1975 in q q e e     at SLAC, detailed studies in understanding the hadronisation process, and hence the energy-momentum profiles of the quark jets, were initiated in 1977 by Feynman and Field. These , a jet  3 like in PETRA, DESY experiments, for more details see [1b]. These g q q flux tube is generated by the electrical field E by Schwinger effect or    e e collision, or as at SLAC, that it produces at the interface between normal phase and the superconducting phase, a decay of the superconductor magnetic field ( B ) generated by the color gluons condensate components like the Meissner effect. The inverse value of the penetration length of B could be equivalent with  W bosons at high densities, and the effective color gluon mass existing in nucleons at QCD. The first evidence for the existence of quarks came in 1968, in deep inelastic scattering experiments at the Stanford Linear Accelerator Center (SLAC). Another's fractions of BHs  merging will becomes electrons and, respectively BH's nuclei of galaxies, these later producing further Universe expansion. BHs  merging the timeline of Universe First of all, I present some of known data. Thus, in [1a] it is therefore assumed, that -potential energy‖ caused by gravitation and -kinetic energy‖ caused by expansion of the Universe are equal to each other (using the relations ct R U  and

How determine the gravitons production due of

2 c M E U U  ): U U U U U E t R G E t c G c M R GM 4 5 5 2 2      , ] [ 10
       G c m
In Inflation models [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF], the scale leaving the horizon at a given epoch is directly related to the number ) ( N of e -folds of slow-roll inflation that occur after the epoch of horizon exit. Indeed, since H

 dN k d  
as of eq.( 46) from [2a], and therefore ) ( ) ln(

 N k k end  , or, ] [m ke k N end 
where end k is the scale leaving the horizon at the end of slow-roll inflation, or usually ] [

1 1 m k k end    , the correct equation being ] [ 1   m e k k N end . When the wavelength ]) [ ( 1 m k  is large compared to the Hubble length ]) [ ( 1 m H 
, the distance that light can travel in a Hubble time becomes small compared to the wavelength, and hence all motion is very slow and the pattern is essentially frozen in. In this new scenario of Universe evolution are reproduced of the known data. Also, along the entire Universe dynamics is verified the Einstein formula

2 mc E 
, or when all the created photons transform in mass of the particles. Since, the FLRW metric of the universe must be of the form ds is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature, or for small commoving time aHc dt 1



, we can consider the distance as

end a a ds L   
, so the volume is given by:

] [ 1 ) ( 3 4 s m c a V
matter  In other words, this model of particle creation by trans-Planckian physics results in a significant part of the present total energy density of matter in the Universe being contained in gravitons with energies P M that is not compatible with the observed behavior of ) (t a . A second example is de Sitter space which contains an event horizon. In this case the temperature T is proportional to the Hubble parameter H , i.e. H T  , such a conclusion being used by author in [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF] to calculate the evolution of Universe. To estimate the horizon entry we use some derivations done in [START_REF] Mehedinteanu | Numerical Analysis Of Pairs Creation By Schwinger Effect In Nucleons And The Beta-Decay Process Acceleration[END_REF]. During Universe evolution [2a], the horizon leave is when

1   leave leave leave H k a , ] [ 10 27 1 1 m H k leave leave      , s c H t leave leave 36 1 10 3 . 3      at the Electroweak epoch
Here, the Hubble constant is defined as resulting from the equations: a we obtain for outside the object (BH, planets, stars etc.)

] [ ] [ 2 2 3 4 3 4 3 2 3 2 2 2 2           m n a c G a c GE a c G a GM s H a a g g   ; (1)
, where: to match the iterations cycle:

g g E n   ;   g P g merging at n n     ;   3 4 3 a M  ; 2 Mc E  ; BH BH m n M    ; 1   H R .
      ; when 1   leave leave leave H k a , kg M U 53 10 2 . 2   ; 2 c m c BH C     ; 62 61 3 1 10 1       P available BH n V H n  , First step
N a H R T k m end end B g        1    .
At BHs merging, the curvature radius R from eq. ( 2) with the number of merging nucleons during which release gravitons as

merging at n  , Sch end r m H R       ] [ 10 3 . 5 3 1
.

The gravitons release from Earth as a gravitational wave

Therefore, the new horizon leave as gravitational wave (GW) is just when the gravitons escape from the Schwarzschild radius (the Universe is a viewed as an inverse big black hole) :

1   leave leave leave H k a , or ] [ 10 4 . 7 3 1 1 m r H k Schw leave leave        .
Now, the new horizon-entry is when the wave length to match the iterations cycle: The strain at Earth surface Now, based on eq. ( 1) we can derive for the G-wave effect in the deformation (strain) of the space-time between Earth and a detector site by using the gravitational pressure due of gravity charges on the area of Schwarzschild radius Schw r , we have:

N a H R T k m end end B g        1    .
18 3 _ 4 2 2 10 24 . 1 3 4               GW end Schw g g Schw a c r n G R r   Separately, ] [ 10 86 . 8 2 3 2 m c M G r Earth Schw     
, we have

       Earth Schw Earth Schw R r R r
, so, in both cases the strain is around

9 10 1 . 1     R r Schw 
, that is near equally with strain as light bending. The average magnitude of the electric field (negative charge) in the event horizon of a micro-black-hole is like that of the model electron given in [START_REF] Abdo | Discovery of TeV Gamma-Ray Emission from the Cygnus Region of the Galaxy[END_REF], and where the inside -trapped‖ photon is similar with the -absorbed‖ photon from thermal energy V in case of

BH 
particle, or in other words the electron is a decaying BH  particle, see below equation ( 4).

