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The walkways were lined with American flags on Monday as more than 100 people arrived at the Greater Pine Island Veterans of Foreign Wars Post #4353 for the annual Memorial Day ceremony. Pine Island VFW Commander Austin White hosted the event.
"Today is a day we need to put aside our disagreements and take time to remember those who are often out of sight but never out of our hearts and minds," Commander Austin White said.
White added a "special touch" to this year's Memorial Day by adding a ceremony about the "13 folds of the U.S. flag." The flag-folding ceremony represents the same religious principles on which our country was originally founded. As the honor guard slowly folded the flag, White recited the symbolic meaning of each of the 13 folds.
During the annual Memorial Day ceremony conducted by the Greater Pine Island Veterans of Foreign Wars Post #4353, the Color Guard demonstrated the proper way to fold a flag.
During the annual Memorial Day ceremony conducted by the Greater Pine Island Veterans of Foreign Wars Post #4353, the Color Guard performed a 21-gun salute.
"We are reminded of each of our heroes every time we see the American flag," White said. "We also honor our fallen by placing an American flag on their casket. When a flag is removed from a casket each of the 13 folds has a profound meaning."
"The first fold of our flag is a symbol of life; the second fold signifies our belief in eternal life; the third fold is made in honor and tribute of the veteran departing our ranks; the fourth fold exemplifies our weaker nature as citizens trusting in God; the fifth fold is an acknowledgment to our country; the sixth fold is for where our hearts lie; the seventh fold is a tribute to our armed forces; the eighth fold is a tribute to the one who entered into the valley of the shadow of death; the ninth fold is an honor to womanhood; the 10th fold is a tribute to father; the 11th fold, represents the seal of King David and King Solomon; the 12th fold represents an emblem of eternity and God the Father, the Son, and Holy Ghost. The last fold, when the flag is completely folded, the stars are uppermost, reminding us of our national motto, 'In God We Trust.'"
"I wanted to remind people what Memorial Day is all about," White said. "We have troops coming home in flag draped coffins and I wanted people to be reminded of the meaning behind the 13 folds of a U.S. flag draped over a service person's casket."
Memorial Day is a federal holiday for remembering the men and women who died while serving in the country's armed forces.
The first Memorial Day, then called Decoration Day, was held on May 30, 1868.
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A quiet residential area of Kato Pafos, the complex comprises 40 cottages, each with private garden and allocated parking. It has a large communal pool surrounded by mature, landscaped gardens.
Spacious, 3 bedroom end of terrace property with an open plan living / dining area and a large kitchen separated by a breakfast bar. Patio doors off the lounge lead out to a large veranda with BBQ area and mature garden. Also a storage room and cloak room located on ground level. On upper floor - 3 double bedrooms with fitted wardrobes and a family bathroom. Master bedroom with en–suite facilities. Indoor staircase leads to large roof terrace.
Property is offered unfurnished and some of the extras include: air condition, white goods and more..
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Q: fisher's exact test (R) - simulated p-value does not vary I have a problem using fisher's exact test in R with a simulated p-value, but I don't know if it's a caused by "the technique" ( R ) or if it is (statistically) intended to work that way.
One of the datasets I want to work with:
matrix(c(103,0,2,1,0,0,1,0,3,0,0,3,0,0,0,0,0,0,19,3,57,11,2,87,1,2,0,869,4,2,8,1,4,3,18,16,5,60,60,42,1,1,1,1,21,704,40,759,404,151,1491,9,40,144),ncol=2,nrow=27)
The resulting p-value is always the same, no matter how often I repeat the test:
p = 1 / (B+1)
(B = number of replicates used in the Monte Carlo test)
When I shorten the matrix it works if the number of rows is lower than 19. Nevertheless it is not a matter of number of cells in the matrix. After transforming it into a matrix with 3 columns it still does not work, although it does when using the same numbers in just two columns.
Varying simulated p-values:
>a <- matrix(c(103,0,2,1,0,0,1,0,3,0,0,3,0,0,0,0,0,0,869,4,2,8,1,4,3,18,16,5,60,60,42,1,1,1,1,21),ncol=2,nrow=18)
>b <- matrix(c(103,0,2,1,0,0,1,0,3,0,0,3,0,0,0,0,0,0,19,869,4,2,8,1,4,3,18,16,5,60,60,42,1,1,1,1,21,704),ncol=2,nrow=19)
>c <- matrix(c(103,0,2,1,0,0,1,0,3,0,0,3,0,0,0,0,0,0,869,4,2,8,1,4,3,18,16,5,60,60,42,1,1,1,1,21),ncol=3,nrow=12)
>fisher.test(a,simulate.p.value=TRUE)$p.value
Number of cells in a and b are the same, but the simulation only works with matrix a.
Does anyone know if it is a statistical issue or a R issue and, if so, how it could be solved?
Thanks for your suggestions
A: I think that you are just seeing a very significant result. The p-value is being computed as the number of simulated (and the original) matrices that are as extreme or more extreme than the original. If none of the randomly generated matrices are as or more extreme then the p-value will just be 1 (the original matrix is as extreme as itself) divided by the total number of matrices which is $B+1$ (the B simulated and the 1 original matrix). If you run the function with enough samples (high enough B) then you will start to see some of the random matrices as or more extreme and therefor varying p-values, but the time to do so is probably not reasonable.
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Sex abuse victims speak publicly after filing lawsuit against Orlando pastor, church
By: Kelly Healey
Updated: May 28, 2019 - 6:36 PM
ORLANDO, Fla. - A year ago, an Orlando pastor was arrested on allegations that he sexually abused a 17-year-old girl, and Tuesday, the teenager, Kenia Gilles and another abuse victim, Jeny Desronvil, held a news conference to help women who might find themselves in a similar situation, an Osborne & Francis Law Firm news release said.
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Billy Leveille was a pastor at the Seventh Day Adventist Church on Rio Grande Avenue, but once the allegations surfaced, he was fired and banned from being a pastor at the congregation, according to a police report.
Gilles and Desronvil have filed a lawsuit against the Florida Conference of Seventh-Day Adventists and Leveille. The women are choosing to identify themselves and speak publicly, the release said.
"Everyone has been making up their own story for us," Gilles said. "I was a junior in high school and you can imagine. I was basically hiding. I changed my number; having to deal with harassment on all types of social media platforms because of what happened."
Watch news conference below:
The pair attended Bethel Eglise Haitienne Des Adventistes Church in Orlando, where Leveille was the senior pastor, the release said. Gilles and Desronvil had attended the church since they were children, and in the summer of 2016, they were appointed to the church's secretarial team; a highly sought-after position within the church, according to the release.
"I've listened to people give their specific opinion of me. If I continued to still listen to those people, I would not be here today," Desronvil said.
The women said it was during that time that Leveille abused them.
Previous story: Orlando pastor accused of having sex with underage church member
"At some point, police were alerted to the abuse and contacted Kenia and her father and interviewed her. They conducted a controlled call with Leveille, which corroborated Kenia's version of events," the release stated.
Attorneys for the victims allege the Florida Conference of Seventh-Day Adventists was negligent in supervising Leveille and others at the church.
"Leveille was charged with unlawful sexual activity with a minor and child abuse. He pleaded guilty and was sentenced to seven years of probation," according to the law firm.
State Attorney Aramis Ayala says she won't run for re-election
11-year-old girl bitten by rattlesnake while at Blue Spring State Park
3 dead, 16 hurt in Japan after man attacks schoolgirls at bus stop
Woman beats wife for leaving marijuana in washing machine, deputies say
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Sex abuse victims speak publicly after filing lawsuit against Orlando…
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Q: How does one use a function to compute the value of a member of a structure using values of other members in C? I have created a struct in C with three members of type int.
I want to use a function to compute the value of the member C by adding the values a and b as shown in my code.
#include <stdio.h>
typedef struct nums
{
int a;
int b;
int c;
} number;
void add (number one);
int main (void)
{
number start = {.c = 3, .b = 4};
add (start);
printf("the sum is %d \n", start.c);
return (0);
}
void add (number one)
{
one.c = one.a + one.b;
}
When I run the code, I get the value of c as 3 instead of 7. How can I solve this problem?
A: Pass argument as a pointer
#include <stdio.h>
typedef struct nums
{
int a;
int b;
int c;
} number;
void add (number *pOne);
int main (void)
{
number start = {.a = 3, .b = 4, .c = 0};
add (&start);
printf("the sum is %d \n", start.c);
return (0);
}
void add (number *pOne)
{
pOne->c = pOne->a + pOne->b;
}
WHY POINTERS?
You pass the address of start as a pointer to add so that you can update start from within add(...). Otherwise, add would update a copy of start stored on the stack that gets popped off on the return of add.
A: Basically in C, if you pass a variable to a function, first the value of the variable is copied and then the function receives it.
So add function modifies the copy, not original one.
If you want to modify the original one in another function, you should pass the address of a variable.
Addresses are also copied when passed, but it's not a problem because the destination isn't copied.
#include <stdio.h>
typedef struct tagNumber {
int a;
int b;
int c;
} Number;
void add(Number* one) {
one->c = one->a + one->b;
}
int main() {
Number start = { .c=3, .b=4 };
add(&start);
printf("the sum is %d\n", start.c);
return 0;
}
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 1,825
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The following time rhEGF (Calbiochem, NORTH PARK, CA) was diluted in media with 1% FBS at desired concentrations, and cells incubated at 37C with 5% CO2 for 5 hours
The following time rhEGF (Calbiochem, NORTH PARK, CA) was diluted in media with 1% FBS at desired concentrations, and cells incubated at 37C with 5% CO2 for 5 hours. of their make use of in combination remedies with various other targeted agents such as for example RNA disturbance (RNAi). This research examines the usage of RNAi and kinase inhibitors for certification of components mixed up in EGFR/AP-1 pathway of Me personally180 cells, and their inhibitory results when evaluated independently or in tandem against multiple the different parts of this essential disease-related pathway. Strategies AP-1 activation was evaluated using an Me personally180 cell range stably transfected using a beta-lactamase reporter gene beneath the control of AP-1 response component following epidermal development aspect (EGF) excitement. Immunocytochemistry allowed for even more quantification of little molecule inhibition on the mobile protein level. RNAi and RT-qPCR tests had been performed to measure the quantity of knockdown with an mRNA level, and immunocytochemistry was utilized to reveal mobile protein amounts for the targeted pathway elements. Results Increased strength of kinase inhibitors was proven by merging RNAi aimed towards EGFR and little molecule inhibitors performing at proximal or distal factors in the pathway. After mobile excitement with evaluation and EGF at the amount of AP-1 activation utilizing a -lactamase reporter gene, a 10C12 flip change or 2.5C3 fold change toward greater potency in the IC50 was observed for MEK-1 and EGFR inhibitors, respectively, in the current presence of RNAi targeting EGFR. Bottom line EGFR pathway elements were experienced as goals for Beloranib inhibition of AP-1 activation using RNAi and little molecule inhibitors. The mix of both of these targeted agencies was proven to raise the efficiency of MEK-1 and EGFR kinase inhibitors, resulting in feasible implications for stopping or conquering medication level of resistance, lowering effective medication doses, and offering new approaches for interrogating mobile signalling pathways. History Cellular processes such as for example proliferation, differentiation, and death are regulated by sign transduction pathways which exert their function through receptor mediated activation commonly. The breakthrough in 1978 the fact that v-Src oncogene was a protein kinase resulted in a "cascade" of analysis into the function of kinases in cell-signalling pathways, and the next Beloranib finding that individual cancer can derive from the experience of non-viral, endogenous oncogenes, a significant part of which code for protein tyrosine kinases (PTKs) [1,2]. The epidermal development aspect receptor (EGFR) is certainly a tyrosine kinase which works as a get good at switch resulting in activation from the transcription aspect, activator protein-1 (AP-1), and various other related pathways. The receptor itself comprises extracellular, transmembrane, and tyrosine kinase domains. Ligand binding elicits a conformational modification from the extracellular area resulting in receptor dimerization and following transphosphorylation of intracellular area tyrosines. The phosphorylated tyrosines become binding sites for sign transducers initiating some kinase actions leading to mobile proliferation and differentiation [3-5]. Aberrant signalling taking place from EGFR leads to its transformation into an oncoprotein, as well as the consequent breakdown of mobile signalling networks qualified prospects to the advancement of malignancies and various other proliferative illnesses. EGFR and its own ligands get excited about over 70% of most malignancies [[4,6], and [7]]. Hidaki, et.al. in the first 1980's uncovered the first protein-kinase inhibitors, and set up the process of changing chemical substance framework to elicit different kinase inhibition specificity [8]. Medication advancement has implemented the lead from the educational community in developing book inhibitory substances at factors along these disease-related pathways. The protein kinase target class may be the second largest band of drug targets behind G-protein-coupled-receptors [3] now. Kinases from the Tyrosine and Serine/Threonine family members have already been targeted by small-molecule inhibitors and monoclonal antibodies effectively, numerous undergoing human clinical trials or launched as therapeutic entities [9-13] successfully. Acquired level of resistance to kinase-targeted anticancer therapy continues to be documented, & most extensively studied with imatinib (Gleevec?), an inhibitor of the aberrant BCR-ABL kinase, in chronic myelogenous leukemia [14]. Resistance has also occurred in EGFR-targeted inhibitor therapy using gefitinib (Iressa?) and erlotinib (Tarceva?). Mutations occurring in the catalytic Beloranib domain of the receptor have been implicated in this resistance, but cannot account for all resistance seen to these small molecule inhibitors, indicating other mechanisms are involved in the resistance seen to date [15,16]. Therefore, multiple strategies will be necessary to overcome the observed resistance to these new molecularly targeted therapies, as well as methods to predict their efficacy. Most kinase inhibitors target the ATP-binding site common to all kinases, and can bind multiple kinases [17]. This generates an inability to predict compound specificity for a particular kinase, and the subsequent need to analyze large numbers of kinases through a screening or profiling approach. Data from these em in vitro /em Rabbit polyclonal to SelectinE assays allow the researcher to predict clinical uses for inhibitors and possible offsite target effects. Studies using purified kinase and substrate are dependent on ATP concentration used, and the apparent Km for ATP can differ between kinases. This can lead to problems in the development of small molecule inhibitors based on competition at the ATP-binding site of a kinase, as the.
doi:10
doi:10.1158/1078-0432.CCR-15-1998. the diagnosis and treatment of AML. In the first portion, we provided some novel insights around the molecular basis of AML, as well as provided an update around the classification of AML. In the second portion, we summarized the results of research on potential molecular therapeutic brokers including monoclonal antibodies, tyrosine kinase/Fms-like tyrosine kinase 3 (FLT3) inhibitors, epigenetic/demethylating brokers, and cellular therapeutic agents. We will also spotlight ongoing research and clinical trials in pediatric AML. Results: We explained clonal evolution and how it changes our view on leukemogenesis, IKK-IN-1 treatment responses, and disease relapse. Pediatric-specific genomic mapping was discussed with a novel diagnostic method highlighted. In the later portion of this review, we summarized the researches on potential molecular therapeutic brokers including monoclonal antibodies, tyrosine kinase/FLT3 inhibitors, IKK-IN-1 epigenetic/demethylating brokers, and cellular therapeutic agents. Conclusion: Gene sequencing techniques should set the basis for next-generation diagnostic methods of AML, and target therapy should be the focus of future clinical research in the exploration of therapeutic possibilities. alterations of slippery malignant cells and Darwinian effects (selection) involving targeting agents. Further study could augment our understanding of the disease process, relapse, and help us in choosing the right therapeutic brokers. "Pediatric-specific" genomic mapping AML accounts for about 20% of pediatric leukemia. Child years AML has a slightly better end result than adult AML, with nearly 60C70% of long-term survival.[9,10,11] Despite considerable variations in treatment techniques, clinical outcomes for child years AML have not improved over the past two decades.[12] Moreover, rigorous chemotherapy is likely to render a substantial proportion of children to experience adverse effects from treatment toxicities.[13] Therefore, new therapeutic strategies are needed for child years leukemia. The fact that some mutations in adult AML are rare or entirely lacking in pediatric AML suggests a different pathogenesis and thus different therapeutic strategy for children. Therefore, the understanding of pediatric-specific genetic alterations is critical for the development of targeted treatment. Reports from the Japanese pediatric leukemia/lymphoma study group have confirmed that much like adult patients with AML, enhancer binding protein (mutations with a lower risk and better prognosis. The actuarial overall survival (OS) at 5 TIE1 years for those with mutations versus no mutations was 83% versus 65%, respectively, with an event-free survival (EFS) of 44% versus 49%, respectively, and a relapse risk (RR) of 64% versus 40%, respectively. It is worth noting that mutations are sensitive to inhibition of the Janus kinase (JAK) pathway, which is usually downstream from your receptor.[18] Therefore, this newly recognized pediatric-specific mutation could also be a potential pediatric-specific therapeutic target. Clinical trials are underway to test the efficacy of JAK inhibitors. An update in diagnostic methods naturally happens following the emergence of new genetic markers. McKerrell mutation. However, the authors also admitted that it would be premature to replace standard cytogenetic screening with Karyogene. Reasons include lack of comprehensiveness (the current panel does not cover some rarer chromosomal rearrangements) and the technical limitations IKK-IN-1 due IKK-IN-1 to the varied level of bioinformatics expertise in medical institutions. New Targets and Therapies Tyrosine kinase/Fms-like tyrosine kinase 3 inhibitors Fms-like tyrosine kinase 3 inhibitors Mutations in status after treatment with sorafenib in combination with chemotherapy.[27] The positive results justify the incorporation of sorafenib into future pediatric AML trials. Midostaurin is a Type III receptor TKI that inhibits FLT3 and other tyrosine kinase receptors.[28] A single-agent clinical trial suggested that despite only a 5% partial remission (PR) rate, midostaurin was able to confer a robust anti-blast response in AML patients, and an additional four patients experienced stable disease.[33] However, only one of the seven AML patients achieved a CR, suggesting the higher selectivity of quizartinib. Third-generation brokers such as crenolanib and gilteritinib are currently in Phase I/II clinical trials, and their therapeutic value in pediatric patients is not yet clear. Additional trials with a larger quantity of samples are currently recruiting patients or are ongoing. Aurora kinase inhibitors The AURKs are serine/threonine kinases that are involved mainly in checkpoint regulation in the cell cycle.[34] Three mammalian AURKs have been identified: AURKA, AURKB, and AURKC. The biological effect of inhibiting AURK in mitosis and its potential IKK-IN-1 clinical significance were first discussed in 2003.[35] Since then, increased consideration to this group has been garnered, and several AURK inhibitors were moved into Phase I/II clinical trials evaluating the treatment of malignancies. To date,.
J Natl Tumor Inst 2000;92(19):1564C72
J Natl Tumor Inst 2000;92(19):1564C72. necessary to keep up with the success and proliferation of GC B-cells, which tolerate significant tension associated with their fast proliferative price, tolerance of somatic hypermutation and oxidative tension(5C7). BCL6 proteins manifestation in GC-derived lymphoma cells needs the strain chaperone Heat surprise proteins 90 (Hsp90), and BCL6 represses its focus on genes in lymphoma cells using Hsp90 like a corepressor proteins(8). Since a commonality among tumors can be their dependency on tension response pathways to keep up their success and proliferation, we postulated that BCL6 expression may be connected in a few genuine way to stress responses in solid tumors. Heat shock element 1 (HSF1) may be the get better at regulator of tension response, regulating the manifestation of heat surprise proteins and additional tension proteins(9). Because HSF1 plays a part in keeping homeostasis after contact with various stressors, it's been implicated in mobile adaptation towards the malignant phenotype(10). Improved HSF1 manifestation has been within many tumor types, and HSF1 depletion leads to reduced cell viability and chemosensitization(11C16). Furthermore, HSF1 is necessary for tumorigenesis and change by several oncogenes including and it is a primary HSF1 focus on gene in tension response, and in doing this, reveal an urgent hyperlink between vertebrate advancement, convergent evolution from the humoral immune system response in various vertebrate organisms, & most critically the explanation for translating BCL6-targeted therapy as a far more specific method of inhibit tension pathways across a wide range of human being tumors. RESULTS can be broadly co-expressed with and connected with unfavorable medical result in solid tumors. Latest reports show that BCL6 can be often indicated in solid tumor cell lines that aren't through the B-cell lineage(2C4). Certainly, we analyzed gene manifestation profiles gathered by TCGA and discovered that is generally overexpressed in lots of solid tumors including breasts, lung, neck and head, esophageal, TIAM1 ovarian and uterine malignancies (Supplementary Fig. 1aCb). Furthermore, high transcript manifestation is connected with reduced progression-free success (PFS) Eplivanserin mixture in at least three common intense tumor types: triple-negative breasts tumor (TNBC), non-small cell lung tumor (NSCLC) adenocarcinoma subtype and gastric adenocarcinoma (GA) (Fig. 1aCc, remaining sections). The risk ratios (HR) (95%CI) had been: 1.74 (1.05 C 2.87), 2.53 (1.94 C 3.30) Eplivanserin mixture and 1.77 (1.46 C 2.06) for TNBC, GA and NSCLC, respectively (Fig. 1aCc). The association of expression with these aggressive tumors may be linked to cellular stress responses clinically. We thus examined the manifestation of the get better at transcriptional regulator of the strain response, transcript manifestation is also connected with reduced PFS in these tumors with an HR of: 1.46 (0.95 C 2.23), 1.90 (1.51 C 2.40) and 1.64 (1.38 C 1.99) for TNBC, NSCLC and GA, respectively (Fig. 1aCc, middle sections). Taking into consideration Eplivanserin mixture a potential hyperlink between tension BCL6 and response, we hypothesized how the same individuals which have poor prognosis connected with high manifestation should be the same individuals with high manifestation. Indeed, manifestation was considerably correlated with manifestation (Supplementary Fig. 1c). Furthermore, separating individuals predicated on high manifestation of both and and low manifestation of both genes created even more powerful HRs between individuals, recommending an additive aftereffect of both genes on PFS (Fig. 1aCc, correct panels). This led us to wonder whether there may be an operating link between BCL6 and HSF1. Open in another window Shape 1. Tumor cells express within an HSF1-reliant way aberrantly.a-c, Kaplan-Meier curves of development free of charge survival of triple-negative breasts tumor (a), lung adenocarcinoma (b) and gastric tumor (c) individuals stratified by or and expression. n, amount of individuals. d, mRNA in heat-shocked cells of mRNA in heat-shocked regular human being adult fibroblasts transfected with nontargeting (siNT) or HSF1 siRNAs (siHSF1) with associated immunoblot for HSF1 (bottom level).
In total, 16 endonuclease inhibitors were found out, which, two inhibited viral replication with negligible cell toxicity
In total, 16 endonuclease inhibitors were found out, which, two inhibited viral replication with negligible cell toxicity. Fine-Tuning NMR Fragment Screening Fragment testing by NMR spectroscopy widely is used in contemporary drug discovery to recognize low molecular pounds compounds that bind to a protein target weakly. to discover substances having the ability to inhibit influenza endonuclease activity and viral replication. Altogether, sixteen endonuclease inhibitors had been found, which, two inhibited viral replication with negligible cell toxicity. Fine-Tuning NMR Fragment Testing Fragment testing by NMR spectroscopy can be trusted in modern medication discovery to recognize low molecular pounds substances that bind weakly to a proteins target. Pressing the limitations of binding detectability in fragment testing by NMR FRAX486 spectroscopy against a model proteinCprotein discussion could prove beneficial to improve strike prices and successes when focusing on additional PPIs by NMR fragment testing. Fragment testing by NMR spectroscopy can be trusted in modern medication discovery to recognize low molecular pounds substances that bind weakly to a proteins target, as an initial step to create a better and stronger drug-like molecule. Sadly, analysts can spend lots of time testing libraries and miss substances that may be extremely guaranteeing still, as FRAX486 fake negatives. This caveat can be frequently exacerbated when focusing on proteinCprotein relationships (PPIs), as useful fragments that could bind to PPI sites may show too fragile affinities to become reliably detected inside a Rabbit polyclonal to ZNF471.ZNF471 may be involved in transcriptional regulation display. Right here, Dias et al. (DOI: 10.1021/ml400296c) possess pushed the FRAX486 limitations of binding detectability in fragment testing by NMR spectroscopy against a magic size PPI. The authors display a revision from the experimental set-ups in the NMR display leads these to save as true strikes three fragments that form section of a high-affinity drug-like chemical substance and that got in any other case escaped binding recognition as fake negatives under regular circumstances. The lessons discovered from this research could prove beneficial to improve strike prices and successes when focusing on additional PPIs by NMR fragment testing. Further Insights on Methuosis Gliobstoma multiforme (GBM) is among the most aggressive mind cancers, displaying limited response to the typical chemotherapy drugs. Additional insight in to the SAR from the specific cell loss of life pathway methuosis is vital for development of the class of substances toward preclinical anticancer tests. Gliobstoma multiforme (GBM) is among the most aggressive mind cancers, displaying limited response to the typical chemotherapy drugs, Gliadel and Temozolomide. It is because tumors harbor genetic mutations that dull the apoptotic process partly. Lately, a true amount of novel cell loss of life pathways distinct from apoptosis have already been discovered. Of particular curiosity is methuosis, seen as a intensive cytoplasmic vacuolization, that leads to lack of membrane integrity and eventual rupturing from the cell. In this presssing issue, Trabbic et al. (DOI: 10.1021/ml4003925) provide further insight in to the SAR of methuosis by indolyl-substituted pyridinylpropenones. The authors display that increasing how big is aliphatic substituents will not decrease vacuolization but considerably decreases cytotoxicity. Such insights on structural requirements necessary for cell FRAX486 loss of life are crucial for development of the class of substances toward preclinical anticancer tests..
The X-ray structure of the PD-1/PD-L1 complex was downloaded from your protein data bank website
The X-ray structure of the PD-1/PD-L1 complex was downloaded from your protein data bank website. and found that PAC-1 a high immune score and M2 TAMs were strongly associated with poor clinical outcomes in patients with MIBC. Further, we analyzed resected samples from 120 patients with MIBC and found that individuals with PD-1-positive TAMs showed a reduction in 5-12 months overall survival and disease-free survival. Additionally, PD-1-positive TAMs showed a significant association with higher programmed death-ligand 1 (PD-L1) expression, the Ki67 index, the pT stage and fewer CD8-positive T cells. Through the co-immunoprecipitation (co-IP) assay of THP-1 derived macrophages, we found that CD68 can bind to PD-1. The binding of CD68 and PD-1 PAC-1 can induce M2 polarization of THP-1 derived macrophages and promote malignancy growth. The anti-CD68 treatment combined with peripheral blood mononuclear cells (PBMC) showed obvious synergy effects on inhibiting the proliferation of T24 cells. Together, these results indicate for the first time that CD68/PD-1 may be a novel target for the prognosis of patients with MIBC. KaplanCMeier analysis and the log-rank test. The prognostic significance of the clinicopathological parameters was analyzed using the chi-square test. The Spearman correlation analysis was used to analyze the correlation between CD68, PD-1, and PD-L1. The relationship of PD-1-positive TAMs CD8-positive T cells was analyzed by Students a LASSO analysis, and recognized 8 survival-associated immune cell types. Further, we derived an immune score for predicting the prognosis of patients with MIBC ( Figures 1A, B ). According to the Kaplan-Meier analysis of the TCGA cohort, PAC-1 patients with MIBC having high immune scores were associated with a significantly reduced 5-12 months OS and DFS outcomes ( 0.0001 and 0.0001, respectively; Figures 1C, D ). The heatmap of the survival, tumor stage, tumor grade, immune score, and the profiles of different immune cells is shown in Physique 1E . Interestingly, M2 TAMs showed a correlation with high immune scores in the heatmap ( Physique 1E ). Therefore, we further analyzed the clinical outcomes for patients with PAC-1 MIBC showing the presence of M2 TAMs in the TCGA cohort. Patients with M2 TAMs showed significantly worse 5-12 months OS and DFS outcomes ( 0.001, respectively; Figures 3B, C ). The 5-12 months OS rate was estimated to be ~33.3% Rabbit polyclonal to TDGF1 in patients with PD-1-positive TAMs, and ~77.78% in patients with PD-1-negative TAMs ( Figure 3B ). Additionally, the 5-12 months DFS rate was ~21.1% in patients with PD-1-positive TAM compared with ~66.7% in patients with PD-1-negative TAM ( Determine 3C ). Open in a separate window Physique 3 (A) Immunohistochemical staining of PD-1-positive TAMs (reddish arrow), PD-1-unfavorable TAMs (purple arrow), strong PD-L1 and poor PD-L1 expression. (B)?Kaplan-Meier analysis of PAC-1 OS in patients with MIBC with PD-1-positive TAMs in the Shanghai General Hospital cohort. (C) Kaplan-Meier analysis of DFS in patients with MIBC with PD-1-positive TAMs in the Shanghai General Hospital cohort. (D) PD-1-positive TAMs showed less CD8-postive T cells nearby. *** 0.001 (Students = 0.027, Table 1 ). Patients with PD-1-positive TAMs were found to exhibit higher pT stages than those without. PD-1-positive TAMs also showed significantly stronger PD-L1 expression ( 0.001), a higher Ki67 index (= 0.003), and worse pathological patterns ( 0.001; Table 1 ). Interestingly, higher PD-L1 expression levels also resulted in poor prognosis of patients with MIBC (data not shown). We investigated the response to cisplatin-based neoadjuvant chemotherapy in patients with PD-1-positive and -unfavorable TAM phenotypes. Patients with MIBC who were administered neoadjuvant chemotherapy in the pT2 stage showed a better prognosis. However, the presence of PD-1-positive and -unfavorable TAM did not improve the response to neoadjuvant chemotherapy ( 0.05, data not shown). Patients with a PD-1-positive TAM phenotype showed a comparatively substandard response to neoadjuvant chemotherapy, which was similar to the 5-12 months OS and DFS outcomes in patients with PD-1-positive TAMs. Intriguingly, PD-1-positive TAMs showed relevance to bladder malignancy related immune response. Based on 120 MIBC patients from Shanghai General Hospital, we found that CD8-positve T cells were comparatively fewer around PD-1-positive TAMs ( 0.001; Physique 3D ), indicating PD-1-positive TAMs could be involved in bladder cancer immune response. In addition, the number of PD-1-positive TAMs showed positive relevance to the PD-L1 expression of bladder malignancy cells (= 0.48, 0.001; Physique 3E ). The Conversation of CD68 and PD-1 Induced TAMs to M2 Polarization Interestingly, when we further analyzed the presence of PD-1-positive TAMs using immunofluorescence staining on FFPE samples, CD68 and PD-1 tended to be expressed synchronously in TAMs ( Physique 4A ). Hence, we conducted an analysis for the correlation of CD68 mRNA expression levels with the PD-1 and PD-L1 mRNA expression levels in the TCGA cohort and found that the expression of CD68 and that of PD-1 and PD-L1 was correlated (= 0.58 and = 0.41, respectively; 0.001 and .
Treatment ought to be individualized predicated on tumor size, symptoms of development and metastasis, and operative risk
Treatment ought to be individualized predicated on tumor size, symptoms of development and metastasis, and operative risk. renal or adrenal tumors with out a comprehensive histopathologic and immunohistochemical evaluation. Because of the potential intense Indocyanine green behavior of the malignancies, timely diagnosis is essential and offers significant therapeutic and prognostic implications incredibly. 1. Intro Epithelioid angiomyolipomas (EAMLs) are uncommon, mesenchymal tumors that participate in the perivascular epithelioid cell neoplasms (PEComas). In addition they talk about some histologic top features of angiomyolipomas (AMLs), however they are primarily made up of epithelioid cells and absence the typical fats tissue component. While AMLs are harmless generally, EAMLs have a tendency to become larger in proportions and may become malignant. They involve the kidneys generally, liver organ, and lungs. Consequently, involvement of additional organs poses a diagnostic problem [1C4]. While sporadic PEComa family members tumors are uncommon incredibly, their occurrence can be higher in individuals with tuberous sclerosis complicated (TSC), a uncommon autosomal dominating disease with imperfect penetrance. TSC can be a syndrome resulting in the introduction of multiple tumors in the retina, pores and skin, kidneys, adrenals, lungs, and additional organs. The approximated world-wide prevalence of TSC can be 1 in 6,000 or 12,000 people [5]. We explain the situation of the 32-year-old gentleman with a brief history of TSC who offered subacute back discomfort and a big intraabdominal mass. The individual was identified as having an initial epithelioid angiomyolipoma/PEComa of the proper adrenal gland with liver organ metastases that was established postsurgery via histological and immunohistochemical evaluation. Indocyanine green To the very best of our understanding, there are less than ten reported instances of EAML arising in the adrenal gland. Furthermore, metastasis towards the liver organ from an initial adrenal EAML continues to be described rarely. 2. Case Demonstration A 32-year-old gentleman shown to the crisis department (ED) having a 1-week background of right-sided lower Indocyanine green back again pain. His health background was significant for TSC. He endorsed exhaustion, unintentional weight lack of around 50 pounds going back 3 months, and night sweats for days gone by weeks to admission previous. He refused any preceding stress, fever, urinary symptoms, hematuria, abdominal discomfort, or adjustments in bowel motions. Past surgical background was unremarkable. He's a lifetime nonsmoker and refused any alcoholic beverages or recreational medication use. Physical examination revealed multiple facial angiolipomas over the nose and cheeks. No enlarged cervical or supraclavicular lymph nodes were found. Respiratory and cardiovascular exams were unremarkable. The abdomen was soft and nondistended, but the right flank was tender to palpation without rebound or guarding. A palpable mass was noted in the right hemiabdomen. Costovertebral tenderness was absent; however, right paraspinal lumbar tenderness was elicited by body movements. Laboratory testing was only remarkable for normocytic anemia with hemoglobin 7.8?g/dL (14C18?g/dl). Urinalysis was normal without blood or red blood cells. Computed tomography (CT) scan of Indocyanine green the abdomen without contrast revealed a right suprarenal vs. renal mass measuring 16??17??20?cm (Figure 1). Areas of necrosis, hemorrhage, and parenchymal calcifications were also noted. These findings were confirmed with a magnetic resonance imaging (MRI) study. The origin of this mass (renal vs. adrenal) was indistinguishable on MRI image due to large tumor burden (Figure 2). There were compression and displacement of the inferior vena cava (IVC) medially, but no obvious IVC invasion. Open in a separate window Figure 1 CT abdomen without contrast showing a large, right suprarenal vs. adrenal mass (arrow). Open Indocyanine green in a separate window Figure 2 MRI of the abdomen showing a large, right abdominal mass from the unclear origin (arrow). Biochemical workup was performed to evaluate whether the mass was of adrenal origin and hormonally active as part of the preoperatory evaluation. Evaluation for metanephrines, normetanephrines, aldosterone, and cortisol overproduction was unremarkable. Subsequently, the patient underwent total right adrenalectomy with en bloc right nephrectomy and resection of regional lymph nodes (Figure 3(a)). Excisional biopsy Rabbit Polyclonal to p38 MAPK of segment 5 of the liver was also performed due to intraoperative finding of two liver nodules. Open in a separate window Figure 3 (a) Necrotic mass involving the adrenal gland and perinephric soft tissue. (b) Malignant angiomyolipoma with large tumor cells with abundant eosinophilic cytoplasm. (c) Melan-A/Mart-1 immunohistochemical stain positive within the tumors cells. Pathology evaluation showed involvement of the adrenal gland and perinephric soft tissue by malignant, large epithelioid cells with abundant pale to eosinophilic cytoplasm, enlarged and irregular nuclei, and conspicuous nucleoli. These cells are.
The thirst column: DAC was added on the first and second day, and anti-leukemia drugs at the 3rd day
The thirst column: DAC was added on the first and second day, and anti-leukemia drugs at the 3rd day. Annexin PI and V staining in cell lifestyle, TUNEL transmitting and assay electron microscopy in pet research. MicroPET was utilized to imaging the tumor in mouse model. Molecular research were executed using microarray appearance analysis, that was utilized to explore linked pathways, and real-time quantitative invert transcription-PCR, western immunohistochemistry and blot, utilized to assess legislation of Wnt/-catenin pathway. Statistical significance among groupings was dependant on one-way ANOVA evaluation accompanied by post hoc Bonferronis multiple evaluation test. Outcomes Among five anti-leukemia agencies in merging with decitabine, the sequential mix of idarubicin and decitabine induced synergistic cell loss of life in U937 cells, and this impact was confirmed in HEL, SKM-1 cells and AML cells isolated from AML sufferers. Importantly, tumor development inhibition within this sequential mixture was found to become greater than in one agent or handles in vivo. Furthermore, sequential mix of the two agencies induced apoptosis and despair Daunorubicin from the Wnt/-catenin pathway in both AML cell lifestyle and animal research. Conclusions The results demonstrated that mix of decitabine and idarubicin had synergistic anti-leukemia results sequentially. These effects were mainly related to demethylation of Wnt/-catenin pathway downregulation and inhibitors of Wnt/-catenin pathway nuclear targets. strong course="kwd-title" Keywords: Decitabine, Idarubicin, Wnt, Severe myeloid leukemia, Myelodysplastic syndromes Launch 5-Aza-2-deoxycytidine (decitabine, DAC), an analog of deoxycytidine, includes a nitrogen group substituted for C-5 from the pyrimidine band [1]. DNA polymerase facilitates the insertion of DAC into DNA through the replication stage of transcription, which upon taking place, network marketing leads to a long lasting mixture with DNA methyltransferase (DNMT). By binding DNMT, DAC decreases the enzymes appearance bioactivity and amounts and causes demethylation of hypermethylated DNA, which induces re-expression of silenced genes [2,3]. As reported previously, low dosages of DAC induce epigenetic modulation, while high dosages have cytotoxic results [4]. Provided the association between DAC-mediated reactivation and hypomethylation of multiple genes, some groups have got looked to the drug because of its essential function in the control of cell proliferation and differentiation [5]. Used, DAC continues to be a highly effective therapy for severe myeloid leukemia (AML) and myelodysplastic syndromes (MDS). Lately, DAC monotherapy was connected with a comparatively low price of complete remission prices in MDS and AML [6-8]. Kantarjianet al. reported within a stage III randomized research of DAC in treatment of 170 MDS sufferers, the entire response price (OR) was 17%, including 9% comprehensive replies [7]. Furthermore, Issa et al. executed a Stage I research of 37 sufferers with AML getting DAC where the OR was 17% [8]. Many groups have attemptedto raise the response price of DAC-based therapy by developing combos remedies [9,10]. More often than not, these took on three types: merging DAC with various other epigenetic modulating agencies, cytotoxic agencies, and using DAC being a biologic response modifier to improve the efficiency of other medications. Because the ramifications of these mixed therapies aren't ideal, it's important to Daunorubicin explore book combinations. In this scholarly study, we have looked into the result of five anti-leukemia medications (idarubicin, IDA; daunorubicin, DNA; aclarubicin, ACLA; thalidomide, THAL; and homoharringtonine, HHT) in conjunction with DAC, provided possibly or sequentially concurrently, Daunorubicin MRC1 on proliferation in a variety of AML cell lines. Strategies and Components Reagents DAC was provided and developed by Pharmachemie BV, Haarlem, holland. HHT was bought from Minsheng Pharmacia (Zhejiang, China). IDA and DNR had been bought from Haizheng Pharmacia (Zhejiang, China). ACLA was bought from Wanle Pharmacia (Shenzhen, China). THAL was bought from Sigma (St. Louis, MO, USA). DAC was utilized soon after dissolving it in phosphate buffer saline (PBS). Various other agents had been dissolved Daunorubicin in PBS and kept at -40C. AML examples Bone marrow examples were gathered during regular diagnostic evaluation after written up to date consent have been attained. Individual disease was characterized using FAB classification, resulting in grouping of individual 1 and individual 3 in AML-M5 category with an increase of than 90% blast cells and individual 2 into AML-M2 category with 80% blast cells; three healthful volunteers were chosen as normal handles. Sufferers mononuclear cells had been.
Clinical improvement is definitely along with a significant reduction in TGF- set alongside the control and pre-treatment group status
Clinical improvement is definitely along with a significant reduction in TGF- set alongside the control and pre-treatment group status. identify the damage. The purpose of this paper can be Moxonidine Hydrochloride to examine enzymatic and nonenzymatic factors involved with catabolism of matrix parts and substances revitalizing their biosynthesis. Consequently, we discuss the adjustments in these elements in body liquids of kids with JIA and their potential diagnostic make use of in the evaluation of disease activity. Understanding the adjustments in ECM parts throughout the child-hood arthritis might provide the intro of both fresh diagnostic equipment and new Moxonidine Hydrochloride restorative strategies in kids with JIA. solid course="kwd-title" Keywords: juvenile idiopathic arthritis, extracellular matrix, proteoglycans, matrix metalloproteinases, reactive air species 1. Intro Juvenile idiopathic arthritis (JIA) may be the most common band of chronic connective cells diseases in kids that is followed by joint framework and function disorders. Clinical symptoms indicating pathological inflammatory procedures in the bones, i.e., discomfort, existence of exudate Moxonidine Hydrochloride or restriction of flexibility, which permit the analysis of JIA, should be present in the individual for at least six weeks. The analysis of JIA, because of its complicated etiopathogenesis, heterogeneity of medical manifestations, and insufficient pathognomonic symptoms, can be a complicated process and is dependant on the assortment of a detailed background from the individual and family members, a physical study of the patient, as well as the efficiency of diagnostic laboratory testing and imaging research Moxonidine Hydrochloride [1,2]. The heterogeneous medical expression of the condition is just about the basis for reputation from the International Little league of Organizations for Rheumatology (ILAR) six subtypes of JIA: Systemic JIA, oligoarticular JIA (including a continual and expanding type), polyarticular JIA (rheumatoid element (RF)-adverse and RF-positive type), enthesitis-related arthritis, psoriatic arthritis and undifferentiated JIA [3,4,5]. Researchers will work on defining fresh JIA classification requirements and different types of the condition [3,6]. Arthropathy builds up in kids with established disorders from the immune system response genetically, even more in people subjected to exterior elements such as for example tension frequently, bacterial attacks (i.e., Mycoplasma pneumoniae, Borrelia burgdorferi, Yersinia enterocolitica, Proteus mirabilis or viral attacks), parvovirus B19, rubella disease, influenza disease, cytomegalovirus, Epstein-Barr disease [7,8,9,10]. The infectious elements, by interfering using the metabolism from the immune system, business lead to the formation of autoantibodies aswell while adjustments in the formation of signaling adhesion and substances substances. As a total result, swelling develops inside the joint constructions, the forming of which can be from the activation of several pro-inflammatory cytokines, including tumor necrosis element (TNF-) and interleukin (IL) we.e.,IL-1, IL-6, IL-8, IL -12, IL-15, IL-17, IL-18 [11,12,13,14]. Pro-inflammatory cytokines result in the damage of articular cartilage, which advances using the duration of JIA, not really compensated from the degree of repair procedures [15,16,17]. These disorders are attributed specifically to adjustments in homeostasis of extracellular matrix the different parts of the connective cells that forms articular cartilage. Extracellular matrix (ECM) can be a multi-component, structured framework that fills the areas between chondrocytes. The cartilage ECM includes collagen proteins primarily, which take into account about two-thirds from the dried out pounds of adult articular cartilage. Type II collagen represents 90% to 95% from the collagen in ECM, while type VI, IX, X, XI, XII, XIV are located in small amounts. The small collagens help form and stabilize the sort II collagen fibril network [18]. Collagen Rabbit Polyclonal to MAGI2 fibrils offer cartilage with tensile power, which depends upon the intensive cross-linking from the collagen. Proteolytic and mechanised harm to the fibrillar network can be thought to be a key, irreversible perhaps, stage in the damage of joint cartilages in arthritis [19]. Furthermore, the cartilage matrixin about one-thirds from the dried out weightis shaped by proteoglycan Moxonidine Hydrochloride (PG) aggregates, including primarily aggrecan and smaller amounts of decorin, biglycan, fibromodulin, lumican or proteoglycan-100. In the framework from the matrix smaller amounts of non-collagen proteins are located, including fibronectin, tenascin, chondronectin, vitronectin, matrilin and thrombospondin [20,21,22,23]. PGs play a particular role in keeping the mechanical-immunological properties of cartilage. PGs are co-formed from the primary protein to which heteropolysaccharide chains of glycosaminoglycans (GAGs) are.
Solitary colonies were hand-picked at 14?days after seeding
Solitary colonies were hand-picked at 14?days after seeding. take action collectively to control ANO1 manifestation and function. Our findings reveal a previously unrecognized mechanism for regulating ANO1 protein levels and determine a potential molecular link between ANO1 rules, epidermal growth element, and additional signaling pathways. and and represents undegraded GST fusion protein or GST. All data in panels are representative of three self-employed biological replicates. ANO1, anoctamin-1; ANO1C, ANO1 C terminus; IgG, immunoglobulin G; SD-2, synthetic defined press deficient in Leu and Trp; SD-4, Synthetic defined media deficient in Leu, Trp, His, and Ura; TRIM, tripartite motif. Physical connection between ANO1 and TRIM23 was further substantiated through pull-down and coimmunoprecipitation (co-IP) assays performed using HEK293T cells expressing epitope-tagged ANO1 and TRIM23: glutathione-also show that the RING domain is not required for ANO1 binding. TRIM23 stabilizes ANO1 and 0.026, n?= 4. caused homozygous deletion of 2?bp in exon 1, indicated by a within the sgRNA region ( 0.007, MRK n?= 5 self-employed biological replicates; ???and ubiquitination system; here, addition of TRIM23 but not TRIM23RING caused ANO1 ubiquitination (Fig.?3and ubiquitination of ANO1 by TRIM23. Panels are representative of 3 to 4 4 independent biological replicates. R, TRIM23RING; ANO1, anoctamin-1; ANO1CV5, V5-tagged full-length ANO1; E1, ubiquitin-activating enzyme; E2, UbcH5c; TRIM, ABX-464 tripartite motif; Ub, ubiquitin. TRIM23 stabilizes ANO1 by knocking out in mice by using CRISPR/Cas9 techniques (Fig.?4exon 2; the place contains a stop codon and an EcoRI site (mice at first instance ((KO) mice (and test utilized for statistical analysis. ANO1, anoctamin-1; DRG, dorsal root ganglion; Ig, immunoglobulin; n, pair quantity of sex-matched littermates; TRIM, tripartite motif. ANO1 is known to be indicated in the dorsal root ganglion (DRG), salivary gland, lung, and heart; thus, we measured ANO1 manifestation in these cells of TRIM23 KO mice by using a validated anti-mouse ANO1 antibody (abdominal53212, Abcam) (Fig.?S1and as well. Functional effect of TRIM23CKOCinduced ANO1 reduction We evaluated the functional effect of TRIM23CKOCinduced ANO1 protein reduction by using the DRG as an example cells: We tested capsaicin-evoked pain sensation in mice because ANO1 has been implicated in DRG-mediated and chemical-induced pain sensation by us as well as others (19, 20). Strikingly, TRIM23 KO more than halved the ABX-464 total licking time in the capsaicin-induced pain-sensation assay, which suggests that TRIM23CKOCinduced ANO1 protein reduction in DRG neurons influences chemical-/the transient receptor potential?vanilloid?1-induced pain sensation (Fig.?4transcription ABX-464 in TRIM21 KO mice. 0.033 (n?= 7 for the salivary gland, n?= 5 for the heart). For mRNA level, denotes IgG; the denotes TRIM21 doublet?also detected in Input. denotes undegraded GST or GSTCANO1. ?) and ANO1 (?) interact in ZR-75-1 cells. are representative of three self-employed biological replicates. ANO1, anoctamin-1; ANO1CV5, V5-tagged full-length ANO1; ANO1C, ANO1 C terminus; Ig G, immunoglobulin G; TRIM, tripartite motif; Xpress-TRIM21, Xpress-tagged TRIM21. Notably, in contrast to TRIM23, TRIM21 moderately decreased ANO1 manifestation and concurrently improved ANO1 ubiquitination to a limited degree (Fig.?5and below). TRIM21 destabilizes ANO1 and by using is replaced having a GFP reporter): TRIM21 KO improved ANO1 expression nearly 3-collapse in the salivary gland and by 40% in the heart (Fig.?6, and transcription and, by extrapolation, TRIM21 protein expression in specific tissues. Here, GFP manifestation in the salivary gland, heart, and lung, but not in the DRG, indicated potential TRIM21 protein manifestation in the three tested tissues other than the DRG (Fig.?6, and and Fig.?S4). Therefore, the lack of switch in ANO1 manifestation in the DRG of TRIM21 KO mice can ABX-464 be attributed to the absence of TRIM21 manifestation (Fig.?S4transcription. However, our results (Fig.?6, and and Fig.?S4) suggest that TRIM21 downregulates ANO1 manifestation inside a tissue-specific manner. Next, we examined the functional effect of TRIM21CKOCinduced ANO1 upregulation. As before (Fig.?4and ?and4D).4D). This result agrees with earlier results acquired in Te11.
While described in earlier studies [10], [51], [52], PI3K and smad activation differs in their subcellular location
While described in earlier studies [10], [51], [52], PI3K and smad activation differs in their subcellular location. to phosphorylate c-Raf, ultimately resulting in Erk activation. Activation of Erk was necessary for TGF- induced fibroblast replication. In addition, Erk phosphorylated the linker region of nuclear localized smads, resulting in increased half-life of C-terminal phospho-smad 2 and 3 and increased duration of smad target gene transcription. Together, these data show that in mesenchymal cell types the TGF-/PI3K/Pak2/Raf/MEK/Erk pathway regulates smad signaling, is critical for TGF–induced growth and is a part of an integrated signaling web made up of multiple interacting pathways rather than discrete smad/non-smad pathways. Introduction Transforming Growth Factor (TGF-) is the prototypic member AZD1208 of a family of structurally related cytokines that control a myriad of cellular functions. TGF- elicits its cellular responses by signaling through a receptor complex of serine/threonine kinase type I (TRI) and type II (TRII) receptors [1], [2]. Ligand binding induced transmission transduction through this receptor complex results in receptor mediated (R-) smad2 and/or smad3 phosphorylation. This phosphorylation at the C-terminal SSXS motif of smad2/3 allows them to complex with the common mediator (Co-) smad4 [3], [4], translocate into the nucleus, and regulate target gene expression AZD1208 [5], [6]. Although both mesenchymal and epithelial cells contain the canonical TGF-/smad signaling cascade, epithelial cell types are growth inhibited, whereas mesenchymal cells are growth stimulated by TGF- suggesting a fundamental mechanistic difference in TGF- signaling between cell types, supplimental to the smad signaling cascade. This has lead to the nomenclature of smad and non-smad or smad-dependant and impartial signaling cascades. There have been a number of these non-smad signaling pathways explained including Erk, Jnk, ROCK, and more recently, p21-activated kinase-2 (Pak2; [7]C[11]). In phenotypically normal cell lines (neither virally transformed nor cancer derived), TGF- regulation of Pak2 activity was found to be stimulated through cdc42/Rac1 and inhibited by Merlin/Erbin [10], [11]. Pak2 is usually specifically activated by TGF- only in mesenchymal cells, as the result of phosphatidylinositol 3-kinase (PI3K) activation and may be associated with TGF- AZD1208 activation of Ras [10], [12], [13]. Conversely, normal epithelial cells appear to inhibit Pak2 activation through an failure to activate PI3K and/or by directly inhibiting Pak2 through Merlin/Erbin [11]. Functionally, PAKs regulate apoptosis, cell motility and cytoskeletal rearrangement [14]. Relevant to this study, Paks have been implicated in mitogen activated protein kinase/extracellular transmission regulated kinase (MAPK/Erk) signaling cascades as potential MAP kinase kinase kinase kinases [15] by regulating the activity of both c-Raf and MEK1 [16], [17]. Classically, with tyrosine kinase receptors, activation of Ras [18], [19] results in activated Raf, which activates MEK1/2, followed by Erk activation. However, Ras independent mechanisms of Erk activation have been explained for both erythropoietin (Epo; [20]) and platelet derived growth factor (PDGF; [21]), suggesting different pathways lead to Erk activation. Although cross-talk between Erk and smad signaling was explained over a decade ago [7], [18], [22], the relationship and mechanism by which this occurs is still unknown. Within the linker region domains of smad2 and smad3 are several potential Erk phosphorylation sites [23], [24]. However, these same sites have also been implicated in smad regulation by the cyclin dependent kinases, CDK8 and 9 [25]. The phosphorylated linker region, has also been shown to both inhibit smad nuclear translocation and signaling [18], [24], [26]C[28] and enhance smad mediated transcriptional activity [7], [23], [25], two mutually exclusive functions. To address this controversy, in this study we further refine the mechanism for cell type specific TGF- activation of Erk. We show that via PI3K, Pak2 activation results in Erk activation in untransformed cells with endogenous levels of transmission transduction proteins. We also show that this activated AZD1208 Erk phosphorylates smads within their linker regions, resulting in the maintenance of smad mediated transcriptional activation, thus demonstrating integration of the Erk and smad pathways, both under the direct control of TGF-. Materials and Methods Cell Culture All cell lines used were managed in high glucose Dulbecco's Modified Eagle Medium (DMEM; Invitrogen, Carlsbad, CA) and purchased from American Type Culture Collection repository (Mannassas, VA; NIH-3T3, CRL-1658; Mv1Lu, CCL-64; HEK-293A, CRL-1573; NMuMG, CRL-1636). The murine embryonic fibroblast cell collection, AKR-2B, was produced in DMEM supplemented with 5% Fetal Bovine Serum (FBS; PAA Labs Inc, Etobicoke, ON)), while NIH-3T3 cells were produced in DMEM supplemented with 10% Newborn Calf Serum (NBCS; Invitrogen, Carlsbad, CA). Pak2 flox/flox MEF parental cell collection and the Cre/Pak2 knockout derivative (kind gift of Dr. Jonathan Chernoff, Fox Chase Cancer Centre, OH) were managed CLG4B in DMEM supplemented with 10% FCS, as were Mv1Lu epithelial cells, while NMuMG growth media also contained 10 g/ml bovine Insulin (Sigma Biochemicial, St. Louis, MO) and 5 ng/ml EGF (Cell Signaling Technologies; Pickerington, ON). All buffer salts, bovine serum albumin (BSA) and acrylamide were purchased from ThermoFisher Biotechnology. Protein Analysis Mesenchymal cell lines were plated 24 h prior to serum depletion (0.1% NBCS/DMEM).
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Sport January 3, 2022 <January 3, 2022
Florian Hempel lost in the third round
The victory over world number five Dimitri van den Berg was a thrilling victory for Florian Hempel. The former handball player could not build success – failed because of an Australian.
No one expected it. After winning the German duel against Martin Schindler in the first round (3-0), Florian Hempel also put in a great performance in his second match. Interactingly subdued Belgium's Dimitri van den Berg 3: 1.
But the Cologne-born man was unable to build on his victory over Belgium in the third round on Monday. They lost 4-1 to Raymond Smith of Australia. Throughout the battle, Hempel did not find his way, but Smith did well.
++ Third round match between Hemble and Smith in a live ticker for reading ++
5:55 pm: game over! Florian Hempel lost to Raymond Smith in the third round. The Australian moves to the 16th round, where they meet the winner of the fight between Steve Lennon (Ireland) and Mervyn King (England).
5. Set – If Smith wins this match, he will advance to the 16th round. Hempel finds nothing in this decision leg. German 100 does not complete. Raymond Smith then threw a "double 20" and won the duel! 2: 3
5. Set – Now Smith is starting well! These are really big darts from Australia. In this leg, "Coleshe Jung" has no chance. 2: 2
5. Set – Hempel now averages over 90 for the first time, but not 36, which is why Smith now has a chance of winning a leg. But he didn't even hit a "double 16". The German then won the "Double 9". 2: 1
5. Set – Smith finished "Double 8", equalized with his throw-off leg. 1: 1
5. Set – Hempel is under a lot of pressure right now. Still, he started well in the first quarter of the fifth set. He finished an 89, and finally he hit the "Bulls Eye"! That's all there is to it 1: 0.
See also Live Stream, TV Channel, UK Start Time, Undercard, Golovkin Fight Results TONIGHT
17:39: This would be very difficult for the German Florian Hempel. Raymond Smith needs to win a set to advance to Round 16.
4. Set – Hempel hit 180! Can he get a tie on this set now? No! Smith finished with 24 runs and won the leg. 1: 3
4. Set – Smith is now back on set. But this time Hempel opposes! When he had double problems before, he now hit a "double 8" and so shortened. 1: 2
4. Set – Incidentally, Hempel increased his average in the previous three sets. This makes it clear that he finds his game great. But it does not help him on this leg. "Double 10" Smith won this break – set to win. 0: 2
4. Set – Exclamation mark for Australians! He countered with 180 runs from the start. But then Smith was weak, 5 not finished! However, Hempel could not take advantage of this. Raymond Smith finished second. Bitter to Hempel! 0: 1
3. Set – Great start for the leg. Hempel bowled 180 and Smith struck back with 140 runs. The German now came into the game and clearly won on foot. So he won this set, which is now only 1: 2. Hempel is back! 3: 2
3. Set – Hempel looks intimidating, and does not have good body language. Can he come here again? In the set, Smith threw 3-0. But Smith missed the "Bulls Eye". Hempel finished with a 36 and drew! Break again to compensate. 2: 2
3. Set – The Australian starts the third 180 of the game. Hempel failed to win the leg with a "double 5". Smith refused to take it and won the next quarter. 1: 2
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3. Set – Smith does not allow anything to burn. With "Double 12" he won his first quarter in the third set. 1: 1
3. Set – Hempel kicked off again. At the start of the set, Smith could not find his best game yet. Is that an opportunity for Germany? Yes! Hempel completed a 32 and won the leg number one. 1: 0
5:10 pm: For the Germans, it was tightening. Florian Hempel also lost the second set and must now win to avoid slipping into a state of despair.
2. Set – Is Smith retaliating now? Hempel loses the thread again, and the Australian takes advantage of it. The second set was for Smith. 2: 3
2. Set – Hempel 51 must be completed. He does it with colors. Quickly in the second set between Hemp and Smith 2: 2. Now the first judge in the second set.
2. Set – Florian Hempel works! In the second set, the third quarter barely won. Very important re-break from Hempel! 1: 2
2. Set – Smith is very good here. So far he has played at an average of 104. Smith missed the first break chance, but Hempel was unable to capitalize. Smith then completed the "Double 10", winning his 20th consecutive fifth consecutive victory. 0: 2
2. Set – Now Smith is on fire. Again he puts the exclamation point 140 twice each time. The first leg of the second set was for the strong Australian. 0: 1
4:56 pm: The first set went to Australian Raymond Smith. You need three more people to win the game. Hempel has yet to find his way into his game.
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1 set – Finding a better way to the Australian game. He throws a "high finish" straight and wins the first set against the throw-off. 1: 3
1 set – Both threw a few triples in the third quarter. Hempel has yet to find his game, with the Australian getting stronger and throwing 180. He took number three and got the first break. 1: 2
1 set – Now Smith is on fire. The second quarter clearly wins. Compensation. 1: 1
1 set – Florian Hempel starts, so there is a small gain. The Germans scored 180 in the first quarter. But Smith is also competing well. However, the hempel leg decides for itself! Very good start. 1: 0
4.43 pm: Now it's Florian Hempel's turn. He comes up with the song "Coleshe Jung".
4.42 pm: let's go. Australian Raymond Smith made his stage debut.
4.33 pm: Michael Smith won against William O'Connor. Let's start Florian Hempel's game now.
4:15 pm: In a few minutes, when the match between Michael Smith and William O'Connor is over, Hempel and Smith will be on stage.
4:10 pm: The winner of the duel will meet the winner of the duel between Steve Lennon (Ireland) and Mervyn King (England) on Thursday afternoon. The two will not meet until noon tomorrow.
4:05 pm: With the victories over Jamie Hughes (3: 1, Round One) and Devon Peterson (3: 0, Round Two), the Australian handled his previous job with ease.
4 pm: Welcome to the Darts Live Ticker! In the fight against Raymond Smith, Florian Hempel has a great chance to make an exciting move to the 16th round of the World Cup.
Calvin Andrus
Travel fan. Freelance analyst. Proud problem solver. Infuriatingly humble zombie junkie.
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/other-math\/CLONE-547b8018-14a8-4d02-afd6-6bc35a0864ed\/chapters-1-8-cumulative-review-exercises-page-621\/18","text":"# Chapters 1-8 - Cumulative Review Exercises - Page 621: 18\n\n$L$\n\n#### Work Step by Step\n\nThey talk about volume, and the most reasonable unit of volume of quantity of oil to a car is $L$.\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2020-02-28 23:02:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.29011034965515137, \"perplexity\": 1837.6538800646651}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875147647.2\/warc\/CC-MAIN-20200228200903-20200228230903-00502.warc.gz\"}"}
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{"url":"https:\/\/ai.stackexchange.com\/questions\/11633\/does-everyone-still-use-discount-rates","text":"# Does everyone still use discount rates?\n\nIn Section 10.4 of Sutton and Barto's RL book, they argue that the discount rate $$\\gamma$$ has no effect in continuing settings. They show (at least for one objective function) that the average of the discounted return is proportional to the undiscounted average reward $$r(\\pi)$$ under the given policy.$$^*$$ They then advocate using average rewards rather than the usual returns of the discounted setting.\n\nI've never encountered someone using average rewards (and no discounting) in the wild, though. Am I just ignorant of some use case, or is pretty much everyone sticking to discounting anyways?\n\n$$r(\\pi)=\\sum_s \\mu_\\pi (s) \\sum_a \\pi(a|s) \\sum_{s',r}p(s',r|s,a)r$$\n\n$$\\mu_\\pi$$ is the stationary state distribution while following policy $$\\pi$$.\n\n$$^*$$Their proof did use the fact that the MDP was ergodic. I'm not sure how often that assumption holds in practice.\n\n\u2022 I have never actually seen a continuing task formulation used \"in the wild\". Possibly that explains something: episodic tasks are a good enough framework for most real world RL problems? \u2013\u00a0tahsmith Jul 12 '19 at 8:35","date":"2021-08-04 17:17:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 7, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8018419146537781, \"perplexity\": 1269.6930152041593}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046154878.27\/warc\/CC-MAIN-20210804142918-20210804172918-00178.warc.gz\"}"}
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Het Lichtfestival Gent is een sinds 2011 terugkerend lichtfestival in Gent. Het vindt plaats in de maanden januari of februari. Gedurende de festivalperiode kunnen bezoekers een parcours door de stad volgen met verschillende lichtsculpturen, projecties en installaties van hedendaagse nationale en internationale kunstenaars.
Geschiedenis
Gent startte met een lichtplan in 1998, ontworpen door het lichtbureau van Roland Jéol uit Lyon. De stad ging lichtaccenten aanbrengen bij belangrijke gebouwen, monumenten, verkeerassen, parken en pleinen. Het lichtplan van Gent werd bekroond met een aantal prijzen: de City-People-Light Award in 2004 en de Auroralia Award in 2012.
De stad plaatste ook lichtkunstwerken. Zo is er het kunstwerk 'Ai Nati Oggi' van Alberto Garutti sinds 2011. De lantaarns op het Sint-Veerleplein zijn verbonden met alle kraamklinieken van de Stad Gent. Telkens er een baby geboren wordt in een van de Gentse ziekenhuizen, lichten de lantaarns op.
Tijdens de inhuldiging van de sfeerverlichting aan het Hof van Beroep op het Koophandelsplein werd het lichtfestival in Gent aangekondigd. Het was de bedoeling van het lichtfestival om het lichtplan van Gent in de schijnwerpers te zetten. De eerste editie van het lichtfestival vond plaats in 2011. Door groot succes was in 2012 het volgende festival. Echter, door de hoge kostprijs werd er besloten het festival hierna driejaarlijks te houden.
Het kunstwerk 'Blauwe Vogels' of 'Les oiseaux de Mr. Maeterlinck' werd in 2016 definitief geïnstalleerd aan de Predikherenlei. Het werk maakte eerder deel uit van het lichtfestival in 2012.
In 2021 werd beslist de aangekondigde editie uit te stellen naar het najaar vanwege de Covid-19 pandemie. De vijfde editie ging door met een langer loopparcours, alle installaties in openlucht en verplichte mondmaskers tijdens het hele parcours. Omwille van de tiende verjaardag van het eerste festival keerden vier publieksfavorieten uit de vorige vier festivals terug.
Cijfers
Zie ook
Amsterdam Light Festival
GLOW Festival
Externe link
Homepage van het Lichtfestival Gent
Evenement in Gent
Lichtkunst
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Philly Rapper Tierra Whack Featured On Beyoncé's Lion…
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J Cole, Meek Mill, and More Performing at…
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DNA is hitting the streets of Philadelphia and beyond, asking what's the word on the street is on various…
Meek Mill Freestyles Over "Back To Back" Beat
Meek Mill made a radio appearance on Funkmaster Flex's show in New York Monday night and did the ultimate ironic…
Meek Mill Responds To Jay Z's Tweet &…
Meek Mill celebrated the release of his highly anticipated album Championships on November 30th, and he took a helicopter back…
Meek Mill Spills The Tea on New Movie…
Meek Mill took to his Instagram page to share the tracklisting for Swizz Beatz new album, "Poison". While promoting the…
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| 7,145
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Кристиан Шенк фон Таутенбург (; * 18 декември 1599, Дрезден; † 3 август 1640, Таутенбург) е шенк и фрайхер на Таутенбург близо до Йена, Фрауенприсниц и Долен-Требра, господар на Тона в Тюрингия.
Произход и наследство
Той е син на Буркхард Шенк фон Таутенбург (* 19 юли 1566; † 2 септември 1605) и съпругата му Агнес фон Еверщайн от Померания (* 1576; † 27 ноември 1636), вдовица на граф Ернст VII фон Хонщайн (1562 – 1593), дъщеря на граф Лудвиг III фон Еверщайн-Наугард (1527 – 1590) и Анна фон Мансфелд-Хинтерорт († 1583). Внук е на Георг Шенк фон Таутенбург (1537 – 1579) и Магалена фон Глайхен-Рембда († 1571).
Сестра му София Шенк фон Таутенбург († 1636) се жени за граф Фридрих Алберт фон Золмс-Зоневалде и Поух (1592 – 1615) и втори път през 1618 г. за фелдмаршал граф Волфганг III фон Мансфелд-Фордерорт (1575 – 1638).
Фамилията притежава от 1427 г. също съседното господство Фрауенприсниц. Шенките фон Таутенбург купуват през 1631 г. господството Тона.
Децата му и 22-годишната му съпруга Доротея Сибила Ройс-Гера-Плауен умират на 25 ноември 1631 г. На 12 май 1638 г. дворецът му във Фрауенприсниц изгаря.
Кристиан Шенк фон Таутенбург умира на 40 години на 3 август 1640 г. в Таутенбург и е погребан на 20 септември 1640 г. във фамилната гробница в църквата във Фрауенприсниц в Саксония-Ваймар. С него измира тюрингската линия на шенките фон Таутенбург и господството е взето от Курфюрство Саксония.
Фамилия
Кристиан Шенк фон Таутенбург се жени на 12 юни 1627 г. в Гера за Доротея Сибила Ройс-Гера-Плауен (млада линия) (* 7 октомври 1609, Гера; † 25 ноември 1631, Таутенбург), дъщеря на Хайнрих II Ройс-Гера-Плауен (1572 – 1635) и Магдалена II фон Шварцбург-Рудолщат (1580 – 1652).
Литература
Christian August Vulpius: Kurze Übersicht der Geschichte der Schenken von Tautenburg, im Journal: Die Vorzeit, Jena 1821
Geschichte der Schenken von Tautenburg. In: Ruinen oder Taschenbuch zur Geschichte verfallener Ritterburgen und Schlösser: nebst ihren Sagen, Legenden und Mährchen, Verlag Lechner 1834, Volume 3, S. 161 – 176
Johann Christoph Friderici: Historia pincernarum Varila Tautenburgicorum ex monumentis ineditis atque scritporibus coaeris eruta, Verlag Fischer, Jena 1722
Detlev Schwennicke, Europaische Stammtafeln, New Series, Vol. I/3, Tafel 361., Vol. XVII, Tafel 85.
Източници
Външни препратки
Schenk von Tautenburg, zeno.org
Германска аристокрация
Родени в Дрезден
|
{
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| 4,147
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Eoconodon és un gènere extint de mesonic triisodòntid que visqué durant el Danià a Nord-amèrica.
Enllaços externs
A new species of Eoconodon (Triisodontidae, Mammalia) from the San Juan Basin, New Mexico
Mesonics del Paleocè
Triisodòntids
|
{
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| 4,407
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\section{Introduction}
The Cauchy distribution is a statistical model with a heavy-tailed symmetric distribution.
We cannot define its expected value and its variance, and it has no moment generating function, due to its heavy tails.
It also appears in physics, and is called the Lorentz distribution alternatively.
Characterizations of probability distributions are interesting in itself and useful when we choose a suitable statistical model.
Various characterizations of the Cauchy distribution have been considered by many authors \cite{Arnold1979,Arnold1990, Bell1985, Chin2020, Dunau1987, Hamedani1993, Hassenforder1988, Knight1976b, Knight1976a, Letac1977, Menon1962, Menon1966, Norton1983, Obretenov1961, Ramachandran1970, Williams1969, Yanushkevichius2007,Yanushkevichius2014}.
This paper proposes yet another type of characterizations of the Cauchy distribution.
Our characterizations concern integral transforms, specifically, the M\"obius and Mellin transforms.
The M\"obius and Mellin transforms of the Cauchy distribution have somewhat simpler forms than the characteristic function of it, that is, the Fourier transform of it.
Our proofs utilize some basic facts of complex analysis and functional analysis.
Furthermore, our approach immediately yields characterizations of the circular Cauchy distribution and the mixture Cauchy model.
This paper adopts McCullagh's parametrization of the Cauchy distribution \cite{McCullagh1996}.
Let $i$ be the imaginary unit.
For a complex number $\gamma$, let $\textup{Re}(\gamma)$ and $\textup{Im}(\gamma)$ be the real and imaginary parts of $\gamma$ respectively.
We denote the distribution with density function
\[ p(x; \gamma) := \frac{\textup{Im}(\gamma)}{\pi} \frac{1}{|x - \gamma|^2}, \ x \in \mathbb{R}, \]
by $C(\gamma), \ \gamma \in \mathbb{H}$, where we let $\mathbb{H} := \{x+yi : y > 0\}$.
This paper is organized as follows.
In Section 2, we give a characterization of the Cauchy distribution by the M\"obius transforms with an application to a characterization of the circular Cauchy distribution.
In Section 3, we give a characterization of the Cauchy distribution by the Mellin transforms with an application to a characterization of the mixture Cauchy model.
\section{Characterization by M\"obius transforms}
Hereafter the symbol $E$ denotes the notation of the expectation of random variables and we denote the complex conjugate of a complex number $\gamma$ by $\overline{\gamma}$.
\begin{Thm}\label{Mobius}
Let $X$ be a real-valued random variable such that there exists $\alpha \in \mathbb{H}$ such that
\[ \alpha = \frac{E\left[\dfrac{X}{X - \overline{\gamma}}\right]}{E\left[\dfrac{1}{X - \overline{\gamma}}\right]}\]
for every $\gamma$ in a subset $D$ of $\mathbb{H}$ having a limit point in $\mathbb{H}$.
Then,
$X$ follows $C(\alpha)$.
\end{Thm}
Let $C_0 (\mathbb{R})$ be the set of continuous functions vanishing at infinity.
The following is standard.
\begin{Lem}\label{Cc}
Let $f \in C_0 (\mathbb{R})$.
Then,
\[ \lim_{b \to +0} \sup_{a \in \mathbb{R}} \left| f(a) - \int_{\mathbb{R}} f(x) \frac{b}{\pi((x-a)^2 + b^2)} dx \right| = 0. \]
\end{Lem}
\begin{proof}
We have that
\[ \left|f(a) - \int_{\mathbb R} f(x) \frac{b}{\pi((x-a)^2 + b^2)} dx \right| \le \int_{\mathbb R} | f(a+t) - f(a) | \frac{b}{\pi(t^2 + b^2)} dt. \]
Since $f \in C_0 (\mathbb{R})$, $f$ is uniformly continuous on $\mathbb{R}$, that is,
it holds that
for every $\epsilon > 0$, there exists $\delta > 0$ such that for every $t \in [-\delta, \delta]$,
\[ \sup_{a \in \mathbb{R}} | f(a+t) - f(a) | \le \epsilon. \]
By this and the fact that $\displaystyle \int_{\mathbb{R}} \frac{b}{\pi((x-a)^2 + b^2)} dx = 1$,
we have that
\[ \int_{\mathbb R} \sup_{a \in \mathbb{R}} | f(a+t) - f(a) | \frac{b}{\pi(t^2 + b^2)} dt \le \epsilon + 2 \|f\|_{\infty} \int_{\mathbb{R} \setminus [-\delta, \delta]} \frac{b}{\pi(t^2 + b^2)} dt, \]
where $\| f \|_{\infty}$ denotes the supremum norm of $f$.
Since $\dfrac{b}{\pi(t^2 + b^2)}$ is increasing as a function of $b \in \mathbb{R} \setminus [-\delta, \delta]$,
by applying the Lebesgue dominated convergence theorem,
we have that
\[ \lim_{b \to +0} \int_{\mathbb{R} \setminus [-\delta, \delta]} \frac{b}{\pi(t^2 + b^2)} dt = 0. \]
\end{proof}
For $\gamma \in \mathbb{H}$,
we let $\phi_{\gamma} : \mathbb{H} \to \mathbb{D}$ be the function defined by
\[ \phi_{\gamma}(z) := \frac{z - \gamma}{z - \overline{\gamma}}, \ \ z \in \mathbb{H}, \]
which is a M\"obius transform and could be regarded as a certain generalization of the Cayley transform.
\begin{proof}[Proof of Theorem \ref{Mobius}]
We have that for every $\gamma \in D$,
\begin{equation}\label{Mobius-mean}
E\left[ \phi_{\gamma}(X) \right] = \phi_{\gamma}(\alpha).
\end{equation}
By the residue theorem,
we see that \eqref{Mobius-mean} holds for $X$ following $C(\alpha)$.
Let $\mu$ be the Borel probability measure on $\mathbb{R}$ induced by $X$.
Then,
\begin{equation}\label{eq-compare}
\int_{\mathbb{R}} \phi_{\gamma}(x) \mu(dx) = \int_{\mathbb{R}} \phi_{\gamma}(x) \nu(dx),
\end{equation}
where $\gamma \in D$ and we let
\[ \nu(dx) := \frac{\textup{Im}(\alpha)}{|x - \alpha|^2} dx. \]
Let $$F_{\mu}(a+bi) := \frac{1}{\pi} \int_{\mathbb{R}} \frac{1}{x - (a+bi)} \mu(dx), \ a+bi \in \mathbb{H}.$$
This is holomorphic on $\mathbb{H}$.
Let $U_{\mu}(a+bi)$ and $V_{\mu}(a+bi)$ be the real and imaginary parts of $F_{\mu}(a+bi)$ respectively.
By replacing $\mu$ with $\nu$, we define $F_{\nu}(a+bi)$, $U_{\nu}(a+bi)$ and $V_{\nu}(a+bi)$ in the same manner.
By comparing the real and imaginary parts of \eqref{eq-compare},
we have that
for every $\gamma = a+bi \in D$,
\[ \int_{\mathbb{R}} \frac{b}{\pi((x-a)^2 + b^2)} \mu(dx) = \int_{\mathbb{R}} \frac{b}{\pi((x-a)^2 + b^2)} \nu(dx), \]
and
\[ \int_{\mathbb{R}} \frac{x-a}{\pi((x-a)^2 + b^2)} \mu(dx) = \int_{\mathbb{R}} \frac{x-a}{\pi((x-a)^2 + b^2)} \nu(dx). \]
Therefore,
$F_{\mu} = F_{\nu}$ on $D$.
By applying the identity theorem for holomorphic functions \cite[Theorem 10.18]{Rudin1987},
$F_{\mu} = F_{\nu}$ on $\mathbb{H}$.
By this and Fubini's theorem,
\[ \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{b}{\pi((x-a)^2 + b^2)} f(x) dx \mu(da) = \int_{\mathbb{R}} V_{\mu}(x+bi) f(x) dx \]
\[ = \int_{\mathbb{R}} V_{\nu}(x+bi) f(x) dx = \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{b}{\pi((x-a)^2 + b^2)} f(x) dx \nu(da). \]
By Lemma \ref{Cc}, we have that
\[ \int_{\mathbb{R}} f(a) \mu(da) = \lim_{b \to +0} \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{b}{\pi((x-a)^2 + b^2)} f(x) dx \mu(da), \]
and
\[ \int_{\mathbb{R}} f(a) \nu(da) = \lim_{b \to +0} \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{b}{\pi((x-a)^2 + b^2)} f(x) dx \nu(da). \]
Thus we have that
\[ \int_{\mathbb{R}} f(a) \mu(da) = \int_{\mathbb{R}} f(a) \nu(da). \]
Since $\mu$ and $\nu$ are both regular, by the Riesz-Markov-Kakutani theorem \cite[Theorem 6.19]{Rudin1987},
we have that $\mu = \nu$, which means that $X$ follows the Cauchy distribution with parameter $\alpha$.
\end{proof}
\begin{Rem}\label{upper}
Let $\overline{\mathbb{H}}$ be the closure of $\mathbb{H}$, that is, $\overline{\mathbb{H}} := \{x+yi : y \ge 0\}$.
Let $F_X (\gamma) := E\left[\dfrac{X}{X - \overline{\gamma}}\right] / E\left[\dfrac{1}{X - \overline{\gamma}}\right]$.
Since
\[ F_X (\gamma) - \overline{F_X (\gamma)} = \dfrac{E\left[ \dfrac{(X-a)^2}{(X-a)^2 + b^2} \right]E\left[ \dfrac{1}{(X-a)^2 + b^2}\right] - E\left[ \dfrac{X-a}{(X-a)^2 + b^2}\right]^2}{\left|E\left[\dfrac{1}{X - \gamma}\right]\right|^2}i,\]
where we let $\gamma = a+bi$,
it holds that for every $\gamma \in \mathbb{H}$, $F_X (\gamma) \in \overline{\mathbb H}$, and furthermore, it holds that
$F_X (\gamma) \in \mathbb H$ if and only if the distribution of $X$ is not a point mass.
It holds that
$\gamma_n$ is the maximal likelihood estimator of Cauchy samples $\{x_1, \cdots, x_n\}$,
if and only if
$\gamma_n = F_X (\gamma_n)$
where the expectation is taken with respect to $\displaystyle \frac{1}{n} \sum_{i=1}^{n} \delta_{x_i}$.
\end{Rem}
The circular Cauchy distribution, also known as the wrapped Cauchy distribution, appears in the area of directional statistics.
It is a distribution on the unit circle and is connected with the Cauchy distribution via M\"obius transforms.
Such connection is considered by \cite{McCullagh1996}.
Let $\mathbb D := \{z \in \mathbb{C} : |z| < 1\}$.
The circular-Cauchy distribution $P^{\textup{cc}}_w$ with parameter $w \in \mathbb{D}$ is the continuous distribution on $[0, 2\pi)$
with density function
\[ \frac{1}{2\pi} \frac{1 - |w|^2}{|\exp(ix) - w|^2}, \ \ x \in [0, 2\pi). \]
We remark that $\phi_{\gamma}$ is a bijection between $\mathbb H$ and $\mathbb D$, and furthermore its inverse is given by
\[ \phi_{\gamma}^{-1}(w) = \frac{\gamma - \overline{\gamma}w}{1 - w}, \ w \in \mathbb{D}. \]
We can extend the domain of $\phi_{\gamma}$ to $\overline{\mathbb H}$.
$\phi_{\gamma}$ defines a bijection between $\mathbb{R}$ and $\{z \in \mathbb{C} : |z| =1, z \ne 1\}$.
If a random variable $X$ follows the circular-Cauchy distribution $P^{\textup{cc}}_{w}$,
then, $\phi_{\gamma}^{-1}(\exp(iX))$ follows the Cauchy distribution with parameter $\phi_{\gamma}^{-1}(w)$.
Therefore, by computations with Theorem \ref{Mobius},
we have that
\begin{Cor}
Let $X$ be a $[0, 2\pi)$-valued random variable such that there exists $w \in \mathbb{D}$ such that
\[ w = \frac{E\left[ \dfrac{\exp(iX)}{ 1 - \eta \exp(iX)} \right]}{E\left[ \dfrac{1}{1 - \eta \exp(iX)} \right]} \]
for every $\eta$ in a subset $\widetilde D$ of $\mathbb{D}$ having a limit point in $\mathbb{D}$.
We also assume that $P(X \ne 0) = 1$.
Then, $X$ follows the circular-Cauchy distribution $P^{\textup{cc}}_{w}$.
\end{Cor}
Let $\overline{\mathbb{D}}$ be the closure of $\mathbb{D}$, that is, $\overline{\mathbb{D}} := \{z : |z| \le 1\}$.
Let
$G_X (\eta) := E\left[ \dfrac{\exp(iX)}{ 1 - \eta \exp(iX)} \right]/E\left[ \dfrac{1}{1 - \eta \exp(iX)} \right]$.
Assume that $P(X \ne 0) = 1$.
Then, for every $\gamma \in \mathbb{H}$,
$\phi^{-1}_{\gamma} \left( G_X (\eta)\right) = F_Y (\phi^{-1}_{\gamma} ( \eta ))$ where we let $Y := \phi_{\gamma}^{-1}(\exp(iX))$.
Then, by Remark \ref{upper}, it holds that for every $\eta \in \mathbb{D}$, $G_X (\eta) \in \overline{\mathbb D}$, and furthermore,
$G_X (\eta) \in \mathbb D$ if and only if the distribution of $X$ is not a point mass.
\section{Characterization by Mellin transforms}
We define the logarithm for complex numbers as follows.
For $z = r \exp(i \theta) \in \mathbb H$ where $r > 0$ and $-\pi \le \theta < \pi$,
we let
\[ \log z := \log r + i \theta. \]
This is holomorphic on $U := \mathbb{C} \setminus \left\{z \in \mathbb{C} | \textup{Re}(z) \le 0, \textup{Im}(z) = 0 \right\}$.
Then,
\begin{equation}\label{def-log}
\log x = \log |x| + i \pi \mathbf{1}_{(-\infty, 0)}(x), \ \ x \in \mathbb{R} \setminus \{0\},
\end{equation}
where $\mathbf{1}_{(-\infty, 0)}$ denotes the indicator function of $(-\infty,0)$.
For every $a \in \mathbb{C}$, we let
\[ z^{a} := \exp(a \log z), \ \ \ z \in U. \]
For every $a \in \mathbb{C}$, we let $0^a := 0$.
This definition is also adopted for $a = 0$.
We remark that $x^{a}$ is {\it not} a real number if $x < 0$ and $a \in \mathbb{R} \setminus \mathbb{Z}$.
For example, $(-8)^{1/3} = -2 \exp(i \pi/3)$.
In this paper, we call $E[X^a]$ the Mellin transform of the random variable $X$.
We deal with the powers of {\it negative} numbers by allowing the powers to be {\it complex-valued}.
In this point, our definition of the powers of random variables is different from the one given in Zolotarev \cite[(3.0.4)]{Zolotarev1986}.
\begin{Thm}\label{power}
Let $X$ be a real-valued random variable such that $E\left[|X|^{\delta}\right] < \infty$ for some $\delta > 0$.
If it holds that $E[X^a] = \gamma^a, a \in D$ for a subset $D \subset (0, \delta)$ having a limit point in $(0, \delta)$ and some $\gamma \in \mathbb{H}$,
then, $X$ follows $C(\gamma)$.
\end{Thm}
Our proof of this assertion depends on Galambos and Simonelli \cite[Theorem 1.19]{galambos2004}.
However their definition of the Mellin transform of random variables is somewhat different from ours, so we need some arguments.
\begin{proof}[Proof of Theorem \ref{power}]
\begin{Lem}\label{F-hol}
Let $J := \left\{x+y i : 0 < x < \delta \right\}$ and $ \overline{J}$ be the closure of $J$.
Let $f(a) := E[X^a], a \in \overline{J}$.
Then, $f$ is well-defined and continuous on $\overline{J}$ and holomorphic on $J$.
\end{Lem}
\begin{proof}
Since $|X^a| = |X|^{\textup{Re}(a)}$ and $\textup{Re}(a) \in [0, \delta]$,
we see that $f$ is well-defined and continuous on $\overline{J}$.
Let $a \in J$ and $h \ne 0$.
We remark that $E[X^a] = E\left[X^a, \ X \ne 0\right]$.
Then we have that
\[ \frac{f(a+h) - f(a)}{h} - E[X^a \log X, X \ne 0] = E\left[X^a \left( \frac{X^h - 1}{h} - \log X \right), \ \ X \ne 0 \right]. \]
If $X \ne 0$, then,
\[ \left| X^a \left(\frac{X^h - 1}{h} - \log X \right) \right| \le |X|^{\textup{Re}(a)} \cdot |h| |\log X|^{2} \exp(|h \log X|) \]
\[ \le |h| |X|^{\textup{Re}(a)} \left( |\log |X||^{2} + \pi^2 \right) \exp\left(|h| \left( |\log |X|| + \pi \right) \right). \]
If $|X| \ge 1$ and $|h| \le (\delta - \textup{Re}(a))/2$, then,
\[ \left| X^a \left( \frac{X^h - 1}{h} - \log X \right) \right| \le \delta \exp(\delta \pi) |X|^{\frac{\delta + \textup{Re}(a)}{2}} ((\log |X|)^{2} + \pi^2). \]
By the assumption,
\[ E\left[ |X|^{\frac{\delta + \textup{Re}(a)}{2}} \left(1 + (\log |X|)^{2} \right), \ |X| \ge 1 \right] < +\infty. \]
If $|X| \le 1$ and $|h| \le \textup{Re}(a)/2$, then,
\[ \left| X^a \left( \frac{X^h - 1}{h} - \log X \right) \right| \le \delta \exp(\delta \pi) |X|^{\frac{\textup{Re}(a)}{2}} ((\log |X|)^{2} + \pi^2). \]
By the assumption and the fact that $\displaystyle \lim_{x \to +0} x^{\beta} \log x = 0$ for every $\beta > 0$,
\[ E\left[ |X|^{\frac{\textup{Re}(a)}{2}} \left(1 + (\log |X|)^{2} \right), \ |X| \le 1 \right] < +\infty. \]
By the Lebesgue dominated convergence theorem,
\[ E\left[X^a \left( \frac{X^h - 1}{h} - \log X \right), \ \ X \ne 0 \right] \to 0, \ h \to 0. \]
\end{proof}
\begin{Lem}\label{F-rep}
\[ f(a) = \gamma^a, \ \ a \in \overline{J}. \]
\end{Lem}
\begin{proof}
Let $\widetilde f(a) := \gamma^a, \ a \in \overline{J}$.
This is holomorphic on $J$.
By the assumption of Theorem \ref{power},
it holds that
$\widetilde f(a) = f(a), \ a \in D$.
By Lemma \ref{F-hol}, $f$ is holomorphic on $J$.
Hence, by the identity theorem for holomorphic functions,
$\widetilde f(a) = f(a), \ a \in J$.
Since $f$ and $\widetilde f$ are both continuous on $\overline{J}$,
we have the assertion.
\end{proof}
\begin{Lem}\label{g-exp-pre}
Let
\begin{equation}\label{g-def}
g(a) := E\left[X^a, \ X > 0 \right] + i E\left[(-X)^a, \ X < 0 \right].
\end{equation}
Then, $g$ is well-defined and continuous on $\overline{J}$ and holomorphic on $J$.
Furthermore,
\begin{equation}\label{g-strip}
g(a) = r^a \left(\cos(a \theta) - \frac{\sin(a\theta)}{\sin(a\pi)} \cos(a\pi) + i \frac{\sin(a\theta)}{\sin(a\pi)} \right), \ a \in J.
\end{equation}
\end{Lem}
We remark that \eqref{g-def} is equivalent to the definition of the Mellin transform of $X$ in \cite[Section 1.3]{galambos2004}.
\begin{proof}
Since $E[|X|^{\delta}] < +\infty$,
$g$ is well-defined and continuous on $\overline{J}$.
If $0 < a < \delta$, then,
\begin{equation}\label{f-re-im}
f(a) = E\left[X^a, \ X > 0 \right] + E\left[(-X)^a, \ X < 0 \right] \cos(a \pi) + i E\left[(-X)^a, \ X < 0 \right] \sin(a \pi).
\end{equation}
If $0 < a < \delta$, then, by Lemma \ref{F-rep},
\begin{equation}\label{x-nega}
E\left[(-X)^a, \ X < 0 \right] = r^a \frac{\sin(a\theta)}{\sin(a\pi)},
\end{equation}
where we let $\gamma = r \exp(i \theta)$.
Hence, as a function of $a$,
$E\left[(-X)^a, \ X < 0 \right]$ is holomorphic on $J$.
By using Lemma \ref{F-rep}, we have that
\[ E\left[X^a, \ X > 0 \right] + E\left[(-X)^a, \ X < 0 \right] \cos(a \pi) = r^a \cos(a\theta), \ 0 < a < \delta. \
Therefore we have that as a function of $a$,
$E\left[X^a, \ X > 0 \right]$ is holomorphic on $J$ and
\begin{equation}\label{x-posi}
E\left[X^a, \ X > 0 \right] = r^a \left(\cos(a\theta) - \frac{\sin(a\theta)}{\sin(a\pi)}\cos(a\pi) \right).
\end{equation}
By \eqref{f-re-im}, \eqref{x-nega} and \eqref{x-posi},
we have \eqref{g-strip}.
\end{proof}
Now we return to the proof of Theorem \ref{power}.
Since $\sin(a\pi) \ne 0$ for $a \in \{yi : y \ne 0\}$ and $\displaystyle \lim_{a \to 0} \frac{\sin(a\theta)}{\sin(a\pi)} = \frac{\theta}{\pi}$,
we could continuously extend the function $g$ in \eqref{g-strip} to the left boundary of $J$, which is the imaginary axis $\{yi : y \in \mathbb{R}\}$.
If $X$ follows the Cauchy distribution $C(\gamma)$, then, by the residue theorem,
\[ E[X^a] = \frac{\textup{Im}(\gamma)}{\pi} \int_{\mathbb{R}} \frac{x^a}{|x - \gamma|^2} dx = \gamma^a, \ a \in J. \]
Hence if we define $g$ for $X$ following the Cauchy distribution $C(\gamma)$ in the same manner as in \eqref{g-def},
then we have the same expression for $g$ as in \eqref{g-strip},
and in particular, they are identical with each other on the imaginary axis $\{yi : y \in \mathbb{R}\}$.
Now Theorem \ref{power} follows from \cite[Theorem 1.19]{galambos2004}.
\end{proof}
We also have the following claim which is similar to Theorem \ref{power}.
\begin{Thm}\label{power-2}
Let $X$ be a real-valued random variable such that $P(X = 0) = 0$ and $E\left[|X|^{\delta}\right] + E\left[|X|^{-\delta}\right] < \infty$ for some $\delta > 0$.
If $E[X^a] = \gamma^a, a \in D$ for a subset $D \subset (-\delta, \delta)$ having a limit point in $ (-\delta, \delta)$ and some $\gamma \in \mathbb{H}$,
then, $X$ follows $C(\gamma)$.
\end{Thm}
The proof of Theorem \ref{power-2} goes in the same manner as in the proof of Theorem \ref{power}.
\begin{Cor}
Let $X$ be a real-valued random variable such that $P(X = 0) = 0$ and $E\left[|X|^{\delta}\right] + E\left[|X|^{-\delta}\right] < \infty$ for some $\delta > 0$. Then, \\
(i) If $E\left[ X^{1/k_n}\right]^{k_n} = \gamma \in \mathbb{H}$ for an infinite increasing sequence $(k_n)_n$,
then, $X$ follows $C(\gamma)$. \\
(ii) If $0 < P(X < 0) < 1$ and $E[(\log X)^n] = E[\log X]^n$ for every $n \in \mathbb{N}$,
then, $X$ follows $C(E[\log X])$.
\end{Cor}
We remark that if $0 < P(X < 0) < 1$ and $P(X = 0) = 0$, then, $0 < P(X > 0) < 1$, and furthermore, $X$ is non-atomic.
\begin{proof}
Assertion (i) follows from Theorem \ref{power-2}.
(ii) We remark that for $p \in (-\delta, \delta)$,
\[ E[\exp(|p \log X|)] \le \exp(|p| \pi) E[\exp(|p| |\log |X||)] \le E\left[|X|^{-|p|} + |X|^{|p|}\right] < +\infty. \]
By the Lebesgue convergence theorem and the assumption, we have that for $p \in (-\delta, \delta)$,
\[ E[X^p] = E\left[ \sum_{n=0}^{\infty} \frac{p^n (\log X)^n}{n!}\right] = \sum_{n=0}^{\infty} \frac{p^n E\left[ (\log X)^n\right]}{n!} = \exp(p E[\log X]). \]
By \eqref{def-log}, we have that
\[ \exp(E[\log X]) = \exp(E[\log |X|]) \exp(i \pi P(X < 0)). \]
By the assumption, we have that $\sin(\pi P(X < 0)) > 0$.
Hence, it holds that $\exp(E[\log X]) \in \mathbb{H}$.
Now apply Theorem \ref{power}.
\end{proof}
It might be interesting to consider sufficient conditions for $E[(\log X)^n] = E[\log X]^n$ for every $n \in \mathbb{N}$.
It is not sufficient that $E[(\log X)^2] = E[\log X]^2$.
For example, if we consider the expectation with respect to
$$\mu = \frac{1}{3} \left( \delta_{\{-1\}} + \delta_{\{\exp(\pi/\sqrt{3})\}} + \delta_{\{\exp(-\pi/\sqrt{3})\}} \right),$$
then, we have that $E[(\log X)^2] = E[\log X]^2$.
We can also give a characterization for the mixture Cauchy model.
If the probability density function is given by
\[ \frac{1-t}{\pi} \frac{\sigma_1}{(x-\mu_1)^2 + \sigma_1^2} + \frac{t}{\pi} \frac{\sigma_2}{(x-\mu_2)^2 + \sigma_2^2}, \]
for some $0 < t < 1$ and $(\mu_1, \sigma_1) \ne (\mu_2, \sigma_2)$,
then, we call the model the mixture Cauchy model $C\left(t; \mu_1 + \sigma_1 i; \mu_2 + \sigma_2 i\right)$.
See Lehmann \cite[pp.480-481]{Lehmann1999} for mixture models of location-scale families.
We can show the following in the same manner as in the proof of Theorem \ref{power}.
\begin{Cor}
Let $X$ be a real-valued random variable such that $E\left[|X|^{\delta}\right] < \infty$ for some $\delta > 0$.
If $E[X^a] = (1-t) \gamma_1^a + t \gamma_2^a, \ a \in D$ for a subset $D \subset (0, \delta)$ having a limit point in $(0, \delta)$ and some $t \in (0,1)$ and $\gamma_1, \gamma_2 \in \mathbb{H}$,
then, $X$ follows the mixture Cauchy model $C(t; \gamma_1; \gamma_2)$.
\end{Cor}
\noindent{\it Acknowledgements} \ The author appreciates the referee for careful reading of the manuscript and giving helpful comments.
The author was supported by JSPS KAKENHI 19K14549.
\bibliographystyle{amsplain}
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\section{Introduction}
In this talk we reconsider \cite{Kondo12,KSFNS13} a massive Yang-Mills theory \cite{YM54} without the Higgs field \cite{Higgs66}.
