prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the main focus of the research paper?

Read the following text and answer briefly.

Context 1: Paper Info

Title: Nuclear Liquid-Gas Transition in the Strong Coupling Regime of Lattice QCD
Publish Date: 28 Mar 2023
Author List: J Kim (from Institute for Advanced Simulation (IAS-4), Forschungszentrum Jülich), P Pattanaik (from Fakultät für Physik, Bielefeld University), W Unger (from Fakultät für Physik, Bielefeld University)

Figure

FIG  1 Typical 2-dimension configuration at β = 1 0, at non-zero quark mass, temperature, chemical potential The black dots are monomers, the blue lines are dimers, the red arrows are baryon loop segments (or triplets g b + f b = ±3 if adjacent to a non-trivial plaquette), and the green squares are plaquette occupations ±1 The actual configurations are 3+1-dimensional 
FIG 2 Chiral susceptibility on a 2 4 volume for various quark masses, as a function of the bare anisotropy γ (with aT = γ 2 /2), analytic results from enumeration compared to numerical data from simulations via the worm algorithm 
FIG 3 Various observables in the µB-T plane on a 2 4 volume at amq = 0 1 The back-bending of the first order transition at temperatures below aT = 0 5 in all observables is an artifact of the small volume, and vanishes in the thermodynamic limit The temperature aT = 1/2 corresponds to the isotropic lattice here 
FIG  4  The chiral condensate (left) and the baryon density (right) for quark mass m = 1 5 as a function of the chemical potential and for various temperatures 
FIG  7  ∆f at amq = 0 2 as a function of chemical potential and β the on a 6 3 × 4 lattice
FIG  8  Baryon mass from ∆E as a function of the quark mass amq, and contributions from different dual variables: monomers, dimers and baryon segments 
FIG  9  Baryon density for volume 4 3 × 8 in the full µB − mq plane, illustrating the strong quark mass dependence of the onset to nuclear matter 
FIG  10 Baryonic observables on various volumes in the first order region amq = 1 5 Vertical bands indicate the mean and error of the nuclear transition 
FIG  12  Left: Extrapolation of the pseudo-critical values of µB for the various volumes into the thermodynamic limit Right: Critical baryon chemical potential for different quark masses The first order transition region is shown in blue, the crossover region is shown in red and the range for critical end point is marked in black 
FIG  17  Nuclear interaction scaled with baryon mass As the quark mass increases, it tends to zero 
FIG  18  Critical baryon chemical potential and baryon mass from different approaches 
Parameters for the Monte Carlo runs to determine the nuclear transition at strong coupling, with statistics after thermalization 

abstract

The nuclear liquid-gas transition from a gas of hadrons to a nuclear phase cannot be determined numerically from conventional lattice QCD due to the severe sign problem at large values of the baryon chemical potential  In the strong coupling regime of lattice QCD with staggered quarks, the dual formulation is suitable to address the nuclear liquid gas transition 
We determine this first order transition at low temperatures and as a function of the quark mass and the inverse gauge coupling β  We also determine the baryon mass and discuss the nuclear interactions as a function of the quark mass, and compare to mean field results  It is known from experiments that at low temperatures, there is a phase transition between dilute hadron gas and dense nuclear matter as the baryon chemical potential increases 
This transition is of first order and terminates at about T c = 16 MeV in a critical end point  The value of the chemical potential µ 1st B at zero temperature is given roughly by the baryon mass m B , where the difference of µ 1st B −m B is due to nuclear interactions  For a review on nuclear interactions see  
As the nuclear force between baryons to form nuclear matter is due to the residual strong interactions between quarks and gluons, it should be accurately described by QCD  We choose to study the nuclear transition and nuclear interaction via lattice QCD , with its Lagrangian being a function of the quark mass and the inverse gauge coupling 
In order to understand the nature of the transition, it is helpful to study its dependence on these parameters  However, at finite baryon density, lattice QCD has the infamous sign problem which does not allow us to perform direct Monte Carlo simulations on the lattice  Various methods have been proposed to overcome the numerical sign problem, but they are either limited to µ B /T 3 or can not yet address full QCD in 3+1 dimensions in the whole µ B − T plane , in particular the nuclear transition is out of reach 
An alternative method is to study lattice QCD via the strong coupling expansion  There are two established effective theories for lattice QCD based on this: (1) the 3-dim  effective theory for Wilson fermions in terms of Polyakov loops, arising from a joint strong coupling and hopping parameter expansion , the dual representation for staggered fermions in 3+1 dimensions, with dual degrees of freedom describing mesons and baryons 
Both effective theories have their limitations: is limited to rather heavy quarks (but is valid for large values of β) whereas ( ) is limited to the strong coupling regime β 1 (but is valid for any quark mass)  We study lattice QCD in the dual formulation, both at infinite bare gauge coupling, β = 0, and at leading order of the strong coupling expansion in the regime β < 1, which is far from the continuum limit 
But since strong coupling lattice QCD shares important features with QCD, such as confinement, and chiral symmetry breaking and its restoration at the chiral transition temperature, and a nuclear liquid gas transition, we may get insights into the mechanisms, in particular as the dual variables give more information in terms of its world lines, as compared to the usual fermion determinant that depends on the gauge variables 
To establish a region of overlap of both effective theories, we have chosen to perform the Monte Carlo simulations in the dual formulation extending to rather large quark masses  This paper is organized as follows: in the first part we explain the dual formulation in the strong coupling regime, in the second part we provide analytic results based on exact enumeration and mean field theory, in the third part we explain the setup of our Monte Carlo simulations and present result on the m q -and β-dependence of the nuclear transition 
Since the strong coupling regime does not have a well defined lattice spacing, we also determine the baryon mass am B to set the parameters of the grand-canonical partition function, aT and aµ B , in units of am B   We conclude by discussing the resulting nuclear interactions, and compare our findings with other results 

Staggered action of strong coupling QCD and its dual representation

In the strong coupling regime, the gauge integration is performed first, followed by the Grassmann integration to obtain a dual formulation  This was pioneered for the strong coupling limit in and has been extended by one of us to include gauge corrections   The sign problem is mild in the strong coupling limit and still under control for β < 1, where we can apply sign reweighting 
The dual degrees of freedom are color-singlet mesons and baryons, which are point-like in the strong coupling limit, and become extended about a lattice spacing by incorporating leading order gauge corrections  The partition function of lattice QCD is given by where DU is the Haar measure, U ∈ SU(3) are the gauge fields on the lattice links (x, μ) and { χx , χ x } are the unrooted staggered fermions at the lattice sites x 
The gauge action S G [U] is given by the Wilson plaquette action and the staggered fermion action S F [ χ, χ, U] is: where the gauge action depends on the inverse gauge coupling β = 2Nc g 2 and the fermion action depends on the quark chemical potential aµ q which favors quarks in the positive temporal direction, and the bare quark mass am q  
First we consider the strong coupling limit where the inverse gauge coupling β=0 and hence the gauge action S G [U] drops out from the partition function in this limit  The gauge integration is over terms depending only on the individual links (x, μ) so the partition function factorizes into a product of one-link integrals and we can write it as:
with z(x, μ) the one-link gauge integral that can be eval-uated from invariant integration, as discussed in , where we write the one-link integral in terms of new hadronic variables: Only terms of the form (M (x)M (y)) k x, μ (with k x,μ called dimers which count the number of meson hoppings) and B(y)B(x) and B(x)B(y) (called baryon links) are present in the solution of the one-link integral 
The sites x and y = x + μ are adjacent lattice sites  It remains to perform the Grassmann integral of the fermion fields χ, χ  This requires to expand the exponential containing the quark mass in Eq  (4) (left), which results in the terms (2am q M (x)) nx (with n x called monomers)  To obtain non-vanishing results, at every site, the 2N c Grassman variables χ x,i and χx,i have to appear exactly once, resulting in the Grassmann constraint (GC):
where n x is the number of monomers, k x,μ is the number of dimers and the baryons form self-avoiding loops x,μ , which due to the constraint cannot coexist with monomers or dimers  With this, we obtain an exact rewriting of the partition function Eq  ( ) for N c = 3, in terms of integer-valued dual degrees of freedom {n, k, }:
where the sum over valid configurations has to respect the constraint (GC)  The first term in the partition function is the contribution from dimers and the second term is the contribution from monomers  The weight factor w( ) for each baryon loop depends on the baryon chemical potential µ B = 3µ q and induces a sign factor σ( ) which depends on the geometry of :
Here, ω is the winding number of the loop   The total sign factor σ( ) ∈ {±1} is explicitly calculated for every configuration  We apply sign reweighting as the dual formulation has a mild sign problem: baryons are non-relativistic and usually have loop geometries that have a positive signs  The dual partition function of the strong coupling limit is simulated with the worm algorithm (see Section III A) and the sign problem is essentially solved in this limit 