4 0 6   c E     , it results ] [ 10 9 . 2 50 C N E     , for BH  particle of ] [ 10
m R  , s c R c H t end end 12 1 10 3 . 3      , with ] [ 10 20 1 1 m H k H EW laeve leae       we found 2 . 48  N
to match the iterations cycle:

N a H R T k m end end B g        1    . 3 10 9   T , ] [ 10 2 . 2 7 _ m BH C     , Hz k c end 8 1 10    
The number of gravitons that has been released following BHs  merging is only In other words the contribution to the space-time deformation is due of , all correspond to the situation at cosmic microwave background radiation (CMBR) time. Since, in the allsky map (pattern) of the CMB appear some voids, these can represent the BHs  merging (clustering) which emit gravitational radiation, and the rest is due of the thermal radiation at Recombination as following of the photon decoupling.

   

The formation of the Primordial BHs galaxies nuclei by BHs

 merging Thus, following author model for LGO measurement interpretation [START_REF] Mehedinteanu | The Quantization of the Gravitational Charges At the Origin of the Gravitational Waves From a Binary Black Hole Merger[END_REF]

] [ 111m R  , s c R c H t end end 7 1 10 3 . 3       , with ] [ 10 2 . 2 7 _ 1 1 m k H BH C laeve leae         we found 5 . 20  N
to match the iterations cycle:

N a H R T k m end end B g        1    . 4 10 7 . 1   T , ] [ 10 26 . 1 7 _ m mgBH C     , Hz k c end 6 1 10    
The number of gravitons that has been released following BHs  merging into primordial galaxies is only

N a H R T k m end end B g        1    , K T 9 10   , ] [ 10 8 . 1 6 _ m mgBH C    , Hz k c end 3 1 10     
The number of gravitons that has been released during BHs  merging into a primordial galaxies BH nuclei is

59 26 _ _ 10 10    mgBHs GW GW BH mg E n
, these generate the increase of curvature radius of the object (Universe) to R . To note, that this process it seems to be equivalent with the use of the cosmological constant in CDM  model (but is not necessary to explicit it use-Einstein was right!) .

The strain at LIGO site Now, based on eq. ( 1) we can derive for the G-wave effect in the deformation (strain) of the space-time between BH and LIGO site by using the gravitational pressure due of gravity charges on the area of Schwarzschild radius Schw r of BH, respectively with the values calculated in section 5, we have:

72 3 _ _ 4 2 2 10 6 . 1 3 4               GW mgBH end Schw g g Schw a c r n G R r  
, where J a

           LIGO Schw LIGO Schw R r R r
, so, in both cases at (LIGO laser) the strain is around

36 10 3 . 1     R r Schw 
, which is undetectable.

The Confinement into nucleons

In the dual-superconductor ( B E  ) picture for the QCD vacuum, the squeezing of theelectric flux ( E ) between quarks (as decayed from BH  particles)is realized by the dual Meissner effect, as the result of condensation(as a solenoidal electric current) of soft photons as color gluons (bosons) radiated by BH  particles, which is the dual version of the electric charge as the Cooper pair. The order parameter is the vacuum expectation value of the creation operator of gluons (like a condensate under a critical temperature c T !), such as e e for Cooper pairing of electrons in superconductors. The color confinement is based on the analogy between the superconductor and the QCD vacuum. In the superconductor, magnetic field is excluded due to the Meissner effect, which is caused by Cooper-pair condensation. As the result, the magnetic flux is squeezed like the Abrikosov vortex in the type II superconductor [START_REF] Suganuma | Dual Higgs Mechanism for Quarks in Hadrons[END_REF]. On the other hand, the color-electric flux is excluded in the QCD vacuum, and therefore the squeezed colorflux tube is formed between color sources. Thus, these two systems are quite similar, and can be regarded as the dual version each other. The color-electric field is then excluded in the QCD vacuum through the dual Meissner effect, and is squeezed between color sources to form the hadron flux tube. Therefore, in case of superconductors an external magnetic flux decreases exponentially into the superconductor, penetrating a distance of the order  . This distance is called the London penetration depth. In a dual superconductor the roles of magnetic and electric fields are exchanged and the Meissner effect tries to expel electric field lines. Quarks and antiquarks carry opposite color charges, and for a quark-antiquark pair the 'electric' field lines run from the quark to the antiquark. If the quark-antiquark pair is immersed in a dual superconductor, then the electric field lines get compressed to a flux tube. The energy associated to the tube is proportional to its length, and the potential energy of the quark-antiquark is proportional to their separation. As resulting from [1a] this Lorenz force appears to be necessary in order to equilibrate the gravity charges (gravitons) embedded in the quarks viewed as ex-micro-black-holes, and these gravitons themselves -deforms‖ the space-time following Einstein-Friedman equation. So, in this paper we will continue to follow the development of references [8a; 8b; 23a; 23b], especially, as regarding the single flux-tube solution in the dual Ginzburg-Landau (DGL) theory. Hadrons, along with the valence quarks ) ( v q (white) that contribute to their quantum numbers, contain virtual quark-antiquark ) ( q q pairs known as sea quarks ) ( s q , see figure 1.b. Sea quarks ( R R ) form when a gluon of the hadron's color field splits; this process also works in reverse in that the annihilation of two sea quarks produces a gluon. Free particles have a color charge of zero: baryons are composed of three quarks, but the individual quarks can have red (R), green (G), or blue (B) charges, or negatives. ) of valence quarks ( u ), (G is the gluon field strength tensor). These are "colorless" combinations. Top: Color charge has "ternary neutral states" as well as binary neutrality (analogous to electric charge).