A motivation of this research stems from some nonperturbative phenomena caused by strong interactions.
\begin{itemize}
\item[(i)]
Confinement and Green functions--- The deep infrared behaviors of the gluon and ghost Green functions are believed to be intimately connected to color confinement in QCD \cite{KO79,Gribov78}.
In the Landau gauge, the decoupling solution \cite{decoupling,FMP09,BGP10} for the gluon and ghost propagators is currently supported rather than the scaling solution \cite{scaling} by recent numerical simulations on large lattices in three and four spacetime dimensions \cite{decoupling-lattice}.
Quite recently, it has been shown \cite{TW11} that the decoupling solution for the gluon and ghost propagators can be well reproduced from a low-energy effective model of a massive Yang-Mills theory, which is a special case of the Curci-Ferrari (CF) model \cite{CF76}.
This feature is not restricted to the Landau gauge and is common to manifestly Lorentz covariant gauges, e.g., the maximal Abelian gauge \cite{MCM06}, as pointed out and demonstrated in \cite{Kondo11}.
We can ask how color confinement in QCD is understood from the CF model.
\item[(ii)]
Glueball mass spectrum---
A glueball should be constructed from the fundamental degrees of freedom of QCD, i.e., quark, gluon and ghost.
For instance, the potential model of \cite{Cornwall82} identifies glueballs with bound states of massive gluons. They are described simply by introducing a naive mass term for gluons,
$\frac12 M^2 \mathscr{A}_\mu \cdot \mathscr{A}^\mu$, which however breaks the Becchi-Rouet-Stora-Tyutin (BRST) symmetry.
We ask how we can introduce a BRST-invariant mass term for gluons to establish a firm field theoretical foundation for treating glueballs, which will enable us to answer how precisely the mass and spin of the resulting glueballs are related to those of the constituent gluons.
\item[(iii)]
Vacuum condensates---
Besides gauge-invariant vacuum condensates represented by $\langle \bar \psi \psi \rangle$ with mass dimension-three and $\langle \mathscr{F}_{\mu\nu}^2 \rangle $ with mass dimension four, which are very important to characterize the nonperturbative vacuum of QCD, there might exist an extra dimension two condensate. In fact, such a lower dimensional vacuum condensate is needed from the phenomenological point of view.
However, such a condensate cannot be constructed from gauge-invariant local composite operators in the framework of the local field theory.
A BRST-invariant vacuum condensate of mass dimension two has been constructed in \cite{Kondo01,KMSI02}. However, it is just on-shell BRST invariant.
Can we construct an off-shell BRST invariant version of vacuum condensate of mass dimension two?
\end{itemize}
Another motivation of studying the CF model comes from the field theoretical interest, since the massive Yang-Mills theory without the Higgs field has an unsatisfactory aspect as a quantum field theory.
Renormalizability \cite{tHooft71,tHooft71b} is an important criterion for a quantum field theory to be a calculable and predictable theory.
In addition, physical unitarity \cite{tHooft71,tHooft71b,KO78,DV70,SF70,Boulware70,CF76b,Ojima82,BSNW96,KG67,DTT88,RRA04} is another important criterion for a quantum theory of gauge fields to be a meaning theory, which prevents unphysical particles from being observed.
In view of this, we remind the readers of the well-known facts:
\begin{itemize}
\item[(i)]
The massless Yang-Mills theory satisfies both renormalizability and physical unitarity \cite{tHooft71,KO78}.
\item[(ii)]
The massive Yang-Mills theory in which local gauge invariance is spontaneously broken by the Higgs field and the gauge field acquires the mass through the Higgs mechanism satisfies both renormalizability and physical unitarity
\cite{tHooft71b}.
\end{itemize}
In fact, the unified theory of Glashow-Weinberg-Salam for the electromagnetic and weak interactions based on the spontaneous symmetry breaking: $SU(2)_L \times U(1)_Y \rightarrow U(1)_{EM}$ predicted the massive gauge bosons $W^+, W^-$, and $Z^0$ which have been discovered in the mid-1980s, and the remaining Higgs particle is about to be discovered.
However, in all the models proposed so far as the massive Yang-Mills theory without the Higgs fields (in which the local gauge symmetry is not spontaneously broken), it seems that renormalizability and physical unitarity are not compatible with each other.
See \cite{DTT88,RRA04} for reviews and \cite{BFQ} for later developments.
Indeed, the CF model has been shown to be renormalizable \cite{CF76b, BSNW96}, whereas the CF model does not seem to satisfy physical unitarity according to \cite{CF76b,Ojima82,BSNW96}.
Although the CF model is not invariant under the usual BRST transformation, it can be made invariant by modifying the BRST transformation. But, the modified BRST transformation is not nilpotent.
It is known that nilpotency is the key property to show physical unitarity in the usual massless Yang-Mills theory,
since the unphysical states form the BRST quartets and the cancellations occur among the quartets (Kugo-Ojima quartet mechanism) \cite{KO79,KO78}.
It is not so clear if nilpotency is necessary to recover physical unitarity in the massive case.
The physical unitarity of the CF model will be discussed in the perturbative and a nonperturbative framework in forthcoming papers \cite{Kondo12}.
\section{The Curci-Ferrari model and the modified BRST transformation}
In order to look for a candidate of the massive Yang-Mills theory without the Higgs field, we start from the usual massless Yang-Mills theory in the most general Lorentz gauge formulated in a manifestly Lorentz covariant way.
The total Lagrangian density is written in terms of the Yang-Mills field $\mathscr{A}_\mu$, the FP ghost field $\mathscr{C}$, the antighost field $\bar{\mathscr{C}}$ and the NL field $\mathscr{N}$.
As a candidate of the massive Yang-Mills theory without the Higgs field,
we add the ``mass term'' $\mathscr{L}_m$:
\begin{subequations}
\begin{align}
\mathscr{L}^{\rm{tot}}_{m\rm{YM}} =& \mathscr{L}_{\rm{YM}} + \mathscr{L}_{\rm{GF+FP}} + \mathscr{L}_{m} ,
\\
\mathscr{L}_{\rm{YM}} =& - \frac{1}{4} \mathscr{F}_{\mu \nu} \cdot \mathscr{F}^{\mu \nu} ,
\\
\mathscr{L}_{\rm{GF+FP}} =& \frac{\alpha}{2} \mathscr{N} \cdot \mathscr{N} + \frac{\beta}{2} \mathscr{N} \cdot \mathscr{N}
+ \mathscr{N} \cdot \partial^{\mu} \mathscr{A}_{\mu}
- \frac{\beta}{2} g \mathscr{N} \cdot (i \bar{\mathscr{C}} \times \mathscr{C})
\nonumber\\
& + i \bar{\mathscr{C}} \cdot \partial^{\mu} \mathscr{D}_{\mu}[\mathscr{A}] \mathscr{C}
+ \frac{\beta}{4} g^2 (i \bar{\mathscr{C}} \times \mathscr{C}) \cdot (i \bar{\mathscr{C}} \times \mathscr{C})
\nonumber\\
=& \mathscr{N} \cdot \partial^{\mu} \mathscr{A}_{\mu} + i \bar{\mathscr{C}} \cdot \partial^{\mu} \mathscr{D}_{\mu}[\mathscr{A}] \mathscr{C}
+ \frac{\beta}{4} ( \bar{\mathscr{N}} \cdot \bar{\mathscr{N}} + \mathscr{N} \cdot \mathscr{N})
+ \frac{\alpha}{2} \mathscr{N} \cdot \mathscr{N}
,
\\
\mathscr{L}_{m} =& \frac{1}{2} M^2 \mathscr{A}_{\mu} \cdot \mathscr{A}^{\mu} + \beta M^2 i \bar{\mathscr{C}} \cdot \mathscr{C} ,
\end{align}
\end{subequations}
where $\alpha$ and $\beta$ are parameters corresponding to the gauge-fixing parameters in the $M \rightarrow 0$ limit,
$
\mathscr{D}_{\mu}[\mathscr{A}] \mathscr{C}(x)
:= \partial_{\mu}\mathscr{C}(x) + g \mathscr{A}(x) \times \mathscr{C}(x)
$,
and
\begin{equation}
\bar{\mathscr{N}} :=-\mathscr{N}+gi\bar{\mathscr{C}} \times \mathscr{C} .
\end{equation}
The $\alpha=0$ case is the CF model with the coupling constant $g$, the mass parameter $M$ and the parameter $\beta$.
In the Abelian limit with vanishing structure constants $f^{ABC}=0$, the FP ghosts decouple and the CF model reduces to the Nakanishi model \cite{Nakanishi72}.
In what follows, we restrict our considerations to the $\alpha=0$ case.
In the $\alpha=0$ case, $\mathscr{L}_{\rm YM} + \mathscr{L}_{\rm GF+FP}$
is constructed so as to be invariant
under both the usual BRST transformation and anti-BRST transformation:
\begin{align}
\begin{cases}
{\boldsymbol \delta} \mathscr{A}_{\mu}(x) = \mathscr{D}_{\mu}[\mathscr{A}] \mathscr{C}(x) \\
{\boldsymbol \delta} \mathscr{C}(x) = -\frac{g}{2} \mathscr{C}(x) \times \mathscr{C}(x) \\
{\boldsymbol \delta} \bar{\mathscr{C}}(x) = i \mathscr{N}(x) \\
{\boldsymbol \delta} \mathscr{N}(x) = 0 \\
\end{cases} ,
\label{BRST}
\quad
\begin{cases}
\bar{\boldsymbol \delta} \mathscr{A}_{\mu}(x) = \mathscr{D}_{\mu}[\mathscr{A}] \bar{\mathscr{C}}(x) \\
\bar{\boldsymbol \delta} \bar{\mathscr{C}}(x)
= -\frac{g}{2} \bar{\mathscr{C}}(x) \times \bar{\mathscr{C}}(x) \\
\bar{\boldsymbol \delta} \mathscr{C}(x) = i \bar{\mathscr{N}}(x) \\
\bar{\boldsymbol \delta} \bar{\mathscr{N}}(x) = 0
\end{cases}
.
\end{align}
Indeed, it is checked that
\begin{align}
{\boldsymbol \delta} \mathscr{L}_{\rm{YM}} = 0 ,
\quad
{\boldsymbol \delta} \mathscr{L}_{\rm{GF+FP}} = 0 ,
\quad
\bar{\boldsymbol \delta} \mathscr{L}_{\rm{YM}} = 0 ,
\quad
\bar{\boldsymbol \delta} \mathscr{L}_{\rm{GF+FP}} = 0 .
\end{align}
This is not the case for the mass term $\mathscr{L}_{m}$, i.e.,
\begin{align}
{\boldsymbol \delta} \mathscr{L}_{m} \ne 0 .
\end{align}
Even in the presence of the mass term $\mathscr{L}_{m}$, however, the total Lagrangian $\mathscr{L}_{\rm mYM}^{\rm tot}$ can be made invariant by modifying the BRST transformation \cite{CF76}:
$\delta_{\rm BRST}'=\lambda {\boldsymbol \delta}'$ with a Grassmannian number $\lambda$ and
\begin{align}
\begin{cases}
{\boldsymbol \delta}' \mathscr{A}_{\mu}(x) = \mathscr{D}_{\mu}[\mathscr{A}] \mathscr{C}(x) \\
{\boldsymbol \delta}' \mathscr{C}(x)
= -\frac{g}{2} \mathscr{C}(x) \times \mathscr{C}(x) \\
{\boldsymbol \delta}' \bar{\mathscr{C}}(x) = i \mathscr{N}(x) \\
{\boldsymbol \delta}' \mathscr{N}(x) = M^2 \mathscr{C}(x)
\end{cases}
.
\end{align}
The modified BRST transformation deforms the BRST transformation of the NL field and reduces to the usual BRST transformation in the limit $M \rightarrow 0$.
It should be remarked that
${\boldsymbol \delta}' \mathscr{L}^{\rm tot}_{\rm mYM}=0$ follows from
\begin{align}
0 = {\boldsymbol \delta}' (\mathscr{L}_{\rm GF+FP} + \mathscr{L}_{m}) ,
\label{req}
\end{align}
while
\begin{equation}
{\boldsymbol \delta}' \mathscr{L}_{m} \ne 0,
\quad
{\boldsymbol \delta}' \mathscr{L}_{\rm GF+FP} \ne 0.
\end{equation}
Similarly, the total action is invariant under a modified anti-BRST transformation $\bar{{\boldsymbol \delta}}'$
defined by
\begin{align}
\begin{cases}
\bar{\boldsymbol \delta}' \mathscr{A}_{\mu}(x) = \mathscr{D}_{\mu}[\mathscr{A}] \bar{\mathscr{C}}(x) \\
\bar{\boldsymbol \delta}' \bar{\mathscr{C}}(x)
= -\frac{g}{2} \bar{\mathscr{C}}(x) \times \bar{\mathscr{C}}(x) \\
\bar{\boldsymbol \delta}' \mathscr{C}(x) = i \bar{\mathscr{N}}(x) \\
\bar{\boldsymbol \delta}' \bar{\mathscr{N}}(x) = - M^2 \bar{\mathscr{C}}(x)
\end{cases}
,
\end{align}
which reduces to the usual anti-BRST transformation in the limit $M \to 0$.
It is sometimes useful to give another form:
\begin{align}
{\boldsymbol \delta}' \mathscr{\bar N}(x) = g \mathscr{\bar N}(x) \times \mathscr{C}(x) - M^2 \mathscr{C}(x) ,
\quad
\bar{{\boldsymbol \delta}}' \mathscr{N}(x) = g \mathscr{N}(x) \times \bar{\mathscr{C}}(x) + M^2 \bar{\mathscr{C}}(x) .
\end{align}
Moreover, the path-integral integration measure
$\mathcal{D} \mathscr{A} \mathcal{D} \mathscr{C} \mathcal{D} \bar{\mathscr{C}} \mathcal{D} \mathscr{N}$
is invariant under the modified BRST transformation.
Indeed, it has been shown in \cite{Kondo12} that the Jacobian associated to the change of integration variables $\Phi(x) \to \Phi'(x) =\Phi(x)+ \lambda {\boldsymbol \delta}' \Phi(x)$ for the integration measure is equal to one.
However, the modified BRST transformation violates the nilpotency when $M \not= 0$:
\begin{align}
\begin{cases}
{\boldsymbol \delta}' {\boldsymbol \delta}' \mathscr{A}_{\mu}(x) = 0 , \\
{\boldsymbol \delta}' {\boldsymbol \delta}' \mathscr{C}(x) = 0 , \\
{\boldsymbol \delta}' {\boldsymbol \delta}' \bar{\mathscr{C}}(x) = i {\boldsymbol \delta}' \mathscr{N}(x)
= i M^2 \mathscr{C}(x) \ne 0 , \\
{\boldsymbol \delta}' {\boldsymbol \delta}' \mathscr{N}(x) = M^2 {\boldsymbol \delta}' \mathscr{C}(x)
= - M^2 \frac{g}{2} \mathscr{C}(x) \times \mathscr{C}(x) \ne 0 .
\end{cases}
\end{align}
The nilpotency is violated also for the modified anti-BRST transformation when $M \not= 0$:
In the limit $M \to 0$, the modified BRST and anti-BRST transformations reduce to the usual BRST and anti-BRST transformations and become nilpotent.
\section{Defining a massive Yang-Mills field}
We require the following properties to construct a non-Abelian massive spin-one vector boson field $\mathscr{K}_{\mu}(x)$ in a nonperturbative way:
\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{(\theenumi)}
\begin{enumerate}
\item
$\mathscr{K}_{\mu}$ has the modified BRST invariance (off mass shell):
\begin{equation}
{\boldsymbol \delta}' \mathscr{K}_{\mu} = 0 .
\end{equation}
\item
$\mathscr{K}_{\mu}$ is divergenceless (on mass shell):
\begin{equation}
\partial^{\mu} \mathscr{K}_{\mu} = 0 .
\end{equation}
\item
$\mathscr{K}_{\mu}$ obeys the adjoint transformation under the color rotation:
\begin{equation}
\mathscr{K}_{\mu}(x) \to U \mathscr{K}_{\mu}(x) U^{-1} , \quad U = \exp[i \varepsilon^A Q^A] ,
\end{equation}
\end{enumerate}
which has the infinitesimal version:
\begin{equation}
\delta \mathscr{K}_{\mu}(x) = \varepsilon \times \mathscr{K}_{\mu}(x) .
\end{equation}
The field $\mathscr{K}_\mu$ is identified with the non-Abelian version of the physical massive vector field with spin one, as ensured by the above properties.
Here (i) guarantees that $\mathscr{K}_{\mu}$ belong to the physical field creating a physical state with positive norm.
(ii) guarantees that $\mathscr{K}_{\mu}$ have the correct degrees of freedom as a massive spin-one particle, i.e., three in the four-dimensional spacetime, i.e., two transverse and one longitudinal modes, excluding one scalar mode.
(iii) guarantees that $\mathscr{K}_{\mu}$ obey the same transformation rule as that of the original gauge field $\mathscr{A}_{\mu}$
We observe that the total Lagrangian of the CF model is invariant under the (infinitesimal) \textbf{global gauge transformation} or \textbf{color rotation} defined by
\begin{align}
&\delta \Phi(x) := [\varepsilon^C i Q^C, \Phi(x)] = \varepsilon \times \Phi(x) ,
{\rm for} \quad
\Phi=\mathscr{A}_{\mu}, \mathscr{N}, \mathscr{C} , \bar{\mathscr{C}} , \\
&\delta \varphi(x) := [\varepsilon^C i Q^C, \varphi(x)] = - i \varepsilon \varphi(x) ,
\end{align}
where $\varphi$ is a matter field.
The conserved Noether charge $Q^A := \int d^3x \mathscr{J}^{0,A}_{\rm color}$ obtained from the color current $\mathscr{J}^0_{\rm color}$ is called the \textbf{color charge}
and is equal to the generator of the color rotation.
It has been shown \cite{Kondo12} that such a field $\mathscr{K}_{\mu}$ is obtained by a nonlinear but local transformation from the original fields $\mathscr{A}_\mu$, $\mathscr{C}$, $\bar{\mathscr{C}}$ and $\mathscr{N}$ of the CF model:
\begin{align}
\mathscr{K}_\mu :=& \mathscr{A}_\mu - M^{-2} \partial_\mu \mathscr{N}
- gM^{-2} \mathscr{A}_\mu \times \mathscr{N}
\nonumber\\
&+ gM^{-2} \partial_\mu \mathscr{C} \times i\bar{\mathscr{C}}
+ g^2 M^{-2} (\mathscr{A}_\mu \times \mathscr{C}) \times i \bar{\mathscr{C}} .
\label{K}
\end{align}
In the Abelian limit or the lowest order of $g$, $\mathscr{K}_{\mu}$ reduces to the Proca field for massive vector:
\begin{equation}
\mathscr{K}_{\mu} \to \mathscr{A}_{\mu} - \frac{1}{M^2} \partial_{\mu} \mathscr{N} := U_{\mu} .
\end{equation}
It should be remarked that $U_\mu$ is invariant under the Abelian version of the modified BRST, but it is not invariant under the non-Abelian modified BRST transformation.
The new field $\mathscr{K}_{\mu}$ is converted to a simple form:
\begin{equation}
\mathscr{K}_{\mu}(x) = \mathscr{A}_{\mu}(x) + \frac{1}{M^2} i {\boldsymbol \delta}' \bar{{\boldsymbol \delta}}' \mathscr{A}_{\mu}(x) .
\label{K2}
\end{equation}
It has been explicitly shown in \cite{Kondo12} that the field $\mathscr{K}_{\mu}$ defined by (\ref{K}) or (\ref{K2}) satisfies all the above properties.
The field $\mathscr{K}_{\mu}$ plays the role of the non-Abelian massive vector field
and is identified with a non-Abelian version of the spin-one massive vector field.
Equation (\ref{K}) gives a transformation from $\mathscr{A}_{\mu}, \mathscr{N}, \mathscr{C}$ and $\bar{\mathscr{C}}$ to $\mathscr{K}_{\mu}$.
As an application of the above result, we can construct a mass term which is invariant simultaneously under the modified BRST transformation, Lorentz transformation and color rotation:
\begin{equation}
\frac{1}{2} M^2 \mathscr{K}_{\mu}(x) \cdot \mathscr{K}^{\mu}(x) .
\end{equation}
This can be useful as a regularization scheme for avoiding infrared divergences in non-Abelian gauge theories.
Moreover, we can obtain a dimension-two condensate which is modified BRST invariant, Lorentz invariant, and color-singlet:
\begin{equation}
\langle \mathscr{K}_{\mu}(x) \cdot \mathscr{K}^{\mu}(x) \rangle .
\end{equation}
This dimension-two condensate is off-shell (modified) BRST invariant and should be compared with the dimension-two condensate proposed in \cite{Kondo01,KMSI02} which is only on-shell BRST invariant:
\begin{equation}
\Big\langle \frac12 \mathscr{A}_{\mu}(x) \cdot \mathscr{A}^{\mu}(x) + \beta \mathscr{C}(x) \cdot \mathscr{\bar C}(x) \Big\rangle .
\end{equation}
\section{Perturbative violation of physical unitarity}
In \cite{KSFNS13} , we have checked in a new perturbative treatment whether or not the CF model satisfies the physical unitarity.
Then we have confirmed the violation of the physical unitarity in the perturbative treatment and we have clarified the reason in the massive Yang-Mills theory without the Higgs field.
The perturbative analysis for the physical unitarity imposes a restriction on the valid energy together with the parameter of the CF model:
$E^2 < 4\beta M^2$
in order to confine unphysical modes (ghost, antighost, scalar mode).
However, $\beta=0$ is not allowed in this scenario.
It should be remarked that even the modified BRST (and anti-BRST) invariant quantity depends on a parameter $\beta$ in the $M \not= 0$ case.
This should be compared with the $M=0$ case, in which $\beta$ is a gauge-fixing parameter and the BRST-invariant quantity does not depend on $\beta$, which means that the physics does not depend on $\beta$ in the $M=0$ case.
This is not the case for $M \not= 0$ \cite{Lavrov12}.
\section{Possible nonperturbative restoration of physical unitarity}
The conclusion obtained in this work still leaves a possibility that the nonperturbative approach can modify the conclusion.
In a subsequent paper, indeed, we will propose a scenario in which the physical unitarity can be recovered in the CF model thanks to the FP conjugation invariance.
Indeed, we will show that the norm cancellation is automatically guaranteed from the Slavnov-Taylor identities if the ghost-antighost bound state exists.
In this way, the physical unitarity can be recovered in a nonperturbative way.
To show the existence of the bound state of ghost and antighost, the Nambu-Bethe-Salpeter equation is to be solved. This is a hard work to be tackled in subsequent papers.
{\it Acknowledgements}:\
This work is supported by Grant-in-Aid for Scientific Research (C) 24540252 from the Japan Society for the Promotion of Science (JSPS).
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 7,610
|
Q: ASP.NET - How to display image in browser without saving image in temp? I have a question- How to display image in browser without saving image in Temp folder? Is it even possible? My code have to reading images from database and displaying images in website. I actually try with converting data from database and I don't know what I want to do.
I try also with "imageHandlers", "FileStream", "Base64StringToBitmap" and nothing still works...
Please write example code or modify my code.
private void LoadImages()
{
ImageButton imageButton = new ImageButton();
string constr = ConfigurationManager.ConnectionStrings["DefaultConnection"].ConnectionString;
using (SqlConnection conn = new SqlConnection(constr))
{
using (SqlCommand cmd = new SqlCommand())
{
cmd.CommandText = "select Id, Name, Data from tblFiles WHERE email = @CurrentUser";
cmd.Parameters.Add("@CurrentUser", SqlDbType.NVarChar);
cmd.Parameters["@CurrentUser"].Value = User.Identity.Name;
cmd.Connection = conn;
conn.Open();
using (SqlDataReader sdr = cmd.ExecuteReader())
{
if (sdr.HasRows)
{
sdr.Read();
string fileName = sdr["Name"].ToString();
FileInfo fi = new FileInfo(fileName);
byte[] byte_image_string = ((byte[])sdr["Data"]);
string image_string = Convert.ToBase64String((byte[])sdr["Data"]) + fi.Name;
imageButton.Height = Unit.Pixel(100);
imageButton.Style.Add("padding", "5px");
imageButton.Width = Unit.Pixel(100);
imageButton.Click += new ImageClickEventHandler(imageButton_Click);
Panel1.Controls.Add(imageButton);
System.Drawing.Image newImage;
if (byte_image_string != null)
{
using (MemoryStream stream = new MemoryStream(byte_image_string))
{
newImage = System.Drawing.Image.FromStream(stream);
//I want here display image to browser without saving
//string newPhoto = "";
//newImage.Save(newPhoto);
imageButton.ImageUrl = "data:image/jpg;base64," + newPhoto;
}
}
conn.Close();
}
}
}
}
}
My example image code from database:
0xFFD8FFE000104A46494600010100000100010000FFE1018C45786966000049492A0008000000020031010200070000002600000069870400010000002E00000000000000476F6F676C6500000500009007000400000030323230099007000B0000007000000086920700080100007B00000002A00400010000006F02000003A0
A: This is how I do it in one of my projects: Look src part
<p><a href='<%#"TheObject.aspx?O=" + Eval("ID") %>'><img runat="server" visible='<%#Eval("AttachmentID") != DBNull.Value %>' class="objPicture1" alt='<%Eval("Title") %>' width="340" height="260" src='<%#"~/Attachment.aspx?ID=" + Eval("AttachmentID")%>' /></a></p>
In Attachment.aspx you have this code:
protected void Page_Load(object sender, EventArgs e)
{
Guid objID = //take the ID of the object ?ID=""
DataRow attachmentRow = //fetch DataRow of the Attachment from Database
if (attachmentRow == null)
return;
Response.ContentType = attachmentRow["ContentType"].ToString();// value of this is image/gif or image/jpeg and etc.
Response.BinaryWrite((byte[])attachmentRow["Data"]); // Data is type image in my case
Response.End();
}
A: It looks like you are creating Base64String in wrong way. You are appending file name to it that should not work. just try following.
string image_string = Convert.ToBase64String((byte[])sdr["Data"]);
Assign directly it to ImageUrl. no need to use MemoryStream here.
imageButton.ImageUrl = "data:image/jpg;base64," + image_string;
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,864
|
{"url":"https:\/\/tex.stackexchange.com\/questions\/461009\/axis-label-is-not-displayed-and-adding-labels-to-stacked-bar-chart","text":"# Axis label is not displayed and adding labels to stacked bar chart\n\nI have made this stacked bar chart but I can't get my x-axis label to be shown even though I have added them in the codes. Also, can anyone tell me how can I add labels to my stacked bar chart like in the picture ? Thanks!\n\n\\documentclass[10pt,twoside,a4paper,fleqn]{report}\n\\usepackage{xcolor,pgfplots}\n\\begin{document}\n\\begin{figure}[ht!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\nxbar stacked,\nlegend style={\nlegend columns=4,\nat={(xticklabel cs:0.5)},\nanchor=north,\ndraw=none\n},\nxlabel= Percentage,\nylabel= Environmental indicators,\nytick=data,\nxtick=data,\naxis y line*=none,\naxis x line*=bottom,\ntick label style={font=\\footnotesize},\nlegend style={font=\\footnotesize},\nlabel style={font=\\footnotesize},\nxtick={0,10,20,30,40,50,60,70,80,90,100},\nwidth=.9\\textwidth,\nbar width=6mm,\nyticklabels={GWP, ODP, POCP, AP, EP(T), EP(FW), EP(M), ADP, CED},\nxmin=0,\nxmax=100,\narea legend,\ny=8mm,\nenlarge y limits={abs=0.625},\n]\n[Construction,fill=Construction]\ncoordinates\n{(7.9,0) (8.8,1) (20.2,2) (28.7,3) (20.5,4) (44.7,5) (17.8,6) (46.6,7) (3.9,8)};\n[Operational,fill=Operational]\ncoordinates\n{(89.7,0) (89.5,1) (76.3,2) (64.5,3) (76.3,4) (50.1,5) (70.5,6) (37.9,7) (95.4,8)};\n[Maintenance,fill=Maintenance]\ncoordinates\n{(1.0,0) (1.5,1) (2.8,2) (6.4,3) (2.9,4) (4.8,5) (3.1,6)(15.3,7) (0.6,8)};\n[Disposal,fill=Disposal]\ncoordinates\n{(1.3,0) (0.2,1) (0.7,2) (0.3,3) (0.3,4) (0.5,5) (8.6,6)(0.1,7) (0.1,8)};\n\n\\legend{Construction, Operational, Replacement, Disposal}\n\\end{axis}\n\\end{tikzpicture}\n\\caption{Life cycle phase contribution analysis per impact category.\\\\Categories from bottom to the top are Global Warming Potential (\\ac{GWP}), Ozone Depletion Potential (\\ac{ODP}), Photochemical Ozone Creation Potential (\\ac{POCP}), Acidification Potential (\\ac{AP}), Terrestrial Eutrophication Potential \\ac{EP(T)}, \\ac{EP(FW)}, \\ac{ADP} and \\ac{CED}.}\n\\label{conventional}\n\\end{figure}\n\\end{document}\n\n\u2022 Thanks for providing a MWE. But, please edit your code so that it produces the erroneous result that you describe. As it is, it is not compilable. For instance you have \\[ where it should just be a [ for optional parameters. \u2013\u00a0Peter Grill Nov 20 '18 at 20:51\n\u2022 Thanks for the reply. I hope it's correct now. \u2013\u00a0thesilencer Nov 20 '18 at 21:49\n\u2022 Did you try compiling it? \u2013\u00a0Peter Grill Nov 20 '18 at 22:21\n\u2022 I'm not quite sure what's the mistake here because I extracted the table from my thesis and it's not exactly the whole thing. Did i miss any package that you can kindly enlighten me on? \u2013\u00a0thesilencer Nov 20 '18 at 23:26\n\u2022 I get errors when I compile this.Please put this in a separate document and make sure it compiles. For instance, the Construction style is not defined, neither is the Construction color. \u2013\u00a0Peter Grill Nov 20 '18 at 23:34\n\nIt is somewhat hard to write an answer because this requires some fair amount of avoidable extra work. I ended up defining the missing colors somehow, and inventing a command \\ac. Now to your questions:\n\n1. You cannot see the xlabel since you put the legend on top. I moved the legend down.\n2. You can add numbers with percentages using nodes near coords. I made a guess which numbers you might be after, and of course this guess is most likely wrong, but perhaps will allow you to modify things in such a way that you get what you want.\n\nResult\n\n\\documentclass[10pt,twoside,a4paper,fleqn]{report}\n\\usepackage{xcolor,pgfplots}\n\\pgfplotsset{compat=1.16}\n\\definecolor{Construction}{RGB}{184,36,33}\n\\definecolor{Operational}{RGB}{233,171,100}\n\\definecolor{Maintenance}{RGB}{124,174,255}\n\\definecolor{Disposal}{RGB}{185,220,165}\n\\newcommand{\\ac}[1]{\\textcolor{red}{#1}}\n\\begin{document}\n\\begin{figure}[ht!]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[clip=true,\nxbar stacked,\nlegend style={\nlegend columns=4,\nat={(xticklabel cs:0.5)},yshift=-5mm,\nanchor=north,\ndraw=none\n},\nytick=data,\nxtick=data,\naxis y line*=none,\naxis x line*=bottom,\nxlabel= Percentage,\nylabel= Environmental indicators,\ntick label style={font=\\footnotesize},\nlegend style={font=\\footnotesize},\nlabel style={font=\\footnotesize},\nxtick={0,10,20,30,40,50,60,70,80,90,100},\nwidth=.9\\textwidth,\nbar width=6mm,\nyticklabels={GWP, ODP, POCP, AP, EP(T), EP(FW), EP(M), ADP, CED},\nxmin=0,\nxmax=100,\narea legend,\ny=8mm,\nenlarge y limits={abs=0.625},\nvisualization depends on=x \\as \\rawx,\nnodes near coords={\\pgfmathprintnumber{\\pgfplotspointmeta}\\%},\nevery node near coord\/.style={xshift=\\rawx*1.5pt+0.5cm,text=white},\n]\n[Construction,fill=Construction]\ncoordinates\n{(7.9,0) (8.8,1) (20.2,2) (28.7,3) (20.5,4) (44.7,5) (17.8,6) (46.6,7) (3.9,8)};\n[Operational,fill=Operational,nodes near coords={}]\ncoordinates\n{(89.7,0) (89.5,1) (76.3,2) (64.5,3) (76.3,4) (50.1,5) (70.5,6) (37.9,7) (95.4,8)};\n[Maintenance,fill=Maintenance,nodes near coords={}]\ncoordinates\n{(1.0,0) (1.5,1) (2.8,2) (6.4,3) (2.9,4) (4.8,5) (3.1,6)(15.3,7) (0.6,8)};\n[Disposal,fill=Disposal,nodes near coords={}]\ncoordinates\n{(1.3,0) (0.2,1) (0.7,2) (0.3,3) (0.3,4) (0.5,5) (8.6,6)(0.1,7) (0.1,8)};\n\\legend{Construction, Operational, Replacement, Disposal}\n\\end{axis}\n\\end{tikzpicture}\n\\caption{Life cycle phase contribution analysis per impact category.\nCategories\nfrom bottom to the top are Global Warming Potential (\\ac{GWP}), Ozone Depletion\nPotential (\\ac{ODP}), Photochemical Ozone Creation Potential (\\ac{POCP}),\nAcidification Potential (\\ac{AP}), Terrestrial Eutrophication Potential","date":"2020-02-21 22:57:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8135228157043457, \"perplexity\": 9114.974218881733}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875145538.32\/warc\/CC-MAIN-20200221203000-20200221233000-00371.warc.gz\"}"}
| null | null |
{"url":"http:\/\/mathhelpforum.com\/advanced-statistics\/121685-marginal-pdf-s.html","text":"# Math Help - Marginal pdf's\n\n1. ## Marginal pdf's\n\nplz help to me, actually I cant find out how to got ans.\n\nthe pdf is given as\nf(x, y) = 2, if (x,y) \u0454 R\nelse 0\n\nthere is marginal pdf's of X and Y\n\n= 2y\n\nthen\n\nE(X) = integral 0 to 1, x2(1-x)dx = 1\/3\n\nnow can you describe this stepwise\n\nE(X) = mathemetical expectation\n\nbecoz I cant analyse it\n\n2. Originally Posted by yog782004\nplz help to me, actually I cant find out how to got ans.\n\nthe pdf is given as\nf(x, y) = 2, if (x,y) \u0454 R\nelse 0\n\nthere is marginal pdf's of X and Y\n\n= 2y\n\nthen\n\nE(X) = integral 0 to 1, x2(1-x)dx = 1\/3\n\nnow can you describe this stepwise\n\nE(X) = mathemetical expectation\n\nbecoz I cant analyse it\nIf the original of this question was in English please post the exact wording.\n\nCB\n\n3. we need to know where f(x,y)=2\nThe support cannot be all of R2.\nprobably some triangle with x<y or y<x\n\n4. ## Intigral\n\nHi Captain\nHi mathengle\n\nHere I attach one you can find it,\nactually it is example of my book and cant analyse these calculation\n\n5. I'm not sure what you want to know.\nThe defintion of the mean of X is\n\n$E(X)=\\int_{-\\infty}^{\\infty}xf(x)dx$\n\nYou first need to find the density and the support (where it is not zero).\nThen integrate.\n\nLikewise $E(X^2)=\\int_{-\\infty}^{\\infty}x^2f(x)dx$\n\n6. thanks mathegle\n\nI get somethink from your suggetion, actualy I m not sure in intrigral operation and calculation, thats y i hav this confusion.","date":"2015-08-30 07:32:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 2, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7802612781524658, \"perplexity\": 4286.603561802478}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440644064919.24\/warc\/CC-MAIN-20150827025424-00105-ip-10-171-96-226.ec2.internal.warc.gz\"}"}
| null | null |
Hey guys, i got a problem with Fifa World cup 2006 for GC. I open the .iso file and i can't see anything, just a black screen, and it shows me 0fps, 70 or more VPS, and 200% speed. But i still can't see anything. How can i fix it? I prefer this rather than the PC version because it has better resolution and graphics. Thank you all.
Update your profile so that the people who look there don't waste their time suggesting that you use the latest build when you already are.
1) Right click on the game in the game list and enable the "mmu speed hack".
2) Make sure your game file is good and not corrupt - dump the game again if needed.
Oh, sorry. I'll update my profile now... Uhmmm. Ok I will try it. My game is Ntsc and all the other games i have, are PAL. I think i should get the game in PAL version.
Edit: I enabled "mmu speed hack" and it works just fine. Solved. Thank u so much!
|
{
"redpajama_set_name": "RedPajamaC4"
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| 9,168
|
\section{Introduction}
This is the second paper of our project \cite{NZ21+}, which aims
to study the relationship between copies of a given substructure
and the eigenvalues of a graph. In this article, we study
the supersaturation problem of 4-cycles under the eigenvalue condition.
The study of 4-cycles plays an important role in the history of
extremal graph theory. The extremal number of $C_4$ (i.e.,
a 4-cycle), denoted by $ex(n,C_4)$, is defined to be the maximum
number of edges in a graph which contains no 4-cycle as a
subgraph. The study of $ex(n,C_4)$ can be at least dated
back to Erd\H{o}s \cite{E38} eighty years ago.
A longstanding conjecture of Erd\H{o}s and Simonovits \cite{ES84}
(see also \cite[p.~84]{CG98}) states that every graph
on $n$ vertices and at least $ex(n,C_4)+1$ edges contains
at least two copies of 4-cycles when $n$ is large. Very recently,
He, Ma and Yang \cite{Y21+} announced this conjecture
does not hold for the cases $n=q^2+q+2$ where $q=4^k$ is large.
The original supersaturation problem of subgraphs in
graphs focuses on the following function:
for a given graph $H$ and for integers $n,t\geq1$,
$$h_{H}(n,t) = min\{\#H: |V(G)|=n, |E(G)|=ex(n,H)+t\},$$
where $ex(n,H)$ is the Tur\'an function of $H$.
Establishing a conjecture of Erd\H{o}s, Lov\'asz
and Simonovits \cite{LS83} proved that
$h_{C_3}(n,k)\geq k\lfloor\frac{n}{2}\rfloor$ for all
$1\leq k<\lfloor\frac{n}{2}\rfloor$. But He et al.'s result
tells us $h_{C_4}(n,1)=1$ for some positive integers $n$.
This means that supersaturation phenomenon of $C_4$
is quite different from the cases of triangles \cite{LS83}.
On the other hand, counting the copies of 4-cycles
plays a heuristic important role in measuring
the quasirandom-ness of a graph (see Chap. 9 in \cite{AS08}).
As an important case of spectral Zarankiewicz problem,
Nikiforov \cite{N10} proved that every $n$-vertex
$C_4$-free graph satisfies that
$\lambda(G)\leq \frac{1}{2}+\sqrt{n-\frac{3}{4}}$ where $\lambda(G)$
is the spectral radius of $G$, and
the earlier bound of Babai and Guiduli \cite{BG08}
gives the correct order of the main term.
As the counterpart of these results, we consider sufficient
eigenvalue condition (in terms of the size of a graph) for the
existence of 4-cycles. A pioneer result can be found in \cite{N09}.
\begin{thm}[\!\!\cite{N09}]\label{Thm:N09}
Let $G$ be a graph with $m$ edges, where $m\geq10$. If $\lambda(G)\geq\sqrt m$
then $G$ contains a 4-cycle, unless $G$ is a star (possibly with some isolated vertices).
\end{thm}
Recently, Theorem \ref{Thm:N09} was extended by the following.
\begin{thm}[\!\!\cite{ZLS21}]\label{Thm:ZLS21}
Let $r$ be a positive integer and $G$ be a
graph with $m$ edges where $m\geq16r^2$. If
$\lambda(G)\geq\sqrt m$, then $G$ contains a
copy of $K_{2,r+1}$, unless $G$ is a star
(possibly with some isolated vertices).
\end{thm}
Let $B_r$ be an \emph{$r$-book}, that is, the
graph obtained from $K_{2,r}$ by adding one edge
within the partition set of two vertices.
Very recently, Nikiforov \cite{N21} proved that,
if $m\geq (12r)^4$ and $\lambda(G)\geq\sqrt m$, then
$G$ contains a copy of $B_{r+1}$, unless $G$ is a
complete bipartite graph (possibly with some isolated vertices).
This result further extends above two theorems and
solves a conjecture proposed in \cite{ZLS21}.
The central topic of this article is the following spectral radius version
of supersaturation problem
of $4$-cycles :
\begin{prob}\label{Prob:Counting4-cycles}
Let $f(m)$ be the minimum number of copies of 4-cycles over
all labelled graph $G$ on $m$ edges with $\lambda(G)>\sqrt{m}$.
Give an estimate of $f(m)$.
\end{prob}
Till now, the only counting result related to Problem \ref{Prob:Counting4-cycles}
is a consequence of Theorem \ref{Thm:ZLS21}.
Note that $K_{2,r+1}$ contains $\frac{r(r+1)}2$ 4-cycles for $r=\frac{\sqrt{m}}4.$
Theorem \ref{Thm:ZLS21} implies that $f(m)\geq \frac{m}{32}$,
unless $G$ is a star (possibly with some isolated vertices).
One may ask for the best answer to Problem \ref{Prob:Counting4-cycles}.
In this paper, we make the first progress towards this problem.
\begin{thm}\label{Thm:Mainresult}
Let $m\geq3.6\times 10^9$ be a positive integer. Then
$f(m)\ge \frac{m^2}{2000}$, unless the graph $G$ is
a star (possibly with some isolated vertices).
\end{thm}
Throughout the left part, we also define $f(G)$ to be the number
of copies of 4-cycles in a graph $G$.
\begin{pro}\label{Prop}
$f(m)\leq \frac{(m-1)(m-2\sqrt{m})}8.$
\end{pro}
\begin{proof}
Let $s=\sqrt{m}+1$ and
$K_s^+$ be the graph obtained from the complete graph
$K_s$ by adding $m-{s \choose 2}$ pendent edges to one vertex of $K_s$.
Clearly, $\lambda(K_s^+)\geq \lambda(K_s)=\sqrt{m}$.
However, observe that $K_s^+$ contains ${s\choose 4}$ copies of $K_4$
and every $K_4$ contains three copies of 4-cycles.
Consequently, $f(K_s^+)=3{s\choose 4}=\frac{(m-1)(m-2\sqrt{m})}8.$
\end{proof}
Together with Theorem \ref{Thm:Mainresult} and
Proposition \ref{Prop}, one can easily find that $f(m)=\Theta(m^2)$.
Let us introduce some necessary notation and terminologies.
Let $G $ be a graph with vertex set $V(G)$ and edge set $E(G)$.
For a vertex $u\in V(G),$ we denote by $N_G(u)$ the set
of neighbors of $u$, and by $d_G(u)$ the degree of $u$.
The symbol $G-v$ denotes the
subgraph induced by $V(G)\backslash \{v\}$ in $G$.
The paper is organized as follows. In Section \ref{Sec:Preliminaries},
we shall give some necessary preliminaries and prove a key lemma.
We present a proof of our main theorem in Section \ref{Sec:Mainproof}.
We conclude this article with one corollary and some open problems
for further study.
\section{Preliminaries}\label{Sec:Preliminaries}
In this section, we introduce some lemmas, which will be used in
the subsequent proof.
The first lemma is known as Cauchy's Interlace Theorem.
\begin{lem}[\cite{BH12}]\label{Lem:CauchyInter}
Let $A$ be a symmetric $n\times n$
matrix and $B$ be an $r\times r$ principal submatrix of $A$ for some $r<n$. If
the eigenvalues of $A$ are $\lambda_1\geq \lambda_2\geq\cdots \geq\lambda_n$, and the eigenvalues of $B$ are
$\mu_1\geq \mu_2\geq\cdots\geq \mu_r$, then
$\lambda_i\geq \mu_i\geq \lambda_{i+n-r}$ for all $1\leq i\leq r$.
\end{lem}
The following inequality is due to Hofmeistar.
\begin{lem}[\!\!\cite{H88}]\label{lem2}
Let $G$ be a graph of order $n$ and $M(G)=\sum_{u\in V(G)}d^2_G(u)$.
Then
\begin{eqnarray}\label{eq1}
\lambda(G)\geq\sqrt{\frac1n M(G)},
\end{eqnarray}
with equality if and only if $G$ is either regular or bipartite semi-regular.
\end{lem}
\begin{lem}[\!\!\cite{LL09}]\label{lem3}
Let $G$ be a graph of order $n$ and size $m$. Then
\begin{eqnarray}\label{eq2}
f(G)=\frac18\sum_{i=1}^n\lambda_i^4+\frac m4-\frac14M(G),
\end{eqnarray}
where $\lambda_1,\ldots,\lambda_n$ are the eigenvalues of
$G$ with $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n.$
\end{lem}
The following result is well-known \cite{N70,BFP08}. A short
proof can also be found in \cite{N17}.
\begin{lem}[\!\!\cite{N70,BFP08,N17}]\label{lem5}
Let $G$ be a bipartite graph with $m$ edges, where $m\geq1$. Then
$\lambda(G)\leq\sqrt{m},$
with equality if and only if $G$ is a complete bipartite
graph (possibly with some isolated vertices).
\end{lem}
We need prove the last lemma. A proof of its one special
case that $m\leq n-2$ can be found in \cite[Lemma~2.4]{IS02}.
\begin{lem}\label{Lem:degreesum}
Let $G$ be a graph with $m$ edges. Then
$M(G)\leq m^2+m.$
\end{lem}
\begin{proof}
Let $G$ be an extremal graph with the maximum $M(G)$ and $n:=|G|$.
Let $V(G)=\{u_1,\ldots,u_{n}\}$, and $d_i:=d_G(u_i)$ for each $u_i\in V(G)$.
We may assume that $d_1\geq\cdots\geq d_{n}\geq1$.
If there exists some integer $i\geq2$ such that $u_iu_1\notin E(G)$,
then we choose a vertex $u_j\in N_G(u_i)$ and define
$G':=G-u_iu_j+u_iu_1$.
Now $d_{G'}(u_1)=d_1+1,$ $d_{G'}(u_j)=d_j-1$ and $d_{G'}(u_k)=d_G(u_k)$ for
each $k\in\{2,\ldots,n\}\setminus\{j\}.$
Consequently,
\begin{eqnarray*}
M(G')-M(G)=(d_1+1)^2+(d_j-1)^2-d_1^2-d_j^2=2d_1-2d_j+2\geq2,
\end{eqnarray*}
a contradiction. Thus, $N_G(u_1)=V(G)\setminus\{u_1\}$,
and so $d_1=n-1$.
Now let $e(G-u_1)$ be the number of edges in $G-u_1$.
Clearly, $e(G-u_1)=m-d_1$.
If $e(G-u_1)=0$, then $G\cong K_{1,m}$,
and so $\sum_{i=1}^{n}d^2_i=m^2+m$, as desired.
In the following, we assume $e(G-u_1)\geq1$.
If $e(G-u_1)\leq d_1-2$,
then $d_i+d_j\leq e(G-u_1)+3\leq d_1+1$ for each $u_iu_j\in E(G-u_1)$.
Now let $G'=G-u_iu_j+u_1u_0$, where $u_iu_j\in E(G-u_1)$ and
$u_0$ is a new vertex adjacent only to $u_1$ in $G'$.
Then
\begin{eqnarray*}
M(G')-M(G)\!\!&=&\!\! (d_1+1)^2+1+(d_i-1)^2+(d_j-1)^2-d_1^2-d_i^2-d_j^2\\
\!\!&=&\!\! 2(d_1-d_i-d_j)+4.
\end{eqnarray*}
It follows that $M(G')>M(G)$, a contradiction.
Therefore, $e(G-u_1)\geq d_1-1$.
Now let $e(G-u_1)=k$ and define a new graph $G':=K_{1,d_1+k}$.
Then $k\geq d_1-1$ and $e(G')=d_1+k=e(G)=m$.
Note that $n=d_1+1$ and $2k=2e(G-u_1)=\sum_{i=2}^{n}(d_i-1)$.
Hence, $2kd_1\geq \sum_{i=2}^{n}d_i^2-d_1^2=M(G)-2d_1^2.$
It follows that
\begin{eqnarray*}
M(G')-M(G)=
(k+d_1)^2+(k+d_1)-M(G)
\geq k^2-d_1^2+(k+d_1)\geq0,
\end{eqnarray*}
as $k\geq d_1-1$.
Thus, $M(G)\leq M(G')=m^2+m$.
This proves Lemma \ref{Lem:degreesum}.
\end{proof}
\section{Proof of Theorem \ref{Thm:Mainresult}}\label{Sec:Mainproof}
In this section, we give a proof of Theorem \ref{Thm:Mainresult}.
We would like to point out that the techniques used in the left part
are completely different from \cite{NZ21+}.
\subsection{A key lemma}
We first prove a key lemma.
\begin{lem}\label{lem7}
Let $G$ be a graph of size $m\geq1.8\times10^9$ and
$X$ be the
Perron vector of $G$ with component $x_u$ corresponding to $u\in V(G)$.
If $\lambda(G)\ge
\sqrt{m}$ and $x_ux_v>\frac1{9\sqrt{m}}$ for any $uv\in E(G)$, then $f(G)\ge \frac{m^2}{500}$ unless
$G$ is a star (possibly with some isolated vertices).
\end{lem}
\begin{proof}
We may assume that $\delta(G)\geq1$, where $\delta(G)$ is the minimum degree of $G$.
Then $G$ is connected (otherwise, we can find an edge $uv$ with $x_ux_v=0$).
By Perron-Frobenius theorem, $X$ is a positive vector.
Let $A=\{u\in V(G): x_u>\frac1{3\sqrt[4]{m}}\}$ and $B=V(G)\setminus A.$
Clearly, $B$ is an independent set.
Now suppose that $f(G)<\frac{m^2}{500}$
and set $\lambda:=\lambda(G)$.
We will prove a series of claims.
\begin{claim}\label{cl1}
We have $\delta(G)\geq2$ unless $G\cong K_{1,m}$.
\end{claim}
\begin{proof}
Assume that there exists a vertex $u\in V(G)$ with $d_G(u)=1$ and $N_G(u)=\{\bar{u}\}$.
Then $x_ux_{\bar{u}}=\frac{x_{\bar{u}}^2}\lambda\leq \frac{x_{\bar{u}}^2}{\sqrt{m}}.$
Since $x_ux_{\bar{u}}>\frac1{9\sqrt{m}}$, we have $x_{\bar{u}}>\frac13.$
Let $u^*\in V(G)$ with $x_{u^*}=\max_{v\in V(G)}x_v$.
Then $x_{u^*}>\frac13$.
Now let $S:=N_G(u^*)$, $T:=V(G)\setminus (S\cup\{u^*\})$, and
$N_S(v)=N_G(v)\cap S$ for a vertex $v\in V(G)$.
Moreover, we partite $S$ into three subsets $S_1$, $S_2$ and $S_3$,
where $S_1=\{v: \frac14<x_v\leq x_{u^*}\},$ $S_2=\{v: \frac16<x_v\leq \frac14\}$,
and $S_3=\{v: 0<x_v\leq \frac16\}$.
Choose a vertex $u\in S_1$ arbitrarily.
By Cauchy-Schwarz inequality,
\begin{eqnarray}\label{eq3}
(\lambda x_u)^2=\Big(\sum_{v\in N_G(u)}x_v\Big)^2\leq d_G(u)\sum_{v\in N_G(u)}x_v^2\leq d_G(u)(1-x_u^2).
\end{eqnarray}
Since $x_u>\frac14$ and $\lambda\geq\sqrt{m}$, we have $d_G(u)\geq \frac m{15}.$
If $|N_T(u)|\leq \frac{m}{450}$, then $|N_S(u)|\geq\frac m{15}-\frac{m}{450}-1\geq \frac{m}{15.6}$,
and thus $G$ contains a copy of $K_{2,\lceil\frac m{15.6}\rceil}$.
Hence, $G$ contains at least ${\lceil\frac m{15.6}\rceil\choose 2}$ ($\geq \frac{m^2}{500}$)
quadrilaterals, a contradiction.
Therefore, $|N_T(u)|\geq \frac{m}{450}$ and $|N_S(u)|<\frac{m}{15.6}$.
Now let $S^*=\{v\in S: x_v<\frac1 {108}\}$, $T^*=\{v\in T: x_v<\frac1 {108}\}$ and
$V'=(S\setminus S^*)\cup(T\setminus T^*)$.
Since $X$ is a unit vector, we have $|V'|\leq 108^2.$
By Cauchy-Schwarz inequality,
\begin{eqnarray}\label{eq0}
\sum_{v\in V'}x_v
\leq \sqrt{|V'|\sum_{v\in V'}x_v^2}\leq \sqrt{|V'|}\leq108.
\end{eqnarray}
Consequently,
$$\sum_{v\in N_S(u)}x_v=\sum_{v\in N_{S\setminus S^*}(u)}x_v+\sum_{v\in N_{S^*}(u)}x_v
\leq 108+\frac 1{108}|N_{S^*}(u)|.$$
Recall that $|N_{S^*}(u)|\leq |N_{S}(u)|\leq\frac{m}{15.6}$ and $x_{u^*}>\frac13$.
It follows that
\begin{eqnarray*}
\sum_{v\in N_S(u)}x_v\leq (324+\frac 1{36}|N_{S^*}(u)|)x_{u^*}
<\frac 1{36}\cdot\frac m{15}x_{u^*}.
\end{eqnarray*}
On the other hand, note that $|N_{T^*}(u)|\geq |N_T(u)|-108^2\geq\frac{m}{525}$
and $x_v<\frac1{108}<\frac1{36}x_{u^*}$ for any $v\in T^*$. Then
\begin{eqnarray*}
\sum_{v\in N_T(u)}x_v
&<&\sum_{v\in N_{T\setminus T^*}(u)}x_{u^*}+\sum_{v\in N_{T^*}(u)}\frac 1{36}x_{u^*}
\leq|N_T(u)|x_{u^*}-\frac {35}{36}\cdot\frac{m}{525}x_{u^*}\\
&=&|N_T(u)|x_{u^*}- \frac 1{36}\cdot\frac m{15}x_{u^*}.
\end{eqnarray*}
It follows that $\sum_{v\in N_{S\cup T}(u)}x_v<|N_T(u)|x_{u^*}.$
Let $e(S,T)$ be the number of edges from $S$ to $T$,
and $e(S)$ be the number of edges within $S$. Then
\begin{eqnarray}\label{eq4}
\sum_{u\in S_1}\sum_{v\in N_{S\cup T}(u)}x_v<e(S_1,T)x_{u^*}.
\end{eqnarray}
Secondly, consider a vertex $u\in S_2$ arbitrarily.
Note that $x_u>\frac16$ and $\lambda\geq\sqrt{m}$.
Then (\ref{eq3}) gives
$d_G(u)\geq \frac m{35}.$
Since $S^*\subseteq S_3$ and $x_{u^*}-x_u>\frac13-\frac14=\frac1{12}$,
we have $$\sum_{v\in N_{S^*}(u)}x_v\leq \frac 1{108}|N_{S_3}(u)|
\leq\frac19|N_{S_3}(u)|(x_{u^*}-x_u),$$
and by (\ref{eq0}) we have
$\sum_{v\in N_{S\setminus S^*}(u)}x_v\leq \sum_{v\in V'}x_v\leq108$. Then
\begin{eqnarray}\label{eq5}
\sum_{v\in N_S(u)}\!\!x_v=\sum_{v\in N_{S\setminus S^*}(u)}x_v+\!\!\!\!\sum_{v\in N_{S^*}(u)}\!\!x_v
\leq 108+\frac 19|N_{S_3}(u)|(x_{u^*}-x_u).
\end{eqnarray}
If $|N_{S^*}(u)|\geq \frac m{72}$,
then $|N_{S_3}(u)|\geq |N_{S^*}(u)|\geq\frac m{72}$.
Since $x_{u^*}-x_u>\frac1{12}$, it follows from (\ref{eq5}) that
$\sum_{v\in N_S(u)}x_v<|N_{S_3}(u)|(x_{u^*}-x_u)$, and thus
\begin{eqnarray}\label{eq6}
\sum_{v\in N_{S\cup T}(u)}x_v<|N_{S_3}(u)|(x_{u^*}-x_u)+|N_T(u)|x_{u^*}.
\end{eqnarray}
If $|N_{S^*}(u)|\leq\frac m{72}$, then
$$|N_{T^*}(u)|\geq d_G(u)-|N_{S^*}(u)|-108^2>\frac m{72}.$$
Hence,
\begin{eqnarray*}
\sum_{v\in N_{T^*}(u)}x_v\leq|N_{T^*}(u)|\cdot\frac1{108}<
|N_{T^*}(u)|x_{u^*}-108,
\end{eqnarray*}
as $x_{u^*}>\frac13.$
It follows that $\sum_{v\in N_T(u)}x_v<|N_T(u)|x_{u^*}-108.$
Combining with (\ref{eq5}), we can also obtain (\ref{eq6}).
Therefore, in both cases we have
\begin{eqnarray}\label{eq7}
\sum_{u\in S_2}\sum_{v\in N_{S\cup T}(u)}x_v<e(S_2,S_3)x_{u^*}+e(S_2,T)x_{u^*}-\sum_{u\in S_2}|N_{S_3}(u)|x_u.
\end{eqnarray}
Thirdly, we consider an arbitrary vertex $u\in S_3$.
Since $x_{u^*}>\frac13$, we have $x_v\leq \frac16<\frac12x_{u^*}$ for each $v\in N_{S_3}(u)$.
Thus, $\sum_{u\in S_3}\sum_{v\in N_{S_3}(u)}x_v\leq e(S_3)x_{u^*}$,
with equality if and only if $e(S_3)=0$.
Therefore,
\begin{eqnarray}\label{eq8}
\sum_{u\in S_3}\sum_{v\in N_{S\cup T}(u)}x_v\leq e(S_3,S_1)x_{u^*}
+e(S_3)x_{u^*}+e(S_3,T)x_{u^*}+\sum_{u\in S_3}\sum_{v\in N_{S_2}(u)}x_v.
\end{eqnarray}
Notice that $$\sum_{u\in S_2}|N_{S_3}(u)|x_u=\sum_{u\in S_3}\sum_{v\in N_{S_2}(u)}x_v.$$
Combining with (\ref{eq4}), (\ref{eq7}) and (\ref{eq8}), we have
\begin{eqnarray}\label{eq9}
\sum_{u\in S}\sum_{v\in N_{S\cup T}(u)}x_v\leq (e(S)
+e(S,T))x_{u^*},
\end{eqnarray}
where if equality holds then $S_1\cup S_2=\varnothing$ and $e(S_3)=0$,
that is, $e(S)=0$.
Furthermore,
we can see that
\begin{eqnarray*}
\lambda^2x_{u^*}
=\sum_{u\in S}\sum_{v\in N_G(u)}\!\!\!x_v=|S|x_{u^*}+\sum_{u\in S}\sum_{v\in N_{S\cup T}(u)}\!\!\!x_v
\leq(|S|+e(S)+e(S,T))x_{u^*}\leq mx_{u^*}.
\end{eqnarray*}
Since $\lambda\geq \sqrt{m}$, the above inequality holds in equality, that is, $\lambda=\sqrt{m}$.
Therefore, $m=|S|+e(S)+e(S,T)$, and (\ref{eq9}) holds in equality (hence $e(S)=0$).
This implies that $G$ is a bipartite graph.
By Lemma \ref{lem5}, $G$ is a complete bipartite graph.
Since $f(G)<\frac{m^2}{500}$, $G$ can only be a star.
This completes the proof.
\end{proof}
In the following, we may assume that $G\ncong K_{1,m}$.
Then by Claim \ref{cl1}, $\delta(G)\geq2$.
\begin{claim}\label{cl2}
$|A|\leq9\sqrt{m}$.
\end{claim}
\begin{proof}
Recall that $x_u>\frac1{3\sqrt[4]{m}}$ for each $u\in A$.
Thus $\sum_{u\in A}x_{u}^2>\frac{|A|}{9\sqrt{m}}$, and hence
$|A|\leq 9\sqrt{m}\sum_{u\in A}x_{u}^2\leq9\sqrt{m}.$
\end{proof}
\begin{claim}\label{cl3}
Let $|G|=\frac m2+b$. Then $-\frac m{125}\leq b\leq|A|$.
\end{claim}
\begin{proof}
Set $\lambda':=\lambda_{|G|}$. Note that $\lambda\geq \sqrt{m}$.
By Lemmas \ref{lem2} and \ref{lem3},
\begin{eqnarray}\label{eq10}
f(G)\geq\frac18(\lambda^4+\lambda'^4)-\frac14M(G)
\geq\frac18(\lambda^4+\lambda'^4)-\frac{|G|}4\lambda^2\geq\frac18\lambda'^4-\frac b4\lambda^2.
\end{eqnarray}
If $b<-\frac m{125}$, then $$f(G)\geq-\frac b4\lambda^2\geq\frac{m}{500}\lambda^2\geq\frac{m^2}{500},$$
a contradiction.
Thus, $b\geq-\frac m{125}$.
On the other hand, recall that $e(B)=0$ and $\delta(G)\geq2$,
then
$$m\geq e(B,A)\geq 2|B|=2(|G|-|A|)=2(\frac m2+b-|A|).$$
Thus, $b\leq |A|$, as desired.
\end{proof}
\begin{claim}\label{cl4}
$\Delta(G)\leq \frac2{15}m,$ where $\Delta(G)$ is the maximum degree of $G$.
\end{claim}
\begin{proof}
We know that $\sum_{i=1}^{|G|}\lambda_i^2=2m$. Thus, $\lambda^2=\lambda_1^2\leq 2m$.
Combining with (\ref{eq10}) and $b\leq|A|\leq9\sqrt{m}$, we have
\begin{eqnarray}\label{eq11}
f(G)\geq\frac18\lambda'^4-\frac b4\lambda^2\geq\frac18\lambda'^4-9m^{\frac32}.
\end{eqnarray}
Now if there exists some $u\in V(G)$ with $d_G(u)>\frac2{15}m$, then
$$|N_B(u)|\geq d_G(u)-|A|>\frac2{15}m-9\sqrt{m}.$$
Since $e(B)=0$, $G$ contains $K_{1,|N_B(u)|}$ as an induced subgraph.
By Lemma \ref{Lem:CauchyInter},
$$\lambda'\leq -\sqrt{|N_B(u)|}<-\sqrt{\frac2{15}m-9\sqrt{m}},$$
and by (\ref{eq11}) we have
\begin{eqnarray*}
f(G)\geq\frac18\lambda'^4-9m^{\frac32}\geq \frac18\Big(\frac2{15}m-9\sqrt{m}\Big)^2-9m^{\frac32}>\frac{m^2}{500},
\end{eqnarray*}
for $m\geq 1.8\times10^9$.
We have a contradiction.
Therefore, $\Delta(G)\leq \frac2{15}m.$
\end{proof}
\begin{claim}\label{cl5}
Let $B^*=\{u\in V(G): d_G(u)=2\}$. Then $B^*\subseteq B$ and
\begin{eqnarray}\label{eq12}
\frac m2+3(b-|A|)\leq |B^*|\leq \frac m2.
\end{eqnarray}
\end{claim}
\begin{proof}
Let $u\in B^*$ and $N_G(u)=\{u_1,u_2\}$. Then $\lambda x_{u}=x_{u_1}+x_{u_2}\leq 2$.
Since $\lambda\geq \sqrt{m}$, we have $x_u\leq\frac{2}{\sqrt{m}}<\frac 1{3\sqrt[4]{m}},$
and so $u\in B$.
Recall that $e(B)=0$. Thus, $e(B^*)=0$, and $m\geq e(B^*,A)\geq 2|B^*|$.
This gives $|B^*|\leq \frac m2.$
On the other hand, note that $|B|=|G|-|A|=\frac m2+b-|A|$,
then
\begin{eqnarray*}
m\geq e(B,A)\geq2|B^*|+3(|B|-|B^*|)=\frac32m+3(b-|A|)-|B^*|.
\end{eqnarray*}
It follows that $|B^*|\geq\frac m2+3(b-|A|).$
\end{proof}
\begin{claim}\label{cl6}
Let $A^*=\{v\in N_G(u): u\in B^*\}$. Then $A^*\subseteq A$ and $|A^*|\leq24$.
\end{claim}
\begin{proof}
Since $e(B)=0$, we have $N_G(u)\subseteq A$ for any $u\in B^*$.
Thus, $A^*\subseteq A$.
Furthermore, we will see that $\frac1{25}<x_v^2\leq\frac{2}{17}$ for each $v\in A^*$.
Let $v$ be an arbitrary vertex in $A^*$.
By Cauchy-Schwarz inequality,
\begin{eqnarray*}
(\lambda x_v)^2=\Big(\sum_{u\in N_G(v)}x_u\Big)^2
\leq d_G(v)\sum_{u\in N_G(v)}x_u^2\leq d_G(v)(1-x_v^2)
\leq \frac2{15}m(1-x_v^2),
\end{eqnarray*}
as $\Delta(G)\leq \frac 2{15}m$.
Since $\lambda\geq\sqrt{m}$, we have $x_v^2\leq \frac2{17}.$
If there exists a vertex $v\in A^*$ with $x_v^2\leq\frac1{25}$,
then by the definition of $A^*$, we can find a vertex $u\in N_{B^*}(v)$.
Clearly, $$\lambda x_u\leq x_v+\sqrt{\frac2{17}}\leq\frac15+\sqrt{\frac2{17}}<\frac59.$$
Consequently,
$$x_ux_v<\frac 1\lambda\cdot\frac 59\cdot\frac15\leq \frac1{9\sqrt{m}},$$
which contradicts the condition of Lemma \ref{lem7}.
Therefore, $x_v^2>\frac1{25}$ for any $v\in A^*$, and so $|A^*|\leq24.$
\end{proof}
\begin{claim}\label{cl7}
Let $V'':=(A\setminus A^*)\cup (B\setminus B^*)$.
Then $|V''|\leq\frac m{60}$ and $e(V'')\leq \frac m{20}.$
\end{claim}
\begin{proof}
Recall that $|A\cup B|=|G|=\frac m2+b$.
Combining with (\ref{eq12}), we obtain that
$|V''|\leq|G|-|B^*|\leq 3|A|-2b.$
Moreover, by Claims \ref{cl2} and \ref{cl3}, we have
$|A|\leq9\sqrt{m}$ and $b\geq-\frac m{125}$. Thus,
$|V''|\leq 27\sqrt{m}+\frac 2{125}m\leq\frac m{60}.$
Now we estimate $e(V'')$.
Again by $|A|\leq9\sqrt{m}$, $b\geq-\frac m{125}$ and (\ref{eq12}), we have
\begin{eqnarray*}
e(A^*,B^*)=2|B^*|\geq m+6(b-|A|)\geq m-\frac6{125}m-54\sqrt{m}.
\end{eqnarray*}
It follows that
$e(V'')\leq m-e(A^*,B^*)\leq\frac6{125}m+54\sqrt{m}\leq \frac m{20}.$
\end{proof}
Now we give the final proof of Lemma \ref{lem7}.
For convenience, let $d'(u):=|N_{V''}(u)|$ for each $u\in V''$.
Note that $e(V'',B^*)=0$. Thus by Claim \ref{cl6}, $$d_G(u)\leq d'(u)+|A^*|\leq d'(u)+24$$
for each vertex $u\in V''$.
Consequently,
\begin{eqnarray}\label{eq13}
\sum_{u\in V''}d^2_G(u)\leq \sum_{u\in V''}(d'(u)+24)^2=96e(V'')+24^2|V''|+\sum_{u\in V''}d'^2(u).
\end{eqnarray}
Since $e(V'')\leq \frac m{20}$, by Lemma \ref{Lem:degreesum} we have
$\sum_{u\in V''}d'^2(u)\leq \frac{m^2}{400}+\frac m{20}.$
Combining this with Claim \ref{cl7} and (\ref{eq13}), we have
\begin{eqnarray}\label{eq14}
\sum_{u\in V''}d^2_G(u)
\leq 96\cdot\frac m{20}+24^2\cdot\frac m{60}+\frac{m^2}{400}+\frac m{20}<\frac {m^2}{225}.
\end{eqnarray}
On the other hand, by Claim \ref{cl4}, $\Delta(G)\leq \frac 2{15}m$, and so
$$\sum_{u\in A^*}d^2_G(u)\leq|A^*|(\Delta(G))^2\leq \frac {96}{225}m^2$$ (as $|A^*|\leq24$).
Moreover, by Claim \ref{cl5} $|B^*|\leq \frac m2$, and thus
$$\sum_{u\in B^*}d^2_G(u)=4|B^*|\leq 2m.$$
Combining with (\ref{eq14}), we get
\begin{eqnarray*}
M(G)=\sum_{u\in V''\cup A^*\cup B^*}d^2_G(u)\leq \frac{1}{225}m^2+\frac{96}{225}m^2+2m<\frac{100}{225}m^2=\frac49m^2.
\end{eqnarray*}
Now by Lemma \ref{lem3}, we have
\begin{eqnarray*}
f(G)\geq \frac18\lambda^4-\frac14M(G)\geq \frac18m^2-\frac1{9}m^2=\frac1{72}m^2>\frac1{500}m^2,
\end{eqnarray*}
a contradiction.
This completes the proof.
\end{proof}
\subsection{Nikiforov's deleting small eigenvalue edge method}
Over the past decades, Nikiforov developed some novel tools and techniques for
solving problems in spectral graph theory (see \cite{N11}). One is the method we called ``deleting
small eigenvalue edge method", or ``The DSEE Method". Generally speaking, an edge $xy\in E(G)$
is called a \emph{small eigenvalue edge}, if $x_ux_v$ is small where $x_u,x_v$ are
Perron components.
By using this method, Nikiforov \cite{N09} successfully proved the following results, of which
some original ideas appeared in \cite{N11} earlier:
\begin{itemize}
\item Every graph on $m$ edges contains a 4-cycle if $\lambda(G)\geq\sqrt{m}$ and $m\geq10$,
unless it is a star with possibly some isolated vertices (see Claim 4 in \cite[pp.~2903]{N09});
\item Every graph on $m$ edges satisfies that the booksize $bk(G)>\frac{\sqrt[4]{m}}{12}$ if $\lambda(G)\geq\sqrt{m}$,
unless it is a complete bipartite graph with possibly some isolated vertices
(see \cite{N21}, this confirmed a conjecture in \cite{ZLS21}).
\end{itemize}
One main ingredient in the proof of Theorem \ref{Thm:Mainresult} is using this method.
\subsection{Proof of Theorem \ref{Thm:Mainresult}}
Now we are ready to give the proof of Theorem \ref{Thm:Mainresult}.
\vspace{2mm}
\noindent
{\bf Proof of Theorem \ref{Thm:Mainresult}.}
Let $G$ be a graph with $e(G)=m$ and $\lambda(G)\geq\sqrt{m}.$
By using the Nikiforov DESS Method \cite{N21},
we first construct a sequence of graphs.
\vspace{2mm}
\noindent
(i) Set $i:=0$ and $G_0:=G$.\\
(ii) If $i=\lfloor\frac m2\rfloor,$ stop.\\
(iii) Let $X=(x_1,x_2\ldots,x_{|G_i|})^T$ be the Perron vector of $G_i$.\\
(iv) If there exists $uv\in E(G_i)$ with $x_ux_v\leq\frac1{9\sqrt{e(G_i)}}$,
set $G_{i+1}:=G_i-uv$ and $i:=i+1$.\\
(v) If there is no such edge, stop.
\vspace{2mm}
Assume that $G_k$ is the resulting graph of the graph sequence
constructed by the above algorithm.
Then $k\leq\lfloor\frac m2\rfloor.$
We can obtain the following two claims.
\begin{claim}\label{cl8}
$\lambda(G_{i+1})\geq\sqrt{m-i-1}$
for each $i\in \{0,1,\ldots,k-1\}$.
\end{claim}
\begin{proof}
Let $X$ be the Perron vector of $G_i$ with component $x_u$ corresponding to $u\in V(G_i)$.
Then, there exists $uv\in E(G_i)$ with $x_ux_v\leq\frac1{9\sqrt{e(G_i)}}$.
Thus,
$$\lambda(G_{i+1})\geq X^TA(G_{i+1})X=X^TA(G_{i})X-2x_ux_v\geq\lambda(G_i)-\frac2{9\sqrt{e(G_i)}}.$$
Hence,
$$\lambda(G_0)\leq \lambda(G_1)+\frac2{9\sqrt{m}}\leq\cdots
\leq \lambda(G_{i+1})+\sum_{j=0}^{i}\frac2{9\sqrt{m-j}}.$$
It follows that
\begin{eqnarray}\label{eq15}
\lambda(G_{i+1})\geq\lambda(G_0)-\frac{2(i+1)}{9\sqrt{m-i-1}}\geq\sqrt{m}-\frac{2(i+1)}{9\sqrt{m-i-1}}.
\end{eqnarray}
This implies that $\lambda(G_{i+1})\geq\sqrt{m-i-1}$, as $i+1\leq k\leq\lfloor\frac m2\rfloor$.
\end{proof}
Now we may assume that all isolated vertices are removed from each $G_i$,
where $i\in \{0,1,\ldots,k\}$.
\begin{claim}\label{cl9}
$G_k$ cannot be a star unless $G_k=G_0\cong K_{1,m}$.
\end{claim}
\begin{proof}
Suppose to the contrary that $k\geq1$ while $G_k$ is a star.
Since $e(G_k)=m-k$, we have $G_k\cong K_{1,m-k}$.
Let $u_0$ be the central vertex of $G_k$ and $u_1,\ldots,u_{m-k}$ be the leaves.
We now let $G_k=G_{k-1}-uv$ and $X$ be the Perron vector of $G_{k-1}$.
If $uv$ is a pendent edge incident to $u_0$, say $uv=u_0u_{m-k+1}$,
then $$\lambda(G_{k-1})=\sqrt{e(G_{k-1})}=\sqrt{m-k+1}$$ and
$\lambda(G_{k-1})x_{u_i}=x_{u_0}$ for $i\in\{1,2,\ldots,m-k+1\}$.
Hence, $\|X\|_2=\sum_{i=0}^{m-k+1}x_{u_i}^2=2x_{u_0}^2$, which gives
$x_{u_0}^2=\frac12$.
It follows that $$x_{u_0}x_{u_{m-k+1}}=\frac{x^2_{u_0}}{\sqrt{e(G_{k-1})}}>\frac{1}{9\sqrt{e(G_{k-1})}},$$
which contradicts the definition of $G_k$.
If $uv$ is an isolated edge or a pendent edge not incident to $u_0$,
then $G_{k-1}$ is bipartite but not complete bipartite.
By Lemma \ref{lem5}, $\lambda(G_{k-1})<\sqrt{e(G_{k-1})}$,
which contradicts Claim \ref{cl8}.
Now we conclude that $uv$ is an edge within $V(G_k)\setminus\{u_0\}$,
say $uv=u_1u_2$,
then $x_{u_1}=x_{u_2}$ and $\lambda(G_{k-1})x_{u_1}=x_{u_0}+x_{u_2}$.
Hence, $x_{u_1}=\frac{x_{u_0}}{\lambda(G_{k-1})-1}<\frac12x_{u_0},$
as $\lambda(G_{k-1})\geq\sqrt{m-k+1}$ by Claim \ref{cl8}.
Consequently,
$$\lambda^2(G_{k-1})x_{u_0}=\sum_{i=1}^{m-k}\lambda(G_{k-1})x_{u_i}
=(m-k)x_{u_0}+(x_{u_1}+x_{u_2})<(m-k+1)x_{u_0}.$$
It follows that $\lambda(G_{k-1})<\sqrt{m-k+1}$,
which also contradicts Claim \ref{cl8}.
\end{proof}
Now we finish the final proof of Theorem \ref{Thm:Mainresult}.
Assume that $G$ is not a star. Then $G_k$ is not a star by Claim \ref{cl9};
moreover, $\lambda(G_k)\geq\sqrt{m-k}=\sqrt{e(G_{k})}$ by Claim \ref{cl8}.
If $k<\lfloor\frac m2\rfloor$, then
$x_ux_v>\frac1{9\sqrt{e(G_k)}}$ for any edge $uv\in E(G_k)$.
Since $e(G_k)=m-k>\frac m2,$
by Lemma \ref{lem7} $f(G_k)\geq \frac {((e(G_k))^2}{500}>\frac {m^2}{2000},$
and so $f(G)>\frac {m^2}{2000},$
as desired.
If $k=\lfloor\frac m2\rfloor$,
then by (\ref{eq15}) we have
\begin{eqnarray*}
\lambda(G_k)\geq\sqrt{m}-\frac{2k}{9\sqrt{m-k}}\geq \sqrt{m}-\frac{m}{9\sqrt{\frac m2}}=\Big(1-\frac{\sqrt{2}}9\Big)\sqrt{m},
\end{eqnarray*}
and so $$\lambda^4(G_k)\geq(1-\frac{\sqrt{2}}9)^4m^2=0.5047m^2>0.504m^2+4m.$$
On the other hand, by Lemma \ref{Lem:degreesum},
$$M(G_k)\leq (e(G_k))^2+e(G_k)=\lceil\frac m2\rceil^2+\lceil\frac m2\rceil\leq 0.25m^2+2m.$$
Thus by Lemma \ref{lem3},
\begin{eqnarray*}
f(G_k)\geq \frac18\lambda^4(G_k)-\frac14M(G_k)>\frac18(0.504-0.5)m^2=\frac1{2000}m^2,
\end{eqnarray*}
and so $f(G)>\frac {m^2}{2000}.$
This completes the proof. $\hfill\blacksquare$
\section{Concluding remarks}
We do not try our best to optimize the constant ``$\frac{1}{2000}$"
in Theorem \ref{Thm:Mainresult}. So it is natural to pose the following problem:
\begin{prob}
Determine $\lim\limits_{m\rightarrow \infty} \frac{f(m)}{m^2}$.
(We think that the upper bound in Proposition \ref{Prop} is close to the truth.)
\end{prob}
By Theorem \ref{Thm:Mainresult} and an inequality $\lambda(G)\geq \frac{2m}n$ due to Collatz and Sinogowitz \cite{CS57},
we deduce the following.
\begin{thm}
Let $G$ be a graph on $n$ vertices and $m$ edges. If
$m>\max\{\frac{n^2}{4},3.6\times 10^9\}$,
then $G$ contains $\frac{n^4}{32000}$
copies of 4-cycles.
\end{thm}
On the other hand, we would like to mention the following conjecture.
\begin{conj}[Conjecture 5.1 in \!\!\cite{ZLS21}]
Let $k\geq 2$ be a fixed positive integer and $G$ be a graph of sufficiently large size $m$ without
isolated vertices. If $\lambda(G)\geq \frac{k-1+\sqrt{4m-k^2+1}}{2}$, then
$G$ contains a cycle of length $t$ for every $t\leq 2k+2$,
unless $G=S_{\frac{m}{k}+\frac{k+1}{2},k}$.
\end{conj}
When $k=1$, the above conjecture reduces to Nikiforov's result (Theorem \ref{Thm:N09}).
Let $B_{r,k}$ be the join of an $r$-clique with an independent set of size $k$.
We conclude this note with a
new conjecture appeared in \cite{LFL} which extends Theorem \ref{Thm:N09}.
\begin{conj}[Conjecture 1.20 in \!\!\cite{LFL}]
Let m be large enough and G be a $B_{r,k}$-free graph with m edges. Then
$\lambda(G)\leq \sqrt{(1-\frac{1}{r})2m}$, with equality if and only if $G$ is a complete bipartite
graph for $r=2$, and $G$ is a complete regular $r$-partite graph for
$r\geq 3$ with possibly some isolated vertices.
\end{conj}
|
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{"url":"https:\/\/zbmath.org\/?q=an:1112.05078","text":"## Trees with large paired-domination number.(English)Zbl\u00a01112.05078\n\nA paired-dominating set of a graph is a dominating set of vertices whose induced subgraph contains a perfect matching. If $$G$$ is a graph without isolated vertices, then it is easy to see that the paired-domination number of $$G$$ is bounded above by twice the domination number of $$G$$. The authors present a constructive characterization of those trees attaining this bound.\n\n### MSC:\n\n 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C05 Trees\n\n### Keywords:\n\ndomination; paired-domination; tree","date":"2022-05-26 14:44:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5193204283714294, \"perplexity\": 625.7927399784958}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662606992.69\/warc\/CC-MAIN-20220526131456-20220526161456-00282.warc.gz\"}"}
| null | null |
\section{Introduction}
Investigations on various nonlinear wave structures that include
solitons/solitary waves, kinks, breathers, shocks, rogue waves, lumps,
dromions, peakons, cuspons, compactons, periodic and elliptic waves, etc.
emerge an interesting avenue among the researchers in the past few
decades~\cite{Yang-book}. This is accomplished through the development of
different analytical methodologies to solve the underlying evolution equations
and numerical simulations running on efficient computation~\cite{ML-book}.
Solitons are one of the inevitable and remarkably stable nonlinear waves that are being a front-runner with multidisciplinary applications starting from
the data transmission networks and nonlinear optics to Bose--Einstein
condensate, plasma systems, and so on~\cite{bgo,bgu,boris,ksa}. The thorough
comprehension of many real-world problems in plasmas, fibre optics, fourth and
fifth states of matter, etc. is significantly advanced by the mathematical and
physical analysis of localized wave solutions and long-time asymptotic
behaviours of these soliton models~\cite{ML-book,Yang-book,bgo,bgu,boris,ksa}.
Further, on contrary to the solitons, rogue waves are another type of nonlinear
entities that are first observed as deep oceanic waves with huge instability
and they are observed in various fields~\cite{akm-rogue}. With certain structural
similarities and connection with unstable rogue waves, stable propagating
dromions and lumps are observed in several higher-dimensional models arising in
the context of metamaterials~\cite{zba}, flexural-gravity waves~\cite{mra},
ferromagnetic film~\cite{xwj}, water waves~\cite{wx1}, superfluids~\cite{djf}
and in hydrodynamics~\cite{fba}.
In the context of water wave theory, the complete study on hydrodynamical and
related physical systems are achieved by exploring several integrable as well
as nonintegrable evolution equations in one and higher
dimensions~\cite{Yang-book}. Integrability is a fascinating property to
characterize any dynamical models in addition to existence of Lax pair and
infinitely many conserved quantities~\cite{ML-book}. The higher dimensional
generalization of the well-known one-dimensional soliton models gives a more
clear picture of the evolutionary behaviours of the associated nonlinear
waves~\cite{hdg,gqx,ylm,lka}.
Integrable soliton models consisting of first-order temporal evolution $u_{t}$ are observed to describe unidirectional wave propagation, that includes Korteweg--de Vries (KdV) and modified-KdV type equations. Further, nonlinear models such as Boussinesq equation, Klein--Gordon equation and Kaup--Kupershmidt equation containing second-order temporal evolution $u_{tt}$ are the manifestation of bidirectional propagation of waves.
Interestingly, nonlinear models with the third-order temporal evolution
$u_{ttt}$ are useful extension~\cite{amw1,amw2,gwa,amw3,gqx1,rip1} in a
variety of physical circumstances where the respective causalities are
preserved~\cite{jgr,csc}. The integrability properties and nonlinear wave
structures in this type of nonlinear models are comparatively less explored.
Recently, Wazwaz introduced the following (3 $+$ 1)-dimensional fifth-order
nonlinear model with third-order temporal and spatiotemporal dispersion
effects~\cite{amw4}:
\begin{equation}
u_{ttt}-u_{txxxx}-12 u_{xt} u_{xx}-8 u_{x}u_{xxt}-4 u_{t}u_{xxx}+(\alpha u_{x}+\beta u_{y}+\gamma u_{z})_{xx}=0. \label{t1}
\end{equation}
Further, the above model is found to be integrable through Painlev\'e
analysis and soliton solutions are obtained using a simplified Hirota
method~\cite{amw4} along with different wave solutions of its one-dimensional
counterpart models~\cite{gqx1,rip1}.
Though solving these nonlinear equations is analytically a challenging task in
general, there are significant developments in formulating powerful tools such
as inverse scattering transforms, Hirota's bilinear method, Darboux and
B\"acklund transforms, Lax pairs, Riemann--Hilbert approach, KP hierarchy
reduction, Gauge transformation, Lie symmetry method,
etc.~\cite{Yang-book,ML-book,bgo,bgu}.
Apart from these, {similarity reduction approach~\cite{cp1,cp2,cp3,cp4}} and
various ancillary methods involving direct algebraic techniques (ODE reduction
approaches), auxiliary equation method, Kudryashov expansion method,
Riccati--Bernoulli sub-ODE method, sinh-Gordon expansion method, cosh-tanh
method, simplest equation method, etc. are utilized widely to get different
classes of travelling wave solutions~\cite{amwb}.
Especially, these methodologies provide a variety of exotic wave patterns,
including solitons, breathers, lumps, dromions, rogue waves, and elliptic
waves, and limitations are arising in utilizing strong tools like inverse
scattering transform, Riemann--Hilbert approach and Darboux transformation due
to the lack of Lax pair guaranteeing complete integrability for many of the
higher-dimensional soliton models. On the advantageous part, the Hirota
bilinear method is an intermediate tool which can be utilized to extract the
localized nonlinear wave solutions to most of the integrable as well as a few non-integrable soliton models and it becomes a widely used tool to
obtain several localized nonlinear wave solutions~\cite{Hirota-book}. {
Through $N$-soliton solutions, utilizing the long-wave limit
method, one can obtain different localized waves including lumps, rogue waves,
breathers and different types of interacting waves. Recently, an improved
long-wave limit method is proposed for obtaining higher-order rogue waves for
the nonlinear Schr\"odinger equation~\cite{abc4}, which can also be extended to
many other integrable models.}
{ The Hirota bilinear method is directly linked with truncated Painlev\'e
approach, where the later is successfully utilized to a variety of constant and
variable coefficient soliton models to identify their integrability property and to obtain various nonlinear wave
structures through auto-B\"acklund and hetero-B\"acklund transformations
including Boussinesq--Burgers system~\cite{cl1}, Whitham--Broer--Kaup-like
system~\cite{cl2}, variable coefficient generalized Burgers system~\cite{cl3}
and the generalized variable-coefficient KdV-modified KdV equation~\cite{cl4}.}
{ Particularly, this Hirota method is successfully used to identify certain
interesting dynamics of multi-lumps and lump chains in the framework of the
KP$1$ model~\cite{abc1,abc2,abc3}}.
{ Interestingly, in addition to the above approaches, some recent
methodologies such as probabilistic approaches~\cite{ch1,ch2} and deep learning
techniques~\cite{ch3,ch4,ch5} attract much interest in understanding different
nonlinear systems in different perspectives.}
{Localized nonlinear waves on variable backgrounds are another important aspect
of study in recent times across multiple disciplines starting from optics
to hydrodynamics systems. However, we can find only a handful amount of works
for such variable background nonlinear waves compared to their counterparts on
constant backgrounds. In the context of nonlinear optics, the effects of
spatially- and temporally-varying backgrounds on solitons, breathers and rogue
waves have revealed interesting outcomes that can be useful to understand the
influence of nonuniform models~\cite{ksjpa20,ksps20,ksa}. Similarly, the
dynamics of nonlinear waves on controllable backgrounds due to non-autonomous
nonlinearities are explored in Bose--Einstein condensates
too~\cite{boris,tk1,tk2,tkan,mani1,mani2}. { Recently, some studies are
reported for nonlinear wave structures in certain higher-dimensional nonlinear
models describing shallow or deep water waves, which include the analyses of
breathers in a variable-coefficient (3$+$1)D shallow water wave
model~\cite{pfh,epjp19}, rogue waves and solitons on spatio-temporally variable
backgrounds in (3$+$1)D Kadomtsev--Petviashvili--Boussinesq system~\cite{ssi},
solitons in an extended (3$+$1)D shallow water wave equation~\cite{jgl}, and
interaction waves in both (3$+$1)D and (4$+$1)D Boiti--Leon--Manna--Pempinelli
models~\cite{pfha,epjp21} to name a few.}
The above studies revealed different exciting results such as amplification,
compression, bending/snaking, tunnelling/cross-over, superposed structures,
etc.~\cite{ksjpa20,ksps20,ksa,boris,tkan,tk1,tk2,mani1,mani2,pfh,epjp19,epjp21,jgl,ssi,pfha}.
So, it is our natural interest to question the possibility of realizing such
phenomena in the above mentioned fifth-order nonlinear model~\eqref{t1}. To the best of
our knowledge, the Painlev\'e integrable model \eqref{t1} under consideration
is only solved for solitary waves on a constant background.
Motivated by the above interesting observations, with a special emphasis on higher-dimensional integrable soliton models, here in this work, our objective is to study the effects of controllable spatial backgrounds on the localized nonlinear waves (lumps and solitons) of the (3 $+$ 1)D fifth-order integrable soliton model with third-order temporal dispersion as given by Eq.~\eqref{t1}. Particularly, we are aimed to demonstrate the dynamics by adopting both exponential and rational localized waves on spatially-varying backgrounds.}
In view of the above perspective, the remaining part of the manuscript is arranged in the following manner. We construct the lump solution of the considered model \eqref{t1} in Sec. \ref{sec2} through its trilinear form and polynomial type seed solution along with a detailed study of its dynamics on constant as well as spatially-varying backgrounds. Section \ref{sec3} deals with the soliton solution of Eq.~\eqref{t1} using Hirota's bilinearization method and their detailed evolutionary dynamics is investigated. In Sec. \ref{sec5}, the bilinear B\"acklund transformation is derived. The final \ref{sec-conclusion} is allotted for conclusions of the present work along with certain future outlook.
\section{Trilinear Equation and Lump Wave Dynamics on Background}\label{sec2}
{ In this section, we aim to analyse the dynamics lump wave arising in the above-considered model (\ref{t1}) on the constant non-vanishing background as well as spatially-varying controllable backgrounds through its exact solution.} For this purpose, first, we implement the following superposed bilinear logarithmic transformation:
\begin{equation}
u(x,y,z,t)=u_{0} (y,z)+\left[{\ln}(f(x,y,z,t)\right]_{x}, \label{t2}
\end{equation}
where $u_{0} (y,z)$ is an arbitrary background which depends on two spatial dimensions and $f (x,y,z,t)$ is the required function to be calculated for constructing the solution of Eq. (\ref{t1}).