Extension to finite β

The leading order gauge corrections O(β) to the strong coupling limit are obtained by expanding the Wilson gauge action Eq  ( ) before integrating out the gauge links  A formal expression is obtained by changing the order of integration (first gauge links, then Grassmann-valued fermions) within the QCD partition function:
With this the O (β) partition function is The challenge in computing Z (1) is to address the SU(N c ) integrals that receive contributions from the elementary plaquette U P   Link integration no longer factorizes, however the tr[U P ] can be decomposed before integration: Integrals of the type J ij with two open color indices -as compared to link integration at strong coupling -have been derived from generating functions
for either J = 0 or for G = U(N c )   The SU(3) result was discussed in , in terms of the dual variables, neglecting rotation and reflection symmetries, there are 19 distinct diagrams to be considered  The resulting partition function, valid to O(β), is with q P ∈ {0, ±1}, and the site weights w x → ŵx , bond weights w b → ŵb and baryon loop weights w → ŵ receive modifications compared to the strong coupling limit Eq  ( ) for sites and bonds adjacent to an excited plaquette q P = 1 
The weights are given in , and are rederived for any gauge group in   The configurations {n, k, , q p } must satisfy at each site x the constraint inherited from Grassmann integration: which is the modified version of Eq  ( ) with q x = 1 if located at the corner of an excited plaquette q p = 0, otherwise q x = 0 
A more general expression that we obtained via group theory and is valid to higher orders of the strong coupling expansion is discussed in terms of tensor networks   A typical 2-dimensional configuration that arises at β = 1 in the Monte Carlo simulations is given in Fig    Note that if a baryon loop enters a non-trivial plaquette, one quark is separated from the two other quarks, resulting in the baryon being extended object, rather being point-like in the strong coupling limit 
The O(β) partition function has been used in the chiral limit to study the full µ B − T plane via reweighting from the strong coupling ensemble  Whereas the second order chiral transition for small values of the aµ B decreased up to the tri-critical point, the first order nuclear transition was invariant: aµ 1st B 1 78(1) at zero temperature has no β-dependence 
For the ratio T (µ B = 0)/µ 1st B (T 0) we found the values 0 787 for β = 0 and 0 529 β = 1, which should be compared to T c / 0 165 for full QCD   However, since reweighting cannot be fully trusted across a first order boundary, direct simulations at nonzero β are necessary  The Monte Carlo technique to update plaquette variables is discussed in Section III A 
In this section, we provide analytic results from exact enumeration for small volumes, and mean field results based on the 1/d expansion, valid in the thermodynamic limit  The main purpose is to compare our Monte Carlo results to these analytic predictions 

Exact enumeration

To establish that our Monte Carlo simulations indeed sample the partition functions Eq  ( ) and Eq  ( ), we have obtained analytic results on a 2 4 volume at strong coupling, and at finite beta in two dimensions on a 4 × 4 volume, comparing O (β) and O β 2 truncations  Our strategy to obtain an exact enumeration of the partition function Z is to enumerate plaquette configurations first, then fixing the fermion fluxes which together with the gauge fluxes that are induced by the plaquettes form a singlet, a triplet or anti-triplet, i e  on a given bond b, g b + f b ∈ {−3, 0, 3}, and last we perform the monomerdimer enumeration on the available sites not saturated by fermions yet by a depth-first algorithm  
At strong coupling, with no plaquettes, g b = 0 and f b are baryonic fluxes  All observables that can be written in terms of derivatives of log(z), such as the baryon density, the chiral condensate, the energy density, and also the average sign, are shown in Fig 

Expectations from mean field theory

Another analytical method to study strong coupling lattice QCD is the mean field approach, where the partition function is expanded in 1 d (d is the spatial dimension) and then a Hubbard-Stratonovich transformation performed   After this procedure, the free energy is a function of temperature T , the chiral condensate σ and chemical potential µ B :
here E[m] is one-dimensional quark excitation energy which is a function of the quark mass m = am q   For N c = 3 and d = 3 we determined the minimum of the free energy with respect to the chiral condensate  This gives us the equilibrium chiral condensate as a function of (T, m, µ B )  The chiral condensate and the baryon density as a function of the baryon chemical potential in lattice units aµ B and for various temperatures at quark mass m = 1 5 is shown in Fig    We have determined the critical temperature to be aT c = 0 23 , which is characterized by an infinite slope of the chiral condensate 
For lower temperatures, there is a clear discontinuity of the chiral con-densate, separating the low density phase from the high density phase  For temperatures above and in the vicinity of aT c the chiral condensate and baryon density has no discontinuity but rapidly changes, corresponding to a crossover transition 
With this method, the phase diagram is plotted for different quark masses in Fig    The second order phase transition in the chiral limit is plotted in solid blue line, the dotted lines show the first order phase transition for different quark masses and the solid red line indicates the critical end point for the different quark masses 
Mean field theory also gives an expression for the pion mass am π and the baryon mass am B : The mean field baryon mass for N c = 3, d = 3 is also plotted in red in Fig    Whereas the baryon mass is around N c in the chiral limit (am B 3 12 for N c = 3), it approximately doubles at m = 3 5 (am B 6 28) which corresponds to the pion mass am π = 4 45, i e  m π /m B = 0 708 
Hence, at around bare mass m = 3 5, the valence quark mass of the baryon corresponds roughly to 1/3 of the chiral limit value of the baryon mass  The first Monte Carlo simulations that could extend in the µ B − T plane was the MDP algorithm , but it required the introduction of the worm algorithm to make substantial progress 
First studies of the worm algorithm applied to the strong coupling limit QCD (with gauge group U(3)) are , and for gauge group SU   Monte Carlo simulations to extend the worm to incorporate leading order corrections were first proposed in   We will shortly review the setup of or Monte Carlo strategy for the nuclear transition, with an emphasis on the challenges to address large quark masses 

Strong Coupling

Without any further resummation, there is a mild sign problem in the dual formulation of lattice QCD in the strong coupling limit  When the average sign σ is not too small (close to zero), it implies that most of the configurations have a positive weight thus allowing us to perform sign reweighting strategies 
In Fig  , ∆f is plotted as a function of the baryon chemical potential and the quark masses  It is seen that ∆f is close to zero for most cases except near the critical chemical potential and for small quark masses, but never exceeds 5 × 10 −4   Hence sign reweighting can be performed in the full parameter space 
The result that the sign problem becomes even milder when increasing the mass is related to the fact that larger critical chemical potentials result in a larger fraction of static baryons (spatial baryon hoppings become rare)  FIG    ∆F at strong coupling as a function of chemical potential and quark mass on a 6 3 × 8 
The sign problem becomes milder as the quark mass increases 

Finite β

All runs at finite β have been obtained for N τ = 4, which corresponds to a moderately low temperature aT = 0 25 compared to the value of the chiral transition aT 1 54  Those simulations were too expensive to attempt N τ = 8 runs, in particular as a higher statistics was required  The spatial volumes are 4 3 , 6 3 and 8 3  
For β values are from 0 0 to 1 0 with step size 0 1, and for am q values from 0 00 to 1 00 with step size 0 01  The values of aµ were chosen close to the nuclear transition, the scanning range is shifted to large values as am q increases  At small quark masses the scanning range is from aµ = 0 4 to 1 0 and for the large quark masses, it is from 0 6 to 1 2 with step size 0 01 
The statistics used for are 15 × 10 4 measurements and between measurement, 40 × N 3 s worm updates 

Residual sign problem

Although it is possible to resum the sign problem at strong coupling with a resummation of baryon and pion world lines, this is not possible when including gauge corrections  In order to compare both sign problems, we kept the original dual formulation to monitor the severity of the sign problem  This is done via the relation
between the average sign σ and the difference of the free energy density ∆f between the full ensemble f and of the sign-quenched ensemble f ||  

Nuclear interactions

We have found that aµ 1st B is very different from the baryon mass  This must be due to strong attractive interactions of nucleons  In contrast to continuum physics, in the strong coupling limit there is no pion exchange due to the Grassmann constraint  Instead, nucleons are point like and hard core repulsive 
However, the pion bath, which is modified by the presence of static baryons, results in an attractive interaction  In , this has been analyzed in the chiral limit using the snake algorithm, and it has been found that the attractive force is of entropic origin  Here, we do not quantify the nuclear interaction via the nuclear potential, but via the difference between critical baryon chemical potential and baryon mass, in units baryon mass, as shown in Fig  , given the am B as measured in Section III C 
This compares better to the 3dim  effective theory  The nuclear interaction is maximal and more than 40% in the chiral limit, which is related to pions being massless: the modification of the pion bath is maximal  We clearly find that the nuclear interaction decreases drastically and almost linearly until it almost approaches zero at about am q = 2 0, corresponding to a pion mass am π = 3 36, see Section II B  The large error bars for larger quark masses, that are due to the subtraction of almost same magnitudes, makes it difficult to extract a non-zero nuclear interaction at the largest quark masses 
In this work, we have determined the baryon mass and the nuclear transition via Monte Carlo: the worm algorithm based on the dual formulation, at finite β equipped with additional updates  All those numerical results and various analytic expressions are summarized in Fig    We find that as the quark mass becomes large, spatial mesons hoppings (i e 
spatial dimers) become rare, which makes this 3+1-dimensional system closer to 1dim  QCD   Also, both the baryon mass and the baryon chemical potential obtained in our dual representation, i e  for staggered fermions, approaches the baryon mass of the 3-dim  effective theory which is based on Wilson fermions 
Another comparison that summarizes the validity of the mean field approach discussed in Section II B is shown in Fig    It is evident that mean field theory has strong deviations for small quark masses, but this discrepancy becomes smaller for larger quark masses  The extension of the study of the nuclear transition to finite inverse gauge coupling β is summarized in Fig  , which shows the β-dependence of aµ c B for various quark masses 
For all quark masses ranging from am q = 0 to am q = 1 0, there is only a very weak β-dependence, confirming the expectation from mean field theory   This works was restricted to isotropic lattices ξ = a/a t = 1, i e  we performed simulations at fixed temperature  Non-isotropic lattices are necessary to vary the temperature at fixed values of β 
This requires to include two bare anisotropies, γ for the fermionic action and γ G for the gauge action  Finite β has only been studied by us in the chiral limit   Clearly, it is interesting to study the location of the nuclear critical point also including higher order gauge corrections and at finite quark mass 
Simulations including O(β 2 ) are under preparation 
Context 2: Weep Not, Child is a 1964 novel by Kenyan author Ngũgĩ wa Thiong'o  It was his first novel, published in 1964 under the name James Ngugi  It was among the African Writers Series  It was the first English language|English novel to be published by an East African  Thiong'o's works deal with the relationship between Africans and white settlers in colonial Kenya, and are heavily critical of colonial rule  Specifically, Weep Not, Child deals with the Mau Mau Uprising, and "the bewildering dispossession of an entire people from their ancestral land " Ngũgĩ wrote the novel while he was a student at Makerere University 