The application of DGL theory to the quark-gluon-plasma (QGP) physics in the ultrarelativistic heavy-ion collisions, or in the early universe, where, a new scenario of the QGP formation via the annihilation of color-electric flux tubes based on the attractive force between them is proposed in [START_REF] Suganuma | Dual Higgs Mechanism for Quarks in Hadrons[END_REF]. The QGP phase is characterized as the deconfinement and the chiral-symmetry restoration. There, for the same flux tubes with opposite flux direction (e.g.

R R  and R R  ), one finds 2 1 Q Q   i.e. 3 2 2 1 e Q Q  
, so that these flux tubes are attracted each other. It should be noted that they would be annihilated into dynamical gluons in this case. For the different flux tubes satisfying 0

2 1  Q Q (e.g. R R  and B B  ), one finds 6 2 2 1 e Q Q  
, so that these flux tubes are attractive. In this case, they would be unified into a single flux tube (similar to G G  flux tube), [8b]. The effect of the potential B is the same as shifting the effective mass

) 1 2 ( 2 4 2 4 2       j c qB c m c m e 
for fermions for each Landau level.

Where

N a H R T k m end end B g        1    .
The

m gluon C qq C            , see below.
Because the magnetic induction of the color magnetic gluons current which is powered by electric field given by a pair of quarks ( 0 H ), , when the induction is

1 15 ] [ 10 2 ) ( c H T B     , ] [ 10 6 23 C N cB E   
In the case of a homogeneous potential directed along the z-axis [START_REF] Grasso | Potential s in the Early Universe[END_REF], the Einstein stressenergy tensor is:

   8 2 2 0 33 22 11 00 B c T T T T B       ; 0 0  i T
, where ] [ , with  e pair giving E as:

2 2 mc mc c e ec T k C C L B             
, and accounting that the inverse of the penetration length

C    .
Also, the interaction energy at interface

B E  is: ] [ 8 2 2 0 J V V B c V vol B vol        , (2.b) 3 8 ) 4 ( 2 C C c C vol V      
, at Compton length equally with the penetration length

   C , that results 3 0 2 ) ( ) (   e C V E   (2.c) With gluons V   as above is obtained ] [ 10 98 . 1 15 T c E B q q   
, where 23 10 9 .

 

q q E with eq. (2.a), that are identically with the above values, indubitable meaning that this force creates the spacetime curvature and this is equilibrated by the gravity charge, see below. With equation ( 4).

4 0 6   c E     , it results ] [ 10 6 . 3 23 C N E     , for q q particle of ] [ 10 7 . 4 16 m C    
, that is near the values calculated before by both methods.

The electrons production

Also, the further Universe expansion could be when the

      
. To mention that only this kind of dataset match the model.

b) A strong prove of the model-the free neutron decay

In the following, we will use some results of section



-which after the release of an electron that it getting the final beta energy as been equally to the out of barrier turning point after the tunneling, and accounting for the nucleons interactions (shell-energy levels). The number of assaults of the barrier, like in Gamow theory [START_REF] Mehedinteanu | On the numerical analysis of decay rate enhancement in metallic environment[END_REF]21] is

inner b a R v n 
; where the velocity is

  s m m v b 8 2 1 10 3 . 2 2    
, where, the inner radius of the barrier is

] [ 10 5 . 3 17 m b R inner    
, see below. For only one of the three vortex-flux tubes ( g q q ) we have:

GeV J m eB 25 ] [ 10 4 9       
, with the above ) (B which is obtained from eq.(1.a) with the resultant potential GeV V 81  , that corresponds to GeV m q q 29  from 4.1a, the energy of the particle for the first Landau level (as above), and we can see that it results to be equally with 3 1  rest mass of the  W , that resulting In case of WKB [START_REF] Mehedinteanu | On the numerical analysis of decay rate enhancement in metallic environment[END_REF], the transmission coefficient is r

Q V m T     2 2
, and the decay constant

T a e n    .
For the thick barrier the transmission coefficient is

b mQ v Qb T   2 2 2     ;
, where, the kinetic energy of the particle after the barrier at b

                            E E E e c m d P cr Compton 2 exp 2 exp exp 3 2  
The emission (transmission through barrier) is sufficient for observation when

cr E E  , with 2 2 1 mc Q  , results C b m cb T    2 2    , or  2 b d b 
. Now, by using the Schwinger effect as in section 2.1 , the number of  W pairs produced inside the nucleon (more inside of the only one resultant flux tube , see figures 1.a; 1.b) due of the potential resultant