{\color{black} Here we wish to emphasize that a simple form of superposed bilinear transformation with a one-dimensional space-varying $\psi(z)$ background is used to study breathers in a variable-coefficient (3+1)D shallow water wave model in \cite{pfh,epjp19}, while in Ref. \cite{pfha,epjp21} multiple interaction solutions for the (3+1)D and (4+1)D Boiti--Leon--Manna--Pempinelli equations with time-varying background $\psi(t)$ is presented. Recently, in Ref.~\cite{ssi}, we have generalized the approach and investigated the effects
of arbitrary spatio-temporal background $\psi(z,t)$ background in the rogue waves and solitons for a (3+1)D Kadomtsev-Petviashvili-Boussinesq (KPB) equation. Next to that, there is another recent work \cite{jgl} on kink-solitons for an extended (3+1)D shallow water wave equation with a similar $\psi(z,t)$ background function. Following these exciting outcomes, in the present work, our aim is to study the influence of pure arbitrary two-dimensional spatial background in the evolution
of lump and soliton, which shows different phenomena as
explained in the forthcoming part of the manuscript. As a future study, one can attempt to apply this and other generalized arbitrary backgrounds (like $\psi(x,y,z,t)$ as hinted in our previous work \cite{ssi}) to different nonlinear wave equations supporting a variety of wave structures that can reveal excellent features and applications.}
Upon substituting the above superposed transformation (\ref{t2}) into the considered model Eq. (\ref{t1}), the following trilinear form is obtained as as follows.
\begin{eqnarray}
&T(f)\Rightarrow& 2f_{t}^3+2 f_{x}[2 f_{xx} f_{xt}+f_{x}(\alpha f_{x}+\beta f_{y}+\gamma f_{z}-2 f_{xxt})]-f[f_{xx}(\beta f_{y}+\gamma f_{z}+2 f_{xxt}) \nonumber\\ && +f_{x}(3\alpha f_{xx}+2\beta f_{xy} +2\gamma f_{xz}-4 f_{xxxt})] +f_{t}[2 (f_{xx}^2-2 f_{x}f_{xxx})+f(f_{xxxx}-3 f_{tt})] \nonumber\\ &&+f^2(\alpha f_{xxx}+\beta f_{xxy}+\gamma f_{xxz}+f_{ttt}-f_{xxxxt})=0. \label{t3}
\end{eqnarray}
{\color{black} Noted that the above trilinear form (\ref{t3}) is surprisingly independent of $u_0(y, z)$ under the superposed transformation (\ref{t2}). This ensures that $u_0(y,z)$ is arbitrary and different types of backgrounds including elliptic functions can be introduced to any possible nonlinear wave structures to the considered fifth-order integrable soliton model (\ref{t1}), once such solutions $f(x,y,z,t)$ satisfy (\ref{t3}). Considering the length of the article, in this section, we demonstrate the algorithm with only the first-order lump solution.}
To construct lump solution, we choose the following quadratic function as seed solution \cite{wx1}
\begin{equation}
f(x,y,z,t)=\theta_{0}+(\alpha_1 x+\beta_1 y+\gamma_1 z+\delta_1 t+\theta_1)^2+(\alpha_2 x+\beta_2 y+\gamma_2 z+\delta_2 t+\theta_2)^2, \label{t4}
\end{equation}
where $\theta_i, \alpha_j, \beta_k, \gamma_l, \delta_m, i=0,1,2, j=k=l=m=1,2$, are the parameters of the lump solution. Substituting (\ref{t4}) in (\ref{t3}), we get a polynomials in $x,y,z$ and $t$. Further, equating to zero all linearly independent terms to zero, after solving the system, we get the following constraints over lump parameters as
\begin{equation}
\alpha_1=0, \alpha_2=\sqrt{\delta_1} \theta_{0}^{1/4}, \beta_1=\dfrac{\delta_1^2-3\delta_2^2-\gamma \gamma_1 \sqrt{\theta_0}}{\beta \theta_0}, \beta_2=\dfrac{\delta_2^3-3 \delta_1^2\delta_2+\gamma \gamma_2 \delta_1 \sqrt{\theta_0}+\alpha \delta_{1}^{3/2}\theta_{0}^{3/4}}{-\beta \delta_1 \sqrt{\theta_0}}.\label{t5}
\end{equation}
Hence, we shall obtain the required form of $f$ from Eq. (\ref{t4}) by substituting the quantities obtained in Eq. (\ref{t5}), from which we can deduce the resultant lump wave solution $u$ from Eq. (\ref{t2}) in a compact form as follows.
\begin{equation}
u(x,y,z,t)=u_0 (y, z)+\frac{2\sqrt{\delta_1} \theta_{0}^{1/4}(\sqrt{\delta_1} \theta_{0}^{1/4} x+\beta_2 y+\gamma_2 z+\delta_2 t+\theta_2)}{\theta_{0}+(\beta_1 y+\gamma_1 z+\delta_1 t+\theta_1)^2+(\sqrt{\delta_1} \theta_{0}^{1/4} x+\beta_2 y+\gamma_2 z+\delta_2 t+\theta_2)^2}. \label{lump-sol}
\end{equation}
The other parameters are given in the Eq. (\ref{t5}). The above obtained lump solution is characterized by seven arbitrary solution parameters ($\delta_1$, $\delta_2$, $\theta_0$, $\theta_1$, $\theta_2$, $\gamma_1$, $\gamma_2$) and three arbitrary system parameters ($\alpha$, $\beta$, and $\gamma$) in addition to the arbitrary spatial background $u_0 (y, z)$. By tuning these arbitrary parameters/functions one can control the nature and evolution of the resulting lump solution. The present solution (\ref{lump-sol}) describes a doubly-localized lump with one peak and one well/hole on a constant background. The nature of the resulting lump wave admits different width, localization and position along different spatio-temporal domains, whereas it exhibits same amplitude throughout the evolution when there exits no background function ($u_0=0$). Note that the free parameters available in the solution (\ref{lump-sol}) play crucial role in controlling the identities of the lump such as the amplitude, width, localization and position, so that a rich categories of lump wave profiles can be constructed accordingly. The amplitude of the lump wave above the zero or non-zero background is proportional to the factor $2\sqrt{\delta_1} \theta_{0}^{1/4}$, while the other parameters help to manipulate other identities of the lump wave. For a clear understanding on the above discussion, we have given graphical demonstration of lump solution (\ref{lump-sol}) along different spatio-temporal domains in Fig. \ref{fig-lump-1}, while its dynamics along spatial dimension $y-z$ at three different time is shown in Fig. \ref{fig-lump-2}. It is important to highlight that the nature of lump wave remains unaltered and it takes different positions due to temporal evolution.
\begin{figure}[h]
\centering\includegraphics[width=0.99\linewidth]{figure1}
\caption{The nature of lump soliton along different spatio-temporal planes without any background $u_0=0$ are given in the 3D plots for the choice $\delta_1=1.0$, $\delta_2=0.5$, $\theta_0=0.5$, $\theta_1=0.3$, $\theta_2=0.7$, $\gamma_1=0.45$, $\gamma_2=0.25$, $\alpha=0.4$, $\beta=0.9$, and $\gamma=1.3$. Especially, doubly-localized lump structure along $x-t$ at $y=z=0.4$ and $y-t$ at $x=z=0.4$, while we obtain line soliton along $z-t$ at $x=y=0.4$. The lower 2D plots represent their corresponding structure at $t=-1.5$ (solid line), $t=0$ (dashed line), and $t=1.5$ (dotted line).}
\label{fig-lump-1}
\end{figure}
\begin{figure}[h]
\centering\includegraphics[width=0.99\linewidth]{figure2}
\caption{The contour plot of the propagation (localization movement) of the lump structure along the two spatial plane $y-z$ at different times $t=-3.4,0,3.4$ and $x=0.4$ without any background for the same choice as in Fig. \ref{fig-lump-1}.}
\label{fig-lump-2}
\end{figure}
\begin{figure}[h]
\centering\includegraphics[width=0.99\linewidth]{figure3}
\caption{The impact of periodic background in the evolution of lump wave along different spatio-temporal planes for $u_0(y,z)=0.35~ \mbox{sn}(0.5 y+0.5 z,0)$ with other parameters as given in Fig. \ref{fig-lump-1}.}
\label{fig-lump-3}
\end{figure} \begin{figure}[h]
\centering\includegraphics[width=0.99\linewidth]{figure4}
\caption{The impact of localized bell-type soliton background in the evolution of lump wave along different spatio-temporal planes for the same choice of parameters as given in Fig. \ref{fig-lump-1} with $u_0(y,z)=0.75~ \mbox{cn}(0.5 y+0.5 z,1)$.}
\label{fig-lump-4}
\end{figure}
\begin{figure}[h]
\centering\includegraphics[width=0.99\linewidth]{figure5}
\caption{The impact of kink-type soliton background in the evolution of lump wave along different spatio-temporal planes for the same choice of parameters as given in Fig. \ref{fig-lump-1} with $u_0(y,z)=0.75~ \mbox{sn}(0.5 y+0.5 z,1)$.}
\label{fig-lump-5}
\end{figure}
\begin{figure}
\centering\includegraphics[width=0.99\linewidth]{figure6}
\caption{The manipulation of lump wave along the spatial domain $y-z$ due to periodic, localized bell-type soliton, combined bell-soliton with periodic, and kink-soliton backgrounds with parameter values as given in Figs. \ref{fig-lump-1}-\ref{fig-lump-5}.}
\label{fig-lump-6}
\end{figure}
Interestingly, when the arbitrary background $u_0(y,z)$ is taken into effect and non-vanishing, the lump wave undergoes various manifestation based on the nature of background variation and results into different wave phenomena. To shed light on the understanding, we have adapted two type of arbitrary spatial background function such as (i) localized soliton type background and (ii) periodic background by incorporating Jacobi elliptic functions as given below.
\begin{eqnarray} u_0(y,z)=p_1 \mbox{sn}(a_1 y+b_1 z,q_1)+p_2 \mbox{cn}(a_2 y+b_2 z,q_2),\label{jacobi} \end{eqnarray}
where $p_j$, $a_j$, and $b_j$ ($j=1,2$) are arbitrary real constants, while $q_1$ and $q_2$ are the elliptic modulus parameters ($0\leq q_1,q_2 \leq 1)$, using which we shall incorporate periodic ($q_1=0$ or $q_2=0$), kink-like soliton ($q_1=1$ and $p_2=0$ ) and bell-type soliton ($q_2=1$ and and $p_1=0$) type backgrounds to the initial lump solution of Eq. (\ref{t1}). The function $u_0(y,z)=p_1 \mbox{sn}(a_1 y+b_1 z,0)$ manifests the lump to appear on the periodic background along $y-t$, whereas one can obtain an interaction between periodic wave and breather along $z-t$ plane. Additionally, the localized soliton type spatial background leads to the superposition lump on soliton along $y-t$, while it results into interacting solitons along $z-t$. However, both type of backgrounds show no impact on the lump along $x-t$ due to the fact that the spatial background $u_0(y,z)$ is considered to be on two dimensions only. For getting further insights to the above arguments, we have graphically shown the influence of both periodic and soliton backgrounds in Figs. \ref{fig-lump-3}-\ref{fig-lump-6}. Especially, we have given the influence of spatially-varying periodic background in Fig. \ref{fig-lump-3} along different spatio-temporal domain, which shows that the lump wave remains unaltered along $x-t$, while it is affected by the background along $y-t$ and transform into interaction with a periodic wave. Interestingly, one can observe the formation of breather along $z-t$ and its interaction with a background periodic wave. In the case of localized bell-type soliton background, we witness the superposition of the lump with solitary wave along $y-t$ and results into an passing-through collision between two solitary waves along $z-t$ as shown in Fig. \ref{fig-lump-4}. Figure \ref{fig-lump-5} depicts the impact of kink-type soliton background on the lump wave along different spatio-temporal planes. Finally, we have shown how the arbitrary controllable backgrounds modulate the lump wave along the spatial domain $y-z$ in Figs. \ref{fig-lump-6}.
\section{Bilinear Formalism and Soliton Dynamics on Background} \label{sec3}
In this section, we construct the solitary wave solution by adopting bilinearization algorithm using Hirota derivatives \cite{Hirota-book} and study their dynamics under different backgrounds of interest. For this purpose, first, by applying the following dimension reducing transformation:
\begin{eqnarray} u(x,y,z,t) = u(\eta , y, z), \end{eqnarray}
where $\eta = (x+\delta t)$, to the original Eq. (\ref{t1}) we obtain
\begin{equation}
\delta u_{\eta \eta \eta \eta \eta}-(\alpha + \delta^3) u_{\eta \eta \eta }- \beta u_{\eta \eta y } - \gamma u_{\eta \eta z}+12\delta (u_{\eta \eta }^2 + u_\eta u_{\eta \eta \eta} )=0. \label{eqn1}
\end{equation}
Further, by using the logarithmic transformation with the superposition of arbitrary functions a
\begin{equation}
u(\eta, y,z) = u_0 (y,z) + \left[\log f(\eta , y, z)\right]_\eta, \label{eqn2}
\end{equation}
the above equation (\ref{eqn1}) can be deduced into the following bilinear form:
\begin{equation}
(\delta D^4_\eta - \chi D_\eta ^2 - \beta D_\eta D_y - \gamma D_\eta D_z) f \cdot f=0, \label{eqn3}
\end{equation}
where $\chi =(\alpha + \delta^3)$. Here $u_0 (y,z)$ is an arbitrary spatial background, while $f(\eta , y, z)$ is the unknown function to be determined. From the above bilinear form (\ref{eqn3}), one can get different types of nonlinear wave solutions by considering appropriate initial seed solution. Here, we obtain the single solitary wave solutions by taking an exponential test function as the initial seed solution and investigate their dynamics on different backgrounds. Further, the following analysis can be extended to explore multiple soliton dynamics on arbitrary backgrounds in a straightforward manner. Note that Wazwaz studied the multi-soliton solution using trilinear form but without any background \cite{amw4}. However, here, our importance is to focus the effectiveness and consequences of arbitrary spatial backgrounds on solitons.
To construct one soliton solution, we choose the initial seed solution as $f=1+ \varepsilon_1 e^{\Omega_1}$, where $\Omega_1=\theta_1 \eta + \phi_1 y + \psi_1 z+ \delta_1$, and
upon its substitution into the bilinear equation \eqref{eqn3} and solving it, we get $\psi_1 = {(\delta \theta_1^3 -\chi \theta_1 - \beta \phi _1)}/{\gamma}$, which results in the following explicit form for $f$:
\begin{equation}
f=1+ \varepsilon_1 e^{\theta_1 \eta + \phi_1 y + (\delta \theta_1^3 - \chi \theta_1 - \beta \phi _1)z/{\gamma}+ \delta_1}. \label{eqn5}
\end{equation}
Thus the resulting one soliton solution can be written from Eq. (\ref{eqn2}) as
\begin{equation}
u(\eta, y,z) = u_0 (y,z) + \frac{\varepsilon_1 \theta_1 e^{\theta_1 \eta + \phi_1 y + (\delta \theta_1^3 - \beta \phi _1 + \chi \theta_1)z/{\gamma}+ \delta_1}}{1+ \varepsilon_1 e^{\theta_1 \eta + \phi_1 y + (\delta \theta_1^3 - \beta \phi _1 + \chi \theta_1)z/{\gamma}+ \delta_1}}. \label{eqn6}
\end{equation}
The above one soliton solution (\ref{eqn6}) contains seven arbitrary constants $\varepsilon_1$, $\theta_1$, $\phi_1$, $\delta_1$, $\alpha$, $\beta$, and $\gamma$ where the first four parameters results exclusively from the solution while the latter three come from the model itself. Apart from these arbitrary constants, we have an additional arbitrary background $u_0 (y,z)$ and it can deliver a rich variety of characteristics across the two spatial dimensions $y$ and $z$. Note that the nature of soliton solution (\ref{eqn6}) we obtained here is nothing but the kink solitons and it carries interesting properties such as amplitude and localization/orientation.
\begin{figure}[h]
\centering\includegraphics[width=0.99\linewidth]{figure7}
\caption{Evolution of kink-solitons along $x-t$ at $y,z=0.2$ (left panels) and $y-t$ at $x,z=0.2$ (middle panels) and anti-kink-soliton along $z-t$ at $y,x=0.2$ (right panels) obtained through the first-order (one) solution (\ref{eqn6}) for the choice $\delta=1.25$, $\delta_1=1$, $\epsilon_0=0$, $\epsilon_1=1.45$, $\theta_1=0.3$, $\phi_1=0.5$, $\alpha=0.4$, $\beta=0.9$, and $\gamma=1.3$. Bottom panels show kink and anti-kink solitons at different time $t=-1.5$ (solid line), $t=0$ (dashed line), and $t=1.5$ (dotted line).}
\label{fig-one-sol-1}
\end{figure}
Especially, the solution (\ref{eqn6}) leads to kink soliton (smooth step-like profile) along $x-t$ and $y-t$ while it supports anti-kink soliton (smooth reverse-step-like profile) along $z-t$ due to different spatio-temporal behaviour.
From the solution, we can understand that the amplitude of these kink soliton is defined by $\varepsilon_1 \theta_1$, whereas its velocity is characterized by $-\delta$, $-\delta\theta_1/\phi_1$, and $-\delta\theta_1\gamma/(\delta \theta_1^3 - \beta \phi _1 + \chi \theta_1)$ along $x-t$, $y-t$, and $z-t$, respectively. One can tune the nature of resulting kink soliton by using the available arbitrary parameters. For an easy understanding, we have depicted the propagation of such first-order solution taking kink and anti-kink soliton forms in Fig. \ref{fig-one-sol-1}. Also, it shows the projected contour plots and line plots along the spatial dimensions at different times.
\begin{figure}[h]
\centering\includegraphics[width=0.9\linewidth]{figure8}
\caption{Nature of wide kink-soliton along $x-t$ at $y,z=0.2$ (left panels), narrow kink-soliton along $y-t$ at $x,z=0.2$ (middle panels) and anti-kink-soliton along $z-t$ at $y,x=0.2$ (right panels) obtained through the first-order (one) solution (\ref{eqn6}) for the choice $\delta=1.25$, $\delta_1=1$, $\epsilon_0=0$, $\epsilon_1=1.45$, $\theta_1=0.3$, $\phi_1=0.5$, $\alpha=0.4$, $\beta=0.9$, and $\gamma=1.3$.}
\label{fig-one-sol-2}
\end{figure}
As discussed in the case of lumps, here in the kink solitons also the arbitrary spatially-varying backgrounds introduces different manifestation of the solitons. Particularly, the periodic background arising for the choice either $p_1\neq 0$, $p_2=0$, $q_1=0$ or $p_1=0$, $p_2\neq 0$, $q_2=0$ influences the kink soliton to transform into a periodically oscillating kink and anti-kink waves as demonstrated in the top panels of Fig. \ref{fig-one-sol-2} in both $y-t$ and $z-t$ planes. On the other hand, the localized $sech$ type background ($p_1=0$, $p_2\neq 0$, $q_2=1$) gives rise to the coexistence of bell soliton on top of the narrowing or steepening kink and anti-kink solitons, which we have shown in the bottom panels of Fig. \ref{fig-one-sol-2}. Also, based on the localization of the kink and bell solitons, this steepening effect gets altered. On contrary to the modulations in $y-t$ and $z-t$ planes, the kink solitons not undergo any significant alteration in their dynamics along $x-t$ plane due to the nature of background function $u_0(y,z)$.
Considering this factor, proceeding further, we can observe different other features in the evolution of kink solitons along the $y-z$ plane which are depicted in Fig. \ref{fig-one-sol-3}. Especially, the anti-kink soliton shown in the top-left panel transforms into periodically oscillating kink solitons when $p_1\neq 0$, $p_2=0$, $q_1=0$ or $p_1=0$, $p_2\neq 0$, $q_2=0$ (top-middle panel), double-periodic kink soliton for $p_1,p_2\neq 0$, $q_1,q_2=0$ (top-right panel), and an elastic interaction of kink and bell type solitons when the background is chosen as $p_1=0$, $p_2\neq 0$, $q_2=1$ (bottom-left panel). Additionally, when the background function is considered as a combined periodic and bell waves, one shall identify their superposed effect on the kink soliton as given in the bottom-middle and bottom-right panel of Fig. \ref{fig-one-sol-3}.
\begin{figure}[h]
\centering\includegraphics[width=0.91\linewidth]{figure9}
\caption{The travelling nature of first-order (one) anti-kink soliton along $y-z$ at different time $t=-8$, $t=0$, and $t=8$ through the solution (\ref{eqn6}) with other parameters as in Fig. \ref{fig-one-sol-1} on zero-background. The influence of (top middle) periodic, (top-right) double-periodic, (bottom-left) bell-type soliton, (bottom-middle)kink-like soliton, and (bottom-right) superposed soliton and periodic type backgrounds on the anti-kink soliton along $y-z$ at $t=0$.}
\label{fig-one-sol-3}
\end{figure}
\section{Bilinear B\"acklund Transformation}\label{sec5}
In this section, we obtained the bilinear B\"acklund transformation of the considered model using the exchange identities of the Hirota's $D$ operator~\cite{Hirota-book,ysh,yxm}. By adapting this alternate approach different types of explicit solutions can be constructed. For this purpose, we introduce the following equation for the functions $f$ and $g$ appearing in the bilinear equation (\ref{eqn3}):
\begin{eqnarray}
&&P=[ ( \delta D_\eta ^4 - \chi D_\eta ^2 - \beta D_{\eta}D_y - \gamma D_\eta D_Z ) g \cdot g ] f^2 - g^2 [ ( \delta D_{\eta}^4 - \chi D_{\eta }^2 - \beta D_{\eta }D_y-\gamma D_\eta D_Z ) f \cdot f ] =0,\qquad \label{eq1}\\
&&\quad =\delta [ (D_{\eta}^4 g \cdot g ) f^2 - g^2(D_\eta ^4 f \cdot f ) ] - \chi [ (D_{\eta}^2 g \cdot g ) f^2 - g^2(D_\eta ^2 f \cdot f ) ] \nonumber\\
&& \qquad - \beta [ (D_{\eta} D_y g \cdot g ) f^2 - g^2(D_\eta D_y f \cdot f ) ] - \gamma [ (D_{\eta} D_z g \cdot g ) f^2 - g^2(D_\eta D_y f \cdot f ) ]=0, \label{eq2}
\end{eqnarray}
where $g$ is real differential function of $\eta, y$ and $z$.
Using the exchange identities for the Hirota $D$ operator \cite{Hirota-book},
\begin{subequations} \begin{align}
(D_{\eta}^2 g \cdot g ) f^2 - g^2 (D_\eta ^2 f \cdot f ) & = 2 D_\eta (D_\eta g \cdot f ) \cdot (fg), \label{eq3a}\\
(D_\eta D_y g \cdot g ) f^2 - g^2 (D_\eta D_y f \cdot f ) & = 2 D_\eta (D_y g \cdot f ) \cdot (fg), \label{eq3b}\\
(D_{\eta}^4 g \cdot g ) f^2 - g^2 (D_\eta ^4 f \cdot f ) & = 2 D_\eta (D_\eta ^3 g \cdot f ) \cdot (fg) - 6 D_\eta (D_\eta ^2 g \cdot f ) \cdot (D_\eta g \cdot f). \label{eq3c}
\end{align} \label{bilid} \end{subequations}
By applying the above bilinear identities (\ref{bilid}) in Eq. \eqref{eq2}, we can arrive at the following form:
\begin{eqnarray}
&&P= \delta [ 2 D_\eta (D_\eta ^3 g \cdot f ) \cdot (fg) - 6 D_\eta (D_\eta ^2 g \cdot f ) \cdot (D_\eta g\cdot f)] - \chi [2 D_\eta (D_\eta g \cdot f )\cdot (fg) ]\nonumber\\
&&\qquad - \beta [2 D_\eta (D_y g \cdot f) \cdot (fg)] - \gamma [2 D_\eta (D_z g\cdot f ) \cdot (fg)] =0. \label{eq4}
\end{eqnarray}
Without loss of generality, assuming that $D_\eta ^2 g \cdot f = \varepsilon _1 fg $, we obtain
\begin{align}
2 D_\eta [(\delta D_\eta ^3 + 3 \varepsilon_1 \delta D_\eta - \chi D_\eta - \beta D_y - \gamma D_z) g \cdot f ](fg) =0. \label{eq5}
\end{align}
Finally, we have the following bilinear B\"acklund transformation:
\begin{subequations}
\begin{eqnarray}
&&u= u_0 (y,z) + (\log f)_\eta, \label{eq6a}\\
&&(\delta D_\eta ^3 + 3 \varepsilon_1 \delta D_\eta -\chi D_\eta - \beta D_y - \gamma D_Z) g \cdot f =0, \label{eq6b}\\
&&(D_\eta ^2 - \varepsilon_1 ) g \cdot f =0. \label{eq6c}
\end{eqnarray}
\end{subequations}
By adopting the above bilinear B\"acklund transformation, we can construct different classes of nonlinear wave solutions on arbitrary spatial background with suitably chosen form of initial functions for $g$ and $f$. For completeness and demonstration purpose, we have constructed hyperbolic and periodic wave solutions in the forthcoming part.
\subsection{Hyperbolic Wave Solution }
We choose the seed functions $f=1$ and $g = \cosh (m_1 \eta +\eta _1 y + p_1 z + r_1 )$ and solving \eqref{eq6b}-\eqref{eq6c}, we get
$p_1 = ({4m_1 ^3 \delta - m_1 \chi - \eta_1 \beta })/{\gamma}.$
This gives us the explicit form of $g$ and $f$ as given below.
\begin{align}
f=1, \quad g =\cosh \left(m_1 \eta + n_1 y + {(4m_1 ^3 \delta - m_1 \chi - \eta_1 \beta )z}/{\gamma}+r_1\right). \label{eq8}
\end{align}
Thus, the final hyperbolic solution of the considered model Eq. (\ref{t1}) is obtained as
\begin{align}
u(x,y,z,t) & =u_0 (y,z) + m_1 \tanh \left( (x+\delta t)m_1+n_1 y + {(4m_1 ^3 \delta - m_1 (\alpha+\delta^3) - \eta_1 \beta )z}/{\gamma}+r_1 \right). \label{eq9}
\end{align}
We can obtain another hyperbolic wave solution, by choosing
$f=1$ and $g (\eta , y , z)=\sinh (m_1 \eta + \eta _1 y + p_1 z +r_1)$,
and solving \eqref{eq6b}-\eqref{eq6c} which would give the same { $p_1$ as above}. Then, the final form of the hyperbolic wave solution of the considered model Eq. (\ref{t1}) is obtained as
\begin{align}
u(x,y,z,t) & =u_0 (y,z) + m_1 \coth \left( (x+\delta t)m_1+n_1 y + {(4m_1 ^3 \delta - m_1 (\alpha+\delta^3) - \eta_1 \beta )z}/{\gamma}+r_1 \right). \label{eq91}
\end{align}
\subsection{Periodic Wave Solution}
Here, we have obtained periodic wave solutions by choosing $f=1$, $g = \cos (m_1 \eta + \eta _1 y + p_1 z +r_1) $, and solving \eqref{eq6b}-\eqref{eq6c}, we get
$p_1 = - ({(4m_1^3 \delta+m_1 \chi + \eta _1 \beta)})/{\gamma}$.
Finally, the resultant periodic wave solution of the considered soliton model (\ref{t1}) is obtained as
\begin{align}
u(x,y,z,t) & =u_0 (y,z) - m_1 \tan \left( (x+\delta t)m_1+n_1 y - {(4m_1 ^3 \delta + m_1 (\alpha+\delta^3) + \eta_1 \beta )z}/{\gamma}+r_1 \right). \label{eq92}
\end{align}
Similarly, we can have alternate form of periodic wave solution of the model (\ref{t1}), by choosing $f=1, g = \sin (m_1 \eta + \eta _1 y + p_1 z +r_1) $, and it can be written as follows.
\begin{align}
u(x,y,z,t) & =u_0 (y,z) + m_1 \cot \left( (x+\delta t)m_1+n_1 y - {(4m_1 ^3 \delta + m_1 (\alpha+\delta^3) + \eta_1 \beta )z}/{\gamma}+r_1 \right). \label{eq93}
\end{align}
It is necessary to highlight the fact that by using different initial seed solutions, the above bilinear B\"acklund transformation can be extended straightforwardly to obtain different other wave solutions. Considering the length of the article, we refrain from presenting such detailed class of solutions and further discussion here. Also, note that the above periodic and hyperbolic solutions exhibit arbitrary spatial background $u_0 (y,z) $ which again give us fruitful dynamics in the underlying wave structures.
\section{Conclusions}\label{sec-conclusion}
{ To summarize the present work, we have investigated an integrable (3 $+$ 1)-dimensional nonlinear equation~\eqref{t1} describing the evolution of {oceanic waves} with higher-order temporal dispersion for the dynamics of the lump and solitons on different spatial backgrounds. Particularly, we have constructed explicit solutions for lump and kink-soliton of the considered model through a systematic analysis using its trilinear and bilinear formulations, respectively, along with suitable forms of polynomial and exponential type initial seed solutions. Our analysis shows that the lump solution results into a well-localized profile with a simultaneous existence of spiking/peak and declining/dip (coupled bright-dark peaks) structure on a constant background. However, the obtained soliton solution leads to a step-like kink and anti-kink patterns along different spatio-temporal domain.
When the arbitrary spatial background function arising in the solution is
incorporated (non-zero), the lump wave results into a number of interesting
wave phenomena. Especially, the periodic and localized type arbitrary spatial
backgrounds generate (i) coexistence of the lump on periodic wave, (ii) breather
formation on periodic background wave, (iii) spiking on top of bell \& kink
solitons, (iv) interaction of line soliton-like nature, and (v) interacting
lump with kink wave. Additionally, the incorporation of these periodic and
localized soliton spatial backgrounds play a crucial role in the transition of
kink solitons into manipulated interaction waves with bell type soliton,
periodic waves, double-periodic waves with substantial changes in their
dynamics. For a better understanding of the resulting dynamics, we have
provided a categorical discussion and clear graphical demonstration for lump
and solitons on both constant and spatially-varying backgrounds. Further, we
have obtained periodic and hyperbolic solutions with arbitrary spatial
backgrounds for the considered model \eqref{t1} through bilinear B\"acklund
transformation. The obtained results will be an important addition along the
context of nonlinear wave manipulation in higher-dimensional models due to
controllable backgrounds. The present investigation shall also be extended to
several other single and multi-component real and complex soliton models
towards enhanced understanding of the dynamical characteristics of exotic
nonlinear waves like rogue waves and dromions on variable
background~\cite{ma1,ma2}. } \\
\setstretch{1.0}
\noindent{\bf Acknowledgments}\\
One of the authors SS acknowledges Ministry of Education (MoE), India, for the financial support through institute (National Institute of Technology, Tiruchirappalli) fellowship. The research work of K Sakkaravarthi is supported by the Young Scientist Training (YST) program of the Asia-Pacific Center for Theoretical Physics (APCTP), Pohang-si, Gyeongsangbuk-do, South Korea. The APCTP is supported by the Korean Government through the Science and Technology Promotion Fund and Lottery Fund.
{ The authors thank the handling editor and anonymous reviewers for providing valuable comments and suggestions that helped immensely to improve the quality of the manuscript.}\\
\noindent{\bf CRediT Author Contribution Statement}\\ {\bf Sudhir Singh}: Conceptualization, Methodology, Writing - Original Draft Preparation, Writing - Review \& Editing. {\bf K. Sakkaravarthi}: Formal Analysis, Investigation, Visualization, Writing - Original Draft Preparation, Writing - Review \& Editing. {\bf K. Murugesan}: Resources, Writing - Review \& Editing, Funding acquisition, Supervision. \\
\noindent{\bf Declaration of Competing Interest}\\ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
\setstretch{0.950}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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How To Use This E-Book
Getting Around the e-Book
This Pocket Guide e-book is designed to give you inspiration and planning advice for your visit to Jamaica, and is also the perfect on-the-ground companion for your trip.
The guide begins with our selection of Top 10 Attractions, plus a Perfect Itinerary feature to help you plan unmissable experiences. The Introduction and History chapters paint a vivid cultural portrait of Jamaica, and the Where to Go chapter gives a complete guide to all the sights worth visiting. You will find ideas for activities in the What to Do section, while the Eating Out chapter describes the local cuisine and gives listings of the best restaurants. The Travel Tips offer practical information to help you plan your trip. Finally, there are carefully selected hotel listings.
In the Table of Contents and throughout this e-book you will see hyperlinked references. Just tap a hyperlink once to skip to the section you would like to read. Practical information and listings are also hyperlinked, so as long as you have an external connection to the internet, you can tap a link to go directly to the website for more information.
Maps
All key attractions and sights in Jamaica are numbered and cross-referenced to high-quality maps. Wherever you see the reference [map], tap once to go straight to the related map. You can also double-tap any map for a zoom view.
Images
You'll find lots of beautiful high-resolution images that capture the essence of Jamaica. Simply double-tap an image to see it in full-screen.
About Berlitz Pocket Guides
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Our wide-ranging travel products – printed travel guides and phrase books, as well as apps and ebooks – offer all the information you need for a perfect trip, and are regularly updated by our team of expert local authors. Their practical emphasis means they are perfect for use on the ground. Wherever you're going – whether it's on a short break, the trip of a lifetime, a cruise or a business trip – we offer the ideal guide for your needs.
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Table of Contents
How To Use This E-Book
Jamaica's Top 10 Attractions
Top Attraction #1
Top Attraction #2
Top Attraction #3
Top Attraction #4
Top Attraction #5
Top Attraction #6
Top Attraction #7
Top Attraction #8
Top Attraction #9
Top Attraction #10
A Perfect Day in Negril
Introduction
Landscape and Vegetation
National Identity
Tourist Attractions
A Brief History
Columbus and the Arrival of Europeans
British Rule
Plantations and Slavery
Emancipation
Independence and Democratic Rule
Politics Today
Historical Landmarks
Where To Go
Montego Bay
Downtown
The Beaches
South of the Bay
East on the Coast Road
Great Houses
Falmouth
Cockpit Country
Discovery Bay and Runaway Bay
St Ann's Bay
Ocho Rios To Annotto Bay
Dunn's River Falls
Rivers and Plantations
Firefly and Annotto Bay
Port Antonio And the East
Port Antonio
West of Port Antonio
East of Port Antonio
Jamaica's Eastern Tip
The Blue Mountains
Kingston and Environs
Downtown
New Kingston
Port Royal
Central Highlands and The South
Spanish Town
Mandeville
Black River
Treasure Beach
Negril and the West
Negril
To the South
What To Do
Sports And Outdoor activities
Beaches and Water Sports
Snorkelling
Diving
Boat Charters, River Tours and Rafting
Sport Fishing
Other Activities
Golf
Walking, Hiking and Cycling
Horse Riding
Spectator Sports
Cricket
Soccer
Horse Racing
Polo
Nightlife and Entertainment
Shopping
Arts and Crafts
Clothing
Coffee
Rum
Cigars
Children's Jamaica
Festivals and Events
Eating Out
What to Eat
Jamaican Cuisine
International Cuisine
What to Drink
Restaurants
Montego Bay
Falmouth
Runaway Bay
Ocho Rios
Port Antonio
Blue Mountains
Kingston
Treasure Beach (South Coast)
Negril
A–Z Travel Tips
A
Accommodation
Airports (see also Getting There)
B
Bicycle Rental
Budgeting for Your Trip
C
Car hire
Climate
Clothing
Crime and Safety
D
Driving
E
Electricity
Embassies and Consulates
Emergencies
G
Gay and Lesbian Travellers
Getting There (see also Airports)
Guides and Tours
H
Health and Medical Care
L
Language
M
Media
Money
O
Opening times
P
Police
Post Offices
Public Holidays
R
Religion
T
Telephones
Time Zones
Tipping
Tourist Information
Transport
V
Visas and Entry Requirements
W
Websites and Internet Access
Y
Youth Hostels
Recommended Hotels
Montego Bay
Runaway Bay
Ocho Rios and Environs
Port Antonio and the East
The Blue Mountains
Kingston Area
treasure Beach (South Coast)
Negril
Jamaica's Top 10 Attractions
Top Attraction #1
Kevin Cummins/Apa Publications
Rose Hall Great House
Some believe that this landmark property is haunted. For more information, click here.
Top Attraction #2
Kevin Cummins/Apa Publications
YS Falls
Fun for all at this tiered waterfall fringed by mature forest. For more information, click here.
Top Attraction #3
Kevin Cummins/Apa Publications
Seven Mile Beach
This stretch of fine sand is considered one of the best beaches in the Caribbean. For more information, click here.
Top Attraction #4
Photoshot
Blue Mountains
Explore this magnificent mountain range on a guided walk. For more information, click here.
Top Attraction #5
Kevin Cummins/Apa Publications
Runaway Bay
One of the most popular resorts on the north coast. For more information, click here.
Top Attraction #6
Kevin Cummins/Apa Publications
The National Gallery
Features a large collection of Jamaican paintings and sculpture. For more information, click here.
Top Attraction #7
Kevin Cummins/Apa Publications
Black River
Home to a variety of birds, fish and crocodiles. For more information, click here.
Top Attraction #8
Kevin Cummins/Apa Publications
Bob Marley Museum
A fascinating insight into the life and work of the reggae icon. For more information, click here.
Top Attraction #9
Kevin Cummins/Apa Publications
Treasure Beach
An ideal place for a relaxing getaway on the south coast. For more information, click here.
Top Attraction #10
Kevin Cummins/Apa Publications
Dunn's River Falls
No visit to Jamaica is complete without a walk up these famous falls. For more information, click here.
A Perfect Day in Negril
7.00am
Early swim
Start the day with a refreshing swim on Seven Mile Beach. You'll have the place to yourself apart from a few joggers and early birds, and the sea will be as calm can be.
8.00am
Breakfast/ lighthouse
Having worked up an appetite, tuck into a full Jamaican breakfast, either at your hotel or at Just Natural restaurant at West End. Then take a morning stroll from the restaurant to the Victorian Lighthouse, from here you get a great view out to sea and along the cliffs.
10.00am
Craft market
Back to Seven Mile Beach by taxi and wander around the craft stalls for a bargain. Pick up some snacks for later – fresh fruit from the women on the beach and delicious patties from Niah's Patties.
11.00am
Taxi tour
Find a reputable taxi driver outside your hotel and agree a rate for a tour of the south coast. Head for Bluefields Bay, via Savanna-la-Mar.
11.30am
Blue Hole
Spend an hour or two at the Blue Hole, jumping 7.5m (25ft) down into the exhilarating water (or use the ladder!), sunbathing around the spring-fed swimming pool, chatting with your hosts over a Red Stripe and maybe lunch if you've phoned in advance. Call in at the Peter Tosh Mausoleum at Belmont, a shrine to the superstar reggae musician.
2.30pm
Black River
Take a boat tour a few miles up the longest river in Jamaica and into the Great Morass, the swamp where the crocodiles live, and fit in a spot of birdwatching before returning to Negril.
5.30pm
Take a sunset cruise
Sail off into the sunset with a rum punch and watch the last flickering rays of the day. You'll be taken to the rocky cliffs at West End, before stopping off at Rick's Café to see intrepid young cliff-jumpers fling themselves off the rocks.
7.30pm
Romantic dinner
Enjoy a romantic candlelit meal at Ivan's Bar & Restaurant at Catcha Falling Star Hotel, nestled on top of Negril's scenic cliffs. Take in the beautiful view while sipping a cocktail or two.
10.00pm
Night fever
Head for The Jungle, the iconic club where you can work up a sweat on one of the four dance floors. Security is tight and the music carries on until the last guest staggers out.
Introduction
The island of Jamaica amazed Christopher Columbus when he visited in 1494 on his second journey to the West Indies. He described it as the 'fairest island' and marvelled at the mountains that 'touched the sky'. Today's visitors will be equally charmed by the warm sunshine, beautiful beaches, rivers and streams that gush from ravines, lush tropical scenery and majestic mountains, as well as the vibrant grassroots culture.
Landscape and Vegetation
The third-largest island in the Caribbean, just south of Cuba, Jamaica is 235km (146 miles) in length and 82km (51 miles) across at its widest point. The island is aligned almost east-to-west in the water so that sunrise wakes the eastern tip, proceeds to caress the length of the island, and kisses the western tip 'good night'. Geographically it is extremely diverse, with a central backbone of high mountains blanketed with a mixture of wet limestone forests and plantations of pine and native hardwood trees, such as mahoe and cedar. These are surrounded by areas of limestone formations, scrub and grassland, coral cliffs and fine sand beaches. Fresh water springs and tropical storms feed 120 rivers and some of the most celebrated waterfalls and cascades on earth.
On land, there is a wealth of animal and bird life. Rare species of butterflies and delicate hummingbirds take to the air, and crocodiles and a few manatees still live in and around large tracts of mangrove swamp in the south.
The island is surrounded by coral reefs, which provide shelter for sea creatures and endless hours of recreation for divers and snorkellers.
Temperatures generally vary only a few degrees around 27°C (80°F), although the heat is tempered by the nearly continuous trade winds that blow across the Atlantic. In the mountains and hills of the interior, the temperature drops with altitude to as low as 3ºC (37°F) on the mist-covered Blue Mountain Peak, the island's highest point at 2,256m (7,402ft).
Bananas: a cash crop
Kevin Cummins/Apa Publications
Much of the land is extremely fertile and produces a range of tropical fruit and vegetable crops such as yams, sweet potatoes and juicy mangoes, providing ample food for the people, as well as cash crops such as bananas, sugar and coffee. Four hundred years ago these crops brought British colonists to rule the land and African slaves to work it, forever changing the landscape and the population.
National Identity
Today's Jamaicans are a mixture of African and English people, with Spanish, Indian and a smattering of Portuguese, Jews, Chinese, Welsh and Irish. The cultures have melded together, giving rise to a fascinating national identity.
Store owner in Ocho Rios
Kevin Cummins/Apa Publications
Since gaining independence in 1962, the black majority has worked to create a country based on confidence from within, working on a principle of pride in oneself and in one's roots. This is so important for the future of the country that the national motto is 'Out of many – one people'.
Like most of the Caribbean islands, Jamaica was originally inhabited by Amerindians who had migrated from South America. The arrival of the Spanish at the end of the 15th century had a cataclysmic effect. Nowadays there is little evidence of the Castilian colonists, nor of the Amerindians they wiped out with their brutal slavery and European diseases – 160 years of Spanish rule have been blotted out by 307 years as a British colony.
Vestiges of the British colonial legacy can still be found, not least in the fact that English is Jamaica's official language: the popularity of cricket is another example. The 13 regional parishes and numerous towns were originally named after British settlements. You can find Manchester, Sheffield and Cambridge in Jamaica, to name but three. However, these British influences have, even from the earliest days of colonial rule, always been tempered and moulded to the Jamaican style. Jamaica has always had a second, 'unofficial' language developed from the early days of slavery. This creole, a mixture of English, African and Spanish words and phrases, is still evolving and is often indecipherable to the outsider. Next to town names derived from Spain and England, you'll also find names such as 'Wait Awhile' and 'Fruitful Vale', derived from the land and lifestyle of Jamaica.
The influence of the United States is now much stronger than that of Britain. Many Jamaicans head to the States for further education, and American economic influence continues to grow: the US dollar is accepted as readily as the Jamaican dollar to pay for goods.
Rastafari Movement
One of the most popular images of Jamaica is that of the Rasta. His mane of dreadlocks and colourful 'tam' hat are instantly recognisable worldwide. Rastafari live by a series of strict rules. They are nonviolent and do not eat meat. Rastas use marijuana as an integral part of their religious experience and do not cut their hair, fearing the same loss of spiritual and physical strength that the biblical Samson experienced.
Members of the Rastafari sect believe themselves to be one of the tribes of Israel, viewing the modern world as 'Babylon', synonymous with evil, and they seek peace with God, whom they believe is in all beings. Their spiritual leader is Haile Selassie, the late emperor of Ethiopia, who was God's messenger on earth – the 'Lion of Judah'.
However, Jamaica still revels in its own identity, which is now internationally recognised through such influential cultural products as the Rastafari movement and reggae music. The Rastafari movement originated in Jamaica in the 1930s and is still predominantly found here. Jamaican music – ska and, especially, reggae – has, since the 1970s, been exported and enjoyed around the world. The strong beat and earthy lyrics seem to symbolise and celebrate the character of this young and lively country. The country's latest accolade is down to Marlon James, the first Jamaican author to win the Man Booker Prize. His novel A Brief History of Seven Killings tells the fictional history of the attempted murder of Bob Marley in 1976. It won the Man Booker in 2015.
Tourist Attractions
Since independence in 1962, tourism has become a major employer and source of income and the island is renowned as one of the top destinations in the Caribbean. While some of the hotels that attracted writers and film stars in the 1950s are still going strong as luxury hideaways, Jamaica has also pioneered the all-inclusive resort catering for the mass market. The best beaches are now home to some fine hotels and large resorts.
It is tempting, and possible, to spend your entire holiday in a resort. Yet to do this is to miss the very essence of what the island is all about. Learn how to do the Jamaican handshake. Taste authentic Caribbean dishes such as ackee and saltfish (Jamaica's national dish), and aromatic hot jerk pork cooked in a pit barbecue. Savour a freshly ground cup of Blue Mountain coffee or enjoy a fine, aged Jamaican rum. Hear the dance hall and reggae music booming from a hundred cranked-up car stereos or the chorus of tiny tree frogs that begins as evening descends. Jamaica is an island with a strong personality that doesn't simply wait in the wings. It comes out to meet you.
Jamaican handshake
The traditional Jamaican handshake – with clenched fists meeting first vertically then horizontally, after which the thumbs touch briefly – signals a parting of mutual understanding and respect.
A Brief History
The earliest signs of people on Jamaica are the remains of the Taino, Amerindians descending from Arawak-speaking people who migrated from the north coast of South America. They travelled to various Caribbean islands along the entire Antillean chain, arriving in Jamaica at the beginning of the 8th century.
The Tainos left an important legacy of rock paintings and carvings in places such as Runaway Caves near Discovery Bay, and shards of pottery found at their settlements near Sevilla la Nueva and Spanish Town have added a little to our knowledge about them. Over 200 Taino sites have been identified, and it is said that when the Spanish arrived in Jamaica there were approximately 100,000 Tainos living on the island. They called Jamaica Xaymaca ('land of wood and water').
Columbus and the Arrival of Europeans
Christopher Columbus first arrived in Jamaica on 5 May 1494. He stayed for only a few days but on his fourth voyage he spent a year stranded here in 1502–3, while his ships were being repaired. However, the island was not settled by the Spanish until 1509. The year before, Columbus's son Diego had been appointed Governor of the Indies by the Spanish monarchy and he made Juan de Esquivel Governor of Jamaica.
In 1510, Esquivel created a base called Sevilla la Nueva near St Ann's Bay, from which he hoped to colonise the rest of the island. The Spanish immediately began subjugating the native Arawak-speaking population, most of whom died under the yoke of oppression and of diseases carried by the Europeans.
The site of Sevilla la Nueva proved unhealthy and mosquito-ridden, and in 1534 the Spanish founded Villa de la Vega, today known as Spanish Town. Pig breeding was the main occupation of these early settlers, but they also planted sugar cane and other crops that required large numbers of labourers. The number of Taino had already fallen dramatically, so the Spanish began to import enslaved people from Africa to work the land; the first Africans arrived in 1517.
Statue of Christopher Columbus, St Ann's Bay
Pete Bennett
The island was never fully exploited by the Spanish. They were much more interested in the gold and other treasures to be found in South America. However, they had to protect the shipping lanes in order to get their treasure home, and this meant keeping hold of as much of the Caribbean (or the 'Spanish Main', as it was then known) as possible. They fortified the more strategic islands, but Jamaica was deemed less important than Cuba or Puerto Rico and, consequently, was poorly protected.
British Rule
In 1654 Oliver Cromwell, Lord Protector of England, dispatched a British fleet to the Caribbean to break the stranglehold of the Spanish. They were repulsed at Hispaniola by a strong Spanish force and decided to take Jamaica as a consolation prize. They sailed into what is now Kingston Bay in May 1655 and sent an ultimatum to the capital. The small Spanish force considered its position and decided to retreat, heading to the north coast and sailing to Cuba. Before they left, they freed their slaves, who fled into the interior of the island.
The Spanish attempted to retake the island in 1658 at the Battle of Rio Bueno but were defeated. Other European powers also began to put pressure on the defending forces and British naval power in the area was badly stretched. Sir Thomas Modyford, the Governor of Jamaica, offered a deal to pirate ships already well established in the area: if the pirates protected British assets, then they were free to harass enemy shipping with impunity. They agreed, and Modyford authorised the pirates to act in the name of the British Crown.
The British flag flies in Kingston
Jamaican National Library
These 'privateers' were welcomed at Port Royal, the English settlement on the southern tip of Kingston harbour, and it quickly developed a reputation as the wickedest city in the world. Plunder was now legitimate business and the city was awash with money and booty from the numerous pirate raids. There was little evidence of religion or of the rule of law. Henry Morgan was chief among the pirate leaders. He and his followers conducted a successful series of bloody raids on Spanish settlements in the Caribbean, culminating in the sacking of Panama, the major city of the Spanish Main.
The Lady Pirates
The pirates who sailed the Caribbean were joined by two women, Mary Read and Anne Bonney, who were said to be as ruthless as their compatriots. They dressed in men's clothing and committed unspeakable atrocities in the name of profit. Captured by the British authorities, they were found guilty of piracy and sentenced to death, but both pleaded 'the belly'. Judges would not kill an unborn child, so both sentences were commuted to life in jail. Mary Read and her young child died of fever only a few months later, but there is no record of what happened to Anne Bonney. Her life after the trial is a mystery.
Henry Morgan
Pete Bennett
In 1670 the Spanish officially ceded Jamaica to Britain as part of the Treaty of Madrid, and the British began a systematic process of settlement, offering land to prospective settlers. They rescinded their agreement with the privateers and began to evict them from Port Royal.
Henry Morgan was offered the post of Lieutenant Governor of the island and charged with driving out his former cohorts. The erstwhile pirate thus became a policeman during the last years of his life. Morgan died in 1688 before his task was complete, but nature finished what he had started: Jamaica suffered a powerful earthquake in 1692, and Port Royal sank into the sea, taking with it many of the treasures stolen from the Spanish.
Plantations and Slavery
As the 18th century began, trade in sugar cane and spices was becoming profitable. Plantation work was labour intensive, but there were few labourers on the island. The decision was made to import a workforce from West Africa, resulting in some 600,000 enslaved Africans being transported to Jamaica over the next few decades. One in five slaves died en route and many more died of disease once on the island. On the back of this cruel system, Jamaica gradually became the biggest sugar producer in the world and a very wealthy island indeed.
Thirteen administrative parishes were created, forming the basis of government that we still see today. The Governor commissioned a representative (or custos) in each parish. Powerful land-owning families organised an Assembly to run the everyday affairs of the island, but many landowners continued to live in Britain, where they exerted tremendous influence in Parliament to protect their Jamaican interests.
Even in these early days there were slaves who fought against the tyranny of the system. The African slaves whom the Spanish had released after 1655 were known as Maroons, from the Spanish word cimarrón (which means 'wild' or 'untamed'). They made their settlements in the hills away from the British but began to attack colonists in a series of raids known as The First Maroon War. British forces suffered constant harassment at their hands and even named a part of the island 'The Land of Look Behind' in recognition of the surprise attacks they suffered. Eventually the British forced the Maroons into more isolated and remote pockets of land.
This war of attrition ended in 1739, when an agreement was reached between the two sides. The Maroons were allowed self-rule in designated areas in return for not helping escaped slaves. This agreement is at the root of Maroon self-government today. The plantation slaves also began organising revolts (the first in 1760), but their situation remained the same and they endured cruel and inhumane treatment.
During the American War of Independence, Jamaica came under threat again from other European powers, which saw Britain's problem to the north as a chance to capture its colonies in the Caribbean. Some islands were taken by the French, but Admiral Rodney saved Jamaica by defeating the French fleet at the Battle of Les Saintes in 1782. Jamaica thereafter became an island of strategic importance for the British, who based a large naval fleet at Fort Charles in Port Royal.
Rodney's riches
Following his decisive victory at the Battle of Les Saintes in 1782, Admiral George Rodney was honoured by the Crown with a barony and a pension of £2,000 a year.
Emancipation
The French Revolution in 1789 sent ripples of discontent through the Caribbean. The French peasants' cry for freedom prompted another Maroon War on Jamaica, after which many Maroons were deported to Nova Scotia. There was, however, a growing abolition movement in Britain. In 1807 Parliament made the trade in slaves illegal, but the powerful sugar lobby ensured that slavery continued on the plantations. The enslaved people were angry and dispirited. Nonconformist churches encouraged the slaves to stand up and take action against injustice. Their intervention guaranteed the popularity of these Christian denominations; Baptist and Adventist churches are still as strong today as in the early 1800s.
Celebrating freedom
Jamaican National Library
The momentum for change was growing, and in 1831 a black lay preacher named 'Daddy' Sam Sharpe led a revolt of 20,000 slaves at Montego Bay. After a campaign of destruction, the authorities assured them that slavery would be abolished. Sharpe and approximately 1,000 other slaves surrendered peacefully, only to be rounded up and publicly executed. This news was met with revulsion in Britain and eventually led to full freedom in 1838.
Unfortunately, being 'free' solved none of the problems suffered by the population. There was no economic infrastructure outside the plantation system, and power remained in the hands of a small minority of white and mixed-race individuals. Meanwhile, Asian labourers took up the work previously carried out by the enslaved Africans; their descendants can still be found on the island, particularly around Little London in the west. As a further blow to the economy, the British Parliament passed the Sugar Equalisation Act in 1846 as part of a new free-trade policy. Jamaica's protected market was effectively gone.
In October 1865 at Morant Bay, there was another uprising, led by Baptist minister Paul Bogle and George Gordon, a mixed-heritage landowner. It brought savage retribution from the authorities, and both leaders were executed, but it prompted the dissolution of the Jamaica Assembly, which was dominated by plantation owners. The island became a Crown Colony ruled directly from London, and over the next few years there were several reforms to its political and social systems.
As the sugar trade declined in importance, economic disaster loomed. Fortuitously, another crop found favour with the industrial world: Jamaica became the island of bananas. The first consignments were exported in 1866 and, within a few years, thousands of tons were being shipped to markets in the US and Britain. The boats carrying the banana crops also fostered a fledgling tourist trade. The first visitors arrived as passengers on them, spending time around Port Antonio.
Marcus Garvey – A National Hero
Born in St Ann's Bay in 1887, Marcus Mosiah Garvey made his mark as a black nationalist, instigating a 'back to Africa' movement. Garvey founded the Universal Negro Improvement Association (unia), advocating black unity and pride, and in 1916 he set up a unia office in New York. His political activities included the establishment of the Black Star Line steamship company and a newspaper, The Negro World, which became a forum for labour grievances. In 1925 Garvey was imprisoned for fraud on what are now considered false charges and later deported from the US. He died in obscurity in London in 1940, but after independence his remains were brought home to Jamaica, and he was inducted as a National Hero.
Still there was little change in conditions for the black majority, who had no economic or political power. The worldwide depression of the 1930s brought a new wave of demonstrations in Jamaica, and a number of individuals emerged to lead the people and pave the way for nationhood: Marcus Garvey called for black self-reliance; in 1938, Norman W. Manley founded the People's National Party (PNP), which found allies in the Jamaica Trades Union Congress and the National Workers Union; Manley's cousin, Sir Alexander Bustamante, formed the Industrial Trade Union and later (1944) the Jamaica Labour Party (JLP). Together, these organisations fought for local rule, which in 1944 resulted in universal voting rights for adults. At the same time, the early years of World War II brought American tourists who were no longer able to travel to Europe on holiday. Jamaica's popularity as a tourist destination was now undeniable.
Independence and Democratic Rule
In the postwar period there continued to be constitutional changes, including self-government for Jamaica in 1959. Britain hoped to create a Federation of Caribbean Islands in the region. Jamaica joined the West Indies Federation in 1958 but withdrew in 1961 following a national referendum.
Since independence in 1962, the political culture of Jamaica, which started out with such confidence and optimism, has been fraught with problems. Violence and corruption have been constant factors in the political process. From 1962 until 1972, the JLP held power. The party's broad aims were to support capitalist policies and to continue close ties with Britain and the rest of the Commonwealth.
In 1972, however, the left-wing PNP was elected with a massive majority. Michael Manley, son of Norman, led the party and pushed for policies that brought Jamaica closer to independent nonaligned countries. Manley was criticised for political links to Fidel Castro's Cuba, foreign investment dried up, wealthy Jamaicans left the island and the economy declined. The uncertain and volatile situation led to gang violence, and Jamaica seemed to be heading for civil war. In 1980 the JLP returned to power following a campaign that ended with hundreds dead. Foreign investment began to trickle back.
Veteran of the times in Kingston
Kevin Cummins/Apa Publications
Politics Today
In 1989 the PNP regained power under Michael Manley, whose policies had changed radically (he now advocated the free market). From 1993, when Manley retired due to poor health, the PNP was led by Percival Patterson, Jamaica's longest serving Prime Minister. He was succeeded in 2006 by Portia Simpson-Miller, the party's first female leader. Her term of office was short-lived, however: the PNP was narrowly defeated in the 2007 elections by the JLP, led by Bruce Golding. Simpson-Miller returned to power in 2011, and on taking up the office suggested that the country should leave the Commonwealth and become a republic. Her term included the celebration of the 50th anniversary of the country's independence.
Historical Landmarks
c.4000–1000 BC The first Amerindians arrive from the Orinoco region of South America.
1494 Christopher Columbus lands on the north coast of Jamaica, claiming it for Spain finding some 200 Taino villages.
1510 First Spanish settlement founded at Sevilla la Nueva.
1517 First boat carrying African slaves arrives at the island.
1655 British forces take the island from the Spanish. The Spanish free slaves, who head to the interior of the island.
1670 The Treaty of Madrid cedes Jamaica to England.
1692 A powerful earthquake destroys the city of Port Royal.
1739 Peace treaty with freed slaves (Maroons), which offers them self-government.
1700s The number of African slaves increases dramatically, with around 250,000 working on Jamaican plantations.
1838 Emancipation of enslaved people.
1865 Morant Bay rebellion seeks better conditions for the liberated slaves. The ringleaders are executed.
1938 Difficult economic conditions lead to the formation of the first trade unions and political parties.
1944 Universal adult suffrage is introduced.
1962 Jamaica declares independence led by the JLP.
1972 Victory at the elections for the left wing Michael Manley and PNP.
1980 The JLP returns to power, led by Edward Seaga.
1989 Michael Manley returns to power with free market policies.
1993 P.J. Patterson of the PNP becomes Prime Minister.
1997 Michael Manley dies. The PNP return for a third term.
2006 P.J. Patterson retires. The PNP elects Portia Simpson-Miller as its first female leader and Jamaica's first female Prime Minister.
2007 After 18 years of PNP government, the JLP wins the elections.
2010 74 killed in Kingston during a state of emergency to arrest a suspected gang leader.
2011 Simpson-Miller's PNP wins a snap general election.
2015 Possession of marijuana for personal use decriminalised.
Where To Go
The island of Jamaica is spectacularly beautiful, from its mountainous interior, lush forests and plentiful streams and rivers to the sandy beaches that frame the coast. You can kick back and relax on the seashore while admiring the view, or be more active and explore the wildlife at the bottom of a coral reef or at the top of a mountain peak. Wherever you go in Jamaica your senses will be bombarded by the sight of glorious landscapes, the pulsating bass tones of the music, the fragrance of brightly coloured tropical fruit and flowers, and the taste of spicy jerk meats and fish or a cold beer on a hot day.
The Blue Mountains
Photoshot
Soon come
Switch to Jamaican time. The phrase 'Soon come' means that things will happen eventually. Don't be in a hurry for anything.
In this chapter, we journey clockwise around the island, starting at the tourist capital of Montego Bay on the northwest coast. Sangster International airport is within relatively easy reach of all the main resort areas on the north and west coasts. Cruise ships also dock at the ports of Montego Bay, Falmouth, Ocho Rios and Port Antonio.
The twisting roads of the rugged interior mean that cross-island journeys to Kingston and the south coast used to take longer than expected, but a major road-building programme has recently changed that. The North Coast Highway and Highway 2000 in the south of the island allow better connections between Kingston, Montego Bay, Ocho Rios and Port Antonio. All parts of the North Coast Highway and most stretches of Highway 2000 are now complete. The Linstead to Moneague segment of Highway 2000 was opened in 2014, while the final 67 km (42 mile) leg should be ready by mid-2016.
Montego Bay
The northern coast of the island has been the major focus of tourist development on Jamaica since the 1970s. Much of the burgeoning development has occurred here, and in some places this has changed the character of the landscape. However, there's no denying that this area has just about everything needed for a perfect holiday, whether you want to do nothing but sit on a beach, dive and snorkel along the coral reefs, enjoy sports, or explore the history and culture of the island.
Fresh fruit for sale on the beach
Kevin Cummins/Apa Publications
Montego Bay, or 'MoBay' as it's called by the locals, is probably the most complete resort area in Jamaica, with its beaches, sports and shopping, along with a large number of hotels that cater to all budgets. The town is only minutes from the Sangster International airport (built as a US Air Force base during World War II), so there is no lengthy transfer to your hotel. The town is surrounded by a host of different sights and sporting facilities. Its disadvantage is a lack of attractiveness: Montego Bay is a rather soulless hodgepodge of sprawling development with no real character. But if you are here simply to have fun, you might not even notice.
'Daddy' Samuel Sharpe
'Daddy' Samuel Sharpe (1801–32) was a literate slave and a lay preacher who lived on the Belvedere Estate, south of Montego Bay. Sharpe encouraged his congregation to lay down their tools until their grievances had been addressed. The resulting slave protest started peacefully at Christmas 1831, but turned violent. It was brutally suppressed, and Sharpe was executed on The Parade, now memorialised as Sam Sharpe Square. It has been argued that the reaction of the British public to the fate of the slaves and the rebellion's leaders led to parliamentary enquiries and the eventual emancipation of Jamaican slaves. Sharpe was honoured as a National Hero in 1975.
Downtown
The resort sits on the east side of the wide bay, with the cruise port on the west. Downtown Montego Bay is located between the two. It is a jumble of loud and boisterous streets, full of people, dogs and goats breathing the fumes of hundreds of buses and cars. Vendors in makeshift shacks sell beer or cigarettes, and oil-barrel barbecues cook jerk chicken and burgers. The town centre is Sam Sharpe Square A map], previously called Charles Square and The Parade, but renamed after the hero of the 1831 slave rebellion who was hanged for his part in the uprising. In one corner of the square are the Cage, an old prison lockup built in 1806 to house drunken sailors or runaway slaves, and the Ring, the site of the once-regular slave auction. The Civic Centre was redeveloped from the ruins of the 1804 court house. Following an extensive refurbishment, it reopened in 2014 as Montego Bay Cultural Centre (Tue–Sun 10am–6pm), home to National Gallery West ([https://nationalgallerywest.wordpress.com) and National Museum West. It also has space for performing arts and houses a bistro and a gift shop.
Local art, Montego Bay
Kevin Cummins/Apa Publications
Nearby, Harbour Street Craft Market B [map] is a constant buzz of activity. This is the place to come to check out the range of local handicrafts and souvenirs available on the island. The windows and doors of over 100 wooden cabins are bedecked with printed sarongs, T-shirts and carved masks. Try your hand at haggling and you're bound to get a better price than you thought.
The Beaches
Head east to the Gloucester Avenue 'Hip Strip' for the beaches and resort life. This is the heart of the action, with some of the busiest bars, loudest music and wildest water sports on the island. As you reach Gloucester Avenue, you'll pass the remains of the Fort Montego, with its small sturdy walls and heavy cannon that guarded the bay for many years. Fort Montego Craft Park is just behind the fort.
Most beaches in MoBay are private, which means you pay a small charge to enter. They are kept neat and tidy, with water sports facilities and areas for changing and showering. The first one along the strip is Walter Fletcher Beach C [map], which is busy at weekends and is home to Aquasol Theme Park.
Further along the strip is Doctor's Cave Beach (daily 8.30am–sunset; www.doctorscavebathingclub.com), the original Montego Bay beach developed in the Edwardian era when sea bathing became a popular pastime throughout the British Empire. It became a centre for wealthy and upper-class visitors and was donated to the town in 1906 by the original owner. It is still as popular as ever and the sand is sublime, but the cave after which the beach was originally named was destroyed in 1932 during a hurricane.
Playing on the floating trampoline at Doctor's Cave Beach
Kevin Cummins/Apa Publications
Cornwall Beach (daily 8am–6pm), another private beach with perfect sand, is behind the St James shopping mall. There is a bar, watersports and beach volleyball facilities.
The cruise port, or Montego Bay Freeport, sits on an outcrop on the west side of the bay. It is a popular stop on cruise itineraries. This area is also home to the Montego Bay Yacht Club (www.mobayyachtclub.com), which hosts a number of yachting regattas through the year. You can hire boats here to take a morning or full day out at sea for sport fishing or just a relaxing jaunt.
Montego Bay Marine Park (Howard Cooke Blvd), established in 1991, covers the whole of Montego Bay from the high tide mark to a depth of 100m and includes two fish sanctuaries where fishing is prohibited. It covers an area of 15.3 sq km (6 sq miles) of reef, sea-grass and mangrove swamps, stretching from Rum Bottle Bay in the west to Tropical Beach by the airport in the east. A number of companies offer underwater tours in glass-bottomed boats or submersible craft, or you can rent snorkel or scuba gear to get a closer look yourself. These can be booked from the offices at Pier 1 (www.pieronejamaica.com), a small marina with a seafood restaurant and an entertainment centre that sits in the middle of the bay between the beaches and the cruise port.
Yachts moored at Montego Bay
Pete Bennett
South of the Bay
South of Montego Bay there are a number of attractions that make enjoyable excursions, if you want to tear yourself away from the beach or book an outing from your cruise ship.
The Barnett Estate 1 map] , with its 18th-century great house, has been owned by the Kerr-Jarrett family since 1755. Nicholas Jarrett arrived on the island in 1655, and the family was at the forefront of economic and political activities on Jamaica for many generations. They once owned almost all the land on which Montego Bay now stands. You can tour the still-operating plantation in a jitney and visit the 18th century Bellefield Great House (tours by appointment: Mon–Thu 10am–3pm; [www.bellefieldgreathouse.com), which offers 'Taste of Jamaica' tours, focusing on Jamaican cuisine.
Near the town of Anchovy, in the hills above Montego Bay, Rocklands Bird Sanctuary 2 [map] (daily 10am–sunset; not recommended for children under five) offers a fascinating close-up encounter with the birds of Jamaica. The sanctuary began almost by accident in the late 1950s when founder Lisa Salmon moved here. She loved the hummingbirds that inhabited the garden and began to feed them so that they became friendly. Sugar water for the hummingbirds and seed for finches and other birds are provided so that you can feed them.
En route for Anchovy
Kevin Cummins/Apa Publications
High in the hills 27km (17 miles) south of Montego Bay is Belvedere Estate 3 [map] (Mon–Sat). It is a working plantation producing a mixed crop of spices and fruits, but it also opens a fascinating window on plantation life during colonial times. The current owners have created an agricultural museum to demonstrate the traditional methods of crop production as well as everyday life on the plantation.
East on the Coast Road
East along the main coast road from the Montego Bay area, there is a string of luxurious resort hotels with facilities such as golf courses and equestrian centres. Two shopping malls at Ironshore and Half Moon Resort have boutiques, gift shops and fast-food outlets. A number of sightseeing attractions lie along this route, which leads to Falmouth and, eventually, to Ocho Rios.
Great Houses
Rose Hall Great House 4 map] (daily 9am–6pm; charge; [www.rosehall.com) is, perhaps, the most infamous house in Jamaica. Set high on a hill above the coast with commanding views, it was started in 1750 by George Ash and named after his wife Rose. The house was completed in 1777–80 by John Palmer, Rose's fourth husband. It was a calendar house, with 365 windows, 52 doors and 12 bedrooms. It later became the home of Annie Palmer when she married into the family; it is Annie who has given the house its fame and reputation. She was allegedly a white witch with potent voodoo powers who had murdered three husbands and an unidentified number of lovers before she herself died under mysterious circumstances. Locals, who believed that the house was haunted by her spirit, buried her nearby so that she could be reunited with her body and rest in peace. It is now believed that her behaviour could have been the result of lead poisoning, from eating her meals off lead plates.
Rose Hall interior
Kevin Cummins/Apa Publications
The house fell into ruin after emancipation. In 1965 it was bought by the Rollins family, who renovated the main building. Rich mahogany wood cut from trees from the surrounding estate was used for new floors and ceilings. The interior has been redecorated with fabrics and furniture dating from the late 1700s to Victorian times. The ballroom has a woven wall-covering that is a reproduction of an original by Philipe de la Salle, which he created for Marie Antoinette.
Greenwood Great House
Greenwood Great House
Further east, Greenwood Great House 5 map] (daily 9am–6pm; [www.greenwoodgreathouse.com) was once the property of one of the wealthiest and most powerful colonial families in Jamaica. The first Barrett family member came to the island with the invading English forces. His descendants were major landowners from the middle of the 16th century and played an important role throughout the colonial history of Jamaica, holding positions of great influence in the judiciary and administrative bodies. Another member of the family was Elizabeth Barrett, who married poet Robert Browning. She was born in England and never came to the island: the responsibility for working the plantation lands fell to her male relatives.
This house, begun in 1780, was only one of the Barrett properties in the area and was built for entertaining rather than for use as a home. It now belongs to the Betton family and has retained many original features and authentic touches. Furniture and art collected over the generations fills the house, but perhaps most fascinating is the collection of original musical instruments and machines used for entertainment before the advent of electricity. The library is the largest of any plantation house in Jamaica, with over 300 volumes, some dating back to 1697.
Just before the town of Falmouth is Falmouth Swamp Safari 6 [map] (tel: 954-3065), home to the indigenous Jamaican crocodile and other tropical creatures. This wetland area covers about 1.6 hectares (4 acres) of mangrove swamp and has been turned into a breeding centre for crocodiles. You can take a guided walking tour of the swamp to see hatchlings, juveniles and adult crocodiles. Scenes from the James Bond film Live and Let Die were filmed here.
Falmouth
Once an important port for the shipment of sugar, molasses and rum, Falmouth has many Georgian buildings dating from the 18th century. The Jamaica National Heritage Trust has declared the whole town a National Monument. It is home to the newest cruise ship port in Jamaica, which can receive the huge Oasis class ships. The cruise ship pier area has been developed as an 18th century concept town with cobbled streets, shops, boutiques, restaurants, bars and shady parks.
Excursions from Falmouth include rafting (daily 8.30am–4.30pm; www.jamaicarafting.com) on the Martha Brae River 7 [map]. The river is 48km (30 miles) long, and the 1.5-hour raft ride covers 5km (3 miles) of navigable river that meanders through the lush countryside, where you can take in the verdant river banks and the peace and quiet. Each 9m (30ft) raft is hand-built by the raft captains to carry two adults. The rafts are made of bamboo from the surrounding countryside and can be used for only four months before they have to be replaced.
Rafting on the Martha Brae River
Jamaican Tourist Board
Overlooking the Martha Brae is the Good Hope Great House 8 map] ([http://goodhopejamaica.com), a fine example of a Georgian plantation house, furnished with antiques and with a commanding view of the surrounding countryside. Tours are available around the house and estate, with its old water wheel, kiln and other sugar mill ruins. Part of the Good Hope Estate is now also a hotel and an adventure park.
Just to the east of Falmouth is a small bay that comes to life at night. Referred to by several names (including 'Glistening Waters' and 'Luminous Lagoon'), it is officially known as Oyster Bay 9 [map]. Once darkness falls, the water in the bay is filled with luminescent micro-organisms that glow when agitated. You can take an evening cruise to watch this fascinating phenomenon and dip a hand in the water to make it happen yourself. The restaurant at the edge of the water is popular for a meal after the boat trip.
Duppies and Obeah
Many Jamaicans believe in the powers of magic and the underworld. Periods of bad luck or ill health are often seen as evidence of witchcraft and spells put on individuals. 'Duppies' are spirits of the dead who come back to earth. Good or evil, they can be manipulated by those still alive for mischief or revenge; 'obeah' is the local term for this type of sorcery. Look for coloured paint around the windows of homes, intended to prevent spirits from entering.
Cockpit Country
In the highlands and hinterlands south of Falmouth is Cockpit Country ) [map], an amazing and almost impenetrable landscape of limestone plateau (or 'karst') pitted with holes and fissures that have created fantastic formations. Deep depressions and high outcrops are blanketed by layers of green vegetation and topped by a lush canopy of trees, making travel difficult and at times dangerous. It is here that the Maroon people chose to live after they had been freed by their Spanish captors in 1655.
Even today there are few roads – this really is one of the last true vestiges of wilderness in Jamaica. Because much of the area is inhospitable to human activity, it is rich in birdlife and rare plant species that have disappeared from other parts of the island. You can find hundreds of caves in the limestone fissures, and some intrepid visitors go 'spelunking', exploring their watery interiors. If you want to investigate the caves take an experienced guide and go well prepared.
Scattered Maroon villages remain, their inhabitants still making a meagre living out of the poor soil. Tours of the village of Accompong and the native Amerindian cave drawings nearby are organised by Cockpit Country Adventure Tours through the South Trelawny Environmental Agency (tel: 876-393-6584; www.stea.net). One of the best times to visit is in early January, when the Maroon people hold a major festival.
Discovery Bay and Runaway Bay
Further east along the coast lies Discovery Bay, said to be the place where Columbus landed in 1494 on his second journey from Spain. The precise location is still disputed, as some say that he landed further along the coastline. Nevertheless, a small park on the roadside at Discovery Bay stands as a tribute to his achievement. Columbus Park ! [map] is built on land donated by the Kaiser Bauxite Company, whose industrial site now dominates the bay. The Park includes an eclectic collection of objects from the history of Jamaica: old railway memorabilia, artefacts from sugar cane processing plants and a banana-tallying machine can all be found here.
The ruins at Columbus Park
Pete Bennett
The mammoth Green Grotto Caves @ map] (daily 9am–4pm; [www.greengrottocavesja.com), just outside Discovery Bay, are easily accessible and safe to explore. The system stretches up to 16km (10 miles) inland and includes Green Grotto, a vast cavern with an underground lake where stalactites are clearly reflected in the crystal clear water, and Runaway Cave. Amerindian paintings, though fading, can still be seen on the walls of the caves. Guided tours include a boat trip on the lake.
Green Grotto Caves
Jamaican Tourist Board
Runaway Bay £ [map] is the appropriately named area of coastline from where the last Spanish governor fled to Cuba as the British invaders closed in. Today Runaway Bay is one of the most popular resorts on the northern coast. A series of hotel complexes has sprung up to take advantage of the fine beaches. The diving and snorkelling opportunities along the reef wall here are said to be the best in Jamaica. Most hotels offer instruction and organised dives out to Ricky's Reef or the Canyon, two major reef areas. There are also a couple of small aircraft lying offshore (relics of drug runners who ran out of luck) that make a fascinating artificial dive site called Ganja Planes.
In the hills south of Runaway Bay is the tiny village of Nine Mile, where the singer Bob Marley was born and spent the early part of his life. Marley's body was brought here after his death and lies in the Bob Marley Mausoleum $ [map] (daily 9am–5pm), where he is buried with his prized guitar. The tombs of his mother and half-brother are also here.
A detailed carving in the Bob Marley Mausoleum
Kevin Cummins/Apa Publications
The surrounding land and the tree under which he sat as a child have been turned into a shrine to the singer, but the ambience is spoiled by the numerous 'guides' and souvenir sellers who crowd your path to the entrance. Inside the compound you'll find genuine Rastafari guides, but here, too, it is very commercial, and not to everyone's taste. The whole site is painted in the bright green, red and yellow Rastafari colours that represent nature, blood and sunshine. The mausoleum lies in a small church with other symbols of Rastafari way of life, including a photograph of Haile Selassie (their spiritual leader) and the lion of Judah depicted in a stained glass window.
St Ann's Bay
To the east is St Ann's Bay, birthplace of the black activist Marcus Garvey. His statue can be found on Main Street, outside the town library. St Ann's Bay is also the site of Sevilla la Nueva % map], the original Spanish settlement on Jamaica, founded in 1509, which sits just to the west of the modern town. It is one of the oldest populated areas on the island, where Amerindian settlements have been found dating back to AD600. The Spanish settlers built a sugar factory here before 1526 and attempted to develop the site, but the persistent fevers contracted from mosquitoes in the swamps forced them to move and create a new capital at Spanish Town in 1538. However, Sevilla la Nueva was not completely abandoned, continuing as a working plantation and rum distillery that were later developed and expanded under British rule. The most obvious remains at the site date from this time. There are vestiges of the rum distillery, cattle pens and a large pimento barbecue for roasting allspice berries. Older remains lie scattered along the shoreline and in shallow water beyond the tidal reach. A recently refurbished museum in the English great house, Seville Great House and Heritage Park (daily 9am–4pm; [www.jnht.com), displays finds from the site and hosts a historical exhibition.
Eight rivers
Although Ocho Rios translates as 'eight rivers' in Spanish, the name is thought to be a mistranslation of the Spanish 'las chorreras', meaning 'river rapids'. There are plenty of waterfalls here but not eight rivers.
Ocho Rios To Annotto Bay
Jamaica's second tourist town is a relatively recent creation. Ocho Rios began in the 1960s when a fishing village was developed with the aim of turning it into a resort. There are several large hotel complexes here and the town is also a popular destination for cruise ships.
Though not the prettiest town on the island, Ochi', as it is known, has beautiful natural attractions nearby and makes a good base for excursions around Jamaica, being within easy reach of Kingston, the Blue Mountains (for more information, click here) and the coast road that leads to Montego Bay (for more information, click here) and Port Antonio (for more information, click here). There is little left of old Ocho Rios: the scant remains of Ocho Rios Fort are probably the oldest and now lie in an industrial area, almost forgotten as the tide of progress has increasingly swept over the town.
The main waterfront area, Turtle Beach A [map], sits in front of the town centre. It is a wide arc of sand, shallow and sheltered. The beach is kept clean and there are facilities and refreshments available. You can hire a boat to take you snorkelling around the reef that runs all along the coast here, just a few hundred metres from the shore. Two massive all-inclusive resorts, the Sunset Jamaica Grande and the Riu Ocho Rios, overlook the beach.
Glass-bottomed boats by Turtle Beach
Kevin Cummins/Apa Publications
Ocho Rios is a shopping mall for cruise-ship passengers. There are a number of expensive jewellery and other duty-free shops, all with goods priced in US dollars (duty-free goods must always be paid for in foreign currency). It's a veritable treasure-trove of quality gems, gold and cigars. Take a look around the incongruous pink Taj Mahal shopping mall or Soni's Plaza in the centre of town. The latter complex, with over 100 shops, probably has the widest choice. There is also a thriving craft market behind the main beach, where you will be able to haggle for locally produced goods, from a T-shirt to a necklace of semi-precious stones. Once the business day stops, there are few bars and restaurants in the town; evening activity tends to focus on the large hotels.
Ocho Rios is surrounded by not only areas of natural beauty but also by landscaped tropical splendour.Nestled on the hillside above the town, Shaw Park Botanical Gardens B [map] (tours daily 8am–4pm; charge), admired among gardening connoisseurs, has wonderful views; it comprises 10 hectares (25 acres) of tropical plants and natural waterfalls that once formed the grounds of the Shaw Park Great House, which later became a hotel.
Dunn's River Falls
This place of fantastic natural beauty and flowing water that epitomises the Amerindian name for Jamaica, Xaymaca ('land of wood and water') is unfortunately a victim of its own popularity. Only 5km (3 miles) west of Ocho Rios, Dunn's River Falls ^ map] (daily 8.30pm–4pm, from 7am on cruise ship days; [www.dunnsriverfallsja.com) is a series of limestone cascades surrounded by overhanging vegetation that carry the water of Dunn's River almost to the sea.
Climbing Dunn's River Falls
Kevin Cummins/Apa Publications
Splashing around in the Falls
Kevin Cummins/Apa Publications
Lines of people, all holding hands, do a slightly wobbly 'conga' to the top, where everyone forgets decorum and gets wet in the pools. You'll be lucky if you have the pools to yourself to enjoy the kind of romantic experience advertised in the tourist brochures as the Falls attract thousands of visitors each year and are packed with tour parties and cruise ship visitors.
Guides are optional, but they are sure-footed and will take care of your camera until you reach the top. Don't forget to take a change of clothing and a towel. There are wooden walkways at the side of the Falls for those who don't want to get drenched.
Rivers and Plantations
The road leading south from Ocho Rios climbs out of the town and twists and turns through a narrow valley of tropical vegetation called Fern Gully & [map]. 5 km (3 miles) of giant cottonwood trees, with their tangle of thick roots, frame varieties of giant fern to create a canopy of fronds and leafy branches over the road. Insects, frogs and birds call together in a cacophony of sound. It feels so humid in the midst of the vegetation that you can imagine being on the set of a prehistoric dinosaur film. The canopy is so thick that very little light penetrates through.
Twenty minutes drive from Ocho Rios, in the parish of St Mary, is White River Valley * map]. This nature retreat offers tubing, horse riding, kayaking and hiking along forest trails and the chance to sample a sumptuous Jamaican meal ([http://chukka.com). The White River runs east from Ocho Rios and marks the boundary between St Mary Parish and St Ann Parish. There are fresh water lagoons, rivers to swim in and picnic spots, as well as a restaurant and souvenir shop.
Ride a camel at Prospect Plantation
Kevin Cummins/Apa Publications
Tubing along White River
Kevin Cummins/Apa Publications
On the main coast road east out of Ocho Rios is Prospect Plantation ( map] ([www.prospect-villas.com), which offers tours by open-air jitney, horse trails and camel-trek trails through crops of coffee, bananas, allspice, plantains, sugar cane and many other crops. The guides at the plantation will give you plenty of information about the natural and introduced flora of the island. Visitors can also visit the butterfly house and feed ostriches.
The Spanish settled the land and grew crops here in the 17th century, but the fine Great House was built by English colonists in the 18th century. The plantation was bought in 1936 by English industrialist Sir Harold Mitchell and became an important focus for diplomatic, political and social activities in Jamaica.
Many important dignitaries have visited the house, including Sir Winston Churchill, and it has become a tradition for trees to be planted to mark each special occasion. The tour passes trees planted by the Royal family of Luxembourg, US civil rights activist Andrew Young and the Duke of Edinburgh, among many others. You can plant your own tree as well. The plantation also houses a private academy for young people, the brainchild of Harold Mitchell. It promotes the ideal of good citizenship through hard work and community service. Many former academy students have gone on to achieve high ranks in the diplomatic and civil services.
A little further east is Harmony Hall , map] (Tue–Sun 10am–5.30pm; [www.harmonyhall.com), a beautiful former Methodist minister's residence built in 1886. The house, which has been home to an art gallery since 1981, has been elegantly preserved with pretty painted wood fretwork and stained shutters. The gallery on the upper floor has a collection of some of the best art and crafts in Jamaica. Original paintings from local and guest artists, ceramics and a collection of imported crafts make it a good place to find a high-quality souvenir. Beneath the gallery is an Italian restaurant. During the season there are many exhibitions and performances in the gardens of the hall.
Intuitive Art
Harmony Hall has a gallery dedicated to Intuitive Art; it includes the work of notable artists such as Evadney Cruickshank, Ras Dizzy, Deloris Anglin, and Michael Parchment.
Firefly and Annotto Bay
The coastal road continues east through the small town of Oracabessa, with its decaying iron fretwork, and on to Galina Point, the most northerly part of Jamaica. Set high on a bluff overlooking the coastline at Galina Point is Firefly ⁄ map] (daily; [www.firefly-jamaica.com), the former home of Noel Coward, dean of British theatre and cinema, and the archetypal Englishman.
Now managed as a museum by Island Outpost on behalf of the Jamaican National Trust, Coward had the house built in 1956 and lived here until his death in 1973. He is buried in the garden, at his favourite spot overlooking the sea. The house is surprisingly small and simple, with one bedroom, a tiny kitchen and a couple of social rooms. What makes it special is its position, with magnificent views of the coastline east toward Port Antonio and southeast to the peaks of the Blue Mountain range.
A sea view near Oracabessa
Kevin Cummins/Apa Publications
The parties Coward held here were legendary. Film stars such as Elizabeth Taylor, Sophia Loren and Charlie Chaplin were entertained with songs at the grand pianos that still sit in the main room. Coward valued his private life, however, and guests were never allowed to stay overnight at Firefly. They were given guest quarters at Blue Harbour on the shore, where Coward lived before building Firefly.
Nearby is the house of another famous person, an author whose fictitious protagonist has taken on an almost real persona. Goldeneye ¤ map] ([www.goldeneye.com) was home to Ian Fleming when he wrote all the James Bond novels. Fleming came to Jamaica in 1942 while serving in British Intelligence, decided to settle here, and bought the property in 1946. Although Bond was noted for his bravery and prowess, he was in fact named after a man of very different talents: Fleming took the name of his '007' hero from the author of the book Birds of the West Indies, which had been researched and written a few years earlier.
Goldeneye and its villas and beach cottages are part of the Island Outpost boutique hotel chain. Alongside is the James Bond Beach Club with changing rooms, restaurant and watersports equipment hire.
Jamaica's Best Beaches
Long Bay (Negril). Seven miles of fine, golden sand gently sloping into shallow water. Low-rise development leaves room for hundreds of palm trees.
Booby Cay (Negril). Lying just off Long Bay, this tiny island provides sand all around its rocky interior.
Doctor's Cave (Montego Bay). The original tourist beach is still as popular as ever, with lots of activities. Come and be sociable.
Lime Cay (Port Royal). Just think 'Robinson Crusoe' and you'll have the right idea. But avoid it at weekends, when it's more like Grand Central Station.
Turtle Beach (Ocho Rios). Everything is in one place, and you get a great view of cruise ships arriving and departing.
Frenchman's Cove (Port Antonio). Fine white sand in sheltered coves, with lots of tropical vegetation. Wander through the coral outcrops to find a private corner.
Long Bay (eastern tip). Just the place for long romantic walks, as rolling waves break on miles of pink sand. Not suitable for swimming because of the dangerous undertow.
Holland Bay (eastern tip). A stretch of fine white sand with not another soul in sight.
Treasure Beach (south coast). With numerous fishing boats, this dark volcanic sand beach is not just for tourists.
Inland from Firefly is Brimmer Hall Plantation ‹ [map] (tours Mon–Fri 9am–4pm), a working plantation of 809 hectares (2,000 acres) that produces a variety of crops, including bananas, coconuts and citrus fruit. It has a beautiful, single-storey great house, made (unusually) of wood and filled with an eclectic collection of furniture from the colonies of the British Empire. The tour, by jitney, shows how the plantation works, with knowledgeable staff to answer questions and give demonstrations of such skills as the correct technique for climbing coconut palms.
Island hibiscus
Kevin Cummins/Apa Publications
The main road continues to hug the north coast, but just before reaching Annotto Bay there is a turn south in the direction of Kingston. Take this route to reach Castleton Botanical Gardens › map] (daily 5.30am–6.30pm, Oct–Feb until 6pm; [www.jnht.com) some 18km (11 miles) inland. The 37 hectares (91 acres) of gardens are set on lands above the Wag Wag River, which twists through a steep and narrow valley.
The gardens were landscaped in 1862 with a large consignment of plants from Kew Gardens in London. Beautiful exotic plants from every corner of the British Empire were subsequently brought here before being transplanted to other gardens on the island. It might not be the oldest, but Castleton is regarded by many as the 'father' of tropical gardens in Jamaica thanks to its work in the propagation and distribution of new plant genera.
Port Antonio And the East
The eastern area of Jamaica is the most tropical and most beautiful part of the island. The high peaks of the Blue Mountains dominate the landscape. This is where lush rainforest mixes with plantations of coffee on the high mountain slopes and meets thousands of banana plants that blanket the coastal plain. The mountains attract moisture sweeping across the Atlantic Ocean and are thus often cloaked in heavy rain clouds that feed the forests and fill numerous streams and rivers. There are few major roads in this area. The main route follows the coastline, circumventing the mountains and leading to some of the least-visited areas of Jamaica that are totally off the tourist track.
The western approach to Port Antonio is characterised by huge groves of banana plants, which in earlier times made the town one of the richest in the Caribbean. All along the northern coast here you will see remains of the old railway line, which once linked the plantations to the port but now mostly provides a place for children to play or animals to graze. The station houses, however, still give an impression of the grandeur of the recent past. The line was closed in 1985, but with the speed at which the native plants have reclaimed the land, it might have been 100 years ago.
Port Antonio
Port Antonio dates from 1723, when the town was called 'Titchfield' after the English estate of the Duke of Portland, who was governor of Jamaica at the time. Expansion began after the 1739 peace treaty with the Maroons, who lived inland south of the site. The area proved unsuitable for sugar cane production, but in 1871 fruit shippers began to take locally grown bananas back to Boston in the United States, and the trade was an immediate success.
Errol Flynn Marina at Port Antonio
Kevin Cummins/Apa Publications
A taste for sugar cane
Kevin Cummins/Apa Publications
In its heyday, Port Antonio was the undisputed 'banana capital of the world', with an additional benefit: the banana boats brought the first tourists to Jamaica. The wealthy visitors travelled out on the empty boat and stayed in the area after the ships took their ripening cargo back to the US or England. Fine hotels catered to the visitors' every need, and the town revelled in the money brought in from abroad.
Those days are long gone, as is the booming banana market: exports from South and Central America broke the Caribbean monopoly in the 1970s. However, Port Antonio harbour still has a buzz of activity, especially in the harvesting season, as all of Jamaica's banana exports leave from here. The manual counting of the 'hands' and 'bunches' of bananas (recounted in Harry Belafonte's Banana Boat Song) was mechanised in the 1960s, but the work of loading the boats is still labour intensive. Developments at Port Antonio include the Errol Flynn Marina, designed to blend in with the town's architecture and the West Harbour's charm. It is the centre for sport fishing, and plays host to the International Blue Marlin Tournament every October, when the harbour is filled with sport-fishing boats from around the Caribbean.
At the market
Kevin Cummins/Apa Publications
Nestled against the Blue Mountains, the town has a beautiful setting. Two wide bays offer natural harbours, and tiny Navy Island sits just offshore. This was the island bought by Errol Flynn when he settled in Port Antonio in 1946; he used it as a garden extension for the large yacht he moored there. His drinking parties were legendary, and he is fondly remembered as a charming rogue. Navy Island is closed to the public but the site is slated for redevelopment.
The headland between the two main bays is called 'The Hill'; here you will find the oldest part of town. The houses of wealthy Port Antonio residents sat away from the bustle of the busy port in a grid of seven or eight streets. This area has fallen into decay, but there are still vestiges of its fine history to be seen. Ornate ironwork now rusts, wooden fretwork moulds and paint peels, yet there remains a beauty about this aging finery. Nearby on St George's Street is St George's Village, a shopping precinct designed as a quaint and quirky take on European architecture.
For a wonderful view of the whole town, take the road up to Bonnie View Plantation Hotel (closed). The twin harbours, Navy Island and the Hill, can be seen from here.
West of Port Antonio
The Rio Grande fi [map], just west of Port Antonio, is the largest river complex on Jamaica, combining a number of tributaries from the Blue Mountains. Rafts have long been a method of transport for local people, who use them to carry bananas down from the upper slopes to the port. Rafting on the Rio Grande was popularised by Errol Flynn and became a 'must' for tourists in the late 1940s – it is still popular today. The lush river valley cuts deep into the heart of the mountains, with sheltered habitats for birds and butterflies. A raft trip here is a more tropical experience than on the Martha Brae River (for more information, click here). Rafters start at Berridale and complete their cruise at Rafters' Rest (St Margaret's Bay). Stop for a snack or a hike along the way.
Rafting on the Rio Grande
Kevin Cummins/Apa Publications
16 km (10 miles) west of Port Antonio are Somerset Falls fl [map] on the Daniel River in Hope Bay. Hidden in the rainforest, the falls plunge through a narrow gorge. Somerset is an old sugar plantation now with lovely gardens on the banks of the river. There are lots of facilities with a swimming pool, river pools and falls. A concrete path to the falls takes you past the ruins of a Spanish aqueduct and Genesis Falls, before reaching Hidden Falls. Here you can get on a boat to go behind the tumbling water into the cave or swim in the pools. The restaurant and bar get lively on Sundays with reggae, dominoes and dancing. Across the road, the river runs over Likkle Portie beach, where there is a lifeguard and a restaurant serving fresh fish meals.
East of Port Antonio
The drive east from Port Antonio offers some of the prettiest views in Jamaica. A series of coral headlands covered in tropical vegetation reach out into the ocean. Beautiful private villas and a small number of fine resort hotels sit proudly on the headlands or nestle in the small bays. Frenchman's Cove ‡ map], a little further east, has a beautiful sandy and shady beach, accessed through the resort (entry charge that can include lunch on the beach; daily 9am–5pm; [www.frenchmanscove.com). Tiny bays of soft sand sheltered by cliffs and cooling vegetation provide a completely different experience from the beaches of Montego Bay. This is the area for romantic private getaways. San San gained a reputation in the days of Errol Flynn for its elegant social scene; today it is an exclusive hideaway with a fine golf course. A small faded sign points the way to Blue Lagoon, a tiny coastal inlet with a freshwater spring just offshore. The freshwater hole is said to be bottomless, although the diver-explorer Jacques Cousteau dived here and measured the depth at 61m (200ft). Beware of touts charging non-existent fees.
Frenchman's Cove
Kevin Cummins/Apa Publications
Boston is best for jerk meat
Kevin Cummins/Apa Publications
Jamaica's Eastern Tip
If you continue along the main coastal road, you'll reach Boston Bay. This small fishing town is the traditional centre of 'jerk', Jamaica's national dish that has a worldwide following. The jerk marinating technique was first developed by the Maroon people as a method of tenderising and cooking their pork. You will smell the roasting meat and aromatic wood fires as you arrive in the village. The fresh pork is cut into 'bellies' and scored to make it easier to cook and serve. It is then covered in the paste that gives jerk its name, placed on a rack over the pit fire and turned every few minutes until it is ready. The marinade is a good deal spicier than you would find in a tourist restaurant, but the meat is wonderfully tender; ask for a bite-sized sample before you buy. Roasted breadfruit with the jerk provides the perfect bland antidote to the spice.
The Maroon community, descendants of proud and tenacious slaves, still live in two isolated pockets on Jamaica. Moore Town and Cornwall Barracks, hidden behind the John Crow Mountains and reached by a road from Port Antonio, make up the nucleus of the eastern group (the western Maroon area is in Cockpit Country, south of Falmouth). The settlements here were founded in 1739 after the peace treaty with the British. Maroon people are very private, still running their own affairs and paying no land taxes to the government. Although their villages don't look very different from the other rural communities on the island, it is the Maroon attitude to life which makes these societies interesting to visit. If you wish to get to know the people, you can arrange to visit them with a guide (for more information, click here).
Maroon festival
On National Heroes Day (third Monday in October) Maroons descend upon Moore Town to honour Nanny, the founder of the town and legendary 18th-century chieftainess of the Windward Maroons.
The coastal road makes its way around the unspoiled eastern tip of Jamaica. The long journey from the major resorts means that few visitors venture this far.
Long Bay ° [map] is one of the longest and most magnificent beaches on the island, nature at its best. There is no development here for two main reasons. First, this sector of coastline is most at risk from the threat of hurricanes as they whip across the Atlantic Ocean and into the Caribbean Sea. Second, the coastal swells here are extremely dangerous, preventing swimming and water sports, although there is surfing for the intrepid. You'll find wooden fishing boats pulled up on the sands and nets hanging out to dry. The beach has fine pink sand; powerful breakers throw sea spray into the air. There are several beach bars that are good for lunch, and where you can sit and admire the dramatic view.
Farther south, near the fishing village of Manchioneal, are Reach Falls · [map] (also known as Reich Falls; Wed–Sun 8.30am–4.30pm, July–Aug daily 8.30am–6pm). There are toilets, changing facilities and concrete steps going down to the falls, but little else. The fresh clear water comes directly down from the mountains of the John Crow National Park and falls into a deep azure pool.
Reach Falls
Alamy
On the easternmost tip of Jamaica stands the isolated Morant Point Lighthouse, built in 1841. With pristine mangrove swamps and the deserted sandy beaches of Holland Bay and Mammee Bay, the landscape is truly magnificent. The land stretches out for miles.
From Morant Point, the road turns west back toward Kingston. There is little to hold the attention here although the area has seen important historical events. Port Morant, a little way west, was the place where Captain Bligh of 'The Bounty' fame first landed breadfruit on Jamaica. The famous mutiny occurred during the first journey, when he refused the crew much-needed water, keeping it instead for the precious plants. Even after all his effort, however, only one plant survived and he had to return with a second cargo. It proved to be worth the effort for Bligh, who received a reward of 1,500 guineas.
Morant Bay is the major settlement in southeast Jamaica; it played a big part in one of the turning points in the history of the island. The Morant Bay rebellion of 1865 was led by Paul Bogle and supported by George William Gordon (after whom Gordon House, the Jamaican seat of Government, is named). The uprising and the violent reaction of the British forces resulted in the destruction of many of the historic buildings in the town, which never really recovered.
Bogle
The historic court house in Morant Bay is a reconstruction of the one burnt down during the 1865 rebellion. Paul Bogle and his brother were hanged from the centre arch of the gutted building. It has not been restored since a second fire in 2007. The new courthouse is located in a modern building at 16 Church Street in St Thomas.
The Blue Mountains
Covering much of the interior of the eastern part of the island are the magnificent Blue Mountains, the highest on Jamaica. There are five major peaks ranging from John Crow Mountain at 1,753m (5,750ft) to Blue Mountain Peak at 2,256m (7,402ft). The mountains are blanketed with thick forests watered by regular tropical downpours from the heavy clouds that surround the high peaks. The blue heat haze that surrounds the mountains and gives them their name can best be seen on warm afternoons, when it is possible to see peak after peak stretching into the distance.
A number of slopes and valleys have remained untouched by man and offer a habitat for rare flora and fauna including the national bird, the streamertail hummingbird (commonly called the doctor bird), and the giant swallowtail butterfly (Papilio homerus), the largest in the Western Hemisphere. The richness of the environment around the Blue Mountains has long been recognised; the Blue Mountains and John Crow National Park º map] ([www.blueandjohncrowmountains.org) was established in 1993 to manage and protect 78,212 hectares (193,292 acres) of land being damaged by illegal loggers and slash and burn farmers.
View over the Blue Mountains
iStock
The best way to view the Blue Mountains is to drive from Buff Bay on the north coast down to Kingston on the B1 highway, although the road can be impassable after heavy rain due to landslides. Always check road conditions before you depart. The interior of the mountain range and the most beautiful parts of the parks are not accessible to vehicles: the best way to experience them is to take a guided walk. There are a variety of routes, which can last from a morning to several days. The hike to Blue Mountain Peak itself is not for the inexperienced and will take a full day; if you want to see the sunrise, start out at 2am to reach the summit in time to greet another Jamaican day (for more information, click here). Whichever option you choose, remember to take some warm clothing, because temperatures here are a few degrees lower than on the coast, even on a sunny day. When the clouds come in, it can feel quite chilly.
In addition to their tropical splendour, the Blue Mountains have slopes which are perfect for growing coffee. Blue Mountain coffee is said by aficionados to be the best in the world. The coffee plantations lie in the humid heights, at altitudes of 915–1,676m (3,000–5,000ft), where soil conditions and the slow growing process (five years from germination to harvesting) produce a fine crop with a high yield. This natural affinity between the Blue Mountains and the coffee bean is amazing, because the first plants are said to have arrived in Jamaica by accident.
Mountain coffee
Coffee beans normally take four months to develop from blossom to harvest, but in the Blue Mountains where the weather is cool, damp and cloudy, they take 10 months, resulting in a harder, larger bean. The sugars in the bean caramelise on roasting, giving the unique flavour.
In 1723 Louis XV of France sent three arabica coffee plants to the French island of Martinique, which lies farther south. A few years later the Governor of Jamaica, Sir Nicholas Lawes, imported a coffee plant and some beans from Martinique and these were the start of the Jamaican coffee industry, the most important business in this part of the island for more than 250 years. Because of the topography and the delicate nature of the plants, much of the work is still done by hand and traditional working practices have endured. Note that coffee grown below 915m (3,000ft) is called Jamaica High Mountain or Jamaica Supreme (or Low Mountain) and is not the same quality.
Kingston and Environs
With one of the largest natural harbours in the world – lying between lush green hills and the Caribbean Sea – Kingston Bay became the perfect site for one of the biggest ports in the Caribbean. Commercial success made Kingston the capital of Jamaica in 1872.