The book is divided into two parts and eighteen chapters  Part one deals mostly with the education of Njoroge, while part two deals with the rising Mau Mau movement 

Plot summary

Njoroge, a little boy, is urged to attend school by his mother  He is the first one of his family able to go to school  His family lives on the land of Jacobo, an African made rich by his dealings with white settlers, namely Mr  Howlands, the most powerful land owner in the area  Njoroge's brother Kamau works as an apprentice to a carpenter, while Boro, the eldest living son, is troubled by his experiences while in forced service during World War II, including witnessing the death of his elder brother  Ngotho, Njoroge's father and a respected man in the surrounding area, tends Mr  Howlands' crops, but is motivated by his passion to preserve his ancestral land, rather than for any compensation or loyalty 

One day, black workers call for a strike to obtain higher wages  Ngotho is ambivalent about participating in the strike because he fears he will lose his job  However, he decides to go to the gathering, even though his two wives do not agree  At the demonstration, there are calls for higher wages  Suddenly, the white police inspector brings Jacobo to the gathering to pacify the native people  Jacobo tries to put an end to the strike  Ngotho attacks Jacobo, and the result is a riot where two people are killed  Jacobo survives and swears revenge  Ngotho loses his job and Njoroge’s family is forced to move  Njoroge’s brothers fund his education and seem to lose respect for their father 

Mwihaki, Jacobo's daughter and Njoroge's best friend, enters a girls' only boarding school, leaving Njoroge relatively alone  He reflects upon her leaving, and realizes that he was embarrassed by his father's actions towards Jacobo  For this reason, Njoroge is not upset by her exit and their separation  Njoroge switches to another school 

For a time, everyone's attention is focused on the upcoming trial of Jomo Kenyatta – a revered leader of the movement  Many blacks think that he is going to bring forth Kenya’s independence  But Jomo loses the trial and is imprisoned  This results in further protests and greater suppression of the black population 

Jacobo and a white landowner, Mr  Howlands, fight against the rising activities of the Mau Mau, an organization striving for Kenyan economic, political, and cultural independence  Jacobo accuses Ngotho of being the leader of the Mau Mau and tries to imprison the whole family  Meanwhile, the situation in the country is deteriorating  Six black men are taken out of their houses and executed in the woods 

One day Njoroge meets Mwihaki again, who has returned from boarding school  Although Njoroge had planned to avoid her due to the conflict between their fathers, their friendship is unaffected  Njoroge passes an important exam that allows him to advance to High School  His village is proud of him, and collects money to pay Njoroge's High School tuition 

Several months later, Jacobo is murdered in his office by a member of the Mau Mau  Mr  Howlands has Njoroge removed from school for questioning  Both father and son are brutally beaten before release and Ngotho is left barely alive  Although there doesn't seem to be a connection between Njoroge's family and the murder, it is eventually revealed that Njoroge's brothers are behind the assassination, and that Boro is the real leader of the Mau Mau  Ngotho soon dies from his injuries and Njoroge finds out that his father was protecting his brothers  Kamau has been imprisoned for life  Only Njoroge and his two mothers remain free, and Njoroge is left as the sole provider of his two mothers  Njoroge fears that he cannot make ends meet; he gives up hope of continuing in school and loses faith in God 

Njoroge asks Mwihaki's for support, but she is angry because of her father’s death  When he finally pledges his love to her, she refuses to leave with him, realizing her obligation to Kenya and her mother  Njoroge decides to leave town and makes an attempt at suicide; however, he fails when his mothers find him before he is able to hang himself  The novel closes with Njoroge feeling hopeless, and ashamed of cowardice 

Characters in Weep Not, Child
 Njoroge: the main character of the book whose main goal throughout the book is to become as educated as possible 
 Ngotho: Njoroge's father  He works for Mr Howlands and is respected by him until he attacks Jacobo at a workers strike  He is fired and the family is forced to move to another section of the country  Over the course of the book his position as the central power of the family weakened, to the point where his self-realization that he has spent his whole life waiting for the prophecy (that proclaims the blacks will be returned their land) to come true rather than fighting for Kenyan independence, leads to his depression 
 Nyokabi and Njeri: the two wives of Ngotho  Njeri is Ngotho's first wife, and mother of Boro, Kamau, and Kori  Nyokabi is his second wife, and the mother of Njoroge and Mwangi 
 Njoroge has four brothers: Boro, Kamau, Kori and Mwangi (who is Njoroge's only full brother, who died in World War II) 
 Boro: Son of Njeri who fights for the Allies in World War II  Upon returning his anger against the colonial government is compounded by their confiscation of the his land  Boro's anger and position as eldest son leads him to question and ridicule Ngotho, which eventually defeats their father's will (upon realizing his life was wasted waiting and not acting)  It is eventually revealed that Boro is the leader of the Mau Mau (earlier alluded to as "entering politics") and murders Mr Howlands  He is caught by police immediately after and is scheduled to be executed by the book's end  It is highly likely that it is also Boro who kills Jacobo 
 Mwihaki: Njoroge's best friend (and later develops into his love interest)  Daughter of Jacobo  When it is revealed that his family killed Jacobo (most likely Boro), Mwihaki distances herself from Njoroge, asking for time to mourn her father and care for her mother 
 Jacobo: Mwihaki's father and an important landowner  Chief of the village 
 Mr  Howlands: A white settler who emigrated to colonial Kenya and now owns a farm made up of land that originally belonged to Ngotho's ancestors  Has three children: Peter who died in World War II before the book's beginning, a daughter who becomes a missionary, and Stephen who met Njoroge while the two were in high school 

Themes and motifs
Weep Not, Child integrates Gikuyu mythology and the ideology of nationalism that serves as catalyst for much of the novel's action  The novel explores the negative aspects of colonial rule over Kenya  Njoroge's aspiration to attend university is frustrated by both the violence of the Mau Mau rebels and the violent response of the colonial government  This disappointment leads to his alienation from his family and ultimately his suicide attempt 

The novel also ponders the role of saviours and salvation  The author notes in his The River Between: "Salvation shall come from the hills  From the blood that flows in me, I say from the same tree, a son shall rise  And his duty shall be to lead and save the people " Jomo Kenyatta, the first prime minister of Kenya, is immortalised in Weep Not, Child  The author says, "Jomo had been his (Ngotho's) hope  Ngotho had come to think that it was Jomo who would drive away the white man  To him, Jomo stood for custom and traditions purified by grace of learning and much travel " Njoroge comes to view Jomo as a messiah who will win the struggle against the colonial government 

See also

Things Fall Apart
Death and the King's Horseman

References

External links
Official homepage of Ngũgĩ wa Thiong'o
BBC profile of Ngũgĩ wa Thiong'o
Weep Not, Child at Google Books

British Empire in fiction
Novels set in colonial Africa
Historical novels
Kenyan English-language novels
Novels by Ngũgĩ wa Thiong'o
Novels set in Kenya
1964 novels
Heinemann (publisher) books
Postcolonial novels
African Writers Series
1964 debut novels



ground_truth: Nuclear liquid-gas transition in lattice QCD.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: When was Weep Not, Child first published?

Read the following text and answer briefly.