) ( 3 g q q vortex u u   of GeV V 80  , results as   E E c E E R cr cr      exp * ) 8 )( ( ) ( 1 3 4 2
, the rate per unit volume of pair creation R in a constant and uniform electric field of strength E , when this electric field E is induced by quarks pairs as ex-Micro-black-holes, or 1



, it results for beta-decay. In our understanding, the created electron takes the energy at the turning point out of the barrier equally with the electron itself for unbounded neutrons, or that of the binding energy of nucleon in isotope nucleus, when it passes the barrier of gluon condensate characterized by an quantum tunneling suppression given as:

a vol n s V V R s R      1 18 10 3 . 2 , where ] 1 [ 10 2 3 71 s m V R   and the volume is b C vol V m V      ] [ 10
  22 10 3 . 7 exp       
, where, as the lifetime of  W being s as the string strength. Thus, the probability (rate) to produce

   e W
, into a more simple way-without the external interactions of the neutron (free-not bounded), is given as: W is the same as to be the necessarily energy to traverse the gluonic barrier, when it decays into  e at the end.

s s s E RV 612 ] [ 582 10 7 . 1 ) exp( 2 1 1 3             , that

The free neutron decay

Consequently, for the   decay process, the energy combines well with the existing one, that releasing an electron which penetrates the barrier:

e e u W W u d           ) 3 3 ( ) 3 2 ( ) 3 3 ( ) 3 1 ( e e e u e e e d          , since    e W
, and

   e W
In case of   decay, it can only happen inside nuclei when the daughter nucleus has a greater binding energy (and therefore a lower total energy) than the mother nucleus. The difference between these energies goes into the reaction of converting a proton into a neutron, a positron and a neutrino and into the kinetic energy of these particles. Thus, an opposite process to the above negative beta decay,   decay of nuclei (only bounded proton) when m  .

          , or, ) 3 3 (

A Correspondence with the recently SM*A*S*H model

Moreover, the SM cannot generate the primordial ination needed to solve the horizon and atness problems of the Universe, as well as to explain the statistically isotropic, Gaussian and nearly scale invariant fluctuations of the CMB. The SM also lacks enough CP violation to explain why the Universe contains a larger fraction of baryonic matter than of antimatter. Aside from these three problems at the interface between particle physics and cosmology, the SM suffers from a variety of intrinsic naturalness issues. In particular, the neutrino masses are disparagingly smaller than any physics scale in the SM and, similarly, the strong CP problem states that the  parameter of quantum chromodynamics (QCD) is constrained from measurements of the neutron electric dipole moment to lie below an unexpectedly small value.

In the new model [START_REF] Ballesteros | Unifying inflation with the axion, dark matter, baryogenesis and the seesaw mechanism[END_REF], [START_REF] Fairbairn | Unifying inflation and dark matter with the Peccei-Quinn field: observable axions and observable tensors[END_REF], [START_REF] Peccei | STRONG CP AND AXIONS[END_REF], [START_REF] Cianci | Exact solutions for Weyl fermions with gravity[END_REF] the solution pointing to a unique new physics scale around a scalar GeV U(1)Y . These charges, Q and Q can decay into SM quarks, other fermions etc., as in my model. , which equilibrate the Coulomb field of quarks, thus diminishing the dipole moment (the strong CP problem) , and preventing the collapsing due of the gluons field, as in the following. Independently, the author in [1c] has obtained the same quantization condition in case of the calculation of nucleons structure. Thus, the ratio of the forces of gravity between two gravitational charges c Gm Planck   In other words, it result that the nuclear force, in fact, is due of the gravitational charges embedded in quarks, which by theirs interaction are emitted gravitons continuously at a very small rate (only 28 10   , see above) sufficiently to deform the spacetime ( m 20 10   , see before), finally impeding the electric and the electromagnetically respectively, collapsing. An experimental prove of ours unifying model is the LGO measurement of gravitational waves, where, to estimate the gravitational waves emitted by the binary BHs is used the Newman-Penrose scalar 4  , it can be shown that one of these scalars-4  in the appropriate frame-encodes the outgoing gravitational radiation of an asymptotically flat system. The identification of the Weyl scalar 4

 with the gravitational radiation content of the spacetime is a result of the peeling theorem, which states that in an appropriate frame the component of the curvature has the slowest fall-off with radius, ) 1 ( r  . To estimate the total energy E emitted in gravitational waves, is used the following formula [START_REF] Pretorius1 | Evolution of Binary Black Hole Spacetimes[END_REF].  is the complex conjugate of 4  , and the surface integrated over is a sphere of constant coordinate radius R (in uncompactified coordinates).

Conclusions

Accounting that the mass of graviton as been of J a c ), that representing the component of the magnetic field (B ). The further Universe expansion is assured by the gravitons released during BHs  merging with theformation of BH's nuclei of galaxies, that fraction deriving from the same number of the initially BH  . In this way it is entirely confirmed the timeline of Universe. A correspondence of our unifying model with the recent SM*A*S*H it was done, with the difference that the gravitons were proved in LIGO experiment, despite the axions particles not yet.