Downtown Kingston
Alamy
Now home to about a quarter of the population of Jamaica, it is a huge city. Modern 'New Kingston', with its office buildings and high rise blocks, is the administrative heart of Jamaica, with government offices, consulates and boutiques. To the northeast lie the foothills of the Blue Mountain range, where wealthy Kingstonians build houses to take advantage of the cooling breezes. The poor live on the flat, dusty plains below, where country life has simply been transplanted to the city. Goats wander the streets and people live in tin shacks with few amenities. They sit up against several other shacks that make up blocks of properties (or 'yards'). The violence and crime that have been a feature of life in Kingston over the years, centre on political and gang rivalries within these yards.
Jamaica's 'Higglers'
You'll meet these persuasive salespeople all over Jamaica, at craft markets or on the beaches. Their goal is to sell you the souvenir that you can't leave Jamaica without. However, since there are no set prices, you must engage in the intricate game of 'haggling' if you want to buy and not get ripped off.
Don't engage in haggling if you are really not interested in buying an article. It only creates bad feelings, and you may feel the sharp edge of a 'patois' tongue. A firm but honest 'No' is better than five minutes of negotiation followed by no sale. Higglers view this as disrespect for them on your part.
As a guideline, aim to start your negotiation at about half the initial price offered. Sale price should generally be about 20 percent lower than the vendor's original offer.
Downtown
Downtown Kingston, once a model of British colonial 'pomp and circumstance', is now surrounded by some of the poorest and most densely populated neighbourhoods in the city. It is not a place to wander around at night. However, during the day the area is full of office workers going about their business and petty crime is no worse than in any other capital city. Take normal precautions.
The city centre developed around the waterfront. Fruit, rum and spices were once transported from the old docks; today the harbour area has been transformed. In 1982, the Jamaica Conference Centre was built; there are also galleries and historical collections that celebrate the culture of the island. The National Gallery ¡ map] (Tue–Thu 10am–4.30pm, Fri 10am–4pm, Sat 10am–3pm, last Sun of the month 11am–4pm; [http://natgalja.org.jm), at 12 Ocean Boulevard, has a comprehensive collection of Jamaican paintings, sculpture and other art, including works from the 1920s; there are many works by Edna Manley (1900–87), one of Jamaica's foremost modern artists, wife of Norman Manley and mother of Michael Manley, both former Prime Ministers. Beside the docks is the recently refurbished Victoria Craft Market, the domain of the famous 'higglers', the assertive women who run the small stalls. The building, constructed in 1872, is a fine example of Victorian colonial architecture.
Exhibit at the National Gallery
Kevin Cummins/Apa Publications
Away from the waterfront, the streets in the city centre feature a number of historic buildings. On Duke Street you will find Headquarters House, built in 1755. The house was selected as the seat of the island legislature in 1872, when the capital was moved from Spanish Town to Kingston. It is now the base of Jamaica National Heritage Trust. Nearby Gordon House, built in 1960, is home to today's legislators; it was named after George William Gordon, leader of the Morant Bay rebellion, who became a member of the Jamaica Assembly and spoke out for the rights of the poor and oppressed.
National Heroes Park, at the north end of Duke Street, used to be a racetrack (you can still make out the shape of the circuit). All of Jamaica's national heroes are buried here with impressive monuments symbolising their lives and achievements: Sir Norman Manley and his artist wife, Edna, Sir Alexander Bustamante, George William Gordon, Sam Sharpe, Marcus Garvey and Nanny of the Maroons, while the north section is reserved for the burial of former prime ministers and other individuals who have contributed to the political and educational development of the country.
Wall of pride
Kevin Cummins/Apa Publications
The Institute of Jamaica (http://instituteofjamaica.org.jm), 10–16 East Street, was founded in 1879 to encourage research in science, art and literature in the true spirit of the Victorian age. It's home to the National Museum of Jamaica, which reopened in 2013 after renovation and expansion with the country's first ever exhibition on the Rastafari movement (http://museums-ioj.org.jm). The complex also houses the Natural History Museum (Mon–Thu 9am–4.30pm, Fri 9am–3.30pm; http://nhmj-ioj.org.jm), entrance on Tower Street, the oldest museum on the island. The herbarium has a collection of over 130,000 specimens of flowering plants, algae, fungi, lichens, mosses and ferns. The National Library (Mon–Thu 9am–5pm, Fri 9am–4pm; www.nlj.gov.jm) next door to the Institute has the largest collection of books, articles and papers on the history of the West Indies and is an important archive.
King Street is the heart of the city centre and the main shopping street. Here is William Grant Park, originally Victoria Park, in reality a small town square that was opened in 1879 with a life-sized statue of Queen Victoria at its centre. In 1977, it was renamed after the black nationalist leader. The Parade, the streets surrounding the park, once heard the marching steps of British soldiers; it was here that slaves were beaten or hanged as punishment for their 'crimes'. Now it is a hive of 'higgler' activity and the hub for bus routes around the city.
New Kingston
New Kingston is the modern commercial centre of the Corporate Area (parishes of Kingston and St Andrew), and is the hub of business, with hotels, fast food places and night spots. Near the Jamaica Pegasus hotel is Emancipation Park, popular with the lunch crowd in the day and joggers in the evening.
On the edge of the district is Devon House (mansion tours Mon–Sat 9.30am–5pm, gardens daily 9.30am–10pm, shops Mon–Sat 10am–6pm, restaurants until 10pm; www.devonhousejamaica.com), built in 1881 as a plantation house for George Steibel, the first black millionaire of Jamaica. The beautiful exterior is complemented by the fine period furniture housed inside. The mansion was renovated in 1967, 1982 and again in 2008. The gardens are a cool place to sit, and the stables and outbuildings have been converted into a lovely courtyard containing attractive shops, cafés, an ice cream parlour and a notable restaurant. Some shops and eating places are open on Sunday.
Devon House
Jamaican Tourist Board
Nearby on Hope Road are Jamaica House, containing the offices of the Prime Minister; Vale Royal, the Prime Minister's official residence; and King's House, home of the Governor General, originally the residence of the Bishop of Jamaica. None of these grand buildings is open to the public.
Tuff Gong Recording Studios used to be located at 56 Hope Road, a small compound where reggae musician, Bob Marley, lived and worked. Since his death it has been transformed into the Bob Marley Museum (guided tours Mon–Sat 9.30am–4pm; www.bobmarleymuseum.com) and managed by the Marley family to protect the memory of his life.
Marley tribute
Kevin Cummins/Apa Publications
The museum has some interesting displays, including Marley's gold records and photographs of activity at the studios. Some of his personal effects can be found in the modest bedroom where he slept.
Port Royal
A spit of land reaches out south of the city across Kingston Bay, sheltering the famous harbour. Called the Palisadoes (after the Spanish word palizada, meaning 'stockade'), this is an arid area of magnificent cacti and margins of mangrove that shelter populations of seabirds. Halfway along the narrow peninsula you'll find Norman Manley International Airport, the main airport of entry for Kingston and the eastern part of the island. At the tip of the Palisadoes is Port Royal.
When the British arrived in the late 1650s they built Fort Cromwell here; it was renamed Fort Charles following the restoration of the British monarchy in 1662. Port Royal, the town surrounding the fort, earned a reputation as the most raucous and debauched city in the Caribbean. With the help of the pirates who made the town their base, Port Royal became a rich city, with the income from sugar and rum combined with stolen Spanish treasure (for more information, click here). After the 1692 earthquake that devastated the city and buried much of its wealth, Port Royal never fully recovered. Some treasures have been salvaged (along with everyday articles such as pewter cutlery and plates); much still lies only a few feet below the waves.
Cannon at Fort Charles
Jamaican Tourist Board
Kingston replaced Port Royal as the commercial centre of the island. However, Fort Charles was rebuilt as a military and naval garrison, and it protected Jamaica and much of the English Caribbean for 250 years until yet another earthquake struck in 1907. The brick fort, home to Lord Horatio Nelson during 1779, still stands proud and 'ship-shape'. The large cannons on the battlements now guard Fort Charles Maritime Museum ™ [map] (daily 9am–4.45pm), which documents the maritime history of Jamaica. Here you can view models both of the fort and of the types of ships that sailed the Caribbean over the centuries.
Jamaica's Predatory Pest
If you travel around the island, you are bound to catch sight of a mongoose running across the road into the undergrowth. This small furry creature was introduced to Jamaica during colonial times to prey on snakes and rats, which were a danger both to the people and to the crops. The mongoose was extremely successful in ridding the island of these two problems. However, it then began to look for other things to eat. It is now considered to be the most populous and vicious pest on the island, preying on domestic chickens as well as eating the eggs and chicks of native wild birds.
Landslides and small quakes have taken their toll on sites at the Fort, and Giddy House is a perfect example of this. The small, square building once stored ordnance, but it has been left at a very precarious angle, sinking back into the sand. The Old Naval Hospital (undergoing restoration) can be found a little farther to the north; its distinctive iron supports were brought to Jamaica in 1819 and were designed to be both earthquake and hurricane proof. The hospital building now houses the National Archaeological and Historical Museum # [map], which displays a fascinating collection of finds from the sunken city of Port Royal.
Residents of the little village of Port Royal make their living from fishing. On weekends, it is popular with families from Kingston who come to enjoy the fresh air or a fried fish dinner at one of the little restaurants that spill out on to the streets. From the marina you can take a boat to Lime Cay, which lies just to the south of Port Royal. This tiny 'desert island' offers the chance to sunbathe on sandy shores or snorkel in clear water and feel a million miles away from Kingston.
Central Highlands and The South
Away from the large towns and tourist resorts, life continues in time-honoured tradition. In central and south Jamaica, numerous small settlements and family farms dot the countryside, where you'll see donkeys tethered at the roadside or trotting along the lanes carrying large baskets. A network of smaller roads that knit the villages together make travelling a real adventure: there are few signposts (and even fewer people) to point the way if you do become lost.
The south coast is peaceful and relatively undeveloped
Kevin Cummins/Apa Publications
The contrast between the landscape of the central highlands and the south coast could not be more marked. The highlands are cool, with green hills rolling through the heart of Jamaica. As you travel south, the landscape changes. Acres of grassland surround coral limestone columns and escarpments. Low-growing acacia trees replace tropical vegetation, with the landscape characterised much more by prairie than by palm trees. The southern-most margins of the island – away from the pressure of human development – are a haven for wildlife.
Spanish Town
The Spanish settlers in 16th-century Jamaica, having tired of the disease-ridden Sevilla la Nueva in the north, looked for a new site for their capital city. They chose the flatlands around the Rio Cobre and, in 1534, established Villa de la Vega, later called St Jago de la Vega. The British captured Jamaica in 1655 and henceforth gave the settlement the rather unimaginative name 'Spanish Town'. As capital of a wealthy colony, Spanish Town (www.spanishtownjamaica.com) had its fair share of fine buildings that included courthouses, administrative offices and official residences. However, the capital was moved in 1872 to Kingston, the commercial heart of Jamaica, and a malaise enveloped Spanish Town from which it never recovered.
Elegant colonial buildings in Spanish Town
Chris Wilkie - Fotolia
The elegant Georgian buildings along the Parade, have fallen into disrepair. The most striking building in the Parade is the white stone edifice that houses the Rodney Memorial ¢ [map], constructed at great expense in gratitude after Admiral Rodney's fleet saved Jamaica by defeating the French at the Battle of Les Saintes in 1782. One wing houses the Jamaica Archives and Records Office, which preserves original documents from throughout the island's history.
On the west side of the square is Old King's House (built in 1762), which was the official residence of the British governor; it was here that the proclamation of emancipation was issued in 1838. It was a fine building was destroyed by fire in 1925. Only the façade is original; the building behind it is modern.
The Jamaican People's Museum of Craft and Technology (Mon–Thu 9.30am–4.30pm, Fri until 3.30pm), a branch of the National Museum of Jamaica, housed in a reconstructed corner of the house, features a model of how the building looked before the fire. The other two sides of this fine Georgian square are taken up by the ruins of the old Court House and what was the House of Assembly, now local government offices.
The Hellshire Hills, south of Spanish Town, come as a surprise to those who think that the tropics can only be lush and green. The landscape here is underscored by limestone and receives fewer than 760mm (30in) of rain per year.
Hellshire
Kevin Cummins/Apa Publications
There is little soil to support plants, resulting in a desert-like landscape of cactus and low scrub trees. The area has become the last haven for many of the native but almost extinct plants, animals and birds of Jamaica. Here you'll find the last few Jamaican iguanas and yellow snakes.
The string of white sand beaches along the coast are popular destinations with Kingstonians on weekends.
Mandeville
Mandeville sits to the west of Spanish Town in the Don Figuero Mountains. Its cool air and pretty setting made it a favourite retreat for colonial families right up to the end of British rule in Jamaica in 1962. They came to spend their weekends here, away from the hot and humid atmosphere of Kingston. Today, it is a favourite place for wealthy Jamaican families for very much the same reason. The town was laid out in 1816 and named after Lord Mandeville, the eldest son of the Duke of Manchester, after whom Manchester Parish was named. The English modelled the town and its buildings on those of their homeland, and one can imagine the village greens, tennis and golf clubs, and grassy verges taken from a typical London suburb.
Cool Manchester
Manchester's cool climate (20°C/70°F in the summer and 16°C/60°F in the winter) appeals to Jamaicans returning to the island after decades living abroad in the UK and North America.
There are a number of interesting attractions lying to the west of Mandeville. Appleton Distillery ∞ [map] (tours Mon–Sat) is situated in rolling hills just to the south of Cockpit Country. Sugar cane was brought to Jamaica by the Spanish in the early 16th century. Much of the crop was exported, but it was treated before being shipped and a by-product of the treatment was molasses, used as a basis for making rum.
Many major sugar factories had a distillery on site, usually producing alcohol for local consumption. Appleton Distillery, in operation since 1749, produces 42,000 litres (74,000 pints) of rum every day. Much of this is 'overproof' rum, the basis of intoxicating rum punches, but the finest rum is aged in casks for up to 30 years to produce a spirit comparable to brandy or cognac.
The distillery offers a tour of the rum plant to learn about its historical production and gives the visitor an opportunity to taste and buy.
In some parts of the island, bamboo was planted along the roadside to provide shelter for people travelling in the heat of the day. The groves of bamboo also created places where slaves would congregate without being seen by their masters. Much of the bamboo has since decayed or been dug up, but Bamboo Avenue § [map], the one remaining section, can be found on the main A2 road between Mandeville and Black River. It is a 4km (2.5-mile) tunnel of bamboo surrounded by sugar cane, with somnolent grazing cattle tethered along its length.
Nearby are YS Falls ¶ map] (Tue–Sun 9.30am–3.30pm; [www.ysfalls.com), found on a working thoroughbred horse stud and cattle ranch that dates from 1684. The water cascades 50m (164ft) over seven tiered falls and has formed two large pools and a small cave system at the base of the second drop.
Cascading falls at YS Falls
Kevin Cummins/Apa Publications
The falls are surrounded by mature native forests and vibrant tropical flowers, but an area of grassy lawn has been created for sunbathing and picnicking. From the ticket office you get a jinty tractor ride to the falls complex, where there are changing rooms, toilet facilities, children's playground, bar, grill and gift shop. You can swim in some of the natural pools, and lifeguards on site will tell you which ones are safe. For more of an adrenaline rush, there is a canopy zip line from the top of the falls to the bottom and river tubing rides, both at extra cost.
It is said that the falls got their name from the initials of the two original landowners, John Yates and Colonel Richard Scott. The cattle and sugar barrels exported from the plantation had these initials branded onto them.
Black River
Once a major port on the south coast, Black River is now a small, sleepy town on the banks of the river from which it took its name. Its industry was the export of red logwood and the dyes of indigo and Prussian blue, which were extremely valuable in Britain. There is still some fine Georgian architecture here, but most visitors come to see the Great Morass Mangrove Swamp • [map], also called Lower Morass. This area, which should not be confused with the Great Morass near Negril, is about 32,375 hectares (80,000 acres) of freshwater and tidal wetlands. The Mangrove Swamp and rush beds are an important habitat for many species of birds and fish, as well as home to a small population of Jamaican crocodiles. Smaller than the Florida species and said to be more docile, they grow to 6m (20ft) in length and can live to an age of 100 years.
Take a tour down Black River
Kevin Cummins/Apa Publications
A Black River man takes a break
Kevin Cummins/Apa Publications
The Black River, at 71km (44 miles), is the longest in Jamaica; it was an arterial route used to transport rum and lumber from the inland plantations. It still provides a living for many families, either from fishing or from the harvesting of bullrushes for basket making. Tours on the river and into the Great Morass start from Black River town. The route takes you into the Mangrove Alley, said to be the quietest place in Jamaica, where you can search out the basking reptiles and native birds that call this place home. Roots, which look like cathedral organ pipes, drop from the higher trees. Your guide will turn off the engine and an eerie silence will envelop the boat. Choose a tour company according to your interests; some guides concentrate on seeing crocodiles while others give a more rounded wildlife tour.
Treasure Beach
The southern coastline of Jamaica has so far resisted the pressure from big developers, partly because it has few main roads. Those who do venture here are rewarded with beautiful scenery and friendly people.
Treasure Beach ª [map] is the only resort area to speak of, with just a handful of hotels stretching across three sandy bays and private coves, well-suited for snorkelling and swimming. There are also cottages for rent, which make this an idyllic spot for walkers and visitors who prefer a quiet getaway to the busy all-inclusive resorts on the north coast.
The pool at Jake's, Treasure Beach
Kevin Cummins/Apa Publications
The local population of St Elizabeth Parish still makes a living from fishing and their wooden boats rest high on the dark, volcanic sand. You'll be able to spend the day relaxing without being hassled. If you want to buy a souvenir, just hail the mobile shop that drives slowly along the road looking for customers. There's very little to do here but chill out.
East of Treasure Beach are some beautiful strips of coast. On one is the Little Ochi seafood restaurant, a simple affair on the beach in the fishing village of Alligator Pond. The government-owned Alligator Hole (also known as Canoe Valley Wetland) on the eastern side of Long Bay has been awarded official status for protection of the last three remaining manatee on Jamaica. Watch out for them being fed by the local conservationists. This pristine region of freshwater and saltwater swamps, edged with limestone cliffs, offers a refuge to this gentle creature as well as to birds and land crabs.
Negril and the West
This area, the furthest from Kingston, lagged behind other parts of Jamaica in modern development, protected from the commercial activity of the east by the limestone landscape of Cockpit Country, making travel and communication difficult. It was an outpost of pirate activity in the 17th and 18th centuries. Today, Negril is at the forefront of the tourist industry.
Heading west from Montego Bay, the main road hugs the coastline. Just a little way out of town is Tryall Estate (www.tryallclub.com). The Georgian Great House has been used as a guest house since the 1930s to supplement falling income. The old plantation has now been transformed into the first (and some say the best) villa resort on the island. The manicured greens of the golf course can be seen on both sides of the main coast road, along with the plantation's water wheel.
On the beach in Negril
Kevin Cummins/Apa Publications
Negril
A pirate hideaway in the 1600s, Negril was rediscovered in the 1960s by the 'children of love' and others looking for an alternative lifestyle. It is now popular with 'spring breakers', US students seeking a party and a good time. Jamaicans say that Negril isn't a place – it's 'a state of mind' where almost anything goes. There are few hippies left today, but the pleasures are still pretty earthy: it's not unusual to see topless sunbathers or catch a faint whiff of 'aromatic' smoke. You're also more likely to see true Rastafari here, along with others who simply enjoy living the image of the religion without abiding by its strict rules. Dreadlocks and tams (colourful knitted hats) are everywhere, along with the passing salutations. Located on the western tip of Jamaica, Negril is also one of the best places in the world for watching the sun go down.
Aloe Vera
As you lie on the beach, you are sure to be offered an aloe vera massage by a passing 'higgler'. You will be told that it will help you develop a golden tan, but be aware that aloe – although an excellent treatment for sunburn – offers no sun-protection whatsoever and your skin will likely burn. Make sure that you use a product with a suitably high SPF while you sunbathe.
To the east, running north, is Long Bay, and Seven Mile Beach q [map], a vast expanse of fantastic sandy beach, while to the west, heading south, is West End, with coral cliffs that drop directly into the clear blue ocean.
Long Bay is 11km (7 miles) of sublime fine sand, gentle azure water and cooling palm trees. It is one of the best beaches in the Caribbean, parts of which are public. Several large resort hotels have been built here but there are also small and intimate, family-run hotels if you want a more personal touch.
Vendors and hair braiders can be found in pink booths along the beach, but any number of 'unofficial' ladies who braid can be found at the cafés and bars.
Paddling at Long Bay
Jamaican Tourist Board
Across Norman Manley Boulevard from the beach is Kool Runnings Water Park w map] (Tue–Sun end of May–Aug; [www.koolrunnings.com), an adventure park with action-packed attractions for all the family, including water slides of all shapes and sizes. At the northern end is the Anancy Village for dry land activities such as go-karting, a bungee trampoline, carousel rides, restaurants and bars.
At the top of Long Bay is Booby Cay, a small island just a short distance offshore. The tree-topped rock surrounded by golden sandy beaches is the archetypal 'desert island' – a great place for snorkelling, sunbathing, or picnics. You can rent a canoe to get there under your own steam or take a leisurely ride in one of the many small ferry boats departing from Long Bay.
The coral cliffs of West End provide a total contrast to Long Bay. Diving and snorkelling are the things to do here among the rocks and caves, and the shimmering waters house some wonderful sea life. West End is also the place to take in the sunset. Tourists flock to Barney's Hummingbird Garden (daily 7.30am–6.30pm, www.barneyshummingbirdgardenjamaica.com), a little piece of blooming paradise full of these colourful birds, and almost everyone heads to Rick's Café e map], (for more information, [click here) perched on the cliff top, to have a drink and set up the camera. While you're waiting, you'll be entertained by the divers who launch themselves from the tops of tiny perches into the azure sea some 9m (30ft) below. They are often joined by courageous tourists, who usually get more applause than the professionals.
Cliff jumping at Rick's Cafe
Kevin Cummins/Apa Publications
Alternatively, a sunset cruise on a catamaran will transport you effortlessly to West End, and you can lie offshore away from the crowds with your rum punch. The cliff road ends at the Victorian-era Negril Lighthouse r [map], which still protects ships passing this rocky promontory.
To the east of Negril is the Great Morass, a wetland area covering around 2,400 hectares (5,900 acres). The wetland and the Royal Palm Reserve t [map] preserves one of the largest swathes of Royal Palm – Jamaica's national plant – left on the island. Numbers have dwindled due to the rapid tourist development, but the palms here are now protected. The wetlands support a large number of birds and land crabs. There have been attempts to drain the wetlands, but this damaged not only the Great Morass but also areas of the coral reef offshore. Both are now officially protected.
To the South
The road along the coastline to the south travels through busy agricultural towns and fishing villages mostly untouched by tourism. The first settlement is Little London, which has a relatively large Indian population that provides the markets with much of Jamaica's fresh produce. Next is Savanna-la-Mar, the capital of Westmoreland Parish, where Mannings School (built in 1738) retains its original, brightly coloured, colonial-style wooden buildings, beautifully preserved – a perfect environment for the children in their smart uniforms. Nearby at Ferris Cross is Paradise Park, a working cattle ranch where you can go horseriding through the grounds and on the beach, but other facilities are closed and the site is up for redevelopment.
South of Savanna-la-Mar the road hugs the coast, here narrow beaches brim with faded wooden pirogue canoes and other boats. This was traditionally one of Jamaica's prime fishing areas, and around Bluefields you'd see local men carrying their catch home. A marine sanctuary in Bluefields Bay was established in 2009 in order to conserve stocks and prevent overfishing, which had posed a serious threat up until then.
Beyond here, at Brighton, is the Blue Hole Mineral Spring y map] (Mon–Thu 9am–11pm, Fri–Sun 9am–midnight, off-season shorter hours; [www.blueholejamaica.com), where there is a 10.5m (35ft) deep blue hole you can swim in and a spring-fed swimming pool.
Nearby at Belmont, is the Peter Tosh Mausoleum u [map]. Tosh, a reggae musician and guitarist in The Wailers, was a committed Rastafari before his murder in 1987.
What To Do
Jamaica cannot claim to have the very best beaches, reefs, or sport fishing in the Caribbean. However, it is indeed one of the best 'all-around' islands in the region, offering a wide range of opportunities for a variety of activities. Under the warm island sun you can enjoy water sports or you can just relax on the sand doing nothing at all. Nightlife includes an abundance of reggae music and dancing, and if you're here at the right time of year you can attend some of the world's biggest music festivals. For those in shopping mode, the choices are many: woodcarvings, colourful clothing, coffee and (of course) rum.
Jewellery on sale at a beach shop, Runaway Beach
Kevin Cummins/Apa Publications
Sports And Outdoor activities
Beaches and Water Sports
Spending the day on the beach taking in the sun is one of the primary reasons tourists visit Jamaica. Every resort area has its own famous beaches with their own particular beauty. Many of the beaches are private, meaning that you must pay admission, but they are well maintained and offer lots of facilities. Negril has the great expanse of Seven Mile Beach on Long Bay and the small 'desert island' of Booby Cay, while Montego Bay has the shorter expanses of Doctor's Cave Beach and Cornwall Beach. In Ocho Rios you will find Turtle Beach, where you can watch the cruise boats docking at the pontoon in the bay. Port Antonio beaches are now the domains of the fine hotels on San San Bay, Frenchman's Cove and Dragon Bay, all of which are tiny coves protected by rocky tropical outcrops. Long Bay on the eastern coast is wonderful because it is remote and uncrowded, but it is not suitable for swimming because of the strong currents. Treasure Beach on the south coast has dark volcanic sand beaches that are home to colourful fishing boats.
At the major resorts, beaches are kept clean and facilities for a range of water sports are readily available. Jet skiing is popular in the sheltered waters near the beaches. If you are adventurous, you can go parasailing, with a boat pulling you along above the beach and waterfront. This is particularly exciting along Long Bay at Negril. A couple of bars have even installed large trampolines offshore, where you can swim out and bounce above the water.
If you do intend to take part in any sporting activity, make sure to check that your travel insurance policy specifically covers it. Some policies have clauses that exclude certain sports.
Snorkelling
Jamaica is particularly good for snorkelling, with many reefs and rocky promontories to explore very close to the shore. There are also a number of shallow areas between reefs that offer a fascinating view of various types of sea life. Beautiful tropical fish in iridescent blues and greens search through the coral for food, and in deeper waters you can spot bigger fish such as rays and nurse sharks. All the major resorts have small boats offering trips to offshore sites if you want to snorkel in deeper water.
Snorkelling in Negril
Pete Bennett
The West End at Negril is ideal for snorkelling. The coral cliffs drop down into a clear azure sea, and there are hundreds of caves and canyons to explore. Montego Bay has the Marine Park with a range of environments. At Doctor's Cave Beach you can snorkel in an area where warm spring water meets the sea, or join a guide to go further out to the Coyoba, Seaworld or Royal reefs where the fish are larger and more varied. Further east, Runaway Bay has fine reefs running parallel to the line of hotels along the beach. Ocho Rios has a wonderful shallow reef running east from Turtle Beach for safe snorkelling.
Jamaica's Top Dive Sites
Negril. Pete Wreck is an old submerged tug boat. Throne Room is a huge cavern with yellow sponges. Sharks' Reef is home to nurse sharks, while you can see eels at Rock Cliff Reef.
Montego Bay. The Marine Park contains underwater walls of coral. Airport Reef and Widowmaker's Cave are two of the most famous sites.
Runaway Bay. Ricky's Reef at a depth of 30m (90ft) is covered in gorgonians and lettuce coral, while Pocket's Reef is a wall dive covered in gorgonians, black coral and sponges. Ganja Planes has the wreckage of aircraft that crashed and are now being colonised by sea creatures.
Ocho Rios. The reef wall drops over 900m (nearly 3,000ft) but comes close to shore, offering nearby dives with a variety of fish and other aquatic life. A deliberately-sunk former minesweeper, Kathryn, is now home to a range of marine life.
Kingston. To dive the sunken city of Port Royal you must obtain special permission. For information contact a local dive operator.
Diving
Much of the northern coast of Jamaica is fringed by areas of deep reef wall that make diving a pleasure. Although some sections of reef have been damaged in recent years, there are still many areas with a wide range of fish and other marine creatures to see. Most of the major resort areas offer diving opportunities and certified training facilities for those who want to learn how to dive. Your hotel may offer certified instruction or guided dives. For more information contact the Jamaica Tourist Board (for more information, click here) for a selection of approved and certified dive operators.
Boat Charters, River Tours and Rafting
If you don't want to get into the water but you'd still like to see aquatic life on the reef, then take a glass-bottomed boat trip. There are a number of companies in all the major resorts. At Negril and Ocho Rios, the boats tie up along the main beach; you can negotiate a price while you sunbathe. The boats in Montego Bay all dock at the same place, so you can compare prices and facilities. Pier 1 has a range of options, from small boats to large semi-submersible craft that will take you under the water in complete comfort. If you are not a confident swimmer, a boat is probably the best way to enter this very different aquatic world.
Visitors can explore Jamaica's inland waters by raft on the Martha Brae River (for more information, click here) or Rio Grande (for more information, click here). Take a trip down the Black River to experience the Great Morass (for more information, click here; for tours contact Irie Safari, tel: 965-2211; http://lostriverkayak.com) and see crocodiles in action, or you can simply get wet and picnic at YS Falls (for more information, click here; tel: 997-6360; http://ysfalls.com), near Mandeville.
Take a trip on a glass-bottom boat from Buccaneer Bay
Kevin Cummins/Apa Publications
Crocodile at Black River
Pete Bennett
Sport Fishing
Sport fishing is also a popular activity. Port Antonio hosts a major international angling tournament each October. Blue-and-white marlin are the prized catch: the waters around Jamaica are especially rich in these magnificent fish and this is the oldest marlin-fishing tournament in the Caribbean. Other fish are plentiful, but the bounty of Jamaican waters is being threatened by overfishing. Respect efforts that are under way to create marine sanctuaries around the island. You will find sport-fishing boats for hire at the marinas in the major resorts: Bay Pointe at Montego Bay, the main beach in Ocho Rios, Morgan's Harbour Hotel at Port Royal in Kingston and Errol Flynn Marina at Port Antonio. You can hire a boat with equipment and crew by the day or half day.
Other Activities
Golf
Jamaica has an excellent range of golf courses, from small nine-hole to championship 18-hole courses. Several important tournaments take place on the island during the year, where you can watch international players from the PGA and the LPGA compete.
Montego Bay has fine courses, professionally designed and maintained in peak condition. The most famous course is at the Tryall Club (www.tryallclub.com), west of Montego Bay, where the greens caress the undulating coastal slopes. To the east of Montego Bay, where the coastal plain is flat and wide and an ideal landscape for golf, there are several large hotels that have courses. The four best are the Half Moon Golf Club (www.halfmoongolf.com), the White Witch course at Ritz Carlton Rose Hall (www.whitewitchgolf.com), the Cinnamon Hill course at the Hilton Rose Hall Resort (www.rosehallresort.com) and Spa and Ironshore Golf and Country Club, all created by internationally acclaimed designers and offering a challenge for all ability levels. These courses are open to the public.
Golf at Sandals, Ocho Rios
Kevin Cummins/Apa Publications
Walking, Hiking and Cycling
Jamaica is a perfect island for walking, hiking or mountain biking, and more and more visitors are looking to get off the beaten track, at least for part of their holiday.
A guide is recommended for a trek to the Blue Mountain Peak or a hike into the Cockpit Country. Several companies organise tours that can be tailored to your needs. The Southern Trelawny Environmental Association (STEA) provides local guides for Cockpit Country tours (tel: 393-6584; www.stea.net). Sun Venture Tours run hiking, caving, cycling, safari, sightseeing, birdwatching and adventure tours island-wide (tel: 960-6685; www.sunventuretours.com). Strawberry Hill Hotel, which is located in Blue Mountains (www.strawberryhillhotel.com), has several guided hikes of different difficulty levels. For mountain biking on footpaths and goat trails, check Blue Mountain Bicycle Tours (tel: 974-7075; www.bmtoursja.com). Blue Mountain Bicycle Tours also organises guided cycling trips down the 460m (1,500ft) drop from Murphy Hill to Dunn's River Falls (see also Bicycle Rental, click here).
For news and information about competitive cycling meets, charity rides and racing, contact the Jamaica Cycling Federation (www.jamaicacycling.com).
The annual Bikeathon Jamaica Challenge is held in May, organised by the Rotary Club of Montego Bay East: the route runs over a six loop, 74km (47-mile) course in Ironshore, Montego Bay (Rose Hall area), with an 18.5km (11.5-mile) recreational race, a kids' race, and a 5km (3 mile) run.
See the island on horseback
Chukka Caribbean Adventures
Horse Riding
There are several places in Jamaica to get in the saddle. Try the facilities at the Rocky Point Stables at the Half Moon Hotel (tel: 953-2286; www.horsebackridingjamaica.com); or Braco Stables, Duncans, Trelawny, 15 minutes drive from Falmouth (tel: 954-0185; www.bracostables.com), which offers a variety of rides and tours.
Spectator Sports
Spectator sports tend to be seasonal. Depending on the time of year, you can attend a range of competitive events.
Cricket
The professional season in Jamaica runs from January to August each year (Jamaica Cricket Association; www.cricketjamaica.org) and international matches are usually played at Sabina Park, Kingston. You might also come across a local game in almost any village. The English introduced cricket to the island, but Jamaican players and spectators bow to nobody in their obvious enthusiasm for the game.
Cricket on the green
Pete Bennett
Soccer
Soccer is hugely popular on Jamaica. The season runs from September to May; the premier league has 12 sides (www.premierleaguejamaica.com) and there are 13 parish leagues. The national squad is known as the Reggae Boyz (www.thereggaeboyz.com). Jamaica last qualified for the World Cup in 1998.
Horse Racing
There is a track at Caymanas Park near Kingston (www.caymanasracetrack.com). Betting is in Jamaican dollars only.
Polo
Matches are played at several places but you can watch an international match at Kingston Polo Club , St Ann Polo Club, Drax Hall (www.stannspoloclub.com) and at Chukka Cove near St Ann's Bay in the north. The polo season runs from January to early August. The Jamaica Polo Association is based at St Ann Polo Club, Drax Hall, St Ann, near Ocho Rios. There is also the Kingston Polo Club, Caymanas Estates, St Catherine, and the Chukka Blue Polo Club, Sandy, Bay, Hanover, near Montego Bay.
Nightlife and Entertainment
Jamaicans love to listen (and dance) to local music. Reggae has been a huge influence on pop music throughout the world. Jamaica has one of the world's most intense grassroots music traditions with a competitive, lucrative recording industry.
Having a ball in Margaritaville, Montego Bay
Kevin Cummins/Apa Publications
You'll find live music in bars and restaurants every night. These will be advertised in the free tourist magazines in your hotel, or out on the street booming from speakers on tops of cars. Negril, Ocho Rios and Montego Bay all have nightclubs that stay open very late.
Many hotels have Jamaican nights where you can watch a dance show and do some dancing yourself. These traditional evenings often feature the rhythms of the wider Caribbean, such as calypso (Trinidad) and merengue (Dominican Republic).
Dance to some traditional beats
Jamaican Tourist Board
The island's biggest music festival is 'Reggae Sumfest' (www.reggaesumfest.com), which is held at a variety of venues in Montego Bay each July and features local and international artists. Ocho Rios also has an annual jazz festival in June (www.ochoriosjazz.com). For a full calendar see www.visitjamaica.com.
Shopping
One thing that you notice about Jamaica is that many shops come to you. You won't be able to walk down the street without someone approaching you with crafts and other commodities. Buying from the street traders means there is no set price, and some people feel uncomfortable about haggling. Bargaining is supposed to be an enjoyable interaction, and nobody can make you buy something that you don't want (for more information, click here).
The major resort towns all have duty-free shopping centres with a range of jewellery, perfume, leather goods and other quality products from around the world. Some of these items can be purchased with savings of up to 30 percent on prices back home, but not everything offers such good value.
Arts and Crafts
Wood carving. The Jamaican people are highly skilled in the art of carving wood. It is one aspect of communal pride that has carried on since colonial times. Woodcarvings are a major souvenir product, and there is a huge range from fine carved pieces to objects in the rough 'naive' style.
You will see natural wood and also a range of colourful productions in the red, yellow and green Rasta colours. Different types of wood have different weights and different finishes. Some of the pieces are extremely lightweight, but the dark lignum vitae wood is heavy and has a beautiful finish when carved. Don't buy articles if the wood still looks green: it has not been allowed to season properly and will split as it dries.
Jewellery. There is also an amazing range of jewellery made from local products and semiprecious stones, but you should be aware that some of the materials used are from protected species. Both tortoiseshell and coral are still on sale. Don't buy them. Not only is it illegal to import these articles back into your home country, but it encourages traders to take more of these endangered living creatures from the sea. Some traders will tell you that the coral jewellery or tortoiseshell they are selling was not taken from the sea but was washed up on the beaches; this is just a sales ploy.
Basket weaving. You will also find a wide variety of basketware made locally from the rushes that can be found in huge beds all around the island. The dried-rush baskets are still used in many households today, and they make a very practical souvenir of your visit to Jamaica.
Art and ceramics. If you want to spend a bit more money on handcrafted goods, there are a number of galleries around the island where you can buy paintings and ceramics by some of the leading artists in Jamaica and the wider Caribbean. Harmony Hall at Ocho Rios is one, and the Half Moon Shopping Centre (just east of Montego Bay) also has a gallery. To see crafts being made, tour the studios of the Wassi Art Pottery Factory, near Ocho Rios.
Clothing
Cool clothing remains a popular choice for shoppers, and Jamaica offers a wide range from designer wear in the boutiques of Kingston to the practical batik sarongs and T-shirts sold in beach stalls. Look out for the 'Reggae to Wear' range.
Coffee
Blue Mountain coffee can be bought and taken home in a number of forms. The roasted beans are sold in small sacks or in vacuum-packed foil containers. The beans can also be ground and then packed in tins or foil packs. Presentation packs (pretty printed bags) add an attractive exterior to the delicious contents. You can buy direct from the growers after a farm tour and a tasting. Worth a visit is the Old Tavern Coffee Estate (tel: 924-2785) run by the Twyman family in Green Hills, Portland, where an informative tour is provided. Remember that High Mountain coffee is not of the same quality as Blue Mountain coffee.
Blue Mountain coffee
Kevin Cummins/Apa Publications
Rum
The drink that sustained a thousand pirates and generations of local people, Jamaican rum is said to be the best in the Caribbean – although other islands may beg to differ. Try before you buy. Appleton Distillery (tel: 963-9216) in St Elizabeth offers a free tasting session as part of its tour, including some mixed-rum drinks that are less alcoholic but equally delicious (for more information, click here). All these products are available throughout the island and at duty-free shops in the airport.
Jamaican rum packs a punch
Kevin Cummins/Apa Publications
Cigars
For over 40 years, Jamaica has had a small-scale industry that produces a range of well-regarded cigars. These can be bought duty-free to take home with you. However, Cuban cigars are also a major business here. Jamaica is only 145km (90 miles) from the south coast of Cuba and imports a full range of what are reputed to be the finest cigars in the world, though they cannot be brought legally back to the USA.
Children's Jamaica
Jamaica is an ideal island for children of all ages. Kids can play for hours at the beach building sandcastles, swimming, or simply splashing in the water. Seven Mile Beach, Long Bay at Negril is perfect for young children, but all the major resorts have clean, safe beaches with good facilities. Older children will enjoy snorkelling, scuba diving or taking a ride on a glass-bottomed boat. Take a trip underwater in a semi-submersible boat at Montego Bay and your children will be captivated by the sea life that lies so close to the shore. Also at Montego Bay is the Aquasol Theme Park offering a whole host of watersports activities.
Negril is the perfect place for children
Kevin Cummins/Apa Publications
Adventurous kids will enjoy a cruise up Black River into the Great Morass Mangrove Swamp to meet crocodiles, which come so close you can almost shake hands with them. At Dunn's River Falls there's excitement for children and adults alike, plus lots of water activities. For a more relaxing kind of fun try floating down the Martha Brae River on a raft.
Most large hotel complexes have children's clubs where kids can spend the whole day enjoying activities and excursions. Conversely, some hotels on Jamaica operate on an 'adults-only' policy.
Cover every inch of young skin with a high factor sun cream, limit kids' time in the sun for the first few days of your holiday and always keep them out of the midday sun. Also make sure that they are well supervised whenever they are near the water.
Festivals and Events
Exact dates vary. If you want to attend a particular event, check with the Jamaica Tourist Board (www.visitjamaica.com).
January Rebel Salute Music Festival; Negril Sprint Triathlon (Long Bay); Air Jamaica Jazz & Blues Festival (Montego Bay).
6 January Accompong Maroon Festival.
February Pineapple Cup Yacht Race (Miami to Montego Bay); Fi Wi Sinting (African heritage festival, Portland); Carnival starts in February, culminates Easter week, main events in Kingston, Negril and Montego Bay.
6 February Bob Marley Birthday Bash.
March West End Reggae Festival (Negril); Fun in the Son (Gospel Festival).
April Montego Bay Yacht Club Easter Regatta.
June Ocho Rios Jazz Festival; Kingston on the Edge Art Festival.
July International Reggae Day Festival (Kingston); Portland Jerk Festival; Reggae Sumfest (international music festival, Montego Bay); Little Ochi Seafood Festival (Alligator Pond, Mandeville); Seville Emancipation Jubilee (heritage festival, St Ann's Bay).
August Mello Go Roun' (festival of performing arts, Kingston).
6 August Independence Day Parade (street carnival featuring junkanoo dancers, Kingston/island wide); Ocho Rios Seafood Festival (Ocho Rios); Breadfruit Festival (Bath, St Ann).
September Falmouth Blue Marlin Tournament.
October Nyammings and Jammins Food Festival (Montego Bay); Port Antonio International Marlin Tournament (for more information, click here) and Port Antonio Local Canoe Tournament; National Heroes Day (island wide); Africa Jamfest (Montego Bay).
November Rastafari Rootzfest; NyamJam Food and Music Festival
December Reggae Marathon and Half Marathon (Negril); Jonkonnu (also known as Junkanoo; street parades and Christmas celebrations across the island); JMMC All Stages Rally Jamaica (motorsports race, Kingston); Milk River Seafood and Jerk Festival
Eating Out
Jamaica is a large and fertile island. Fruits and vegetables grow in abundance on family farms, and the land is grazed by cattle, goats and pigs. The clear waters are full of edible fish as well as lobster, shrimp and other seafood. You will be offered an amazing variety of dishes, all very fresh. The range of eating opportunities across the island is remarkable, from cheap street stalls and beach bars to fine restaurants offering international and 'new Jamaican' cooking.
What to Eat
The island's historical and ethnic heritage has contributed to a unique cuisine: it is a story of African cooking techniques and Indian spices meeting Caribbean ingredients. Jamaican food has a reputation for being spicy but, surprisingly, most of the dishes are tasty but not hot. The heat comes from a sauce found in a little bottle that is always on the table, allowing you to add as much spice as you like – or none at all. This hot sauce is manufactured on the island with a secret recipe based on 'Scotch bonnet' pepper, one of the hottest in the world. A little goes a long way, so start carefully and discover your personal taste level.
You will find tame versions of all Jamaican dishes on the menu at large hotels, which often provide a night of Jamaican cuisine where you can sample a range of dishes along with some Jamaican entertainment.
Jamaican Cuisine
Ackee and saltfish. This dish was once a staple food for the enslaved Africans who were transported to the island, and it is now the official national dish of Jamaica. Ackee, which is native to Ghana in West Africa, is a vegetable now found in great abundance on Jamaica. The ackee is harvested only when it is ripe, as it is poisonous otherwise. It is chopped and cooked until it takes on the appearance of firm scrambled eggs. The enslaved Africans added a small amount of protein-rich salted codfish for a cheap and nutritious way to start the day. Today ackee is often served with other types of fish or with bacon as part of a traditional Jamaican breakfast. It comes with various kinds of carbohydrate such as dumplings called 'Johnny cakes', or perhaps with bammy, a cassava pancake.
Ackee, fresh from the tree
Kevin Cummins/Apa Publications
Meat dishes. Jamaicans always cook their meat well rather than rare, so you won't have to worry about the dangers of undercooked meat. But meat served in local restaurants is chopped into pieces with a cleaver rather than being butchered and trimmed, so beware of sharp pieces of bone which might be present in the prepared dish.
Jerk. The modern national dish of Jamaica is 'jerk', which takes its name from the hot marinade used to season meats or fish. You will find it everywhere from the menus of fine restaurants to beach bars and street barbecue stalls. The dish was invented in Maroon country (near Boston Bay in the east of the island) and was originally used to tenderise pork, which was then cooked slowly and served hot and tender. The marinade became popular across the island for all meat, and today you can eat jerk chicken and even jerk fish. The Boston recipe is a mixture of 21 spices and very piquant indeed.
You can watch jerk pork and chicken being prepared in Boston Bay and then try it for yourself. The meat is freshly butchered (the animals are slaughtered in the mornings under the auspices of health inspectors), then marinated and cooked within hours. You will be served the meat with breadfruit, which has a neutral flavour to cool the palate. In other parts of the island, the jerk ranges in flavour and hotness. In hotels and international restaurants, it can be quite mild; you'll find that Jamaicans snub their noses at such offerings.
Full of flavour
Authentic Jamaican cuisine is flavoursome thanks to the spices and seasonings used to marinate the meat and fish, usually overnight. They can include pimento, allspice, thyme, garlic, cloves, ginger and fiery scotch bonnet (peppers).
Jerk chicken with rice and peas
iStock
Goat curry. There are herds of goats alongside all the highways and byways of Jamaica. Goat curry (referred to as 'curry goat') became part of the Jamaican diet following the arrival of the Indian itinerant workers who came to work the plantations following the abolition of slavery. The curry style has adapted over the generations and is now really more of a flavour than a true Indian method of preparation.
Fish dishes. A most amazing array of fish and shellfish can be found in the waters surrounding Jamaica. You can be guaranteed absolutely fresh seafood because the small boats come in daily with their catch. In many restaurants the 'catch of the day' will be the tastiest and freshest option. It might be tuna, snapper, or kingfish; whatever the choice, it will always be superb. The lobster and conch are also fresh and delicious, although they are seasonal. Different areas of the island specialise in certain types of seafood. Around Bluefields, south of Negril, it is spicy shrimp, and at Middle Quarter you will find Escovitch fish, which is fried and then pickled.
Rice and peas. Most main dishes are accompanied by a side dish of rice and peas. It originated as an inexpensive and nutritious option in colonial times, when it could be served as a meal in itself when money was scarce. The 'peas' (actually red or kidney beans) and the rice are cooked slowly together with a touch of coconut milk.
Vegetables and fruit. Because fresh vegetables in Jamaica are varied and plentiful, you will always be given a generous accompaniment with any main dish. The list includes callaloo (a spinach-like vegetable), yam, breadfruit, pumpkin and potatoes. Starchy vegetables have been a staple of Jamaican diets since the days of slavery when they were needed to provide energy for hard labour.
You can sample the abundant fresh fruits from stalls in the street or from hawkers on the beach. Hotels will have a wonderful selection at breakfast or to finish a meal in the evening. Bananas are obviously popular, but you can also choose from guava, mango, papaya, pineapple and coconut. There is in addition a range of unusual fruits found only in Jamaica. Look out for sweetsop and soursop (rough-skinned fruits, said to be aphrodisiacs and best made into a milky drink) along with the star apple and the ugli (a citrus fruit).
Fresh fruit and vegetables in Spanish Town
Kevin Cummins/Apa Publications
As one of the island's major crops, the banana has a special place in Jamaican cuisine. It is eaten raw but also in many hot desserts. You can have banana fritters and, for a touch of luxury, bananas flambéed in Jamaican rum.
Other hot and cold desserts include tarts and custard, which is traditionally flavoured with coconut cream. Ice creams made with fresh fruit are also extremely refreshing: 'matrimony' is a Jamaican favourite, which mixes orange and star apples with cream. Other local favourites include rum and raisin and grapenut ice cream.
Snacks. Jamaican fast food consists of a number of cheap dishes that are prepared at home or bought at roadside stalls for lunch on the run. 'Patties' are thin oven-baked pastries filled with meat, fish or vegetables. 'Stamp-and-go' are fish fritters, so called because just before being cooked they are flattened with the palm of the hand. These dishes are often served as hors d'oeuvres in hotels or in private homes.
International Cuisine
In addition to serving toned-down versions of local dishes, Jamaica's resorts offer a wide range of international cuisine. There are a number of Italian restaurants all across the island, from those offering quick trattoria-style service to upscale dining establishments with full service. Visitors seeking Mexican and Chinese cooking will find choices as well, and the comforts of American and Continental food are also available. There are branches of international fast-food chains in Kingston, Montego Bay and Ocho Rios if you want a burger or fried chicken.
What to Drink
One advantage of a trip to Jamaica is that you can drink the tap water. You can be assured that food washed in tap water is safe to eat and that ice made from it is safe in your sodas or frozen daiquiris.
Red Stripe
You might hear some older locals asking for a 'policeman' at the bar. Don't be alarmed – they just want a Red Stripe beer. The name was taken from the stripes on the trousers and cap of the Jamaican police uniform.
Beer. Red Stripe, a lager-type beer, has long been associated with Jamaica. It is light and very refreshing on a long, hot Jamaican day. You will find it in every café and bar. However, Jamaicans also have a liking for stout beers, which they like to drink at room temperature. You will find that Dragon Stout and Guinness are widely available. You can order your drink cold if you don't mind the locals having a little joke at your expense.
Rum. The first thing that you will be offered when you arrive at your hotel is a rum cocktail. Appleton, the 'overproof' white rum, is the best-known brand, used as the basis for almost limitless recipes. Whichever rum you choose, be careful because they all 'pack a punch'. Many hotels and bars will have their own special recipes, but most will combine rum with fresh fruit juice, lime, or coconut milk. Rum can also be combined with cream and other flavourings to produce a range of smooth after-dinner drinks. Perhaps the best known liqueur is Tia Maria (produced from the Jamaican coffee bean), which makes the perfect accompaniment to a hot cup of coffee.
Appleton rum
Kevin Cummins/Apa Publications
Non-alcoholic drinks. The choice of fruit juices is huge, and you can find single juices or blends in every bar and restaurant. 'Ting', a refreshing fizzy grapefruit drink, is locally produced. Jamaica also produces a ginger ale which has a little more kick than the standard and is extremely refreshing in the heat of the day. You will find all the internationally recognised brands of fizzy drinks readily available.
Cocoa. Jamaican cocoa beans contain a chemical which is a mild stimulant. The roasted and ground beans or seeds can be used to make a delicious hot drink as well as chocolate.
A cocoa pod contains seeds that make delicious chocolate
Kevin Cummins/Apa Publications
Jamaican coffee. Said to be the best in the world and extremely expensive due to the small crop and high demand, most Blue Mountain coffee is exported, so you might not find it in every establishment on the island. The coffee is extremely mild and low in caffeine, with a hint of natural sweetness.
During the 1960s, the reputation of Blue Mountain coffee suffered because inferior lowland beans began to be blended with quality mountain beans to increase the crop and meet demand. In 1973, the government stepped in to create an official standard for Blue Mountain coffee, thus restoring confidence in the marketplace. Today, only coffee grown at an altitude of 610m (2,000ft) or above can be sold as 100 percent Blue Mountain. You might also discover products advertised as 'blended' Blue Mountain coffee; these will contain at least 20 percent Blue Mountain beans.
Restaurants
We have used the following symbols to give an idea of the price for a three-course meal for one, excluding drinks; tips are extra. Prices are in US dollars:
$$$$ over $50 $$ $20–30
$$$ $30–50 $ below $20
Montego Bay
Houseboat Grill $$–$$$$ Southern Cross Boulevard, Montego Bay, tel: 979-8845, www.thehouseboatgrill.com. Restaurant on a houseboat moored in the Marine Park close to the Freeport, reached by a short pontoon ride. Dinner only, with the bar open from 4.30pm for sunset drinks. Fusion cuisine, offering meat and seafood and some excellent vegetarian options. A glass-bottomed section is fascinating for children.
Margaritaville $–$$ Gloucester Avenue, Montego Bay, tel: 952-4777, www.margaritavillecaribbean.com. Lively sports bar and grill located at the start of the main strip. Roof-top deck with water chute and floating trampoline. Frequent 'special' evenings with themed entertainment. More formal (and more expensive) dining at Marguerite's next door. Also in Ocho Rios and Negril.
MVP Smokehouse $–$$ Bogue Road, Reading St. James, Montego Bay, tel: 622-7198, www.mvpsmokehouse.com. This popular authentic Jamaican restaurant on the outskirts of Montego Bay serves excellent jerk chicken, pork, fish, shrimp and lobster that can be accompanied by signature sauces, sweet potatoes, rice and bammy. The ambience is convivial with relaxing reggae music in the background. Tue–Sun 11am–9pm.
Pier 1 $–$$ Howard Cooke Boulevard, Montego Bay, tel: 952-2452. Picturesquely located on the waterfront, Pier 1 is an open-air seafood restaurant that turns into a vibrant nightclub after 10pm. The menu features sandwiches, wraps, soups as well as some Jamaican-style specials.
The Pelican Grill $$–$$$ Gloucester Avenue, Montego Bay, tel: 952-3171, www.pelicangrillja.com. Long established local favourite that is popular with visitors too. Hearty Jamaican food, including a good breakfast menu, and American specialities. Good value for money. Daily 7am–10pm.
The Pork Pit $ 27 Gloucester Avenue, Montego Bay, tel: 952-1046. Very basic but very good Jamaican food – jerk chicken, pork and ribs sold by weight, served through the kitchen window, with garden gazebos to sit and eat in. Cash only. Sun–Thu 11am–11pm, Fri–Sat 11am–midnight.
Scotchie's $ Falmouth Road, Montego Bay, tel: 953-8041. Casual dining near Rose Hall and the Holiday Inn. Open 11am–11pm, serving some of the best jerk pork, chicken and fish with local accompaniments such as roasted breadfruit, potato, yam or festival. Also has branches in Ocho Rios, Scotchie's Too (near St Ann's Polo Club) and in Kingston, Scotchie's Jerk Center (2 Chelsea Avenue).
FALMOUTH
Glistening Waters $$–$$$ Luminous Lagoon, Rock, tel: 954-3229, www.glisteningwaters.com. Before your meal, take a boat trip at night on the bioluminescent lagoon, or swim in the water for an explosive experience. Good seafood and international as well as Jamaican dishes. Daily 11am–10pm, bar until 1am.
runaway bay
Ultimate Jerk Centre $ Main Street, Discovery Bay, tel: 973-2054. A casual, popular roadside restaurant serving tasty, affordable and authentic Jamaican food. Every Friday night there is a happy hour and music, last Saturday of the month is an Old Hits party with local DJ. They also hold a New Year's Eve party here. Sun–Thu 8.30am–10.30pm, Fri–Sat 8.30am–midnight.
Ocho Rios
Almond Tree $$$$ Hibiscus Lodge Hotel, Main Street, Ocho Rios, tel: 974-2676, www.hibiscusjamaica.com. A hotel restaurant offering great views of the sea and serving international and local cuisine. Inside and open-air dining by candlelight. Live music three nights a week, piano bar. Also excellent Jamaican breakfast.
Evita's Italian Restaurant $$–$$$$ Eden Bower Road, Ocho Rios, tel: 974-2333, www.evitasjamaica.com. Italian food with a touch of Jamaican spice; house specialities are Lasagna Rastafari, One Love Penne and Jerk Spaghetti. Try the Jamaica Bobsled for dessert. The essential place to see-and-be-seen, this is the only restaurant overlooking both Ocho Rios and the sea. Everyone in the music, fashion, or film business has probably eaten here. Mon–Sat 11am–11pm, Sun 1–11pm.
Juici-Beef Patties $ 1 Newlin Street, Ocho Rios. If you love patties, this small place cooks up some of the best on the island. The fillings include beef, chicken, shrimp, lobster, soy and cheese. Be prepared to queue.
Port Antonio
Anna Banana's $$ 7 Folly Road, Port Antonio, tel: 715-6533. A small beach-side restaurant and sports bar located a little way out of the town centre. Decent Jamaican cuisine with seafood specialities and good value for money.
Dickie's Best Kept Secret $$$–$$$$ Port Antonio western outskirts, tel: 809-6276. An unassuming painted shack on a clifftop overlooking the bay, run for many decades by Dickie Butler, who is rumoured to have entertained Errol Flynn, Princess Margaret and Winnie Mandela, among others. Excellent home cooking and quite an experience. Reservations recommended.
Mille Fleurs $$$ Hotel Mocking Bird Hill, Port Antonio, tel: 993-7267, www.hotelmockingbirdhill.com. A creative mix of international and local cuisine that includes a good vegetarian selection. Most produce is locally grown, with some from the restaurant's own organic garden. Located in a beautiful position in the foothills of the Blue Mountains offering spectacular panoramic views. Open for breakfast, lunch and dinner.
Blue Mountains
The Gap Café $$–$$$ Hardware Gap, Newcastle, tel: 361-4192. High up in the hills with wonderful views over Kingston, this 19th-century way station is an ideal place to stop when hiking or touring the Blue Mountains. Open for breakfast, lunch and high tea, with delicious Blue Mountain coffee. Jamaican and Italian food, from curry goat to pizza, or combine the cuisines with jerk chicken pasta. Indoor or outdoor dining, with elegant china and table linen in the restaurant.
Kingston
Cuddy'z $$–$$$ Shops 4–6, New Kingston Shopping Centre, Kingston, tel: 920-8019, www.cuddyzsportsbar.com. Owned by former West Indies bowler Courtney Walsh, this sports bar is popular with sports personalities and the after-work crowd. Friday night is lively with scheduled events. Jamaican and international food or TexMex, lots of choice.
Jade Garden $$$ Shops 54–59, Sovereign Centre, 106 Hope Road, Kingston, tel: 978-3476, 978-3479. Hong Kong chefs prepare traditional Chinese food and the island's two largest saltwater tanks ensure that all seafood is absolutely fresh. There is a choice of over 100 dishes. Views of the Blue Mountains from picture windows are spectacular. Reservations recommended.
Regency Bar & Lounge $$$–$$$$Terra Nova All-Suite Hotel, 17 Waterloo Road, Kingston, tel: 926-2211, www.terranovajamaica.com. One of the best restaurants in town, Regency serves international cuisine with a Caribbean twist. The menu includes numerous seafood dishes, and the wine list is extensive. Open daily bar 11am–2am, restaurant 6.30–10pm.
Tamarind Indian Cuisine $$ 28 Orchard Village Plaza, 18-22 Barbican Road, Kingston, tel: 977-0695, www.tamarindindiancuisine.com. Hearty Indian and Asian fusion dishes in the vicinity of the Bob Marley Museum. The selection is great, the food delicious and the portions are generous - the only downside is that it is often packed.
Treasure Beach (South Coast)
Jack Sprat $–$$$ Treasure Beach, tel: 965-3583. Great beach bar at the western end of Jakes (for more information, click here). Typical wooden shack with music, bar, pizzas, ice cream, catch of the day and lots of specials, such as lobster curry when in season.
Jakes $$–$$$$ Jakes Hotel, Calabash Bay, Treasure Beach, tel: 965-3000, www.jakeshotel.com. The best of Jamaica's spicy cuisine, including saltfish and ackee, rice and peas, fish in coconut milk and escoveitch fish. The catch is always fresh, the vegetables and fruit local. Soups include conch chowder and cream of pumpkin.
Little Ochie $–$$ Alligator Pond Beach, tel: 610-6566. Great setting on the beach with wooden tables under thatch, some made from old fishing boats on stilts. Choose your own fish, lobster or other seafood, have it cooked to order and served with bammy, festival and scotch bonnet chillies (these can be very hot).
Yabba $$$ Treasure Beach Hotel, Treasure Beach, tel: 965-0110, www.jamaicatreasurebeachhotel.com. Jamaican and international cuisine, including fresh seafood, served in a relaxing atmosphere. The vegetables are grown on the owner's farm. Open for breakfast, lunch and dinner.
Negril
Ivan's Bar and Restaurant $$–$$$$ West End Road, Negril, tel: 957-0390, www.catchajamaica.com. Restaurant at the Catcha Falling Star Hotel, lovely location on the cliffs overlooking the water. Open all day but get there at 6pm for the sunset.
Just Natural Restaurant $$ West End Road, Negril, tel: 957-0235. Excellent vegetarian meals from appetizers to desserts. Also offers fresh fish and seafood. Particularly good for authentic Jamaican breakfast with a huge fruit plate and Blue Mountain coffee.
Margaritaville Bar and Grill $$–$$$ Norman Manley Boulevard, Negril, tel: 957-4467, www.margaritavillecaribbean.com. Lively sports bar and grill located centrally on Negril's beach. Convenient for lounging on the beach, with full service available. Frequent 'special' evenings with themed entertainment. Free pick-up service.
Niah's Patties $ Seven Mile Beach, Negril. At the back of the craft market stalls just off the beach, this little shack serves up authentic hand-made patties to fill a hole any time. Several varieties, from lobster to vegetable, with accompanying sauces.
Norma's $$$$ Sea Splash Resort, Negril, tel: 957-4041. Restaurant by the water with a romantic atmosphere. European cuisine with a Jamaican nouvelle flair. Open for dinner.
Presley's Bar & Grill $$$$ West End Road, Negril, tel: 440-9833. Informal, local place offering fresh local seafood and produce. Really tasty food and huge portions. Dinner by reservation so book the day before. Only two tables in Presley's shack tucked in between stalls, across the road from Rockhouse Hotel.
Rick's Café $$–$$$$ West End Road, Negril, tel: 957-0380, www.rickscafejamaica.com. Open for lunch and dinner offering everything from surf 'n' turf to chip 'n' dips. The place to come for sunset watching with local lads diving into the sea from rocks and trees for tips. Live music during the evenings; a lively, happening place.
Treehouse Restaurant $$$$ Norman Manley Boulevard, Negril, tel: 957-4287, http://negriltreehouse.com. Imaginative Jamaican cuisine. Have chicken, pork, fish and seafood cooked any way you like, and pizza. Sunday jazz and breakfast on the beach.
Xtabi Cliff Restaurant $$–$$$ Xtabi Resort, Lighthouse Road, Negril, tel: 957- 0524, www.xtabinegril.com. This place is justly proud of its lobster and meat dishes, chargrilled or cooked any way you like.
A–Z Travel Tips
A
Accommodation
Jamaica has a full range of accommodation, from basic beach shacks to some of the finest resorts in the Caribbean and small family-run hotels can be found both close to the beach and inland in the mountains. In Kingston the best hotels are designed for business travellers, while guest houses are of variable quality and not particularly cheap.
All-inclusive resorts were invented in Jamaica and are found mostly along the north and west coasts. Some hotels accept couples only and others cater for families, but most offer a wide range of sporting activities, wedding and honeymoon packages and excursions.
There are also some boutique hotels where you can be pampered in luxury and enjoy some of the island's most beautiful scenery on the coast and in the mountains. These do not come cheap. Many of the small luxury hotels have been around since the 1950s, attracting movie stars and the glitterati before mass tourism arrived. Recent additions to this sector include the Island Outpost group.
Jamaica also has a wide range of private villas for rent, with or without staff. Contact the Jamaica Association of Villas and Apartments (JAVA), Office #4, Ocean Village Shopping Centre, Ocho Rios, St Ann (tel: 974-2508, www.javavillas.org) for more details.
For general information on rental of cabins in the Blue Mountains, contact the Jamaica Conservation and Development Trust, the local NGO responsible for the management of the Blue and John Crow Mountains National Park (tel: 920-8278/9, www.jcdt.org.jm).
Prices change dramatically between high and low season. Low season is from mid-April to mid-December, and you can make savings of up to 40 percent during this period. High season is extremely busy, so it is important to make reservations well in advance to guarantee the accommodation of your choice.
Airports (see also Getting There)
There are three international airports.
Norman Manley International Airport at Kingston (KIN; tel: 924-8452; www.nmia.aero), serves Kingston and the east of the island; it also caters to international business travellers. Transfer to Kingston takes 20 minutes and is 15km (9.5 miles). There is a bus service, but a taxi direct to your destination is a more sensible option. Transfer time to Port Antonio is around three hours.
Sangster International Airport at Montego Bay (MBJ; tel: 952-3124; www.mbjairport.com), serves the north coast, the west of the island and also handles the charter aircraft that fly to the island. Central Montego Bay is only 5 minutes away and there are taxis outside the terminal building even though many hotels and resorts provide transport for the short transfer. Transfers to other resorts by coach are as follows: Ocho Rios is around 2 hours, Runaway Bay 1.5 hours and Negril one hour.
Ian Fleming International Airport (OCJ; tel: 975-3101; www.ifia.aero), formerly Boscobel Aerodrome, near Ocho Rios, opened in 2011 and serves specialised small charters and private jets up to the size of a Dash 8.
There are also a number of domestic airports: Negril Aerodrome, Ken Jones Airport (Port Antonio) and Tinson Pen (Kingston), which operate transfer flights from the international airports. In Kingston, most domestic flights leave from Tinson Pen.
B
Bicycle Rental
Bicycle rental is a sensible way to see a little more of the area where you are based. It is particularly useful in Negril, where the land is flat. Contact Dependable Bike Rental (tel: 957-4764), in Negril, or Kool Bike Rental (http://koolbikerental.tripod.com) at the Negril Yacht Club for bicycles or motorbikes. Cycles can be rented by the day or by the week. It is also possible to undertake bicycle tours into the Blue Mountains and around Port Antonio, along the relatively quiet roads (for more information, click here).
Budgeting for Your Trip
To help you budget for your trip, here are some prices for the things you will need.
Flights to Jamaica. A charter flight from London in the UK to Montego Bay starts from about £400 including taxes, while a scheduled flight to either Kingston or Montego Bay starts from around £445. These are low season prices and fluctuations occur throughout the year. From the US, the best deals are from Florida, with return tickets on budget airlines, such as Jet Blue, starting from about US$160 excluding tax.
Accommodation. A room can range in price from US$40 in low season for a hideaway such as Ital Rest Cottages (for more information, click here) to US$1,800 per night for a suite in a top-class hotel such as Jamaica Inn (for more information, click here) in high season, depending on whether you opt for room only (EP) or for a fully inclusive luxury resort hotel (AI). For a room in a less expensive hotel, allow US$45–100 per person; for a medium-standard hotel, prices range from US$100–250 per night. All-inclusive resorts and luxury hotels start at over US$200 per day but can rise to more than US$1,000, depending on the facilities and comfort level provided.
Self catering. Rental rates of houses, cottages and studios in resorts start from US$500 per week, but for a villa with fully fitted kitchen and maid service expect to pay over US$1,500 per week, and for the very luxurious (such as those at Blue Lagoon), from US$7,500 per week. These prices can prove to be good value on a per-person basis.
Meals. For lunch in a moderately priced but good establishment allow US$20 per person plus drinks; for dinner, allow US$40 per person plus drinks.
Car rental. Allow around US$28–95 per day depending on the size of car and whether the hire company is local or international. The lower figure is the price for a compact car from a local company in low season, while the higher price is for a compact 4x4 from an international company. Weekly rates are better value. If you want to hire a car with a driver, expect to pay around US$100–180, depending on distance, for a 10-hour day, including fuel.
Local transport. Bus fares are cheap, both in town and for longer distances, if you have the stomach for a journey at speed, often on twisting mountain roads. In Kingston the urban bus fare is US$0.92 and express buses US$1–2.40. However, travelling by bus is not recommended for the fainthearted. Taxis charge about US$5 for a short journey and US$20 for 10 miles, but check the fare beforehand. For long journeys the taxi fare could be more than hiring a car, but you can negotiate a deal with a small taxi company.
Arriving and Departing. An air arrival tax of $20 in addition to an arrival tax of $20 and a departure tax of $20 are usually included in the price of your flight ticket.
C
Car hire
Jamaica is the third-largest of the Caribbean islands, and to see all its delights it is best to hire private transport. The condition of the roads and Jamaican driving habits do create concerns for car hirers (see Driving), but with common sense and care, renting a car should enhance your trip, not spoil it.
The major car rental companies have offices at the two international airports:
Hertz: Kingston (tel: 924-8028, at the airport), www.hertz.com.
Avis: Kingston (tel: 924-8293, at the airport), Montego Bay (tel: 952-0762, at the airport), www.avis.com.jm.
Island Car Rentals is the largest local fleet: Kingston (tel: 926-8861, and at the airport, tel: 924-8075), Montego Bay (tel: 952-7225, at the airport), www.islandcarrentals.com, minimum age 23.
Local companies are more competitively priced than the international companies and provide a similar quality of service. Always satisfy yourself as to the age and condition of the car before confirming the booking. You can specify whether you want a manual or automatic transmission. Many companies make an extra charge for delivering the car to your hotel; this can amount to another day's rental charge.
All national driving licences will be recognised by rental companies. Drivers must have held a licence for at least one year before they can rent. All renters must give a deposit, which ranges from US$500 to US$1,000; if you are under 25 years of age, there will also be a bond to comply with insurance regulations. A credit card is the most sensible method of giving the deposit, although cash can also be used.
In the US, some insurance companies cover hire cars; check to see whether you are covered on your policy or through your credit card before purchasing insurance. Damage waiver is recommended, which will add around US$15 per day to your costs.
In high season it is important to book a car in advance, as demand will be high. In low season you should be able to negotiate a package that will give you a better price, and it can often be better to wait rather than book in advance.
Service stations are open daily and accept cash only (Jamaican dollars or US dollars) for fuel.
Climate
Jamaica is a tropical island. It has virtually no change in seasons, the temperature varying between 25°C and 28°C (77°F and 83°F), although it is cooler in the mountains. Rainfall averages around 198cm (78in) each year and is greatest between August and November, which is considered low season for visitors. However, rain can fall in short, heavy tropical showers at all times of the year, especially in the afternoon. Rainfall varies considerably between the wetter east and the drier west of the island. Hurricane season, which afflicts the whole Caribbean, runs from the beginning of June through to the end of November.
Clothing
Lightweight clothing is sensible throughout the year along the coasts. Many people manage happily with T-shirts and shorts during the day, and wear something a little more formal in the evening. Cotton or other breathable materials are ideal. In the mountains, sweaters are a good idea for evenings or in case of a change in the weather. If you plan to visit interior towns or Kingston, more conservative clothing might be appropriate. Beachwear is acceptable only in the immediate area of the beach and not in shops and banks.
A hat and sunglasses are important, as the sun is very strong, especially in the middle of the day. When you first arrive, always make sure that you have clothing to cover your skin to prevent burning; a lightweight long-sleeved shirt is fine.
Footwear should be light and comfortable: a pair of sandals or flip-flops for the beach, along with a smarter choice for evenings. If you plan to visit the Blue Mountains, a pair of stout shoes or walking boots is essential.
Crime and Safety
Jamaica, and Kingston in particular, have a reputation for crime and violence, but in fact there are few attacks on tourists and the Jamaican countryside has a comparatively low crime rate. Much of the violent crime is confined to about four police districts in Kingston, which are prone to drug gangs and political inter-neighbourhood rivalry. As with any city, visitors are advised to exercise caution.
The use of marijuana, or 'ganja' (as it is known on the island), is not uncommon among Jamaicans of all classes; many smoke it, while others use it as a medicinal herb. Rastafari use it as a sacrament in religious observances. The drug is easily available and most visitors will be offered a supply at some stage during their holiday. Until recently it was strictly illegal to possess or use marijuana, however in February 2015 the parliament passed a law allowing the possession of up to 2oz (57g) for personal use.
The Jamaican authorities have increased security patrols in the resort areas, and you will see the blue uniforms of the 'Tourist Police' on the beaches. Many hotels also employ private security personnel, who patrol beaches and hotel entrances to deter hawkers and others.
Many Jamaican men make a living as impromptu (and definitely unofficial) guides, and they might approach you in the street or on the beach. Use caution in your dealings and use accredited companies only. Do not accept offers to ride in unauthorised taxis (official taxis have red number plates); they will not be insured to carry fare-paying passengers.
Always take out a travel insurance policy and photocopy important documents in case you need to make a claim.
D
Driving
Driving in Jamaica can be an adventure or a worry. The roads are in very bad condition and there is a lot of traffic. You might find cars driving towards you on the wrong side of the road, only to realise that they are avoiding a large pothole on their own side of the street. Always drive with utmost care and be ready to stop at any moment for potholes, animals and people. Cross-country routes, particularly in the area of the Blue Mountains, are prone to flooding or landslides. After periods of rain, you should always check before setting out to be sure that the road is passable; ask bus or truck drivers or employees at your hotel.
The big road-building programme is almost complete. The North Coast Highway was built in stages: Negril–Montego Bay, Falmouth–Ocho Rios, and Ocho Rios–Port Antonio. Highway 2000 in the south connects Kingston with Montego Bay via St Catherine, Manchester, St Elizabeth, Westmoreland and Hanover, and Kingston with Ocho Rios via St Ann. Most sections are now open, while the final 67 km (42 miles) long leg should be ready in mid-2016.
Speed limits and safety. Vehicles drive on the left, and speed limits are 50km/h (30mph) in towns and 80km/h (50mph) in rural areas. Despite this, many Jamaican drivers ignore the speed limits and drive at a dangerous speed. Always drive at a safe pace and allow plenty of time to reach your destination. Roundabouts (or traffic circles) are common. Give way to any traffic from the right at roundabouts.
Road signs feature easily recognisable international symbols. However, you will find that distance signs can be in either miles or kilometres, which can create confusion. The unit of measurement used will always be indicated at the side of the number.
Fuel and service. There are fuel stations open seven days a week in all towns. Always carry out basic checks on a rental vehicle when you take delivery of it and before setting out. Public telephones are rare in the interior; if you do break down, it could be hours before you get help, so always carry a mobile phone. If you have mechanical difficulties, contact your rental company for assistance.
Parking. When parking in towns or near beaches, try to find a car park with some security, and always park with the car in full view. At night, always park in a well-lit location.
E
Electricity
Jamaica operates at 110 volts/50 cycles as standard; current at 220 volts is available in some hotels on the island. Appliances with US and Canadian plugs can be used without adapters, but appliances from the UK and Europe will require one.
Embassies and Consulates
All diplomatic representatives have offices in Kingston.
Canada: High Commission, 3 West Kings House Road, Waterloo Road Entrance, Kingston 10, tel: 926-1500, www.jamaica.gc.ca
France: 13 Hillcrest Avenue, Kingston 6, tel: 946-4000, www.ambafrance-jm-bm.org.
Germany: 10 Waterloo Road, Kingston 10, tel: 926-6728, www.kingston.diplo.de.
UK: High Commission, 28 Trafalgar Road, Kingston 10, tel: 936-0700, http://ukinjamaica.fco.gov.uk/en.
US: 142 Old Hope Road, Kingston 6, tel: 702-6000, http://kingston.usembassy.gov.
Emergencies
In the event of an emergency, call 119 for police and 110 for fire or ambulance and medical services.
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Gay and Lesbian Travellers
Male homosexuality is an offence, punishable by prison in Jamaica. Consequently, homophobia is rife and there is no open gay scene, though LGBT rights are now one of the major political issues in the country.
Getting There (see also Airports)
By air. Flying into Jamaica is an easy option from the US, Canada and Europe. Miami, New York, Atlanta, Chicago and Toronto are all major hubs in North America, with easy connections to other US and Canadian cities. London is the hub for Europe, with easy connections for the UK and Ireland.
The following major airlines fly into Jamaica: Air Jamaica (www.airjamaica.com), Air Canada (www.aircanada.com), American Airlines (www.aa.com), British Airways (www.britishairways.com), Delta (www.delta.com), Jet Blue (www.jetblue.com), Spirit Airlines (www.spiritair.com), and Virgin Atlantic (www.virgin-atlantic.com). Scheduled flights will normally land at Norman Manley International Airport in Kingston. If you will be spending most of your time around Montego Bay or Negril, get a flight to Sangster International Airport at Montego Bay, where the transfer time is much shorter. Many other scheduled airlines and charter companies offer services depending on the time of year, with more services during high season. Most charter flights land at Montego Bay in the north.
Visitors from Australia and New Zealand can travel through either the US or Britain to pick up a connection to Jamaica. Both directions involve long journeys and possibly a stopover en route, so consult an airline specialist for advice about schedules and costs.
By sea. Many tourists visit Jamaica as a port-of-call on a cruise. Montego Bay, Falmouth and Ocho Rios are major cruise destinations, with comprehensive facilities for cruise passengers. Ports are well placed to offer tours to a range of attractions that can be visited on a day ashore.
Guides and Tours
There is a comprehensive tour programme offering visits to sites across the island. These can be booked either through your own tour or cruise company or through the Tourist Board offices. For those who don't want to hire a car, this is an ideal way to see more of Jamaica. Tour companies will pick you up at your hotel and bring you back at the end of the day. Full-day tours often include lunch.
JUTA (Jamaican Union of Travellers Association) provides licensed taxis and tour buses for excursions to all major attractions; there are JUTA branches around the island: 80 Claude Clarke Avenue, Montego Bay, tel: 952-0813, http://jutatoursltd.com; Norman Manley Boulevard, Negril, tel: 957-4620, www.jutatoursnegrilltd.com. JUTA can also arrange individual itineraries. Prices for the same tour do vary, and you can save money by booking directly with JUTA or with the Jamaica Tourist Board, rather than through your own tour operator.
H
Health and Medical Care
Hygiene standards are generally high in Jamaica, and the tap water is drinkable. Mosquitoes can be a problem, especially just after sunset, and cases of Chikungunya virus passed by mosquitos have been confirmed on the island, so cover up or apply insect repellent. Don't step on the spiny sea urchins as you snorkel or dive; the spines will embed themselves in your flesh and the sores can become infected. Go easy on the alcohol, especially in the sunshine, as this can lead to dehydration. Take time to build a tan to avoid sunburn and sunstroke; use a sunscreen with a sufficiently high SPF.
Most hotels have an arrangement with a local doctor who will be on-call for any problem. Each major town on the island has a hospital; however, the nearest hospital to Ocho Rios is at St Ann's Bay, and the closest to Negril is at Savanna-la-Mar.
Always take out comprehensive insurance when you travel to cover unforeseen health emergencies or accidents.
L
Language
English is the official language of Jamaica and is spoken by everyone on the island. However, the local population also uses a Caribbean-English creole language, Patois, when speaking with each other. It originally developed when the Elizabethan English of the British colonists mixed with the West African languages spoken by the African slaves transported to the island. With subsequent additions of English, African, and Spanish vocabulary, Jamaican English has evolved into an everyday medium that is difficult for outsiders to understand.
M
Media
Radio and television. Jamaica has three TV nationwide stations and lots of local and cable channels. It has over a dozen radio stations, some of which are owned by the government. There are also a number of independent local radio stations and some online stations. Most hotels and many bars also receive satellite services, so you'll find BBC World, CNN and ESPN widely available.
Newspapers and magazines. The major national newspapers in Jamaica are the Daily Gleaner and Sunday Gleaner (http://jamaica-gleaner.com) and the Jamaica Observer (www.jamaicaobserver.com), alongside The Star (http://jamaica-star.com), an evening paper.
Money
Currency. The currency of Jamaica is the Jamaican dollar (J$; colloquially called the 'jay'), and there are 100 cents in each dollar. Paper bills are issued in denominations of $50, $100, $500, $1,000 and $5,000; coins are issued in denominations of 1 cent, 10 cents, 25 cents, 50 cents, $1, $5, $10 and $20. The smaller coins, being practically worthless, are being phased out. The US dollar is also widely accepted in shops and restaurants.
Jamaican dollars may be converted to foreign currency at the airport before departure upon presentation of an official exchange receipt. For visitors there are no restrictions on the import or export of foreign currencies, as long as they are declared, but the import or export of local currency is prohibited.
Travellers' cheques and credit cards. Travellers' cheques are widely accepted in Jamaica for cash in banks, for goods in shops, and for hotel and restaurant charges. Credit cards are also widely accepted except for fuel purchases, which must be made with cash (Jamaican or US dollars). If you want to obtain a cash advance with a credit card, you must take your card into a bank and produce photo ID. There are lots of ATMs (cash machines) in Jamaica, accepting a variety of international credit and debit cards.
Currency exchange. Money is changed at hotels, though at a less advantageous rate than in banks. There are also a number of 'Cambio' shops which are official money changers. You must have one official exchange receipt if you want to change money back before you return home. Changing money on the black market is illegal, but it is one of the services offered by street merchants. Beware of being cheated if you decide to use these unofficial money changers.
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Opening times
Banks: 9am–2pm Monday to Thursday; 9am–4pm Friday.
Government offices (including Tourist Board): 8.30am–5pm, Monday to Thursday, 8.30am–4pm Friday.
Shops: 8.30am–4.30 or 5pm Monday to Friday, 8am–4pm Saturday, but this can vary enormously in resort towns and from low to high season.
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Police
Police officers wear navy uniforms with red stripes on their hats and trousers. The emergency phone number for the police is 119. The Jamaica Constabulary Force is based at 101–103 Old Hope Road, Kingston 6.
Post Offices
All major towns have a Post Office. These are open 8am–5pm Monday to Friday. The postal system is notoriously slow, and postcards often take three weeks to reach their destination. Post boxes are red, but unreliable. If you have anything important or urgent to send, it is best to use a commercial carrier.
Public Holidays
Government offices and services are generally closed on the following days:
New Year's Day 1 January
Ash Wednesday
Good Friday
Easter Monday
Labour Day 23 May
Emancipation Day 1 August
Independence Day 6 August
National Heroes Day third Monday in October
Christmas Day 25 December
Boxing Day 26 December
R
Religion
Jamaica is a Christian island, with Protestant denominations in the majority. However, many other major religions are also represented and have places of worship. You will find that Jamaicans always dress very smartly to go to church. One of the significant minorities is the Rastafari movement, a way of life rather than a religion, whose true adherents are said to number fewer than 100,000. With their characteristic dreadlocked hair, they are seen as being almost synonymous with the image of Jamaica. Their influence on the popular culture of the island remains strong.
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Telephones
When calling from abroad, the country code for Jamaica is 876. When making an international call from Jamaica, always dial 00 before the country code. When in Jamaica, you need dial only the seven-digit local number; there are no area codes within Jamaica.
You can rent or buy mobile/cell phones, or buy a local SIM card to put in your own phone, but check that it is unlocked. Check with your home service provider that your phone will work in Jamaica, most will. Also check on your smartphone's connectivity, plus the use of apps, or Skype on a smartphone. Most of these options should work with the right phone. If you can access Wi-Fi, check out internet phone services. Jamaican service providers are LIME (www.lime.com) and Digicel (www.digiceljamaica.com).
Time Zones
Jamaica operates on Eastern Standard Time, which is 5 hours behind GMT; however, it does not switch to daylight saving time. The following chart shows the time in various cities in winter:
Los Angeles New York Jamaica London Sydney
9am noon noon 5pm 3am (next day)
Tipping
Tipping is standard practice throughout the island, except at a few all-inclusive resorts where the 'no tipping' policy is clearly stated. It is common for a service charge to be automatically added to restaurant bills; this should be clearly stated on the menu or on the bill. If not, then a 10 percent to 15 percent tip should be added.
For taxi drivers, tip 10 percent to 15 percent; for porters, J$100–175 per bag; for hotel maids, J$100–175 per day.
Tourist Information
For useful information to help you plan your trip, the Jamaica Tourist Board (www.visitjamaica.com) has offices in the following countries:
US: 5201 Blue Lagoon Drive, Suite 670, Miami, FL 33126 tel: (305) 665-0557; 1-800-526-2422 (toll-free); email: info@visitjamaica-usa.com.
UK: 1–2 Prince Consort Road, London SW7 2BZ, England, tel: (020) 7225-9090; email: mail@visitjamaica.uk.com.
Canada: 303 Eglinton Avenue East, Suite 200, Toronto, Ontario M4P 1L3, tel: (416) 482-7850, 1-800-465-2624 (toll free); email: jtb@visitjamaica-ca.com.
Tourist Offices can be found in the following locations:
Kingston: 64 Knutsford Boulevard, Kingston 5, tel: 929-9200; email: info@visitjamaica.com.
Montego Bay: Tourism Centre, Montego Bay Convention Centre, Rose Hall, St James, Montego Bay, tel: 952-4425.
Transport
The metropolitan areas of Kingston and Montego Bay have an improved bus system. Taxis and bus franchises provide easy commuting to coastal and interior areas of the island.
The tour company JUTA (for more information, click here) operates commercial air-conditioned bus services between the airports and the major resort areas. As an example, a one-way trip from Sangster Airport at Montego Bay to Negril costs US$25 to Negril Beach and US$30 to West End. A private transfer costs US$70 for 2 passengers.
Once you are settled, many restaurants and bars will provide free transportation in the evenings if you eat with them; just give them a call from your hotel.
Taxis are plentiful, but remember to use cars with red number plates: these are registered and properly insured. Always agree on a price for the ride before you get into the taxi, as they do not carry meters. Find out from other travellers what the going rate is for the journey that you want to make.
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Visas and Entry Requirements
Residents of the US and most Commonwealth and European countries do not need a visa to visit Jamaica, but must carry a passport valid for at least six months and a return ticket. Visitors from Canada can enter with a valid passport, naturalisation certificate, or photo ID with birth certificate, but a passport is essential to transit the US.
You should declare any unusual or expensive items (such as cameras or electrical goods) on arrival to assure the authorities that they are for personal use only.
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Websites and Internet Access
A number of websites can provide you with information about Jamaica before you book your trip, including details about hotels and attractions, car rental companies, and general facts and history:
www.visitjamaica.com – official site of the Jamaica Tourist Board
www.go-jamaica.com
All these sites will link you with other useful sites for your trip. Jamaica has clusters of internet cafés, which are widely available in the tourist areas. Many of the larger hotels also offer the use of a computer in a public area or Wi-Fi internet access for visitors with their own mobile device.
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Youth Hostels
There are no youth hostels in Jamaica which are members of the Hostelling International Organisation. Several places call themselves hostels, but are really budget hotels. See www.hosteljamaica.com
Recommended Hotels
In both style and price, there is a wide choice of accommodation in Jamaica. At the upper end of the scale are the large, expensive luxury resorts and boutique hotels offering exclusivity, which are popular with honeymooners and celebrities. There is also a range of standard hotels at all levels. More modest accommodation can be found in small guesthouses and family-run hotels that offer clean rooms but few other facilities. A few historic plantation houses have been converted into hotels for a 'colonial feel'. Whatever your budget and taste, there will be something on the island to suit you.
Jamaica pioneered the all-inclusive hotel, where all your meals, drinks, sporting activities and other services are included in the price. This is the bedrock of mass market tourism on the island. Most of the resorts are in the north coast beach areas of Negril, Montego Bay and Ocho Rios. Some specialise in family holidays, others are for couples only. The resort chains have locations across the island, check their websites for details: Couples, www.couples.com; Decameron, www.decameron.com; Iberostar, www.iberostar.com; Riu, www.riu.com; Sandals, www.sandals.com; Sunset Resorts, www.sunsetresortsjamaica.com; SuperClubs, www.superclubs.com.
The following selection of hotels covers a variety of accommodation options. The categories below indicate prices in US dollars per room, based on double occupancy.
$$$$ over $200
$$$ $150–200
$$ $100–150
$ under $100
Montego Bay
Coyaba Beach Resort and Club $$$$ Little River, Ironshore (8km/5 miles east of Montego Bay at Mahoe Bay), tel: 953-9150, 877-232-3224 (toll-free from US and Canada), www.coyabaresortjamaica.com. Family-owned, 50-room hotel with good facilities and understated traditional elegance on a private, white-sand beach. The large rooms all have balconies with either garden or sea view 'silent' air conditioning, ceiling fans, hairdryers, in-room safes. Private dock with pick-up for fishing and diving charters. Complimentary water sports including kayaking and snorkelling. Land-based activities include a children's playground, tennis courts with visiting professional coach, gym and massage. The restaurant is run by an award-winning chef.
Toby's Resort $$ 1 Kent Avenue, Montego Bay, tel: 952-4370, www.tobyresorts.com. At the end of the 'Hip Strip', this small hotel is convenient for the airport, bars, restaurants and beaches, all within walking distance. The staff are helpful and friendly, the rooms are simple but clean and comfortable with a balcony. There's a good restaurant, bar and pool.
Half Moon, A RockResort $$$$ Rose Hall, Montego Bay, tel: 800-438-7241, 888-830-5974, http://halfmoon.rockresorts.com. Beautifully landscaped gardens and a private bay giving the resort its name. Set in 160 hectares (400 acres) of grounds, with 33 villas, 152 suites and 45 rooms, the hotel has been a luxury destination since 1954. Mahogany furniture, Jamaican paintings, cable TV, air conditioning, mini-bars, hair dryers and in-room safes. The resort has four restaurants, a variety of snack bars, a spa and a shopping village. Land and water sports facilities include squash, tennis, health and fitness centre, equestrian centre and a par-72 championship golf course.
Ridgeway Guest House $ 34 Queen's Drive, Montego Bay, tel: 952-2709, www.ridgewayguesthouse.com. A small, family-run inn with 10 simple rooms in a modern block, built in the garden of the original guest house and within walking distance of the airport. The staff are friendly and can help to arrange excursions, car hire is available on site. Good value accommodation.
Round Hill Hotel and Villas $$$$ John Pringle Drive, Montego Bay, tel: 956-7050, 800-972-2159, www.roundhill.com. A casually elegant hotel set in a former pineapple plantation at the edge of the crystal clear waters of a private white-sand beach. There is a lovely infinity pool, watersports centre with some complimentary activities, diving, deep-sea fishing and yachting can be arranged, tennis with resident pro, spa with fitness centre, jogging/walking path, boutique and shop, free shuttle to main shops. 36 ocean-front rooms designed by Ralph Lauren and 27 individually owned villas with 2–5 bedrooms. The cuisine is international with Jamaican touches.
The Tryall Club $$$$ 20km (12 miles) west of Montego Bay, tel: 956-5660, 800-259-8017 (toll-free from US), www.tryallclub.com. A luxurious seaside villa hideaway with a championship golf course on a 890-hectare (2,200-acre) tropical estate, originally a sugar plantation until 1918 when coconut palms were planted. Presided over by a Georgian great house and situated in manicured gardens and rolling hills, with 2.5km (11⁄2 miles) of coastline and a palm-dotted white sand beach. Privately owned villas with 2–8 bedrooms.
Wexford $$–$$$ 39 Gloucester Avenue, Montego Bay, tel: 952-2854, www.thewexfordhotel.com. Convenient for nightlife and the 'Hip Strip', just across the road from the public beach. 60 rooms and one-bedroom apartments, sea view or garden view overlooking the pool. Tiled floors, air conditioning, TV, phone, balconies, a good option if you want to be in town. Shuttle service and free entry to Aquasol Beach. Dine in Rosella's for Jamaican cuisine.
runaway bay
Club Ambiance $$$$ Main Road, St Ann, Runaway Bay, tel: 973-7795, www.clubambiance.com. Large all-inclusive resort with 100 rooms in five blocks. All rooms have air conditioning and a sea view. There is also a refurbished three-bedroom, private beachfront villa available to rent. Three beaches, one is clothes optional. Nightclub and restaurant on the complex. No guests under 18 years old.
Piper's Cove Resort $–$$ Salem, Runaway Bay, tel: 973-7156. 14 one-bedroom apartments, some with sea view, and six studio apartments with kitchenette, bathroom, private balcony, safety deposit box, cable TV, air-conditioning. There are pleasant gardens and it is popular for weddings. It has a private beach, pool, restaurant and games room.
Ocho Rios and environs
Goldeneye $$$$ Oracabessa, St Mary, tel: 622-9007, 800-OUTPOST, www.goldeneye.com. An exclusive and luxurious retreat created around the former home of James Bond author, Ian Fleming. The resort, which is part of the Island Outpost boutique hotel chain, has 11 beach or lagoon cottages including the Ian Fleming villa, secluded in lush gardens with private coves and beaches. Modern decor as befits a hotel in the Island Outpost chain, with state-of-the-art amenities. All have access to the James Bond Beach Club alongside, with three beaches, a two-storey, open-air restaurant and watersports.
Hibiscus Lodge $$–$$$ 83–87 Main Street, Ocho Rios, tel: 974-2676, 974-2813; www.hibiscusjamaica.com. Rooms perched on top of cliffs surrounded by gardens, with paths and stairways down to the sea, but no beach. The reef just offshore is good for snorkelling. Located a short way out of the centre of Ocho Rios, but within walking distance of the shops and nightlife. The hotel has a good restaurant on site and a bar overlooking the water. Good-sized swimming pool and cliff-top jacuzzi. 26 simple rooms with air conditioning and fans, balconies have a sea view; you pay for the pretty setting rather than the amenities, but good value for Ocho Rios.
Jamaica Inn $$$$ Main Street, Ocho Rios, tel: 974-2514, 800-837-4608, www.jamaicainn.com. This has been one of the island's best hotels since the 1950s, with 47 suites in pretty gardens overlooking a lovely, private beach. This award-winning hotel offers excellent amenities and personal service; it is beautifully designed and maintained. Luxury facilities include spa treatments and delicious gourmet local cuisine.
Rooms on the Beach $$ Main Street, Ocho Rios, tel: 1-877-467-8737, www.roomsresorts.com. A budget option from the SuperClubs chain. 99 rooms in a three-storey beachfront block near the centre of Ocho Rios. Simple air-conditioned rooms. Facilities include kayaking, windsurfing, snorkelling and scuba diving, and there is a pool on the property and a free Wi-Fi internet. Four meal plans to choose from. There is also a branch in Negril.
Sandals Royal Plantation $$$$ Main Street, Ocho Rios, tel: 888-726-3257, in the UK tel: 0800-022-3030, 207-582-9895, www.sandals.co.uk. Seafront resort in operation since the 1950s, with 74 ocean view suites and one exclusive villa, three gourmet restaurants, spa, watersports, scuba diving and little luxuries such as 24-hour room service, beach butler and afternoon tea on the terrace. Bedrooms have comfortable down mattresses and Italian sheets on the mahogany beds. Guests also have access to the Sandals Golf and Country Club, with complimentary green fees and transport to and from the 18-hole championship course.
Port Antonio and the east
Great Huts $–$$$$ Boston Bay, tel: 353-3388, www.greathuts.com. Located on the cliffs of Boston Bay this seafront eco-resort is perfect for a romantic weekend or a holiday with the whole family. Accommodation is in rustic African-style bamboo huts, treehouses and tents furnished with exotic textiles, paintings and carvings, and comfortable bamboo and driftwood beds. Some units have hot water, some share bathrooms and all have electricity. Surfing, snorkelling and yoga facilities. The price includes breakfast and Wi-Fi internet access; restaurant on the property. The Boston Jerk Centre is nearby.
Frenchman's Cove Bed & Breakfast $–$$$$ Frenchman's Cove, Portland, tel: 993-7270, 564-9779, www.frenchmanscove.com. In a beautiful location on an 18-hectare (45-acre) private estate outside Port Antonio, with a white sand beach and freshwater stream. Simply decorated villas of varying sizes nestled on the cliff sides, or rooms and suites in the main house. This long-established resort is slightly dated but quiet and relaxing. Facilities include a beachfront grill-style restaurant and two bars.
Goblin Hill Villas at San San $$$–$$$$ San San, tel: 925-8108, www.goblinhill.com. A lush site set high on a hillside with excellent views over the sea. Self-contained villas of one or two bedrooms with fully equipped kitchens staffed with cook/housekeeper. Freshwater swimming pool and two tennis courts. Complimentary access to Frenchman's Cove; nature trails lead through the gardens with extensive lawns and woods.