Context 3: South Dakota v  Wayfair, Inc  - Harvard Law Review
Fourth Circuit Invalidates Maryland Statute Regulating Price Gouging in the Sale of Generic Drugs 
South Dakota Supreme Court Holds Unconstitutional State Law Requiring Internet Retailers Without In-State Physical Presence to Remit Sales Tax 
Judicial junk, the Court has long thought, is easier to scrap when the erroneous precedent cannot be fixed by Congress, as in constitutional cases 1× 1  See Burnet v  Coronado Oil & Gas Co , 285 U S  393, 405–10 (1932) (Brandeis, J , dissenting); Lee Epstein, William M  Landes & Adam Liptak, The Decision to Depart (or Not) from Constitutional Precedent: An Empirical Study of the Roberts Court, 90 N Y U  L  Rev  1115, 1116 (2015) (“[Justice Brandeis’s] dissenting opinion       now has the status of black letter law ”)  On the flip side, whenever a bad precedent can be corrected by Congress, stare decisis applies with “special force ”2× 2  See Patterson v  McLean Credit Union, 491 U S  164, 172–73 (1989)  The Court, following Justice Brandeis, usually articulates the rule as distinguishing between “constitutional” and “statutory” precedents  See, e g , id  But the distinction is occasionally said to be between “constitutional” and “nonconstitutional cases ” See, e g , Glidden Co  v  Zdanok, 370 U S  530, 543 (1962) (plurality opinion)  Nomenclature aside, the Court has — until now — adhered to Justice Brandeis’s key insight that the important factor is whether or not the mistake may be legislatively corrected  Last Term, in South Dakota v  Wayfair, Inc ,3× 3  138 S  Ct  2080 (2018)  the Court tinkered with this thinking in overruling an outdated dormant commerce clause precedent  Dormant commerce clause decisions technically produce constitutional holdings, but Congress may override them at will 4× 4  See Prudential Ins  Co  v  Benjamin, 328 U S  408, 421–27 (1946)  Under the usual logic of stare decisis, it should take special force to dislodge such precedents  But Wayfair applied the weakened stare decisis of constitutional cases, asserting that the Court must “address a false constitutional premise         whether or not Congress can or will act ”5× 5  Wayfair, 138 S  Ct  at 2096–97 
Emerging from Wayfair is an odd and ominous development in stare decisis doctrine  Odd, because it turns on a formal classification instead of on Congress’s practical ability to fix the problem  Ominous, because the Court’s logic leads far past the dormant commerce clause  Wayfair grants only feeble stare decisis to precedents that set a “constitutional default rule,”6× 6  Id  at 2096 (“While       Congress has the authority to change the physical presence rule, Congress cannot change the constitutional default rule ”)  meaning constitutional decisions that allow for legislative adjustment or override  This new stare decisis analysis makes other precedents setting constitutional default rules more vulnerable — including, perhaps, mainstays of criminal procedure like Miranda v  Arizona7× 7  384 U S  436 (1966)  and Mapp v  Ohio 8× 8  367 U S  643 (1961) 
Since its 1967 decision in National Bellas Hess, Inc  v  Department of Revenue,9× 9  386 U S  753 (1967)  the Court has held that, under the “dormant” or “negative” implication of the Commerce Clause,10× 10  The dormant or negative commerce clause is a judicial derivation from the Commerce Clause “prohibiting States from discriminating against or imposing excessive burdens on interstate commerce without congressional approval,” which “strikes at one of the chief evils that led to the adoption of the Constitution, namely, state tariffs and other laws that burdened interstate commerce ” Comptroller of the Treasury of Md  v  Wynne, 135 S  Ct  1787, 1794 (2015)  states may not compel remote sellers with no physical presence in the state to collect and remit sales taxes 11× 11  See Bellas Hess, 386 U S  at 759–60  In Quill Corp  v  North Dakota,12× 12  504 U S  298 (1992)  the Court refused to overrule the “bright-line, physical-presence requirement” of Bellas Hess, leaning heavily on stare decisis 13× 13  Id  at 317–18  Three Justices joined a concurrence explaining that their decision rested solely “on the basis of stare decisis ” Id  at 320 (Scalia, J , concurring in part and concurring in the judgment)  So the physical presence test remained the law of the land while the internet conquered the earth  Justice Kennedy had joined the Quill majority and Justice Scalia’s concurring opinion emphasizing stare decisis, but by 2015 he had second thoughts  Writing separately in Direct Marketing Ass’n v  Brohl,14× 14  135 S  Ct  1124 (2015)  Justice Kennedy acknowledged that “[t]he Internet has caused far-reaching systemic and structural changes in the economy” and therefore “Quill now harms States to a degree far greater than could have been anticipated earlier ”15× 15  Id  at 1135 (Kennedy, J , concurring)  He concluded with the wish that “[t]he legal system should find an appropriate case for this Court to reexamine Quill and Bellas Hess ”16× 16  Id 
Seldom has a concurring opinion signed by a lone Justice prompted a state to officially declare an emergency  Yet in 2016, in response to Justice Kennedy’s overture, the South Dakota legislature passed a law, S B  106, “to provide for the collection of sales taxes from certain remote sellers       and to declare an emergency ”17× 17  2016 S D  Sess  Laws ch  70 pmbl  217 (codified at S D  Codified Laws § 10-64 (2017))  It required every remote seller to collect and remit sales tax if the seller’s business in South Dakota comprised either a “gross revenue” greater than $100,000 or at least 200 “separate transactions” within one calendar year 18× 18  Id  § 1  Significantly, the law did not apply retroactively 19× 19  Id  § 5  The “emergency” declaration was necessary to give the law immediate effect, for the purpose of “permitting the most expeditious possible review of the constitutionality of this law” by the U S  Supreme Court 20× 20  Id  § 8(8)  As Justice Alito put it, the “South Dakota law [was] obviously a test case ”21× 21  Transcript of Oral Argument at 27, Wayfair, 138 S  Ct  2080 (No  17-494), https://www supremecourt gov/oral_arguments/argument_transcripts/2017/17-494_7lho pdf [https://perma cc/8HYH-VU8N] 
Expeditiously, a group of remote sellers challenged the law  After being sued by South Dakota for refusing to register for the newly required sales tax license, Wayfair, Inc , Overstock com, Inc , and Newegg, Inc  moved for summary judgment in South Dakota circuit court on the grounds that S B  106 was unconstitutional under Quill and Bellas Hess — a point South Dakota conceded, indicating that it was seeking review by the U S  Supreme Court to overturn Quill 22× 22  State v  Wayfair Inc , 2017 SD 56, ¶¶ 9–11, 901 N W 2d 754, 759–60  Accordingly, the South Dakota circuit court granted the motion for summary judgment and South Dakota appealed to the state’s highest court 23× 23  Id  ¶ 12, 901 N W 2d at 760  The South Dakota Supreme Court unanimously affirmed, recognizing that South Dakota’s “arguments on the merits” may be “persuasive” but “Quill remains the controlling precedent ”24× 24  Id  ¶ 18, 901 N W 2d at 761  See generally Recent Case, State v  Wayfair Inc , 2017 SD 56, 901 N W 2d 754 (S D  2017), 131 Harv  L  Rev  2089 (2018) 
The U S  Supreme Court vacated and remanded 25× 25  Wayfair, 138 S  Ct  at 2100  Writing for the Court one last time, Justice Kennedy26× 26  Justices Thomas, Ginsburg, Alito, and Gorsuch joined Justice Kennedy’s opinion  pilloried Quill’s physical presence rule as “arbitrary, formalistic,” “anachronistic,” and “unfair and unjust” to both states and brick-and-mortar retailers 27× 27  Wayfair, 138 S  Ct  at 2092, 2095  After all, the rationale of Quill was that remote sellers lacked a sufficiently “substantial nexus” with the state to justify imposing a duty of tax collection 28× 28  Quill Corp  v  North Dakota, 504 U S  298, 311 (1992) (quoting Complete Auto Transit, Inc  v  Brady, 430 U S  274, 279 (1977))  This was wrong even in the mail-order catalog days of 1967 and 1992, but “the Internet revolution has made [Quill’s] earlier error all the more egregious and harmful ”29× 29  Wayfair, 138 S  Ct  at 2097; see also id  at 2092  The rule deprived the states of billions of dollars, since they could not force remote sellers to collect the tax and consumers hardly ever paid it on their own 30× 30  Id  at 2088 (“[C]onsumer compliance rates are notoriously low ”)  Quill “serve[d] as a judicially created tax shelter” for remote retailers who do a great deal of business online 31× 31  Id  at 2094 
Satisfied that Bellas Hess and Quill were wrongly decided, the Court then jumped the hurdle of stare decisis  The Quill Court had feared upsetting reliance interests 32× 32  Quill, 504 U S  at 317 (“Bellas Hess       has engendered substantial reliance and has become part of the basic framework of a sizable industry ”)  Wayfair shrugged off this concern, noting that “stare decisis accommodates only ‘legitimate reliance interest[s]’”; by contrast, reliance on the physical presence rule was largely due to consumers evading their use-tax obligations 33× 33  Wayfair, 138 S  Ct  at 2098 (alteration in original) (quoting United States v  Ross, 456 U S  798, 824 (1982))  Quill had also appealed to Congress’s ultimate authority over interstate commerce as a reason to abide by a precedent, even if wrongly decided 34× 34  See Quill, 504 U S  at 318–19; id  at 320 (Scalia, J , concurring in part and concurring in the judgment) (“Congress       can change the rule of Bellas Hess by simply saying so ”)  But Wayfair denied that Congress’s ability to change the law was a proper consideration:
While it can be conceded that Congress has the authority to change the physical presence rule, Congress cannot change the constitutional default rule  It is inconsistent with the Court’s proper role to ask Congress to address a false constitutional premise of this Court’s own creation  Courts have acted as the front line of review in this limited sphere; and hence it is important that their principles be accurate and logical, whether or not Congress can or will act in response 35× 35  Wayfair, 138 S  Ct  at 2096–97 
Having dispensed with the physical presence rule, the Court remanded the case to the South Dakota courts to determine in the first instance “whether some other principle in the Court’s Commerce Clause doctrine might invalidate the Act ”36× 36  Id  at 2099  But the Court listed “several features [of South Dakota law] that appear[ed] designed to prevent discrimination against or undue burdens upon interstate commerce ” Id 
Justices Thomas and Gorsuch each filed concurring opinions  Justice Thomas wistfully likened himself to Justice White — who voted for Bellas Hess but against Quill a quarter-century later — and confessed that he “should have joined [Justice White’s dissenting] opinion ”37× 37  Id  at 2100 (Thomas, J , concurring)  Justice Thomas added that the “Court’s entire negative Commerce Clause jurisprudence” is wrong and should be abandoned 38× 38  Id  Justice Gorsuch also wrote separately to express skepticism of the Court’s dormant commerce clause jurisprudence, raising “questions for another day” of whether the doctrine “can be squared with the text of the Commerce Clause, justified by stare decisis, or defended as misbranded products of federalism or antidiscrimination imperatives flowing from Article IV’s Privileges and Immunities Clause ”39× 39  Id  at 2100–01 (Gorsuch, J , concurring) 