  , the Friedmann equations are equivalent to this pair of equations:

  Or, for inside of these objects: at Electroweak epoch or Reheating due of the quantum fluctuations [2a] a huge number of the micro-black holes as Planck particles

  The energy of the graviton becomes at an eventually detector (like LIGO) with the above of particles released as the gravitational wave (like the photons of the light wave) remains equally with the above value curvature radius it results from eq.(1) with the integral (which the starts at the outside of the

  base of BH's nuclei of galaxies production An important contribution to further Universe expansion is given by the BH's nuclei of galaxies which result from another fraction (

  generate the curvature radius of the object (Universe) R .

Figure 1 .

 1 Figure1.a. The mass generation by dual Meissner effect, the  decay for a free neutron and, n -normal vacuum, s -superconductor, QW -quantum well.At Confinement we have for the vortex potentialg q q  3 as from [8a], GeV V C 80 1      

Fig. 1 .

 1 Fig.1.b. Fields due to color charges as in sea quarks ( ) ( R R q q) of valence quarks ( u ), (G is the gluon field strength tensor). These are "colorless" combinations. Top: Color charge has "ternary neutral states" as well as binary neutrality (analogous to electric charge).

  the raw flow consequences squeezing this cromoelectrical flux into a vortex line, followed by forcing an organization into a triangular Abrikosov lattice, see figure1. The penetration of B is

  From [2a], we have the lower critical field:

  The equivalence between the Lorenz force energy which squeezes the electrical field  the interface between normal and superconducting phase we havec E B 

  constant electric field E the separation d must be sufficiently large 2 2mc eEd  Probability for separation d as a quantum fluctuation

3 

 3  corresponds to the height of gluon condensate barrier, due of the phase slip with  sea quarks color flux tubes, see figures 1.a; 1.b. The value of the resulting flux tube it remains as in (4.2.a), respectively of GeV 4 . 0

.

  by emission of an electron and an electron antineutrino to become a proton energy corresponding to cr E is much higher than E , respectively as from eq. understanding this occurs when one of the down quarks ) (d in the neutron ) (udd transforms into an up quark )(u due of interacting with the charge of  W boson of the pair  W , transforming the neutron into a proton ) (uud . In mean time the other part of this pair  W boson decays into an electron and an electron antineutrino Probable the claimed energy of boson 

3 

 3 identically with our case where are generated BHs  by quantum oscillations, that by decaying becomes two Weyl fermions Q and Q in the 3 and representations of SU(3)c and with charges 3 1  and 3 1 under

2 embedded 1

 21 in quarks together with photons (see the electron structure[START_REF] Mehedinteanu | What Is Gravity and How Is Embedded in Mater Particles giving them mass (not the Higgs field)?[END_REF]), and of electromagnetic force between the vortex and the quark flux tube (electric field), see figureIf we consider only an electric Coulomb field of the quark dipole in the middle of the gluons condensate, we obtain an another very important result, namely, the value of fine structure constant s  :

  -at Reheating), the merging of the primordial Micro-black-holes generated as due of Quantum fluctuation at Electroweak epoch, it conducts to a can generate at Confinement epoch inside the nucleons an electrical field ( E ) that condensate the free photons, also as resulting from the radiating of the Micro-black-holes, but as gluons of near the same number ( 96 10 

  -the Hubble length is slowly varying, we

	have	d	ln	k		d	(ln(	aH	))		d	ln	a		dt a 		Hdt	. From the definition Eq. (38) of [2] this
																a
	gives	ln				(	)

Electroweak symmetry breaking-quarks epoch(QGP)