Jamaica Palace Hotel $$$–$$$$ Williamsfield, tel: 993-7720, www.jamaica-palacehotel.com. Large comfortable hotel with 80 suites and rooms with round beds, colourful floral furnishings, air conditioning and bath/shower. Also on the property are an art gallery, a swimming pool and a good restaurant.
Mocking Bird Hill Hotel $$$–$$$$ Port Antonio, tel: 993-7267, www.hotelmockingbirdhill.com. On a hilltop above the town, five minutes' drive from Frenchman's Cove beach. A tranquil hideaway nestled in the verdant foothills of the Blue Mountains, decorated throughout with original art and with an art gallery attached. The hotel promotes environmental awareness, encouraging sustainable development at all levels. The hotel is popular with birdwatchers. Fine restaurant serving nouvelle Caribbean cuisine. Ten white-tiled rooms, with Jamaican hand-crafted bamboo furniture and locally printed fabrics, offer a garden view downstairs; superior rooms upstairs with views of the hillside and the sea beyond.
Strawberry Fields Together! $–$$$$ Robin's Bay, St Mary, tel: 655-0136, in the UK tel: 203-3183-784, www.strawberryfieldstogether.com. Secluded and rustic standard or deluxe cottages sleep 25 dorm-style or 7 couples. There is also a camping area where you can pitch your tent for $15 per night per person. Quiet during the week but busy at weekends with Jamaican families staying over or just here for the day. Private beach with life guard, snorkelling, volleyball, trampoline, table tennis and organised excursions. Large nature reserve. Meal plans available; local cuisine.
Trident Hotel $$$$ Anchovy, Port Antonio, tel: 633-7000, 888-433-526; www.tridentportantonio.com. Luxurious sophistication on the northeast shore. Thirteen glamorous oceanfront villas with modernist interior, terrace, private pool, and broadband internet access. Located on site, Mike's Supper Club is a smart music and dining venue where you can revel in both jazz and Japanese-Jamaican cuisine. The resort amenities also include a private beach, gym and spa.
the Blue Mountains
Forres Park Guest House $–$$$$ Mavis Bank, tel: 927-8275, www.forrespark.com. Entrance on the main road by the Mavis Bank Coffee Factory, rooms are available in the Swiss chalet-style main house or in nearby cosy cabins on the working coffee farm. Trails for hiking and birdwatching, although most of the birds can be seen at daybreak from the balcony. Guides can be arranged for a trip to Blue Mountain Peak. Customised spa treatments will rejuvenate you after your excursions.
Lime Tree Farm $$$$ Tower Hill, Mavis Bank, tel: 446-0230; www.limetreefarm.com. Three cottages on a working coffee farm with views of the Blue Mountains and surrounding valleys. Large bedrooms, bathroom and terrace, spacious enough for a small family. Price includes transfers from Kingston and all meals with a bottle of wine at dinner. Delicious food using local ingredients and herbs. Good hiking and birdwatching.
Kingston Area
Courtleigh Hotel and Suites $$$–$$$$ 85 Knutsford Boulevard, Kingston 5, tel: 936-3570, www.courtleigh.com. In the financial district of New Kingston, convenient for shopping and eating. Rooms have tea/coffee maker, air conditioning, Wi-Fi, dataport, cable TV, hairdryer, with an outdoor swimming pool and fitness room, restaurant and bar.
Grand Port Royal Hotel $$$–$$$$ Port Royal, tel: 967-8494. Situated in Port Royal at the entrance to Kingston Harbour, this hotel offers commanding views of the city skyline and the Blue Mountains. Spacious accommodation in 60 rooms and suites. Facilities include a freshwater swimming pool, gaming lounge, pub and two restaurants, including the acclaimed restaurant next to the marina, which is lovely at night
Indies Hotel $ 5 Holborn Road, Kingston 10, tel: 926-2952, www.indieshotel.com. 15 rather small rooms in two wings overlooking a patio garden. Single, double and triple rooms available, all with bathroom, air conditioning, cable TV and phone. Simple but adequate with good, budget-priced restaurant and bar. Convenient for shops, restaurants, entertainment and sightseeing.
The Jamaica Pegasus $$$$ 81 Knutsford Boulevard, Kingston 5, tel: 926-3691, www.jamaicapegasus.com. Situated in the financial and business district, close to many of the area's foremost attractions. All 300 rooms and suites are equipped with high-speed internet access, satellite TV, hair dryers, safes, complimentary coffee- and tea-making facilities, and balconies with either mountain or pool/ocean view. Non-smoking floors. Restaurants and cafés offer a variety of fare, plus there are bars and evening entertainment.
Strawberry Hill Hotel $$$$ New Castle Road, Irish Town, St Andrew, tel: 944-8400, 800-OUTPOST, www.strawberryhillhotel.com. Romantic, delightful cottages feature traditional 19th-century Jamaican architecture for an authentic colonial atmosphere. Near Kingston and surrounded by the Blue Mountains and extensive gardens, with exceptional panoramic views (including Kingston) and an infinity pool, small but perfect. In-room CD and DVD players with selection of music. Excellent fusion cuisine. Spa treatments, yoga pavilion, plunge pool and sauna encourage health and well-being. The hotel, part of the Island Outpost chain, has won many awards for its architecture and design, and is one of the best places to stay in the whole of the Caribbean.
treasure beach (south coast)
Ital-Rest Cottages $ Treasure Beach, tel: 421-8909. Two simple, rustic, thatched cottages with a mountain or sea view, each with two bedrooms, bathroom and kitchen. The café in the garden has vegan and vegetarian food, music, dominoes and table tennis. Local restaurant close by. Friendly and helpful, 100m/yds to the sea, this is genuine laid-back Jamaica.
Jakes $–$$$$ Calabash Bay, Treasure Beach, tel: 965-3000, www.jakeshotel.com. An eclectic collection of colourful cottages set atop low cliffs in a secluded bay, and a truly special place to stay. Created by painter-photographer-art director Sally Henzell and part of the Island Outpost chain. Each room has a different theme, from Jamaican shack to Mexican pueblo and they vary in size from a single room with a garden view to a 4-bedroom cottage, including delightful, romantic honeymoon suites. All have in-room music equipment and free use of the hotel's extensive music collection. Tropical ceiling fans and mosquito nets maintain the traditional feel. Rooms also in the Henzell's historic home. Two excellent restaurants with the freshest of seafood and other local specialities. Media room with Wi-Fi and computer for guests' use, spa and yoga retreats. Swim at Jack Sprat Beach or in the seawater swimming pool.
Sunset Resort $–$$$$ Calabash Bay, Treasure Beach, St Elizabeth, tel: 965-0143, www.sunsetresort.com. 14 rooms and suites overlook Calabash bay and beach where there are fishing boats pulled up on the sand. American and Jamaican-owned, it is good for family groups as suites can become apartments, and large parties can rent the entire villa. Staff can arrange deep-sea fishing in traditional boats or motor cruisers. Rather odd green astroturf around the pool. Satellite TV, air conditioning, coffee makers, flowery wall decorations, restaurant and lounge.
Negril
The Caves $$$$ Lighthouse Road, West End, tel: 957-0270, www.islandoutpost.com. Part of the Island Outpost hotel chain, this award-winning romantic hotel is perched on the cliffs above the sea. Luxury cottages of thatched wood and stone. Inside the caves are an intimate dining room and part of the spa. Steps lead down to the water where you can snorkel in more caves. Relax with a yoga class or explore the area on bicycles, kayaks or rafts. Several dining options and the Blackwell Rum Bar. No children under 16.
Charela Inn $$–$$$$ Norman Manley Boulevard, Negril, tel: 957-4277, www.charela.com. Rooms on the beach in a long-established, family-run hotel with Jamaican-French owners and French chef, so food is a priority. There is a bakery in the hotel and they have one of the best wine lists on the island. Rooms vary in size and amenities but are well-equipped and three are adapted for wheelchair users.
Rockhouse Hotel $$–$$$ West End, Negril, tel: 957-4373, www.rockhousehotel.com. Commanding a rocky promontory in West End with views of spectacular sunsets, this collection of thatch-roofed villas has a tranquil setting. Cliff-top pool, spa and access to swimming and snorkelling in Pristine Cove more than make up for the lack of a sandy beach. Not suitable for children under 12. 34 rooms, studios and villas.
Seasplash Resort $$–$$$$ Norman Manley Boulevard, tel: 957-4041, in the UK tel 0800-7297-2900, www.seasplash.com. On the narrow part of Negril's 11-km (7-mile) sandy beach, within walking distance of beach bars and restaurants. Rooms and suites are spacious and comfortable, with good fittings and furnishings, all well equipped. Low-season prices drop by one-third from high-season rates and are excellent value. Very good restaurant on site, Norma's (for more information, click here), with steps down to the sea.
Tensing Pen $$$–$$$$ West End, Negril, tel: 957-0387, http://tensingpen.com. 16 luxurious rooms in cottages perched on cliffs with hammocks and lots of private areas for quiet sunbathing and relaxation around the property. Laid back, unpretentious and sociable. Yoga and massage facilities on site, lots of activities offered. Breakfast is included and dinner is served daily using fresh local produce.
Xtabi Resort $–$$$$ Lighthouse Road, West End, Negril, tel: 957-0121, www.xtabi-negril.com. Seafront, poolside or garden cottages, rooms or a suite, on the cliffs offer a choice of simple or luxury accommodation, all with balcony or verandah, some with air conditioning. Steps lead down to caves for snorkelling. Native wood floors and rustic furnishings, outside showers with privacy walls. All rooms have safes, some have kitchenettes and many have refrigerators; the garden rooms are more modern, with tiled floors and air-conditioning. Cliffside bar and restaurant.
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{
"redpajama_set_name": "RedPajamaBook"
}
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O'Hara was one of two residents who recently reported possible wild-animal attacks on their pets to Orange County Animal Services.
The reports prompted the county to urge residents to report all missing or injured pets. In a release, the county noted the possibility of coyotes being involved in the attacks. No one saw the incidents but coyotes were spotted where they occurred.
Managers at the Chapel Hill Country Club, where O'Hara found Gus, said the groundskeeper only guessed coyotes had attacked the little dog and that no one had seen the predators on the course.
"It could have been; it's possible," assistant manager Jeff Earley said. But "people see a fox or a big dog out there, and they think it's a coyote."
A state wildlife expert, however, said based on a description of Gus' injuries and other factors, it's "highly suspected" that a coyote or coyotes attacked him.
"It does sound typical of the types of wounds we see when a coyote attacks," said Colleen Olfenbuttel, the black bear and fur-bearing biologist for the N.C. Wildlife Resources Commission. 'Of course we'll never know for sure."
Three factors suggest a coyote or coyotes attacked O'Hara's dog:
• Gus was bitten in the throat, back and rear legs, consistent with how coyotes attack and begin eating their prey, she said. Domestic dogs tend to "sloppier," being less efficient killers and making lots of bites all over the body.
• Gus was found a distance from his home, suggesting he was dragged, consistent with a coyote removing prey from near people. Dogs don't do that, she said. "They usually just kill it and leave it."
• This is coyote breeding season. The animals become highly territorial before mating and rearing their pups.
"They view dogs as trespassers on their territories," Olfenbuttel said. "Unfortunately some dogs do get attacked and killed."
Co-existence
Coyotes are in all 100 North Carolina counties. They can be killed year round – hunters shot an estimated 27,152 in 2012-13 and trappers another 3,852, Olfenbuttel said – but they adapt quickly to changes in their environment, including threats.
Orange County Animal Services Director Bob Marotto initially said the two recent attacks were in northwest Chapel Hill. But he corrected himself last week. In addition to O'Hara's report off Pinehurst Drive in southeastern Chapel Hill, a fatal attack on a dog was reported near Arboretum and Poinsett drives, in eastern Chapel Hill.
Marotto and an educator with the Humane Society of the United States, who gave a standing-room-only talk on coyotes last year at the Orange County animal shelter, encourage co-existence.
Most coyotes avoid people. If you see one, wave your arms and shout at it. Protect dogs and cats by keeping your animals on leashes or indoors, securing garbage cans and not leaving pet food outdoors.
"It's very understandable," Olfenbuttel said. "People want to open the back door and let their dogs outside."
But to be safe, pet owners should accompany dogs outside or build a 6-foot fence; a coyote can scale anything shorter, she said.
And if you're worried about coyotes, rethink bird feeders. They attract squirrels and other small animals that coyotes eat, she said.
Side of the fairway
O'Hara saw her dog on the side of the fairway just as dawn was breaking and ran to him.
"I reached down to pick Gus up," she said. "He didn't move or make a sound and he was freezing cold, but his eyelid fluttered."
She carried him to the car and drove to her vet, where a veterinarian stabilized him, and then to the vet school, where he spent three hours in surgery.
"His whole body was filled with tubes," she said. "He was so damaged."
A few days later, O'Hara said, she "let him go."
O'Hara did not know there were coyotes in Chapel Hill, but after Gus was attacked she heard from neighbors who had seen coyotes, even several coyotes together.
"I'd seen deer, I'd seen a red fox," she said. "I haven't seen a red fox in more than two years. I imagine the coyotes have gotten them too."
Gus loved to go for rides and kayaking, O'Hara said.
"I'm distraught over losing my boy. ... He was my constant companion."
"I don't want this to happen to anyone else," she said. "And it's going to."
Gus, an 11-year-old West Highland White Terrier, was attacked in January off Pinehurst Drive in Chapel Hill. His owner reported the incident as a possible coyote attack to the Orange County Animal Service Department.
CNN's Brooke Baldwin gets graduation speech advice from Charles Barkley
NAACP rally calls on Price for town hall meeting
90% of Orange County high school teachers are white. This group wants to change that.
By Shelbi Polk
Representatives of several faith communities gathered at the Orange County Schools central office to ask the district to focus on hiring diverse and bilingual staff, as well as model leadership.
MORE CHAPEL HILL NEWS
Chapel Hill-Carrboro City Schools taps former member to fill board vacancy
Southern Season's former executives not guilty of fraud, a federal judge rules
Chapel Hill: Obituaries
Allen H. Barton
Joyce Marie Harper
In Memory of Charles Warren Millard, III
Dr. Kemp Jones
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}
The Jupiter Trojans are a collection of asteroids that lie in a 1:1 mean motion resonance with Jupiter and are confined to two extended swarms centered about the L$_{4}$ and L$_{5}$ Lagrangian points, which lead and trail the planet's motion by an angular distance of $\sim 60$ degrees. Since the first such asteroid was discovered more than a century ago, thousands of Trojans have been confirmed, and the current catalog contains over 6000 objects ranging in size from (624) Hektor, with a diameter of roughly 200 km, to subkilometer-sized objects. Estimates of the total number of Trojans larger than 1 km in diameter range from $\sim 1.0\times10^{5}$ \citep{nakamura} to $\sim2.5\times10^{5}$ \citep{szabo}, corresponding to a bulk mass of approximately $10^{-4}$ Earth masses. These values are comparable with those calculated for main belt asteroids of similar size, making the Trojans a significant population of minor bodies located in the middle Solar System. The orbits of Trojans librate around the stable Lagrangian points with periods on the order of a hundred years and are stable over the age of the Solar System, although long-timescale dynamical interactions with the other outer planets decrease the regions of stability and lead to a gradual diffusion of objects from the Trojan swarms \citep{levisonnature}. Escaped Trojans may serve as an important source of short-period comets and Centaurs, a few of which may have Earth-crossing orbits \citep{marzari}.
Due to their peculiar location and dynamical properties, Trojans lie at the intersection of several of the most important topics in planetary science. The origin and evolution of this population have been a subject of particular interest in recent decades. Early theories proposed a scenario in which the Trojans formed at the same heliocentric distance as Jupiter. In this model, Trojans were created out of the body of planetesimals and dust in the primordial solar nebula that remained after the runaway mass accretion phase of Jupiter and were subsequently stabilized into their current orbits around the Lagrangian points \citep{oldmodel}. However, it has been shown that such \textit{in situ} formation at 5.2~AU cannot explain the presently observed total mass and broad orbital inclination distribution. A recent theory, known as the Nice model, suggests a more complex picture in which the Trojan population originated in a region beyond the primordial orbit of Neptune, and the orbits of Jupiter and Saturn were initially situated much closer to the Sun than they are now \citep{tsiganis}. Through interactions with neighboring planetesimals and perhaps an encounter with a large Neptune-sized body \citep{nesvorny}, the gas giants underwent a rapid migration, crossing resonances and setting off a period of chaotic dynamical alterations in the outer Solar System. It is hypothesized that during this time, the primordial trans-Neptunian planetesimals were disrupted, and a fraction of them were scattered inwards and captured by Jupiter as Trojan asteroids, while the remaining objects were thrown outwards to larger heliocentric distances and eventually formed the Kuiper belt \citep{morbidelli}.
The current understanding of the composition of Trojan asteroids remains incomplete. Visible spectroscopy has shown largely featureless spectra with spectral slopes ranging from neutral to moderately red \citep[e.g.,][]{dotto,fornasier,melita}. Spectroscopic studies of Trojans have also been carried out in the near-infrared, a region that contains absorption bands of materials prevalent in other minor body populations throughout the Solar System, such as hydrous and anhydrous silicates, organics, and water ice \citep[e.g.,][]{emerybrown,dotto,yangjewitt,emery}. These spectra were likewise found to be featureless and did not reveal any incontrovertible absorption signals to within noise levels. As such, models of the composition and surface properties of Trojans remain poorly constrained. However, several authors have noted bimodality in the distribution of various spectral properties: Bimodality in spectral slope has been detected in both the visible \citep{szabo,roig,melita} and the near-infrared \citep{emery}. The infrared albedo of Trojans has also been shown to display bimodal behavior \citep{grav}. These observations indicate that the Trojans may be comprised of two separate sub-populations that categorically differ in their spectroscopic properties.
While future spectroscopic study promises to improve our knowledge of Trojan composition and structure, a study of the size distribution, or as a proxy, the magnitude distribution, may offer significant insight into the nature of the Trojan population. The magnitude distribution preserves information about the primordial environment in which the Trojans were accreted as well as the processes that have shaped the population since its formation, and can be used to test models of the origin and evolution of the Trojans. In particular, an analysis of the distribution of the attested sub-populations may further our understanding of how these sub-populations arose and how they have changed over time. In this paper, we use published photometric and spectroscopic data to categorize Trojans into two sub-populations and compare their individual magnitude distributions. When constructing the data samples, we evaluate and correct for incompleteness to better model the true Trojan population. In addition to fitting the magnitude distributions and examining their behavior, we explore various interpretations of the data.
\section{Trojan data}
Several sources were consulted in compiling the Trojan data samples analyzed in this work. They are described in the following.
\subsection{Selection of Trojan Dat Samples}
The primary data set is comprised of Trojan asteroids listed by the Minor Planet Center (MPC),\footnote{\texttt{www.minorplanetcenter.org} (Accessed 2014 May 10).} which maintains a compilation of all currently-confirmed Trojans. Absolute magnitude ($H$) and orbital parameter values were taken off of Edward Bowell's ASTORB datafile.\footnote{\texttt{http://www.naic.edu/~nolan/astorb.html} (Accessed 2014 April 27).} The resulting data set, referred to in the following as the \textit{main sample}, contains 6037 Trojans. Of these, 3985 are from the L$_{4}$ swarm and 2052 are from the L$_{5}$ swarm, corresponding to a leading-to-trailing number ratio of 1.94. This significant number asymmetry between the two swarms has been widely noted in the literature and appears to be a real effect that is not attributable to any major selection bias from Trojan surveys, at least in the bright end of the asteroid catalog \citep{szabo}. The brightest object in the main sample has an absolute magnitude of 7.2, while the faintest object has an absolute magnitude of 18.4. The vast majority of Trojans in the main sample (4856 objects) have $H\ge 12.5$, with most of these faint asteroids having been discovered within the last 5 years. In the literature, estimates of the threshold magnitude below which the current total Trojan asteroid catalog is complete lie within the range $H \sim 10.5 - 12$. Therefore, it is only possible to adequately analyze the magnitude distribution of faint Trojans if appropriate scaling techniques are invoked to correct for sample incompleteness. These techniques are discussed in Section 3.2.
Another data set used in this work consists of observations from the fourth release of the Moving Object Catalog of the Sloan Digital Sky Survey (SDSS-MOC4). The SDSS-MOC4 contains photometric measurements of more than 470,000 moving objects from 519 observing runs obtained prior to 2007 March. Of these objects, 557 have been identified to be known Trojans listed in the ASTORB file (243 from L$_{4}$ and 314 from L$_{5}$), and will be referred to in the following as the \textit{Sloan sample}. This data sample includes measured flux densities in the \textit{u, g, r, i, z} bands, centered at 3540, 4770, 6230, 7630, and 9130 {\AA}, respectively, and with bandwidths of $\sim 100$ \AA. As discussed in detail by \cite{szabo}, the distribution of the positions of SDSS observing fields through June 2005 in a coordinate system centered on Jupiter indicates that both L$_{4}$ and L$_{5}$ Trojan swarms were well-covered (i.e., the positions of the observing fields cover a wide range of orbital eccentricity and relative longitude values consistent with Trojan asteroids). Those authors identified 313 known Trojans in the SDSS-MOC3 (previous release) and determined that the survey detected all known Trojans within the coverage area brighter than $H=12.3$. Observing runs since then have expanded the coverage of the sky to include new Trojan swarm regions, yielding 244 additional known Trojans. It is expected that the detection threshold of the Sloan survey (i.e., magnitude to which the SDSS has detected all Trojans within its observing fields) in these newly-covered regions is similar to that determined for the previously-covered regions, and therefore, we may consider our Sloan sample to be a reliable subset of the total Trojan population up to $H\sim 12.3$. This means that the detection threshold of the Sloan sample lies at least 1~mag fainter than the completeness limit of the main sample mentioned above. As part of the analysis presented in the next section, we will confirm the detection threshold of the Sloan sample and use it to arrive at a better estimate of the completeness of the main sample.
\subsection{Categorizing Trojans}
Recent observational studies have identified bimodality in the Trojan population with respect to various photometric and spectroscopic quantities. In this work, we used three earlier analyses of Trojans to classify objects into two color populations.
In \cite{emery}, near-infrared (0.7$-$2.5 $\mu$m) spectra of 58 Trojans were collected during four observing runs at the NASA Infrared Telescope Facility and were combined with previously-published spectra of 10 other Trojans. Together, these objects range in magnitude from $H = 7.2$ to $H = 10.7$. For each object, the authors measured the reflectance fluxes in four bands, centered at 0.85, 1.22 (J-band), 1.63 (H-band), and 2.19 (K-band) $\mu$m, from which color indices were calculated using $m_{\lambda1}-m_{\lambda2} = 2.5\log(R_{\lambda2}/R_{\lambda1})$, where $m_{\lambda1}-m_{\lambda2}$ is the color index for two wavelengths, and $R_{\lambda2}/R_{\lambda1}$ is the ratio between the corresponding reflectance fluxes. These color indices quantify the spectral slopes of the Trojans in the near-infrared, with higher index values corresponding to redder spectra. Notably, the plot of the J-K color index versus the 0.85-J color index for the asteroids analyzed is not continuous; rather, there is a distinct break separating a redder group (Group I) from a less red group (Group II). The distribution of the 0.85-H color index likewise shows a clear bimodality, while the H-K histogram is unimodal, suggesting that the difference between the two groups of asteroids is concentrated primarily in the short-wavelength end of the near-infrared spectrum ($\lambda < 1.5$~$\mu$m). Both L$_{4}$ and L$_{5}$ swarms were shown to display similar bimodal behavior, and it was determined that the two identified groups in the analyzed Trojan sample could not have been drawn from a unimodal distribution to a very high confidence level ($>99.99\%$). We included the color indices of 15 additional Trojans (Emery et al., in preparation) for a total of 83 objects, which we categorized into Group I (19 objects) and Group II (64 objects).
\cite{grav} presented thermal model fits for 478 Trojans observed with the Wide-field Infrared Survey Explorer (WISE), which conducted a full-sky survey in four infrared wavelengths: 3.4, 4.6, 12, and 22 $\mu$m (denoted W1, W2, W3, and W4, respectively). Using the survey data, the W1 albedo was computed for each object, and it was shown that the distribution of W1 albedos as a function of diameter is discernibly bimodal for the 66 objects with diameters larger than $\sim$60 km, which corresponds to objects brighter than $H\sim 9.6$; for the smaller (fainter) Trojans, the errors in the measured albedos are much larger, and a clear bimodality was not discernible. Among these 66 large Trojans, 51 have W1 albedo values between 0.11 and 0.18 (Group A), while 15 have W1 albedo values between 0.05 and 0.10 (Group B). Within each group, the albedo values show no dependence on diameter and are tightly clustered, with average separations between adjacent albedo values of 0.001 and 0.004 for Group A and Group B, respectively.
More importantly, when considering the Trojans that are in both the \cite{grav} and the \cite{emery} data sets, one finds that every object in Group A is a member of Group I, and every object in Group B is a member of Group II, with the sole exception of (1404) Ajax, which has high H-K and 0.85-J color indices characteristic of redder Group I objects, but a relatively low W1 albedo value of 0.085. This correspondence between groups categorized with respect to different spectroscopic quantities reinforces the proposal presented by \cite{emery} that the Trojans are comprised of two distinct populations with dissimilar spectral properties and likely different compositions. In particular, we conclude that Group I and Group A are both sampled from one of the two Trojan populations; these objects have redder color indices, and we will refer to this population as the red (R) population. Analogously, Group II and Group B are both sampled from the second Trojan population, which will be referred to as the less red (LR) population, due to the relatively lower near-infrared color indices of its members.
Using the robust and consistent bimodalities observed by \cite{emery} and \cite{grav}, we categorized 93 Trojans as either LR (20 objects) or R (73 objects). However, these population sizes are too small to allow for statistically meaningful statements about the overall Trojan population. Moreover, the faintest object in this group has an absolute magnitude of $H=10.7$, which would restrict our analysis of the Trojan color populations to just the relatively bright objects. In order to expand our categorization of Trojans into color populations, we turned to photometric data from the Sloan survey.
\cite{roig} studied 250 known Trojans from the SDSS-MOC3 and computed spectral slopes from the listed \textit{u, g, r, i, z} band flux densities. The authors noted that the distribution of spectral slopes is bimodal. We expanded on this study, reproducing the spectral slope calculations and including new Trojans listed in the SDSS-MOC4. Following the procedure used in \cite{roig}, we corrected the flux densities using the solar colors provided in \cite{ivezic}: $c_{u-r}=(u-r)-1.77$, $c_{g-r} = (g-r)-0.45$, $c_{r-i}=(r-i)-0.10$, and $c_{r-z}=(r-z)-0.14$. The reflectance fluxes, $F$, normalized to $1$ in the \textit{r} band, were defined as: $F_{u} = 10^{-0.4c_{u-r}}$, $F_{g}=10^{-0.4c_{g-r}}$, $F_{i} =10^{0.4c_{r-i}}$, and $F_{z} = 10^{0.4c_{r-z}}$. The relative errors $\Delta F/F$ were estimated using the second-order approach in \cite{roiggilhutton}:
\begin{equation}\label{error}\Delta F/F = 0.9210\Delta c(1+0.4605\Delta c),\end{equation}
where the color errors $\Delta c$ are computed as the root-squared sum of the corresponding magnitude errors, e.g., $\Delta c_{u-r} = \sqrt{(\Delta u)^{2}+(\Delta r)^{2}}$. The error in $F_{r}$ was estimated using $\Delta c_{r-r} = \sqrt{2}\Delta r$. We discarded all asteroid observations that had a relative error greater than 10\% in any of the fluxes besides $F_{u}$, which usually has larger errors due to the effects of instrument noise in and around the \textit{u}-band. We also considered only asteroids with magnitudes in the range $H<12.3$, over which the Sloan survey is expected to have detected all Trojans within its survey area.
The resulting asteroid set contains 254 objects (114 in L$_{4}$ and 140 in L$_{5}$), 24 of which were included in the \cite{emery} and/or \cite{grav} analyses and previously categorized by spectrum. For each object, the spectral slope $S$ was computed from a linear least-squares fit to a straight line passing through the fluxes $F_{g}$, $F_{r}$, $F_{i}$, and $F_{z}$, taking into account the individual errors $\Delta F$ ($F_{u}$ was not used in this computation, as per \cite{roig}). If an object had multiple observations, the average of the spectral slopes computed for all observations was used. The histogram of spectral slopes is shown in Figure~\ref{spectralslope}. From the plot, the bimodality in the spectral slope distribution is evident.\footnote{In \cite{roig}, it was reported that only objects in the L$_{4}$ swarm showed this bimodality in spectral slope. Our present analysis includes many more asteroids from the SDSS-MOC4, and we observe bimodality in both L$_{4}$ and L$_{5}$ swarms.} By fitting the spectral slope distribution with two Gaussians, we found that one of the two modes is centered at $S = 5.3 \times 10^{-5}$~\AA$^{-1}$, while the other mode is located at higher spectral slopes (i.e., redder colors), with a peak at $S = 9.6 \times 10^{-5}$~\AA$^{-1}$; the best-fit Gaussian distribution functions are plotted in Figure~\ref{spectralslope}. This two-peaked distribution shape is similar to the one presented by \cite{emery} for the H-K color index. In particular, the 24 Trojans in the Sloan sample that have already been categorized into LR and R populations (4 in LR and 20 in R) align with the two modes shown in Figure~\ref{spectralslope}. Therefore, we can say that objects with spectral slope values consistent with the left mode belong to the LR population, while objects with spectral slope values consistent with the right mode belong to the R population. There is some overlap between the two modes, which makes it difficult to categorize all of the Trojans observed by the SDSS into populations. Nevertheless, we may expand our categorization by adopting conservative break-off spectral slope values: All Trojans with $S\le 5.3 \times 10^{-5}$~\AA$^{-1}$ were classified as less red, while all Trojans with $S\ge 9.6 \times 10^{-5}$~\AA$^{-1}$ were classified as red. Using this method, we were able to categorize 151 of the 254 asteroids in the SDSS-MOC4 with $H< 12.3$; 47 objects belong to the LR population, and 104 objects belong to the R population, with the remaining 103 objects being uncategorized.
The estimated 95\% detection flux density thresholds for the \textit{u, g, r, i, z} bands are 22.0, 22.2, 22.2, 21.3, and 20.5, respectively \citep{ivezic}. The average relative band magnitudes for the 151 Trojans in the color populations that were imaged by the SDSS are $u-r = 2.08$, $g-r = 0.62$, $i-r=-0.26$, $z-r = -0.42$ for R objects and $u-r = 2.01$, $g-r = 0.52$, $i-r=-0.18$, $z-r = -0.26$ for LR objects. For an object to be listed on the Moving Object Catalog, it must have detections in at least three bands. The detection threshold in the \textit{z}- and \textit{i}-bands are the lowest. For objects with the same \textit{r}-band magnitude, LR objects are less reflective at longer wavelengths, so for objects with magnitudes near the detection thresholds, there is a bias against LR objects. However, the differences between the relative band magnitudes among the two color populations are not large, and this bias is only expected to affect the objects with absolute magnitudes at the very faint end of our considered range and beyond. Therefore, for our data samples, this effect is minor and is not taken into consideration in our analysis.
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{f1.png}
\end{center}
\caption{Distribution of spectral slopes of all 254 Trojans in the Sloan sample with $H< 12.3$ (solid green), and the distributions of spectral slopes of 24 Trojans classified into the LR and R populations per \cite{emery} and \cite{grav} (blue with diagonal hatching and red with cross hatching, respectively). The best-fit Gaussian distribution functions for the two color populations are shown as black dashed lines.} \label{spectralslope}
\end{figure}
We have compared three photometric and spectroscopic studies of Trojans and determined that the bimodal behaviors observed in all these studies are consistent and indicative of the existence of two separate color populations. Of the 842 objects in the main sample with $H< 12.3$, 478 are in the L$_{4}$ swarm, and 364 are in the L$_{5}$ swarm, which entails a leading-to-trailing number ratio of 1.31. This ratio is notably smaller than the value of 1.94 obtained for the total Trojan catalog, which suggests that there may be major detection biases favoring L$_{4}$ Trojans among the faintest objects. After categorizing the objects in the main sample, we found that 64 objects belong to the LR population, and 157 objects belong to the R population, while the remaining 621 objects were not categorized because they have either not been analyzed by any of the three studies discussed above or have spectral slope values between $5.3 \times 10^{-5}$~\AA$^{-1}$ and $9.6 \times 10^{-5}$~\AA$^{-1}$. In Figure~\ref{colors}, the cumulative magnitude distribution $N(H)$, i.e., the total number of asteroids with absolute magnitude less than or equal to $H$, is plotted for the main sample and the two color populations. The distributions plotted here have not been scaled to correct for incompleteness.
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{f2.png}
\end{center}
\caption{Plot of the unscaled cumulative magnitude distributions for the main Trojan sample (black circles) and the categorized R and LR color populations (red squares and blue triangles, respectively). These data have not yet been corrected for incompleteness.} \label{colors}
\end{figure}
\section{Analysis}
In this section, the magnitude distributions of the Trojan populations are studied. We present best-fit curves to describe the magnitude distributions and compare their behavior.
\subsection{Population Distinctness}
Previously, we classified Trojans into LR and R populations based on various spectroscopic and photometric quantities. While the observed bimodalies indicate that the two populations differ categorically with respect to several spectral properties, the current lack of understanding of Trojan surface composition makes it difficult to use these spectral properties in studying the origin and evolution of Trojans. Moreover, the distinction in spectral properties does not preclude the possibility that the Trojans are simply a mixed population of LR and R objects, with a constant number ratio between the two populations at each magnitude. To determine whether the two color populations are distinct, we must compare the shape of their distributions.
While the LR and R populations are incomplete, there is no reason to believe that one of the two populations is significantly more complete than the other. In particular, the ratio of R to LR objects at each magnitude is not expected to be affected by any major bias (see Section~3.2 for details of our analysis of sample completeness). Since the difference in shape of two magnitude distributions is determined largely by the variation of the number ratio of the two distributions with respect to magnitude, we may test for population distinctness of the Trojan color samples by using the current LR and R populations as plotted in Figure~\ref{colors}, without the need to scale up both populations to correct for incompleteness.
Already from the unscaled cumulative magnitude distributions plotted in Figure~\ref{colors}, one can see that the distributions of the color populations are dissimilar. To analytically examine the distinctness of the LR and R populations, we used the two-sample Kuiper variant of the Kolmogorov$-$Smirnov test \citep[Kuiper$-$KS test;][]{press}. This nonparametric statistic quantifies the likelihood that two data samples are drawn from the same underlying distribution. It evaluates the sum of the maximum distances of one distribution above and below the other and returns a test decision value, $p$, between 0 and 1, which represents the probability that the two data samples are not drawn from the same underlying distribution. The Kuiper$-$KS test is sensitive to differences in both the relative location and the shape of the two cumulative distributions. It is particularly appropriate when dealing with distributions that differ primarily in their tails, as is the case with the Trojan color populations.
Running the Kuiper$-$KS test on the two color populations, we obtained a $p$-value of 0.973. This high test decision value demonstrates that the two color populations are not sampled from a single underlying distribution to a confidence level of 97.3\%. In other words, the LR and R Trojan populations are distinct not only with respect to the spectral properties of their members, but also with respect to their overall size/magnitude distributions.
\subsection{Sample Completeness}
When analyzing a population distribution, it is important to determine and properly correct for any incompleteness in the data sample. To ensure that our curve-fitting adequately models the true Trojan magnitude distribution, we used the Sloan sample to estimate the incompleteness of the main sample and color populations.
As discussed in Section~2.1, the detection threshold of the SDSS within its coverage area is much fainter than the completeness threshold of the overall Trojan catalog. The Sloan survey broadly sampled the orbital parameter space characteristic of both Trojan swarms. Important to our analysis is whether there exists any variation in the magnitude distribution of objects across different regions of the Trojan swarms, since such variation would lead to the total magnitude distribution of the Sloan sample being significantly different from the true total magnitude distribution. Recent studies of Trojans have not observed any discernible correlation between absolute magnitude and eccentricity or inclination in either the leading or the trailing swarm \citep[c.f.,][]{szabo,fernandez}, so it is unlikely that the Sloan sample is characterized by any bias with respect to magnitude. We may therefore consider the Sloan sample to be an accurate scaled-down representation of the overall Trojan population. With the exception of a few bright Trojans, all objects in the data samples have absolute magnitudes given with tenth-place accuracy (e.g., $H = 10.1$); in other words, they are effectively binned into 0.1~mag groups. To evaluate the completeness of our main sample, we examine the ratio $R$ between the cumulative number of objects in the Sloan sample and the cumulative number of objects in the main sample for each 0.1~mag bin. Over the range of magnitudes for which both the main sample and the Sloan sample are complete, $R$ should be roughly constant at some value. As the magnitude increases up to the detection threshold of the Sloan sample, the main sample becomes incomplete and $R$ should increase steadily. At higher magnitudes, past the detection threshold of the Sloan sample, $R$ is expected to decrease, since a large number of faint Trojans have been discovered since the release of the SDSS-MOC4.
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{f3.png}
\end{center}
\caption{Plot of the ratio between the cumulative number of objects in the Sloan sample and the cumulative number of objects in the main sample for various absolute magnitude bins (red circles). The black dashed line indicates the average value $R_{*}$ for bins with $H=10.0 \rightarrow 11.2$.} \label{comparison}
\end{figure}
Figure~\ref{comparison} shows the values of $R$ plotted with absolute magnitude. From the plot, the expected behavior described earlier is evident: For bins with $H< 11.3$, the value of $R$ is roughly constant at $R_{*} = 0.264$, which is the average of $R$ for bins with $H=10.0 \rightarrow 11.2$. (Bright objects were omitted from the average, since the small bin numbers lead to significant scatter in $R$.) At fainter magnitudes, $R$ increases until $H = 12.3$, after which it decreases rapidly. From this, we conclude that every Trojan brighter than $H=11.3$ is contained in the main sample (that is, the total Trojan catalog), while the Sloan sample is complete up to $H=12.3$ (that is, contains an unbiased subsample of Trojans), which confirms the completeness limit estimate given in \cite{szabo}. Using the calculated values of $R$, we can now evaluate the catalog efficiency $\eta_{\mathrm{mpc}}$ of the main sample, i.e., the ratio of the number of Trojans $n_{\mathrm{mpc}}$ currently cataloged by the Minor Planet Center to the true number of Trojans $n_{0}$, in each bin with $H<12.3$. For $H<11.3$, the main sample is complete, so $\eta_{\mathrm{mpc}}$ = 1. For $11.3\le H < 12.3$, we first evaluate the ratio $r(H)$ between the non-cumulative (i.e., differential or bin-only) number of Trojans in the Sloan and main samples for each 0.1~mag bin; the values of $r(H)$ in this interval are greater than the benchmark value of $r_{*} = \bar{r} = 0.29$, where $\bar{r}$ is the average of $r(H)$ over the interval $H=10.0 \rightarrow 11.2$. The catalog efficiency value for each bin is given by $r(H)/r_{*}$. We subsequently fit a fifth-order polynomial through the binned catalog efficiency values over the domain $11.3\le H < 12.3$ to arrive at a smooth functional form $\eta_{1}(H)$. The catalog efficiency can be expressed as a single piecewise-defined function:
\begin{equation}\label{detection}\eta_{\mathrm{mpc}}(H) =
\begin{cases}
1,&\text{for }H<11.3\\
\eta_{1}(H),&\text{for }11.3\le H < 12.3
\end{cases}.\end{equation}
More careful consideration must be made when correcting for incompleteness in the color populations. While the absolute magnitude distribution of Trojans does not appear be dependent on the location in orbital parameter space and would not be affected by the particular locations of observed fields within the Trojan swarms, as discussed earlier, correlations between the color of objects and orbital parameters may lead to biases in the resulting magnitude distributions of the color populations. Most Trojans (621 out of 842) were not categorized as either less red or red, with the brightest unclassified asteroid having $H=9.6$. The majority of objects in our color populations (151 out of 221) were classified using the spectral slope categorization method based off Sloan data. Using data from the SDSS-MOC3, \cite{szabo} and \cite{roig} reported a weak correlation between spectral slope and inclination, with objects at larger inclinations tending to be redder; \cite{fornasier} reported a similar correlation in their study of visible spectral slope and interpreted it as a lack of faint objects with low spectral slope. This color-inclination correlation was found to be the same in both swarms. \cite{szabo} identified a bias in their data: the L$_{5}$ subsample of Trojans had a significantly larger fraction of objects with high inclinations than the L$_{4}$ subsample. In our analysis, such asymmetric coverage would cause the number ratio of R-to-LR L$_{5}$ Trojans to be unrealistically inflated and skew the overall color distributions.
To determine whether a similar bias is present among the 254 objects in the current Sloan sample, we computed the fraction of objects in the SDSS-MOC4 with large inclinations ($i>20^{\circ}$) for the leading and trailing swarms independently. It was found that the fraction is similar for the two swarms (0.24 for L$_{4}$ and 0.22 for L$_{5}$). This means that observing runs since the release of the SDSS-MOC3 have captured more high-inclination regions of the L$_{4}$ swarm, and as a result, the leading and trailing swarms are equally well-sampled in the SDSS-MOC4 data. Therefore, no selection bias with respect to inclination is discernible in the Sloan sample, and we may consider the LR and R color populations defined in Section 2.1 to be a representative subset of the true color composition of the overall Trojan population. In particular, the number ratio of red to less red Trojans in each bin should be approximately the same as the true ratio at that magnitude. We define a categorization efficiency value for each bin, which is the ratio between the number of already-categorized Trojans in the LR and R populations, $n_{LR}(H)+n_{R}(H)$, and the total number of detected Trojans, $n_{\mathrm{det}}(H)$. Over the domain $9.6\le H < 12.3$, where the color classification is incomplete, we followed a similar procedure to that used in deriving the detection efficiency and fitted a polynomial through the categorization efficiency values to obtain a smooth function $\eta_{2}(H)$. We can write the overall categorization efficiency function as
\begin{equation}\label{categorization}\eta_{\mathrm{cat}}(H) =
\begin{cases}
1,&\text{for }H<9.6\\
\eta_{2}(H),&\text{for }9.6\le H < 12.3
\end{cases}.\end{equation}
This categorization efficiency function is the same for both LR and R populations and must be coupled with the detection efficiency function $\eta_{\mathrm{det}}(H)$ for $H\ge 11.3$.
The total efficiency functions for the main sample and color populations, which take into account catalog and/or categorization incompleteness, are given by:
\begin{equation}\label{eta}\eta(H) =
\begin{cases}
\eta_{\mathrm{mpc}}(H),&\text{for the main sample}\\
\eta_{\mathrm{cat}}(H)\times\eta_{\mathrm{mpc}}(H),&\text{for the LR and R populations}
\end{cases}.\end{equation}
We used catalog and categorization efficiency to scale up the data samples so that they approximate the true Trojan population. Similar scaling methods have been employed in the study of the size distribution and taxonomy of main belt asteroids \citep{demeo}. We demonstrate our method with the following example: at $H=11.5$, there are 43 objects in the main sample, 17 of which are also contained in the Sloan sample. The ratio between the number of objects in the Sloan and main samples is $r = 17/43 \approx 0.395$, which yields a catalog efficiency value of $\eta = r_{*}/r \approx 0.73$. Thus, the approximate true number of Trojans with $H=11.5$ is $n_{0}=43/\eta \sim 59$. The scaled and unscaled cumulative magnitude distributions for the main sample and color populations are shown in Figures~\ref{total}-\ref{LR}.
\subsection{Distribution Fits}
Previous analyses of the magnitude distributions of Trojans (see, for example, \cite{jewitt}) have shown that the differential magnitude distribution, $\Sigma(H) = dN(H)/dH$, is well-described by a broken power law with four parameters:
\begin{equation}\label{distribution}\Sigma(\alpha_{1},\alpha_{2},H_{0},H_{b}|H) =
\begin{cases}
10^{\alpha_{1}(H-H_{0})},&\text{for }H< H_{b}\\
10^{\alpha_{2}H+(\alpha_{1}-\alpha_{2})H_{b} - \alpha_{1}H_{0}},&\text{for }H \ge H_{b}
\end{cases},\end{equation}
where there is a sudden change from a bright-end slope $\alpha_{1}$ to a shallower faint-end slope $\alpha_{2}$ at some break magnitude $H_{b}$. $H_{0}$ is the threshold magnitude for which $\Sigma(H_{0}) = 1$ and serves to properly normalize the distribution to fit the data. \cite{jewitt} obtained the slope values $\alpha_{1}=1.1$ and $\alpha_{2}=0.4$ from their study of 257 Trojans, which did not correct for incompleteness in the faint-end distribution. More recent studies of faint Trojans by \cite{szabo} and \cite{yoshida} obtained faint-end slope values of 0.44 and 0.38, respectively.
We fitted the magnitude distributions of the total Trojan sample and the two color populations to the broken power law distribution function in equation~\eqref{distribution} by using a maximum likelihood method similar to the one used in \cite{fraser} for their study of Kuiper belt objects (KBOs). Given a list of Trojan magnitudes and a particular set of parameters for the distribution function to be fitted, this technique defines a likelihood function $L$, which returns the probability that a random sampling of the distribution will yield the data. The maximum likelihood method is well-suited for analyzing data sets like the ones under consideration, since it is robust to small data counts and non-Gaussian statistics, for which typical $\chi^{2}$ fitting methods are inappropriate. Also, other statistical considerations like catalog and categorization efficiency can be easily integrated into the formulation.
The likelihood function used in our fitting takes the form
\begin{equation}\label{likelihood}L(\alpha_{1},\alpha_{2},H_{0},H_{b}| H_{i})\propto e^{-N}\prod_{i}P_{i},\end{equation}
where $H_{i}$ is the absolute magnitude of each detected Trojan, $N$ is the total number of detected objects expected in the magnitude range under consideration, and $P_{i}$ is the probability of having object $i$ with magnitude $H_{i}$ given the underlying distribution function $\Sigma$. Taking into account detection and categorization incompleteness, $N$ is given by
\begin{equation}\label{N}N = \int^{H_{\mathrm{max}}}_{-\infty} \eta(H)\Sigma(\alpha_{1},\alpha_{2},H_{0},H_{b}|H) \,dH,\end{equation}
where $\eta(H)$ is the efficiency function defined in equation~\eqref{eta}, and $H_{max}=12.3$. By including the efficiency function, we ensure that the curves are fitted to the true Trojan distribution, not the incomplete detected Trojan distribution. The probability $P_{i}$ is simply the differential density function evaluated at $H_{i}$, i.e., $P_{i} = \Sigma(\alpha_{1},\alpha_{2},H_{0},H_{b}|H_{i}) $.
The best-fit distribution functions were obtained by maximizing the likelihood function over the four-dimensional parameter space using an affine-invariant Markov chain Monte Carlo (MCMC) Ensemble sampler with 100,000 steps \citep{mcmc}. The optimal parameters and corresponding 1$\sigma$ errors were computed for each distribution. The magnitude distribution of the main sample (all Trojans with $H< 12.3$) is best-fit by $\alpha_{1}=1.11\pm0.02$, $\alpha_{2}=0.46\pm 0.01$, $H_{0}=7.09^{+0.03}_{-0.02}$, and $H_{b}=8.16^{+0.03}_{-0.04}$. The bright-end slope is consistent with the value calculated in \cite{jewitt}, while the faint-slope is steeper than previously-obtained values, due to our correction for incompleteness in the Trojan catalog past $H=11.3$. The L$_{4}$ and L$_{5}$ Trojans were independently analyzed for detection completeness and fitted in a similar fashion. The optimal values of the slopes $\alpha_{1}$ and $\alpha_{2}$ for the two swarm distributions were found to be indistinguishable within calculated uncertainties. This agrees with the results of earlier studies \citep[see, for example,][]{yoshida2} and demonstrates that the leading and trailing Trojan swarms have magnitude distributions that are identical in shape, differing only in overall asteroid number.
The magnitude distributions of the color populations were both individually fitted to a broken power law. The optimal parameters for the R population magnitude distribution are $\alpha_{1}=0.97^{+0.05}_{-0.04}$, $\alpha_{2}=0.38\pm0.02$, $H_{0}=7.24^{+0.05}_{-0.07}$, and $H_{b}=8.70^{+0.08}_{-0.11}$, while for the LR population magnitude distribution, they are $\alpha_{1}=1.25^{+0.09}_{-0.04}$, $\alpha_{2}=0.52^{+0.03}_{-0.01}$, $H_{0}=7.77^{+0.04}_{-0.09}$, and $H_{b}=8.15^{+0.06}_{-0.10}$. Figures~\ref{total}-\ref{LR} show the cumulative magnitude distributions for the main sample and the color populations, along with the best-fit curves that describe the true distributions. In each plot, the lower distribution is the cumulative count for the unscaled data set, and the upper distribution is the approximate true distribution, scaled to correct for catalog and/or categorization incompleteness (as described in Section 3.2).
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{f4.png}
\end{center}
\caption{Plot depicting the scaled (white squares) and unscaled (black circles) cumulative magnitude distributions for the total Trojan population, along with the best-fit curve describing the true Trojan cumulative distribution.} \label{total}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{f5.png}
\end{center}
\caption{Plot depicting the scaled (magenta squares) and unscaled (red circles) cumulative magnitude distributions for R population, along with the best-fit curve describing the true cumulative distribution.} \label{R}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{f6.png}
\end{center}
\caption{Plot depicting the scaled (cyan triangles) and unscaled (blue circles) cumulative magnitude distributions for LR population, along with the best-fit curve describing the true cumulative distribution.} \label{LR}
\end{figure}
\section{Discussion}
The analysis of the Trojan magnitude distributions in the previous section, which utilized the most current asteroid catalog and corrected for catalog incompleteness, presents the most accurate picture to date of the true Trojan population up to the cutoff magnitude $H=12.3$.
Using the distribution fits calculated for both the total and the two color populations, we can try to reach a better understanding of the origin and evolution of the Trojans, and in particular, the nature of the two color populations.
The most notable feature of the magnitude distributions is the transition from a steep power-law slope to a shallower slope at $H\sim8 - 9$. Previous studies of the total Trojan magnitude distribution \citep[e.g.,][]{morbidelli} have suggested that the broken power-law shape separates the population into two groups: Objects with magnitudes brighter than the break magnitude are described by a power-law slope that reflects the primordial accretion processes that created the original Trojan population. On the other hand, objects with magnitudes fainter than the break magnitude form a sub-population that has reached collisional equilibrium and is mostly comprised of collisional fragments of larger objects. It was demonstrated in pioneering work by \cite{dohnanyi} that the magnitude distribution of a small body population that evolves solely through self-collisions attains an equilibrium power-law slope of $\alpha_{*} \sim 0.5$ when collisional equilibrium is achieved, regardless of the initial shape of the distribution. The faint-end slope of the total Trojan magnitude distribution that we obtained by fitting the data is $\alpha_{2} = 0.46\pm0.01$, which is consistent with the canonical collisional equilibrium slope. In relation to the history of the Trojan population, there arises the question of whether the sharp roll-over to a shallower faint-end slope in the currently-observed population is a consequence of collisional evolution after the Trojans were emplaced in their current orbits around Jupiter, or a result of collisional interactions in the primordial trans-Neptunian region prior to emplacement. Several authors have modeled the collisional evolution of Trojans and determined that the observed broken power-law distribution is best reproduced when assuming that a break was present at the time of emplacement \citep[c.f.,][] {marzari,deelia}. Furthermore, these studies have shown that the intrinsic collision probabilities characteristic of the Trojan swarms are insufficiently high to have brought about any significant collisional evolution among objects with magnitudes brighter than the break. Thus, the currently-observed bright-end distribution reflects the shape of the primordial size distribution of large Trojans at the time of emplacement.
A more peculiar aspect of the Trojans is the magnitude distributions of the color populations - in particular, the difference between the faint-end slopes of the R and LR populations ($0.38\pm0.02$ and $0.52^{+0.03}_{-0.01}$, respectively). The Kuiper$-$KS test demonstrated that the magnitude distributions of the color populations are remarkably distinct, which indicates that the two populations likely formed in different places before being emplaced into the Trojan regions. While the fitted bright-end slopes are different, the distinction is most apparent in the faint-end portion of the distributions. (Running the Kuiper$-$KS test on just the bright-end portions of the color distributions yielded intermediate $p$-values, which are inconclusive as a metric for population distinctness.) A hypothesis that posits a scenario in which the two color populations arose from different regions in the primordial trans-Neptunian disk would be able to explain the different bright-end slopes, which are determined primarily by the accretion environment. However, in light of the interpretation that the faint-end portion of the broken power-law distributions is a result of collisional evolution, the significant difference between the faint-end slopes poses a challenge. One possible explanation would be that just as different accretion environments can lead to different bright-end slopes, non-uniform collisional dynamics in the primordial trans-Neptunian disk could have resulted in the color populations experiencing different early collisional histories owing to their different formation regions. Various areas of the primordial disk may have been characterized by a wide range of impact velocities and intrinsic collision probabilities. In such a model, the currently-observed discrepancy between the faint-end slopes would be a relic of the pre-emplacement collisional evolution of the two color populations. Indeed, very little is known about the nature of the early Solar System, so one could not exclude this possibility.
That said, the fact that the overall Trojan population is characterized by a faint-end slope so close to the canonical collisional equilibrium slope suggests that perhaps there is another explanation in which the two color populations experienced a similar collisional evolution within the primordial trans-Neptunian disk and were emplaced with similar faint-end slopes. In this case, the different faint-slopes would be explained by positing a mechanism that converts R objects to LR objects, hence flattening the faint-end slope of the R population, while simultaneously steepening the faint-end slope of the LR population.
Previous laboratory work has shown that irradiation of surfaces rich in terrestrial bitumens and other organic compounds, which tend to have a characteristic red color, leads to the flattening of the spectral slope and a resulting less red color \citep{moroz,kanuchova}. However, since the incident radiation flux on the surface of a spherical body scales in tandem with size, this flattening effect is expected to be the same across the full range of Trojan sizes and hence does not explain the discrepant faint-end slopes observed in the magnitude distributions of the color populations. Furthermore, the timescale for flattening the spectrum of a R Trojan is much smaller than the time that has elapsed since emplacement and formation \citep{melita09}, so if irradiation is the sole mechanism for converting R objects to LR ones, one would not expect any R objects to remain. In \cite{melita09}, an additional mechanism is proposed whereby minor cratering events disrupt the spectrally flattened irradiation crust and excavate underlying material, which the authors of that work posit as being red in color, consistent with that of typical surfaces rich in complex organic materials. The added contribution of cratering leads to irradiated LR objects becoming R objects once again through resurfacing, thereby preventing all the R objects from turning into LR objects. However, the characteristic collisional timescale and, correspondingly, the timescale of resurfacing decrease with decreasing asteroid size, while the rate of irradiation is the same for all objects, as mentioned earlier. Therefore, the resurfacing of Trojans through cratering becomes more effective at returning irradiated LR objects to R objects when one goes to smaller sizes. This would lead to a relative excess of R objects at faint magnitudes, which is the opposite of what is evident in the observed color distributions.
In this work, we suggest an alternative explanation for the discrepancy in faint-end slopes and examine the possibility that the fragments resulting from a catastrophic shattering impact on a R object are LR. In other words, we hypothesize that R and LR Trojans have more or less identical interiors, differing only in the spectroscopic properties of their outer surfaces, and that the destruction of red objects is the primary mechanism by which R objects become LR, thereby resulting in a relative depletion of red Trojans in the range of sizes for which shattering collisions have been significant. To assess the viability of this conversion hypothesis, we ran a series of simple numerical simulations that model the collisional evolution of the Trojan population since emplacement. The mechanics of our algorithm are similar to those used in previous studies of Trojan collisions \citep[c.f.,][]{marzari}. Earlier works have shown that the overall Trojan$-$Trojan collisional frequency among large objects with $H > 9$ is very low ($\ll$ 1 Gyr$^{-1}$). This means that most of the collisional activity is concentrated in the faint-end of the magnitude range, and that the magnitude distribution of bright objects is expected to remain almost unchanged over the age of the Solar System. Therefore, we only considered initial magnitude distributions that are broken power-laws of the form described in equation~\eqref{distribution} with a bright-end distribution identical to that of the currently-observed population ($\alpha_{1} = 1.11$, $H_{0} = 7.09$, and $H_{b} = 8.16$). For the initial faint-end slope, we considered values ranging from 0.45 to 0.55, in increments of 0.01. Objects in the initial population with absolute magnitudes in the range $H=7\rightarrow 23$ were divided into 50 logarithmic diameter bins using the conversion formula $D = 1329\times 10^{-H/5}/\sqrt{p_{v}}$, where we have assumed a uniform geometric albedo of $p_{v} = 0.04$ \citep{fernandez}.
The initial color populations were constructed by taking constant fractions of the total population across all bins; based on the calculated 0.1~mag bin number ratios between R and LR Trojans in the bright-end portion of our data, we considered initial R-to-LR number ratios, $k$, ranging from 4 to 5, in increments of 0.5. The collisional evolution was carried out over 4~Gyr in 100,000 time steps of length $\Delta t= 40000$. At each time step, the expected number of collisions $N_{\mathrm{coll}}$ between bodies belonging to any pair of bins is given by
\begin{equation}N_{\mathrm{coll}}=\frac{1}{4}\langle P\rangle N_{\mathrm{tar}}N_{\mathrm{imp}}\Delta t (D_{\mathrm{tar}}+D_{\mathrm{imp}})^{2},\end{equation}
where $N_{\mathrm{tar}}$ and $N_{\mathrm{imp}}$ are the number of objects in a target bin with diameter $D_{\mathrm{tar}}$ and an impactor bin with diameter $D_{\mathrm{imp}}$, respectively; $\langle P\rangle = 7.35 \times 10^{-18}~\mathrm{yr}^{-1}~\mathrm{km}^{-2}$ is the intrinsic collision probability for Trojan$-$Trojan collisions and was approximated by the weighted average of the probabilities calculated by \cite{delloro} for L$_{4}$ and L$_{5}$ Trojans, taking into account the currently-observed number asymmetry between the two swarms. For a target bin with diameter $D_{\mathrm{tar}}$, only impactor bins with diameters satisfying the condition $D_{\mathrm{imp}} \ge D_{\mathrm{min}}$ were considered, where $D_{\mathrm{min}}$ is the minimum impactor diameter necessary for a shattering collision and defined as \citep{bottke}
\begin{equation}D_{\mathrm{min}}=\left(\frac{2Q_{D}^{*}}{V_{\mathrm{imp}}^{2}}\right)^{1/3}D_{\mathrm{tar}},\end{equation}
where $V_{\mathrm{imp}}=4.6~\mathrm{km}~\mathrm{s}^{-1}$ is the weighted average of the L$_{4}$ and L$_{5}$ impact velocities calculated by \cite{delloro}, and $Q_{D}^{*}$ is the strength of target. In our algorithm, we utilized a size-dependent strength scaling law based off one used by \cite{durda} in their treatment of collisions among small main-belt asteroids:
\begin{equation}\label{scaling}Q_{D}^{*} = c\cdot 10\cdot(155.9D^{-0.24} + 150.0D^{0.5} + 0.5D^{2.0})~\mathrm{J\,kg}^{-1},\end{equation}
where a parameter $c$ was included to adjust the overall scaling of the strength and varied in increments of 1 from 1 to 10 in our test trials.
Our model tracked the collisional evolution of the two color populations separately and computed the number of collisions between objects of the same color, as well as collisions involving objects of different colors. For each time step, the number of collisions between all relevant pairs of bins was calculated, and the corresponding target and impactor numbers were subtracted from their respective bins. In all cases, regardless of the color of the target and/or impactor, the collisional fragments were redistributed into LR bins, thereby modeling the conversion of R objects to LR fragments through shattering. After running simulations for all possible values of the parameters ($\alpha_{2}$, $k$, $c$), we found that a large number of test runs yielded final total and color magnitude distributions that were consistent with the observed distributions analyzed in Section 3. To determine which run best reproduced the calculated faint-end slopes, we compared the simulation results directly with the fitted distribution curves. The test run that resulted in the best agreement with the data had an initial total distribution with faint-end slope $\alpha_{2}=0.47$, a strength scaling parameter $c = 6$, and began the collisional time integration with a R-to-LR bin number ratio $k = 4.5$. Plots comparing the final simulated distributions from this test run with the observed data are shown in Figure~\ref{fig:simulations}.
Although the simulations did not take into account other processes that may have affected the Trojan asteroids (e.g., dynamical dissipation), several conclusions about the evolution of Trojans can be made. First, the similarity between the initial test distributions that yielded good agreement with the data and the present-day total magnitude distribution indicates that collisional evolution has not played a major role in the post-emplacement development of the Trojan population, at least in the magnitude range we have considered in this work. In fact, our simulations are consistent with there being only 1 or 2 major collisions (involving asteroids with $D>100$~km) in the past 4~Gyr. To date, the only incontrovertible asteroid family that has been detected among the Trojans is the Eurybates family \citep{broz}, which shows that the currently-observed bright-end distribution is largely identical to the bright-end distribution of the primordial Trojan population at the time of emplacement. Second, the R-to-LR collisional conversion model has yielded simulated final color distributions that match the currently-observed color magnitude distributions well. This model is also supported by photometric data from members of the Eurybates family, all of which have very low spectral slope values that are consistent with LR objects \citep{fornasier}. Thus, the conversion hypothesis offers a feasible explanation for the curious faint-end slope discrepancy between the R and LR populations.
The R-to-LR conversion model assessed here is attractive because it has some basis in recent work on KBOs, which, in the Nice model, arise from the same body of material as the Jupiter Trojans. The Kuiper belt is comprised of several sub-populations, among which are the so-called ``red'' and ``very red'' small KBOs \citep{fraser,peixinho}. A recent hypothesis describes a scenario in which KBOs formed in the trans-Neptunian disk at a range of heliocentric distances \citep{brown}. During formation in the primordial disk, all of these objects would have accumulated a mix of rock and volatile ices of roughly cometary composition. After the disk dissipated, the surface ices on these bodies began sublimating from solar radiation, leading to differential sublimation of individual ice species based on the location of the object. Whether a particular volatile ice species on the surface of these objects is retained or sublimates away is dependent on the volatility of the ice species and the temperature of the region where the object resides. As a result, for each ice species, there would have existed some threshold heliocentric distance for which objects at greater heliocentric distances would have retained that ice species on their surfaces, while those that formed closer in would have surfaces that were completely depleted in that ice species. Irradiation of surface ices would lead to significant darkened irradiation mantle, which serves to protect ices embedded deeper down from sublimation and the further action of irradiation. Therefore, the hypothesis in \citet{brown} argues that the presence or absence of one particular volatile ice species may be the key factor in producing the observed bimodality in color among the small KBOs: Objects that retained that volatile ice species on their surfaces formed a ``very red'' irradiation mantle, while those that lost that volatile ice species from their surfaces formed a ``red'' irradiation mantle.
If the LR and R Jupiter Trojans were drawn from the same two sources as the ``red'' and ``very red" KBOs, any exposed volatile ices on the surface would have evaporated away during the process of emplacement to smaller heliocentric distances. In our hypothesis, we posit that the more intense irradiation at $\sim 5$~AU flattens the spectral slope of the irradiation mantles that formed prior to emplacement. As a result, the Trojans that formed a ``red'' irradiation mantle would be left with surfaces that appear relatively less red, while those that formed at greater heliocentric distances and developed a ``very red'' irradiation mantle would end up with the relatively redder surfaces characteristic of R Trojans. When a Trojan shatters during a catastrophic impact, the irradiation mantle on the surface would disintegrate and any newly-exposed volatile ices in the interior (including, crucially, the particular species responsible for the formation of the ``very red'' irradiation mantle) would sublimate away within a relatively short timescale. Thus, if one assumes that LR and R Trojans have similar interior compositions, the fragments resulting from the shattering of a R Trojan would indeed be spectroscopically identical to those that would result from shattering a LR Trojan. Subsequent irradiation of these pristine fragments would eventually raise the spectral slope slightly, but not to the extent as would result if volatile ices were retained on the surface. As a consequence, in the range of magnitudes for which collisions are significant, shattering events since emplacement would have gradually depleted the number of R Trojans while simultaneously enriching the number of LR Trojans.
Ultimately, the nature of the Trojans and the source of their bimodal color distribution may involve a complex interplay between several different physical processes. A full understanding of the origin of this color bimodality and the mechanisms that have shaped the Trojan color populations hinges upon better knowledge of the composition and chemistry of these objects, which may be obtained in the future with higher-quality spectroscopic observations.
\begin{figure}
\centering
\subfigure[Comparison plot showing the final total distribution generated by the simulation (dashed black line) and the observed total distribution of Trojans, scaled to correct for incompleteness (white circles).]{
\includegraphics[width=0.4\textwidth]{f7a.png}
\label{fig:total}
}
\subfigure[Same as (a), but for the R (red dashed line and squares) and LR (blue dash-dotted line and triangles) color distributions.]{
\includegraphics[width=0.4\textwidth]{f7b.png}
\label{fig:color}
}
\subfigure[Plot comparing the best-fit distribution curves computed from the observed data (solid lines) with the final distribution curves generated by the simulation (dashed liens) for the total, R, and LR populations.]{
\includegraphics[width=0.4\textwidth]{f7c.png}
\label{fig:fits}
}
\caption{Comparison between the results from the best test run ($\alpha_{1}=1.11$, $\alpha_{2} = 0.47$, $H_{0}=7.09$, $H_{b} = 8.16$, $c = 6$, $k = 4.5$) and the observed Trojan magnitude distributions.}\label{fig:simulations}
\end{figure}
\section{Conclusion}
In this paper, we have examined the magnitude distributions of the two color populations that make up the Jupiter Trojans. Earlier spectroscopic and photometric studies in the visible, near-infrared, and infrared were compared and shown to be consistent with one another, confirming the existence of two separate populations of Trojans whose members differ categorically with respect to various spectral properties. Using primarily spectral slope values calculated from the SDSS-MOC4 photometric data, we were able to categorize 221 Trojans with absolute magnitudes less than 12.3 into the R and LR color populations. In the process of compiling the data samples and evaluating for catalog and categorization incompleteness, we concluded that the current Trojan catalog is complete to $H=11.3$, while the SDSS is likely to have detected all Trojans in its coverage area with $H<12.3$.
Using the Kuiper$-$KS test, we demonstrated that the two color populations have magnitude distributions that are distinct to a high confidence level. Fitting the distributions to a broken power law, we found that both the bright-end ($\alpha_{1}^{\mathrm R}=0.97^{+0.05}_{-0.04}$ versus $\alpha_{1}^{\mathrm LR}=1.25^{+0.09}_{-0.04}$) and the faint-end ($\alpha_{2}^{\mathrm R}=0.38\pm0.02$ versus $\alpha_{2}^{\mathrm LR}=0.52^{+0.03}_{-0.01}$) power-law slopes are different, with the most evident distinction in the faint-end portion of the magnitude distribution. Meanwhile, the total Trojan magnitude distribution is characterized by power-law slopes that are largely consistent with previously-published values ($\alpha_{1}=1.11\pm0.02$ and $\alpha_{2}=0.46\pm0.01$). The distinctness of the R and LR magnitude distributions suggests that the color populations likely formed in different regions of the primordial debris disk. The discrepancy between the faint-end slopes in particular may indicate that the color populations underwent different collisional evolutions before being emplaced into their current orbits. By running simulations of Trojan self-collisions, we have shown that this discrepancy is also consistent with a scenario in which the R objects differ from the LR objects only by the presence of a thin outer irradiation crust, and the color populations were emplaced with similar faint-end slopes. Subsequent shattering collisions could have led to the observed divergence of the faint-end slopes as all collisional fragments would be spectroscopically less red. Future study of Trojan asteroid spectra and composition promises to further our understanding of the origin and evolution of the two color populations.
\section*{Acknowledgements}
This work had its inception at the ``In Situ Science and
Instrumentation for Primitive Bodies'' study funded by the W.M. Keck
Institute for Space Studies. The authors were supported by NASA Grant NNX09AB49G. The authors also thank an anonymous reviewer for constructive comments that helped to improve the manuscript.
\small
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,731
|
\section{Introduction}\label{se-1}
A graph $G$ considered throughout this paper is simple with vertex set $V(G)$ and edge set $E(G)$, where $n(G)=|V(G)|$ denote the \emph{order} and $|E(G)|=m(G)$ the \emph{size} of $G$.
The set of the neighbors of a vertex $v\in V(G)$ is denoted
by $N_{G}(v)$, and the degree of $v$ is denoted by $d_G(v)$ (or $d(v)$ for short).
For $U\subseteq V(G)$, let $G[U]$ be the subgraph induced by $U$.
The \emph{adjacency matrix} of a graph $G$ is defined as $A(G)=(a_{ij})$, where $a_{ij}=1$ if vertices $i$ and $j$ are adjacent, and $a_{ij}=0$ otherwise.
The largest eigenvalue of $A(G)$ is called the \emph{spectral radius} of $G$, denoted by $\rho(G)$.
A subset $S\subseteq V(G)$ is called an \emph{independent set} of $G$ if there is no edge between every vertex in $S$.
The \emph{independence number} of $G$, denoted by $\alpha(G)$, is the maximum cardinality of an independent set in $G$.
Let $\mathbb{G}_{n,\alpha}$ be the set of connected graphs with order $n$ and independence number $\alpha$.
Through out this paper,
we call a graph is a \emph{minimizer graph} if it attains the minimum spectral radius over all graphs in $\mathbb{G}_{n,\alpha}$ for given $k=n-\alpha$.
As usual,
let $K_n$, $K_{1,n-1}$ and $P_n$ be respectively the complete graph, the star and the path of order $n$.
The research on the maximum or minimum spectral radius among connected graphs with a given graph invariant has been studied extensively.
The extremal graphs with the maximum spectral radius for a given clique number, chromatic number, matching number, diameter, independence number and so on, are studied in \cite{111,61,153,72,143,Stevanovic,Lu}.
Compared to the maximum spectral radius,
characterizing the graphs with the minimum spectral radius is difficult.
The results for giving diameter is only determined for some special values.
For examples, the graph $G$ of the minimum spectral radius with small diameter $diam(G)\in \{1, 2, 3, 4\}$ are determined in \cite{36,8}, with large diameter $diam(G)\in \{n-1,n-2,n-3, \lfloor n/2\rfloor\}$ in \cite{154}, $diam(G)=n-4$ in \cite{166},
simultaneously, $diam(G)\in \{n-4,n-5\}$ in \cite{37}, and $ diam(G)\in\{n-6,n-7,n-8\}\cup [\frac{n}{2}, \frac{2n-3}{3}]$ in\cite{87}.
In general, characterizing the graphs of the minimum spectral radius is still an open problem \cite{Stevanovic}.
Particularly, Stevanovi\'{c} \cite{Stevanovic} pointed out that determining the graph with the minimum spectral radius among connected graph with independence number $\alpha$ appears to be a tough problem.
In fact, the corresponding results for the independence number are less compared to the diameter, here we list them in Tab.\ref{result} for references.
\begin{table}[h]
\centering
\footnotesize
\begin{tabular*}{13.5cm}{p{40pt}|p{25pt}|p{130pt}|p{25pt}|p{97pt}}
\hline
References&$\alpha$&The minimizer graph&Year&Author
\\\hline
\cite{Xu}& $1$& the complete graph $K_n$&2009& Xu, Hong, Shu and Zhai\vspace{0.1cm}\\
\cite{Xu}&$2$& $F(\lceil\frac{n}{2}\rceil,\lfloor\frac{n}{2}\rfloor)$ (see Fig.\ref{fig-W-D})& 2009&Xu, Hong, Shu and Zhai\vspace{0.1cm}\\
\cite{Du}&$3$& $P(\frac{n}{3},3)$ for $3|n$, where $\frac{n}{3}\ge15$& 2013&Du and Shi\vspace{0.1cm}\\
\cite{Du}&$4$& $P(\frac{n}{4},4)$ for $4|n$, where $\frac{n}{4}\ge24$& 2013&Du and Shi\vspace{0.1cm}\\ \hline
\cite{Xu}&$\lceil\frac{n}{2}\rceil$& the path $P_n$&2009& Xu, Hong, Shu and Zhai\vspace{0.1cm}\\
\cite{Xu}&$\lceil\frac{n}{2}\rceil+1$&
$\left\{\begin{array}{ll}
W_n& \mbox{ if $n$ is odd}\\
D_n& \mbox{ if $n$ is even}
\end{array}\right.$ (see Fig.\ref{fig-W-D})&2009& Xu, Hong, Shu and Zhai\vspace{0.1cm}\\
\cite{Lou}& $n-4$& Theorem 1.2& 2022&Lou and Guo\vspace{0.1cm}\\
\cite{Xu}&$n-3$&
Theorem 3.2
&2009& Xu, Hong, Shu and Zhai\vspace{0.1cm}\\
\cite{Xu}& $n-2$& $T(\lceil\frac{n-3}{2}\rceil,\lfloor\frac{n-3}{2}\rfloor)$ (see Fig.\ref{fig-W-D})& 2009&Xu, Hong, Shu and Zhai\vspace{0.1cm}\\
\cite{Xu}&$n-1$& the star $K_{1,n-1}$& 2009&Xu, Hong, Shu and Zhai\\\hline
\end{tabular*}
\caption{\small Some results}\label{result}
\end{table}
In Tab.\ref{result} the first four lines list the results for small independence number $\alpha\in \{1,2,3,4\}$, and the others for large $\alpha\in \{\lceil\frac{n}{2}\rceil, \lceil\frac{n}{2}\rceil+1, n-4, n-3,n-2,n-1\}$.
There leaves a large unknown range for $\alpha$ to research.
Let $T^*$ be the minimizer graph of $\mathbb{G}_{n,\alpha}$ for given $k=n-\alpha$.
In this paper, we restrict on $\alpha\ge\lceil\frac{n}{2}\rceil$ to characterize the minimizer graph $T^*$ and determine its spectral radius $\rho(T^*)$.
Recently, for $\alpha\ge\lceil\frac{n}{2}\rceil$, Lou and Guo \cite{Lou} gave a general result that the graph with minimum spectral radius in $\mathbb{G}_{n,\alpha}$ is a tree.
Based on this result, we further give the structural features for the minimizer graph, and then provide of a constructing theorem for it.
Consequently we completely determine the minimizer graphs in $\mathbb{G}_{n,\alpha}$ along with their spectral radii for any given $k=n-\alpha\le \frac{n}{2}$.
Our article is organized as follows.
In Section 2, we give several necessary lemmas and concepts as well as notations.
In Section 3 we characterize the structure of the minimizer graph in $\mathbb{G}_{n,\alpha}$ for $k=n-\alpha\le \frac{n}{2}$ in detail (see Theorem \ref{thm-minimizer-main}).
In Section 4, we give a construction theorem for the minimizer graphs in $\mathbb{G}_{n,\alpha}$ together with their spectral radii for $k=n-\alpha\le \frac{n}{2}$ ( see Theorem \ref{thm-minimizer-spectral-radius}).
In Section 5, as an application, we determine the minimizer graphs in $\mathbb{G}_{n,\alpha}$ for $\alpha=n-1,n-2,n-3,n-4,n-5,n-6$ along with their spectral radii, the first four results are known in \cite{Xu,Lou} and the last two are new.
\section{Preliminaries}
In the section, we cite some useful lemmas and notations for the later use.
\begin{lem}[\cite{Cvetkovic1}]\label{lem-subgraph-radius}
If $H$ is the subgraph of a connected graph $G$, then $\rho(H)\le \rho(G)$. Particularly, if $H$ is proper then $\rho(H)< \rho(G)$.
\end{lem}
Let $G$ be a connected graph. By the Perron-Frobenius theorem, there is unique positive unit eigenvector of $A(G)$ corresponding to $\rho(G)$, which is called the \emph{ Perron vector} of $G$, and denoted by $\mathbf{x}$,
the entry of $\mathbf{x}$ corresponding to vertex $u$ is denoted by $x_u$.
\begin{lem}[\cite{Wu}]\label{lem-perron-entry-radius}
Let $u$, $v$ be two distinct vertices of a connected graph $G$. Suppose $w_1,w_2,...,w_t$ $(t\ge1)$ are some vertices of $N_G(v)\setminus N_G(u)$ and $\mathbf{x}$ is the Perron vector of $G$. Let $G'=G-\{vw_i\mid i=1,2,...,t\}+\{uw_i\mid i=1,2,...,t\}$. If $x_u\ge x_v$ then $\rho(G)<\rho(G')$.
\end{lem}
An \emph{internal path} of $G$ is a sequence of vertices $v_1,v_2,...,v_s$ with $s\ge 2$ such that:\\
(i) the vertices in the sequence are distinct (except possibly $v_1=v_s$),\\
(ii) $v_i$ is adjacent to $v_{i+1}$ ($i=1,2,...,s-1$),\\
(iii) the vertex degrees satisfy $d(v_1)\ge3$, $d(v_2)=\cdots=d(v_{s-1})=2$ (unless $s=2$) and $d(v_s)\ge3$.
\begin{figure}[h]
\centering
\unitlength 0.85mm
\linethickness{0.4pt}
\footnotesize
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(154,27.831)(0,0)
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\put(3,20){$K_{\lceil\frac{n}{2}\rceil}$}
\put(25,20){$K_{\lfloor\frac{n}{2}\rfloor}$}
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\put(139,25){\circle*{1.5}}
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\put(143,16){\circle*{1.5}}
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\put(139,25){\line(1,0){9}}
\put(128,16){$\ldots$}
\put(146,16){$\ldots$}
\put(125,15){$\underbrace{}_{\lceil\frac{n-3}{2}\rceil}$}
\put(144,15){$\underbrace{}_{\lfloor\frac{n-3}{2}\rfloor}$}
\put(93,23){$u$}
\put(110,23){$v$}
\put(100,2){$D_n$}
\put(60,2){$W_n$}
\put(8,2){$F(\lceil\frac{n}{2}\rceil,\lfloor\frac{n}{2}\rfloor)$}
\put(128,2){$T(\lceil\frac{n-3}{2}\rceil,\lfloor\frac{n-3}{2}\rfloor)$}
\end{picture}
\caption{\footnotesize{Some minimizer graphs}}\label{fig-W-D}
\end{figure}
\begin{lem}[\cite{Hoffman}]\label{lem-subdividing-internal-radius}
Suppose that $G\not= W_n$ (see Fig.\ref{fig-W-D}) and $uv$ is an edge on an internal path of $G$. Let $G_{uv}$ be the graph obtained from $G$ by the subdivision of the edge $uv$. Then $\rho(G_{uv})<\rho(G)$.
\end{lem}
\begin{lem}[\cite{eigenspace}]\label{vector}
Let $G$ be a connected graph with the adjacent matrix $A(G)$ and let the vector $\mathbf{0}<\mathbf{y}\in \mathbb{R}^{n(G)}$ (i.e., any entries of $\mathbf{y}$ is non-negative and $\mathbf{y}\not=\mathbf{0}$). If $A(G)\mathbf{y}\le \lambda\mathbf{y}$ then $\rho(G)\le \lambda$.
\end{lem}
Denote by $L(T)$ the set of all leaves of a tree $T\not=P_{2t}$,
and let $L(P_{2t})=\{u\}$, where $u$ is one end vertex of $P_{2t}$.
\begin{lem}[\cite{Lu}]
For every tree $T$ , there exists a maximum independent set $S(T)$ of $T$ such that
$L(T )\subseteq S(T )$.
\end{lem}
Later we always assume that $S(T)$ is the maximum independent set of $T$ such that $L(T )\subseteq S(T )$.
\begin{lem}[\cite{Lou}]\label{lem-diam}
Let $G$ be a connected graph with order $n$ and $\alpha(G)\ge \lceil \frac{n}{2}\rceil$. Then $diam(G)\le 2(n-\alpha(G))$.
\end{lem}
Let $G=(A,B)$ be a bipartite graph with $A=\{u_1,\ldots,u_t\}$ and let $G'=G\circ (l(u_1),\ldots,l(u_t))$ be a graph obtained from $G$ by joining $u_i$ with $l(u_i)\ge0$ new pendant vertices for
$1\le i \le t$.
Particularly, we denote $G'=G\circ l\mathbf{1}_{A}$ if $l(u_1)=\cdots=l(u_t)=l$ is a constant.
The following lemma is crucial for our main result.
\begin{lem}[\cite{Csikvari}]\label{lem-bipartite}
Let $G=(A,B)$ be a bipartite graph and let $G'=G\circ l\mathbf{1}_{A}$. Then $\rho(G')=\sqrt{\rho^2(G)+l}$.
\end{lem}
\section{The structure of the minimizer graph in $\mathbb{G}_{n,\alpha}$ }
In this section, we first cite several lemmas in \cite{Lou} that describe the structure and properties for minimizer graph, and then we provide a series of propositions which together give some new characters of the minimizer graph's structure. We first quote a nice result from \cite{Lou} which determines the shape of the minimizer graph.
\begin{lem}[\cite{Lou}, Theorem 1.2]\label{thm-extremal-graph-tree}
Let $G$ be the minimizer graph in $\mathbb{G}_{n,\alpha}$, where $\alpha\ge \lceil\frac{n}{2}\rceil$. Then $G$ is a tree.
\end{lem}
A vertex $v$ of a tree $T\not=P_{n},D_n$ (see Fig.\ref{fig-W-D}) is called a \emph{control vertex} if $d(v)\ge3$ and $T-v$ contains at least $d(v)-1$ paths.
Particularly, $v_2$ and $v_{2t}$ are defined as the control vertices for $P_{2t+1}=v_1v_2\cdots v_{2t}v_{2t+1}$,
$v_2$ and $v_{2t}$ the control vertices for $P_{2t}=v_1v_2\cdots v_{2t}$,
$v$ and $u$ the control vertices for $D_{n}$ shown in Fig.\ref{fig-W-D}.
Additionally, denote $L(P_{2t})=\{v_1\}$.
A path $P$ of $T$ is called \emph{control path} if its two ends are control vertices.
Particularly, if a tree $T$ contains only one control vertex, then the control path of $T$ is $P_1$.
It is clear that a star $K_{1,n-1}$, $P_{n}$ and $D_{n}$ have unique control path as well as $W_n$, and the diameter path of $T$ must contain a proper control path.
Under the assumption of $\lceil\frac{n}{2}\rceil+2\le \alpha(T)\le n-2$, the authors defined in \cite{Lou} the concepts that a \emph{branching vertex} $u\in T$ is a vertex with $d(u)\ge 3$ and an \emph{end branching vertex} is such a branching vertex $u$ which does not lie on any path between other two branching vertices. Obviously, an end branching vertex must be control vertex, but not vice versa. The control vertex extends the concept of end branching vertex to $P_{n}$, $D_n$ and $K_{1,n-1}$ because of $\alpha(P_{n})=\lceil\frac{n}{2}\rceil$, $\alpha(D_{n})=\lceil\frac{n}{2}\rceil+1$ and $\alpha(K_{1,n-1})=n-1$. The authors in \cite{Lou} gave three propositions about end branching vertex, and here we summarize them in the following lemma.
\begin{lem}[\cite{Lou}, Propositions 3.1, 3.2 and 3.3]\label{pro-branching-vertices}
Let $T$ be a minimizer graph in $\mathbb{G}_{n,\alpha}$, where $\lceil\frac{n}{2}\rceil+2\le \alpha\le n-2$. Then \\
(i) $T$ has at least two end branching vertices,\\
(ii) Every end branching vertex only attaches some leaves (by another words, end branching vertex $u$ attaches exactly $d(u)-1$ leaves),\\
(iii) Let $v_0$, $v_q$ be two end branching vertices of $T$ and $P_{q+1}=v_0v_1\cdots v_q$ be a path that connects $v_0$ and $v_q$. Then $q$ is even and $v_i\in S(T)$ for odd $i\in[1, q-1]$.
\end{lem}
Lemma \ref{pro-branching-vertices} can be extended as the following lemma.
\begin{lem}\label{pro-end-branching}
Let $T^*$ be a minimizer graph in $\mathbb{G}_{n,\alpha}$ for $ \alpha\ge \lceil\frac{n}{2}\rceil$, and $P_{q+1}=v_0v_1\cdots v_q$ be a control path of $T^*$. Then\\
(i) Every control vertex $u$ of $T^*$ attaches at least $d(u)-1$ leaves,\\
(ii) $q$ is even and $v_i\in S(T^*)$ for odd $i\in[1, q-1]$.
\end{lem}
\begin{proof}
From Lemma \ref{pro-branching-vertices}, (i) and (ii) hold for $\lceil\frac{n}{2}\rceil+2\le \alpha\le n-2$. It remains to consider $\alpha=n-1$, $\lceil\frac{n}{2}\rceil$ and $\lceil\frac{n}{2}\rceil+1$. According to the results listed Table \ref{result}, if $\alpha(T^*)\in \{n-1,\lceil\frac{n}{2}\rceil,\lceil\frac{n}{2}\rceil+1\}$ then $T^* $ would be one of $K_{1,n-1}$, $P_n$, $D_n$ or $W_n$. From Fig.\ref{fig-W-D} we see that either one of their control vertices can only hang leaves and thus (i) holds.
Additionally, the control path of $K_{1,n-1}$, $P_n$, $D_n$ and $W_n$ are all unique by definition.
It is easy to verify that (ii) holds for $T^*\in \{K_{1,n-1}, P_n, D_n, W_n\}$.
\end{proof}
In what follows we always denote by $T^*$ a minimizer graph in $\mathbb{G}_{n,\alpha}$ for $ \alpha\ge \lceil\frac{n}{2}\rceil$ as assumed as in Lemma \ref{pro-end-branching}.
\begin{lem}\label{pro-path-1}
Let $v'v$ be an edge of a tree $T$, and $v$ attach with a pendant path $P=vy_1y_2\cdots y_s$ for $s\ge 2$.
For even $t\le s$, let $T'$ be a graph obtained from $T$ by replacing $v'v$ with a path $v'x_1x_2\cdots x_tv$ and simultaneously deleting $t$ vertices $y_{s-t+1},y_{s-t+2},...,y_s$. If one of $v$ and $v'$ is in $S(T)$, then $n(T')=n(T)$ and $\alpha(T')=\alpha(T)$.
\end{lem}
\begin{proof}
Let $T_1$ be the graph obtained from $T$ by replacing $v'v$ with a path $v'x_1x_2\cdots x_tv$ and $T_2$ a graph obtained from $T_1$ by deleting vertices $y_{s-t+1},y_{s-t+2},...,y_s$. Then $T'=T_2$ and, obviously $n(T)=n(T')$.
We first verify $\alpha(T_1)=\alpha(T)+\frac{t}{2}$.
Without loss of generality, assume that $v\in S(T)$ and then $v'\not\in S(T)$.
Thus $S(T)\cup\{x_1,x_3,...,x_{t-1}\}$ is an independent set of $T_1$ and so
$\alpha(T_1)\ge\alpha(T)+\frac{t}{2}$.
For the maximum independent set $S(T_1)$, let $S_1=S(T_1)\cap \{x_1,x_2,...,x_t\}$ and $S_2=S(T_1)\cap V(T)$.
Then $\alpha(T_1)=|S(T_1)|=|S_1|+|S_2|$.
Since $x_1x_2\cdots x_t$ is an induced path in $T_1$, it can produce at most $\frac{t}{2}$ independent vertices and so $|S_1|\le \frac{t}{2}$.
If $|S_2|\le |S(T)|$ then $\alpha(T_1)=|S(T_1)|\le \alpha(T)+\frac{t}{2}$. Otherwise $|S_2|> |S(T)|$, it implies that $v', v\in S_2$ ( because if there is at most one of $v'$ and $ v$ belonging to $S_2$ then $S_2$ will be an independent set of $T$ which contracts $|S_2|> |S(T)|$ ).
In this situation, we have $|S_2|=\alpha(T)+1$ and $|S_1|\le \frac{t}{2}-1$, and so $\alpha(T_1)=|S_1|+|S_2|\le \alpha(T)+\frac{t}{2}$. It follows that $\alpha(T_1)=\alpha(T)+\frac{t}{2}$.
Next, by considering whether $y_{s-t}\in S(T_1)$ or not, we can verify $\alpha(T_2)=\alpha(T_1)-\frac{t}{2}$. In fact, $s-t$ has the same parity with $s$, if $y_{s-t}\in S(T_1)$ then $y_{s-t+2}, y_{s-t+4},...,y_s\in S(T_1)$; if $y_{s-t}\not\in S(T_1)$ then $y_{s-t+1}, y_{s-t+3},...,y_{s-1}\in S(T_1)$. It leads to our conclusion.
\end{proof}
\begin{pro}\label{pro-path}
Let $P_{q+1}=v_0v_1\cdots v_q$ be a control path of $T^*$. Then $v_i$ does not attach pendant paths of length more than one for $ 1\le i \le q-1$.
Moreover, $v_i$ does not attach any leaf for odd $i$.
\end{pro}
\begin{proof}
Obviously, our result holds if $T^*=P_n$. It is clear that $T^*\not=P_n$ has a control vertex of degree at least $3$. Without loss of generality, we assume that $v_0$ is the control vertex with $d(v_0)\ge3$.
Suppose to the contrary that $v_i$ attaches a pendant path $P =v_iy_1\cdots y_s$ with $s\ge2$. For an even $2\le t\le s$, let $T_1$ be a graph obtained from $T^*$ by replacing $v_{i-1}v_i$ with a path $v_{i-1}x_1x_2\cdots x_tv_i$. Note that $d_{T^*}(v_0), d_{T^*}(v_i)\ge 3$, the edge $v_{i-1}v_i$ belongs to an internal path of $T^*$ that is some sub-path of $P_{q+1}$, thus we have $\rho(T_1)<\rho(T^*)$ from
Lemma \ref{lem-subdividing-internal-radius}.
Let $T_2$ be a graph obtained from $T_1$ by deleting vertices $y_{s-t+1},y_{s-t+2},...,y_s$. Notice that there is exactly one of $v_{i-1}$ and $v_i$ belonging to $S(T^*)$ according to Lemma \ref{pro-end-branching}(ii), we have $n(T_2)=n(T^*)$ and $\alpha(T_2)=\alpha(T^*)$ by Lemma \ref{pro-path-1}. Hence $T_2\in \mathbb{G}_{n,\alpha}$.
Since $T_2$ is a subgraph of $T_1$, we have $\rho(T_2)<\rho(T_1)$ from Lemma \ref{lem-subgraph-radius},
and thus $\rho(T_2)<\rho(T^*)$ which contracts the minimality of $T^*$.
Particularly, we have $v_i\in S(T^*)$ for odd $i$ by Lemma \ref{pro-end-branching}(ii). It implies that
$v_i$ does not attach any leaf for odd $i$.
\end{proof}
\begin{lem}\label{split}
Let $G$ be a connected graph with the Perron vector $\mathbf{x}=(x_u\mid u\in V(G))$ and $v\in V(G)$.
Suppose that $N_G(v)=\{w_1,w_2,...,w_t\}$ ($t\ge3$) and $x_{w_1}=\min_{w\in N_G(v)} \{x_w\}$.
Let $G'$ be a graph obtained from $G$ by replacing $v$ with two new vertices $v'$, $v''$ and meanwhile adding new edges $v'w_1, v'w_2,...,v'w_s$ and $\ v''w_1, v''w_{s+1},\ v''w_{s+2},...,v''w_t$ for some $s \in[2, t-1]$ (see Fig.\ref{graph transformation-v}).
If $vw_1$ is a cut edge of $G$ then $\rho(G')\le \rho(G)$,
with equality if and only if
$t=3$ and $x_{w_1}=x_{w_2}=x_{w_3}$.
\end{lem}
\begin{figure}[h]
\centering
\footnotesize
\unitlength 1mm
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\begin{picture}(128,28)(0,0)
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\put(25,0){\footnotesize$G$}
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\end{picture}
\caption{\footnotesize{Graph $G$ and $G'$ in Lemma \ref{split} }}
\label{graph transformation-v}
\end{figure}
\begin{proof}
Since $vw_1$ is a cut edge, let $G_v$ and $G_{w_1}$ be two components of $G-vw_1$ containing $v$ and $w_1$, respectively.
Similarly, denote by $G'_{w_1}$ the component of $G'-v'w_1-v''w_1$ containing $w_1$, and $G'_{v',v''}$ the remaining part containing $v'$ and $v''$.
It is clear that $G'_{v',v''}-\{v',v''\}=G_v-v$ and $G'_{w_1}=G_{w_1}$.
Now we define a vector $\mathbf{y}=(y_u\mid u\in V(G'))$, where
$$y_u=
\left\{\begin{array}{ll}
x_v &\mbox{if $u\in \{v',v''\}$,}\\
x_u &\mbox{if $u\in G'_{v',v''}-\{v',v''\}$,}\\
2x_u &\mbox{if $u\in G'_{w_1}$.}
\end{array}\right.$$
Clearly, $\mathbf{y}> \mathbf{0}$.
In what follows we will consider the $u$-entry of $A(G')\mathbf{y}$. First of all, from our definition $(A(G')\mathbf{y})_u=\rho(G)y_u$ if $u\not=v',v'',w_1$.
For $u=v'$, we have
\begin{equation}\label{eq-v'}
\begin{array}{ll}
(A(G')\mathbf{y})_{v'}=\sum_{i=1}^s y_{w_i}&=2x_{w_1}+\sum_{i=2}^s x_{w_i}\\
&=\sum_{i=1}^t x_{w_i}+(x_{w_1}-\sum_{i=s+1}^t x_{w_i})\\
&=\rho(G)x_v+(x_{w_1}-\sum_{i=s+1}^t x_{w_i})\\
&\le \rho(G)x_v\ \ \ \ ( \mbox{due to $x_{w_1}=\min_{w\in N_G(v)} \{x_w\}$} )\\
&=\rho(G)y_{v'}.
\end{array}\end{equation}
Similarly, we have
\begin{equation}\label{eq-v''}
(A(G')\mathbf{y})_{v''}=y_{w_1}+\sum_{i=s+1}^t y_{w_i}=\rho(G)x_v+(x_{w_1}-\sum_{i=2}^s x_{w_i})\le \rho(G)x_v= \rho(G)y_{v''}.\end{equation}
Let $N_{G}(w_1)=\{v,u_1,...,u_p\}$ and then $N_{G'}(w_1)=\{v',v'',u_1,...,u_p\}$. We have
$$
(A(G')\mathbf{y})_{w_1}=y_{v'}+y_{v''}+\sum_{i=1}^p y_{u_i}=x_v+x_v+\sum_{i=1}^p 2x_{u_i}
=2\rho(G)x_{w_1}=\rho(G)\cdot(2 x_{w_1})
=\rho(G)y_{w_1}.$$
By above discussions, we have $A(G')\mathbf{y}\le \rho(G)\mathbf{y}$.
By Lemma \ref{vector}, we have $\rho(G')\le \rho(G)$.
The equalities (\ref{eq-v'}) and (\ref{eq-v''}) hold (i.e., $A(G')\mathbf{y}=\rho(G)\mathbf{y}$) if and only if $t=3$ and $x_{w_1}=x_{w_2}=x_{w_3}$.
Note that $G'$ is connected and the vector $\mathbf{y}$ is positive, $\mathbf{y}$ is an eigenvector of $A(G')$ corresponding to the spectral radius $\rho(G')$ if $A(G')\mathbf{y}=\rho(G)\mathbf{y}$.
Thus
$\rho(G')= \rho(G)$ if and only if $t=3$ and $x_{w_1}=x_{w_2}=x_{w_3}$.
It completes the proof.
\end{proof}
\begin{pro}\label{pro-middle-branch-vertex}
Let $P_{q+1}=v_0v_1\cdots v_q$ be a control path of $T^*$.
Then $d_{T^*}(v_i)=2$ for odd $i\in[1, q-1]$.
\end{pro}
\begin{proof}
Suppose to the contrary that $d_{T^*}(v_i)\ge3$ for some odd $i\in[1, q-1]$.