Chief Justice Roberts dissented 40× 40  Justices Breyer, Sotomayor, and Kagan joined the Chief Justice’s dissent  Surprisingly, the dissenting Justices “agree[d] that Bellas Hess was wrongly decided, for many of the reasons given by the Court ”41× 41  Wayfair, 138 S  Ct  at 2101 (Roberts, C J , dissenting)  The dispute between the majority and the dissent turned entirely on the principles and application of stare decisis  Chief Justice Roberts argued that whether or how to reverse Quill should be left to Congress, which “has the flexibility to address these questions in a wide variety of ways” and “can focus directly on current policy concerns rather than past legal mistakes ”42× 42  Id  at 2104  He also pointed to the “baffling” burdens of compliance with the idiosyncratic tax codes of “[o]ver 10,000 jurisdictions,” particularly for small businesses, and doubted that new “software” — the majority’s proposed solution to this mess43× 43  Id  at 2098 (majority opinion) (“Eventually, software that is available at a reasonable cost may make it easier for small businesses to cope with these problems ”)  — would soon solve the problem 44× 44  Id  at 2103–04 (Roberts, C J , dissenting)  In Bellas Hess, the Court reasoned that the dormant commerce clause protects interstate business from being “entangle[d]       in a virtual welter of complicated obligations to local jurisdictions ” Nat’l Bellas Hess, Inc  v  Dep’t of Revenue, 386 U S  753, 759–60 (1967)  The dissent replied that the Court “vastly underestimate[d] the skill of contemporary man and his machines ” Id  at 766 (Fortas, J , dissenting)  The dispute in Wayfair over whether software is up to the task effectively reprised the old debate from Bellas Hess, only this time couched as part of the stare decisis inquiry’s concern for reliance interests rather than as a matter of dormant commerce clause doctrine  While Wayfair acknowledged that “[c]omplex state tax systems could have the effect of discriminating against interstate commerce,” 138 S  Ct  at 2099, the Court remarked that “[t]he physical presence rule is a poor proxy” for an inquiry into any actual burdens imposed on interstate commerce, id  at 2093 
Chief Justice Roberts emphasized that a “heightened form of stare decisis”45× 45  Wayfair, 138 S  Ct  at 2102 (Roberts, C J , dissenting)  applies when “Congress       can, if it wishes, override this Court’s decisions with contrary legislation ”46× 46  Id  at 2101 (first citing Michigan v  Bay Mills Indian Cmty , 134 S  Ct  2024, 2036 (2014) (tribal sovereign immunity); then citing Kimble v  Marvel Entm’t, LLC, 135 S  Ct  2401, 2409 (2015) (statutory interpretation); and then citing Halliburton Co  v  Erica P  John Fund, Inc , 134 S  Ct  2398, 2411 (2014) (judicially created doctrine implementing a judicially created cause of action))  In Quill, the Chief Justice noted, the Court had taken to heart that “Congress may be better qualified” and “has the ultimate power to resolve” the question47× 47  Id  at 2102 (quoting Quill Corp  v  North Dakota, 504 U S  279, 318 (1992))  while Justice Scalia had “recogniz[ed] that stare decisis has ‘special force’ in the dormant Commerce Clause context due to Congress’s ‘final say over regulation of interstate commerce ’”48× 48  Id  (quoting Quill, 504 U S  at 320 (Scalia, J , concurring in part and concurring in the judgment))  Moreover, “[i]f stare decisis applied with special force in Quill, it should be an even greater impediment” afterward since Quill effectively “tossed [the ball] into Congress’s court ”49× 49  Id  (alteration in original) (quoting Kimble, 135 S  Ct  at 2409); cf  Bay Mills, 134 S  Ct  at 2039 n 12 (“When we inform Congress that it has primary responsibility over a sphere of law, and invite Congress to consider a specific issue within that sphere, we cannot deem irrelevant how Congress responds ”)  Because the Court invited Congress to act and then “suddenly chang[ed] the ground rules, the Court may have waylaid Congress’s consideration of the issue ”50× 50  Wayfair, 138 S  Ct  at 2102–03 (Roberts, C J , dissenting) 
In Wayfair, the Court applied the flimsier form of stare decisis to a precedent that could have been overruled by Congress  It did so in the context of a dormant commerce clause case, but Wayfair’s logic extends to all constitutional default rules — that is, constitutional decisions that Congress remains free to change  Not only does Wayfair deviate from the Court’s decades-old stare decisis analysis, it also imperils other precedents that set constitutional default rules 
The Court’s reasoning in Wayfair departs from its prior stare decisis analysis  In 1932, Justice Brandeis posited that stare decisis must bend “in cases involving the Federal Constitution, where correction through legislative action is practically impossible ”51× 51  Burnet v  Coronado Oil & Gas Co , 285 U S  393, 406–07 (1932) (Brandeis, J , dissenting)  The Court has long since adopted his argument,52× 52  See, e g , Smith v  Allwright, 321 U S  649, 665 (1944)  as well as its corollary — that stare decisis commands “special force in the area of statutory interpretation” where “Congress remains free to alter what [the Court has] done ”53× 53  Patterson v  McLean Credit Union, 491 U S  164, 172–73 (1989)  For normative evaluations of heightened stare decisis for statutory precedents, see generally Einer Elhauge, Statutory Default Rules: How to Interpret Unclear Legislation 211–23 (2008); and William N  Eskridge, Jr , Overruling Statutory Precedents, 76 Geo  L J  1361, 1364–1409 (1988)  Justice Brandeis’s logic demands that dormant commerce clause cases, where Congress is free to act, be granted the weightier stare decisis 54× 54  Scholars have noted the curious fact that Justice Brandeis included many dormant commerce clause cases as examples of overruled constitutional precedents  See, e g , Earl M  Maltz, Commentary, Some Thoughts on the Death of Stare Decisis in Constitutional Law, 1980 Wis  L  Rev  467, 468–469, 469 n 11  One explanation for this is that Justice Brandeis sought the authority of Chief Justice Taney’s dictum that the Court’s “opinion upon the construction of the Constitution is always open to discussion” — which referred to the dormant commerce clause  See Burnet, 285 U S  at 408 n 3 (Brandeis, J , dissenting) (quoting The Passenger Cases, 48 U S  (7 How ) 283, 470 (1849) (Taney, C J , dissenting))  In Chief Justice Taney’s time, it was thought that Congress could not override the Court’s dormant commerce clause decisions, see Cooley v  Bd  of Wardens, 53 U S  (12 How ) 299, 321 (1852), so the context of Chief Justice Taney’s dictum does not conflict with Justice Brandeis’s theory of stare decisis  The Court applied this reasoning in Quill, as Chief Justice Roberts underscored 55× 55  Wayfair, 138 S  Ct  at 2102 (Roberts, C J , dissenting) 
Yet the Wayfair majority refused to consider Congress’s authority to legislate as a relevant factor for stare decisis 56× 56  Even Justice Kennedy’s earlier opinion in Direct Marketing contemplated judicially overruling Quill, conspicuously neglecting a possible legislative solution  See supra p  278  The Court even insisted that to do so “is inconsistent with the Court’s proper role,” since Quill embodied “a false constitutional premise of th[e] Court’s own creation ”57× 57  Wayfair, 138 S  Ct  at 2096 (emphasis added)  This refusal breaks from the practical Brandeisian wisdom that has guided the Court’s treatment of precedent for the better part of a century  The point is not that stare decisis should have ultimately propped up Bellas Hess yet again, as Wayfair’s dissenting Justices maintained  After all, a realistic approach that is alert to each branch’s institutional capacities might have led to the conclusion that Congress was actually ill-equipped to overrule Quill  In this vein, the Court could have sensibly pointed out that Congress is unlikely to stick its neck out with a tax hike (or a look-alike) from which only the states would benefit 58× 58  For two practical arguments to this effect, see Brian Galle, Essay, Kill Quill, Keep the Dormant Commerce Clause: History’s Lessons on Congressional Control of State Taxation, 70 Stan  L  Rev  Online 158, 160–62 (2018), https://review law stanford edu/wp-content/uploads/sites/3/2018/03/70-Stan -L -Rev -Online-158-Galle pdf [https://perma cc/22YP-P4V5]; Edward A  Zelinsky, The Political Process Argument for Overruling Quill, 82 Brook  L  Rev  1177, 1191–92 (2017)  Indeed, South Dakota advanced such practical arguments in its brief 59× 59  See Petitioner’s Brief at 54, Wayfair, 138 S  Ct  2080 (No  17-494) (“Congress has little incentive to act here because it would be (or appear to be) authorizing new or greater tax collections from its constituents, while receiving none of the revenue in return ”)  More generally, the Court might have discussed the limits of the states’ influence in the federal system as a reason not to wait for congressional intervention, a topic it has debated on other occasions 60× 60  See Richard H  Pildes, Institutional Formalism and Realism in Constitutional and Public Law, 2013 Sup  Ct  Rev  1, 30–32; see also Galle, supra note 58, at 159 (“Congress is not a trustworthy guardian of state fiscal power, making continuing judicial involvement a more appealing prospect ”)  Or it could have argued that new facts on the ground — namely, the blast of e-commerce that hit like a comet after Quill — overpowered stare decisis of any force, special or plain 61× 61  Two recent studies of stare decisis highlighted the physical presence rule as exemplifying a precedent that may reasonably be overruled due to changed facts  See Bryan A  Garner et al , The Law of Judicial Precedent 364–65 (2016); Randy J  Kozel, Settled Versus Right: A Theory of Precedent 112–13 (2017)  It should be noted that the authors of The Law of Judicial Precedent classify the physical presence rule as a constitutional precedent for stare decisis purposes, thus anticipating the Court’s misstep in Wayfair  Garner et al , supra, at 354–65  Because even statutory precedents may sometimes be overruled,62× 62  See Patterson v  McLean Credit Union, 491 U S  164, 173–74 (1989) (discussing justifications for overruling statutory precedents)  Contra Lawrence C  Marshall, “Let Congress Do It”: The Case for an Absolute Rule of Statutory Stare Decisis, 88 Mich  L  Rev  177 (1989)  the Court could have killed Quill without first planting its constitutional kiss of death 63× 63  Cf  Thomas R  Lee, Stare Decisis in Historical Perspective: From the Founding Era to the Rehnquist Court, 52 Vand  L  Rev  647, 704 (1999) (“Justice Brandeis’       memorable prose has since become a mandatory part of the burial rite for any constitutional precedent ”) 
The Court resisted such arguments  Instead, Wayfair reasoned that Congress’s total ability to correct an erroneous decision counts for nothing when the Court gets the Constitution wrong  That such a theory sprouts from a case like Wayfair, which repudiated a “formalistic distinction,”64× 64  Wayfair, 138 S  Ct  at 2092  is ironic  Wayfair’s stare decisis analysis resorts to the formalism of making constitutional a “magic” word65× 65  See Transcript of Oral Argument, supra note 21, at 12  rather than asking whether Congress can step in 