	or When immediately, The number of gravitons that has been released following BHs J GeV a EW BH 17 10 10 11 11      , that results with  (nucleons) merging is
	3 c  10 27   10 20 2 ] [ m 1 H       10 by iteration for ] [ m ; t 10 N 29 3 . 3      10 26 10 6 . 3 10  s 58 16 ; R 2 .  ; in eq. (2); 10 5 . 6 20    41 [ 26 10 8 . 8 10 2 . 3    n       BH g merging 3 8 G  1 H leave  92 . 0    2 leave a a     1  EW a   k ; only  at n  , and that generate the ] [ 10 2 . 9 1 19    m k end ] m  ;  ] [ 10 7 . 1 27 m l l P C     ; 92 . 0  EW a ; where 96 10    P P g n n leave end end  curvature radius of the object ( r  ).  Schw as a -fix‖ number of ; Now, the horizon-entry is when the wave length N e k k   ; ] [ 31 1   m k ,; the
																									g
	end gravitons escaped (an inverse process of a black-hole ) from micro-blacks holes (the end end end scale factor arrives at H k a  , the frequency is Hz k c 9 1 10 3 . 9      , and
	fraction for matter the Compton length 0   25 . C  decaying till the Confinement, see below, and which deforms the spacetime. ) created in Universe as the Planck particles during theirs totally ] [ 10 2 . 8 17 _ 0 m c m BH nucleon g      , for J nucleons 10 10    it In both steps the number of gravitons that has been released following BHs  merging is results GeV a a end end nucleon BH P 2 . 2 10 6 . 3 10 10 10             ; it results
	only _ end a found n BH at N   merging 1 , and from eq. (2)    26 17 10 96 10 10     BH g n   1 r H Sch 26 end   7 . 35 	   8 10 8 .  68 10  3 , and these generate the curvature radius ] [ m , s c H t end end 11 1 10 9 . 2      , we
	of the object			R		d		20 10 	.
																									H
	In other words the contribution to the space-time deformation is due of
	n	EW			68 10		2 n					6 10	, for two	BHs 	merging.
			H																		P
	To mention that only this data set match the model.
	The			BH 	particles decay to QGP;	m		4 . 1		22  10	kg		9 . 7		4 10	GeV	, or
																									QGP
			BH 		11 10		a	end		ee		9 . 7	4 10 	GeV		5  10  26 . 1	J	,	k	end		k	leave e		N	;
	k	1 end 				9 . 7	10  14 	] m [	,	a	end	_	QGP		6 10 1  27 .	,	C 		ee		3 . 2	10  21 	[	] m	, with eq. (2)
	R		2 . 1		10	7 	m	, and	H	1  	10 7 	[	] m	,	t		H	1 	c		3 . 3	16  10 	s	,
																									end	end	end
	H	1 		k	1 					H	1 		10 20 	[	] m	we found	N		15	.	88	to match the iterations cycle:
				leae					laeve								EW
	m				 		k		T				R		H	1		a		N	.
			g																B		end	end
	The number of gravitons that has been released following	BHs 	merging is only
	n		at		merging			 	BH 	n	10 g		26	  1 . 26 		5 96 10 10 	10		26			9 . 7		74 10	, and these generate the
	curvature radius of the object R .
	In other words the contribution to the space-time deformation is due of
	n	QGP			9 . 7	74 10 	2 n		9 . 7	12 10 	, for two BHs 	merging.
			H																					P
	61 10 To mention that only this dataset match the model. 9 70 10 10    J J E n P U P  with BH 10 19   	a	end	ee 		10 11	GeV		J 17	, it results
	with	k		1 				H	1 		10 27 	] m [	by iteration for	N		5 . 2	; in eq. (2);	k		2 . 9	10 19 	[	m	1 	]	;
									leave							leave	end
	7 10 8  2 . Another proof-the light bending by Earth  ee end a ; ] [ 10 18 1 m H    ; t 3 . 3  The number of graviton in case of Earth is	s U 27  10  n 	; 10 R 26  	5 . 6 .  2 3 	[ 58 20  10 10 	] m ;	;
	l					l					5 . 1 2	10  27 	[	] m	;	g
	C					P
	Second step GM U Earth 				2 . 3 	32 10	J
								R
												Earth

  number of gravitons that has been released following

	BHs  , and these generate the merging is only In other words the contribution to the space-time deformation is due of     79 26 10 96 26 10 9 . 2 10 10 4 . 3 10 10          BH g merging at n n   curvature radius of the object R . 17 79 10 9 . 2 2 10 9 . 2      P g qq H n n , for two BHs  merging. Thus, from the overall balance the number of gravitons consumed to push the space time is 35 36 17 6 12 10 10 10 9 . 2 10 7 . 5 10 9 . 7 8 1           P n . In other words, at Confinement it remains at leas 1-2 gravitons of GeV J C nucleon 1 10 10 10 26        . To mention that only this data set match the model. From [2a], we have 0 H -an -external‖ electro-magnetic field of a dipole created by the pair u u (the chromoelectrical colors field) ] [ 10 33 . 8 4 24 3 0 0 0 C N r e d E H       ,where fm r 05 . 0  -is the electrical flux tube radius, fm d 48 . 0  -the distance between
	the two quarks charges, this is in fact equilibrated by the gluons field , and respectively,
	from eq. (2.a;2.b) at a more deep penetration
	7 . 4	10	16	6 . 8	10	17	[	]

  Thus, with the graviton in my model, alias axion in SMASH model, that of an energy

		g	a c end  		2	26  10 	J	(when it born !) ; when	a	leave		k	leave	H	leave		1	, of mass
	M U			43 10 	kg	;
	m c BH    C 	c	2			; and with		BH 		10 11	a EW		10 11	GeV		J 17	alias  from SMASH, it
	results with	k	1 leave 		H	1 leave 		10 27 	] m [	by iteration for	N		16	.	2	; in eq. (2);
	k end			2 . 9	10 19 	[	m	1 	]	;	a	end	ee		92 . 0	;	H	1  	10 20 	[	] m	;	t		3 . 3	29  10 	s	;
	R		5 . 6		] 20 m [  10	;	l C		7 . 1	] 27 m [  10 	;	a	EW		92 . 0	; where the remnant number of
	gravitons into a quark		n	g		9 . 0	, and the number of gravitons that has been released
	following two Weyl fermions of quarks tubes, viewed by ours, as BHs 	merging is only
	n	at		merging		 	BH 	n	10 g		26	  17 	10 8 . 0 	26			5		10		28	, and these generate the curvature radius
	inside the nucleon		R		20 10 


David Martínez Martínez

Fabien Seyfert

Martine Olivi

Johann Sence

Stéphane Bila

François Torres

Thierry Monédière

Bernard Jarry

Aurélien Perigaud

Bouchra Frigui

D Martínez

F Torrès

A Périgaud

Optimized synthesis of a microwave equalizer for matching a high performance small antenna

I. INTRODUCTION

The deployment of electronic systems devoted to communication, localization, or supervision has to answer an increasing demand in terms of integration density and flexibility, both in frequency and coverage. The flexibility in frequency can be simultaneous with multi-band devices or selective with reconfigurable devices. The flexibility of coverage allows directing the beam through the target in order to optimize the link budget.