Then we can label $N_{T^*}(v_i)=\{ w_{1},\ldots, w_{s} ,v_{i-1},v_{i+1}\}$ with $s\ge1$.
Let $\mathbf{x}$ be the Perron vector of $T^*$.
Without loss of generality, we may assume that $x_{w_1}=\min_{w\in N_{T^*}(v_i)} \{x_w\}$.
Let $T_2$ be a graph obtained from $T^*$ by replacing $v_i$ with two new vertices $v'_i$, $v''_i$ and meanwhile adding edges $v'_iw_1,v'_iw_2,...,v'_iw_s$ and $ v''_iw_1,v_i''v_{i-1}, v''_iv_{i+1}$.
Clearly, $n(T_2)=n(T^*)+1$.
By Lemma \ref{pro-end-branching}(ii), we have $v_i\in S(T^*)$ for odd $i\in[1, q-1]$.
Thus $v_{i-1},v_{i+1}\notin S(T^*)$ and $w_t\notin S(T^*)$ for $t\in [1,s]$.
It follows that
$S(T_2)=(S(T^*)\setminus \{v_i\})\cup \{v'_i,v''_i\}$ and so $\alpha(T_2)=\alpha(T^*)+1$.
Hence $T_2\in \mathbb{G}_{n+1,\alpha+1}$.
Since $v_{i}w_1$ is a cut edge of $T^*$, we have $\rho(T_2)\le \rho(T^*)$
by Lemma \ref{split}.
Let $T_3$ be a graph obtained from $T_2$ by deleting a leaf. Thus $T_3\in \mathbb{G}_{n,\alpha}$. By Lemma \ref{lem-subgraph-radius}, we have $\rho(T_3)< \rho(T_2)\le\rho(T^*)$, a contradiction.
It completes the proof.
\end{proof}
\begin{pro}\label{cor-diam}
$diam(T^*)$ is even expect of $T^*=P_n$. Moreover, we have $diam(T^*)=2,4$ and $n-1$ for $\alpha=n-1, n-2$ and $\lceil\frac{n}{2}\rceil$, respectively, and $6\le diam(T^*)\le 2(n-\alpha)$ for $\alpha\in [\lceil\frac{n}{2}\rceil+1, n-3]$.
\end{pro}
\begin{proof}
First of all, from Tab.\ref{result}, we know that $T^*=K_{1,n-1}$ if $\alpha=n-1$, and so $diam(T^*)=2$. If $\alpha=n-2$ then $T^*=T(\lceil\frac{n-3}{2}\rceil,\lfloor\frac{n-3}{2}\rfloor)$, and so $diam(T^*)=4$ (see Fig.\ref{fig-W-D}). If $\alpha=\lceil\frac{n}{2}\rceil$ then $T^*=P_n$, and so $diam(T^*)=n-1$. Next we suppose that $\alpha\in [\lceil\frac{n}{2}\rceil+1, n-3]$.
Clearly, there is a diameter path $P=u_0u_1\cdots u_{D-1}u_D$ that connects the two leaves $u_0$ and $u_{D}$ of $T^*$, which, we know, includes a control path $P'=u_1\cdots u_{D-1}$, where $D=diam(T^*)$.
By Lemma \ref{pro-end-branching}(ii), $D-2$ is even and so $D$ is even.
First we have $diam(T^*)\le 2(n-\alpha)$ by Lemma \ref{lem-diam}.
On the other hand, note that the diameter path $P$ contains two control vertices, we have $diam(T^*)\ge4$.
It suffices to verify $diam(T^*)\not=4$.
Suppose to the contrary,
the diameter path becomes $P=u_0u_1u_2u_3u_4$.
Thus $u_1$ and $u_3$ are two control vertices. It implies that $u_0$, $u_4\in S(T^*)$, moreover $u_2 \in S(T^*)$ by Lemma \ref{pro-end-branching}(ii) and $u_1,u_3\notin S(T^*)$.
Recall that $\alpha\le n-3$, we have $|V(T^*)\setminus S(T^*)|=n-\alpha\ge 3$. Thus, apart from $u_1$ and $u_3$ there exists at least one neighbor of $u_2$, say $w\notin S(T^*)$. Thus $d_{T^*}(u_2)\ge3$, it contradicts Proposition \ref{pro-middle-branch-vertex}.
\end{proof}
For each $x\in V(T^*)\setminus L(T^*)$, according to Lemma \ref{pro-end-branching}(i) and Proposition \ref{pro-path}, there exists a control path $P= v_0v_1\cdots v_q$ (not necessarily unique) such that $x$ is a vertex of $P$, i.e., $x=v_i$ for some $0\le i\le q$.
According to Lemma \ref{pro-end-branching}(ii) and Proposition \ref{pro-middle-branch-vertex} we have known that
$$
\left\{\begin{array}{ll}
\mbox{ $q$ is even}&\\
\mbox{ $v_i\notin S(T^*)$}& \mbox {if $i$ is even}\\
\mbox{ $v_i\in S(T^*)$ and $d_{T^*}(v_i)=2$ } & \mbox {if $i$ is odd}\\
\end{array}\right.
$$
It is clear that the labelling of $x=v_i$ depends on the choice of $P$, however the parity of its subscript $i$ is independent with $P$. So we can define such a vertex $x=v_i$ \emph{odd } if $i$ is odd, and \emph{even } otherwise. Thus $V(T^*)\setminus L(T^*)$ is partitioned as even vertices, denoted by $V_2^*$, and odd vertices, denoted by $V_1^*$.
It is clear that $S(T^*)=L(T^*)\cup V_1^*$ and so $V_1^*$ is independent. Moreover, any two vertices $x_1$ and $x_2$ in $V_2^*$ cannot be adjacent since otherwise edge $x_1x_2$ must belong to some control path, thus $V_2^*$ is also independent. It is clear that $V_1^*\cup V_2^*$ induces a subtree of $T^*$, i.e., $T^*[V_1^*\cup V_2^*]=T^*\setminus L(T^*)$, which is called the \emph{main tree} of the minimizer graph $T^*$ and denoted by $T^*_-$.
Notice that the leaves of $T^*$ can only hang at some vertices of $V_2^*$ according to Proposition \ref{pro-path}, the leaves of $T^*_-$ must be the control vertices of $T^*$, and the degree of vertex in $V_1^*$ is unchanged in $T^*_-$. We immediately have the following proposition.
\begin{pro}\label{eq-degree1}
$T^*_{-}=T^*\setminus L(T^*)$ is a tree with bipartite partition $(V_1^* ,V_2^*)$, and
$$d_{T^*_{-}}(v)=\left\{\begin{array}{ll}
d_{T^*}(v)=2& \mbox{ if $v\in V^*_1$},\\
d_{T^*}(v)-l(v)& \mbox{ if $v\in V^*_2$},
\end{array}\right.$$
where $l(v)$ is the number of the leaves in $T^*$ attached at $v$.
\end{pro}
\begin{pro}\label{lem-main-tree}
$|V^*_2|=n-\alpha$, $|V^*_1|=n-\alpha-1$, $n(T^{*}_{-})=2(n-\alpha)-1$ and $|L(T^*)|=2\alpha-n+1$.
\end{pro}
\begin{proof}
According to above discussions, we have $V(T^*)=L(T^*)\cup V^*_1\cup V^*_2$ and $S(T^*)=L(T^*)\cup V^*_1$ are two partitions, which gives $|V^*_2|=|V(T^*)|-|S(T^*)|=n-\alpha$.
Since $T^{*}_{-}=( V^*_1, V^*_2)$ is bipartite, we have $m(T^{*}_{-})=\sum_{v\in V^*_1} d_{T^*_{-}}(v) =2|V^*_1|$ by Proposition \ref{eq-degree1}. Note that $T^{*}_{-}$ is a tree, we have
\begin{equation}\label{eq-n-1}
n(T^{*}_{-})=m(T^{*}_{-})+1=2|V^*_1|+1.\end{equation}
On the other hand, it is clear that
\begin{equation}\label{eq-n-2}
n(T^{*}_{-})=|V^*_2|+|V^*_1|=n-\alpha+|V^*_1|.\end{equation}
Combining (\ref{eq-n-1}) and (\ref{eq-n-2}), we obtain
$|V^*_1|=n-\alpha-1$ and $n(T^{*}_{-})=2(n-\alpha)-1$. Thus $|L(T^*)|=|S(T^*)|- |V^*_1|=\alpha-(n-\alpha-1)=2\alpha-n+1$.
\end{proof}
For $\alpha=\lceil\frac{n}{2}\rceil$, we have $T^*=P_{n}$. Note that $T^*_{-}=T^*\setminus L(T^*)$. By the choice of $L(P_{n})$, $T^*_-=P_{n-2}$ for odd $n$ and $T^*_-=P_{n-1}$ for even $n$. Thus $diam(T^{*}_{-})=2(n-\lceil\frac{n}{2}\rceil)-2$ is even. Moreover,
from Lemma \ref{pro-end-branching}(i) and Proposition \ref{cor-diam},
we have the following result.
\begin{pro}\label{main-tree-diam}
$diam(T^{*}_{-})$ is even, moreover $diam(T^*_{-})=0$ and $2$ for $\alpha=n-1$ and $n-2$, respectively, and $4 \le diam(T^*_{-}) \le2(n-\alpha)-2$ for $\alpha\in [\lceil\frac{n}{2}\rceil, n-3]$.
\end{pro}
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\caption{\footnotesize The graph $T^*_{-}$}\label{main}
\end{figure}
Let $P=v_0v_1\cdots v_q$ be a control path of $T^{*}$,
which is also a path in main tree $T^*_-$ connecting two leaves $v_0$ and $v_q$.
Let $X_0$ and $Y_0$ be respectively the sets of odd and even vertices in $P$ and clearly $T^{*}_{-}[X_0\cup Y_0]=P$.
Let $\tilde{X}_0= V^*_1\setminus X_0$ and $\tilde{Y}_0= V^*_2\setminus Y_0$.
The induced subgraph $H=T^{*}_{-}[\tilde{X}_0\cup \tilde{Y}_0]=T^{*}_{-}\setminus P$ is described in Fig.\ref{main}.
By Proposition \ref{eq-degree1}, odd vertex $v\in X_0$ can only join two even vertices on the path $P$, we have $E_{T^*_{-}}(X_0,\tilde{Y}_0)=\emptyset$ and thus $N_{T^*_{-}}(\tilde{Y}_0)\subseteq \tilde{X}_0$ because $T^*_{-}=(V_1^* ,V_2^*)$ is bipartite (see Fig.\ref{main}).
Now we consider the induced subgraph $H=T^{*}_{-}[\tilde{X}_0\cup \tilde{Y}_0]$.
Recall that $d_{T^{*}_{-}}(x)=2$ for any odd vertex $x\in \tilde{X}_0$, its two neighbors $y$ and $y'$ belong to $V_2^*=Y_0\cup \tilde{Y}_0$.
Obviously, $y$ and $y'$ can not simultaneously belong to $Y_0$, since otherwise $T^*_{-}$ will form a cycle containing $x$, $y$ and $y'$.
Denote by $X_{1} $ the subset of $\tilde{X}_0$ consisting of the vertex $x\in \tilde{X}_0$ such that its one neighbor $y'\in Y_0$ and another $y\in \tilde{Y}_0$.
Then $\tilde{X}_0\setminus X_{1}$ consists of vertices in $\tilde{X}_0$ whose two neighbors are in $\tilde{Y}_0$ (see Fig.\ref{main}).
Let $X_{1}=\{x_{1},x_{2},...,x_{t}\}$ and $Y_{1}=N_H(X_{1})\subseteq \tilde{Y}_{0}$ (see Fig.\ref{main}). Clearly, $x_{i}$ has two neighbors $y'_{i}\in Y_0$ and $y_{i}\in Y_1$.
For $i\not=j$, we see that $y_{i}\not=y_{j}$ since otherwise $T^*_-$ has a cycle containing $x_{i}$ and $x_{j}$. It implies that $|X_{1}|=|Y_{1}|$ (see Fig.\ref{main}).
In addition, the $t$ vertices of $X_{1}$ belong to distinct components of $H$ since otherwise the component, say containing $x_{1}$ and $x_{2}$, will join at the vertices of $P$ and form a cycle in $T^*_{-}$. Thus $H$ has at least $t$ components. Next
we recursively define $X_i$ and $Y_i$ for $i=2,3,...$ (refer to Fig.\ref{main}):
$$\begin{array}{ll}
&X_2=N_H(Y_1)\setminus X_1\subseteq \tilde{X}_0, \ \ Y_2=N_H(X_2)\setminus Y_1\subseteq \tilde{Y}_0,\ \ ...\\
&X_j=N_H(Y_{j-1})\setminus X_{j-1}\subseteq \tilde{X}_0,\ \ Y_j=N_H(X_j)\setminus Y_{j-1}\subseteq \tilde{Y}_0, \ \ ...
\end{array}$$
Note that the leaves of $T^*_{-}$ are the control vertices of $T^*$ and included in $\tilde{Y}_0$. Each $Y_j\subset \tilde{Y}_0$ would contain some leaves of $T^*_{-}$. Without loss of generality, we can assume that $Y_h$ contains only leaves of $T^*_-$ (see Fig.\ref{main}).
It follows that
$\tilde{X}_0=\bigcup^h_{s=1}X_{s} \mbox{ and } \tilde{Y}_0=\bigcup^h_{s=1}Y_{s}$
are two partitions.
Since $T^*_{-}$ is connected and $X_1$ is a cut set, for any vertex $ v\in\tilde{X}_0\cup \tilde{Y}_0$ there exists exactly one vertex $x \in X_1$ such that $v$ and $x$ are connected by one path in $H$. Thus, $H$ has $|X_{1}|$ components. Moreover, for $1\le s \le h$ we know that each $x\in X_{s} $ has degree $d_H(x)=2$ and its one neighbor $y'\in Y_{s-1}$ and another $y\in Y_s$ (see Fig.\ref{main}). Such neighbors $y'$ and $y$ are uniquely determined by $x$ and thus it induces two maps $\sigma': x\longrightarrow y'=\sigma'(x)$ and $\sigma: x\longrightarrow y=\sigma(x)$ for $x\in \tilde{X}_{0}$. It is easy to see that $\sigma$ is a bijection, the induced subgraph $T^*_{-}[X_{s},Y_{s}]$ consists of independent edges, denoted by $M_s=\{x\sigma(x)\mid x\in X_{s} \}$ and so $|X_{s}|=|Y_{s}|$.
Thus, the edge set $E(T^*_{-})$ has the partition:
$$E(T^*_{-})=E(P)\ \bigcup\ (\cup^h_{s=1} M_s)\ \bigcup\
( \cup^h_{s=1} E(X_{s},Y_{s-1}) ),$$
where $E(X_{s},Y_{s-1})=\{x\sigma'(x) \mid x \in X_s\}$ for $s=1,..., h$.
Let $E'= \cup^h_{s=1} E(X_{s},Y_{s-1})=\{x\sigma'(x) \mid x\in \tilde{X}_{0}\}$.
Then $|E'|=\sum^h_{s=1}|X_{s}|=|\tilde{X}_0|$.
Recall that $|X_0|=\frac{q}{2}$ and $|V^*_1|=n-\alpha-1$, we have
$|\tilde{X}_0|=|V^*_1\setminus X_0|=n-\alpha-1-\frac{q}{2}$.
So $h\le \sum^h_{s=1}|X_{s}|=|\tilde{X}_0|=n-\alpha-1-\frac{q}{2}$.
On the other hand, for each $y_h\in Y_h$, we have $d_{T^*_-}(y_h,P)=2h$, where $d_{T^*_-}(y_h,P)$ is the minimum distant between $y_h$ and $v$ for any $v\in V(P)$.
Hence $d_{T^*_-}(y_h,P)+\frac{q}{2}=2h+\frac{q}{2}\le diam(T^*_-)$, and so
$h\le \lfloor\frac{diam(T^*_-)}{2}-\frac{q}{4}\rfloor$.
Summarizing the above discussion, we obtain the following results that are described by the above symbols.
\begin{pro}\label{lem-matching}
Let $P=v_0v_1\cdots v_q$ be a control path of $T^{*}$,
$X_0$ and $Y_0$ be the sets of odd and even vertices in $P$, respectively. Let
$X_s$ and $Y_s$ be defined above for $s=1, ..., h$
(see Fig.\ref{main}). Then\\
(i) $h\le \min\{ \ n-\alpha-1-\frac{q}{2}, \lfloor\frac{ diam(T^*_-)}{2}-\frac{q}{4}\rfloor \ \}$. Particularly,
if the control path $P$ of $T^{*}$
is a diameter path of $T^*_-$, then $h\le \min\{ \ n-\alpha-1-\frac{diam(T^*_-)}{2}, \lfloor\frac{ diam(T^*_-)}{4}\rfloor \ \}$,\\
(ii) $V^*_1=\bigcup^h_{s=0} X_s$ and $V^*_2=\bigcup^h_{s=0} Y_s$ are two partitions,\\
(iii) $T^*_{-}[X_0\cup Y_0]=P$, and $T^*_{-}[X_s\cup Y_s]=M_s$ is an edge independent set for $1\le s\le h$, \\
(iv) The edge set $E(T^*_{-})$ has the partition:
$E(T^*_{-})=E(P)\ \bigcup\ (\cup^h_{s=1} M_s)\ \bigcup\ \{x\sigma'(x) \mid x\in \cup^h_{s=1} X_s\}$.
Moreover, the number of edges in the three partitions is $q$, $n-\alpha-1-\frac{q}{2}$ and $n-\alpha-1-\frac{q}{2}$, respectively,\\
(v) The induced subgraph $H=T^*_{-}[(\cup^h_{s=1} X_s) \bigcup (\cup^h_{s=1} Y_s)]$ has the unique perfect matching $\cup^h_{s=1} M_s$ and $|X_1|$ components.
\end{pro}
Given a minimizer graph $T^*\in \mathbb{G}_{n,\alpha}$, usually its main tree $T^*_-$ is not unique in which we may take a control path
as $P=v_0v_1\cdots v_d$ for $d=diam(T^*_-)$. By Proposition \ref{lem-matching}(iv), we know that $T^*_-$ is composed of $P$, $X_s$ and $Y_s$ along with the edges between them. To exactly, the edges between $X_s$ and $Y_s$ is an edge independent set $M_s=\{x_{s1}y_{s1},x_{s2}y_{s2},...,x_{st_s}y_{st_s}\}$ determined by
$\sigma:x_{si}\longleftrightarrow y_{si}$ for $i=1,2,...,t_s$. The edges between $X_s$ and $Y_{s-1}$ are labelled as $x_{sj}y_{s-1,j'}$ which is determined by $\sigma':x_{sj}\longrightarrow y_{s-1,j'}$.
By using $\sigma'$ we now define a map $\sigma^*:M_s\longrightarrow M_{s-1}$ by $\sigma^*(x_{sj}y_{sj})=x_{s-1,j'}y_{s-1,j'}$ ( if $s=1$ then we define $\sigma^*(x_{1j}y_{1j})=v_{s_j-1}v_{s_j}$ for some even vertex $v_0,v_d\not=v_{s_j}\in V(P)$ ).
In this way, $\sigma^*$ determines main tree $T^*_-$, and thus $T^*_-$ can be written by $T^*_-= T(d; M_1, M_2,...,M_h;\sigma^*)$.
By Proposition \ref{main-tree-diam} and Proposition \ref{lem-matching}(i)(iv) we can determine the parameters as in (\ref{eq-t})
\begin{equation}\label{eq-t}
\left\{\begin{array}{ll}
d=2(n-\alpha)-2 \mbox{ if $\alpha\in [ n-2, n-1]$}; 4\le d\le 2(n-\alpha)-2 \mbox{ if $\alpha\in [\lceil\frac{n}{2}\rceil, n-3]$}\\
h\le \min\{ n-\alpha-1-\frac{d}{2}, \lfloor\frac{ d}{4}\rfloor \}\\
\sum^h_{s=1} |M_s|=n-\alpha-1-\frac{d}{2}\\
\sigma^*:M_s\longrightarrow M_{s-1},\ 1\le s \le h
\end{array}\right.
\end{equation}
All such main trees satisfying (\ref{eq-t}) we collect in the set $\mathcal{T}^*_-(n,\alpha)$.
For example,
let $\alpha=n-1$ ($n-2$ or $n-3$), (\ref{eq-t}) becomes $d=0$ ($2$ or $4$), $h=0$ and $M_s=\emptyset$.
So $\mathcal{T}^*_-(n,n-1)=\{P_1\}$, $\mathcal{T}^*_-(n,n-2)=\{P_3\}$ and $\mathcal{T}^*_-(n,n-3)=\{P_5\}$ (see Fig.\ref{main-tree}),
which are just the results of Theorems 3.1 and 3.2 in \cite{Xu}.
For $\alpha=n-4$, (\ref{eq-t}) becomes
$$\left\{\begin{array}{ll}d\in \{4,6\}\\
h\le \min\{ 3-\frac{d}{2}, \lfloor\frac{ d}{4}\rfloor \}\\
\sum^h_{s=1} |M_s|=3-\frac{d}{2}\\
\sigma^*:M_s\longrightarrow M_{s-1},\ 1\le s \le h
\end{array}\right.\Rightarrow
\left\{\begin{array}{ll}d=4\\
h= 1\\
\sum^h_{s=1} |M_s|=1\\
\sigma^*:M_s\longrightarrow M_{s-1},\ 1\le s \le h
\end{array}\right.
\mbox{ or }
\left\{\begin{array}{ll}d=6\\
h=0\\
M_s=\emptyset
\end{array}\right.
$$
If the former occurs then $\sigma^*(x_{11}y_{11})=v_1v_2$ and we obtain a main tree $F_1=T(4; \{x_{11}y_{11}\}; \sigma^*)$; if the later occurs we obtain a main tree $F_2=T(6; \emptyset; \sigma^*)=P_7$. Thus
$\mathcal{T}^*_-(n,n-4)=\{F_1,F_2\}$ (see Fig.\ref{main-tree}),
and it is just the result of Theorem 1.3 in \cite{Lou}.
\begin{figure}[h]
\centering
\footnotesize
\unitlength 0.9mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(146,24.5)(0,0)
\put(26,0){$P_3$}
\put(50,6.5){$v_0$}
\put(56,6.5){$v_2$}
\put(62,6.5){$v_4$}
\put(57,0){$P_5$}
\put(24,11){\circle*{1.5}}
\put(30,11){\circle*{1.5}}
\multiput(27,22)(-.03370787,-.12359551){89}{\line(0,-1){.12359551}}
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\put(51,11){\circle*{1.5}}
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\put(63,11){\circle*{1.5}}
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\put(22,9){\dashbox{1}(10,4)[cc]{}}
\put(22,20){\dashbox{1}(10,4)[cc]{}}
\put(49,9){\dashbox{1}(16,4)[cc]{}}
\put(49,20){\dashbox{1}(16,4)[cc]{}}
\put(45,21){$X_0$}
\put(45,10){$Y_0$}
\put(18,21){$X_0$}
\put(18,10){$Y_0$}
\put(2,10){$Y_0$}
\put(6,0){$P_1$}
\put(6,9){\dashbox{1}(4,4)[cc]{}}
\put(8,11){\circle*{1.5}}
\put(85,25){$v_{1}$}
\put(89,6.5){$v_{2}$}
\put(90,0){$F_1$}
\put(83,11){\circle*{1.5}}
\put(89,11){\circle*{1.5}}
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\put(95,11){\circle*{1.5}}
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\put(81,9){\dashbox{1}(16,4)[cc]{}}
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\put(99,9){\dashbox{1}(4,4)[cc]{}}
\put(104,21){$X_1$}
\put(104,10){$Y_1$}
\put(100,25){$x_{11}$}
\put(100,6.5){$y_{11}$}
\put(77,21){$X_0$}
\put(77,10){$Y_0$}
\put(134,0){$F_2$}
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\put(120,21){$X_0$}
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\end{picture}
\caption{\footnotesize The main trees}\label{main-tree}
\end{figure}
Recall that $T^*_-=T^*\setminus L(T^*)$ and the leaves of $T^*$ can only hang at some even vertices of $T^*$, to exactly $u\in V^*_2$ hangs $l(u)$ leaves (we specify $l(u)=0$ if $u$ hangs no any leaf).
Denote by $l_{V_2^*}=(l(u)\mid u\in V_2^*)$, we conclude that $T^*$ can be presented by $T^*=T^*_{-}\circ l_{V_2^*}$. It naturally follows the result.
\begin{thm}\label{thm-minimizer-main}
Let $T^*$ be the minimizer graph in $\mathbb{G}_{n,\alpha}$, where $\alpha\ge \lceil\frac{n}{2}\rceil$. Then there exists $T^*_{-}\in \mathcal{T}^*_-(n,\alpha)$ such that
$T^*=T^*_{-}\circ l_{V_2^*}$, where $T^*_{-}=(V_1^*,V_2^*)$ and $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ satisfying $\sum_{u\in V_2^*}l(u)=|L(T^*)|=2\alpha-n+1$.
\end{thm}
Given $\alpha=n-k$ for $1\le k\le \frac{n}{2}$, let $\mathbb{G}_{n,\alpha}^k=\{G\in \mathbb{G}_{n,\alpha}\mid n-\alpha=k\}$. For $T^*\in \mathbb{G}_{n,\alpha}^k$, Theorem \ref{thm-minimizer-main} tell us that $T^*=T^*_{-}\circ l_{V_2^*}$. $T^*_{-}$ is usually not unique and determined by (\ref{eq-t1})
\begin{equation}\label{eq-t1}
\left\{\begin{array}{ll}
d= 2k-2 \mbox{ if $k\in [1,2]$};
4\le d\le 2k-2 \mbox{ if $k\in [3,\frac{n}{2}]$}\\
h\le \min\{k-1-\frac{d}{2}, \lfloor\frac{ d}{4}\rfloor \}\\
\sum^h_{s=1} |M_s|=k-1-\frac{d}{2}\\
\sigma^*:M_s\longrightarrow M_{s-1},\ 1\le s \le h&
\end{array}\right.
\end{equation}
We collect all such $T^*_{-}$ in the set $\mathcal{T}^*_-(k)$. Moreover, by Proposition \ref{eq-degree1}, $T^*_{-}=(V_1^*,V_2^*)$ is bipartite ( refer to Fig.\ref{main}) satisfying $d_{T^*_{-}}(v)=d_{T^*}(v)=2$ for $v\in V_1^*$ and the leaves of $T^*_{-}$ are included in $V_2^*$ that are just the control vertices of $T^*$. Additionally, $|V_2^*|=k$ and $|V_1^*|=k-1$ by Proposition \ref{lem-main-tree}. The leaf sequence $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ satisfies $\sum_{u\in V_2^*}l(u)=|L(T^*)|=2\alpha-n+1=n-2k+1$, which will be further characterized in detail in next section.
\section{Constructing Theorem for the minimizer graphs of $\mathbb{G}_{n,\alpha}^k$}
In this section, we will give a constructing theorem for the minimizer graph $T^*\in \mathbb{G}_{n,\alpha}^k$. According to Theorem \ref{thm-minimizer-main}, $T^*$ comes from its main tree $T^*_-=(V_1^*,V_2^*)$ by adding a leaf sequence $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ on $V_2^*$. Denote by $\bar{l}=\lfloor\frac{|L(T^*)|}{|V^*_2|}\rfloor=\lfloor\frac{n-2k+1}{k}\rfloor$ the average value of $l_{V_2^*}$.
There exists a unique $0\le r \le k-1$ such that $n-2k+1\equiv r$ ( $\mbox{mod } k$ ), i.e., $n+1\equiv r$ ( $\mbox{mod } k$ ) ( Note that henceforth we always adhere to this choice for $r$ ), we have
$|L(T^*)|=n-2k+1=k\bar{l}+r$,
$n(T^*)=n=2k-1+k\bar{l}+r$ and \begin{equation}\label{ll-eq}
\frac{n-3k+2}{k}\le
\bar{l}=\frac{|L(T^*)|-r}{k}=\frac{n-2k+1-r}{k}
\le\frac{n-2k+1}{k}.
\end{equation}
\begin{lem}\label{lem-bound}
For given $1\le k=n-\alpha\le \frac{n}{2}$, let $T^*$ be the minimizer graph in $\mathbb{G}^k_{n,\alpha}$. Then $\rho(T^*)<\sqrt{\bar{l}+5}$, where $\bar{l}=\lfloor\frac{n-2k+1}{k}\rfloor$.
\end{lem}
\begin{proof}
Let $P_{2k-1}$ be a path with bipartite partition $(X,Y)$ such that $X$ contains two end vertices of $P_{2k-1}$. Then $|X|=k$ and $|Y|=k-1$. We now construct a tree
$T=P_{2k-1}\circ (\bar{l}+1)\mathbf{1}_{X}$. Since $\rho(P_{2k-1})=2\cos\frac{\pi}{2k}<2$,
by Lemma \ref{lem-bipartite} we have $$\rho(T)=\sqrt{\rho^2(P_{2k-1})+\bar{l}+1}<\sqrt{\bar{l}+5}.$$
On the other aspect, recall that $|L(T^*)|=k\bar{l}+r$, where $0\le r \le k-1$, we have $$|L(T)|=|X|(\bar{l}+1)=k(\bar{l}+1)=|L(T^*)|+k-r>|L(T^*)|.$$
Thus we can construct a subtree $T'$ by deleting $k-r$ leaves from $T$ such that $|L(T')|=|L(T^*)|$. Moreover,
$$\begin{array}{ll}
n(T')&=n(T)-(k-r)\\
&=|P_{2k-1}|+|X|(\bar{l}+1)-(k-r)\\
&=2k-1+ k\bar{l}+r\\&=n, \\
\alpha(T')&=\alpha(T)-(k-r)\\
&=|Y|+|X|(\bar{l}+1)-(k-r)\\
&=k-1 +k\bar{l}+r\\&=n-k.
\end{array}
$$
Thus $T'\in \mathbb{G}^k_{n,\alpha}$.
Since $T'$ is a subgraph of $T$, by Lemma \ref{lem-subgraph-radius} we have
$\rho(T^*)\le\rho(T')<\rho(T)<\sqrt{\bar{l}+5}$.
The proof is completed.
\end{proof}
By exchanging the position of $P_{2k-1}$ and $W_{2k+3}$ (see Fig.\ref{fig-W-D}) in the proof of Lemma \ref{lem-bound}, we get the following result.
\begin{lem}\label{cor-bound}
For given $1\le k=n-\alpha\le \frac{n}{2}$, let $T^*$ be the minimizer graph in $\mathbb{G}^k_{n,\alpha}$ and let $\bar{l}=\lfloor\frac{n-2k+1}{k}\rfloor$.
If $|L(T^*)|\le k\bar{l}+4$, then $\rho(T^*)\le \sqrt{\bar{l}+4}$.
\end{lem}
\begin{proof}
Let $W_{2k+3}=(X,Y)$ (see Fig.\ref{fig-W-D}) be a bipartite graph on $2k+3$ vertices such that $X$ does not contain any pendant vertices of $W_{2k+3}$.
Then $|X|=\lceil\frac{2k+3-4}{2}\rceil=k$.
Now we construct a tree $W=W_{2k+3}\circ \bar{l}\mathbf{1}_{X}$.
It's well known that
$\rho(W_{2k+3})=2$.
By Lemma \ref{lem-bipartite}, we have $$\rho(W)=\sqrt{\rho^2(W_{2k+3})+\bar{l}}=\sqrt{\bar{l}+4}.$$
Since $|L(T^*)|\le k\bar{l}+4$, from Proposition \ref{lem-main-tree} we have $$n(T^*)=n(T^*_{-})+|L(T^*)|\le 2k-1+k\bar{l}+4=n(W_{2k+3})+|X|\cdot \bar{l}= n(W).$$
Then we can construct a subtree
$W'$ by deleting $n(W)-n(T^*)$ leaves from $W$ such that $n(W')=n(T^*)=n$,
and so $$
\alpha(W')=\alpha(W)-(n(W)-n(T^*))=(n(W)-|X|)-(n(W)-n(T^*))=n-k.$$
Thus $W'\in \mathbb{G}^k_{n,\alpha}$.
Since $W'$ is a subtree of $W$, by Lemma \ref{lem-subgraph-radius} we obtain $\rho(T^*)\le\rho(W')\le \rho(W)\le\sqrt{\bar{l}+4}$.
The proof is completed.
\end{proof}
By Lemmas \ref{lem-bound} and \ref{cor-bound}, we can estimate the value of $l(u)$.
\begin{lem}\label{cor-s-range}
For given $1\le k=n-\alpha\le \frac{n}{2}$, let $T^*$ be the minimizer graph in $\mathbb{G}^k_{n,\alpha}$ and let $\bar{l}=\lfloor\frac{n-2k+1}{k}\rfloor$, $r=n-2k+1-\bar{l}k$.
For $u\in V^*_2$, we have
\begin{equation}\label{ll-eq-1}\bar{l}+r-2k+2-d_{T^*_-}(u) \le l(u)\le \bar{l}+4-d_{T^*_{-}}(u).\end{equation}
Particularly, if $0\le r\le 4$ then
\begin{equation}\label{ll-eq-2}\bar{l}+r-k+1-d_{T^*_-}(u)\le l(u)\le \bar{l}+3-d_{T^*_{-}}(u).\end{equation}
\end{lem}
\begin{proof}
By Lemma \ref{lem-bound}, we have $d_{T^*}(u)=\rho^2(K_{1,d_{T^*}(u)})< \rho^2(T^*)<\bar{l}+5$ and for each $u \in V^*_2$,
\begin{equation}\label{ll-eq-3} l(u)=d_{T^*}(u)-d_{T^*_-}(u)\le \bar{l}+4-d_{T^*_-}(u).\end{equation}
On the other hand, note that $|L(T^*)|=n-2k+1=k\bar{l}+r$, we have
$$\begin{array}{ll}
l(u)&=|L(T^*)|-\sum\limits_{u\not=v\in V^*_2}l(v)\\
&\ge k\bar{l}+r-\sum\limits_{u\not=v\in V^*_2}(\bar{l}+4-d_{T^*_-}(v))\\
&=k\bar{l}+r-(k-1)(\bar{l}+4)+\sum_{v\in V^*_2}d_{T^*_-}(v)-d_{T^*_-}(u)\\
&=k\bar{l}+r-(k-1)(\bar{l}+4)+m(T^*_-)-d_{T^*_-}(u)\\
&=k\bar{l}+r-(k-1)(\bar{l}+4)+2(k-1)-d_{T^*_-}(u)\\
&=\bar{l}+r-2k+2-d_{T^*_-}(u)
\end{array}$$
which together with (\ref{ll-eq-3}) leads to (\ref{ll-eq-1}).
If $0\le r\le 4$ then $|L(T^*)|\le k\bar{l}+4$. By Lemma \ref{cor-bound}, we have $d_{T^*}(u)=\rho^2(K_{1,d_{T^*}(u)})< \rho^2(T^*)\le\bar{l}+4$. Thus the corresponding (\ref{ll-eq-3}) becomes \begin{equation}\label{eq-upper-1}
l(u)=d_{T^*}(u)-d_{T^*_-}(u)\le \bar{l}+3-d_{T^*_-}(u).\end{equation}
As the same arguments as above, from (\ref{eq-upper-1}) we get (\ref{ll-eq-2}).
It completes the proof.
\end{proof}
According to Proposition \ref{lem-matching}(iii), we have
\begin{equation}\label{eq-degree}
d_{T^*_-}(u)\le k-1 \ \mbox{ for $u\in V^*_2$}.\end{equation}
From (\ref{ll-eq-1}), (\ref{ll-eq}) and (\ref{eq-degree}) we have
\begin{equation}\label{eq-5-0}
\begin{array}{ll}
l(u)&\ge\bar{l}+r-2k+2-d_{T^*_-}(u)\\
&=\frac{n-2k+1-r}{k}+r-2k+2-d_{T^*_-}(u)\vspace{0.1cm}\\
&\ge \frac{n-2k+1-r}{k}+r-2k+2-(k-1)\vspace{0.1cm}\\
&=\frac{n-3k^2+k+ 1+(k-1)r}{k}.
\end{array}
\end{equation}
Now we define $n_0=3k^2-k-1-(k-1)r$
for a given $k\ge 1$
and
\begin{equation}\label{n-k}
\ell_{n,k}=\frac{n-3k^2+k+ 1+(k-1)r}{k}=\frac{n-n_0}{k}\ \mbox{ for $n\ge n_0$}.\end{equation}
Then we have $n_0+1=3k^2-k-kr+r$, and so $n_0+1\equiv r$ ( $\mbox{mod } k$ ).
Recall that $n+1\equiv r$ ( $\mbox{mod } k$ ), we have $n\equiv n_0$ ( $\mbox{mod } k$ ) and $\ell_{n,k}$ is an integer.
Based on the above notions and symbols we give a constructing theorem for the minimizer graph in $\mathbb{G}_{n,\alpha}^k$.
\begin{thm}\label{thm-minimizer-spectral-radius}
For given $1\le k=n-\alpha\le \frac{n}{2}$,
let $n+1\equiv r$ ( $\mbox{mod } k$ ) and $n_0=3k^2-k-1-(k-1)r$, where $0\le r\le k-1$. For $n\ge n_0$, $T^*$ is a minimizer graph in $\mathbb{G}_{n,\alpha}^k$ if and only if there exists a minimizer graph $T_{n_0}\in \mathbb{G}_{n_0,n_0-k}^k$ such that $T^*=T_{n_0}\circ \ell_{n,k} \mathbf{1}_{V_2^*}$ and $\rho(T^*)=\sqrt{\rho^2(T_{n_0})+\ell_{n,k}}$, where $\ell_{n,k}$ is defined by (\ref{n-k}).
\end{thm}
\begin{proof}
Let $T^*$ be a minimizer graph in $\mathbb{G}_{n,\alpha}^k$.
By Theorem \ref{thm-minimizer-main},
we have
$T^*=T^*_{-}\circ l_{V_2^*}$ for some $T^*_{-}\in \mathcal{T}^*_-(k)$ and leaf sequence $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ satisfying $\sum_{u\in V_2^*}l(u)=n-2k+1$, moreover $l(u)$ satisfies (\ref{ll-eq-1}) or (\ref{ll-eq-2}) by Lemma \ref{cor-s-range}.
Since $n\ge n_0$ and $k\ge 1$, we see that $0\le \ell_{n,k}\le l(u)$ for $u\in V_2^*$. Thus we can construct a graph $G$ obtained from $T^*$ by deleting $\ell_{n,k}$ leaves at each $u\in V_2^*$, i.e., $G=T^*_{-}\circ (l(u)-\ell_{n,k}\mid u\in V_2^*)$.
Note that $|V_2^*|=k$, we have $n(G)=n-\ell_{n,k}|V_2^*| =n_0$ and $\alpha(G)=\alpha-\ell_{n,k}|V_2^*|=\alpha-(n-n_0)=n_0-k$.
Thus $G\in \mathbb{G}_{n_0,n_0-k}^k$. On the other hand,
it is clear that $T^*=G\circ \ell_{n,k}\mathbf{1}_{V_2^*}$. By Lemma \ref{lem-bipartite}, we have
\begin{equation}\label{mm-eq-1}
\rho(T^*)=\sqrt{\rho^2(G)+\ell_{n,k}}.\end{equation}
Notice that $\ell_{n,k}$ is a determined number related with $n$ and $k$ and independent with the choice of $G$, from (\ref{mm-eq-1}) we see that $G $ must be a minimizer graph in $\mathbb{G}_{n_0,n_0-k}^k$. The necessity holds.
Conversely, if there exists a minimizer graph $T_{n_0}\in \mathbb{G}_{n_0,n_0-k}^k$ such that $T^*=T_{n_0}\circ \ell_{n,k} \mathbf{1}_{V_2^*}$, then $n(T^*)=n_0+\ell_{n,k}|V_2^*|=n$ and $\alpha(T^*)=\alpha(T_{n_0})+\ell_{n,k}|V_2^*|=n_0-k+n-n_0=n-k=\alpha$. Thus $T^*\in\mathbb{G}_{n,\alpha}^k$ and $\rho(T^*)=\sqrt{\rho^2(T_{n_0})+\ell_{n,k}}$ by Lemma \ref{lem-bipartite}.
Since $T_{n_0}$ is a minimizer graph in $\mathbb{G}_{n_0,n_0-k}^k$ and $\ell_{n,k}$ is a determined number, $T^*$ is a minimizer graph in $\mathbb{G}_{n,\alpha}^k$. The sufficiency holds.
\end{proof}
\begin{remark}
According to Theorem \ref{thm-minimizer-spectral-radius}, the minimizer graph $T_{n_0}\in \mathbb{G}_{n_0,n_0-k}^k$ can be used to construct the minimizer graph of $\mathbb{G}_{n,\alpha}^k$. We call $T_{n_0}$ the \emph{kernel} of minimizer graph $T^*\in \mathbb{G}_{n,\alpha}^k$ if $T^*=T_{n_0}\circ \ell_{n,k} \mathbf{1}_{V_2^*}$. Given $k$, $n_0=3k^2-k-1-(k-1)r$ is a constant, we can simply found the kernel of minimizer graph among $\mathbb{G}_{n_0,n_0-k}^k$ by computer. Therefore, Theorem \ref{thm-minimizer-spectral-radius} completely determines the minimizer graph in $\mathbb{G}_{n,\alpha}^k$ and its spectral radius.
\end{remark}
\section{Construction for the minimizer graphs in $\mathbb{G}^k_{n,\alpha}$ for $1\le k \le 6$}
Theorem \ref{thm-minimizer-spectral-radius} completely characterizes the minimizer graphs of $\mathbb{G}_{n,\alpha}^k$ for $1\le k\le \frac{n}{2}$.
As an application, we will use a consistent approach to find the minimizer graphs of $\mathbb{G}^k_{n,\alpha}$ for $k= 1,2,3,4,5,6$. The results for $k=1,2,3,4$ were made in \cite{Xu,Lou} by different methods and other two are new.
Given $1\le k\le \frac{n}{2}$,
let $n+1\equiv r$ ( $\mbox{mod } k$ ) and $n\ge n_0=3k^2-k-1-(k-1)r$, where $0\le r\le k-1$ and $n+1\equiv n_0+1\equiv r$ ( $\mbox{mod } k$ ).
By Theorem \ref{thm-minimizer-spectral-radius}, each minimizer graph $T^*\in\mathbb{G}_{n,\alpha}^k$ can be presented by $T^*=T_{n_0}\circ \ell_{n,k} \mathbf{1}_{V_2^*}$, where the kernel $T_{n_0}$ is a minimizer graph in $ \mathbb{G}_{n_0,n_0-k}^k$.
By Theorem \ref{thm-minimizer-main}, $F^*=T_{n_0}$ can be presented by $F^*=F^*_{-}\circ l_{V_2^*}$, where $F^*_{-}=(V_1^*,V_2^*)\in \mathcal{T}^*_-(k)$ and $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ satisfying $\sum_{u\in V_2^*}l(u)=n_0-2k+1$.
In addition, it follows from Lemma \ref{cor-s-range} that $l_{V_2^*}$ is also restricted by (\ref{ll-eq-1}) or (\ref{ll-eq-2}).
Note that $n_0$ is small for $k=1,2,3,4$ and $\mathcal{T}^*_-(1)=\{P_1\}$, $\mathcal{T}^*_-(2)=\{P_3\}$,
$\mathcal{T}^*_-(3)=\{P_5\}$, $\mathcal{T}^*_-(4)=\{F_1,F_2\}$ (see Fig.\ref{main-tree}).
By using computer, we can simply choose the minimizer graphs in $\mathbb{G}_{n_0,n_0-k}^k$ by selecting proper $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ and finally get these minimizer graphs $T_{n_0}$ and then obtain $T^*\in\mathbb{G}_{n,\alpha}^k$ for $k=1,2,3$ and $4$.
For instance, let $k=3$, we have $n+1\equiv r$ ( $\mbox{mod } 3$ ) for some $0\le r\le2$, and $n\ge n_0=23-2r$.
If $r=0$ then $n_0=23$.
By Theorem \ref{thm-minimizer-main}, $T_{n_0}=F^*=F^*_{-}\circ l_{V_2^*}$ is a minimizer graph in $\mathbb{G}_{23,20}^3$, where $F^*_{-}=(V_1^*,V_2^*)\in \mathcal{T}^*_-(3)=\{P_5\}$ and $l_{V_2^*}=(l(v_0), l(v_2), l(v_4))$ satisfying $l(v_0)+ l(v_2)+l(v_4)=18$.
Additionally, $l_{V_2^*}$ is also restricted by (\ref{ll-eq-2}).
By searching the proper $l_{V_2^*}$, we find that the minimizer graph achieves at $l_{V_2^*}=(l(v_0), l(v_2), l(v_4))=(7,4,7)$, i.e., $F^*=P_5\circ(7,4,7)$ is the minimizer graph in $\mathbb{G}_{23,20}^3$.
Note that $\ell_{n,k}=\frac{n-23}{3}$ from (\ref{n-k}), by Theorem \ref{thm-minimizer-spectral-radius} we obtain the minimizer graph
$$\begin{array}{ll}T^*&=T_{n_0}\circ \ell_{n,k}\mathbf{1}_{V_2^*}\vspace{0.1cm}\\&= F^*\circ (\frac{n-23}{3},\frac{n-23}{3},\frac{n-23}{3})\vspace{0.1cm}\\
&=P_5\circ(7+\frac{n-23}{3},4+\frac{n-23}{3},7+\frac{n-23}{3})\vspace{0.1cm}\\
&=P_5\circ(\frac{n-2}{3},\frac{n-11}{3},\frac{n-2}{3})
\end{array}$$
in $\mathbb{G}_{n,\alpha}^3$ if $n+1\equiv 0$ ( $\mbox{mod } 3$ ) (see the fourth line in Tab.\ref{table-min-1-4}). Similarly we can find the minimizer graphs in $\mathbb{G}_{n,\alpha}^3$ for $n+1\equiv 1,2 $ ( $\mbox{mod } 3$ ) (see the fifth, sixth lines in Tab.\ref{table-min-1-4}).
As the same as above we can find all the minimizer graphs $T^*\in\mathbb{G}_{n,\alpha}^k$ along with their parameters ( including spectral radius ), which are listed in Tab.\ref{table-min-1-4} for $k=1,2,3,4$, respectively. They are just all the minimizer graphs obtained in \cite{Xu,Lou}.
\begin{table}[H]
\footnotesize
\caption{\small The minimizer graphs \\\footnotesize ( $n+1\equiv r$ ( $\mbox{mod } k$ ) and $n\ge n_0=3k^2-k-1-(k-1)r$ )}
\centering
\renewcommand{\arraystretch}{1.3}
\begin{tabular*}{14.7cm}{p{2pt}|p{3pt}|p{7pt}|p{95pt}|p{28pt}|p{19pt}|p{125pt}|p{40pt}}
\hline
$k$&$r$&$n_0$& the kernel $T_{n_0}\in \mathbb{G}_{n_0,n_0-k}^k$ &$\rho^2(T_{n_0})$&$\ell_{n,k}$& the minimizer graph $T^*\in \mathbb{G}_{n,\alpha}^k$
& $\rho(T^*)$\\\hline
$1$ &$0$&$1$&$P_{1}$ &$0$&$n-1$ &$P_1\circ (n-1)=K_{1,n-1}$&$\sqrt{n-1}$\\\hline
\multirow{2}*{$2$}&$0$&$9$&
$P_3\circ(3,3)$&$5$
&$\frac{n-9}{2}$&
$P_3\circ(\frac{n-3}{2},\frac{n-3}{2})$
&$\sqrt{\frac{n+1}{2}}$\\\cline{2-8}
&$1$&$8$&$P_3\circ(2,3)$&
$\frac{7+\sqrt{5}}{2}$& $\frac{n-8}{2}$& $P_3\circ(\frac{n-4}{2},\frac{n-2}{2})$ &$\sqrt{\frac{n-1+\sqrt{5}}{2}}$\\\hline
\multirow{3}*{$3$}&$0$&$23$&
$P_5\circ(7,4,7)$& $7+\sqrt{3}$&$\frac{n-23}{3}$&
$P_5\circ(\frac{n-2}{3},\frac{n-11}{3},\frac{n-2}{3})$ &$\sqrt{\frac{n-2+3\sqrt{3}}{3}}$\\\cline{2-8}
&$1$&$21$&$P_5\circ(6,4,6)$&$8$&$\frac{n-21}{3}$& $P_5\circ(\frac{n-3}{3},\frac{n-9}{3},\frac{n-3}{3})$ &$\sqrt{\frac{n+3}{3}}$\\\cline{2-8}
&$2$&$19$&$P_5\circ(5,4,5)$&$6+\sqrt{2}$&$\frac{n-19}{3}$&
$P_5\circ(\frac{n-4}{3},\frac{n-7}{3},\frac{n-4}{3})$ &$\sqrt{\frac{n-1+3\sqrt{2}}{3}}$\\\hline
\multirow{7}*{$4$}&\multirow{2}*{$0$}&\multirow{2}*{$43$}&
$F_1\circ(10,6,10,10)$& \multirow{2}*{$12$}&
\multirow{2}*{$\frac{n-43}{4}$}&
$F_1\circ(\frac{n-3}{4},\frac{n-19}{4},\frac{n-3}{4},\frac{n-3}{4})$
&
\multirow{2}*{$\sqrt{\frac{n+5}{4}}$}\\
&&&
$F_2\circ(10,8,8,10)$&&&$F_2\circ(\frac{n-3}{4},\frac{n-11}{4},\frac{n-11}{4},\frac{n-3}{4})$ &
\\\cline{2-8}
&$1$&$40$&$F_1\circ(9,6,9,9)$&
$\frac{19+\sqrt{13}}{2}$&$\frac{n-40}{4}$&
$F_1\circ(\frac{n-4}{4},\frac{n-16}{4},\frac{n-4}{4},\frac{n-4}{4})$ &$\sqrt{\frac{n-2+2\sqrt{13}}{4}}$\\\cline{2-8}
&\multirow{3}*{$2$}&\multirow{3}*{$37$}&
$F_2\circ(8,7,7,8)$&\multirow{3}*{$\frac{19+\sqrt{5}}{2}$}
&\multirow{3}*{$\frac{n-37}{4}$}&
$F_2\circ(\frac{n-5}{4},\frac{n-9}{4},\frac{n-9}{4},\frac{n-5}{4})$
&\multirow{3}*{$\sqrt{\frac{n+1+2\sqrt{5}}{4}}$}\\
&&&$F_2\circ(9,6,7,8)$&&&$F_2\circ(\frac{n-1}{4},\frac{n-13}{4},\frac{n-9}{4},\frac{n-5}{4})$&\vspace{0.1cm}\\
&&&$F_2\circ(9,6,6,9)$&&&
$F_2\circ(\frac{n-1}{4},\frac{n-13}{4},\frac{n-13}{4},\frac{n-1}{4})$ &\\ \cline{2-8}
&$3$&$34$&$F_1\circ(8,3,8,8)$&$\frac{15+\sqrt{21}}{2}$
&$\frac{n-34}{4}$&$F_1\circ(\frac{n-2}{4},\frac{n-22}{4},\frac{n-2}{4},\frac{n-2}{4})$ &$\sqrt{\frac{n-4+2\sqrt{21}}{4}}$\\
\hline
\end{tabular*}\label{table-min-1-4}
\end{table}
\begin{remark}
From Tab.\ref{table-min-1-4}, one can see that for $k=4$ the minimizer graph is not unique if $n+1\equiv r$ ( $\mbox{mod } 4$ ) with $r=0$ and $2$.
\end{remark}
Let $k=n-\alpha=5$. Here we determine the minimizer graph $T^*$ of $\mathbb{G}^5_{n,\alpha}$ in three steps.
{\flushleft\bf Step I.} In the step one, we will find main trees in $\mathcal{T}^*_-(5)$. Now (\ref{eq-t1}) becomes
$$\left\{\begin{array}{ll}d\in \{4,6,8\}\\
h\le \min\{ 4-\frac{d}{2}, \lfloor\frac{ d}{4}\rfloor \}\\
\sum^h_{s=1} |M_s|=4-\frac{d}{2}\\
\sigma^*:M_s\longrightarrow M_{s-1}\\
\ \ \mbox{ for } 1\le s \le h
\end{array}\right.\Rightarrow
\left\{\begin{array}{ll}d=4\\
h= 1\\
\sum^h_{s=1} |M_s|=2\\
\sigma^*:M_s\longrightarrow M_{s-1}\\
\ \ \mbox{ for } 1\le s \le h
\end{array}\right.,\
\left\{\begin{array}{ll}d=6\\
h= 1\\
\sum^h_{s=1} |M_s|=1\\
\sigma^*:M_s\longrightarrow M_{s-1}\\
\ \ \mbox{ for } 1\le s \le h
\end{array}\right.
\mbox{ or }
\left\{\begin{array}{ll}d=8\\
h=0\\
M_s=\emptyset
\end{array}\right.
$$
If the first occurs then $\sigma^*(x_{11}y_{11})=\sigma^*(x_{12}y_{12})=v_1v_2$, which leads to a main tree $F^5_1=T(4; \{x_{11}y_{11}, x_{12}y_{12}\}; \sigma^*)$ (see Fig.\ref{fig-main-5});
if the second occurs then $\sigma^*(x_{11}y_{11})=v_1v_2$, which produces a main tree $F^5_2=T(6; \{x_{11}y_{11}\}; \sigma^*)$ (see Fig.\ref{fig-main-5}); if the later occurs we obtain a main tree $F^5_3=T(8; \emptyset; \sigma^*)=P_9$ (see Fig.\ref{fig-main-5}). Thus
$\mathcal{T}^*_-(5)=\{F^5_1,F^5_2,F^5_3\}$.
\begin{figure}[h]
\centering
\footnotesize
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\caption{\footnotesize The main trees for $k=5$}\label{fig-main-5}
\end{figure}
{\flushleft\bf Step II. } In the step two, let $n+1\equiv r$ ( $\mbox{mod }5$ ) and $n\ge n_0=3k^2-k-1-(k-1)r=69-4r$, where $0\le r\le 4$, we will determine the kernel $T_{n_0}$ of minimizer graph $T^*\in \mathbb{G}^5_{n,\alpha}$ according to the different $r$.
Since the kernel $T_{n_0}$ is also
the minimizer graph in $\mathbb{G}_{n_0,n_0-5}^5$, by Theorem \ref{thm-minimizer-main}
there exists some main tree $T_-^*=(V_1^*,V_2^*)\in \mathcal{T}^*_-(5)$ such that
$T_{n_0}=T^*_{-}\circ l_{V_2^*}$ with leaf sequence $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ satisfying $|L(T_{n_0})|=\sum_{u\in V_2^*}l(u)=n_0-9=60-4r$. It remains to find $l(u)$ of $T_{n_0}$ for $u\in V_2^*$.
Let $\bar{l}=\lfloor\frac{|L(T_{n_0})|}{|V_2^*|}\rfloor=\lfloor\frac{60-4r}{5}\rfloor$.
First we suppose that $r=0$. Then $n_0=69$ and $\bar{l}=12$.
From (\ref{ll-eq-2}), $(l(u)\mid u\in V_2^*)$ satisfies
\begin{equation}\label{eq-sum-1}
\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=60\\
8-d_{T^*_-}(u)\le l(u)\le 15-d_{T^*_{-}}(u) \mbox{ for any $u\in V_2^*$}\end{array}\right.\end{equation}
where $d_{T^*_-}(u)$ can be determined by a main tree selected from the set $\mathcal{T}^*_-(5)=\{F^5_1,F^5_2,F^5_3\}$.
If $T^*_-=F^5_1=(V^*_1,V^*_2)$, from Fig.\ref{fig-main-5} we see that $V^*_2=\{v_0,v_2,v_4, y_{11}, y_{12}\}$ and $d_{F^5_1}(v_{0})=d_{F^5_1}(v_{4})=d_{F^5_1}(y_{11})=d_{F^5_1}(y_{12})=1$
and $d_{F^5_1}(v_{2})=4$.
Bringing these values into (\ref{eq-sum-1}), we get $38$ solutions for $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ which is collected in the set $\mathcal{L}_{F^5_1}$ and listed in Tab.\ref{tab-solution}.
By Theorem \ref{thm-minimizer-main}, $\mathbb{F}^5_1=\{F^5_1\circ l_{V_2^*}\mid l_{V_2^*}\in \mathcal{L}_{F^5_1}\} $ contains $38$ possible minimizer graphs in $\mathbb{G}_{n_0,n_0-5}^5$.
Comparing their spectral radii, we obtain $T^5_1=F^5_1\circ (13,8,13,13,13)$ with minimum spectral radius $\sqrt{13+\sqrt{5}}$ among $\mathbb{F}^5_1$ (see the first line in Tab.\ref{table-1}).
\begin{table}[H]
\footnotesize
\centering
\begin{tabular*}{14.6cm}{p{72pt}p{72pt}p{72pt}p{72pt}p{72pt}}
\hline
\multicolumn{5}{c}{$(\ l(v_0),\ l(v_2), \ l(v_4),\ l(y_{11}), \ l(y_{12})\ )$}\\ \hline
$(14, 4,14, 14, 14)$& $(13,5, 14, 14, 14)$&$(12,6, 14, 14, 14)$& $(13,6, 13, 14, 14)$& $(11,7, 14, 14, 14)$\\
$(12,7, 13, 14, 14)$& $(13,7, 13, 13, 14)$& $(10,8, 14, 14, 14)$& $(11, 8, 13, 14, 14)$& $(12,8, 12, 14, 14)$\\ $(12, 8, 13, 13, 14)$& \bm{$(13, 8, 13, 13, 13)$}& $(9, 9, 14, 14, 14)$& $(10,9, 13, 14, 14)$& $(11,9, 12, 14, 14)$\\ $(11,9, 13, 13, 14)$& $(12,9, 12, 13, 14)$& $(12,9, 13, 13, 13)$& $(8,10, 14, 14, 14)$ & $(9,10, 13, 14, 14)$\\ $(10,10, 12, 14, 14)$& $(10,10, 13, 13, 14)$ &$(11,10, 11, 14, 14)$& $(11,10, 12, 13, 14)$& $(11,10, 13, 13, 13)$\\ $(12,10, 12, 12, 14)$& $(12,10, 12, 13, 13)$ & $(7,11, 14, 14, 14)$& $(8,11, 13, 14, 14)$& $(9,11, 12, 14, 14)$\\ $(9,11, 13, 13, 14)$& $(10,11, 11, 14, 14)$& $(10,11, 12, 13, 14)$& $(10,11, 13, 13, 13)$ & $(11,11, 11, 13, 14)$\\ $(11,11, 12, 12, 14)$& $(11,11, 12, 13, 13)$& $(12,11, 12, 12, 13)$&&\\ \hline
\end{tabular*}
\caption{\small The set $\mathcal{L}_{F^5_1}$}\label{tab-solution}
\end{table}
If $T^*_-=F^5_2=(V^*_1,V^*_2)$, from Fig.\ref{fig-main-5} we see that $V^*_2=\{v_0,v_2,v_4, v_6, y_{11}\}$ and $d_{F^5_2}(v_{0})=d_{F^5_2}(v_{6})=d_{F^5_2}(y_{11})=1$,
$d_{F^5_2}(v_{2})=3$ and $d_{F^5_2}(v_{4})=2$. As similar as above,
we get $200$ solutions of (\ref{eq-sum-1}) which is collected in $\mathcal{L}_{F^5_2}$. Thus $\mathbb{F}^5_2=\{F^5_2\circ l_{V_2^*}\mid l_{V_2^*}\in \mathcal{L}_{F^5_2}\} $ contains $200$
possible minimizer graphs, by comparing spectral radii we get $T^5_2=F^5_2\circ (13,10,11,13,13)$ with minimum spectral radius $3.9068$ among $\mathbb{F}^5_2$ (see the second line in Tab.\ref{table-1}).
If $T^*_-=F^5_3=(V^*_1,V^*_2)$, from Fig.\ref{fig-main-5} we have $V^*_2=\{v_0,v_2,v_4, v_6, v_8\}$ and $d_{F^5_3}(v_{0})=d_{F^5_3}(v_{8})=1$,
$d_{F^5_3}(v_{2})=d_{F^5_3}(v_{4})=d_{F^5_3}(v_{6})=2$.
Similarly, we can put 170 solutions of (\ref{eq-sum-1}) in $\mathcal{L}_{F^5_2}$ and find that $T^5_3=F^5_3\circ (13,11,12,11,13)$ is the minimizer graph with spectral radius $\sqrt{\frac{27+\sqrt{13}}{2}}$ among $\mathbb{F}^5_3=\{F^5_3\circ l_{V_2^*}\mid l_{V_2^*}\in \mathcal{L}_{F^5_3}\}$ (see the third line in Tab.\ref{table-1}).
Finally, by comparing the spectral radii of $T^5_1$, $T^5_2$ and $T^5_3$, we get
$T_{n_0}=T^5_1=F^5_1\circ (13,8,13,13,13)$ is the minimizer graph in $\mathbb{G}_{69,64}^5$ with respect to $r=0$.
Follow the same procedure as $r=0$, we can obtain the minimizer graph $T_{n_0}$ in $\mathbb{G}_{n_0,n_0-5}^5$ with $n_0=69-4r$ for $r=1,2,3$ and $4$, which are all listed in Tab.\ref{table-1}, respectively.
That is $T_{n_0}=F^5_1\circ(13,8,13,13,13)$, $F^5_3\circ(12,11,10,11,12)$ and $F^5_3\circ(12,9,10,9,12)$ if $r=0,1,2$ and $3$, respectively, and $T_{n_0}=F^5_1\circ(10,4,10,10,10)$, $F^5_2\circ(10,6,8,10,10)$ or $F^5_3\circ(10,8,8,8,10)$ if $r=4$.
\begin{table}[H]
\footnotesize
\caption{\small The kernel $T_{n_0}$ for $k=5$}
\centering
\renewcommand{\arraystretch}{1.25}
\begin{tabular*}{15.00cm}{p{2pt}|p{3pt}|p{6pt}|p{135pt}|p{13pt}|p{13pt}|p{120pt}|p{35pt}}
\hline
$k$& $r$& $n_0$& the condition (\ref{eq-sum-1})
& $T^*_-$ & \#
& graph $T^5_i$ & $\rho(T^5_i)$ \\ \hline
\multirow{15}*{5}&\multirow{3}*{0}&\multirow{3}*{$69$}&
\multirow{3}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=60\vspace{0.08cm}\\8-d_{T^*_-}(u)\le l(u)\le 15-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^5_1$&$38$&
\bm{$F^5_1\circ(13,8,13,13,13)=T_{n_0}$}& \bm{$\!\!\sqrt{13+\!\sqrt{5}}$}\vspace{0.15cm}\\ \cline{5-8}
&&&&$F^5_2$&$200$&$F^5_2\circ(13,10,11,13,13)$& $3.9068$ \vspace{0.15cm}\\ \cline{5-8}
&&&&$F^5_3$&$170$&$F^5_3\circ(13,11,12,11,13)$& $\sqrt{\frac{27+\sqrt{13}}{2}}$\vspace{0.08cm} \\
\cline{2-8}
& \multirow{3}*{1}&\multirow{3}*{$65$}&
\multirow{3}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=56\vspace{0.08cm}\\8-d_{T^*_-}(u)\le l(u)\le 14-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^5_1$&$27$&$F^5_1\circ(12,8,12,12,12)$& $\sqrt{\frac{25+\sqrt{17}}{2}}$\vspace{0.08cm}\\ \cline{5-8}
&&&&$F^5_2$&$130$&$F^5_2\circ(12,9,11,12,12)$& $3.8090$ \vspace{0.15cm}\\ \cline{5-8}
&&&&$F^5_3$&$110$&\bm{$F^5_3\circ(12,11,10,11,12)=T_{n_0}$}& \bm{$3.8054$}\vspace{0.15cm} \\
\cline{2-8}
&\multirow{3}*{2}&\multirow{3}*{$61$}&
\multirow{3}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=52\vspace{0.08cm}\\8-d_{T^*_-}(u)\le l(u)\le 13-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^5_1$&$18$&$F^5_1\circ(12,4,12,12,12)$& $\sqrt{\frac{21+\sqrt{41}}{2}}$ \\ \cline{5-8}
&&&&$F^5_2$&$80$&$F^5_2\circ(12,7,9,12,12)$& $3.7003$ \vspace{0.15cm}\\ \cline{5-8}
&&&&$F^5_3$&$66$&\bm{$F^5_3\circ(12,9,10,9,12)=T_{n_0}$}& \bm{$3.6980$}\vspace{0.15cm} \\
\cline{2-8}
&\multirow{3}*{3}&\multirow{3}*{$57$}&
\multirow{3}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=48\vspace{0.08cm}\\8-d_{T^*_-}(u)\le l(u)\le 12-d_{T^*_{-}}(u)
\end{array}\right.$}&$F^5_1$&$12$&\bm{$F^5_1\circ(11,4,11,11,11)=T_{n_0}$}& \bm{$\!\!\sqrt{10+\!\sqrt{8}}$}\vspace{0.1cm}\\ \cline{5-8}
&&&&$F^5_2$&$46$&$F^5_2\circ(11,6,9,11,11)$& $3.5845$ \vspace{0.15cm}\\ \cline{5-8}
&&&&$F^5_3$&$38$&$F^5_3\circ(11,9,8,9,11)$& $3.5820$\vspace{0.15cm} \\
\cline{2-8}
&\multirow{3}*{4}&\multirow{3}*{$53$}&
\multirow{3}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=44\vspace{0.08cm}\\8-d_{T^*_-}(u)\le l(u)\le 11-d_{T^*_{-}}(u)
\end{array}\right.$}&$F^5_1$&$7$&\bm{$F^5_1\circ(10,4,10,10,10)=T_{n_0}$}& \bm{$2\sqrt{3}$} \vspace{0.15cm}\\ \cline{5-8}
&&&&$F^5_2$&$24$&\bm{$F^5_2\circ(10,6,8,10,10)=T_{n_0}$}& \bm{$2\sqrt{3}$} \vspace{0.15cm}\\ \cline{5-8}
&&&&$F^5_3$&$19$&\bm{$F^5_3\circ(10,8,8,8,10)=T_{n_0}$}& \bm{$2\sqrt{3}$}\vspace{0.15cm} \\
\hline
\multicolumn{8}{l}{\begin{tabular}{@{}l@{}}\# indicates the number of the solutions of $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ satisfying the condition (\ref{eq-sum-1}) in the fourth
\\ column for given $T^*_-\in \mathcal{T}^*_-(5)$.\end{tabular} }
\end{tabular*}\label{table-1}
\end{table}
{\flushleft\bf Step III. } In the step three, we will determine the minimizer graph $T^*$ in $\mathbb{G}^5_{n,\alpha}$ for any
$n\ge n_0=69-4r$ with $0\le r\le 4$.
By Theorem \ref{thm-minimizer-spectral-radius} we have $T^*=T_{n_0}\circ \ell_{n,5} \mathbf{1}_{V_2^*}$ and $\rho(T^*)=\sqrt{\rho^2(T_{n_0})+\ell_{n,5}}$, where $\ell_{n,5}=\frac{n-n_0}{5}=\frac{n-(69-4r)}{5}$.
For $r=0$, we know $T_{n_0}=F^5_1\circ(13,8,13,13,13)$ and
$\rho^2(T_{n_0})=13+\sqrt{5}$.
Note that $\ell_{n,5}=\frac{n-69}{5}$,
we have
$$\begin{array}{ll}T^*&=T_{n_0}\circ \ell_{n,5} \mathbf{1}_{V_2^*}=T_{n_0}\circ \frac{n-69}{5} \mathbf{1}_{V_2^*}\vspace{0.2cm}\\
&=F^5_1\circ(13+\frac{n-69}{5} ,8+\frac{n-69}{5} ,13+\frac{n-69}{5} ,13+\frac{n-69}{5} ,13+\frac{n-69}{5} )\vspace{0.2cm}\\&=
F^5_1 \circ(\frac{n-4}{5},\frac{n-29}{5},\frac{n-4}{5},\frac{n-4}{5},\frac{n-4}{5})=T^*_{n,0}\ \ \mbox{(see Fig.\ref{111})}
\end{array}$$
is the minimizer graph in $\mathbb{G}_{n,n-5}^5$ with $\rho(T^*)=\sqrt{13+\sqrt{5}+\ell_{n,5}}=\sqrt{\frac{n-4}{5}+\sqrt{5}}$.
As in the case of $r=0$, we can obtain the minimizer graph $T^*=T^*_{n,r}$ along with the spectral radius for $r=1,2,3$ and $4$, respectively, which are shown in
Fig.\ref{111} and omit the specific calculations. Finally we summarize these results in the following Theorem \ref{thm-n-5}.
\begin{thm}\label{thm-n-5}
Let $T^*$ be the minimizer graph in $\mathbb{G}^5_{n,\alpha}$ and let $n+1\equiv r$ ( $\mbox{mod }5$ ), where $0\le r \le 4$. For $n\ge 69-4r$, we have
$$T^*=\left\{
\begin{array}{ll}
T^*_{n,0}=F^5_1 \circ(\frac{n-4}{5},\frac{n-29}{5},\frac{n-4}{5},\frac{n-4}{5},\frac{n-4}{5})& \mbox {if $r=0$,}\vspace{0.15cm}\\
T^*_{n,1}=F^5_3 \circ (\frac{n-5}{5},\frac{n-10}{5},\frac{n-15}{5},\frac{n-10}{5},\frac{n-5}{5})& \mbox {if $r=1$,}\vspace{0.15cm}\\
T^*_{n,2}=F^5_3 \circ (\frac{n-1}{5},\frac{n-16}{5},\frac{n-11}{5},\frac{n-16}{5},\frac{n-1}{5})& \mbox {if $r=2$,}\vspace{0.15cm}\\
T^*_{n,3}=F^5_1 \circ (\frac{n-2}{5},\frac{n-37}{5},\frac{n-2}{5},\frac{n-2}{5},\frac{n-2}{5})& \mbox {if $r=3$,}\vspace{0.15cm}\\
T^*_{n,4}=\left\{\begin{array}{ll}F^5_1\circ(\frac{n-3}{5},\frac{n-33}{5},\frac{n-3}{5},\frac{n-3}{5},\frac{n-3}{5})
\vspace{0.15cm}\\
F^5_2 \circ (\frac{n-3}{5},\frac{n-23}{5},\frac{n-13}{5},\frac{n-3}{5},\frac{n-3}{5})\vspace{0.15cm}\\
F^5_3 \circ (\frac{n-3}{5},\frac{n-13}{5},\frac{n-13}{5},\frac{n-13}{5},\frac{n-3}{5})
\end{array}\right. & \mbox {if $r= 4$,}\\
\end{array}\right.$$
where minimizer graphs $T^*_{n,0}$, $T^*_{n,1}$,..., $T^*_{n,4}$ are described in Fig.\ref{111}.
Moreover, the spectral radius of $T^*$ is $\rho(T^*_{n,0})=\sqrt{\frac{n-4}{5}+\sqrt{5}}$, $\rho(T^*_{n,1})=\sqrt{\frac{n-5}{5}+2.4812}$, $\rho(T^*_{n,2})=\sqrt{\frac{n-6}{5}+2.6751}$, $\rho(T^*_{n,3})=\sqrt{\frac{n-7}{5}+\sqrt{8}}$ and $\rho(T^*_{n,4})=\sqrt{\frac{n+7}{5}}$, respectively.
\end{thm}
\begin{figure}[h]
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\end{picture}
\caption{\footnotesize The minimizer graphs for $k=5$}\label{111}
\end{figure}
In the rest of this section, let $k=n-\alpha=6$, we briefly
repeat the steps of I, II and III, as in the previous proofs of Theorem \ref{thm-n-5}, to determine the minimizer graphs in $\mathbb{G}^6_{n,\alpha}$.
{\flushleft\bf Step I.} To find $\mathcal{T}^*_-(6)$, we need consider (\ref{eq-t1}) in case of $k=6$, which leads four possibilities:
$$
\footnotesize
\left\{\begin{array}{ll}d\in \{4,6,8,10\}\\
h\le \min\{ 5-\frac{d}{2}, \lfloor\frac{ d}{4}\rfloor \}\\
\sum^h_{s=1} |M_s|=5-\frac{d}{2}\\
\sigma^*:M_s\longrightarrow M_{s-1}\\
\ \ \mbox{ for } 1\le s \le h
\end{array}\right.\!\!\Rightarrow
\left\{\begin{array}{ll}d=4\\
h= 1\\
\sum^h_{s=1} |M_s|=3\\
\sigma^*:M_s\longrightarrow M_{s-1}\\
\ \ \mbox{ for } 1\le s \le h
\end{array}\right.\!\!\!\!,\
\left\{\begin{array}{ll}d=6\\
h= 1\\
\sum^h_{s=1} |M_s|=2\\
\sigma^*:M_s\longrightarrow M_{s-1}\\
\ \ \mbox{ for } 1\le s \le h
\end{array}\right.\!\!\!\!,\
\left\{\begin{array}{ll}d=8\\
h= 1\\
\sum^h_{s=1} |M_s|=1\\
\sigma^*:M_s\longrightarrow M_{s-1}\\
\ \ \mbox{ for } 1\le s \le h
\end{array}\right.
\!\!\!\!\mbox{or}
\left\{\begin{array}{ll}d=10\\
h=0\\
M_s=\emptyset
\end{array}\right.
$$
The first leads to $F^6_1=T(4; \{x_{11}y_{11}, x_{12}y_{12},x_{13}y_{13}\}; \sigma^*)$ (see Fig.\ref{fig-main-6});
the second leads two main trees $F^6_2=T(6; \{x_{11}y_{11}, x_{12}y_{12}\}; \sigma_1^*)$ and
$F^6_3=T(6; \{x_{11}y_{11}, x_{12}y_{12}\}; \sigma_2^*)$ (see Fig.\ref{fig-main-6});
the third leads $F^6_4=T(6; \{x_{11}y_{11}\}; \sigma_3^*)$ and
$F^6_5=T(6; \{x_{11}y_{11}\}; \sigma_4^*)$ (see Fig.\ref{fig-main-6});
the later leads to $F^6_6=T(10; \emptyset; \sigma^*)=P_{11}$ (see Fig.\ref{fig-main-6}). Thus
$\mathcal{T}^*_-(6)=\{F^6_1,F^6_2,F^6_3,F^6_4,F^6_5,F^6_6\}$.
\begin{figure}
\centering
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\caption{\footnotesize{All main trees for $k=6$}}\label{fig-main-6}
\end{figure}
{\flushleft\bf Step II. } Let $n+1\equiv r$ ( $\mbox{mod }6$ ) and $n\ge n_0=3k^2-k-1-(k-1)r=101-5r$, where $0\le r\le 5$. To determine the kernel $T_{n_0}$ of minimizer graph $T^*\in \mathbb{G}^6_{n,\alpha}$,
we need to traverse the main tree $T_-^*=(V_1^*,V_2^*)\in \mathcal{T}^*_-(6)$ to determine the kernel $T_{n_0}$.
To exactly
$T_{n_0}=T^*_{-}\circ l_{V_2^*}$ and its leaf sequence $l_{V_2^*}=(l(u)\mid u\in V_2^*)$ satisfying the condition
\begin{equation}\label{eq-con-6}
\left\{\begin{array}{ll}|L(T_{n_0})|=\sum_{u\in V_2^*}l(u)=n_0-11=90-5r,&\vspace{0.1cm}\\
\bar{l}+r-5-d_{T^*_-}(u)\le l(u)\le \bar{l}+3-d_{T^*_{-}}(u)& \mbox{ if $0\le r \le 4$,}\vspace{0.1cm}\\
\bar{l}-5-d_{T^*_-}(u)\le l(u)\le \bar{l}+4-d_{T^*_{-}}(u)&\mbox{ if $r=5$,}
\end{array}\right.\end{equation}
where $\bar{l}=\lfloor\frac{|L(T_{n_0})|}{|V_2^*|}\rfloor=\lfloor\frac{90-5r}{6}\rfloor$.
As the same process as $k=5$, for each $0\le r\le 5$
we first obtain the set $\mathcal{L}_{F^6_i}$, whose elements are $l_{V_2^*}$ satisfying (\ref{eq-con-6}) for the main tree $T^*_-=F^6_i\in \mathcal{T}^*_-(6)$
( \# of Tab.\ref{table-1-6} indicates the number of elements in $\mathcal{L}_{F^6_i}$). Let $\mathbb{F}^6_i=\{F^6_i\circ l_{V_2^*}\mid l_{V_2^*}\in \mathcal{L}_{F^6_i}\}$ for $i=1,...,6$.
Next, by comparing the spectral radii of graphs in $\mathbb{F}^6_i$, we can get $T^6_i$ (see the seventh column in Tab.\ref{table-1-6}) with the minimum spectral radius among $\mathbb{F}^6_i$. Finally, by comparing the spectral radii of $T^6_i$ ($i=1,...,6$), we obtain the kernel $T_{n_0}$ shown in the seventh column of Tab.\ref{table-1-6}.
{\flushleft\bf Step III. }
Let $n\ge n_0=101-5r$, where $0\le r\le 5$ and $\ell_{n,6}=\frac{n-n_0}{6}=\frac{101-5r}{6}$.
By Theorem \ref{thm-minimizer-spectral-radius} we have $T^*=T_{n_0}\circ \ell_{n,6} \mathbf{1}_{V_2^*}$ and $\rho(T^*)=\sqrt{\rho^2(T_{n_0})+\ell_{n,6}}$,
where
the kernel $T_{n_0}$ is the minimizer graph determined in Step II shown in Tab.\ref{table-1-6} according to different $0\le r\le 5$.
Finally, we can state our result for $k=6$ in the following Theorem \ref{thm-n-6}.
\begin{table}[H]
\footnotesize
\caption{\small The kernel $T_{n_0}$ for $k=6$}
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular*}{15.6cm}{p{5pt}|p{5pt}|p{8pt}|p{135pt}|p{10pt}|p{15pt}|p{130pt}|p{38pt}}
\hline
$k$& $r$& $n_0$& the condition (\ref{eq-con-6})
& $T^*_-$ & \#
& graph $T^6_i$ & $\rho^2(T^6_i)$ \\ \hline
\multirow{44}*{6}
&\multirow{6}*{0}&\multirow{6}*{$101$}&
\multirow{6}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=90\vspace{0.08cm}\\10-d_{T^*_-}(u)\le l(u)\le 18-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^6_1$&
$60$
&
$F^6_1\circ(16,10,16,16,16,16)$&
$16+\sqrt{6}$
\\ \cline{5-8}
&&&&$F^6_2$&
$165$
&$F^6_2\circ(16,11,15,16,16,16)$&
$18.4370$
\\ \cline{5-8}
&&&&$F^6_3$&
$243$
&\bm{$F^6_3\circ(16,13,13,16,16,16)=T_{n_0}$}&
\bm{$17+\sqrt{2}$}
\\
\cline{5-8}
&&&&$F^6_4$&
$791$
&\bm{$F^6_4\circ(16,13,14,15,16,16)=T_{n_0}$}&
\bm{$17+\sqrt{2}$}
\\
\cline{5-8}
&&&&$F^6_5$&
$495$
&$F^6_5\circ(16,15,12,15,16,16)$&
$18.4309$
\\
\cline{5-8}
&&&&$F^6_6$&
$651$
&\bm{$F^6_6\circ(16,15,14,14,15,16)=T_{n_0}$}&
\bm{$17+\sqrt{2}$}
\\
\cline{2-8}
&\multirow{6}*{1}&\multirow{6}*{$96$}&
\multirow{6}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=85\vspace{0.08cm}\\10-d_{T^*_-}(u)\le l(u)\le 17-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^6_1$&
$42$
&
$F^6_1\circ(16,5,16,16,16,16)$&
$\frac{27+\sqrt{69}}{2}$
\\ \cline{5-8}
&&&&$F^6_2$&
$120$
&$F^6_2\circ(16,8,13,16,16,16)$&
$17.6378$
\\ \cline{5-8}
&&&&$F^6_3$&
$154$
&$F^6_3\circ(16,10,11,16,16,16)$&
$17.6579$
\\
\cline{5-8}
&&&&$F^6_4$&
$496$
&$F^6_4\circ(16,10,14,13,16,16)$&
$17.6323$
\\
\cline{5-8}
&&&&\multirow{6}*{$F^6_5$}&
\multirow{6}*{$330$}
&\bm{$F^6_5\circ(15,14,11,13,16,16)=T_{n_0}$}&\multirow{6}*{\bm{$\frac{33+\sqrt{5}}{2}$}}\\
&&&&&&\bm{$F^6_5\circ(15,14,11,14,15,16)=T_{n_0}$}&\\
&&&&&&\bm{$F^6_5\circ(16,13,11,13,16,16)=T_{n_0}$}&\\
&&&&&&\bm{$F^6_5\circ(15,14,12,13,16,15)=T_{n_0}$}&\\
&&&&&&\bm{$F^6_5\circ(15,14,12,14,15,15)=T_{n_0}$}&\\
&&&&&&\bm{$F^6_5\circ(16,13,12,13,16,15)=T_{n_0}$}&\\
\cline{5-8}
&&&&\multirow{4}*{$F^6_6$}&
\multirow{4}*{$396$}
&\bm{$F^6_6\circ(15,14,13,14,14,15)=T_{n_0}$}&\multirow{4}*{\bm{$\frac{33+\sqrt{5}}{2}$}}\\
&&&&&&\bm{$F^6_6\circ(15,14,13,14,13,16)=T_{n_0}$}&\\
&&&&&&\bm{$F^6_6\circ(15,14,14,13,13,16)=T_{n_0}$}&\\
&&&&&&\bm{$F^6_6\circ(16,13,13,14,13,16)=T_{n_0}$}&\\
\cline{2-8}
&\multirow{6}*{2}&\multirow{6}*{$91$}&
\multirow{6}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=80\vspace{0.08cm}\\10-d_{T^*_-}(u)\le l(u)\le 16-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^6_1$&
$29$
&
$F^6_1\circ(15,5,15,15,15,15)$&
$13+\sqrt{14}$
\\ \cline{5-8}
&&&&$F^6_2$&
$84$
&$F^6_2\circ(15,8,12,15,15,15)$&
$16.7443$
\\ \cline{5-8}
&&&&$F^6_3$&
$95$
&\bm{$F^6_3\circ(15,10,10,15,15,15)=T_{n_0}$}&
\bm{$15+\sqrt{3}$}
\\
\cline{5-8}
&&&&$F^6_4$&
$296$
&\bm{$F^6_4\circ(15,10,13,12,15,15)=T_{n_0}$}&
\bm{$15+\sqrt{3}$}
\\
\cline{5-8}
&&&&$F^6_5$&
$210$
&$F^6_5\circ(15,12,11,12,15,15)$&
$16.7491$
\\
\cline{5-8}
&&&&$F^6_6$&
$236$
&\bm{$F^6_6\circ(15,12,13,13,12,15)=T_{n_0}$}&
\bm{$15+\sqrt{3}$}
\\
\cline{2-8}
&\multirow{6}*{3}&\multirow{6}*{$86$}&
\multirow{6}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=75\vspace{0.08cm}\\10-d_{T^*_-}(u)\le l(u)\le 15-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^6_1$&
$19$
&
\bm{$F^6_1\circ(14,5,14,14,14,14)=T_{n_0}$}&
\bm{$\frac{25+\sqrt{45}}{2}$}
\\ \cline{5-8}
&&&&$F^6_2$&
$56$
&$F^6_2\circ(14,7,12,14,14,14)$&
$15.8664$
\\ \cline{5-8}
&&&&$F^6_3$&
$54$
&$F^6_3\circ(14,9,10,14,14,14)$&
$15.8830$
\\
\cline{5-8}
&&&&$F^6_4$&
$166$
&$F^6_4\circ(14,10,11,12,14,14)$&
$15.8878$
\\
\cline{5-8}
&&&&$F^6_5$&
$126$
&$F^6_5\circ(14,12,9,12,14,14)$&
$15.8750$
\\
\cline{5-8}
&&&&$F^6_6$&
$126$
&$F^6_6\circ(14,12,11,12,12,14)$&
$15.8969$
\\
\cline{2-8}
&\multirow{6}*{4}&\multirow{6}*{$81$}&
\multirow{6}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=70\vspace{0.08cm}\\10-d_{T^*_-}(u)\le l(u)\le 14-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^6_1$&
$12$
&
\bm{$F^6_1\circ(13,5,13,13,13,13)=T_{n_0}$}&
\bm{$15$}
\\ \cline{5-8}
&&&&$F^6_2$&
$35$
&\bm{$F^6_2\circ(13,7,11,13,13,13)=T_{n_0}$}&
\bm{$15$}
\\ \cline{5-8}
&&&&$F^6_3$&
$30$
&\bm{$F^6_3\circ(13,9,9,13,13,13)=T_{n_0}$}&
\bm{$15$}
\\
\cline{5-8}
&&&&$F^6_4$&
$86$
&\bm{$F^6_4\circ(13,9,11,11,13,13)=T_{n_0}$}&
\bm{$15$}
\\
\cline{5-8}
&&&&$F^6_5$&
$70$
&\bm{$F^6_5\circ(13,11,9,11,13,13)=T_{n_0}$}&
\bm{$15$}
\\
\cline{5-8}
&&&&$F^6_6$&
$66$
&\bm{$F^6_6\circ(13,11,11,11,11,13)=T_{n_0}$}&
\bm{$15$}
\\
\cline{2-8}
&\multirow{6}*{5}&\multirow{6}*{$76$}&
\multirow{6}*{$\left\{\begin{array}{ll}\sum_{u\in V_2^*}l(u)=65\vspace{0.08cm}\\5-d_{T^*_-}(u)\le l(u)\le 14-d_{T^*_{-}}(u)
\end{array}\right.$}&
$F^6_1$&
$83$
&
\bm{$F^6_1\circ(12,5,12,12,12,12)=T_{n_0}$}&
\bm{$\frac{23+\sqrt{29}}{2}$}
\\ \cline{5-8}
&&&&$F^6_2$&
$220$
&$F^6_2\circ(12,7,10,12,12,12)$&
$14.2080$
\\ \cline{5-8}
&&&&$F^6_3$&
$364$
&$F^6_3\circ(12,8,9,12,12,12)$&
$14.2361$
\\
\cline{5-8}
&&&&$F^6_4$&
$1211$
&$F^6_4\circ(12,9,10,10,12,12)$&
$14.2470$
\\
\cline{5-8}
&&&&$F^6_5$&
$715$
&$F^6_5\circ(12,10,9,10,12,12)$&
$14.2283$
\\
\cline{5-8}
&&&&$F^6_6$&
$1001$
&$F^6_6\circ(12,10,10,11,10,12)$&
$14.2831$
\\
\hline
\end{tabular*}\label{table-1-6}
\end{table}
\begin{thm}\label{thm-n-6}
Let $T^*$ be a minimizer graph in $\mathbb{G}^6_{n,\alpha}$ and let
$n+1\equiv r$ ( $\mbox{mod }6$ ), where $0\le r \le 5$. For $n\ge 101-5r$, all the minimizer graphs in $\mathbb{G}^6_{n,\alpha}$ and their spectral radii are listed in Tab.\ref{table-k-6},
in which $F^6_i$ are shown in Fig.\ref{fig-main-6} for $i=1,2,...,6$.
\end{thm}
\begin{table}[H]
\footnotesize
\caption{\small The minimizer graph $T^*$ and its spectral radius ( $n+1\equiv r$ ( $\mbox{mod }6$ ) )}
\centering
\renewcommand{\arraystretch}{1.35}
\begin{tabular*}{15.2cm}{p{2pt}|p{144pt}|p{36pt}||p{2pt}|p{136pt}|p{36pt}}
\hline
$r$& $T^*$& $\rho(T^*)$&$r$& $T^*$& $\rho(T^*)$\\ \hline
\multirow{3}*{$0$}&
$F^6_3\!\circ\!(\frac{n-5}{6},\frac{n-23}{6},\frac{n-23}{6},\frac{n-5}{6},\frac{n-5}{6},\frac{n-5}{6})$
&\multirow{3}*{\!\!\!$\sqrt{\frac{n+1}{6}\!+\!\!\sqrt{2}}$}&
\multirow{3}*{$2$}&
$F^6_3\!\circ\!(\frac{n-1}{6},\frac{n-31}{6},\frac{n-31}{6},\frac{n-1}{6},\frac{n-1}{6},\frac{n-1}{6})$
&\multirow{3}*{\!\!\!$\sqrt{\frac{n-1}{6}\!+\!\!\sqrt{3}}$}
\\
&$F^6_4\!\circ\!(\frac{n-5}{6},\frac{n-23}{6},\frac{n-17}{6},\frac{n-11}{6},\frac{n-5}{6},\frac{n-5}{6})$&
&&$F^6_4\!\circ\!(\frac{n-1}{6},\frac{n-31}{6},\frac{n-13}{6},\frac{n-19}{6},\frac{n-1}{6},\frac{n-1}{6})$&\\
&$F^6_6\!\circ\!(\frac{n-5}{6},\frac{n-11}{6},\frac{n-17}{6},\frac{n-17}{6},\frac{n-11}{6},\frac{n-5}{6})$&
&&$F^6_6\!\circ\!(\frac{n-1}{6},\!\frac{n-19}{6},\frac{n-13}{6},\frac{n-13}{6},\frac{n-19}{6},\frac{n-1}{6})$&\\
\cline{1-6}
\multirow{10}*{$1$}&
$F^6_5\!\circ\!(\frac{n}{6}\!-\!1,\frac{n}{6}-2,\frac{n}{6}-5,\frac{n}{6}-3,\frac{n}{6},\frac{n}{6})$
&\multirow{10}*{\!\!\!$\sqrt{\frac{n}{6}+\!\frac{1+\sqrt{5}}{2}}$}&
\multirow{2}*{$3$}&
\multirow{2}*{$F^6_1\!\circ\!(\frac{n-2}{6},\frac{n-56}{6},\frac{n-2}{6},\frac{n-2}{6},\frac{n-2}{6},\frac{n-2}{6})$}
&\multirow{2}*{$\!\!\!\sqrt{\frac{n\!-11}{6}\!+\!\!\frac{\sqrt{45}}{2}}$}\\
&$F^6_5\!\circ\!(\frac{n}{6}\!-\!1,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!5,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!1,\frac{n}{6})$&
&&&\\\cline{4-6}
&$F^6_5\!\circ\!(\frac{n}{6},\frac{n}{6}-3,\frac{n}{6}-5,\frac{n}{6}-3,\frac{n}{6},\frac{n}{6})$&
&\multirow{6}*{$4$}&
$F^6_1\!\circ\!(\frac{n-3}{6},\frac{n-51}{6},\frac{n-3}{6},\frac{n-3}{6},\frac{n-3}{6},\frac{n-3}{6})$
&\multirow{6}*{$\sqrt{\frac{n+9}{6}}$}
\\
&$F^6_5\!\circ\!(\frac{n}{6}\!-\!1,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!4,\frac{n}{6}\!-\!3,\frac{n}{6},\frac{n}{6}\!-\!1)$&
&
&$F^6_2\!\circ\!(\frac{n-3}{6},\frac{n-39}{6},\frac{n-15}{6},\frac{n-3}{6},\frac{n-3}{6},\frac{n-3}{6})$&\\
&$F^6_5\!\circ\!(\frac{n}{6}\!-\!1,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!4,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!1,\frac{n}{6}\!-\!1)$&
&&$F^6_3\!\circ\!(\frac{n-3}{6},\frac{n-27}{6},\frac{n-27}{6},\frac{n-3}{6},\frac{n-3}{6},\frac{n-3}{6})$&\\
&$F^6_5\!\circ\!(\frac{n}{6},\frac{n}{6}-3,\frac{n}{6}-4,\frac{n}{6}-3,\frac{n}{6},\frac{n}{6}-1)$&
&
&$F^6_4\!\circ\!(\frac{n-3}{6},\frac{n-27}{6},\frac{n-15}{6},\frac{n-15}{6},\frac{n-3}{6},\frac{n-3}{6})$&\\
&$F^6_6\!\circ\!(\frac{n}{6}\!-\!1,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!3,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!1)$&
&
&$F^6_5\!\circ\!(\frac{n-3}{6},\frac{n-15}{6},\frac{n-27}{6},\frac{n-15}{6},\frac{n-3}{6},\frac{n-3}{6})$&\\
&$F^6_6\!\circ\!(\frac{n}{6}\!-\!1,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!3,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!3,\frac{n}{6})$&
&&$F^6_6\!\circ\!(\frac{n-3}{6},\!\frac{n-15}{6},\frac{n-15}{6},\frac{n-15}{6},\frac{n-15}{6},\frac{n-3}{6})$&\\
\cline{4-6}
&$F^6_6\!\circ\!(\frac{n}{6}\!-\!1,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!2,\frac{n}{6}\!-\!3,\frac{n}{6}\!-\!3,\frac{n}{6})$&&\multirow{2}*{$5$}&
\multirow{2}*{$F^6_1\!\circ\!(\frac{n-4}{6},\frac{n-46}{6},\frac{n-4}{6},\frac{n-4}{6},\frac{n-4}{6},\frac{n-4}{6})$}
&\multirow{2}*{$\!\!\!\sqrt{\frac{n\!-7}{6}\!+\!\!\frac{\sqrt{29}}{2}}$}\\
&$F^6_6\!\circ\!(\frac{n}{6},\frac{n}{6}-3,\frac{n}{6}-3,\frac{n}{6}-2,\frac{n}{6}-3,\frac{n}{6})$&&&&\\
\hline
\end{tabular*}\label{table-k-6}
\end{table}
\begin{remark}
The key to determining a minimizer graph $T^*$ is to determine its kernel, which is construct from a main trees in the set $\mathcal{T}_{-}^*(k)$. The number of the main trees increase as $k $ increases, and we see from known results, in particular our Theorem \ref{thm-n-5} and Theorem \ref{thm-n-6}, that every main tree in $\mathcal{T}_{-}^*(k)$ generates at least one minimizer graph.
\end{remark}
\section{Conclusion}
Theoretically, Theorem \ref{thm-minimizer-spectral-radius} together with Theorem \ref{thm-minimizer-main} completely characterize the minimizer graph and its spectral radius in $\mathbb{G}_{n,\alpha}^k$ when $k=n-\alpha \le\frac{n}{2}$,
moreover we give a general method for determining the minimizer graphs in three specific steps.
However, the characterization of minimizer graph and its spectral radius for $k=n-\alpha >\frac{n}{2}$ is still an open problem.
It is clear from our results that, on the one hand, although the minimizer graph $T^*=T_{n_0}\circ \ell_{n,k} \mathbf{1}_{V_2^*}$ and its spectral radius $\rho(T^*)=\sqrt{\rho^2(T_{n_0})+\ell_{n,k}}$ have uniform expressions depending on its kernel $T_{n_0}$, the minimizer graphs are generally not unique and their tree structures are diverse as increases with $k$.
On the other hand, the representation of the minimizer graph depends on the classification of $n$ by mod $k$.
The determination of the minimizer graph and the calculation of its spectral radius have a certain complexity.
Given $k$, the kernel $T_{n_0}$ is the minimal graph in $\mathbb{G}_{n_0,n_0-k}^k$, where $n_0=3k^2-k-1-(k-1)r$, and when $k$ is small (e.g., $k=1, 2,... ,6$) we can simply determine $T_{n_0}$ from the structural features of the minimizer graph obtained in this paper,
thus giving minimizer graphs of arbitrary order $n\ge n_0$. However, as $k$ increases $n_0$ grows by the square order of $k$, the kernel of the minimizer graph can only be found with the help of a computer.