Moreover, the Court’s new thinking on stare decisis threatens other constitutional default rules  Wayfair now stands for the proposition that a “constitutional default rule” — a term the Court apparently lifted from South Dakota’s reply brief on the merits66× 66  Reply Brief at 22, Wayfair, 138 S  Ct  2080 (No  17-494) (“Congress is polarized, which makes it critical       to get the constitutional default rule right ”)  — gets only weakened stare decisis  To appreciate why this holding matters, it is worth exploring the concept and scope of constitutional default rules  Contract theory describes default rules as legal rules that the parties may “contract around ”67× 67  See, e g , Ian Ayres & Robert Gertner, Filling Gaps in Incomplete Contracts: An Economic Theory of Default Rules, 99 Yale L J  87, 87 (1989)  Although “constitutional default rule” could be read broadly to include a variety of actors and contracting mechanisms,68× 68  See John Ferejohn & Barry Friedman, Toward a Political Theory of Constitutional Default Rules, 33 Fla  St  U  L  Rev  825, 826 (2006) (“When we speak of default rules in constitutional law, we typically are talking about specifications of ways the government can act (or modify its behavior) to get around a constitutional prohibition ”)  the Court’s use of the term for purposes of stare decisis may be narrowly defined as judicial precedents of constitutional law that “are ultimately subject to congressional control ”69× 69  Gillian E  Metzger, Congress, Article IV, and Interstate Relations, 120 Harv  L  Rev  1468, 1525 (2007) (describing judicially enforceable “constitutional default rules imposing obligations on the states in the name of union [that] are ultimately subject to congressional control”)  The dormant commerce clause is a paradigmatic constitutional default rule because what the Court does today Congress may undo tomorrow  Justice Scalia declared this fact “[t]he clearest sign that the negative Commerce Clause is a judicial fraud,” for “[h]ow could congressional consent lift a constitutional prohibition?”70× 70  Comptroller of the Treasury of Md  v  Wynne, 135 S  Ct  1787, 1808 (2015) (Scalia, J , dissenting)  But that’s what a constitutional default rule is  The Court has allowed Congress to overturn its dormant commerce clause cases since 1891 71× 71  See In re Rahrer, 140 U S  545, 560–62 (1891) 
Dormant commerce clause cases are not the only constitutional default rules  Professor Laurence Tribe’s treatise identifies two others 72× 72  1 Laurence H  Tribe, American Constitutional Law § 6-35 (3d ed  2000)  And in a groundbreaking article, Professor Henry Monaghan revealed “a substructure of substantive, procedural, and remedial rules” forming “a constitutional common law subject to amendment, modification, or even reversal by Congress ”73× 73  Henry P  Monaghan, The Supreme Court, 1974 Term — Foreword: Constitutional Common Law, 89 Harv  L  Rev  1, 2–3 (1975); see also Susan R  Klein, Identifying and (Re)Formulating Prophylactic Rules, Safe Harbors, and Incidental Rights in Constitutional Criminal Procedure, 99 Mich  L  Rev  1030 (2001) (further developing Monaghan’s theory in criminal procedure context)  What follows is a list of six lines of cases beyond the dormant commerce clause that may be fairly described as constitutional default rules  The first two are drawn from Tribe’s treatise while the next four are found in Monaghan’s article:
(1) State Taxation of Federal Instrumentalities: States may not tax instrumentalities of the federal government74× 74  McCulloch v  Maryland, 17 U S  (4 Wheat ) 316, 436 (1819)  — unless Congress consents 75× 75  See, e g , Helvering v  Gerhardt, 304 U S  405, 411 n 1 (1938) (“Congress may curtail an immunity which might otherwise be implied or enlarge it beyond the point where, Congress being silent, the Court would set its limits ” (citations omitted))  One court has described such judicial decisions as setting a “constitutional default rule ” United States v  Delaware, 958 F 2d 555, 560 n 9 (3d Cir  1992) (“[W]e must decide the constitutional default rule for this type of tax, fully aware that Congress could decide at any time to reverse our decision statutorily ”)  (2) Article I, Section 10 Cases: Article I, Section 10 provides that certain prohibitions on the states may be waived by Congress 76× 76  See U S  Const  art  I, § 10, cls  2–3  The Court has taken note of this when considering whether to overrule, for instance, an Import-Export Clause precedent 77× 77  See Hooven & Allison Co  v  Evatt, 324 U S  652, 668 (1945) (“In view of the fact that the Constitution gives Congress authority to consent to state taxation of imports and hence to lay down its own test for determining when the immunity ends, we see no convincing practical reason for abandoning the test which has been applied for more than a century        ”), overruled on other grounds by Limbach v  Hooven & Allison Co , 466 U S  353 (1984)  In Michelin Tire Corp  v  Wages, 423 U S  276 (1976), the Court left open the question whether “Congress may authorize, under the Import-Export Clause, an exaction that it could not directly impose under the Tax Clause ” Id  at 301 n 13  Metzger, however, argues that the Import-Export Clause is free of other clauses’ limits on congressional power  See Metzger, supra note 69, at 1500 & n 120  (3) Bivens Cases: In Bivens v  Six Unknown Named Agents of Federal Bureau of Narcotics,78× 78  403 U S  388 (1971)  the Court held that a violation of the Fourth Amendment gives rise to a right to sue for damages 79× 79  Id  at 397  But the Court has also held that “[s]uch a cause of action may be defeated       when       Congress has provided an alternative remedy which it explicitly declared to be a substitute for recovery directly under the Constitution and viewed as equally effective ”80× 80  Carlson v  Green, 446 U S  14, 18–19 (1980)  (4) Miranda Cases: The Miranda Court famously “encourage[d]” Congress and the states to explore alternative “procedures which are at least as effective in apprising accused persons of their right of silence and in assuring a continuous opportunity to exercise it ”81× 81  Miranda v  Arizona, 384 U S  436, 467 (1966)  In Dickerson v  United States, 530 U S  428 (2000), the Court struck down a congressional attempt to effectively abolish Miranda, holding that “Miranda announced a constitutional rule that Congress may not supersede legislatively ” Id  at 444  But Dickerson also stood by Miranda’s “invitation for legislative action” to replace Miranda with an adequate substitute  Id  at 440; see also Michael C  Dorf & Barry Friedman, Shared Constitutional Interpretation, 2000 Sup  Ct  Rev  61 (discussing legislative alternatives to Miranda)  (5) The Police Lineup Case: In United States v  Wade,82× 82  388 U S  218 (1967)  the Court created an exclusionary rule for evidence obtained from a police lineup in violation of the Sixth Amendment right to counsel but acknowledged that it could be replaced by “[l]egislative or other regulations       which eliminate the risks of abuse ”83× 83  Id  at 239  (6) The Exclusionary Rule Cases: Mapp v  Ohio made the Fourth Amendment “exclusionary rule” binding on the states,84× 84  367 U S  643, 655 (1961)  yet Congress is thought to have the power to replace it 85× 85  See Bivens v  Six Unknown Named Agents of Fed  Bureau of Narcotics, 403 U S  388, 422–24 (1971) (Burger, C J , dissenting) (inviting Congress to replace the Fourth Amendment exclusionary rule); Harold J  Krent, How to Move Beyond the Exclusionary Rule: Structuring Judicial Response to Legislative Reform Efforts, 26 Pepp  L  Rev  855, 864–71 (1999) 
All of the above are arguably constitutional default rules set by the Court that remain, to one degree or another, open to congressional revision  The list could be longer or shorter, depending on which default rules the Court will view as constitutional86× 86  A shorter list could be produced by whittling away at the constitutional status of the cases identified by Monaghan  While the Court has held that Miranda is a constitutional decision, Dickerson, 530 U S  at 444, some of the other cases may be viewed as nonconstitutional  See, e g , Collins v  Virginia, 138 S  Ct  1663, 1675–80 (2018) (Thomas, J , concurring) (arguing that Mapp is “nonconstitutional,” id  at 1678 n 5); Richard H  Fallon, Jr  et al , Hart and Wechsler’s The Federal Courts and the Federal System 775–77 (7th ed  2015) (discussing whether Bivens is constitutionally required)  Conversely, a longer list might include any constitutional right that can be waived by a party  See, e g , Daniel A  Farber, Another View of the Quagmire: Unconstitutional Conditions and Contract Theory, 33 Fla  St  U  L  Rev  913, 918 (2006) (describing the Eleventh Amendment as “just a contractual default rule that the states are free to barter away”)  Such a list might also include various constitutionally inspired judicial presumptions  See, e g , Jack Goldsmith & John F  Manning, The President’s Completion Power, 115 Yale L J  2280, 2299 (2006) (describing the Chevron presumption of delegated interpretive power to administrative agencies as “a constitutionally inspired default rule”); Nicholas Quinn Rosenkranz, Federal Rules of Statutory Interpretation, 115 Harv  L  Rev  2085, 2097–98 (2002) (describing clear statement rules as “constitutional default rules” reversible by Congress)  Many other decisions could likely be characterized as constitutional default rules; the list above is only an initial stab  and on how it will answer open questions about congressional authority over certain constitutional provisions 87× 87  See, e g , Thomas v  Wash  Gas Light Co , 448 U S  261, 272 n 18 (1980) (plurality opinion) (leaving unresolved whether Congress may limit constitutional full faith and credit obligations); White v  Mass  Council of Constr  Emp’rs, Inc , 460 U S  204, 215 n 1 (1983) (Blackmun, J , concurring in part and dissenting in part) (leaving unresolved “whether Congress may authorize       what otherwise would be a violation” of the Privileges and Immunities Clause); 1 Tribe, supra note 72, § 6-35, at 1243–44 (arguing that Congress cannot override judicial constructions of the Privileges and Immunities Clause); Metzger, supra note 69, at 1486–89 (arguing the opposite)  But the takeaway is clear: weaker stare decisis for constitutional default rules  Pre-Wayfair, one would have thought that stare decisis applies with special force to such precedents, given congressional power to set them straight  Not anymore  Why? Because it is improper to “ask Congress to address a false constitutional premise of th[e] Court’s own creation ”88× 88  Wayfair, 138 S  Ct  at 2096  The Latin for Wayfair’s doctrine is not stare decisis, which should reflect a realistic, working relationship between the legislative and judicial branches  It is mea culpa 
In its zeal to update the Constitution for “the Cyber Age,”89× 89  Id  at 2097  the Court deleted Congress from stare decisis doctrine in constitutional cases  The Court had better options  It could have left Quill on Congress’s doorstep, as the dissent argued  Or it could have justified overruling Quill notwithstanding the special force of stare decisis  Instead, the Court reasoned that it doesn’t matter whether Congress is willing and able to do the job: a constitutional mess calls for a judicial clean-up crew  For constitutional default rules — a category of decisions embracing the dormant commerce clause and sweeping far beyond — Wayfair’s new theory of stare decisis makes the Court’s precedents less sticky and Congress less relevant 