The increase of both efficiency and compactness requires co-designing elementary functions (antenna, filter, amplifier...), which allows reducing interconnection stages and taking into account the interactions between individual elements [START_REF] Troubat | Mutual synthesis of combined microwave circuits applied to the design of a filter-antenna subsystem[END_REF], [START_REF] Armengaud | Design of a ka-band mmic filtering lna with a metamorphic hemt technology for a space application[END_REF], [START_REF] Demirel | Codesign of a pa-antenna block in silicon technology for 80-ghz radar application[END_REF], leading to optimized sub-systems. In the frame of this work, the demonstration will be given with the co-design of dual band filters together with a RHCP (right-hand circular polarization) dual-band antenna for improving the total efficiency of a receiver for GNSS (Global Satellite Navigation System) radio navigation systems.

II. DUAL-BAND RHCP ANTENNA

The antenna design is described in details in [START_REF] Fezai | Low-profile dual-band circularly polarized microstrip antenna for gnss applications[END_REF]. The structure, depicted in Fig. 1, consists of a microstrip patch antenna with two orthogonal excitation ports and four slots, providing a dual-band circular polarization behaviour when the ports are combined with 90 offsets. The 2-port antenna is designed considering severe specifications in terms of gain, axial ratio and surface occupation. In order to fulfil all specifications, it comes difficult to match the antenna over the two passbands. The scattering parameters are given in Fig. 2. The antenna is connected to a coupler for combining the 2 excitation ports, as shown in Fig. 3. Since the antenna is not perfectly matched in the two passbands (L2 + E6 is the first passband and L1 is the second passband), a non-negligible part of the signal is dissipated in the load connected to the isolated port. Consequently, the total efficiency is degraded in the passband edges as shown in Fig. 4. In order to improve the total efficiency, matching filters have to be designed and optimized with respect to antenna scattering parameters.

III. A COMPLEX EQUALIZATION PROBLEM

The problem of matching is stated from the de-embedding point of view. It consist on the design of the global system equalizer + load directly with a set of conditions to ensure that the load is included in such a system and can be extracted afterwards [START_REF] Bode | Network Analysis and Feedback Amplifier Design, ser. Bell Telephone Laboratories series[END_REF], [START_REF] Fano | Theoretical Limitations on the Broadband Matching of Arbitrary Impedances[END_REF]. To simplify the design, we perform a first order rational approximation of the antenna in each band. For a load of degree 1, these conditions become just a constraint on the integral of the reflection coefficient (in logarithmic form). In Fig. 6 we can see an example of equalization for a load of degree 1 (the surface under both lines must be the same). For instance, as in the previous example, the reflection of global system (dashed line in Fig. 6) is computed as:

|S 11 | 2 = |T | 2 + α 2 |T | 2 + β 2 (1)
while the transmission coefficient takes the form:

|S 21 | 2 = β 2 -α 2 |T | 2 + β 2 (2)
where T is a reference filtering function (here we take just the Tchebyshev polynomial) of the desired degree and α, β are positive constant with β ≥ α.

It can be proved that β = β(α) is a function that provides the right surface under the curve of return losses for every α. Moreover for any filtering function T , the level of return losses in the passband has only one maximum with respect to α.

By taking T as the Tchebyshev polynomial of degree N, and being s the desired surface, we obtain: The return losses obtained after equalization can be computed as:

β(α) = sinh N arcsinh sinh arcsinh(α) N + H (3)
H = s 2π sin π 2N (4)
|RL| 2 = cosh 2 (N arcsinh(x)) cosh 2 (N arcsinh(x -H)) (5)
x

= H + sinh 1 N arcsinh(α) (6)
Finally, after the extraction of the load the equalized filter is obtained along with the length of the transmission line at the output port (Fig. 5). It should be noted that only the equalizer for the first polarization is shown here along with the part of the antenna corresponding to the first polarization. Nevertheless the antenna is symmetrical and the cross-polarization can be neglected. Thus another identical equalizer would be required for the second polarization.

IV. GAIN EQUALIZATION

With the aim to compute the effective transmission of the system when losses are considered, we work with a 2-ports model of the antenna where the second port represent the transmitted signal regarding the first polarization (efficiency). The transmission coefficient of the filter cascaded with the 2-ports antenna takes the following expression:

S C 21 = S F 21 S A 31 1 -S F 22 S A 11 (7)
where the scattering parameters S A ij refers to the antenna and the parameters S F ij to the equalizer including the transmission line at the output port.

The objective now in is to maximize the transmission (S C 21 ) with a given requirement on the rejection in the stopband. With the condition over the rejection, the problem remains:

M ax S F M in f ∈I |S C 21 | -γ i S C 21 (f si ) (8)
where I represents the passband, f si the frequencies to reject and γ a positive weight. This is a local MaxMin optimization that converges extremely fast if the equalizing filter for the first order approximation of the load is taken as initial point.

V. DUAL-BAND FILTER

In order to perform a dual-band equalization, a dual-band filter is designed by implementing a duplexed solution and performing a fix point algorithm over both channels. In each iteration of the fix-point algorithm, the equalizer of one of the channels is designed by taking as load the system composed by the junction, the antenna and the equalizer of the other channel. Below we show a detailed implementation of this fix-point algorithm:

1) Compute the single-band equalized for the first band (Fig. 5).