As Stevanovi\'{c} pointed out in \cite{Stevanovic}, determining the graph with the minimum spectral radius among connected graph with independence number $\alpha$ appears to be a tough problem.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,425
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VY Canis Majoris är en röd hyperjätte som är en av de största kända stjärnorna i Vintergatan räknat i diameter. Den lyser 300 gånger starkare än solen. Varje år förlorar VY Canis Majoris omkring 30 gånger jordens massa i stoft och gas.
VY Canis Majoris magnitud varierar mellan 6,5 och 9,6.
Referenser
Externa länkar
Röda superjättar
Röda hyperjättar
Stjärnbilden Stora hunden
Halvregelbundna variabler
HD-objekt
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,973
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Solar carport inverter installation requires pre-planning and safety precautions
By Kelsey Misbrener | April 13, 2020
American Solar Power
Installers have plenty of options when it comes to choosing an inverter for a solar carport project since they aren't required to comply with rapid shutdown codes. Still, string inverters are the favorite choice for these installations for a number of reasons.
Los Angeles-based installer American Solar Power (ASP) prefers to install string inverters in carport arrays because they're compact and simple. The company tries to stay away from string inverters that must be paired with optimizers to operate, because then the installation and O&M plan become more complicated.
"The wire management is challenging on the optimizers," said Edwin Baranian, owner of ASP. "It just has a lot of tedious extra work to polish up the project."
Steven Roseborough, director of commercial operations at ASP, said optimizers are also usually the first piece of a solar project to fail. Since they're mounted underneath the modules, swapping these out on carport installations typically requires renting a scissor lift to reach them.
String inverter maintenance usually only requires a ladder for ASP unless the inverter is broken and needs to be replaced. In most cases, maintenance workers can climb a ladder to the inverter, hook up a laptop to troubleshoot, then swap out faulty fuses or terminals.
Replacing a whole, heavy unit takes more time and money, so Baranian suggests that system owners install reliable, well-reviewed inverters to try to ensure longevity.
American Solar Power installed a 430-kW carport solar project for Tesoro Apartments in Los Angeles using Fronius string inverters.
"If I have any advice, it would be to use equipment that has a track record of operation," Baranian said.
In addition to maintenance and mounting considerations, installers should choose inverter brands that are proven to perform well in outdoor conditions.
"The products that can handle really low temperatures as well as very high temperatures without effectively exceeding their ability to operate, those are the ones you want to install," said Michael Mills-Price, head of inverter and energy storage business at PV Evolution Labs (PVEL). PVEL published its first PV Inverter Scorecard in 2019.
Most inverters installed in carport arrays will be shaded by the structure, unlike some roof-mounted inverters, but will still be exposed to weather and higher daily operational temperatures than those sheltered in a garage.
Along with natural wear and tear on the inverter, installers must take precautions to make sure people don't damage them, accidentally or intentionally.
Sunworks installed this 950-kW solar carport at a senior living community in Lincoln, California.
"When they're out in the open like that and you've got anything that can be flipped, switched or turned or anything like that, it's nice to get it up out of the way," said Don Peek, director of commercial design for California installation company Sunworks. "I mean, anybody determined can go up and still mess with things, but most people don't."
Mills-Price said the ideal mounting solution for string inverters would be at the ground level in a locked cabinet, but usually there isn't space for that in the parking lot layout. Most installers instead mount string inverters high enough on the steel beams that people can't touch them.
Peek said the majority of inverter manufacturers are conscious of vandalism risks and design secure mounts and hidden bolts so inverters can't be ripped off poles.
"If you make it too easy to steal, it'll disappear," Peek said.
Manufactured mounting points
Although installers can use the same bracket as in a garage installation to mount string inverters on poles, mounting on steel is much different than wood.
"I've seen our guys installing these inverters and trying to drill through I-beam steel and just burning through bits," said Darren Kelly, manager of business development and marketing at Baja Carports. "It's something they're not used to. They're used to drilling into wood, mount that thing and get out of there. Now they're drilling into steel."
Baja Carports offers a way to avoid that onsite hassle. The company consults with installers during the carport manufacturing process, plots exactly where inverters will go on the poles, and pre-drills mounting holes at the factory.
"On steel, everything's harder," Kelly said. "Anyone could drill into wood no problem. But on steel, we have to consider where everything goes so that it can be done in the shop versus in the field."
Nontraditional carport inverter choices
If the installation is big enough for central inverters to make financial sense, Mills-Price said this inverter type can be a great option for carport installs.
American Solar Power's 150-kW Bobrick carport project in Los Angeles uses a Solectria central inverter.
"You can put a single inverter to cover the three or four carports depending on how they're laid out, tie them all into one location. Central inverters are ground-mounted by nature, and you'd have one service location," he said.
Central inverters would need to be protected by some type of gate or fencing for safety, but maintenance could be performed at ground-level.
Mills-Price said even microinverters can be a good choice for carport installations. Since they're usually mounted to the back of the panel or to the racking itself, there's no need to find a place for them on the poles.
"If your carport's a little complicated, a little spread out or something like that, a microinverter might be a good choice," Sunworks' Peek said. "It separates the strings a little bit more."
Microinverters also offer superior monitoring since they're module-level power electronics. Peek said O&M is not much more difficult with micros than strings. Since most of Sunworks' carport installations are 12 ft high or less, a 10-ft ladder is all his team needs to swap out microinverters or string inverters.
Still, Peek prefers string inverters on carport installations since they're simple but still give installers the reliability of multiple inverters on a larger system.
"It's kind of the good medium between the two extremes, and it's cost effective too," Peek said.
No matter what type of inverter installers choose for carport installations, picking one that's proven to work best outside and taking steps to prevent vandalism will ensure the longest lifespan possible.
Tips for mounting solar inverters on commercial rooftops
Kelsey Misbrener
Kelsey is managing editor of Solar Power World and host of the Contractor's Corner podcast.
Burns & McDonnell installs 60 MWh of stand-alone energy storage in Texas
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,580
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You are at:Home»Music»No 'Po' Boyz:' The Davisson Brothers Band finding success abroad
No 'Po' Boyz:' The Davisson Brothers Band finding success abroad
By Christina Fuoco-Karasinski on August 30, 2018 Music
This has been a whirlwind 12 months for rural West Virginia country stars the Davisson Brothers Band.
Singer/rhythm guitarist Donnie Davisson and Chris Davisson, along with longtime friends drummer Aaron Regester and bassist Rus Reppert, toured coast to coast and across the ocean in support of their single "Po' Boyz," which hit No. 2 on the Australian country charts.
There's nothing poorly about this band. They're finding success just being themselves.
"We stay true to ourselves and make sure we are as authentic as we can," says Chris Davisson, lead and slide guitarist. "The video (for 'Po' Boyz') was shot in West Virginia in the Appalachian Mountains. We filmed in the farm where we hang out, the mountains where we fish and hang out. We kept it real."
Music is in the Davisson brothers' blood. Their father is a longtime musician and the boys were making a living with music at age 12.
"We've never done anything else," says Davisson, who was a multisport athlete in high school.
"Po' Boyz" is found on the band's latest album, Fighter, recorded with producer Keith Stegall (Alan Jackson, Zac Brown Band). The Davisson Brothers recorded the basic tracks live in the recording studio, creating a project that can be reproduced in a concert setting.
"We intentionally tried not to overdo anything," Davisson says. "Less is more with us."
Rolling Stone Country hailed the Davisson Brothers Band as one of the 10 new country artists you need to know. Davisson isn't quite sure what the band is doing right.
"Nobody has the formula for a hit song and why they work," he says. "We just try to be ourselves. Rural America and Rural Australia just dig it."
To watch the video for "Po' Boyz," visit http://bit.ly/2xaz2j.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,142
|
One of the most repeated problems I am asked to resolve, is the tricky problem of black spot, most commonly found in the bathroom, and kitchen, where exists the ideal conditions of moisture and warmth to create the black spot fungus. This fungus is not uncommon in bedrooms, or even living rooms, and can have disastrous results on clothes and soft furnishings!
The principal cause is a lack of fresh air, the problem is compounded by central heating, double glazing, and draft excluders, ironically all recent so-called improvements, to the modern home! I know there is a conflict of interest, when we are constantly told to maximise insulation, install double glazing, and add central heating, to greatly improve the comfort and the thermal values of our homes. However that shouldn't mean abandoning the provision of valuable fresh air, vital for the good health of our families and our homes!
Bathing, showering, cooking, washing and drying clothes, making tea and coffee, these are all major contributing factors to water condensing within the home, and this moisture needs ventilating properly to avoid fungus problems. A few simple tips, to try and improve things. If it is too cold to have the bathroom window open whilst showering / bathing, open it immediately afterwards, and leave the bathroom door ajar, open a couple of windows around the house or flat to encourage fresh air to flow throughout. Try avoiding hanging washing on radiators, for a flat lacking in space, try hanging washing over the bath, with the extractor on to remove some of the moisture created. Fitting a quality extractor (vented to the outside) is a useful tool, to help any property with condensation issues, but if the problem persists, you may have to consider introducing a proper mechanical air control system, in order to cure the problem in the long term!
From experience, I found that the problems of condensation tend to be considerably worse, in smaller confined spaces, the smaller the bathroom or kitchen, the bigger the problem. Another common cause, albeit one we have to accommodate now with our very busy lives, is the tumble drier. The best method of using a tumble dryer regularly, is to insure the machine is vented via an outside wall, which is sometimes not easy in a small flat. Alternatively if using a condensing dryer, follow the instructions carefully, clean and change the filter regularly. If after leaving doors and windows open, throughout your property, the problem persists, you may want to consider using a de-humidifier in your home, which works like a vacuum cleaner sucking in air and removing the moisture from it. (You can hire a de-humidifier, from a hire shop for a trial period to see if this eradicates the problem before making a purchase). This fungus problem, not only makes your home look bad, there is normally a terrible musty smell, and an increased possibility of respiratory illness, so it's a problem that needs to be taken seriously, and addressed as soon as possible!
A Tommy tip for cleaning off black spot fungus is to make a solution of bleach, hot water, and liquid soap or cleaning fluid, then apply with a sponge, turning over and using the green pan scrubber side to apply a little elbow grease, which normally removes the fungus if it is not too well entrenched.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,568
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\subsection{Modern data assimilation and hybrid modeling in geosciences}
In this case study, we focus on image data fusion (blending) methods as optimal approaches for overcoming individual sensor's limitations and combining different multiresolution datasets. Blending Landsat and MODIS has been the preferred sensor combination in the literature and enabled predicting gap free surface reflectances \cite{gao2006blending,gevaert2015comparison} at Landsat spatial resolution (30 m). Both missions provide long time series of data with a high degree of consistency. The MODIS sensor, on board of Terra and Aqua platforms, provides global observations and a daily revisit cycle at a cost of having a coarse spatial resolution (250-1000m). This resolution clearly limits its utility for fine-scale environmental applications, but on the other hand, MODIS high temporal resolution allows tracking rapid land-cover changes while maximizes the possibility of having cloud-free observations. We have also capitalized on using these two sensors, especially because the proposed approach has been specifically designed to exploit past temporal information to improve the results.
Here, we focus on a KF logic method named HIghly Scalable Temporal Adaptive Fusion Model (HISTARFM). This algorithm was implemented in the Google Earth Engine (GEE) cloud computing platform \cite{gorelick2017google} and consists of a bias-aware Bayesian data assimilation scheme \cite{dee1998data}. HISTARFM uses two Kalman estimators operating simultaneously to reduce the amount of noise and decrease possible biases (if present) in the predicted Landsat spectral reflectances. The first estimator is an optimal interpolator (a special case of the Kalman filter with no dynamic model) that produces estimates of Landsat reflectance values for a given time by combining a Landsat climatology (mean monthly values considering many years) and linearly blended Landsat and MODIS spectral information. The second coupled estimator is an additional Kalman filter which is in charge of correcting dynamically possible biases of the reflectance produced (if present) by the first estimator. The next figure shows an example of the results of the HISTARFM algorithm over an area with massive gaps due to cloud contamination and sensor malfunctioning.
\begin{figure*}[h!]
\subfigure[Original RGB composite with Landsat data ]{\includegraphics[width=.487\textwidth]{Figuras/Gaps.PNG}\label{fig:twitter}}
\subfigure[HISTARFM gap filled RGB composite]{\includegraphics[width=.49\textwidth]{Figuras/nogaps.PNG}\label{fig:twitter-sentiment}}
\caption{Differences between the original Landsat data and the gap filled data set processed with the the proposed data assimilation approach. Both images correspond with a cropland area in Texas state (US) for the date May 2010.}
\end{figure*}
HISTARFM takes advantage of the GEE platform, this enables to process huge amounts of data significantly faster than other approaches available in the literature. The validation of the proposed method over 1050 sites spread out over the conterminous United States indicated the feasibility of the method and provided satisfactory results. The relative mean errors remained below 2$\%$ (in all spectral bands) and the relative mean absolute errors and relative root mean squared errors ranged between low to moderate (10-20$\%$) depending on the spectral band. Moreover, the high degree of agreement between the validation errors and the predicted uncertainties by HISTARFM indicated the utility of this information for error propagation purposes.
\subsection{Multitemporal and multisensor gap-filling in remote sensing}
The presence of gaps in EO data limits their applicability in a number of applications that need continuous data. Standard techniques for gap filling temporal series such as linear or cubic interpolation, or auto-regressive functions fail to reconstruct sharp transitions or long data gaps. Also, they are not able to infer information from other collocated sensors measuring the same biophysical variable, which is the setting found, for instance, when harmonizing data from multiple satellites into consistent climate data records of Essential Climate Variables (ECVs)~\cite{GCOS}. Another challenging setting for standard gap-filling approaches is the fusion of collocated microwave and optical observations for cloud-free estimates of vegetation descriptors, which needs to exploit the relationships between the two~\cite{Pipia19}. In this section, we show how we can efficiently deal with the spatio-temporal gaps of collocated satellite-based observations by employing a multi-output Gaussian Process model based on the Linear Model of Corregionalization (LMC)~\cite{Alvarez11}. The method allows learning the relationships among the different sensors and build an across-domain kernel function to transfer information across the time series and do predictions with associated confidence intervals on regions where no data are available.
\begin{figure}[t!] \centering
\includegraphics[width=0.95\textwidth]{Figuras/compo_dahra_inkscape.pdf}
\caption{Results of the application of the multi-output LMC-GP gap-filling technique at DAHRA validation site (1 station). Top left: site location and land use map. Top right: time series of \textit{in-situ} (black-lines), and satellite-based soil moisture estimates from SMOS, ASCAT and AMSR2 (orange dots denote the training data and green lines the predictions). The blue rectangle indicates the time period that is represented in the bottom figure. The bottom-left dashed rectangle exemplifies how the method reconstructs long data gaps in AMSR2 based on no-rain information from the other two sensors, assigning a higher uncertainty when no training data is available. The bottom-right dashed rectangle points out a specific rainfall event that was captured only by SMOS and is accounted for in the reconstruction of ASCAT and AMSR2 time series.}
\label{fig:dahra}
\end{figure}
We illustrate the procedure using soil moisture time series from three spaceborne microwave sensors, which are integrated in the ESA Climate Change Initiative (CCI) soil moisture product \cite{Dorigo2017}: the ESA's SMOS L-band radiometer, the AMSR2 C-band radiometer on-board JAXA's GCOM-W1, and the ASCAT C-band scatterometer on-board Eumetsat MetOp satellites. The temporal period of study is 6 years, starting in June 2010. Each product presents different observational gaps due to the presence of Radio Frequency Interferences at their operating frequency or high uncertainty in their inversion algorithm (e.g. presence of snow masking observations, dense vegetation, or high topography). The problem we face here is that we need a gap-filling methodology able to handle several outputs together and force a ``sharp'' reconstruction of the time series so that fast dry-down and wetting-up dynamics are preserved (avoid smoothing). We show the application of the LMC multi-output GP regression at two {\em in-situ} soil moisture networks from the \href{https://ismn.geo.tuwien.ac.at}{International Soil Moisture Network}: REMEDHUS in Spain (17 stations~\cite{Sanchez12}), and DAHRA in Senegal (1 station~\cite{dahra}). In terms of temporal coverage, they are representative of best-case (REMEDHUS), and wort-case (DAHRA) scenarios, where best satellite coverage is of 91\% and of 64\% of the study period, respectively.
Results of the application of the proposed LMC-GP over DAHRA are shown in Figure ~\ref{fig:dahra}, together with the original satellite time series and the {\em in-situ} data as a benchmark. It can be seen that the reconstructed soil moisture time series follow closely the original time series, capturing the wetting-up and drying-down events and filling the missing information (e.g. see the peak in October 2014 which was captured only by SMOS and is reproduced by the three reconstructed time series). Also, predictions have associated uncertainties related to the availability of training data for each specific sensor. It is remarkable that for AMSR2 the reconstructed time series back-propagate to dates before the satellite was launched (shown here for illustration purposes), yet they look very consistent with the real satellite data. Given the soil moisture products present no-data in different time and space locations, the method allows to provide predictions at all time stamps where at least there is one satellite measurement, maximizing therefore the spatio-temporal coverage of the data sets.
\begin{table}[!ht]
\small
\caption{Mean error (ME) [m$^3$m$^{-3}$], unbiased RMSE (ubRMSE) [m$^3$m$^{-3}$] and Pearson's correlation (R) for the original and reconstructed satellite time series against {\em in-situ} measurements from REMEDHUS and DAHRA networks. Variable `gaps' reports the percentage of days that were gap-filled in the reconstructed series.}
\vspace{-0.5cm}
\begin{center}
\small \renewcommand{\tabcolsep}{0.1cm}
\begin{tabular}{c|cccc|cccc}
\toprule
& \multicolumn{4}{c|}{\bf{REMEDHUS}} & \multicolumn{4}{c}{\bf DAHRA } \\
& ME & ubRMSE & R & gaps[\%] & ME & ubRMSE & R & gaps[\%] \\
\midrule
SMOS & -0.032 & 0.003 & 0.81 & - & -0.0143 & 0.001 & 0.79 & - \\
SMOS rec & -0.033 & 0.003 & 0.81 & 8.58 & -0.014 & 0.001 & 0.78 & 54.14 \\
\midrule
ASCAT & 0.002 & 0.004 & 0.79 & - & 0.071 & 0.002 & 0.70 & - \\
ASCAT rec & -0.001 & 0.004 & 0.78 & 23.68 & 0.064 & 0.002 & 0.70 & 36.15 \\
\midrule
AMSR2 & 0.118 & 0.005 & 0.86 & - & 0.026 & 0.002 & 0.73 & - \\
AMSR2 rec & 0.084 & 0.005 & 0.81 & 52.24 & 0.019 & 0.001 & 0.79 & 77.26 \\
\bottomrule
\end{tabular}
\end{center}
\label{table:sm_stats}
\end{table}
Statistical scores from comparison with {\em in-situ} data at the two sites of the original and reconstructed time series are shown in Table~\ref{table:sm_stats}. The analyses show that Pearson's correlation coefficient (R), mean error (ME) and unbiased root-mean-squared error (ubRMSE) with respect to {\em in-situ} data remain within reasonable bounds and are not affected to a high degree by the reconstruction, even in the more extreme case of DAHRA. These results provide confidence in the proposed technique and their potential to mitigate the effect of missing information in satellite-based observational records.
\subsection{Natural hazard prediction and data fusion}
Natural hazards cause thousands of deaths and inflict tremendous societal damage every year. To demonstrate the catastrophic influence of such hazards, between 2005 and 2014, 700,000 people were killed, and 1.7 billion people were affected worldwide by disasters\footnote{https://www.unisdr.org/}. This clearly demonstrates the importance of developing accurate and efficient mapping, modeling, and prediction techniques to reduce the catastrophic impact of natural hazards.
The success of information fusion techniques has revolutionized the challenging and vitally important applications of modeling natural hazards \cite{AP1}. In this context, the fusion of multiple sources of data using ML techniques has been reported as an effective tool to greatly contribute to increasing the accuracy of the prediction models \cite{AP2, AP3,P20,P21}.
To provide a bird's-eye view over the variety of data fusion models for hazard modeling, a number of relevant research studies have been summarized in Table \ref{table:Hazard}, which provides an overview of the key contributions, investigated data sources, and the tackled hazard type. The fusion of the multiple data sources, e.g., satellite imaging, radar data, laser point clouds, UAV images, weather stations, crowdsourcing data, social media, and GIS have shown their advantages to greatly enhance the robustness and performance of hazard detection and avoidance systems, leading to a safer planetary anytime, anywhere.
\begin{table}[!ht]
\scriptsize
\vspace{12pt}
\caption{Notable Research studies on data fusion techniques for hazard modeling .}
\centering
\begin{tabular}{| p{3.0cm} | p{5.5cm} | p{5.0cm}| p{1.4cm}|}
\hline
\textbf{Research Studies} & \textbf{Contribution} & \textbf{Data fusion} & \textbf{Hazard type} \\
\hline
Shi et al. (2019) \cite{AP4} & An enhanced flexible spatiotemporal data fusion model for prediction & Fusion of Landsat and MODIS satellite data & Landslide \\
\hline
Shanet al. (2019) \cite{AP5} & High-rate real-time GNSS seismology and early warning system & Fusion of displacement and acceleration seismology data & Earthquake \\
\hline
Yang et al. (2019) \cite{AP6} & Susceptibility mapping using the B-GeoSVC model and Hierarchical Bayesian method & Fusion of regional and local information & Landslide \\
\hline
Feng et al. (2019) \cite{AP7} & Integration of remotely sensed drought factors for accurate drought prediction & Fusion of thirty remotely-sensed drought factors from the Tropical Rainfall Measuring Mission (TRMM) and MODIS & Drought \\
\hline
Zou et al. (2019) \cite{AP8} & Wildfire smoke simulations and observations for regional hazard prediction & Fusion of PM2.5 air pollution & Wildfire \\
\hline
Knipper et al. (2019) \cite{AP9} &
Evapotranspiration prediction for irrigation management & Fusion of evapotranspiration time-series retrievals from multiple satellite platforms & Drought \\
\hline
Guerriero et al. (2019) \cite{AP10} &
Flood hazard mapping in convex floodplain: Multiple probability models fusion, bank threshold, and levees effect spatialization & Fusion of LiDAR-derived high-resolution topography and ground-based measurements & Flood \\
\hline
Lee and Tien (2018) \cite{AP11} & Probabilistic Framework for disaster prediction & Fusion of multiple sources, including physical sensors measuring environmental quantities and big data from social sensors & Disaster \\
\hline
Azmi and Rüdiger (2018) \cite{AP12} & Validating the data fusion‐based drought index (DFDI) and recalibrating the regional drought thresholds for increasing the predictive accuracy. & Fusion of drought index & Drought \\
\hline
Alizadeh and Nikoo (2018) \cite{AP13} & A fusion-based methodology for drought prediction & Fusion of diverse remotely sensed data & Drought \\
\hline
Pastick et al. (2018) \cite{AP14} & Spatiotemporal analysis of dryland ecosystems & Data fusion of Landsat-8 and Sentinel-2, and MODIS & Drought \\
\hline
Li and Fan (2017) \cite{AP15} & Accurate landslides hazard prediction & Fusion of remote sensing data and UAV & Landslide \\
\hline
Zhuo and, Han (2017) \cite{AP16} & Accurate drought prediction & Fusion of remotely sensed data and weather stations & Drought \\
\hline
Rosser et al. (2017) \cite{AP17} & Rapid and accurate flood inundation mapping & Fusion of social media, remote sensing, and topographic information & Flood \\
\hline
Renschler and Wang (2017) \cite{AP18} & Accurate flood prediction & Fusion of multi-source GIS, hydraulic modeling based on remote sensing data, and LiDAR & Flood \\
\hline
Hillen (2017) \cite{AP19} & Hazard prediction and mitigation & Fusion of geo-information and mobility data & Geohazards and flood \\
\hline
\end{tabular}
\label{table:Hazard}
\vspace{24pt}
\end{table}
Among the variety of geohazards, flood and drought modeling and prediction is regarded as a very complex phenomenon which is known to be among the least understood natural hazards due to its multiple causing reasons or contributing factors operating at different temporal and spatial scales. Fortunately, the application of data fusion in flood and drought modeling has evolved significantly compared to other hazards \cite{AP19,AP20,AP21}. To demonstrate the effectiveness of ML-based information fusion techniques for the challenging application of flood prediction, in the following, we provided a dedicated case study.
\subsubsection{Flood prediction by integrating remote sensing and weather station data}
Floods are increasingly recognized as a frequent natural hazard worldwide. Increasing the accuracy of flood susceptibility mapping is of utmost importance for efficient land use management, policy analysis, and the advancement of the mitigation structures to optimally reduce the devastation.
\textbf{Introduction to flood susceptibility mapping: }
Hazard susceptibility mapping of flood is essential for mitigation due to their higher destructive power in a short period. ML-based methods are among the most popular methods used for accurate mapping of the flood hazard \cite{AP25}. Several comparative studies in the literature reported promising results using ML methods \cite{AP26, AP27}. Along with employing the hybridization and ensemble techniques for improving the accuracy of ML models, the data fusion techniques have shown promising results. The aim of this case study is to demonstrate the performance of a data fusion approach to integrate information obtained by weather stations, land survey, and satellite data to improve the accuracy of the flood superstability mapping.
\textbf{Materials and methods:}
The study area is Gorganroud Basin, located in the northwest of Iran between latitudes of 36º 25' to 38º 15' N and longitudes of 56º 26' to 54º 10' E. In this case study, the data fusion is conducted by locating the flooded and non-flooded points and identifying the inundated regions using Sentinel-2 satellite images. Due to the lack of recorded location of flood occurrences, the inundation areas are identified using the Modified Normalized Difference Water Index (\red{MNDWI}) of Sentinel-2. Radiometrically calibrated and terrain corrected Sentinel-2 Level-1C dataset is stored within GEE to support the free cloud computing facilities for this case study \cite{AP28}. Fig. \ref{fig:statAP2} shows the study area. The inundated area is extracted by MNDWI during a period from March and April of 2019 when the flood affected the region (Fig. \ref{fig:statAP3}). Furthermore, feature selection using simulated annealing and modeling through RF is used to identify the hazard areas. The validation is done using hit and miss analysis.
\begin{figure*}
\centering
\includegraphics[width=0.5\linewidth]{Figuras/Picture10.png}
\vspace{-1.2em}
\caption{Location of the case study; distribution of the weather stations in the basin.}
\label{fig:statAP2}
\end{figure*}
After identifying the inundated area, the number of 368 flash-flood locations were randomly considered from the inundated points and their locations were confirmed through field surveys. For modeling, the inundated points were considered as the dependent variable and used for modeling with RF.
\begin{figure*}
\centering
\includegraphics[width=0.7\linewidth]{Figuras/Picture11.png}
\vspace{-1.2em}
\caption{a) Flooded and non-flooded points (b) inundated regions according to Sentinel-2.}
\label{fig:statAP3}
\end{figure*}
\textbf{Modeling:}
After preparing the predictand flood/non-flood locations as input and the predicting variables as output, the model is developed where the values of 0 and 1 were assigned to the non-flood and flood occurrence locations, respectively. From the whole dataset, 70\% of the data is considered for training while the remaining 30\% of the data is used for testing. A 10-fold cross-validation methodology was used to train the ML models. The results of the hazard modeling using RF is shown in Fig. \ref{fig:statAP4}.
\begin{figure*}
\centering
\includegraphics[width=0.7\linewidth]{Figuras/Picture15.png}
\vspace{-1.2em}
\caption{Hazard sustainability mapping of flood using RF.}
\label{fig:statAP4}
\end{figure*}
\subsection{Fusion of Earth data and Social media}
Earth data is routinely used to infer many aspects of Earth. Since most Earth data are generated using satellites or airborne platforms, they are often limited to a birds-eye perspective. Given the high sensor quality and non-invasiveness of remote sensing, such birds-eye Earth data have become a primary driver for understanding the morphological structure of Earth, including applications ranging from biology \cite{buchanan2009delivering,giri2011status}, environmental and social sciences \cite{skidmore2003environmental,turner2003methodological}, cartography and mapping \cite{huang2018urban}, among others. Unfortunately, this non-invasiveness of the birds-eye perspective implies some limitations in the semantic concepts one can distinguish. This difference is manifested in the distinction of land-cover mapping from land-use mapping or as well between building function (e.g., what people are supposed to do in this location) and activity estimation (e.g., what people are observed to do in reality). While the first one is what a satellite or airborne platform can sense, the second one is even more important to many applications. In this context, additional data and measurements can greatly help resolve these ambiguities. Such additional (covariate) data sources include base map data including information like building footprints, street networks, cadastral information, historical information or and data contributed by citizens, for example, through using social media. Social media information is envisioned to augment the birds-eye view with ground-level features like images or text originating from a certain location or more abstract anthropogenic signals correlated to population density, wealth, or other spatial distributions of interest.
In general, there are three categories of how location-based social media can be analyzed and fused with ESS data: one is based on the metadata. In computer science, metadata comprises all data that describes the data at hand, usually including information on time, identity, place, size, and user profiles. The second category of exploiting social media is based on text mining, trying to understand and relate the message content to the Earth. Finally, the third category considers social media images and video sequences. Similarly to the text case, many images in social networks do not relate to the location they are sent from. Hopefully, the totality of images and videos in a certain area tells something about the area or one can detect which messages are related to a given location. All three aspects of social media require different data mining, ML and data fusion approaches as they all pose very different challenges to the overall system.
\subsubsection{Spatial Statistics of Metadata}
\begin{figure*}[!ht]
\subfigure[Global Twitter Density]{\includegraphics[width=.49\textwidth]{Figuras/twitter-density.png}\label{fig:twitter}}
\subfigure[Positive Sentiment of Twitter messages around New York]{\includegraphics[width=.49\textwidth]{Figuras/twitter-sentiment.png}\label{fig:twitter-sentiment}}
\caption{Illustrations of Social Media Statistics on the example of Twitter public stream data.}
\end{figure*}
The most traditional technique relies on spatial summary statistics of messages. One assumes that the patterns in which messages are generated are related to interesting factors which influence ESS parameters of interest. For example, social media users are typically quiet while sleeping and have a detectably higher activity during the day \cite{zheng2018survey}. That is, messages that originate from a certain location late in the evening as well as in the early morning with a break of a few hours might indicate some home or sleep location, definitely a spatial information. Fusing with overhead imagery, we might be able to distinguish between a hotel (large building, social media activity very low during late night) and a residential building (small building, much green around, social media activity low during late night). In addition, the type of user or intention of the message is reflected in metadata to some extent. Marketing and job announcements of corporations, for example, tend to use the full capacity of social media containing as many hashtags as possible to reach many users, and contain many words exploiting the full capacity of the message. It has been shown that simple features extracted from social media metadata alone can increase the convergence speed and final quality of a deep learning model predicting urban land use modeled after local climate zones paradigm by a significant margin \cite{Leichter18}. Figure \ref{fig:twitter} depicts spatial density of Twitter data measured as the radius of a $k$-nearest-neighbor environment of each point. As can seen, Twitter usage is tightly connected to many socio-demographic factors. This relation has been discussed in \cite{li2013spatial} as well.
\subsubsection{Text Mining}
Text mining comprises a set of techniques in which natural language text is transformed into a numeric representation that captures some semantics of the text. Many techniques have been introduced in the past that are based on character and word frequencies including TF-IDF ranking of documents in keyword search \cite{lott2012survey}, or sparse text mining based on word-document matrices including topic modelling and Latent Dirichlet Allocation (LDA) \cite{chen2019experimental}. However, the limitation of these methods has usually been to find a good vocabulary for a given task that does not contain meaningless words and pre-processing text in order that different forms of words are detected as the same word in these methods. When it comes to spatial data, however, language is gradually changing with location and might jump to a completely different tongue when crossing borders, complicating this process even more. In the deep learning area, text embeddings have become feasible which are a bit less dependent on these basic, but very hard language pre-processing techniques, and exploit very large datasets to cover the variations automatically \cite{manna2019effectiveness,tang2014learning}. These map a vocabulary into a Euclidean space of chosen dimension in a way such that semantically similar words end up near each other. Doing this in a certain way leads to the surprising property that the addition in the vector space becomes semantically sensible with some limitations \cite{Boja16}. But generally, one aims to become able to perform {\em semantic operations} in the latent space as follows:
\begin{align*}
\text{King - Man + Woman} & \approx \text{Queen}\\
\text{Paris - France + Germany} & \approx \text{Berlin}
\end{align*}
Such numeric representations have then been used in deep learning using long short-term memories (LSTMs) \cite{wang2015unified} in order to allow for the deep learning system to analyze sentences taking care to important semantic words like negations. In the last months, advanced language models based on transformer architectures like BERT and GPT2 have shown impressive language understanding and generation capacities. Their careful application to social media in the context of ESS is a promising direction for future research. In relation to ESS, text embeddings have been successfully applied to distinguish residential and commercial buildings \cite{igarssbtclassification}. In addition, standard challenge problems of text mining such as sentiment analysis can as well bring interesting spatial information to life. Sentiment analysis has been well-researched in the natural language domain and in the computer science and ML domain in the context of learning from sparsely labeled streams \cite{iosifidis2019sentiment}. The idea is to assign a rating ranging from negative to positive to a text capturing the emotion it represents. Figure \ref{fig:twitter-sentiment} depicts the distribution of positive sentiment tweets around New York, which seems to be somehow skewed towards commercial centers.
\subsubsection{Image Mining and Multimedia Analysis}
In a similar vein to text mining, the whole body of image analysis and computer vision research can be applied to social media images too. This can help collect information about the Earth surface from a ground-level view, assuming that there are enough images taken at the location from where they are sent. Similar to the text case, images attached to social media messages might not be related to the location where they are posted. Consequently, the multimedia information available from social media is extremely noisy and one must take this into account \cite{igarssmutualinformation}.
Social media images have been used to extract statistics such as object counts essentially augmenting message metadata and opening the field for applications of spatial statistics. However, images can as well be directly used in order to classify, for example, land-use classes \cite{dlr126371,groundaerialfusion}. In this case, however, causal relations usually break and the system will be ``right for the wrong reason'', a typical side-effect of overfitting. In general, the social media stream will contain images that might not be representative for the actual context of the post, where this context information can be the physical surroundings as well as the activity (e.g., dining) or the socio-demographic context (e.g., rich vs. poor). An end-to-end ML system can learn to exploit all various types of context in the images and this abstract context is related to the location of origin at least in the sense that a user has been in a given context in this location. Consequently, it is not surprising that social media images -- though they look like they do not relate significantly to the surroundings as they often do not depict the surroundings -- reveal a lot of social, spatial, and economic information and can, therefore, help augment typical ESS questions \cite{cao2018integrating}.
\subsubsection{Summary}
Social media comprises a nice yet challenging data source for Earth observation. While it is obvious that social media correlates to human activities and is dense in urban regions, it is surprising that the content of the messages and their metadata can be used to distinguish traditional land cover and land use ambiguities in remote sensing imagery. However, methods of learning from such noisy data streams with such sparse labels are still in its infancy and need additional breakthroughs in unsupervised ML, natural language processing, and spatial data science in order to provide full potential.
\section{Introduction}
The Earth is a highly complex, dynamic, and networked system where very different physical, chemical and biological processes interact, to form the world we know \cite{Kump04,Stanley05,Brasseur05}. The description of such a complex system needs of the integration of different disciplines such as Physics, Chemistry, Mathematics and other applied sciences, leading to what has been coined as Earth System Science (ESS) \cite{Jacobson00}. The analysis of the Earth system involves studying interacting processes occurring in several spheres (atmosphere, hydrosphere, cryosphere, geosphere, pedosphere, biosphere, and magnetosphere) as well as the anthroposphere. Earth system science provides the physical basis of the world we live in, with the final objective of obtaining a sustainable development of our society\footnote{\href{https://www.un.org/sustainabledevelopment/sustainable-development-goals/}{https://www.un.org/sustainabledevelopment/sustainable-development-goals/}}.
Traditionally, Earth system models characterize processes and their relations by encoding the known physical knowledge. This involves deriving models from first principles, which typically involves physics-based mechanistic modeling. Such Earth system models are complex constructs, and can be designed at different scales, but provide and encode the fundamental basis for understanding, forecasting, and modeling.
Models are often confronted with observations for refinement and improvement. In the last five decades, the field of Earth Observation (EO) from space has allowed monitoring and modelling the processes on the Earth surface, and their interaction with the atmosphere, by obtaining quantitative measurements and estimations of geo-bio-physical variables, and has permitted to detect extremes, changes, and anomalies. By combining EO data from stations, sensors, and ancillary model simulations, we can now monitor our planet with unprecedented accuracy, spatially explicitly and temporally resolved.
In the last decade, however, we have witnessed two important changes: the big data and the Machine Learning (ML) revolutions, which have impacted many areas of Science and Engineering, but also the Earth sciences \cite{Reichstein19}. Both combined are leading to paradigm shifts in the way we now study the Earth system:
\begin{itemize}
\item {\em The big data revolution.} Nowadays, we observe and model the Earth with a wealth of observations and data measured from a plethora of different sensors measuring states and physical variables at unprecedented spatial, spectral, and temporal resolutions. They include remote sensing systems, mounted on satellites, airplanes, and drones, but also {\em in-situ} observations (increasingly from autonomous sensors) at, and below, the surface and in the atmosphere. Data from numerical, radiative transfer and climate models, and from reanalysis, are also very effective for tackling specific problems in Earth science. The data deluge is ever-growing in volume (hundreds of petabytes already), speed (estimated around 5 Pb/yr), variety (in sampling frequencies, spectral ranges, spatio-temporal scales and dimensionality), and uncertainty (from observational errors to conceptual inconsistencies).
\item {\em The Machine Learning revolution.} Besides, ML techniques have emerged as a fundamentally effective way to model and extract patterns out of big data in a (semi)automatic manner. Earth science has been also impacted by such revolution in many different ways. For example, machine learning models are now routinely used to predict and understand components of the Earth system: 1) classification of land cover types, 2) modelling of land-atmosphere and ocean-atmosphere exchange of greenhouse gases, 3) detection of anomalies and extreme events, and 4) causal discovery have greatly benefited from ML approaches. ML has been also used to complement physical models developed in the last 50 years, which are now able to assimilate the measured data and yield more accurate estimates/predictions of the evolution of the system (or parts thereof), such as the Atmosphere or the Ocean.
Hybrid modeling and physics-aware machine learning are nowadays emerging fields too, and promise data-driven and physically consistent models for the future study of the Earth system.
\end{itemize}
All in all, the access to such unprecedented big data sources, increased computational power, and the recent advances in ML offer exciting new opportunities for expanding our knowledge about the Earth system from data. EO is now at the center of the data processing pipeline. Either data assimilation, canonical ML, or advanced hybrid modeling needs to successfully exploit at maximum the diversity and complementarity of the different data sources. The use of Information Fusion is thus essential to obtain robust models in this discipline, with physical coherence and high accuracy, able to describe the different processes the Earth system.
In any information fusion process involving EO data, heterogeneous data coming from very different sources are considered. It is thus understood that we are able to extract descriptors (features, covariates) that describe them, following a specific problem-dependent procedure, so these features can be further processed by applying algorithms. Depending on the problem tackled, these data features can present different spatio-temporal resolution, high dimension, or any other characteristic that makes their direct processing using algorithms (ML in this case) difficult. The different processes and methods for data fusion in EO can be broadly classified depending on the level where the fusion is done: at a sub-feature (data transformations to harmonize sources), at the feature level (direct fusion of data sets (or their resulted features) into the ML algorithms), or at the decision level (different processing paths with fusion at the decision level).
In this paper, we review the current state of information fusion data-driven algorithms based on ML techniques, for problems in EO data analysis. The proposed review and perspective paper has a clear practical intention: we first describe the most relevant previous works in the field, structuring the description into different applications of EO. We also describe the most relevant and important data sets and sources for EO problems, i.e., what are the most important data sources currently available for EO studies, and how a researcher can obtain them. Finally, a carefully selected set of case studies will complete the work by illustrating in a practical way different aspects of ML information fusion for EO problems. We conclude the paper with a general outlook and discussion about the near future of this particular field.
Following the objective of this paper, we have structured this work into three large blocks after this introduction, and a final section of conclusions, discussion and outlook. First, we provide a complete and updated literature review of ML information fusion in EO, including previous reviews and overview works, a taxonomy of the field and a review of illustrative previous works related to ML for information fusion. The second block of the paper is devoted to describe the existing data sources useful for EO, including satellite data, in-situ observations and different models applied to EO problems. The third block of the paper presents different case studies focused on ML information fusion in real EO problems. The final part of the paper consists of a discussion, conclusion, and perspective section, where the current challenges on the field, recent trends and possible future research are outlined.
\section{Machine Learning information fusion in Earth observation: a comprehensive literature review}
Earth observation is a huge research area, involving extremely different problems, applications and cases in all the spheres of the Earth system. The challenge of summarizing all the work that have been done in the whole area is therefore unmanageable. We, however, focus on ML information fusion for EO, which alleviates somehow the difficulty, though it is still very hard to summarize all the work on this field, and there are different ways to tackle this task. We have decided to structure this section based on applications and problems faced with ML information fusion in EO problems. We have therefore based this discussion of previous works on existing applications and problems tackled with ML information fusion. First, just to show the huge previous work carried out in the whole EO area in the last years, we discuss some previous reviews and overview works in topics somehow related to EO, but covering more specific or partial aspects, not directly related to ML information fusion. We present afterwards a taxonomy of ML information fusion approaches, and finally we carry out then a complete description of previous works dealing with ML information fusion in EO problems, classified by different problems or application areas.
\subsection{Previous reviews and overviews in EO}
The importance of Earth modeling and the current interest in related applications have led to a huge amount of research work in the last years, some of them focused on information fusion techniques, in different areas of the topic. The interest is of a such magnitude that a considerably high number of reviews and overviews have been published, the large majority within the last 5 years. Table~\ref{table:summary} summarizes the body of literature in the intersection of ML, Earth sciences, and information fusion applications.
The first review article focused on information fusion for Earth observational data is \cite{Wald00}. This seminal review paper summarized the main concepts related to information fusion in a general, coarse grain, approach. Another early review on topics involving Earth sciences is the one in \cite{Cherkassky06}, mainly focused on a broad description of computational intelligence methods applied to this field, with special emphasis on the importance of data fusion. The subsequent review articles are much more specific, focused on detailed parts of ESS or on data source/methods. Many of these reviews deal with different aspects of remote sensing. Regarding this, \cite{Torabzadeh14} has presented a review of fusion spectroscopy images and laser systems for forest ecosystem characterization, while \cite{Chen15} has recently analyzed the state-of-the art of spatio-temporal fusion models for remote sensing, by means of a comparison among the most important existing ones. Also within the framework of remote sensing, \cite{Gomez15} has presented a review on multi-modal classification techniques for remote sensing images, and \cite{Schmitt16} presented a review specifically focused on data fusion techniques and algorithms for remote sensing.
Image fusion methods and algorithms have been fully described in several recent review papers too: \cite{Ghassemian16} presented a general review of image fusion for remote sensing, and \cite{Garzelli16} carried out a review on image fusion from the super-resolution paradigm. In \cite{Yokoya17,PG22}, the main methods and algorithms for hyperspectral and multispectral data fusion in remote sensing have been reviewed, whereas in \cite{Gham19} the basis, state-of-the-art and challenges of multisource and multitemporal data fusion algorithms are discussed.
Two recent review papers deal with Big Data methods in Earth sciences: \cite{Guo16} is focused on big data techniques from satellite data sources, whereas \cite{Gibert18} introduces environmental data science algorithms and methods, in a wide number of ESS applications. Finally, some reviews on deep learning with focus on Earth sciences have been recently published, such as \cite{Ball17,Ma19,PLi2019,Yuan20}, which describe the most recent deep learning approaches in remote sensing applications, or \cite{Zhang18} with a broader perspective in the big data area. Finally, it is worth mentioning the perspective paper \cite{Reichstein19} which, focused on describing the most important challenges and future directions in data-driven ESS, suggest that multisource information fusion and hybrid modeling will play a fundamental role in the near future.
\subsection{A taxonomy of ML information fusion approaches}
A variety of information fusion schemes have been proposed in the context of EO. Broadly speaking, information fusion is concerned on the multisource data combination and support decision making. Each fusion method is designed for a specific problem so it is challenging, if not impossible, to define a full taxonomical overview of the field. The main building blocks, however, have to do with the exploitation of i) disparate inputs, ii) data (pre)processing approach, iii) fusion mechanism, and iv) outputs post-processing. Actually, fusion approaches are usually named depending on the type of modalities, so a simple taxonomy of fusion problems can be defined in terms of {\em when, at what level, and how} the fusion is done. This is why we distinguish between the following types of fusion approaches:
\begin{enumerate}
\item {\em sub-feature level}, which usually involve different spatial-temporal scales fused following appropriate transforms of the data with the aim to harmonize sources into a common ``multidimensional grids'';
\item {\em feature level}, where a direct fusion of the data sets is simply stacked and fed into the ML of choice. This direct stacking can be more sophisticated if optimal feature combinations, and data source transformations, are learned from data to end up stacking feature representations;
\item {\em decision level}, where one performs different processing paths for each modality, followed by fusion at the decision level. This assumes that the outputs can be combined to improve the achieved accuracy. Different methods exist here that optimally operate on the combination of output activation functions.
\end{enumerate}
The best way to understand such differences is however to present different real examples in the literature, as follows.
\subsection{Literature review}
In this section we review the most important previous works on ML information fusion in EO problems. In terms of ML algorithms, the field mainly exploits either {\em classifiers} (of land use or land cover), or {\em anomaly, target and change detection algorithms} (for screening or identifying one class of interest and discard the rest), or {\em regression methods} (to estimate a particular variable of interest from either sensory data mounted on satellites, airborne or drones). We have structured this section in different subsections by application area or problem type, taken into account the most usual problems in which ML information fusion techniques have obtained significant results.
\subsubsection{Surface temperature}
The accurate estimation of surface temperature (inland and sea) from different sensors, both grounded and satellites, is extremely important in a number of problems, including agricultural studies, energy balance, land desertification and climate change applications, among others. In the literature, it is possible to find different studies applying data driven and ML techniques together with information fusion methods to estimate surface temperature. In \cite{Ortiz12} a feature level information fusion algorithm which hybridizes local atmospheric variables information from a ground station with synoptic information from numerical models is proposed for temperature prediction at Barcelona airport, Spain. An ensemble of Support Vector Regression (SVR) algorithms is used to carry out the information fusion and to obtain the temperature prediction. In \cite{Moosavi15}, a hybrid wavelet ML feature level fusion approach is proposed to obtain high-resolution land surface temperature, mixing Landsat 8 thermal bands and MODIS (moderate-resolution imaging spectroradiometer) pixels. Wavelets Support Vector Regression, adaptive network-based fuzzy inference system (ANFIS) and neural networks are the artificial intelligence methods tested in this problem. In \cite{Xia19}, a problem of high-spatiotemporal-resolution land surface temperature reconstruction is tackled, by applying a weighted combination kernel-based and fusion methods, in order to improve the spatial and temporal resolutions of satellite images. This method can be classified as a sub-feature level approach. Specifically, MODIS and Landsat 8 datasets have been considered for the experimental evaluation of the proposed method, obtaining more accurate images than the kernel method on its own. Recently, in \cite{Zhang20} a feature level information fusion process from different sources (measuring points) and a gated neural network was proposed to estimate the sea temperature at Bohai Sea, China.
\subsubsection{Droughts and water quality}
Closely related with surface temperature estimation, the analysis of drought and water quality using ML and data fusion techniques has been recently proposed. In \cite{Park17}, a high resolution soil moisture drought index was proposed. This index is based on measurements of the Advanced Microwave Scanning Radiometer on the Earth Observing System over the Korea, improved by MODIS and Tropical Rainfall Measuring Mission satellite sensors information, which was used to carry out a high-resolution feature level downscaling with a Random Forest (RF) algorithm to 1 Km measurements. In \cite{Yao17} a SVR algorithm was proposed to obtain an accurate estimation of Evapotranspiration by fusion of three process-based Evapotranspiration algorithms: MOD16, PT-JPL and SEMI-PM, to produce a feature level information fusion approach.
In \cite{Alizadeh18}, the study of drought events was carried out by applying new fusion approaches from high-resolution satellite and reanalysis data at feature level. The work in \cite{Feng19} is focused on determining whether the fusion of several remotely-sensed drought factors could be effectively used for monitoring drought events in Australia. This problem is tackled as a regression task with information fusion at feature level, where three ML approaches have been tested, RF, SVR and artificial neural networks.
Regarding ML fusion methods for water quality studies, in \cite{Dona15} a Genetic Programming approach is applied to fusion data from different satellite sources, such as MODIS or Landsat Thematic Mapper. The objective is to generate daily estimates of different water quality parameters such as chlorophyll-{\em a} concentrations or water transparency, for a freshwater lake (Albufera) in Valencia, Spain. Also, \cite{Jiang18} proposes a general framework based on computer vision feature representation, for the fusion of multisource spatiotemporal data for hydrological modeling (sub-feature level information fusion), following by the application of artificial neural networks and SVR algorithms. Finally, a review of fusion methods in water quality can be found in \cite{Sagan20}.
\subsubsection{Cloud detection and classification}
Cloud detection and classification is an EO area where ML information fusion has been successfully applied. In \cite{Liu18} a Convolutional Neural Network (CNN) has been applied to a problem of cloud classification using ground-based images. This proposal uses a two-stream structure which contains the vision subnetwork and multimodal subnetwork. In another layer of the network (fusion layer), the visual and multimodal features from the two subnetworks are extracted and then integrated using a weighted strategy. This is therefore a decision level method. The results have been tested in a specific database of multimodal ground-based clouds. In \cite{Li19} a cloud detection method based on CNN was proposed. The idea is to extract multi-scale and high-level spatial features by using a symmetric encoder-decoder module. The feature maps of multiple scales are then passed to a multi-scale feature fusion module, designed to fuse the features of different scales for the output in another decision level process. A final binary classifier is able to obtain the final cloud and cloud shadow mask. The method was validated on a large number of optical satellite images around the world, with different spatial resolutions ranging from 0.5 to 50 m.
\subsubsection{Land use applications}
Land use is another important area of interest in EO where information fusion has emerged lately. Recently, there have been many different works on land use tasks which combine different data sources or fusion algorithms with ML, in order to obtain robust approaches with optimal performance. For example, \cite{Wang16} proposes the fusion of multispectral HJ1B imagery (from China's HJ-CCD B satellite) and ALOS (Advanced Land Observing Satellite) PALSAR L-band (Phased Array type L-band Synthetic Aperture Radar) data for land cover classification, using sub-feature level information fusion, Support Vector Machine (SVM) and RF algorithms.
In \cite{Puttina17} an investigation focused on extraction of buildings from middle and high resolution satellite images is carried out, by fusing the information of different spectral indices with ML algorithms and sub-feature level information fusion techniques.
In \cite{Guy18} a deep CNN paradigm has been applied to a problem of automatic-land use classification from satellite images. Besides, in \cite{Lu19} a cellular automata -- Markov model is proposed in order to generate land use images from fusing images belonging to different years. This approach considers a sub-feature level information fusion mechanism, and has been successfully tested in reconstructing land use images in Hefei (China), from satellite data from the last 30 years. In \cite{Rasaei19} a feature level approach for fusing soil data coming from legacy soil surveys with direct soil information from remote sensing images was proposed. In \cite{P197}, several decision level and feature level fusion approaches were developed to tackle the problem of local climate zones classification based on a multitemporal and multimodal dataset, including image (Landsat 8 and Sentinel-2) and vector data (from OpenStreetMap) using ensemble classifiers and deep learning approaches. Finally, in \cite{Shaharum20} a feature level fusion of information from Google Earth and ML algorithms, including SVMs and regression trees, was proposed for a problem of palm oil mapping in Malaysia.
\subsubsection{Image classification and segmentation}
Image classification is another field closely related to very different remote sensing applications, where ML information fusion has been successfully applied. In \cite{Alajlan12} a decision level method which combines SVM and fuzzy C-means clustering for fusing hyperspectral images information is proposed. In this approach, the SVM is used to generate a spectral-based classification map, whereas the fuzzy C-means is used to provide an ensemble of clustering maps. In \cite{Ghamisi17} a feature level fusion of hyperspectral and light detection and ranging (LiDAR) images is proposed, under the hypothesis that LiDAR provide a source of complementary information, which can be really useful to improve classification of hyperspectral data. Specifically, the derived features from the two sources are fused via either feature stacking or graph-based feature fusion, and the fused features are fed to a deep CNN with logistic regression to produce the final classification map. The fusion of hyperspectral data and LiDAR is also treated in \cite{Khoda15}, and mixing hyperspectral data, LiDAR and other data sources with CNN in \cite{Xu18}. Finally, in \cite{Liang18} an unsupervised feature extraction method based on deep multiscale spectral-spatial feature fusion for hyperspectral images classification is proposed. The method is based on pre-trained filter banks and on a new unsupervised cooperative sparse autoencoder method to fuse together the deep spatial feature and the raw spectral information (sub-feature level fusion).
\subsubsection{Renewable Energy}
Renewable energy resources are fully related to ESS, since the main renewable sources (wind, solar, ocean) are fully conditioned by atmospheric or oceanic conditions. Due to their intermittent intrinsic nature, renewable energy sources present difficulties to be integrated in the energy mix, and usually need prediction techniques to this end. Interestingly, the main renewable energies are affected by climate change, which produces a redistribution of the renewable resources. In general, the study of renewable resource prediction is a hot topic in which the fusion of different sources of information has also been explored. In fact, information fusion in renewable energy has been mainly exploded in solar energy prediction systems. Solar energy resource prediction is fully connected to the prediction of clouds, which is a difficult problem from the meteorological point of view, in which information from different data sources is able to improve the prediction systems.
In \cite{Mellit08}, several neural network architectures with different input data sources were proposed to problems of solar radiation. More recently, in \cite{Salcedo14} a solar radiation prediction problem from different sources data was proposed. Data from in-situ measurement and from the GFS model were the inputs for a temporal Gaussian Process algorithm, which obtained better results than alternative ML algorithms in the problem. Another work dealing with data from different sources in a problem of solar radiation prediction is \cite{Mazorra16}, which proposed the prediction of intra-day solar radiation by means of data from irradiance measurements, satellite data and weather prediction models using artificial neural networks. Results in two different points of the Canary islands were reported. In \cite{Urraca20} an study on the evaluation of global horizontal irradiance from two different Reanalysis projects (ERA5 and COSMO-REA6) is carried out. This work uses ground and satellite-based data to estimate the accuracy of these reanalysis in obtaining horizontal irradiance. The results obtained show that Reanalysis data have important absolute error when estimating the irradiation, due to an inefficient cloud cover estimation. However, reanalysis products are an important data source for prediction and estimation problems in renewable energy, useful for information fusion with other sources, as in \cite{Babar20}, where data from ERA5 reanalysis are hybridized with satellite measurements at feature level, and all the information is processed with a RF algorithm in order to obtain an accurate estimation of solar irradiance at high latitudes.
\subsubsection{Model-data integration and assimilation}
Any optical remote sensing data present significant amounts of noise and missing data due to clouds, cloud shadows, and aerosol contamination, which difficult its use in any subsequent application. In addition, optical remote sensing sensors are hampered by a limited temporal, spectral or spatial resolutions \cite{thenkabail2018advanced}. As an example, medium spatial resolution sensors, such as Landsat or the Sentinel 2, have low temporal resolutions (16 and 8 day revisit cycle respectively) causing missing values continue to be one of the major limitations for their operational use, especially in areas with moderate to high cloud occurrence. The mitigation of undesired inherent data noise and minimizing the amount of missing data present are mandatory tasks in almost any application, since they are incompatible with a robust remote monitoring framework of earth's surface.
Because of the importance of this topic, the available scientific literature is rich in methods to deal with these issues, and solutions vary significantly with the different levels of sophistication \citep{kandasamy2013comparison}. Temporal, spatial, spatio-temporal, and blending (sensor fusion) approaches have been very valuable tools to reduce noise and recover missing pixels information, being data fusion methods very interesting approaches for overcoming individual sensor's limitations and combining different multiresolution datasets.
Most of data fusion approaches need to compute complex spatial operations to account for inhomogeneities within coarser spatial resolution pixels \citep{gao2006blending}. These operations are computationally demanding allowing only the application of these algorithms to small areas. In the context of modern data assimilation approaches, Sedano \textit{et al.} \citep{sedano2014Kalman} introduced a pixel-based method with a Kalman filter (KF) \citep{Kalman1960new} to fuse time series of MODIS and Landsat vegetation indices. This KF implementation obtained satisfactory results but also allowed to account for realistic uncertainties in its calculations. Moreover, the KF does not require explicit parameter tuning and it scales well in large scale applications due to its pixel-based nature.
\subsubsection{Unstructured domain data fusion}
The number of EO platforms for capturing remotely sensed data has been exponentially increased, ranging from an ever-growing number of satellites in orbit and planned for launch, to new platforms for capturing fine spatial resolution data such as unmanned aerial vehicles (UAVs). Moreover, a great interest has been recently dedicated to the new sources of ancillary data, to name a few, social media, crowd sourcing, scraping the internet and so on \cite{P7,P9}. These data have a very different modality to remote sensing data, but may be related to the subject of interest and, consequently, may be found useful with respect to specific problems. For example, social media data can provide local and live/real-time information suitable for accurate monitoring of our living environment \cite{P192}, in particular in a variety of applications relevant to smart cities \cite{P193,P194}, emergency and environmental hazards \cite{P195,P196}, among others.
\subsubsection{Other applications}
There are other applications of ML algorithms in EO which exploit data fusion to improve their results. One of the first works on exploiting data fusion from multi-sensor sources with ML was \cite{Fisher98}, where a problem of ionograms inversion is tackled using data fusion techniques with neural networks. More recently, there have been alternative specific applications, such as \cite{Carro17} where a problem of total ozone in column prediction is tackled with SVMs and information fusion from different data sources, such as numerical models, ground stations and satellite data. In \cite{Du19} a problem of eddy detection from different satellite sensors images was faced, using a deep learning approach for multi-scale feature fusion, followed by a SVM algorithm.
In \cite{Kattenborn15} the combination of data from multiple sensors was used to improve the estimation of forest biomass. Data from interferometric and photogrammetric based predictors, in combination with hyperspectral predictors were used, by applying ML algorithms such as RF, Generalized Additive Models or boosted algorithms. These approaches were applied to estimate biomass in a temperate forest near Karlsruhe, Germany. In \cite{Effrosy18} different ML classifiers were applied to a problem of detecting seagrass presence/absence and distinguishing seagrass families in the Mediterranean, from fusion of seagrass presence data and other external environmental variables. As a final application of ML information fusion in ESS, in \cite{Li18b}, a problem of environmental event sound recognition is tackled by means of a staked CNN.
\begin{table}[!ht]
\scriptsize
\vspace{12pt}
\caption{Summary of Information Fusion works that use Machine Learning in Earth observation applications. For each group of applications (first column) we have summarized the main points concerning the type / level of information fusion (second column) and the different ML techniques used (on the third one). The last column groups the corresponding references for any set of applications.}
\centering
\begin{tabular}{| p{2.5cm} | p{7.5cm} | p{4.0cm}| p{1cm}|}
\hline
\textbf{Approach and / or Application} & \textbf{Type of data fusion / Level of Information Fusion} & \textbf{Machine Learning algorithms} & \textbf{Ref.} \\
\hline
General reviews on ESS & Fundamentals of Information Fusion to properly process Earth data sets. \cite{Wald00}. Information Fusion at several levels applied to climate, weather, geophysical and hydrologic problems \cite{Cherkassky06}. & Neural, Fuzzy and Evolutionary Computation \cite{Cherkassky06} & \cite{Wald00,Cherkassky06} \\
\hline
Reviews on applications of remote sensing that focus on different data/level fusion. & Need for information fusion at different levels in deep learning approaches to remote sensing in EO applications \cite{Ball17,Ma19,Zhang18}. Fundamentals and challenges of multisource and multitemporal data fusion algorithms \cite{Gham19,Guo16}. Comparison of the four most relevant spatio-temporal fusion models \cite{Chen15}. Review on different types of multi-modal image fusion for classification at subpixel level, pixel level, feature level, and decision level \cite{Gomez15}. Review on data fusion approaches for remote sensing \cite{Schmitt16}. Discussion of super-resolution solutions for spatio-temporal fusion and pan-sharpening \cite{Garzelli16}. Review on spectral and spatial information fusion for hyperspectral data classification \cite{PG22}. Analysis of hyperspectral and multispectral data fusion at several levels \cite{Yokoya17,PLi2019}. Hyperspectral fusion of images at decision level \cite{Alajlan12,Ghamisi17,Khoda15,Xu18}. Multiscale spectral-spatial feature fusion for hyperspectral images at sub-feature level fusion \cite{Liang18}. & Neural Networks \cite{Gomez15,Ghassemian16,Ghamisi17}, Extreme Learning Machines \cite{Gham19}, Support Vector Machines \cite{Gomez15,Ghassemian16,Yokoya17,Gham19}, Deep learning \cite{Gomez15,Ghassemian16,Gham19,PLi2019,Yuan20,Ball17,Ma19,Zhang18,PG22}, Fuzzy C-means Clustering \cite{Alajlan12}, Convolutional Neural Networks \cite{Ghamisi17,Xu18,PG22}, Unsupervised cooperative sparse auto-encoder method \cite{Xu18}. & \cite{Ball17,Ma19,Zhang18,Gham19,Guo16,Chen15,Gomez15,Garzelli16,Yokoya17,PLi2019,Yuan20,Alajlan12,PG22,Ghamisi17,Xu18,Liang18} \\
\hline
Combining deep learning and process-based approaches for Earth System Science & Multi-source, high-dimensional, multi-scale, complex spatio-temporal, interrelated data \cite{Reichstein19}. & Discusses deep-learning challenges in ESS. Suggests that future models should integrate process-based and machine learning approaches & \cite{Reichstein19} \\
\hline
Surface Temperature & Information fusion is carried out at feature level \cite{Ortiz12,Moosavi15,Zhang20} and sub-feature level \cite{Xia19}. & Support Vector Regression \cite{Ortiz12}, ANFIS \cite{Moosavi15}, kernel methods \cite{Xia19}, Neural Networks \cite{Zhang20}. & \cite{Ortiz12,Moosavi15,Xia19,Zhang20}. \\
\hline
Droughts events & Information fusion is carried out at feature level \cite{Park17,Yao17,Alizadeh18,Feng19}. & Random Forest \cite{Park17}, Support Vector Regression, Neural Networks \cite{Yao17,Alizadeh18,Feng19}. & \cite{Park17,Yao17,Alizadeh18,Feng19}. \\
\hline
Water quality & Different fusion data from satellites \cite{Dona15}, sub-feature level information fusion \cite{Jiang18}. & Genetic Programming \cite{Dona15}, Neural Networks and Support Vector Regression \cite{Jiang18}, review of recent techniques \cite{Sagan20}. & \cite{Dona15,Jiang18,Sagan20} \\
\hline
Cloud detection and classification & Image fusion at several levels. & Convolutional Neural Networks \cite{Liu18,Li19} & \cite{Liu18,Li19}. \\
\hline
Land use & Multispectral satellite images, at sub-feature level information fusion \cite{Wang16,Puttina17,Guy18,Lu19}. Feature level approach for fusing soil images \cite{Rasaei19}. Feature level approach for palm oil mapping \cite{Shaharum20}. & Support Vector Machines \cite{Wang16,Shaharum20}, Random Forests algorithms \cite{Wang16}, Regression trees \cite{Shaharum20}, Deep Convolutional Neural Networks \cite{Guy18} in a cellular automata -- Markov \cite{Lu19}. & \cite{Wang16,Puttina17,Guy18,Lu19,Rasaei19,Shaharum20}. \\
\hline
Forest parameters estimation. & Combination of airborne laser scanning and imaging spectroscopy data at data level \cite{Torabzadeh14}. Fusion of seagrass presence data and other external environmental variables \cite{Effrosy18}. & Support Vector Machines \cite{Torabzadeh14}. Random Forests, Generalized Additive Models or boosted algorithms \cite{Kattenborn15}. & \cite{Torabzadeh14,Kattenborn15,Effrosy18}. \\
\hline
Local climate zones classification & Several decision level and feature level fusion approaches, based on a multitemporal and multimodal images. & Ensemble classifiers and deep learning approaches & \cite{P197}. \\
\hline
Solar radiation prediction in renewable energy & Irradiance measurements, reanalysis products and satellite data \cite{Mellit08,Salcedo14,Mazorra16,Babar20}. & Neural Networks \cite{Mellit08,Mazorra16}, Gaussian Process algorithm \cite{Salcedo14}, Random Forest \cite{Babar20}. & \cite{Mellit08,Salcedo14,Mazorra16,Babar20} \\
\hline
Model-data integration and assimilation & Pixel-based method to fuse time series of MODIS and Landsat vegetation indices. & Kalman filter (KF) & \cite{sedano2014Kalman} \\
\hline
Other applications & Ionograms inversion \cite{Fisher98}, Total ozone atmospheric content \cite{Carro17}, Eddies detection at sea \cite{Du19}, Forest Biomass estimation \cite{Kattenborn15}, Seagrass presence \cite{Effrosy18}, Environmental sound recognition \cite{Li18b} & Neural networks \cite{Fisher98}, Support Vector Machines \cite{Carro17,Du19}, Random Forest \cite{Kattenborn15}, Convolutional Neural Networks \cite{Li18b} & \cite{Fisher98,Carro17,Du19,Kattenborn15,Effrosy18,Li18b} \\
\hline
\end{tabular}
\label{table:summary}
\vspace{24pt}
\end{table}
\section{Data sources and models for Earth observation}\label{Data_sources}
The study of the ESS and its many components is based on observational data from very different sources, and very different atmospheric and climate models and simulations, among other data sources (such as social media or socio-economic data, sometimes fused with them). Every minute, millions of sensors collect data from the whole planet. Their processing, fusion with physic-based models, analysis and study is in the core of many works on EO-related problems. In this section, we describe the most important data sources and models currently available for studies on EO. We also provide the complete reference to the majority of these data sources, so the interested reader knows where they can be obtained. We have structured the section into subsections describing satellite, in-situ (ground-based, atmosphere and marine observations sources), forecasting models and finally reanalysis projects.
\subsection{Satellite observations}
Attending to the type of {\em energy sources} involved in the data acquisition, two main kinds of remote sensing imaging instruments can be distinguished: either {\em passive} optical remote sensing, which relies on solar radiation as the illumination source \cite{Danson95,Richards99,Ustin04,Liang04,lillesand08}, or {\em active} sensors, where the energy is emitted by an antenna towards the Earth's surface and the energy scattered back to the satellite is measured \cite{Mott07,Wang08}. Some examples of passive sensors are infrared, charge-coupled devices, radiometers, passive microwave, and multi and {hyperspectral} sensors \cite{shaw02}. On the other hand, in {Radar} systems, such as {Real Aperture RAR (RAR)} or {Synthetic Aperture Radar (SAR)}, are examples of systems for active remote sensing. Figure~\ref{fig:sat_comparison} shows some characteristics (spatial, spectral, and temporal resolutions) of the main available optical and microwave satellite sensors.
\begin{figure}[h!]
\centerline{\includegraphics[height=4.1cm]{./Figuras/msi_comparison} \hspace{0.5cm}
\includegraphics[height=4.1cm]{./Figuras/resolution}
\hspace{0.5cm}
\includegraphics[height=4.1cm]{./Figuras/LEAVE_satellites}}
\vspace{-0.25cm}
\caption{{\em Left:} Performance comparison of the main air- and space-borne multi- and
hyperspectral systems in terms of spectral and spatial resolution.
{\em Middle:} Evolution of the spatial-spectral resolution through the years.
{\em Right:} Spatial and temporal coverage for some optical and microwaves satellite sensors.
Credits: http://www.enmap.de/.
}
\label{fig:sat_comparison}
\end{figure}
\subsection{In-situ observations: ground, atmosphere, and ocean}
\subsubsection{Ground-based observations}\label{Ground-based}
Ground-based observations have been traditionally the most basic source of atmospheric data, especially before the satellite Era. Currently, ground-based observations and satellites are mainly assimilated by numerical and climatic models, so its importance is still very high in EO applications, mainly in meteorological and climatological studies.
\begin{itemize}
\item The European Climate Assessment \& Dataset project (ECA\&D) dataset \cite{Klein02,ECAD}. The ECA\&D database consists of daily station series of different meteorological/climatological variables: daily mean temperature, precipitation, sea level pressure or wind speed, among others. A gridded version with daily temperature, precipitation and pressure fields is also available for this database.
\item The Climatic Research Unit (CRU) temperature datasets \cite{CRU}. These databases, usually known as CRU and HadCRUT4, are global temperature datasets, providing gridded temperature anomalies across the world.
\item The Global precipitation data from the Global Precipitation and Climatology Center (GPCC) \cite{GPCC}. The GPCC provides gridded quality controlled station data. Another source of precipitation data is the database from the Global Precipitation Climatology Project (GPCP) \cite{GPCP}, which provides monthly precipitation dataset from 1979 to the present, combining observations and satellite precipitation data into $2.5^{\circ} \times 2.5^{\circ}$ global grids.
\item The Global Historical Climatology Network-Monthly (GHCN-M) database \cite{GHCN}. The CHCN-M provides gridded land precipitation and temperature anomalies on a $5^{\circ} \times 5^{\circ}$ basis for the entire globe, from 1900 to 2015. This database is useful for climate monitoring activities, including calculation of global land surface temperature anomalies and trends.
\item The Goddard Institute for Space Studies (GISS) database \cite{GISS}. The GISS surface temperature database provides a measure of the changing global surface temperatures, with monthly temporal resolution since 1880. The data is available on an $2.5^{\circ} \times 2.5^{\circ}$ and two smoothing levels of 250km and 1200km smoothing. A land only version is also available. The dataset is continuously updated, and it is necessary to take into account that there are missing data values within this database.
\end{itemize}
There are other gridded ground-based databases at NOAA Earth System Research Laboratory (ESRL), which can be explored and downloaded from \cite{ESRL}. When it comes to large data volumes, the U.S. Government's open data initiative \href{http://www.data.gov}{DATA.GOV://www.data.gov} collects and harmonizes all kind of social, economical, environmental, remotely-sensed datasets.
\subsubsection{Observations and simulations of land-atmosphere interactions}
Monitoring the land-atmosphere interactions is currently done thanks to a global network of continuous measurements. This is called the \href{https://fluxnet.fluxdata.org/}{FLUXNET} which is a global network of micro-meteorological tower sites that use eddy covariance methods to measure the exchanges of carbon dioxide, water vapor, and energy between the biosphere and atmosphere. FLUXNET is a global ``network of regional networks'' that serves to provide an infrastructure to compile, archive and distribute data for the scientific community. The large-scale measurement network, FLUXNET integrates site observations of these fluxes globally and provides detailed time series of carbon and energy fluxes across biomes and climates \cite{Baldocchi2008}. The data have been used to perform local or regional studies, but also to upscale globally the fluxes: move from point-based flux estimates to spatially explicit gridded fields of carbon and energy fluxes with machine learning and information fusion techniques~\cite{Tramontana16bg,Jung18fluxcom}.
An interesting initiative on land and atmosphere data harmonization is the Earth System Data Lab (ESDL) platform, \href{http://earthsystemdatalab.net/}{ESDL}, which curates a big database with more than 40 variables to monitor the processes occurring in our Planet. They are grouped in three data streams (land surface, atmospheric forcings but also socio-economic data~\cite{Mahecha19esdc}) and allow running algorithms in the web platform.
\subsubsection{Atmospheric observations}\label{atmos-based}
Atmospheric soundings data \cite{RUC,UWYO} are useful instruments to obtain an instant state of the atmosphere, i.e. a vertical profile of the atmosphere at a single point in time and above a particular position on Earth. Usually, a small instrument package called {\em radiosonde} is embedded into to a weather balloon, which is released from the surface and usually reaches the troposphere. The radiosonde is able to measure different properties of the atmosphere such as the vertical profile of temperature, dew point, wind speed and direction, among others, as it ascends. Atmospheric soundings data are very useful tools for EO, mainly in meteorological problems, specially in those related to local phenomena prediction, such as convection initialization or cloud formation, etc.
\subsubsection{Marine observations}\label{marine-based}
Marine-based data sources are also freely available in many cases, and contribute to the study of different parts of the ESS. One of the main sources of physical oceanographic/Meteorology data is the data base of the National Data Buoy Center of the USA (NDBC) \cite{NDBC}. This database contains freely available data from dozens of ocean buoys located at the Atlantic Ocean, Caribbean Sea, West Coast of the USA and Alaskan Gulf. There are other important data sources for marine observations, such as the Pan-European Infrastructure for Ocean \& Marine Data Management \cite{PEIOMDM} in Europe or Australian Ocean Data Network \cite{OpenAccAus} for the Asia-Pacific region.
Over the last decade, the development of cutting-edge robotic technology has dramatically demonstrated the potential of autonomous observations to overcome the issue of data scarcity. For example, the ground-breaking Argo international program has set up an array of more than 3500 profiling floats that provide measurements of temperature and salinity profiles from the surface down to 2000 m below sea level every 10 days. As the first-ever global in-situ ocean-observing network in the history of oceanography, \href{http://www.argo.ucsd.edu/}{Argo} provides a crucial complement to satellite systems, thus enabling observation, understanding and prediction of the ocean's functioning and its role in the Earth's climate.
\subsection{Numerical weather models}\label{Forecasting_models}
Weather prediction models, also known as numerical weather models, solve systems of differential equations (the Navier-Stokes equations, energy, mass and linear momentum conservation) to obtain the future state the atmosphere. Specifically, fluid motion, thermodynamics, radiative transfer, and atmospheric chemistry are taken into account, using a coordinate system which divides the whole planet into a grid. Thus, wind speed, heat transfer, solar radiation, relative humidity, among other variables are calculated within each grid node, and the interactions with neighboring nodes are then used to estimate the atmospheric evolution for the future.
A few global forecasting models are run in the world, using current weather observations relayed from radiosondes, weather satellites and other observing systems as inputs (data assimilation process). Processing the vast datasets and obtaining the solution of the system of differential equations previously mentioned in each node of the global grid require of powerful supercomputers. Even in this case, the forecast skill of current numerical weather models extends to only about six days, due to the non-linear and chaotic nature of the atmosphere. Some of the global forecasting models currently in operation are:
\begin{itemize}
\item The Global Forecast System (GFS) is produced by the National Centers for Environmental Prediction (NCEP) \cite{Kanamitsu89,Kanamitsu91}.
\item The Global Environmental Multiscale Model (GEM), often known as the CMC model in North America, is an integrated forecasting and data assimilation system developed in the Recherche en Pr\'evision Num\'erique (RPN), Meteorological Research Branch (MRB), and the Canadian Meteorological Centre (CMC) \cite{Cote98,Cote98b,Yeh02}.
\item The Navy Global Environmental Model (NAVGEM) is a global numerical weather prediction computer simulation run by the United States Navy's Fleet Numerical Meteorology and Oceanography Center \cite{Hogan14}.
\item The Integrated Forecasting System (IFS), is the global numerical model run by the European Center for Medium-range Weather Forecast (ECMWF) \cite{ECMWF}.
The Unified Model (UM) is a numerical model of the atmosphere developed by the Met. Office (UK) \cite{Walters17}. It can be used for both weather and climate applications.
The ARPEGE (Action de Recherche Petite Echelle Grande Echelle) model is the operational numerical weather model at M\'et\'eo France \cite{Deque94}. This system was developed and it is currently maintained in collaboration with the ECMWF.
\end{itemize}
There are some works which have evaluated and compared the performance of these and other weather numerical models in different EO application contexts \cite{Buizza05,Perez13}. Recently, the possibility of using ML algorithms as alternative techniques for global weather forecasting has also been discussed \cite{Dueben18}.
\subsubsection{Reanalysis projects}\label{sec:Reanalysis}
A {\em Reanalysis} project is a methodology that combines existing past observations, by applying data assimilation techniques, with modern numerical weather models. Reanalysis projects usually extend over several decades and cover the entire planet (global) or extended regions (regional), being a very useful tool for obtaining a comprehensive picture of the state of the Earth system, which can be used for meteorological and climatological studies. There are several reanalysis projects currently in operation, and some others which were the precursors of the current ones.
Two of the first reanalyses projects in operation were the NCEP/NCAR Reanalysis (Reanalysis-1) \cite{Kalnay96}, a global reanalysis of atmospheric data spanning 1948 to present and the NCEP/DOE Reanalysis (Reanalysis-2) project \cite{Kanamitsu02}, spanning 1979 to present. The Climate Forecast System Reanalysis (CFSR) \cite{Saha10} was an effort to generate an uniform, continuous, and best-estimate record of the state of the ocean-atmosphere interaction for use in climate monitoring and diagnostics. It is a global reanalysis, spanning data from January 1979 through March 2011. ERA-Interim \cite{ERA_Interim} is a global atmospheric reanalysis developed by the ECMWF. It covers from 1979, continuously updated in real time. The spatial resolution of the data set is approximately 15 km, on 60 vertical levels from the surface up to 0.1 hPa. ERA-Interim provides 6-hourly atmospheric fields on model levels, pressure levels, potential temperature and potential vorticity, and 3-hourly surface fields. ERA-5 \cite{ERA5} is the latest reanalysis project from the ECMWF. This reanalysis covers the Earth on a 30km grid and resolves the atmosphere using 137 levels from the surface up to a height of 80km. The Modern-Era Retrospective analysis for Research and Applications (MERRA) dataset was released in 2009 \cite{Rienecker11}. MERRA data span the period from 1979 through February 2016 and were produced on a $0.5^{\circ} \times 0.66^{\circ}$ grid with 72 layers. MERRA was used to drive stand-alone reanalyses of the land surface (MERRA-Land) and atmospheric aerosols (MERRAero). It is also worth mentioning the Japanese 25-year Reanalysis (JRA-25) \cite{JRA25} which is the first long-term global atmospheric reanalysis undertaken in Asia, and it covers the period 1979-2004.
There are also Regional reanalysis, such as the North American Regional Reanalysis (NARR)\cite{Mesinger06}. It is a regional reanalysis of North America containing temperatures, winds, moisture, soil data, and many other parameters. The high-resolution reanalysis system COSMO-REA6 has been developed based on the NWP model COSMO \cite{Bollmeyer15}. This is a regional reanalysis system for Continental Europe. This reanalysis data set currently covers the period 1995-2016 and it is currently in operation.
The European Reanalysis and Observations for Monitoring project (EURO4M) \cite{Euro4M} is a EU funded project that provides timely and reliable information about the state and evolution of the European climate.
Finally, note that there are several works focused on direct comparison of several reanalysis for the evaluation of different meteorological phenomena \cite{Hodges11,Bao12,Chaudhuri14}.
\section{Case studies on ML information fusion in Earth observation}
In this section we show empirical evidence of the performance of different ML fusion algorithms, working at different fusion levels in real practical EO problems. We discuss here four case studies: first, we present a problem of gap filling of several soil moisture time series from multiple microwave satellite data, working at different resolutions. In the second case study, we describe different algorithms that blend heterogeneous satellite sources in the optical range at different spatial and temporal scales in the Google Earth Engine platform. The third case study shows how natural hazards can be better predicted and modelled with ML data fusion. Finally, we introduce a case study where we discuss the fusion methodologies of EO and social media, stressing the importance and of unstructured data in EO, and the difficulty of their management.
\input{case_study_gapfilling.tex}
\input{case_study_assimilation.tex}
\input{case_study_natural_hazards.tex}
\input{case_study_socialmedia.tex}
\section{Conclusions, discussion and outlook}
In this article we have reviewed the state-of-the-art on Machine Learning (ML) information fusion for Earth observation (EO). The article, with a clear practical application, has been structured on literature review, data-sources and models description and some selected case studies where ML fusion information has obtained excellent performance in real problems. We have shown that the amount and diversity of the available data sources for EO has made information fusion a key step for successful real applications, with high societal, economical and environmental implications. We anticipate a huge impact in the upcoming years given the ever-growing increase, improvement, and ubiquity in sensory systems used in the EO field. The temporal, spatial and spectral resolutions are increasing dramatically, higher resolution models are now available, and social networks data promise to complement the view of the processes occurring in all spheres of the Earth system. We have also shown that ML approaches have the ability to blend and extract knowledge from these data, obtaining excellent results in a large number of EO problems and applications.
There is an important challenge that the EO community has to face: the scalability of algorithms in the big data era. Many interesting approaches exist nowadays (high performance computing platforms, more efficient and sparse algorithms, green AI methodologies, etc.), and we expect much more advances on this in the near future. Besides this more technical problem, we can identify two important stepping stones: how to extract information in highly unstructured data, and how to achieve understanding through information fusion. On the one hand, the platforms and sources of information may vary considerably in multiple dimensions. For example, the types of properties sensed and the spatial and spectral resolutions of the data, and sometimes the temporal resolution of the corresponding sensor. This also applies even to the sensors that are mounted on the same platform, or that are part of the same satellite configuration. The rapid increase in the number and availability of data with an enormous amount of heterogeneity, and non-stationary properties, creates serious challenges for their effective and efficient processing. For a particular problem at hand, there may exist multiple remote sensing and ancillary datasets which leads to a dilemma: how best to integrate multiple datasets, probably unstructured and non-stationary, for maximum utility? It is currently one of the main challenges in EO. On the other hand, and perhaps more important, we have the issue of problem understanding through information fusion. Blending the information provided by different sources and ancillary systems not only can be an efficient way of improving the performance of ML algorithms in specific problems, but also a way of improving the understanding of the systems, usually as part of larger structures, and providing this way a better physical interpretation of the results obtained. The challenge is here to combine adequately sources of information which can provide very different perspectives to the problem, or even different possible interpretations depending on the case, and which improve the performance of the ML algorithms at the same time.
To tackle the previous challenges, we identify several exciting venues that open in the immediate future of ML information fusion in Earth sciences and EO in particular. A promising approach is about designing ML models that incorporate {\em domain knowledge} internally in a more sophisticated approach to data assimilation. Two methodological approaches have potential in facilitating the fusion of data-driven and arbitrary data sources and also physical models: probabilistic programming and differentiable programming. Probabilistic programming allows for accounting of various uncertainty aspects in a formal but flexible way, which allows in principle to account for data and model uncertainties, inclusion of priors and constraints coming from ancillary data and/or process-based (theory-driven) modeling. Differentiable programming, on the other hand, allows for efficient optimization of arbitrary losses owing to automated differentiation, which might help make the large, nonlinear and complex modeling more tractable. Attaining hybrid physics-aware ML will allow us advancing in improved modeling in terms of consistency and interpretability.
Understanding, especially from heterogeneous data, is harder than predicting, and ML algorithms are currently only mere (yet powerful) interpolation techniques: they excel in fitting arbitrary functional data relations but do not have a clear notion of the underlying causal relations. Despite the great predictive capabilities of current methods, there is still little actual {\em learning} in the ML information fusion pipeline. In this context, {\em causal inference} methods aim at discovering and explaining the causal structure of the underlying system. When interventions in the system are not possible, {\em observational causal inference} comes into play. Observation-based causal discovery aims at extracting potential causal relationships from multivariate datasets, and goes beyond the commonly adopted correlation approach, which merely obtains associations between variables. Today the science of ``causal inference'' is sufficiently advanced to unravel relations between multiple coupled variables beyond correlations even in the presence of non-linearities and non-stationarities. Observational causal inference could make a decisive change in the way we process, analyze and understand multi-source and multi-sensory data.
\section*{Acknowledgments}
This research has been partially supported by the Ministerio de Econom\'{i}a y Competitividad of Spain (Grant Ref. TIN2017-85887-C2-2-P). GCV work is supported by the European Research Council (ERC) under the ERC-CoG-2014 SEDAL Consolidator grant (grant agreement 647423).
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\section*{Introduction}
Despite extensive implications for diverse fluid dynamical systems and more than a century of research, the transition from pressure-driven laminar flow to inertial turbulence in even the simplest geometries remains one of the major unresolved problems in fluid mechanics. Elaborate experiments have recently shed new light on the nature of this transition by measuring the decay and splitting of local turbulent domains/clusters (puffs) in pipe flows and have determined the critical Reynolds number -- ratio of inertial to viscous forces -- at which the transition occurs \cite{Hof2011,Hof2015}. Short-range interactions between the locally turbulent puffs, which feed on surrounding laminar flow as an absorbing state, drive a continuous transition to a fully turbulent flow. Recent experimental evidence from channel and circular Couette flows \cite{Sano2015,Hof2016} together with direct numerical simulation studies and predator-prey models \cite{Shih2015}, have provided evidence that the transition at the critical Reynolds number is characterised by the directed percolation universality class.
Strikingly, here we show that for a profoundly distinct class of turbulence at {\em zero}-Reynolds number, the transition in a channel can also be characterised by the emergence of spontaneous puffs created by microscopic activity of biological fluids. Even for this zero-Reynolds number class of turbulent-like flows, we find that the critical exponents belong to the directed percolation universality class.
Zero-Reynolds number turbulence is established through continuous energy injection from the constituent elements of an active fluid in many biological systems, including bacterial suspensions \cite{Dombrowski2004,Julia2012,Aranson2012,Aranson2014,Sano2016bacteria,Wioland2016}, cellular monolayers \cite{Benoit2012,ourSM2015,Sano2016cells}, or sub-cellular filament/motor protein mixtures \cite{Dogic2012,Francesc2016}.
Although the inertia is negligible (Reynolds number $\sim 10^{-6}$) in such systems, active turbulence is characterised by a highly disordered distribution of vortices \cite{Frey2015,Giomi2015}.
However, meso-scale turbulence in living fluids possesses a characteristic vortex length scale, which distinguishes it from scale-invariant inertial turbulence \cite{Heidenreich2015}, and it is considered a new class of turbulent flow \cite{Julia2012,Jorn2013,Frey2015}.
To study the transition to meso-scale turbulence, we computationally solve the continuum equations of active nematics in micro-channels, which have successfully reproduced the patterns of bacterial ordering in bulk \cite{Sano2016bacteria} and in confinement \cite{Volfson2008}, the flow structure and correlation lengths of microtuble bundles \cite{Dogic2012,ourprl2013,Francesc2016} and the flow patterns of dividing cells \cite{ourSM2015} (see Supplementary Information for the details of the model).
In this zero-Reynolds number regime, the transition to turbulence occurs by increasing the amount of local energy injection (activity) in the living fluids. In a confined environment, the activity leads to spontaneous symmetry breaking and the generation of unidirectional flow \cite{Joanny2005}, which is followed by an oscillatory regime characterised by distorted streamlines \cite{Giomi2012,Julicher2016}, upon increasing the activity. Further increase in the activity leads to the emergence of a stable lattice of velocity vortices throughout the channel [\fig{fig:flows}(a)], and this transitions to meso-scale turbulence at higher activities [\fig{fig:flows}(b)]. The emergence of the intermediate vortex-lattice in active matter has been observed experimentally in motility assays of microtubles \cite{Chate2012}, in bacterial suspension in a channel confinement \cite{Wioland2016}, and also numerically by hydrodynamic screening of activity-induced flows due to frictional damping \cite{ourNComm2016}. In stark contrast to inertial turbulence, the Reynolds number is irrelevant here and the transition between flow regimes is governed by the dimensionless {\it activity number} $\mathrm{A} = \sqrt{\zeta h^2/K}$ [\fig{fig:contP}(a)]. This parameter characterises the ratio of the channel height $h$, which here is equivalent to the hydrodynamic screening length, to the characteristic activity-induced length scale $\ell_{a}=\sqrt{K/\zeta}$, that represents the relative importance of the intrinsic activity $\zeta$ and the orientational elasticity $K$ of the nematic fluid.
The marked difference between the various flow states is clearly seen in the structure of the vorticity. Therefore, in order to characterise the transition between the regimes, we measure the distribution of the local enstrophy $\epsilon$, averaged across the channel. This quantity represents the strength of vortices in the flow, and has also been used for determining the nature of inertial turbulence. The vortex-lattice state possesses a well defined peak in the enstrophy [\fig{fig:contP}(b)]. As the active flow transitions at higher activities, the enstrophy distribution broadens demonstrating that vorticity cascades down into meso-scale turbulence.
The gradual disappearance of the peak in the enstrophy distribution [\fig{fig:contP}(b)] suggests a continuous transition from the vortex-lattice to meso-scale turbulence. But how does the active turbulence develop from the vortex-lattice?
Figure \ref{fig:flows}(c) shows a snapshot of the vorticity field in a long channel close to the transition. The vortex-lattice predominantly occupies the entire channel. Locally, however, we can identify regions of the channel where vortex pairs split into smaller non-ordered vortices [\fig{fig:flows}(d)]. This coexistence of the global vortex-lattice and clusters of local active turbulence controls the transition to turbulence in the channel. We term these localized domains of non-ordered vorticity {\it active puffs}, in analogy to the inertial puffs observed in the experiments on scale-invariant turbulence in long tubes \cite{Hof2011}. Unlike the inertial puffs that are externally initiated by perturbations to the flow field (such as induced pressure jumps), active puffs arise spontaneously due to the innate active forcing of the flow.
The emergence and dynamics of active puffs is clearly characterised in the space-time kymograph of enstrophy [\fig{fig:puff}]. At initial times, the entire channel is in the absorbing vortex-lattice state. However, depending on the activity number, active puffs spontaneously spread through the channel length [\fig{fig:puff},~Fig.~S.1]. A puff can split, giving birth to new puffs, while for every given moment there is a finite chance that an active puff decays back to the absorbing state [\fig{fig:puff}(a)] or that a new puff is spontaneously born. Above some critical activity number, this competition between decaying to the absorbing state and splitting produces a statistical steady-state in which active puffs coexist with the vortex-lattice. The coexistence results in a well-defined turbulence fraction within the channel [\fig{fig:puff}(b)]. At the highest activities, the active flow approaches the fully turbulent state, with active puffs ultimately occupying the entire channel [\fig{fig:puff}(c)].
We thus measure the turbulence fraction, the area fraction occupied by active puffs in the channel, as a function of the activity number [\fig{fig:turbfraction}(a)]. Well below the critical point, active puffs have a short lifetime and rarely split [\fig{fig:puff}(a)], leading to a negligible turbulence fraction in the steady-state [\fig{fig:turbfraction}(a)]. However, as the critical value of the activity is approached, puff decay becomes less likely and splitting time decreases substantially [\fig{fig:puff}(b)]. Above the critical point, the puff population does not die out, producing a steady-state, non-zero turbulence fraction [\fig{fig:puff}(c)], and we find the turbulence fraction continuously increases with a power-law dependence $\sim(A-A_{cr})^\beta$ [\fig{fig:turbfraction}(a)]. We measure the exponent to be $\beta=0.275 \pm 0.043$, which closely matches the universal exponent of the $(1+1)$ directed percolation process ($\beta=0.276$) and is in agreement
with the value that has recently been measured for inertial turbulence in Couette flow ($\beta=0.28 \pm 0.03$) \cite{Hof2016}. This is striking as it draws a parallel between the zero-Reynolds number meso-scale turbulence in living fluids, which possesses a characteristic vorticity length scale, and high Reynolds number inertial turbulence, which is scale-invariant.
Since the internal activity can spontaneously create active puffs from the absorbing vortex-lattice state with a small probability, the transition corresponds to directed percolation with spontaneous site activation, as in a weak external field \cite{Hinrichsen2000}, while the transition to inertial turbulence maps to the zero field limit (see Supplementary Information).
To scrutinize the critical behaviour at the transition point, we further measure the spatial and temporal distributions of vortex-lattice gaps in the absorbing state (see Supplementary Information).
These distributions of the absorbing state characterise correlations of the active puffs\cite{Julia1981} and obey power laws with exponents $\mu_{\perp},~\mu_{||}$ for space and time, respectively [\fig{fig:turbfraction}(b),~(c)]. The temporal exponent is measured to be $\mu_{||}=1.84\pm0.04$ and the spatial exponent is $\mu_{\perp}=1.8 \pm 0.1$. These values also correspond to the exponents for $(1+1)$ directed percolation ($\mu_{||}=1.84,~\mu_{\perp}=1.748$)\cite{Hinrichsen2000}. The values of the critical exponents obtained from our measurements for meso-scale turbulence in a channel and for $(1+1)$ directed percolation with spontaneous site activation are summarised in Table~\ref{tab} and are compared with the experimentally measured exponents for the inertial turbulence in simple shear experiments in one dimensional geometries \cite{Hof2016}. It would not be unexpected that directed percolation universality class will continue to hold in higher dimensions as in experiments on inertial turbulence in passive liquid crystals \cite{Sano2007,Sano2009} and in channel flows \cite{Sano2015}.
Our findings present a first concrete connection between turbulence in living fluids and classical scale-invariant turbulence, beyond a superficial visual similarity, by showing that the transitions to these two profoundly distinct types of spatio-temporal disorder in channel flows belong to the same universality class, namely that the critical behaviour is represented by a directed percolation process. This opens new possibilities for further investigation of the nature of meso-scale turbulence and using tools from non-equilibrium statistical mechanics to explain critical behaviours in biological systems. Future research should investigate the transitions between ordered flow states and applicability of the directed percolation universality class in higher dimensions and in complex biological fluids.
\newpage
\section*{Methods}
\subsection*{Active nematohydrodynamics simulations}
The spatiotemporal evolution of a living fluid is described by active nematohydrodynamics equations based on the theory of liquid crystals. This formulation has been extensively applied to biological systems including bacterial suspensions \cite{Volfson2008}, microtuble/motor protein mixtures \cite{Dogic2012,Giomi2013,ourprl2013} and cellular monolayers \cite{Julicher2008,ourSM2015}. The total density $\rho$ and the velocity field ${\bf u}$ of the active matter obey the incompressible Navier-Stokes equations
\begin{align}
\vec{\nabla}\cdot\vec{u} &=0,\label{eqn:cont}\\
\rho\left(\partial_t + \vec{u}\cdot\vec{\nabla}\right) \tens{u} &= \vec{\nabla}\cdot\tens{\Pi},
\label{eqn:NS}
\end{align}
where $\tens{\Pi}$ is the stress tensor. While several studies of meso-scale turbulence have characterised the dynamics of the flow using only the velocity field as the relevant order parameter \cite{Julia2012,Jorn2013}, an additional order parameter field is required to account for the orientational order of active fluids. This is particularly important since several experiments have now established the existence and pivotal role of the orientational order in the dynamics of bacterial suspensions \cite{Volfson2008,Sano2016bacteria}, microtuble bundles \cite{Dogic2012,Francesc2016}, assemblies of fibroblast cells \cite{Silberzan2014}, and more recently in stem cell cultures \cite{Sano2016cells}. To account for the macroscopic orientational order of microscopic active and anisotropic particles, the nematic tensor $\tens{Q}=\frac{3q}{2}( \vec{n}\vec{n}-\tens{I}/3)$ is considered, where $q$ denotes the coarse-grained magnitude of the orientational order, $\vec{n}$ is the director, and $\tens{I}$ the identity tensor. The nematic tensor evolves as
\begin{align}
\left(\partial_t + \vec{u}\cdot\vec{\nabla}\right) \tens{Q} - \tens{S} &= \Gamma \tens{H},
\label{eqn:lc}
\end{align}
where $\Gamma$ is a rotational diffusivity and the co-rotation term
\begin{align}
\tens{S} = &\left(\lambda \tens{E} + \tens{\Omega}\right)\cdot\left(\tens{Q} + \frac{\tens{I}}{3}\right) + \left(\tens{Q} + \frac{\tens{I}}{3}\right) \cdot \left(\lambda \tens{E} - \tens{\Omega}\right) \nonumber\\
&\quad - 2 \lambda \left(\tens{Q} + \frac{\tens{I}}{3}\right)\left( \tens{Q} : \vec{\nabla} \vec{u}\right),
\label{eqn:cor}
\end{align}
accounts for the response of the orientation field to the extensional and rotational components of the velocity gradients, as characterised by the strain rate $\tens{E}=(\vec{\nabla}\vec{u}^{T}+\vec{\nabla}\vec{u})/2$ and vorticity $\tens{\Omega}=(\vec{\nabla}\vec{u}^{T}-\vec{\nabla}\vec{u})/2$ tensors, and weighted by the tumbling parameter $\lambda$. The relaxation of the orientational order is determined by the molecular field,
\begin{align}
\tens{H} &= -\frac{\partial \mathcal{F}}{\partial \tens{Q}} + \vec{\nabla}\cdot\frac{\partial \mathcal{F}}{\partial( \vec{\nabla}\tens{Q})}, \label{eqn:molpot}
\end{align}
where $\mathcal{F}=\mathcal{F}_{b}+\mathcal{F}_{el}$ denotes the free energy.