ground_truth: Weep Not, Child was first published in 1964.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What did Justice Kennedy argue about Quill in Direct Marketing Ass'n v. Brohl?

Read the following text and answer briefly.

Context 4:  The process is repeated for all the couplings obtaining then a fully connected adjacency matrix  The results here presented are obtained with $\lambda = 0 01$ 
   For the minimization we have used the MATLAB routine \emph{minFunc\_2012}\cite{min_func}  
      
      \begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{Jor11_2D_l2_JR_soJR_TPJR}
    	\caption{Top panels: instances of single site coupling reconstruction for the case of $N=64$ XY spins on a 2D lattice with ordered $J$ (left column) and bimodal distributed $J$ (right column)  
    	
    	Bottom panels: sorted couplings }
    	\label{PL-Jor1}
    \end{figure}

   
To produce the data by means of numerical Monte Carlo simulations a system with $N=64$  spin variables  is considered on a deterministic 2D lattice with periodic boundary conditions  
Each spin has then connectivity $4$, i e , we expect to infer an adjacency matrix with $N c = 256$ couplings different from zero  
The dynamics of the simulated model is based on the Metropolis algorithm and parallel tempering\cite{earl05} is used to speed up the thermalization of the system 
The thermalization is tested looking at the average energy over logarithmic time windows and
the acquisition of independent configurations
starts only after the system is well thermalized 

   For the values of the couplings we considered two cases: an ordered case, indicated in the figure as  $J$ ordered (e g , left column of Fig  \ref{PL-Jor1}) where the couplings can take values $J_{ij}=0,J$, with $J=1$, 
   and a quenched disordered case, indicated in the figures as  $J$ disordered (e g , right column of Fig  \ref{PL-Jor1})
   where the couplings can take also  negative values, i e , 
    $J_{ij}=0,J,-J$, with a certain probability   The results here presented were obtained with bimodal distributed $J$s: 
   
    $P(J_{ij}=J)=P(J_{ij}=-J)=1/2$   The performances of the PLM have shown not to depend on $P(J)$  
   
    We recall that in Sec  \ref{sec:plm} we used the temperature-rescaled notation, i e , $J_{ij}$ stands for $J_{ij}/T$  
   
    To analyze the performances of the PLM, in Fig  \ref{PL-Jor1} the inferred couplings, $\mathbb{J}^R_{\rm inf}$, are shown on top of the original couplings,  $\mathbb{J}^R_{\rm true}$ 
     The first figure (from top) in the left column shows  the $\mathbb{J}^R_{\rm inf}$ (black) and the $\mathbb{J}^R_{\rm tru}$ (green) for a given spin
     at temperature $T/J=0 7$ and number of samples  $M=1024$  PLM appears to reconstruct the correct couplings, though zero couplings are always given a small inferred non-zero value  
     In the left column of Fig   \ref{PL-Jor1},  both the $\mathbb{J}^R_{\rm{inf}}$ and the $\mathbb{J}^R_{\rm{tru}}$ are sorted in decreasing order and plotted on top of each other  
     We can clearly see that $\mathbb{J}^R_{\rm inf}$ reproduces the expected step function  Even though the jump is smeared, the difference between inferred couplings corresponding to the set of non-zero couplings 
     and to the set of zero couplings can be clearly appreciated 
     Similarly, the plots in the right column of Fig  \ref{PL-Jor1} show the results obtained for the case with  bimodal disordered couplings, for the same working temperature and number of samples  
     In particular, note that the algorithm infers half positive and half negative couplings, as expected 
     
     
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{Jor11_2D_l2_errJ_varT_varM}
\caption{Reconstruction error $\mbox{err}_J$, cf  Eq  (\ref{eq:errj}), plotted as a function of temperature (left) for three values of the number of samples $M$ and  as a function $M$ (right) for three values of temperature in the ordered system, i e , $J_{ij}=0,1$  
The system size is $N=64$ }
\label{PL-err-Jor1}
\end{figure}

In order to analyze the effects of the number of samples and of the temperature regimes, we plot in Fig  \ref{PL-err-Jor1} the reconstruction error, Eq  (\ref{err}), as a function of temperature for three different sample sizes $M=64,128$ and $512$  
The error is seen to sharply rise al low temperature, incidentally, in the ordered case, for  $T<T_c \sim 0 893$, which is the Kosterlitz-Thouless transition temperature of the 2XY model\cite{Olsson92}  
 However, we can see that if only $M=64$ samples are considered, $\mbox{err}_J$ remains high independently on the working temperature  
 In the right plot of Fig  \ref{PL-err-Jor1},  $\mbox{err}_J$ is plotted as a function of $M$ for three different working temperatures $T/J=0 4,0 7$ and $1 3$  As we expect, 
 $\mbox{err}_J$  decreases as $M$ increases  This effect was observed also with mean-field inference techniques on the same model\cite{Tyagi15} 

To better understand the performances of the algorithms, in Fig  \ref{PL-varTP-Jor1} we show several True Positive (TP) curves obtained for various values of $M$ at three different temperatures $T$  As $M$ is large and/or temperature is not too small,  we are able to reconstruct correctly all the couplings present in the system (see bottom plots) 
The True Positive curve displays how many times the inference method finds a true link of the original network  as a function of the index of the vector of sorted absolute value of reconstructed couplings $J_{ij}^{\rm inf}$  
The index $n_{(ij)}$ represents the related spin couples $(ij)$  The TP curve is obtained as follows: 
first  the values $|J^{\rm inf}_{ij}|$ are sorted in descending order and  the spin pairs $(ij)$ are ordered according to the sorting position of $|J^{\rm inf}_{ij}|$  Then,
     	a cycle over the ordered set of pairs $(ij)$, indexed by $n_{(ij)}$, is performed, comparing with the original network coupling $J^{\rm true}_{ij}$ and verifying whether it is zero or not  The true positive curve is computed as
\begin{equation}
\mbox{TP}[n_{(ij)}]= \frac{\mbox{TP}\left[n_{(ij)}-1\right] (n_{ij}-1)+ 1 -\delta_{J^{\rm true}_{ij},0}}{n_{(ij)}}
\end{equation}
As far as $J^{\rm true}_{ij} \neq 0$, TP$=1$  As soon as the true coupling of a given $(ij)$ couple in the sorted list is zero, the TP curve departs from one  
In our case, where the connectivity per spin of the original system is  $c=4$ and there are $N=64$ spins, we know that we will have  $256$ non-zero couplings   
     	If the inverse problem is successful, hence, we expect a steep decrease of the TP curve when  $n_{ij}=256$ is overcome 

In Fig  \ref{PL-varTP-Jor1}
it is  shown that,  almost independently of $T/J$, the TP score improves as $M$ increases  Results are plotted for three different temperatures, $T=0 4,1$ and $2 2$, with increasing number of samples $M = 64, 128,512$ and $1024$ (clockwise)  
We can clearly appreciate the improvement in temperature if the size of the data-set is not very large: for small $M$, $T=0 4$ performs better  
When $M$ is high enough (e g , $M=1024$), instead, the TP curves do not appear to be strongly influenced by the temperature 

\begin{figure}[t!]
	\centering
	\includegraphics[width=1\linewidth]{Jor11_2D_l2_TPJR_varT_varM}
	\caption{TP  curves  for 2D short-range ordered $XY$ model with $N=64$ spins at three different values of $T/J$ with increasing - clockwise from top - $M$ }
	\label{PL-varTP-Jor1} 
\end{figure} 

\subsection{$XY$ model with complex-valued couplings}
For the complex $XY$ we have to contemporary  infer $2$ apart coupling matrices,  $J^R_{i j}$ and $J^I_{i j}$   As before, a system of $N=64$ spins is considered on a 2D lattice 
For the couplings we have considered both ordered and bimodal disordered cases 
In Fig  \ref{PL-Jor3}, a single row of the matrix $J$ (top) and the whole sorted couplings (bottom) are displayed for the ordered model (same legend as in Fig  \ref{PL-Jor1}) for the real, $J^R$ (left column), and the imaginary part, $J^I$   

\begin{figure}[t!]
	\centering
\includegraphics[width=1\linewidth]{Jor3_l2_JRJI_soJRJI_TPJRJI}
	\caption{Results related to the ordered complex XY model with $N=64$ spins on a  2D lattice  Top: instances of  single site reconstruction for the real, JR (left column), and
		the imaginary, JI (right column), part of $J_{ij}$  Bottom: sorted values of JR (left) and JI (right) }
		