2) The impedance in the second band of the first channel equalized shunt with the antenna become the load for the second channel (Z 2 L in Fig. 7a). Then a new equalizer is made for the second band.

3) This process is iterated until the changes in each band become negligible. We obtain a diplexer with two singleband equalizers (Fig. 7b). Finally, in order to compute the input junction, since the input load impedance is constant (50Ω) and the channels are not too close to each other, we can just compute the length of the input lines such as at the centre of each band, the adjacent channel is seen as an open circuit. Fig. 7c shows the final schematic for the dual-band equalizer where the coupling matrices of both filter are shown in ( 9) and (10) and the lengths of the transmission lines 1 and 2 are 17mm and 46mm respectively.

VI. PHYSICAL STRUCTURE SYNTHESIS

The filter synthesis indicates that, to achieve the dual-band filtering function, it is necessary to implement two duplexed two poles filters. The strongest constraint in this study is the relative bandwidth of the two combined filters which is approximately 27%. In order to build these two filters, it is necessary to choose a technology and a topology adapted to wide band filtering. The strongest constraint in this study is the relative bandwidth of the two combined filters which is approximately 27%. An usual solution is then planar quarter wavelength resonators filter. Indeed, this type of filter presents the advantage to exhibit strong couplings meanwhile it does not resonate until 3 times the initial resonant frequency.

The main drawback of this type of structure is that it requires electric wall at one of its ends, that is generally ensured by one or more via(s) connected to ground.

However, this point does not require any additional manufacturing step to be addressed since vias are already necessary to build the coplanar feeding-lines of the antenna. Each filter is firstly designed according to the parameters provided by the synthesis defined above. To meet the requirements of the coupling matrix, there are some design rules to follow. To begin with, the diagonal matrix terms are adjusted by the resonator lengths. Then, the off-diagonal terms are set up by the gap between each resonator. Finally, owing to the strong input/output couplings, it is not possible to perform them by proximity. The solution is then to couple them directly with the feeding lines. Their positions determine the values of the couplings but they also create frequency shifts. To counter-balance these effects, a capacitive effect has to be added to each resonator. This is achieved by adding a short length line at the end of each one.

Because of parasitic couplings, filters are firstly tuned by fitting EM responses with the objective curves extracted from coupling matrix consigned in (9) and (10). To take into account the capacitive and inductive effect of the meanders and corners in the junctions, as well as the variation on the line impedance due to the electric walls, each filter was tuned separately but inside the duplexed structure with the initial guess for the length of the transmission lines. The output ports of both filters are then connected together to obtain a diplexer as shown in Fig. 7b. Slight adjustments were necessary to adjust the response of the physical structure to fit the ideal one (whether it be with radiated losses or with return losses) but also to correct the phase with some line lengths corrections (Tr. line 1 and Tr. line 2). The last step is to compute the line lengths at the input of each filter (Tr. line 3 and Tr. line 4). These lengths are deduced from the EM response of the diplexer by looking at the phase of the reflection provided by each channel in the other band.

VII. RESULTS

The results obtained by the simulation are shown in Fig. 9 compared with the theoretical ones. The full-wave simulation response fits well with the one calculated by the mathematical synthesis, both in magnitude and phase in spite of a small variation on the phase of the transmission coefficient that has no influence on the result. After connecting the antenna, we obtain the result shown in Fig. 10. It can be seen that, although the efficiency level in the passbands (L2, E6 and L1) is flatter, the level is lower. This is due to the losses in the equalizer filters and should not be seen as a drawback since the received signal must be filter in any case and here the filtering is done by the equalizer.

ACKNOWLEDGEMENT

This work is performed in the frame of a public research program (ANR ASTRID COCORAM).

VIII. CONCLUSION

A dual-band equalizer was build to improve the total efficiency of a dual-band RHCP antenna. The minimum efficiency of the antenna (at the upper cut-off frequency of the E6 band) was improved from 36% to (-) %. Moreover, since the equalizer is composed of two channel filters matched to the antenna, a filtering chain is no longer required with the advantage in size that this entails.

Finally, a prototype has been designed in microstrip technology to prove the feasibility of this approach, obtaining a response that agrees perfectly with the circuital model and provides the expected result in terms of efficiency.

Fig. 1 .

 1 Fig. 1. Dual-band RHCP antenna integrated a microstrip slotted patch

Fig. 3 .

 3 Fig. 3. Schematic of the RHCP antenna connected to a hybrid coupler for combining the two ports with 90 phase difference.

Fig. 4 .

 4 Fig. 4. Total efficiency of the dual-band RHCP antenna connected to the hybrid coupler.

1 Fig. 5 .

 15 Fig. 5. Equalization chain.

Fig. 6 . 4 1

 64 Fig. 6. Load of degree 1 along with the result of the equalization.

Fig. 7 .

 7 Fig. 7. Different stages in the design of the dual-band equalizer.

Fig. 8 .

 8 Fig. 8. Layout of the dual-band filter.

Fig. 9 .

 9 Fig. 9. Response of the dual-band microstrip equalizer.

Fig. 10 .

 10 Fig. 10. Obtained efficiency vs original.