	\label{PL-Jor3}
\end{figure}
 
 
  \section{PLM with Decimation}
 \label{sec:res_dec}
 

\begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_tPLF_varT_varM}
    	\caption{Tilted Pseudolikelyhood, ${\cal L}_t$, plotted as a function of decimated couplings  Top: Different ${\cal L}_t$ curves obtained for different values of $M$ plotted on top of each other  Here $T=1 3$  The black line indicates the expected number of decimated couplings, $x^*=(N (N-1) - N c)/2=1888$  As we can see, as $M$ increases, the maximum point of ${\cal L}_t$ approaches $x^*$  Bottom: Different ${\cal L}_t$ curves obtained for different values of T with $M=2048$  We can see that, with this value of $M$, no differences can be appreciated on the maximum points of the different  ${\cal L}_t$ curves }
   	\label{var-$t$PLF}
   \end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[width=1\linewidth]{Jor1_dec_tPLF_peak_statistics_varM_prob eps}
    	\caption{Number of most likely decimated couplings, estimated by the maximum point of $\mathcal{L}_t$, as a function of the number of samples $M$  We can clearly see that the maximum point of $\mathcal{L}_t$ tends toward $x^*$, which is the right expected number of zero couplings in the system }  
    	\label{PLF_peak_statistics}
    \end{figure}
      
      For the ordered real-valued XY model we show in Fig  \ref{var-$t$PLF}, top panel, the outcome on the tilted pseudolikelyhood, $\mathcal{L}_t$ Eq  \eqref{$t$PLF}, of the progressive decimation: from a fully connected lattice  down to an empty lattice  The figure shows the behaviour of $\mathcal{L}_t$ for three different data sizes $M$  A clear data size dependence of the maximum point of  $\mathcal{L}_t$, signalling the most likely value for decimation, is shown  For small $M$ the most likely number of  couplings is overestimated and for increasing $M$ it tends to the true value, as displayed in Fig  \ref{PLF_peak_statistics}   In the bottom panel of Fig  \ref{var-$t$PLF} we display instead different 
 $\mathcal{L}_t$ curves obtained for three different values of $T$ 
  Even though the values of $\mathcal{L}_t$ decrease with increasing temperature, the value of the most likely number of decimated couplings appears to be quite independent on $T$ with $M=2048$ number of samples 
In Fig  \ref{fig:Lt_complex} we eventually display the tilted pseudolikelyhood for a 2D network with complex valued ordered couplings, where the decimation of the real and imaginary coupling matrices proceeds in parallel, that is, 
when a real coupling is small enough to be decimated its imaginary part is also decimated, and vice versa 
One can see that though the apart errors for the real and imaginary parts are different in absolute values, they display the same dip, to be compared with the maximum point of $\mathcal{L}_t$ 
     
       \begin{figure}[t!]
    	\centering
      	\includegraphics[width=1\linewidth]{Jor3_dec_tPLF_new}
      	\caption{Tilted Pseudolikelyhood, ${\cal L}_t$, plotted with the reconstruction errors for the XY model with $N=64$ spins on a 2D lattice  These results refer to the case of  ordered and complex valued couplings  The full (red) line indicates ${\cal L}_t$  The dashed (green) 
      		and the dotted (blue) lines show the reconstruction errors (Eq  \eqref{eq:errj}) obtained for the real and the imaginary couplings respectively  We can see that both ${\rm err_{JR}}$ and ${\rm err_{JI}}$ have a minimum at $x^*$ }
          	\label{fig:Lt_complex}
    \end{figure}

\begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_JR_soJR_TPJR}
   	\caption{XY model on a 2D lattice with $N=64$ sites and real valued couplings  The graphs show the inferred (dashed black lines) and true couplings (full green lines) plotted on top of each other  The left and right columns refer to the
   		 cases of ordered and bimodal disordered couplings, respectively  Top figures: single site reconstruction, i e , one row of the matrix $J$  Bottom figures: couplings are plotted sorted in descending order }  
   	\label{Jor1_dec}
   \end{figure}
   
\begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{Jor3_dec_JRJI_soJRJI_TPJRJI}
    	\caption{XY model on a 2D lattice with $N=64$ sites and ordered complex-valued couplings 
    		The inferred and true couplings are plotted on top of each other  The left and right columns show the real and imaginary parts, respectively, of the couplings  Top figures refer to a single site reconstruction, i e , one row of the matrix $J$  Bottom figures report the couplings sorted in descending order }
    	\label{Jor3_dec}
    \end{figure}
    
       







       \begin{figure}[t!]
     	\centering
     	\includegraphics[width=1\linewidth]{MF_PL_Jor1_2D_TPJR_varT}
     	\caption{True Positive curves obtained with the three techniques: PLM with decimation, (blue) dotted line,  PLM with $l_2$ regularization, (greed) dashed line, and mean-field, (red) full line   These results refer to real valued ordered couplings with $N=64$ spins on a 2D lattice  The temperature is here $T=0 7$ while the four graphs refer to different sample sizes: $M$ increases clockwise }
     	\label{MF_PL_TP}
     \end{figure}
    
    \begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{MF_PL_Jor1_2D_errJ_varT_varM}
    	\caption{Variation of reconstruction error, ${\rm err_J}$, with respect to temperature as obtained with the three different techniques, see Fig  \ref{MF_PL_TP}, for four different sample size:  clockwise from top   $M=512,1024, 2048$ and $4096$ } 
    	\label{MF_PL_err}
    \end{figure}
    
     Once the most likely network has been identified through the decimation procedure, we perform the same analysis displayed in Fig  \ref{Jor1_dec}  for ordered and then quenched disordered real-valued couplings
and in Fig  \ref{Jor3_dec} for  complex-valued ordered couplings   In comparison to the results shown in Sec  \ref{sec:res_reg},
  the PLM with decimation leads to rather cleaner results  In Figs  \ref{MF_PL_err} and \ref{MF_PL_TP} we compare the performances of the PLM with decimation in respect to ones of the PLM with $l_2$-regularization  These two techniques are also analysed in respect to a mean-field technique previously implemented on the same XY systems\cite{Tyagi15} 
  
    For what concerns the network of connecting links, in Fig  \ref{MF_PL_TP} we compare the TP curves obtained with the three techniques  The results refer to the case of ordered and real valued couplings, but similar behaviours were obtained for the other cases analysed  
  The four graphs are related to different sample sizes, with $M$ increasing clockwise  When $M$ is high enough, all techniques reproduce the true network  
  However, for lower values of $M$ the performances of the PLM with $l_2$ regularization and with decimation drastically overcome those ones of the previous mean field technique  
  In particular, for $M=256$ the PLM techniques still reproduce the original network while the mean-field method fails to find more than half of the couplings  
  When $M=128$, the network is clearly reconstructed only through the PLM with decimation while the PLM with $l_2$ regularization underestimates the couplings  
  Furthermore, we notice that the PLM method with decimation is able to clearly infer the network of interaction even when $M=N$ signalling that it could be considered also in the under-sampling regime $M<N$   
 
 
In Fig  \ref{MF_PL_err} we compare the temperature behaviour of the reconstruction error 
In can be observed that for all temperatures and for all sample sizes  the reconstruction error, ${\rm err_J}$, (plotted here in log-scale) obtained with the PLM+decimation is always smaller than 
that one obtained with the other techniques  The temperature behaviour of ${\rm err_J}$ agrees with the one already observed for Ising spins in \cite{Nguyen12b} and for XY spins  in \cite{Tyagi15} with a mean-field approach:  ${\rm err_J}$ displays a minimum around $T\simeq 1$ and then it increases for very lower $T$; however,
 the error obtained with the PLM with decimation is several times smaller  than the error estimated by the other methods 



 
 

     
     \section{Conclusions}
     \label{sec:conc}


Different statistical inference methods have been applied to the inverse problem of the XY model 
After a short review of techniques based on pseudo-likelihood and their formal generalization to the model we have tested their performances against data generated by means of Monte Carlo numerical simulations of known instances
with diluted, sparse, interactions 

The main outcome is that the best performances are obtained by means of the  pseudo-likelihood method combined with decimation  Putting to zero (i e , decimating) very weak bonds, this technique turns out to be very precise for  problems whose real underlying interaction network is sparse, i e , the number of couplings per variable does not scale with number of variables 
The PLM + decimation method is compared to the PLM + regularization method, with $\ell_2$ regularization and to a mean-field-based method  The behavior of the quality of the network reconstruction is analyzed by looking at the overall sorted couplings and at the single site couplings, comparing them with the real network, and at the true positive curves in all three approaches  In the PLM +decimation method, moreover, the identification of the number of decimated bonds at which the tilted pseudo-likelihood is maximum allows for a precise estimate of the total number of bonds  Concerning this technique, it is also shown that the network with the most likely number of bonds is also the one of least reconstruction error, where not only the prediction of the presence of a bond is estimated but also its value 

The behavior of the inference quality in temperature and in the size of data samples is also investigated, basically confirming the low $T$ behavior hinted by Nguyen and Berg \cite{Nguyen12b} for the Ising model  In temperature, in particular, the reconstruction error curve displays a minimum at a low temperature, close to the critical point in those cases in which a critical behavior occurs, and a sharp increase as temperature goes to zero  The decimation method, once again, appears to enhance this minimum of the reconstruction error of almost an order of magnitude with respect to other methods 
 
The techniques displayed and the results obtained in this work can be of use in any of the many systems whose theoretical representation is given by Eq  \eqref{eq:HXY} or Eq  \eqref{eq:h_im}, some of which are recalled in Sec  \ref{sec:model}  In particular, a possible application can be the field of light waves propagation through random media and the corresponding problem of the  reconstruction of an object seen through an opaque medium or a disordered optical fiber \cite{Vellekoop07,Vellekoop08a,Vellekoop08b, Popoff10a,Akbulut11,Popoff11,Yilmaz13,Riboli14} 

 	



ground_truth: Quill harmed states more than anticipated due to the Internet.