We use the Landau-de Gennes bulk free energy \cite{DeGennesBook},
\begin{align}
\mathcal{F}_{b} = \frac{A}{2}\tens{Q}^2 + \frac{B}{3}\tens{Q}^3 + \frac{C}{4}\tens{Q}^4,
\end{align}
and $\mathcal{F}_{el} = \frac{K}{2} (\vec{\nabla}\tens{Q})^2$, which describes the cost of spatial inhomogeneities in the order parameter, assuming a single elastic constant $K$.
In addition to the viscous stress $\tens{\Pi}^\textmd{visc} = 2 \eta \tens{E}$, \eq{eqn:NS} must account for contributions to the stress $\tens{\Pi}$ from the nematic elasticity and the activity. The nematic contribution to the stress is
\begin{align}
\tens{\Pi}^{\text{elastic}}=&-P\tens{I} +2 \lambda(\tens{Q} + \tens{I}/3) (\tens{Q}:\tens{H})\nonumber\\
&\quad-\lambda \tens{H}\cdot(\tens{Q} + \frac{\tens{I}}{3}) - \lambda (\tens{Q} + \frac{\tens{I}}{3})\cdot \tens{H}\nonumber\\
&\qquad-\tens{\nabla}\tens{Q}:\frac{\partial\mathcal{F}}{\partial(\tens{\nabla}\tens{Q})} + \tens{Q}\cdot\tens{H} - \tens{H}\cdot\tens{Q},
\end{align}
which includes the pressure $P$ \cite{Berisbook}. The active contribution to the stress takes the form $\tens{\Pi}^\textmd{act} = -\zeta \tens{Q}$ \cite{Sriram2002}, such that any gradient in $\mathbf{Q}$ generates a flow field, with strength determined by the activity coefficient, $\zeta$.
The equations of active nematohydrodynamics (\eq{eqn:cont}-\ref{eqn:lc}) are solved using a hybrid lattice Boltzmann and finite difference method \cite{Davide2007,Suzanne2011,ourpta2014}. Discrete space and time steps are chosen as unity and all quantities can be converted to physical units in a material dependent manner \cite{Cates2008, Henrich2010, ourprl2013}.
Simulations are performed with the parameters $A=0$, $B=0.3$, $C=-0.3$, $\Gamma=0.34$, $K=0.04$, $\lambda=0.3$, $\rho=1$, and $\mu=2/3$, in lattice Boltzmann units. These specific parameters are chosen to match those previously used to quantitatively probe velocities fields of the active nematohydrodynamics of kinesin/microtuble bundles in experiments {\cite{Dogic2012}}.
We use a channel with a height $h=25$ and length $L=3000$. No-slip boundary conditions are applied to channel walls and periodic boundary conditions are used at the channel extremities. The results reported here are for homogeneous boundary conditions for the director field on the channel walls. In addition, we have performed simulations with homeotropic and weak anchoring boundary conditions and find that the transitions described in the main text are independent of the anchoring boundary conditions on the walls.
\subsection*{Directed percolation model with spontaneous activation}
To examine the behaviour of the (1+1) directed percolation universality class, we utilized the Domany-Kinzel cellular automaton \cite{DomanyKinzel1984} and chose probabilities to correspond to bond directed percolation in the presence of a weak external conjugated field.
We define a diagonal square lattice with empty sites corresponding to the absorbing phase (the vortex lattice state in confined active flows) and occupied sites corresponding to the activated phase (active puffs of meso-scale turbulence).
At time $t$ each site is occupied with some probability $p_2$ if both backward sites (at time $t-1$) are occupied, with probability $p_1$ if only one backward site is occupied, and with probability $p_0$ for spontaneous site activation if neither backward site is occupied.
Bond directed percolation in the presence of a weak external $h$ is recovered with the choice $p_2 = p_1 \left(2 - p_1\right)$ \cite{Lubeck2006} and $p_0=h\neq0$ \cite{Lubeck2002}.
In the confined active nematic, we find that $p_0$ is small but has a non-zero value since spontaneous puff creation is observed.
Our directed percolation simulations employ periodic boundary conditions and a lattice size of $10^4$ sites in the spatial dimension to coincide with the lattice Boltzmann system. Data is obtained from $10^3$ runs of $5\times10^3$ time steps each.
We consider $p_0=\left\{ 0, 10^{-9}, 10^{-8}, 10^{-7}, 10^{-6} \right\}$ and find the critical probability $p=0.64470\pm 0.00002$ as expected (0.6447001(1) \cite{Jensen1996,Hinrichsen2000}).
Comparing directed percolation with spontaneous activation simulations to the kymographs in the main text suggests that active puff creation is unlikely, and so we use $p_0 = 10^{-7}$ as a reasonable estimate.
Measuring $N_\perp$ and $N_\parallel$, as for the lattice Boltzmann simulations, supplies the critical exponents reported in the main text.
\subsection*{Calculating the turbulence fraction}
To obtain the active puffs, the enstrophy field $\varepsilon(x,y,t)=\vec{\Omega}\cdot\vec{\Omega}$ is calculated from the vorticity field $\vec{\Omega}(x,y,t)$. The enstrophy field is averaged across the channel $\epsilon(x,t)=\left<\varepsilon(x,y,t)\right>_y$ to obtain space-time diagrams (kymographs) of the enstrophy. As described in the main text, the channel-averaged enstrophy in the vortex-lattice phase shows regular periodic oscillations, while local active turbulence domains (the active puffs) exhibit fluctuating, noisy enstrophy signals (Fig. S1(a)).
To relate the height-averaged enstrophy signal to the onset of meso-scale turbulence, we perform image processing on the kymographs. Each kymograph is Fourier transformed in both time and space.
The primary peaks are masked in reciprocal space-time to produce the kymographs without the structured oscillations of the periodic, background of the vortex lattice (Fig. S1(b)).
To quantify the turbulence fraction within the channel, the existence and extent of turbulent active puffs must be automatically measured.
Local active puffs are detected from the unmasked kymographs by dividing the height-averaged enstrophy signal into small time intervals going from $t_i$ to $t_{i+n}$. For every binned time interval, the discretized temporal autocorrelation function $c_k(x)=\frac{1}{n-1}\sum_{j=1}^{n-k}(\epsilon_{j}(x)-\bar{\epsilon})(\epsilon_{j+k}(x)-\bar{\epsilon})$ is calculated, where $k$ runs over time intervals from $0$ to $n$, $\epsilon$ is the averaged enstrophy over the discretised signal sample, and $\bar{\epsilon}$ is the enstrophy signal averaged over all time and space.
For every $x$ point and for every binned time interval, we determine if the autocorrelation function periodic or non-periodic, indicating that the point belongs to the absorbing vortex lattice state, or an activated turbulent region, respectively. By averaging these intervals, the turbulence fraction is obtained.
The spatial interval distribution $N_\perp$ of the absorbing state (the vortex-lattice) is measured from the processed kymographs by recording the length intervals between active puff regions for fixed temporal coordinates. Similarly, the time interval distribution $N_\parallel$ between puffs is found by recording the temporal duration of the absorbent state regions for fixed spatial coordinates. At short spatial intervals $L$, $N_\perp$ exhibits oscillations, which represent the characteristic size of the repeating vortex-lattice state.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,137
|
require "rubygems"
ENV["RAILS_ENV"] ||= "test"
require "rails/application"
require File.expand_path("../../config/environment", __FILE__)
require "rspec/rails"
require "rspec/autorun"
require "capybara/rspec"
require "sidekiq/testing"
include Warden::Test::Helpers
Dir[Rails.root.join("spec/support/**/*.rb")].each { |f| require f }
ActiveRecord::Migration.check_pending! if defined?(ActiveRecord::Migration)
Capybara.default_wait_time = 30
RSpec.configure do |config|
config.fixture_path = "#{::Rails.root}/spec/fixtures"
config.infer_base_class_for_anonymous_controllers = false
config.order = "random"
config.include Devise::TestHelpers, type: :controller
config.include Warden::Test::Helpers, type: :feature
config.include FactoryGirl::Syntax::Methods
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 500
|
Q: IIS 8.5 FTP using IIS Users Can someone please provide me with an up-to-date step-by-step guide on how to add ftp publishing securely using SSL to a website in IIS 8.5 web server using IIS Users.
I then want to connect to the website using FileZilla.
*
*I have correctly installed ftp using windows firewall. All rules are enabled.
*I have installed IIS Service Management and have created an IIS user (IISuser with a password).
*I have tried selecting my site and creating an ftp publishing but every time I try to connect to the server with the IIS User through FileZilla after authenticating the SSL by trusting it in FileZilla I get a 503 invalid host name.
I want to create an IIS user to access the ftp, instead of a user on the server because I want to ensure maximum amount of security - if I create another user with limited access, they can still remote into the server, rather than just an IIS user being able to access the web server (IIS).
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 410
|
\section{Introduction}
One of the most intriguing discoveries at the Relativistic Heavy Ion Collider (RHIC) is the strongly coupled quark gluon plasma (QGP)\cite{Gyulassy:2004zy} created in the heavy ion collisions. There have been great experimental efforts on the quantitative study of various properties of QGP in terms of both energy loss and transverse momentum $P_T$ broadening effects\cite{Gyulassy:1993hr, Baier:1996kr, Baier:1996sk, Baier:1998kq, Zakharov:1996fv}. For example, as a clear indication of a jet quenching effect due to large energy loss, a large suppression of the single hadron spectra in the high $P_T$ region in central $AuAu$ collisions has been observed\cite{Adams:2003kv, Adams:2003im, Adler:2003qi}. In addition, RHIC\cite{Adler:2002tq} has also observed that the back-to-back hadron correlations for moderate $P_T$ disappear for central $AuAu$ collisions. Although one can attribute this effect to both energy loss and $P_T$ broadening effects, it is believed that the normalized angular correlation around $\Delta \phi \sim \pi$ is mostly due to medium transverse momentum broadening with $\Delta \phi$ being the azimuthal angle difference between the trigger hadron and the associate hadron.
In fact, it was shown in the Baier-Dokshitzer-Mueller-Peigne-Schiff (BDMPS) approach\cite{Baier:1996kr, Baier:1996sk, Baier:1998kq} that the energy loss and $P_T$ broadening effects are related through the following formula $-\frac{dE}{dx}\simeq \frac{\alpha_s N_c}{4} \hat{q} L $, where $\hat{q} L$ represents the typical transverse momentum squared that a parton acquires in the medium of length $L$ . Here, $\hat q$ is the so-called jet-quenching parameter which depends on the density of the QGP medium. Therefore, one would expect that the energy loss effect should be tied together with the transverse momentum broadening effects in heavy ion experiments.
Since the commencement of the LHC, similar suppression of single hadron spectra\cite{CMS:2012aa, Abelev:2012hxa} and inclusive jets \cite{Aad:2012vca} has also been found in $PbPb$ collisions, which implies that similar jet quenching effects persist in the LHC regime. In the meantime, approximately a factor of two suppression of the back-to-back dihadron correlation with $ 8 \, \textrm{GeV}<P_{T, trig} <15 \, \textrm{GeV}$\cite{Aamodt:2011vg} in central heavy ion collisions also suggests the presence of significant medium effects.
Nevertheless, the dijet measurements conducted by CMS and ATLAS at the LHC\cite{Chatrchyan:2011sx, Aad:2010bu} seem to be a bit puzzling at first sight. On one hand, they observed striking dijet asymmetries in central $PbPb$ collisions which is consistent with the jet quenching effect\cite{Qin:2010mn}. Since the dijet asymmetry strongly depends on the transverse energy difference of the dijet system, this observable is not as sensitive to the $P_T$ broadening of jets as the angular correlation. On the other hand, there is no trace of significant angular decorrelation found in the same dijet measurement. As a matter of fact, the normliazed angular distribution in central $PbPb$ collisions is almost the same as the one measured in $pp$ collisions for $\Delta \phi > 2$.
From the theoretical point of view, there are mainly two competing contributions to the correlation (decorrelation) of the dijet angular distribution in high energy heavy ion collisions, namely, the Sudakov effect and the medium induced $P_T$ broadening (For the normalized angular distribution as shown in Ref.~\cite{Chatrchyan:2011sx}, one expects that the energy loss effect is not very important.). The Sudakov effect, also known as the parton shower, has been an important topic of QCD studies for several decades. It normally occurs due to large amounts of gluon radiation in hard processes, such as high invariant mass Drell-Yan lepton pair production process as well as the $W$ and $Z$ boson production\cite{Collins:1984kg}. Especially, recent studies\cite{Banfi:2008qs, Mueller:2012uf, Mueller:2013wwa, Sun:2014gfa} in several areas of QCD have shown that it is important to perform the Sudakov resummation in order to obtain a consistent description of back-to-back dijet angular correlations in hard processes. It is also important to mention that the Sudakov factor arises from the incomplete cancellation of real and virtual graphs in high order perturbative calculations if we are measuring the transverse momentum of the high mass Drell-Yan lepton pair (or the transverse momentum of heavy particles) or the momentum imbalance (or the angular correlation) of dijets produced in high energy scattering. If one integrates over the transverse momentum of the produced particle or the azimuthal angle difference of dijets, the Sudakov effect disappears since the real-virtual cancellation becomes more complete after the integration.
In order to quantitatively study $P_T$ broadening effects in back-to-back dijet angular correlation measurements with the presence of medium effects, we need to develop a sophisticated formalism which incorporates Sudakov effects and the medium induced $P_T$ broadening effects, and investigate the interplay of these two effects in different experimental environments.
In general, one expects that the medium effects are absent in $pp$ collisions, and the correlations are solely due to Sudakov effects in the back-to-back dijet configurations. This has led to the successful description\cite{Sun:2014gfa} of the Tevatron ($p\bar p$)\cite{Abazov:2004hm} and the LHC ($pp$)\cite{Khachatryan:2011zj} dijet correlation data. Generally speaking, the larger the collision energy and jet transverse momentum are, the larger the Sudakov effects are. In the case of $pA$\cite{Chatrchyan:2014hqa} and $AA$\cite{Aad:2010bu, Chatrchyan:2011sx} collisions, the produced dijet system can also interact with either the cold nuclear medium or the hot-dense QGP medium, which generates extra transverse momentum broadening effects. In dijet productions at the LHC with the transverse momentum of the leading jet larger than $100 \, \textrm{GeV}$, Sudakov effects dominate over medium effects. Rough estimates give the transverse momentum broadening of the Sudakov effect at the LHC energy for dijet productions with $P_T \sim 100\, \textrm{GeV}$ as $\langle \triangle P_T^2\rangle \sim 100 \, \textrm{GeV}^2$~\cite{Sun:2014gfa}, as opposed to that due to medium effects which is $\langle \triangle P_T^2\rangle \sim \hat q L \sim 10 \, \textrm{GeV}^2$. Note that since the nature of momentum broadening in the transverse direction is the same as a random walk or Brownian motion, which suggests that we should always compare $\langle \triangle P_T^2\rangle$ instead of $\langle |\triangle P_T|\rangle$. This naturally explains why there are no visible medium modifications found for dijet angular correlation measurement in both $pPb$\cite{Chatrchyan:2014hqa} and $PbPb$\cite{Aad:2010bu, Chatrchyan:2011sx} collisions at the LHC, since the corresponding modification in terms of dijet angular distributions is too small to be seen at the LHC. To probe the medium effects through angular correlation measurements, we either need to lower the $P_T$ of the dijet system or measure dihadrons with much lower $P_T$ as in Ref.~\cite{Adler:2002tq, Aamodt:2011vg}. This can significantly reduce the he Sudakov effects. Therefore, as recently pointed out in Ref.~\cite{Mueller:2016gko, Chen:2016vem}, one can also measure medium effects at RHIC through dijets with roughly $P_T \sim 35 \, \textrm{GeV}$ and hadron-jet as well as dihadron correlations.
In this paper we study the transverse momentum distribution of jets produced by a hard scattering in the medium. For explicitness we consider a jet to be produced in the deep inelastic scattering of a transverse virtual photon on a nucleus. We consider in detail two separate cases where (i) the time scale over which the jet is produced, $\tau_q$, is much less than the size, $L$, of the nucleus and (ii) where $\tau_q$ is much greater than $L$ in the target rest frame. The transverse momentum of the jet then comes from various sources, namely, from the hard scattering itself, from radiation not induced by the medium (Sudakov radiation), from multiple scattering of the jet in the medium ($\hat q$) and from radiation induced by the medium (radiative corrections to $\hat q$). In our current discussion we take the transverse momentum of the virtual photon to be zero to minimize the transverse momentum coming from the hard scattering.
Although our discussion is done in the context of cold nuclear matter, a large nucleus, it is straightforward to extend to hot matter simply by changing from the $\hat q$ of cold matter to the $\hat q$ of hot matter. For example the discussion given in Sec.~\ref{lm}, for $\tau_q \ll L$, can be used to describe the imbalance between the transverse momentum of the two jets produced in a hard scattering in heavy ion collisions.
In Ref.~\cite{Mueller:2016gko, Chen:2016vem}, the relative importance to imbalance (the azimuthal angle between the two jets, hadron-jet or dihadrons) of Sudakov emission and medium induced broadening (multiple scattering effects together with medium induced radiation) was analyzed for jets produced in heavy ion collisions. In Sec.~\ref{lm} we include the medium induced radiative contribution, namely, radiative corrections to $\hat q$, to the imbalance. If the $\hat q$ of Sec.~\ref{lm} is taken to be that of hot matter then we have evaluated all the contributions to $\hat q$ included in the analysis of Ref.~\cite{Mueller:2016gko, Chen:2016vem}.
In the case that the transverse momentum broadening is dominated by Sudakov double logarithmic radiation, as in the case of jet production in LHC heavy ion collisions, it is necessary to revisit the evaluation of radiative corrections to $\hat q$ as done in the context of a $\hat q$-dominated broadening. This is done in Sec.~\ref{rcq} where all double logarithmic radiative corrections to $\hat q$ are evaluated.
In Sec.~\ref{smc} we consider small-$x$ deep inelastic scattering where the jet is formed on a time scale long compared to the length of the medium. We begin in Sec.~\ref{fixed} by doing the analysis assuming a fixed coupling and with the photon virtuality in the scaling region of the small-$x$ evolution. Up to an overall constant we are able to get analytic expressions for the jet broadening in (\ref{e37}) or, in the various regions shown in Fig.~\ref{f6}, in (\ref{e39})-(\ref{e41}).
It is interesting to investigate what happens at a fixed amount of the broadening, $k_\perp$, of the jet as one varies the hardness, $Q^2$, from moderate to large values while always assuming that $x$ is small enough that one remains in the scaling region of the small-$x$ evolution. When $\ln \frac{Q^2}{k_\perp^2} <\frac{1}{\sqrt{\alpha_s}}$ Sudakov effects are not visible and the transverse momentum, $k_\perp$, comes completely from small-$x$ evolution and exhibits scaling in (\ref{e39}). As $Q^2$ is increased one gets a scaling behavior with a simple factor giving the Sudakov contribution given by (\ref{e40}). When $\ln \frac{Q^2}{k_\perp^2} > \frac{1}{\alpha_s}$ the scaling behavior is completely destroyed and the transverse momentum distribution is flat reflecting the randomizing effects of the Sudakov radiation. This is exhibited in (\ref{e41}).
In Sec.~\ref{rcs}, running coupling effects are introduced and we no longer suppose that $Q^2$ lie in the scaling region of the small-$x$ evolution. The three different regions of Fig.~\ref{f6} give very similar results as compared to the fixed coupling case. The first region, where $Q^2$ is not so large, shows no Sudakov modification of the spectrum of transverse momentum broadening. The next region of somewhat larger $Q^2$ again has a simple Sudakov factor (see (\ref{e49})) modifying the small-$x$ answer. Finally, the large $Q^2$ region again completely eliminates all $k_\perp$-dependence, as given in (\ref{e56}).
In the case where $\tau_q \gg L$, $\hat q $-effects are not very visible, since they are hidden in the initial distribution for the small-$x$ evolution. In Sec.~\ref{meq} we show explicitly how $\hat q$-effects, and radiative corrections to $\hat q$, come into the initial condition for small-$x$ evolution. If there were no radiative corrections to $\hat q$, the initial condition for small-$x$ evolution is just the scattering matrix for a dipole given by the McLerran-Venugopalan model. If one uses $\hat{q}_t$ rather than $\hat q$ in the MV model initial condition then evolution in the medium is also included and will show up as an enhancement of $Q_s^2$. We conclude and summarize in Sec.~\ref{con}.
\section{Large medium forward jet production in DIS}
\label{lm}
\subsection{The basic formulas}
We begin our discussions of forward jet production in deep inelastic scattering (DIS) on a large nucleus in the case $\tau_q =\frac{2q_+}{Q^2}$ is much less than the length of the medium. For a scattering at impact parameter $b$ in the nucleus, the nuclear medium length is $L=2\sqrt{R^2-b^2}$ with $R$ the nuclear radius. The (transverse) virtual photon initiating the process has momentum $q_\mu$ with $q_\mu =(q_+, q_-= -\frac{Q^2}{2q_+}, q_\perp=0)$. The process is illustrated in Fig.~\ref{f1} where the forward quark (or antiquark) has momentum $k$ and travels a distance $z$ in the medium after its production. In the current situation of $\tau_q/L \ll 1$, this production can take
place on a definite nucleon in the nucleus with that nucleon at a distance $L-z$ from the front face of the nucleus.
In Refs.~\cite{Luo:1993ui, Zhang:2014dya,Kang:2016ron}, similar process has been considered to study the modification of average transverse momentum squared due to the medium effects. In this paper, we focus on the transverse momentum spectrum, where all the relevant QCD dynamics
play important roles.
In this large-$x$ process there is no small-$x$ evolution. However, there is the DGLAP evolution of the quark distribution of the struck nucleon,
the Sudakov effects due to the hard scattering and the measurement of the forward quark, and finally the multiple scattering and medium induced radiation of the outgoing quark. At the moment we do not introduce a cone condition for the produced quark jet nor do we consider the fragmentation of the quark. These can be included accordingly for a complete evaluation of the forward jet electro-production. Our purpose here is to illustrate in a simple context the various effects that may occur in jet production in a medium.
The transverse momentum spectrum of the quark is given by
\begin{equation}
\frac{dN}{d^2 b d^2 k_\perp}= \int \frac{d^2 x_\perp}{(2\pi)^2} e^{-ik_\perp\cdot x_\perp} \rho\, xq_N\left(x, \frac{1}{x_\perp^2 +1/Q^2}\right) \int_0^L dz e^{-\mathcal{E}}, \label{spectrum}
\end{equation}
where
\begin{equation}
\mathcal{E}=\hat{q} x_\perp^2 z/4 +\mathcal{E}_\textrm{Sud}+\mathcal{E}_\textrm{Medium Induced Radiation (MIR)}, \label{tot}
\end{equation}
with the quark transport coefficient
\begin{equation}
\hat {q} =\frac{C_F}{N_c} \frac{4\pi^2 \alpha_s N_c}{N_c^2-1} \rho \, xG(x) \ . \label{qhatdef}
\end{equation}
Here, $\rho$ is the nucleon density and $xG$ the nucleon's gluon distribution, while $xq_N$ the quark distribution of a nucleon should be evaluated at a scale $x_\perp^2$, that is $q_N=q_N \left(x, \frac{1}{x_\perp^2 +1/Q^2}\right)$. When $x_\perp=0$, see below, one gets $q_N$ as the quark distribution at the hard scattering scale. The various terms of Eq.~(\ref{tot}) can be interpreted as follows: $\hat q$ term accounts for multiple scattering as the quark passes through the nucleus; $\mathcal{E}_\textrm{Sud}$ accounts for the real and virtual Sudakov corrections, which are medium independent, induced by the hard scattering; and $\mathcal{E}_\textrm{MIR}$ accounts for gluonic radiative corrections which involve a single scattering in the medium. As we shall see below, the $\hat q$ and $\mathcal{E}_\textrm{MIR}$ terms in (\ref{tot}) can be combined into a more complete $\hat q$, which we shall call $\hat {q}_t \equiv \hat q _{\textrm{total}}$, where
\begin{equation}
\hat {q}_t x_\perp^2 z/4 =\hat {q} x_\perp^2 z/4 +\mathcal{E}_\textrm{MIR}.
\end{equation}
Then the $z$-integral in (\ref{spectrum}) can be done giving
\begin{equation}
\frac{dN}{d^2 b d^2 k_\perp}=\int \frac{d^2 x_\perp}{x_\perp^2} \frac{\rho\, xq_N\left(x, \frac{1}{x_\perp^2 +1/Q^2}\right)}{\pi^2 \hat {q}_t} e^{-ik_\perp\cdot x_\perp} \left(1-e^{-\hat{q}_t x_\perp^2L/4}\right) e^{-\mathcal{E}_{\textrm{Sud}}}. \label{wwquark}
\end{equation}
The right hand side of (\ref{wwquark}) has the form of an unintegrated Weizsacker-Williams quark distribution in analogy with the Weizsacker-Williams (WW) \cite{Kovchegov:1996ty, Kovchegov:1998bi, Kharzeev:2003wz} gluon distribution. We note that
\begin{equation}
\int \frac{dN}{d^2 b d^2 k_\perp} d^2 b d^2 k_\perp =A xq_N.
\end{equation}
The Sudakov factor in (\ref{wwquark}) is naturally included as part of the WW quark distribution since the usual Wilson line of the WW distribution implicitly includes the Sudakov factor, see the discussions below.
\subsection{The Sudakov factor}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=12cm]{f1}
\end{center}
\caption[*]{Forward jet production in DIS on a large nucleus in the large-$x$ region.}
\label{f1}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=12cm]{f2}
\end{center}
\caption[*]{Forward jet production in DIS in dipole model.}
\label{f2}
\end{figure}
In order to evaluate the Sudakov term, and later the $\mathcal{E}_\textrm{MIR}$ term, it is convenient to bring the complex conjugate amplitudes in Fig.~\ref{f1} into the amplitude and view the process as in Fig.~\ref{f2}\cite{Kovchegov:2001sc, Mueller:2012bn}. In Fig.~\ref{f2} we have taken the virtual photon to interact on the front face of the nucleus so that the quark goes through a length $L$ of nuclear matter. We have also added a gauge link at $t=\infty$ to make the process manifestly gauge invariant, and we have indicated a gluon line $l$ which is emitted, and absorbed by the $0_\perp$ and $x_\perp$ quark and antiquark lines. (Emission and reabsorption of $l$ off $0_\perp$ corresponds to a virtual correction to the quark line in the amplitude of Fig.~\ref{f1}. Emission and reabsorption off $x_\perp$ corresponds to a virtual correction to the quark line in the complex conjugate amplitude of Fig.~\ref{f2} while emission off $0_\perp$ ($x_\perp$) and absorption off $x_\perp$ ($0_\perp$) corresponds to a real gluon emission correction to the graph in Fig.~\ref{f1}.)
Now the evaluation of $\mathcal{E}_{\textrm{Sud}}$ is straightforward\cite{Mueller:2012uf, Mueller:2013wwa}
\begin{equation}
\mathcal{E}_{\textrm{Sud}} =2\frac{\alpha_s C_F}{2\pi} \int_{q_+/\left[Q^2 x^2_\perp\right]}^{q_+} \frac{dl_+}{l_+} \int_{1/x_\perp^2}^{\frac{l_+}{q_+}Q^2} \frac{dl_\perp^2}{l_\perp^2} =\frac{\alpha_s C_F}{2\pi} \ln^2\left(Q^2 x^2_\perp\right). \label{sud7}
\end{equation}
The various limits to the $l_\perp^2$ and $l_+$ integration are determined as: (i) The lower limit to the $l_\perp^2$ integration comes from the fact that the softer $l_\perp$-values cancel between emissions (absorptions) off the $0_\perp$ and $x_\perp$ lines. (ii) The upper limit of the $l_\perp^2$-integration comes from the requirement that $\tau_l >\tau_q$. This is shown in some detail in appendix A. The limits on the $l_+$-integration are manifest. The logarithmic contribution given in (\ref{sud7}) comes completely from the virtual contributions as described above. The real emissions serve only to cancel the virtual emissions in the $l_\perp^2 x_\perp^2 \ll 1$ region.
The lifetime, $\tau_l =\frac{2l_+}{l_\perp^2}$, can be either less than $L$ or greater than $L$ in (\ref{sud7}) so that the gluon, $l$, will sometimes exist within the medium. However, the gluon is too close to either the quark $0_\perp$ or antiquark ($x_\perp$) for the interactions with the medium to distinguish, say, the quark-$l$ system from the quark so that medium interactions with the gluon cancel out leaving the Sudakov term medium independent.
It is interesting to note that the Sudakov effects occur when a dipole is created in a medium, as given by (\ref{spectrum}) and illustrated in Fig.~\ref{f2}, however there are no Sudakov effects in dipole nucleus scattering where the $t<0$ and $t>0$ regions occur in a symmetric way and there is no hard reaction to stimulate radiation.
If $Q$ is very large then the typical values of $k_\perp$ for which $\frac{dN}{d^2 b d^2 k_\perp}$ is large will be determined by $\mathcal{E}_{\textrm{Sud}}$ given in (\ref{sud7}) and used in (\ref{spectrum}) rather than by $\hat q$ or $\hat{q}_t$\cite{Mueller:2016gko}. This is the situation for jet azimuthal angle distributions measured in ion-ion collisions at the LHC where Sudakov effects overwhelm $\hat q$ effects
\cite{Mueller:2016gko}. The interplay of Sudakov and $\hat q$ effects in (\ref{spectrum}) is an essential factor for dijet production in
heavy ion collisions. Theoretically, in the case
that Sudakov effects are the dominant broadening effects,
the radiative corrections to $\hat q$ leading to $\hat{q}_t$ changes
from the standard calculations of Refs.~\cite{Liou:2013qya,Wu:2011kc}, which will be discussed in the
following subsection.
\subsection{Radiative corrections to $\hat q$.}
\label{rcq}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=10cm]{f3}
\end{center}
\caption[*]{Radiative correction to dipole-nucleus scattering in dipole model.}
\label{f3}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=10cm]{f4}
\includegraphics[width=6.5cm]{f4p}
\end{center}
\caption[*]{Domains of integration respectively for $K>\tau$ (left panel) and for $0<K\leq\tau$ (right panel).}
\label{f4}
\end{figure}
In the previous evaluation of the radiative corrections (double logarithmic) to $\hat q$~\cite{Liou:2013qya, Wu:2011kc, Blaizot:2014bha, Iancu:2014kga}, one considers gluon emission from a dipole, similar to that in Fig.~\ref{f2}. However, in this case, the gluon interacts with the medium making the effect medium dependent, as a correction to $\hat q$. The effective value of $x_\perp^2$ of the dipole is $x_\perp^2 \sim 1/(\hat q L)=1/Q_s^2$ when transverse momentum broadening is $\hat q$ dominated. If, however, the broadening is Sudakov dominated the value of $x_\perp^2$ will change and a new evaluation is necessary. At lowest order the radiative correction to $\hat q$ is illustrated in Fig.~\ref{f3} and given by
\begin{equation}
\hat{q}_t =\hat q \left(1+ \frac{\alpha_s N_c}{\pi} \int \frac{dl_\perp^2}{l_\perp^2} \int \frac{dl_+}{l_+}\right), \label{qt}
\end{equation}
where the limits of integration have yet to be set. $\hat q$, as earlier, is the quark transport coefficient and we work in the fixed coupling approximation. The limits of integration in (\ref{qt}) are set by the following constraints:
\begin{eqnarray}
\frac{2l_+}{l_\perp^2}&<&L \label{e9} \ ,\\
\frac{2l_+}{l_\perp^2} &<&\frac{l_\perp^2}{\hat q}\label{e10} \ ,\\
\frac{2l_+}{l_\perp^2} &>&r_0 \label{e11}\ ,\\
l_\perp^2 &<&\frac{1}{x_\perp^2} \label{e12} \ ,\\
l_+ &<& q_+ \label{e13}\ .
\end{eqnarray}
The physics meanings of the above constraints are as follows:
(\ref{e9}) is the constraint that the gluon, $l$, be within the medium; (\ref{e10}) is a single scattering requirement, necessary to get a double logarithm; (\ref{e11}) requires that the fluctuation live longer than the proton size, $r_0$; (\ref{e12}) requires that the gluon transverse distance from the dipole is greater than the dipole size, which is necessary for a double logarithm to emerge. In particular, (\ref{e10}) is a stronger requirement than (\ref{e9}) when $l_\perp^2 <\hat q L$, while (\ref{e9}) is the stronger requirement when $\hat q L < l_\perp^2 <1/x_\perp^2$. Much of what follows can also be found in \cite{Iancu:2014sha}. We include this simplified discussion for completeness.
Let us start with $1/x_\perp^2 > \hat{q}L$. Writing (\ref{qt}) more completely and using the constraints of (\ref{e9})-(\ref{e13}), we arrive at,
\begin{equation}
\hat{q}_t -\hat q = \bar{\alpha}_s \hat q \left[\int_{\hat q r_0}^{\hat q L} \frac{d l_\perp^2}{l_\perp^2} \int_{l_\perp^2 r_0}^{(l_\perp^2)^2/\hat{q}}\frac{d l_+}{l_+}+\int_{\hat q L}^{1/x_\perp^2} \frac{d l_\perp^2}{l_\perp^2} \int^{l_\perp^2 L}_{l_\perp^2 r_0}\frac{d l_+}{l_+}\right], \label{e14}
\end{equation}
or
\begin{equation}
\hat{q}_t -\hat q = \bar{\alpha}_s \hat q \ln \frac{L}{r_0} \left(\frac{1}{2} \ln \frac{L}{r_0}+\ln\frac{1}{\hat{q}L x_\perp^2}\right),
\end{equation}
where $\bar{\alpha}_s \equiv \alpha_s N_c/\pi$.
In order to sum the whole series of double logs it is convenient to introduce the following logarithmic variables
\begin{eqnarray}
K&=& \ln \frac{1}{\hat q r_0 x_\perp^2}, \quad K_1=\ln \frac{l_\perp^2}{\hat q r_0}, \\
\tau &=& \ln \frac{L}{r_0}, \quad \tau_1=\ln \frac{l_+}{l_\perp^2 r_0}\ .
\end{eqnarray}
With these notations, Eq.~(\ref{e14}) takes the form
\begin{equation}
\hat{q}_t -\hat q = \bar{\alpha}_s \hat q \int_{0}^{\tau} d\tau_1 \int_{\tau_1}^{K} dK_1 = \bar{\alpha}_s \hat q \left[K\tau -\frac{1}{2} \tau^2 \right]. \label{e18}
\end{equation}
The domain of integration for $K_1$ , $\tau_1$ in (\ref{e18}) is shown in the left panel of Fig.~\ref{f4}. The boundary $\frac{2l_+}{l_\perp^2} =\frac{l_\perp^2}{\hat q}$ given in (\ref{e10}) becomes the boundary $\tau_1 =K_1$ in Fig.~\ref{f4}. It is now straightforward to sum the complete double logarithmic series as
\begin{equation}
\hat{q}_t =\hat q \sum_{n=0}^{\infty} \Delta_n, \label{sum}
\end{equation}
where
\begin{equation}
\Delta_n =\Pi_{i=1}^n \bar{\alpha}_s \int_{0}^{\tau_{i+1}} d\tau_i \int_{\tau_i}^{K_{i+1}} dK_i \, \label{e20}
\end{equation}
with $\tau_{n+1}=\tau$ and $K_{n+1}=K$ in (\ref{e20}). Therefore, we find that $\Delta_n$ obeys
the following equation,
\begin{equation}
\frac{\partial}{\partial \tau}\frac{\partial}{\partial K} \Delta_n (\tau, K) = \bar{\alpha}_s \Delta_{n-1} (\tau, K), \label{e21}
\end{equation}
which, with (\ref{e18}), gives
\begin{equation}
\Delta_n = \frac{\bar{\alpha}_s^n K^{n-1}\tau^n \left[(n+1)K-n\tau\right]}{n! (n+1)!}. \label{e22}
\end{equation}
Using (\ref{e22}), the sum in (\ref{sum}) can be derived~\cite{Iancu:2014sha}
\begin{equation}\label{qtlargeK}
\hat{q}_t =\hat{q} \left[ \frac{1}{\sqrt{\bar{\alpha}_s K\tau }} I_1 (2 \sqrt{\bar{\alpha}_s K\tau }) +\left(1- \frac{\tau}{K}\right) I_2 (2 \sqrt{\bar{\alpha}_s K\tau }) \right].
\end{equation}
In the case of $\hat{q}r_0\leq1/x_\perp^2 \leq \hat{q}L$, i.e., $0\leq K \leq \tau$, the domain of integration for $K_1$ is shown in the right panel of Fig. \ref{f4}. With that, we find that Eq.~(\ref{qt}) can be written as
\begin{equation}
\hat{q}_t -\hat q = \bar{\alpha}_s \hat q \int_{0}^{K} d\tau_1 \int_{\tau_1}^{K} dK_1 = \frac{1}{2}\bar{\alpha}_s \hat q K^2,
\end{equation}
which is simply given by (\ref{e18}) with $\tau$ being replaced by $K$. Similarly, it is easy to see, for this case, that
\begin{equation}
\Delta_n =\frac{\bar{\alpha}_s^n K^{2n}}{n! (n+1)!}, \quad
\textrm{and} \quad
\hat{q}_t =\hat{q} \frac{1}{\sqrt{\bar{\alpha}_s }K} I_1 (2 \sqrt{\bar{\alpha}_s }K).
\end{equation}
In summary, we have the following results for different $K$ values,
\begin{subequations}
\label{equations}
\begin{align}
\label{eqa}
\hat{q}_t &=\hat{q} &\text{if $K<0$,}\vspace{2mm}\\
\label{eqb}
\hat{q}_t &=\hat{q} \frac{1}{\sqrt{\bar{\alpha}_s }K} I_1 (2 \sqrt{\bar{\alpha}_s }K)&\text{if $0\leq K\leq\tau$,}\vspace{2mm}\\
\label{eqc}
\hat{q}_t &= \hat{q} \left[ \frac{1}{\sqrt{\bar{\alpha}_s K\tau }} I_1 (2 \sqrt{\bar{\alpha}_s K\tau }) +\left(1- \frac{\tau}{K}\right) I_2 (2 \sqrt{\bar{\alpha}_s K\tau }) \right]& \text{if $K>\tau$.}
\end{align}
\end{subequations}
The above results, together with (\ref{sud7}),
give the complete evaluation of $\mathcal{E}$ in (\ref{spectrum}) via (\ref{tot}). (When used in (\ref{spectrum}) the $L$ in $\tau$ should be changed to $z$ the effective path length for the integrand of (\ref{spectrum}).)
The spectrum $\frac{dN}{d^2 b d^2 k_\perp}$ is then given by (\ref{spectrum}) or (\ref{wwquark}) where all the ingredients for evaluating (\ref{spectrum}) or (\ref{wwquark}) are given by (\ref{sud7}) and (\ref{equations}). It is straightforward to include running coupling effects and higher order corrections to the Sudakov term in (\ref{sud7}). (See (\ref{running43}) for running coupling corrections.) However, it is not clear at present how to include running coupling corrections to $\hat{q}_t$ in a resummed way. See Ref.~\cite{Iancu:2014sha} for the state of the art.
More interestingly, following (\ref{e21}) and summing over all $n$, one can in fact write down a double differential evolution equation for $\hat{q}_t $ as follows
\begin{equation}
\frac{\partial}{\partial \tau}\frac{\partial}{\partial K} \hat{q}_t = \bar{\alpha}_s \hat{q}_t, \label{qe}
\end{equation}
which is equivalent to the DGLAP evolution equation for the gluon distribution in the double logarithmic limit. As shown in Ref.~\cite{Gorshkov:1966ht, Kovchegov:2012mbw}, the solution of (\ref{qe}) can be written in terms of superpositions of modified Bessel functions $I_{\nu} (x)$ with coefficients determined by boundary conditions, since $\left(\frac{\tau}{\kappa}\right)^{\frac{\nu}{2}}I_\nu(2\sqrt{\bar\alpha_s\tau\kappa})$ for arbitrary $\nu$ is a solution to (\ref{qe}). For example, given the boundary conditions $\hat{q}_t|_{K=0}=\hat{q}_t|_{\tau=0}=\hat{q}$, one can find $ \hat{q}_t = \hat{q} I_0(2 \sqrt{\bar{\alpha}_s K\tau })$ which is equivalent to the usual DGLAP double logarithmic solution for gluon distributions.\footnote{This is natural since it is known that $\hat{q}$ is proportional to target gluon distributions by definition.} Furthermore, it is straightforward to check that (\ref{qtlargeK}) or (\ref{eqc}) is the solution to (\ref{qe}) given boundary conditions $\hat{q}_t|_{K=0}=\hat{q} (1-\frac{\bar{\alpha}_s}{2} \tau^2 )$ and $\hat{q}_t |_{\tau=0}=\hat{q}$. This indicates that the evolution equation of $\hat{q}_t$ in the double logarithmic limit is also given by (\ref{qe}) with particular boundary conditions which reflect the information of the target medium such as length $L$ and multiple scatterings. Therefore, it seems that we can obtain the full results in (\ref{equations}) by continuing the solution (\ref{eqc}) to (\ref{eqb}) at $K=\tau$ then to (\ref{eqa}) at $K=0$.
\section{Small-$x$ forward jet production in DIS}
\label{smc}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=10cm]{f5}
\end{center}
\caption[*]{Radiative correction to dipole-nucleus scattering in dipole model in the small-$x$ limit.}
\label{f5}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=10cm]{f6}
\end{center}
\caption[*]{Three regions of the transverse momentum spectrum as a function of $\ln\frac{Q^2}{k_\perp^2}$ in case $\tau_q\gg L$.}
\label{f6}
\end{figure}
We now turn to the limit opposite to that of the large medium considered in Sec. \ref{lm}, namely the case where $\tau_q \gg L$. Here the process has the virtual photon splitting into a quark-antiquark dipole which then further evolves before passing over the nucleus. The process is illustrated in Fig.~5 where evolution of the ($x_{\perp 1}, 0_\perp$) and ($x_{\perp 2}, 0_\perp$) dipoles in the amplitude and complex conjugate amplitude are not explicitly shown, nor are the interactions with the target nucleus shown.
\subsection{The forward jet spectrum; fixed coupling analysis}
\label{fixed}
The forward quark (or antiquark) jet spectrum coming from the scattering of a transverse virtual photon is usually written as \cite{Mueller:1999wm}
\begin{eqnarray}
\frac{dN}{d^2b d^2 k_\perp} &=&\sum_f e^2_f \frac{Q^2 N_c}{32\pi^6} \int d^2 x_{1\perp }d^2 x_{2\perp }e^{-ik_\perp \cdot (x_{1\perp}-x_{2\perp})} \int_0^1 dz \left[z^2+(1-z)^2\right] \times \notag \\
&&\nabla_{x_{1\perp}} K_0 \left[\sqrt{Q^2 x_{1\perp}^2 z(1-z)}\right]\cdot \nabla_{x_{2\perp}} K_0 \left[\sqrt{Q^2 x_{2\perp}^2 z(1-z)}\right] \left[1+S(x_{1\perp} -x_{2\perp})-S(x_{1\perp})-S(x_{2\perp})\right]. \label{smallxe24}
\end{eqnarray}
The different factors in (\ref{smallxe24}) are straightforward to understand: the $\nabla K_0$ factors are the quark-antiquark wavefunctions of the virtual transverse photons in the amplitudes and complex conjugate amplitude; $\left[z^2+(1-z)^2\right]$ is the splitting function of the photon. For the various $S$-matrices the combination in (\ref{smallxe24}) guarantees that there be at least one interaction in the amplitude and in the complex conjugate amplitude. The normalization is
\begin{equation}
\int d^2b d^2k_\perp \frac{dN}{d^2b d^2 k_\perp} =\sum_f e^2_f \left[ xq_A^f (x, Q^2)+x\bar{q}_A^f (x, Q^2)\right]. \label{norm}
\end{equation}
However, (\ref{smallxe24}) is missing a Sudakov factor. One often says that DIS scattering is given in terms of a dipole scattering amplitude times the virtual photons' quark-antiquark wave functions. That is true of (\ref{norm}) with $\frac{dN}{d^2b d^2 k_\perp}$ given by (\ref{smallxe24}). However if a jet is measured rather than integrated over, as in (\ref{norm}), a Sudakov factor \cite{Mueller:2013wwa}
\begin{equation}
\textrm{Sudakov} = e^{-\frac{\alpha_s C_F}{2\pi} \ln^2 \left[Q^2(x_{1\perp}-x_{2\perp})^2+1\right]} \label{sud26}
\end{equation}
should be inserted in the integrand in (\ref{smallxe24}). (The $1$ in $Q^2(x_{1\perp}-x_{2\perp})^2+1$ is included to make the $x_{1\perp} \to x_{2\perp}$ limit, as occurs in (\ref{norm}), non-singular.) Now inserting (\ref{sud26}) into the integrand of (\ref{smallxe24}) and changing the variables of integration, we will get
\begin{eqnarray}
\frac{dN}{d^2b d^2 k_\perp} &=&\sum_f e^2_f \frac{Q^2 N_c}{32\pi^6} \int d^2 x_{1\perp }d^2 x_{\perp } \int_0^1 dz e^{-ik_\perp \cdot x_{\perp}}\left[z^2+(1-z)^2\right] \nabla_{x_{1\perp}} K_0 \left[\sqrt{Q^2 x_{1\perp}^2 z(1-z)}\right] \notag \\
&&\cdot \nabla_{x_{1\perp}} K_0 \left[\sqrt{Q^2 (x_{1\perp}-x_{\perp})^2 z(1-z)}\right] e^{-\frac{\alpha_s C_F}{2\pi} \ln^2 (Q^2 x_{\perp}^2)}\left[1+S(x_{\perp})-2S(x_{1\perp})\right]. \label{sudx27}
\end{eqnarray}
While it appears difficult to do the $x_{1\perp}$-integration in (\ref{sudx27}) exactly, it is clear that $|x_{1\perp}|\sim |x_\perp|$ dominates the leading power contribution of the integral and that $z\sim \frac{1}{Q^2 x_{\perp}^2}$ or $1-z \sim \frac{1}{Q^2 x_{\perp}^2}$ so that
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\mathcal{C}\int \frac{d^2 x_\perp}{\pi x_\perp^2} e^{-ik_\perp\cdot x_\perp} e^{-\frac{\alpha_s C_F}{2\pi} \ln^2 (Q^2 x_{\perp}^2)} T(x_\perp, Y) \label{e28}
\end{equation}
where $1-S=T$ and $Y=\ln\frac{1}{x_{\textrm{Bj}}}$. $\mathcal{C}$ is a constant in the sense that it has no $k_\perp$-dependence but it will depend on the form of $T$, that is in the scaling region it will depend on the anomalous dimension giving the scaling behavior. As an illustration, let us suppose that the energy is high enough that $k_\perp$ can be taken in the scaling region \cite{Kwiecinski:2002ep, Iancu:2002tr}
\begin{equation}
T(x_\perp )=\left[ Q_s^2 x_\perp^2\right]^{1-\lambda_0}.
\end{equation}
It is straightforward to get
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =2\mathcal{C}\int_0^{\infty} \frac{d x_\perp}{ x_\perp} \left(Q_s^2 x_\perp^2\right)^{1-\lambda_0} J_0(k_\perp x_\perp) e^{-\frac{\alpha_s C_F}{2\pi} \ln^2 (Q^2 x_{\perp}^2)}
\end{equation}
or
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =2\mathcal{C} \left(\frac{Q_s^2}{Q^2}\right)^{1-\lambda_0}\int_0^{\infty} \frac{d x_\perp}{ x_\perp} J_0(k_\perp x_\perp) e^{-\frac{\alpha_s C_F}{2\pi} \ln^2 (Q^2 x_{\perp}^2)+(1-\lambda_0) \ln (Q^2 x_{\perp}^2)}.
\end{equation}
Changing variables to $z=\ln (Q^2 x_{\perp}^2)$ one gets
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\mathcal{C} \left(\frac{Q_s^2}{Q^2}\right)^{1-\lambda_0}\int_{-\infty}^{+\infty} dz J_0\left(\frac{k_\perp}{Q} e^{z/2} \right)e^{-\frac{\alpha_s C_F}{2\pi} z^2 +(1-\lambda_0)z }. \label{e32}
\end{equation}
Although the $z$ integration has be written as going from $-\infty$ to $\infty$ in (\ref{e32}), the effective range can be taken as $z< \ln \frac{Q^2}{k_\perp^2}
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\mathcal{C} \left(\frac{Q_s^2}{Q^2}\right)^{1-\lambda_0}\int_{-\infty}^{\ln \frac{Q^2}{k_\perp^2}} dz e^{-\frac{\alpha_s C_F}{2\pi} z^2 +(1-\lambda_0)z }. \label{e33}
\end{equation}
It is now straightforward to write (\ref{e33}) in terms of the error function $Erf(x)$ by rewriting (\ref{e33}) as
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\mathcal{C} \left(\frac{Q_s^2}{Q^2}\right)^{1-\lambda_0}e^{\frac{(1-\lambda_0)^2 \pi}{2\alpha_s C_F}} \sqrt{\frac{2\pi}{\alpha_s C_F}} \int_{-\infty}^{\sqrt{\frac{\alpha_s C_F}{2\pi}} \left[\ln \frac{Q^2}{k_\perp^2}-\frac{(1-\lambda_0) \pi}{\alpha_s C_F}\right]} dw e^{-w^2 }, \label{e34}
\end{equation}
where we have introduced
\begin{equation}
w=\sqrt{\frac{\alpha_s C_F}{2\pi}} \left[z-\frac{(1-\lambda_0) \pi}{\alpha_s C_F}\right].
\end{equation}
Using
\begin{equation}
Erf (x) =\int_0^x dt e^{-t^2},
\end{equation}
one has
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\mathcal{C} \left(\frac{Q_s^2}{Q^2}\right)^{1-\lambda_0}e^{\frac{(1-\lambda_0)^2 \pi}{2\alpha_s C_F}} \sqrt{\frac{2\pi}{\alpha_s C_F}} \left\{ Erf \left[\sqrt{\frac{\alpha_s C_F}{2\pi}} \left(\ln \frac{Q^2}{k_\perp^2}-\frac{(1-\lambda_0) \pi}{\alpha_s C_F}\right)\right] +\frac{\sqrt{\pi}}{2} \right\}. \label{e37}
\end{equation}
One can write approximate results for the three regions shown in Fig.~\ref{f6} as follows:
\begin{eqnarray}
\left.\frac{dN}{d^2b d^2 k_\perp}\right|_{\circled{1}}&=&\frac{\mathcal{C}}{1-\lambda_0}\left(\frac{Q_s^2}{k_\perp^2}\right)^{1-\lambda_0}, \label{e39} \\
\left.\frac{dN}{d^2b d^2 k_\perp}\right|_{\circled{2}}&=&\frac{\mathcal{C}}{1-\lambda_0}\left(\frac{Q_s^2}{k_\perp^2}\right)^{1-\lambda_0} e^{-\frac{\alpha_s C_F}{2\pi} \ln^2 \frac{Q^2}{k_\perp^2}}, \label{e40}\\
\left.\frac{dN}{d^2b d^2 k_\perp}\right|_{\circled{3}}&=&\pi \mathcal{C} \sqrt{\frac{2}{\alpha_s C_F}} e^{\frac{(1-\lambda_0)^2 \pi}{2\alpha_s C_F}} \left(\frac{Q_s^2}{Q^2}\right)^{1-\lambda_0}. \label{e41}
\end{eqnarray}
Equations (\ref{e39}-\ref{e41}) are approximate equations, accurate away from the boundaries of their respective regions. To get an accurate evaluation at the boundary of regions $\circled{2}$ and $\circled{3}$ one must use (\ref{e37}). Equations (\ref{e39}) and (\ref{e40}) have a smooth transition between regions $\circled{1}$ and $\circled{2}$ so that (\ref{e40}) can be used in both regions. Also, if $\ln \frac{Q^2}{Q_s^2} <\frac{(1-\lambda_0) \pi}{\alpha_s C_F}$ region $\circled{3}$ does not exist and (\ref{e40}) again becomes the relevant formula.
In region $\circled{1}$ Sudakov effects are very small and the "normal" scaling result holds as indicated in (\ref{e39}). When $k_\perp^2$ decreases, one moves to region $\circled{2}$ where Sudakov effects appear in a very simple way, modifying the geometric scaling formula. Finally, if $\ln \frac{Q^2}{Q_s^2}$ is large enough, when $\ln \frac{Q^2}{k_\perp^2}$ gets larger than $\frac{(1-\lambda_0) \pi}{\alpha_s C_F}$ the $k_\perp$ of the jet does not come mainly from small-$x$ evolution and all $k_\perp$-dependence has disappeared due to the randomizing effects of Sudakov radiation.
\subsection{The forward jet spectrum; running coupling}
\label{rcs}
Now we shall repeat the discussion given in Sec.~\ref{fixed} but using a running QCD coupling and without the assumption that $Q^2$ is in the scaling region of the small-$x$ evolution. Not too much changes from our earlier analysis and much of the discussion of Sec.~\ref{fixed} can be directly taken over to the running coupling case. The main change is the modification of (\ref{sud26}). Now Sudakov effects take the form
\begin{equation}
\ln \textrm{Sud.} =- \frac{C_F}{\pi} \int_{1/x_\perp^2}^{Q^2} \frac{dl_\perp^2}{l_\perp^2} \alpha_s (l_\perp^2) \int^{q_+}_{q_+\frac{l_\perp^2}{Q^2}} \frac{dl_+}{l_+}.
\end{equation}
Using $\alpha_s =\frac{1}{b\ln l_\perp^2/\Lambda^2}$, one finds
\begin{equation}
\ln \textrm{Sud.}=-\frac{C_F}{\pi b}\left[ \ln \frac{Q^2}{\Lambda^2} \ln\frac{\ln \frac{Q^2}{\Lambda^2} }{\ln \frac{1}{x_\perp^2\Lambda^2}}-\ln Q^2 x_\perp^2\right] \label{running43}
\end{equation}
which now replaces (\ref{sud26}). Except for the Sudakov factor, (\ref{sudx27}) and (\ref{e28}) still hold. It is straightforward to write (\ref{running43}) in terms of $z=\ln Q^2 x_\perp^2 $ as
\begin{equation}
\ln \textrm{Sud.}=\frac{C_F}{\pi b}\left[ \ln \frac{Q^2}{\Lambda^2} \ln\left(1-\frac{z}{ \ln\frac{Q^2}{\Lambda^2}}\right)+z\right] \label{e44}
\end{equation}
and, expanding the logarithm and using $\alpha_s (Q)=\frac{1}{b\ln Q^2/\Lambda^2}$, to get
\begin{equation}
\ln \textrm{Sud.}=-\frac{C_F}{\pi b^2 \alpha_s (Q)} \sum_{n=2}^{\infty}\frac{\left(zb\alpha_s (Q)\right)^n}{n} \label{sudex}
\end{equation}
or
\begin{equation}
\ln \textrm{Sud.}=-\frac{\alpha_s (Q) C_F}{2\pi }z^2 +\cdots \label{sudexpand}
\end{equation}
in case $zb\alpha_s (Q)$ is small.
The regions $\circled{1}, \circled{2}, \circled{3}$ in Fig. ~6 are essentially unchanged except for the value of $\ln \frac{Q^2}{k_\perp^2}$ separating regions $\circled{2}$ from $\circled{3}$. Let us begin near $\ln \frac{Q^2}{k_\perp^2}=0$, the left-most region in Fig.~6 where (\ref{e33}) now becomes
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\mathcal{C} \int_{0}^{\ln \frac{Q^2}{k_\perp^2}} dz e^{-\frac{\alpha_s C_F}{2\pi} z^2 +\ln T(z, Q, Y) } \label{e47}
\end{equation}
where $Y$, in $T(z, Q, Y)$, is $Y=\ln\frac{1}{x_{\textrm{Bj}}}$. When $\ln \frac{Q^2}{k_\perp^2}$ is not too large $z\alpha_s (Q) \ll 1 $ and we have used (\ref{sudexpand}) as the Sudakov factor. So long as $\frac{\alpha_s C_F}{2\pi} \ln^2 \frac{Q^2}{k_\perp^2} <1 $, region $\circled{1}$, the Sudakov factor in (\ref{e47}) may be dropped and
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\mathcal{C} \int_{0}^{\ln \frac{Q^2}{k_\perp^2}} dz \, T(z, Q, Y). \label{e48}
\end{equation}
If $Q$ were in the scaling region, $T(x_\perp )=\left[ Q_s^2 x_\perp^2\right]^{1-\lambda_0}$, (39) would emerge but, in any case, in region $\circled{1}$ the result for $\frac{dN}{d^2b d^2 k_\perp}$ is the usual result with Sudakov effects being negligible.
As one moves into regions $\circled{2}$, (\ref{e47}) remains valid so long as $\ln \frac{Q^2}{k_\perp^2}$ is not too large. As in (\ref{e48}) the $z$ integration is dominated by the upper end, $z\sim \ln \frac{Q^2}{k_\perp^2}$, of the integral so that
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} \sim e^{-\frac{\alpha_s (Q) C_F}{2\pi} \ln^2 \frac{Q^2}{k_\perp^2}} T(x_\perp =\frac{1}{k_\perp}). \label{e49}
\end{equation}
If $k_\perp$ is in the scaling region then (\ref{e40}) will emerge. Here the Sudakov correction is a simple factor times the usual result without including Sudakov effects. The transition between $\circled{2}$ and $\circled{3}$ is determined by
\begin{equation}
\frac{d}{dz} \left( \ln \textrm{Sud.} +\ln T\right) =0.
\end{equation}
Using (\ref{e44}) this gives the equation
\begin{equation}
\frac{zb\alpha_s (Q)}{1-zb\alpha_s(Q)} =\frac{\pi b}{C_F} \frac{1}{T} \frac{\partial T}{\partial z}.\label{e51}
\end{equation}
(In case $T=(Q_s^2 x_\perp^2)^{1-\lambda_0}=\left(\frac{Q_s^2}{Q^2}\right)^{1-\lambda_0} e^{(1-\lambda_0)z}$, the scaling region, and if one used (\ref{sudexpand}) the boundary $\ln \frac{Q^2}{k_\perp^2}=\frac{(1-\lambda_0)\pi}{\alpha_s C_F}$ shown in Fig.~6 would emerge.)
Without assuming that the boundary between regions $\circled{2}$ and $\circled{3}$ is in the scaling region we can parametrize $T$ as
\begin{equation}
T\propto (x_\perp^2)^{1-\lambda_{\textrm{eff}}} \sim e^{(1-\lambda_{\textrm{eff}})z}, \label{e52}
\end{equation}
where $\lambda_{\textrm{eff}}$ may depend on $Y$ and on $x_\perp$. Assuming the $z$ dependence of $\lambda_{\textrm{eff}}$ is small, a reasonable assumption, using (\ref{e51}) and (\ref{e52}) gives
\begin{equation}
\ln \frac{Q^2}{k_\perp^2} =\frac{(1-\lambda_{\textrm{eff}}) \pi}{C_F \alpha_s(Q) \left[1+\frac{\pi b (1-\lambda_{\textrm{eff}})}{C_F}\right]}\equiv z_0 \label{e53}
\end{equation}
as the boundary between regions $\circled{2}$ and $\circled{3}$. (The boundary between $\circled{2}$ and $\circled{3}$ as given by (\ref{e53}) is to the left of the end point, $\ln \frac{Q^2}{Q_s^2}$, in Fig.~\ref{f6} so long as $\ln \frac{Q^2}{\Lambda^2} >\left[1+\frac{\pi b (1-\lambda_{\textrm{eff})}}{C_F}\right] \ln \frac{Q_s^2}{\Lambda^2}$, which we suppose to be the case.)
In region $\circled{3}$ all $k_\perp$-dependence is lost because the Sudakov factor cuts off the $z$ integration before the upper limit, $\ln \frac{Q^2}{k_\perp^2}$, is reached. It is in this upper limit that the $k_\perp$-dependence resides. Generalizing (\ref{e47}) to read
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\mathcal{C}\int_0^{\ln \frac{Q^2}{k_\perp^2}} dz e^{\ln \text{Sud} +\ln T}\label{e54}
\end{equation}
or, using (\ref{e44})
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} =\frac{\mathcal{C}}{b\alpha_s}\int_0^{1} dw e^{\frac{C_F}{\pi b^2 \alpha_s} \left[\ln(1-w)+w\right]}T(w, Q, Y) \label{e55}
\end{equation}
where $b\alpha_s z =w$. Doing the $w$-integration by integrating about the saddle point (\ref{e53}) gives
\begin{equation}
\frac{dN}{d^2b d^2 k_\perp} = \frac{\pi \mathcal{C}}{1+\frac{\pi b (1-\lambda_{\textrm{eff}})}{C_F}}\sqrt{\frac{2}{\alpha_s C_F}}e^{\frac{C_F}{\pi b^2 \alpha_s} \left[\ln(1-w_0)+w_0\right]}T(z_0, Q, Y) \label{e56}
\end{equation}
with $w_0=b\alpha_s z_0$ with $z_0$ given by (\ref{e53}). Eq.~(\ref{e41}) is recovered if one only keeps the quadratic term in $w_0$ in the exponential in (\ref{e56}) and if one takes a scaling solution for $T$.
\subsection{Medium effects}
\label{meq}
In the case of $\tau_q \ll L$, the medium effects, i.e., the multiple scattering and radiative corrections interacting with the nucleus, led to explicit nuclear medium effect summarized in (\ref{equations}) which can directly affect the spectrum and, if $Q^2$ is not too large, compete with Sudakov effects when (\ref{spectrum}) or (\ref{wwquark}) is used to evaluate the spectrum. In the small-$x$ limit such medium effects must be hidden in the $T$ in (\ref{e48}) or (\ref{e49}). The usual way to incorporate multiple scattering effects into $T$ is to use the McLerran-Venugopalan (MV) model
\begin{equation}
T_{\textrm{MV}} (x_\perp) =1-e^{-Q^2_s(\textrm{MV})x_\perp^2/4} \label{mv}
\end{equation}
as the initial condition for the evolution of $T(x_\perp, Y)$ using the Balitsky-Kovchegov (BK) equation. The evolution is done from $Y_0$ to $Y=\ln\frac{1}{x_{Bj}}=\ln s x_\perp^2$ and where $Y_0$ is determined by requiring that the coherence of the dipole, $x_\perp$, be the nuclear length $L=2\sqrt{R^2-b^2}$ with $R$ the nuclear radium and $b$ the impact parameter. One easily determines $Y_0$ is
\begin{equation}
Y_0 =\ln LM
\end{equation}
with $M$ the nucleon mass.
What is missing in the above discussion is evolution in the medium. The $Q^2_s(\textrm{MV})$ in (\ref{mv}) is given by
\begin{equation}
Q^2_s(\textrm{MV}) = \hat q L
\end{equation}
with $\hat q$ given in (\ref{qhatdef}). One can incorporate evolution in the medium, evolution below $Y_0$, simply by replacing $Q^2_s(\textrm{MV})$ in (\ref{mv}) by $Q^2_{s\, \textrm{initial}}=Q_{s \, \textrm{in}}^2$ where
\begin{equation}
Q_{s \, \textrm{in}}^2 =\hat{q}_t L
\end{equation}
with $\hat{q}_t$ given in (\ref{equations}). Now, in contrast to $Q^2_s(\textrm{MV})$, $Q_{s \, \textrm{in}}^2 $ has a strong dependence on the dipole size and additional dependence on medium length due to double logarithmic corrections. Thus the initial condition, at $Y_0$,
\begin{equation}
T_{\textrm{in}} (x_\perp) =1-e^{-Q^2_{s\, \textrm{in}}x_\perp^2/4} \label{in}
\end{equation}
should include evolution in the medium, at least in the fixed coupling limit. In the weak coupling limit $\bar{\alpha}_s \to 0$, $Q_{s \, \textrm{in}}^2 \to Q^2_s(\textrm{MV})$ ($\hat{q}_t \to \hat q$) and the above initial condition reduces to the MV initial condition since all the medium evolution is negligible. In general, we believe (\ref{in}) is an improved initial condition for BK-JIMWLK \cite{Balitsky:1995ub, Kovchegov:1999yj, JalilianMarian:1997gr, Iancu:2001ad} evolution compared to (\ref{mv}).
\section{Conclusion}
\label{con}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=8cm]{qmlarge.pdf}\hspace{1cm}\includegraphics[width=8cm]{qmsmall.pdf}
\end{center}
\caption[*]{Left figure corresponds to large medium case with $\tau_q\equiv 2q_+/Q^2 \ll L$: The red shaded region shows when in-medium radiation contributes to transverse momentum broadening. The cancellation of the double logarithmic contributions from Fig.~\ref{fig:nlo} is indicated. The Sudakov contribution, coming from graph B in Fig.~\ref{fig:nlo}, is shown as the green shaded region in the upper right hand corner. Here the formation time $\tau_l\equiv \frac{2l^+}{l_\perp^2}$. Right figure indicates the shock wave limit with $\tau_q \gg L$: In-medium radiation and multiple scattering, shown in the shaded region, now gives the initial condition, at $\tau_l=L$ or $Y_0=\ln LM$ with $M$ the nucleon mass, for BK evolution. The region where Sudakov suppression effects come from is indicated in the upper right hand part of the figure. Regions where $A+B$ of Fig.~\ref{fig:nlo} and $B+C$ of Fig.~\ref{fig:nlo} cancel are shown. Above the line $\tau_l=L$ and below the line $\tau_l=\tau_q$ is the region of BK evolution. }
\label{qm2}
\end{figure}
Through the one-loop calculation of quark jet production in DIS on a large nucleus by allowing one extra gluon radiation, we integrate over the full phase space of the radiated gluon, and show that the medium induced broadening effects can be separated from the conventional Sudakov effects coming from parton showers in the vacuum in both large medium and shock wave cases, which are summarized in the left and right phase space plots in Fig.~\ref{qm2}, respectively. As shown in Fig.~\ref{qm2}, the transverse momentum broadening due to Sudakov effects and medium induced radiation as well as the small-$x$ evolution (the small-$x$ evolution is absent in the former case since $x_{\textrm{Bj}}$ is taken to be large) come from different regions of the phase space of the radiated gluon.
A similar calculation for the process of producing a gluon jet in DIS with a gluonic current\cite{Kovchegov:1998bi, Mueller:1999wm} can also be performed, which leads to a similar result as the quark jet production considered in this paper. In this case, the color factor of the Sudakov double logarithm is $C_A$ instead of $C_F$, and the medium effects is taken care of by the WW gluon distribution with the possible corresponding small-$x$ evolution\cite{Dominguez:2011gc} and the gluonic $\hat q$ in the initial condition. Based on these examples, we argue that there should be a factorization between the medium $P_T$ broadening effects and the Sudakov effects in general for hard processes, since the Sudakov effects normally arise from vacuum soft-collinear gluon radiation and they only depend on the virtuality of the considered hard processes and the measured transverse momentum.
The separation of Sudakov effects and medium broadening effects allows us to have a more sophisticated framework to compute the medium $P_T$ broadening effects in hard processes especially the dijet or dihadron productions in heavy ion collisions where Sudakov effects are not negligible. We can further use these processes as probes to quantitatively extract the values of $\hat q$ at RHIC and the LHC energies (see e.g., Refs.~\cite{ Mueller:2016gko, Chen:2016vem}).
\begin{acknowledgments}
This work was supported in part by the U.S. Department of Energy under the contracts DE-AC02-05CH11231 and DE-FG02-92ER40699, and by the NSFC under Grant No.~11575070 and No.~11521064. This material is also based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Number DE-SC0004286 (BW). B. W. would like to thank Yuri Kovchegov for useful and informative discussions. Three of the authors, A. H. M, B. W. and B. X., would like to thank Dr. Jian-Wei Qiu and the nuclear theory group at BNL for the hospitality and support during their visit when this work was finalized.
\end{acknowledgments}
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{"url":"http:\/\/mathhelpforum.com\/statistics\/67305-estimate-probability-success-given-k-successes-n-trials.html","text":"# Math Help - Estimate probability of success given k successes in n trials\n\n1. ## Estimate probability of success given k successes in n trials\n\nI'm trying to estimate the probability of success when I'm given data where a certain number of successes (k) occur in a certain number of trials (n).\n\nIOW, instead of trying to figure out the chances of k successes given n trials and a probability (p) (whcih I know how to do). I'm trying to estimate p when given k successes and n trials. An important note is that all potential values for p between 0 and 1 are equally likely.\n\nI can't seem to wrap my head around all of the Bayesian inference stuff I'm reading. I understand exactly what needs to be done involving the integral calculus, I just can't do it. Integration by parts just seems to lead to a never ending process.\n\nAny help?\n\n2. Originally Posted by Capt Vee\nI'm trying to estimate the probability of success when I'm given data where a certain number of successes (k) occur in a certain number of trials (n).\n\nIOW, instead of trying to figure out the chances of k successes given n trials and a probability (p) (whcih I know how to do). I'm trying to estimate p when given k successes and n trials. An important note is that all potential values for p between 0 and 1 are equally likely.\n\nI can't seem to wrap my head around all of the Bayesian inference stuff I'm reading. I understand exactly what needs to be done involving the integral calculus, I just can't do it. Integration by parts just seems to lead to a never ending process.\n\nAny help?\nTry Tabular Integration, near the bottom of this page: Integration by parts - Wikipedia, the free encyclopedia\n\nIt speeds up things a lot!\n\n3. Originally Posted by Capt Vee\nI'm trying to estimate the probability of success when I'm given data where a certain number of successes (k) occur in a certain number of trials (n).\n\nIOW, instead of trying to figure out the chances of k successes given n trials and a probability (p) (whcih I know how to do). I'm trying to estimate p when given k successes and n trials. An important note is that all potential values for p between 0 and 1 are equally likely.\n\nI can't seem to wrap my head around all of the Bayesian inference stuff I'm reading. I understand exactly what needs to be done involving the integral calculus, I just can't do it. Integration by parts just seems to lead to a never ending process.\n\nAny help?\nHi Capt Vee,\n\nI sounds like you want a Binomial proportion confidence interval. If you simply want a method for computing a confidence interval (rather than a derivation), you can find some methods described here:\n\nBinomial proportion confidence interval - Wikipedia, the free encyclopedia\n\n4. If this is a Bernoullit trial, try using the Binomial Model.\nIf it is, let me know. I\"ll elaborate.","date":"2015-03-31 13:01:54","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9027175307273865, \"perplexity\": 272.84269710222975}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-14\/segments\/1427131300578.50\/warc\/CC-MAIN-20150323172140-00249-ip-10-168-14-71.ec2.internal.warc.gz\"}"}
| null | null |
Support is often seen as a costly, necessary evil that often gets in the way of profitability ! Done wrong, it can be exactly that - a time and money sink that no-one really wants to deal with.
However, do it right, and it can be a hugely valuable asset for your business, generating vast quantities of insight, building a solid foundation of loyal customers, and enhancing your brand experience immeasurably.
We have huge experience in the Call Center market. We know what has worked and what hasn't worked. We know the pitfalls, we know the teething pains, we have been there first-hand during the early days of organic-funded environments, through to being part of a market-leading organization famed for it's scale. We have done this for tiny startups through to the world's largest and most influential brands. Throughout all of this, we have gathered invaluable knowledge and experience on what makes things really tick from a customer experience perspective, but also from a call center perspective.
Test Studio - a ring-fenced customer support environment that allows a brand to safely test new methodologies, technologies, processes, data architectures and services. This is a low-risk way for companies to trial initiatives, managed with a proof-of-concept mindset, without the inherent risks of rolling out into a live operational environment. We still deal with real customers, and deal with real issues, but the primary intent of the operation is to prove new initiatives and deliver learning-outcomes prior to full launch.
Launch Studio - a support environment that is geared to help a brand through the launch period of a product or service. Specializing in seeding the digital channels with knowledge, interacting across all relevant channels, working closely with the product teams to generate content, and gearing effectively for an uncertain volume of interactions. This is a short-term environment that can complete the launch activities and then hand over to another environment, or can be extended into a full Support Studio if required.
Support Studio - a self-contained outsourced support team. The focus of the support studio is to deliver outstanding service to the end-customer, in a manner representative of the brand values. Offering an omni-channel approach, the Studio can manage customers using one or more channels such as social media, online services, email, phone, forums and chat. Studios can be configured in a client's premises, a ThreeDotZero facilityor in a distributed home-working model.
In all of the above options, data is a key source of value. We ensure that the activities are underpinned by excellent data management and knowledge management. This is then ordered, interpreted, and used to communicate vital information back into the client on a continual basis.
Customer Support environments are a gold mine for interesting, important and invaluable information about your customers, your products and how your business is performing. Too often, organizations are blind to that fact, and do not tap into the rich data that is being generated on a daily basis from those environments.
Data is wasted, ignored, or just misunderstood. This data is known as Dark Data and is the key to generating insights that can lead to company-wide performance improvements, more certainty on investment or operational decisions, and more often than not, information about product feature enhancements.
At 3DZ we really love to throw a light on this Dark Data and can help companies understand more about what is being generated, what is not being generated, and what the data is telling you.
|
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| 8,640
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Name | Type | Description | Notes
------------ | ------------- | ------------- | -------------
**replacement** | **String** | Sequence of characters to replace words within a transcript when reported by a detector | [optional]
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 4,418
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Amphipterygidae är en familj av trollsländor. Amphipterygidae ingår i överfamiljen Calopterygoidea, ordningen trollsländor, klassen egentliga insekter, fylumet leddjur och riket djur. Enligt Catalogue of Life omfattar familjen Amphipterygidae 11 arter.
Kladogram enligt Catalogue of Life:
Källor
Externa länkar
Trollsländor
Amphipterygidae
|
{
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| 4,872
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{"url":"https:\/\/www.ques10.com\/p\/34424\/a-4-stoke-cycle-ci-engine-develops-11kw-per-cylind\/","text":"Question: A 4-stoke cycle C.I. engine develops 11KW per cylinder while running at 1800 rpm and using fuel oil of $32^o$ API.\n0\n\nA 4-stoke cycle C.I. engine develops 11KW per cylinder while running at 1800 rpm and using fuel oil of 32\u00b0 API. Fuel injection occupies 32\u00b0C of crank travel and takes place through a fuel injections orifice 0.47 mm diameter with a flow coefficient of 0.9. Fuel is injection at a pressure of 1118.2 bar in to combustion chamber where the pressure is 31.38 bar. Estimate the quantity of fuel injected in kg\/Kwh.\n\nSubject : IC Engines\n\nTopic: Chap 3: Compression Ignition Engines\n\nDifficulty: Medium\n\nmumbai university ic(65) \u2022 158 views\n modified 6 weeks ago by Ankit Pandey \u2022 70 written 11 months ago by\n0\n\nP = 11 kW\n\nN = 1800 rpm\n\nD = 0.47 mm\n\n$\\Theta$ = 32\u00b0\n\n$P_{i}$ = 118.2\n\n$P_{c}$ = 31.38\n\n$C_{df}$ = 0.9\n\nSpecific gravity of fuel,\n\nDuration of Injection\n\nVelocity of fuel,\n\nVolume of fuel Injected\/Cycle,\n\nNozzle orifice area,","date":"2019-10-18 07:33:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.36908262968063354, \"perplexity\": 14216.61570394183}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986677964.40\/warc\/CC-MAIN-20191018055014-20191018082514-00074.warc.gz\"}"}
| null | null |
You haven't bookmarked any ski area.
Unfortunately only 3 skiareas can be bookmarked. Please remove one.
Always up to date. On this page, you'll find all the information you need about snow conditions at Val Senales (Schnalstal) Ski Area. Our webcams give you an up-to-the-minute look at the actual weather conditions of the ski area any time you like.
High temperatures between 16° and 25°.
Click on "remember ski area" to get all important information about your bookmarked areas.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 7,354
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Biba 2012: Cooper Gay appoints head of risk management
By Newsdesk2012-05-17T10:15:00+01:00
Dalton steps up from claims director role
Cooper Gay & Co has announced the appoitment of Peter Dalton as head of risk management, with immediate effect.
Dalton started his insurance career in 1987, joining the Cooper Gay marine claims team in 2001. He was appointed claims director in 2006. In his new role, he will work alongside the existing CGSC Group's risk management team, with responsibility for developing the existing framework and strategies.
Cooper Gay chief executive Sam Hovey said: "Having worked with Peter for a number of years, it is true to say he is widely recognised for his insurance and operational knowledge, and I am delighted that he has accepted the new position.
"His leadership of the claims department proved to us that Peter is capable of implementing new strategies and initiatives, both in the UK and globally; expanding his responsibilities was the natural progression."
BIBA News
Cooper Gay hires Gordon Newman as chief exec
Newman replaces Sam Hovey who has resigned from the company
Lightyear Capital is frontrunner to buy Cooper Gay stake
Broker seeking private equity investment to expand
Cooper Gay returns to profit in 2011
Lloyd's broker made £4.5m profit after tax, compared with 2010's £381,000 loss
|
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\section{Introduction}\label{sec:intro}
Let $\alpha>0$ and consider the bilinear Bochner-Riesz mean defined by
\begin{eqnarray*}\label{def:bbr}\mathcal B^{\alpha}_R(f,g)(x)=\int_{{\mathbb {R}}^{n}}\int_{{\mathbb {R}}^{n}}\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)^{\alpha}_{+}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi ix\cdot(\xi+\eta)}d\xi d\eta,
\end{eqnarray*}
where $R>0$ and $f,g\in \mathcal{S}({\mathbb {R}}^n), n\geq 1$. Here $\hat f$ denotes the Fourier transform of $f$ given by $\hat{f}(\xi)=\int_{{\mathbb {R}}^n}f(x)e^{-2\pi ix.\xi}dx$ and $\mathcal{S}({\mathbb {R}}^n)$ denotes the space of Schwartz class functions.
We refer to ~\cite{BGSY, JLV, JS, JSK, LW} for the study of $L^p$ boundedness properties of the bilinear Bochner-Riesz means. In this paper we are concerned with the maximal and square functions associated with the bilinear Bochner-Riesz means. The maximal function associated with the bilinear Bochner-Riesz means $\mathcal B^{\alpha}_R(f,g)(x)$ is defined by
$$\mathcal B^{\alpha}_*(f,g)(x)=\sup_{R>0} |\mathcal B^{\alpha}_R(f,g)(x)|.$$
The maximal function $\mathcal B^{\alpha}_*(f,g)(x)$ plays a key role in addressing the issue of almost everywhere convergence of the bilinear Bochner-Riesz means $\mathcal B_R^{\alpha}(f,g)(x)$ as $R\rightarrow \infty$. We refer to~\cite{JL, JS} for recent results on $L^p$ boundedness of the maximal function $\mathcal B^{\alpha}_*(f,g)$ for a wide range of $\alpha$ and exponents $p_1,p_2, p$.
The bilinear analogue of Stein's square function for Bochner-Riesz means is recently introduced and studied in~\cite{CKSS}. This is defined by
\begin{eqnarray*}
\mathcal G^{\alpha}(f,g)(x)&:=&\left(\int_0^{\infty}|\frac{\partial}{\partial R}\mathcal{B}_R^{\alpha+1}(f,g)(x)|^2 RdR \right)^{1/2}\\
&=&\left(\int_{0}^{\infty}|\mathcal K^{\alpha}_R\ast (f\otimes g)(x,x)|^2\frac{dR}{R}\right)^{1/2},
\end{eqnarray*}
where $\widehat{\mathcal K^{\alpha}_R}(\xi,\eta)=2(\alpha+1)\frac{|\xi|^2+|\eta|^2}{R^2}
\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)^{\alpha}_+$ and $\otimes$ denotes the tensor product.
Note that in the spatial variables the kernel (in the sense of vector-valued operators) of $\mathcal G^{\alpha}$ is given by
$${\mathcal K^{\alpha}_R}(y_1,y_2) = c_{n+\alpha}R^{2n-2}\Delta\left(\frac{ J_{\alpha+n} (|(Ry_1,Ry_2)|)} {|(Ry_1,Ry_2)|^{\alpha+n}}\right),~y_1,y_2\in {\mathbb {R}}^n.$$
Here $J_{\alpha+n}$ denotes the Bessel function of order $\alpha+n$.
The index $\alpha=n-\frac{1}{2}$ is called the critical index for the bilinear Bochner-Riesz problem. Motivated by the problem of linear Bochner-Riesz means at the critical index (which is $\frac{n-1}{2}$ for the linear case) and recent developments in the direction of bilinear Bochner-Riesz problem, see for example \cite{JS,JL,CKSS}, in this paper we investigate weighted boundedness of $\mathcal B^{n-\frac{1}{2}}_*$ and $\mathcal G^{n-\frac{1}{2}}$. Also, the issue of end-point boundedness for both the operators is addressed. We invite the reader to \cite{Christ,St,SW,ana} for results on the linear Bochner-Riesz problem at critical index.
We need to introduce some notation in order to state the results. Let $1\leqslant p_{1}, p_{2}< \infty$ and $p$ be such that $
\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$
\begin{definition}\cite[Definition 3.5]{Ler1} (Bilinear weights)
Let $\vec{p}=(p_{1},p_{2})$. For a given pair of weights $\vec{w}=(w_{1},w_{2})$, set $
v_w:=\prod_{i=1}^{2} w_{i}^{p/p_{i}}$.
We say that $\vec{w}\in A_{\vec{P}}$ if
\begin{equation*}
[\vec{w}]_{A_{\vec{P}}}:= \sup_{Q} \Big(\frac{1}{|Q|}\int_{Q}v_w\, dx \Big) \prod_{j=1}^{2}\Big(\frac{1}{|Q|}\int_{Q} w_{j}^{1-p'_{j}} \,dx\Big) ^{p/{p_{j}^\prime}} <\infty.
\end{equation*}
When $p_j=1$, $\big(\frac{1}{|Q|}\int_Qw_j^{1-p'_j}\big)^{1/p'_j}$ is understood as $(\inf_Q w_j)^{-1}$. Here $Q$ denotes a cube in ${\mathbb {R}}^n$ with sides parallel to coordinate axes. The quantity $[\vec{w}]_{A_{\vec{P}}}$ is referred to as the bilinear $A_{\vec{P}}$ characteristic of the bilinear weight
$\vec{w}$.
\end{definition}
The following are the main results of this paper.
\begin{theorem} \label{mainthm} Let $T=\mathcal {B}_*^{n-\frac{1}{2}}~~\text{or}~\mathcal G^{n-\frac{1}{2}}$. Then $T$ is bounded from $L^{p_{1}}(\omega_{1})\times L^{p_{2}}(\omega_{2})\rightarrow L^{p}(v_{\omega})$ for all bilinear weights $\vec{\omega}\in A_{\vec{P}}$ with $1<p_{1}, p_{2} \leq \infty$ and $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}$.
\end{theorem}
Further, we show that both the operators fail to satisfy weak-type estimates at the end-point $(1,1,\frac{1}{2})$.
\begin{proposition}\label{endpoint:sf}
The bilinear square function $\mathcal G^{n-\frac{1}{2}}$ is unbounded from $L^1({\mathbb {R}}^n)\times L^1({\mathbb {R}}^n)$ to $L^{\frac{1}{2},\infty}({\mathbb {R}}^n), n\geq 1$.
\end{proposition}
In the case of maximal function we get a stronger result at the end-point $(1,1,\frac{1}{2})$. First, observe that in view of bilinear transference principle it is sufficient to work with the operator defined for functions on the unit cube $Q_n=[-\frac{1}{2},\frac{1}{2})^n$. Let us use the same notation to denote the operator in the periodic case as in the case of ${\mathbb {R}}^n$. We have the following.
\begin{theorem}\label{div}
Let $n\geq 1$. There exists an integrable function $f$ on $Q_n$ and a positive measure set $E$ of $Q_n$ such that
$$\limsup_{R\to\infty}|\mathcal B^{{n-\frac{1}{2}}}_R(f,f)(x)|=\infty$$
for almost every $x\in E$.
\end{theorem}
In particular, we get that $\mathcal B_*^{n-\frac{1}{2}}$ is unbounded from $L^1({\mathbb {R}}^n)\times L^1({\mathbb {R}}^n)$ to $L^{\frac{1}{2},\infty}({\mathbb {R}}^n), n\geq 1$. These results give us a complete picture of $L^p$ boundedness properties of operators $\mathcal B_*^{n-\frac{1}{2}}$ and $\mathcal G^{n-\frac{1}{2}}$.
\subsection*{Organization of the paper} In Section~\ref{sec:mfexam} we prove Theorem~\ref{div}. The proof of Theorem~\ref{mainthm} establishing the weighted estimates for $\mathcal B^{n-\frac{1}{2}}_*$ and $\mathcal G^{n-\frac{1}{2}}$ is presented in~sections~\ref{max:sec} and \ref{sec:sqr} respectively. The issue of end-point isse for $\mathcal G^{n-\frac{1}{2}}$ is discussed in Section~\ref{sec:sqrexam}.
\section{End-point estimates for the maximal function \texorpdfstring{$\mathcal B^{n-\frac{1}{2}}_*$}{B}}\label{sec:mfexam}
In this section we prove Theorem~\ref{div}. We will make use of the ideas presented in [\cite{Bochner}, \cite{SW} page 267] where the corresponding result is proved for the maximal Bochner-Riesz function in the linear case. We exploit their method and make suitable modifications to it to address the bilinear problem.
The main idea is to estimate the maximal function acting on $L^1$ functions which peak at the origin. This is verified for the Dirac mass first. More precisely, first we show that $\mathcal B^{n-\frac{1}{2}}_R(\delta_0,\delta_0)$, as $R\rightarrow \infty$ becomes unbounded for almost all $x\in Q_n$, where $\delta_0$ is the Dirac mass at the origin. This is proved in Lemma~\ref{Lemma:Dirac}. Later, we complete the proof in two steps. In the first step, with the help of the estimate~\eqref{Lemma:Dirac}, we replace one of the Dirac masses by a suitable $L^1$ function on $Q_n$. This step is then used to replace the other Dirac mass by the same $L^1$ function to achieve the desired result.
\begin{lemma}\label{Lemma:Dirac}
Let $\delta_0$ be the Dirac mass at the origin in ${\mathbb {R}}^n, n\geq 1$. Then
\begin{eqnarray}\label{Dirac} \limsup_{R\to\infty}|\mathcal B^{n-\frac{1}{2}}_R(\delta_0,\delta_0)(x)|=\infty~
\end{eqnarray}
for almost every $x\in Q_n$.
\end{lemma}
\begin{proof}
Consider the set
$$S =\{x \in {\mathbb {R}}^{n}: \{1\}\cup\{|(x,x)-m|: m\in{\mathbb Z}^{2n}\}\text{ is linearly independent over rationals}~ {\mathbb Q}\}.$$
Note that the complement of $S$ is a set of measure zero in ${\mathbb {R}}^n$. Let $x_0\in(Q_n\setminus\{0\})\cap S$. Let $K_R^{\alpha}(x,y)$ denote the kernel of the bilinear Bochner-Riesz operator $\mathcal B^{\alpha}$, i.e., $\hat{K}_R^{\alpha}(\xi,\eta)=(1-\frac{|m_1|^2+|m_2|^2}{R^2})^{\alpha}_{+}$, $(m_1,m_2)\in {\mathbb Z}^{2n}$. By the Poisson summation formula, we get
\begin{equation}\label{kernel}
K_R^\alpha((x_0,x_0))=C_\alpha R^{2n}\sum_{m\in{\mathbb Z}^{2n}}\frac{J_{n+\alpha}(2\pi R|(x_0,x_0)-m|)}{(R|(x_0,x_0)-m|)^{n+\alpha}}.
\end{equation}
Recall the asymptotics for the Bessel functions
\begin{eqnarray*}
J_{n+\alpha}(2\pi R|(x_0,x_0)-m|)&=&\frac{e^{2\pi iR|(x_0,x_0)-m|}e^{-i(\frac{\pi}{2}(n+\alpha)+\frac{\pi}{4})}+e^{-2\pi iR|(x_0,x_0)-m|}e^{i(\frac{\pi}{2}(n+\alpha)+\frac{\pi}{4})}}{\pi\sqrt{R|(x_0,x_0)-m|}}\\
&&+~O((R|(x_0,x_0)-m|)^{-\frac{3}{2}}).
\end{eqnarray*}
Note that the infinite series in~\eqref{kernel} converges absolutely for $\alpha>n-\frac{1}{2}$. Recall that we are concerned with the estimates when $\alpha=n-\frac{1}{2}$, but we cannot take $\alpha\to n-\frac{1}{2}$ in the equation above. However, by taking an average over the parameter $R$, we can take $\alpha\to n-\frac{1}{2}$ and get the following estimate at $\alpha=n-\frac{1}{2}$.
\begin{eqnarray*}
\frac{1}{T}\int_1^T K_R^{n-\frac{1}{2}}((x_0,x_0))e^{2\pi i\lambda R}dR&=& C_n \sum_{m\in{\mathbb Z}^{2n}}\frac{e^{-i(\frac{\pi}{2}(2n-\frac{1}{2})+\frac{\pi}{4})}}{|(x_0,x_0)-m|^{2n}}\left(\frac{1}{T}\int_1^T e^{2\pi iR(\lambda+|(x_0,x_0)-m|)}dR\right) \\
&&+~ C_n \sum_{m\in{\mathbb Z}^{2n}}\frac{e^{i(\frac{\pi}{2}(2n-\frac{1}{2})+\frac{\pi}{4})}}{|(x_0,x_0)-m|^{2n}}\left(\frac{1}{T}\int_1^T e^{-2\pi iR(\lambda-|(x_0,x_0)-m|)}dR\right)\\
&&+~ C_n \sum_{m\in{\mathbb Z}^{2n}}O\left(\frac{1}{|(x_0,x_0)-m|^{2n+1}}\right)\frac{1}{T}\int_1^T \frac{dR}{R}.
\end{eqnarray*}
In the equation above note that if $\lambda\neq\pm|(x_0,x_0)-m|, m\in{\mathbb Z}^{2n}$, then all the terms on the right hand side vanish as $T\rightarrow \infty$. Moreover, if $\lambda=\pm|(x_0,x_0)-m|$ for some $m\in{\mathbb Z}^{2n}$, the right hand side converges (as $T\rightarrow \infty$) to
$$C_n \frac{e^{\pm in\pi}}{|(x_0,x_0)-m|^{2n}}.$$
Consider the set $\Lambda_{x_0}=\{|(x_0,x_0)-m|:m\in{\mathbb Z}^{2n}\}$ and enumerate it as $\Lambda_{x_0}=\{\lambda_1,\lambda_2,\lambda_3,...\},$
where $\lambda_1<\lambda_2<\lambda_3<...$ and $\sum\limits_{j=1}^\infty\frac{1}{\lambda_j^{2n}}=\infty$. With this choice and notation we have that
\begin{align*}
\lim_{T\to\infty}\frac{1}{T}\int_1^T K_R^{n-\frac{1}{2}}((x_0,x_0))e^{2\pi i\lambda R}dR=\left\{
\begin{array}{ll}
C_n \frac{e^{in\pi}}{\lambda_j^{2n}} & \mbox{if } \lambda=\lambda_j \\
0 & \mbox{if } \lambda\neq\pm\lambda_j \\
C_n \frac{e^{-in\pi}}{\lambda_j^{2n}} & \mbox{if } \lambda=-\lambda_j
\end{array}
\right.
\end{align*}
Since $(x_0,x_0)\in S$, the set $\{1\}\cup\{\lambda_1,\lambda_2,\lambda_3,...\}$ is linearly independent over the rationals $\mathbb Q$ and hence no expression of the form $\pm\lambda_{j_1},\pm\lambda_{j_2},...,\pm\lambda_{j_s}$ can be equal to an integer. This gives us that
$$\lim_{T\to\infty}\frac{1}{T}\int_1^T K_R^{n-\frac{1}{2}}((x_0,x_0))\prod_{j=1}^N \left[1+\frac{e^{-in\pi}e^{2\pi i\lambda_j R}+e^{in\pi}e^{-2\pi i\lambda_j R}}{2}\right]dR=C_n\sum_{j=1}^N\frac{1}{\lambda_j^{2n}}$$
Note that the assumption
$$\sup_{R\geq1}|K_R^{n-\frac{1}{2}}((x_0,x_0))|\leq A_{x_0}<\infty$$
will yield
\begin{eqnarray*}
C_n\sum_{j=1}^N\frac{1}{\lambda_j^{2n}}&=& \lim_{T\to\infty}\frac{1}{T}\int_1^T K_R^{n-\frac{1}{2}}((x_0,x_0))\prod_{j=1}^N\left[1+\frac{e^{-in\pi}e^{2\pi i\lambda_j R}+e^{in\pi}e^{-2\pi i\lambda_j R}}{2}\right]dR\\
&\leq& A_{x_0} \frac{1}{T}\int_1^T\prod_{j=1}^N\left[1+\frac{e^{-in\pi}e^{2\pi i\lambda_j R}+e^{in\pi}e^{-2\pi i\lambda_j R}}{2}\right]dR=A_{x_0}
\end{eqnarray*}
This contradicts the choice that $\sum\limits_{j=1}^\infty\frac{1}{\lambda_j^{2n}}=\infty$. Therefore, for $x\in S\cap Q_n$ we get that
$$\sup_{R\geq1}|K_R^{n-\frac{1}{2}}(x,x)|=\infty.$$
This completes the proof of Lemma~\ref{Lemma:Dirac}.
\end{proof}
Next, we show that Dirac masses in Lemma~\ref{Lemma:Dirac} can be replaced with suitable $L^1$ functions. This part is done in two steps as follows.
\subsection*{Step I:}
In this step we will show that in the estimate \eqref{Dirac} we can replace one of the Dirac masses with an $L^1$ function so that the estimate holds on a set of positive measure.
Let $\Phi\in\mathcal{S}({\mathbb {R}}^n)$ be a radial function such that $\hat{\Phi}$ is non-negative and supported in the unit ball of ${\mathbb {R}}^n$ with $\int_{{\mathbb {R}}^n}\hat{\Phi}(\xi)d\xi=1$. Given $\epsilon>0$ define
$$\phi_\epsilon(x)=\frac{1}{\epsilon^n}\sum_{m\in{\mathbb Z}^n}\hat{\Phi}\left(\frac{x+m}{\epsilon}\right).$$
The Poisson summation formula yields
$$\phi_\epsilon(x)=\frac{1}{\epsilon^n}\sum_{m\in{\mathbb Z}^n}\hat{\Phi}(\frac{x+m}{\epsilon})=\sum_{m\in{\mathbb Z}^n}{\Phi}(\epsilon m)e^{2\pi im\cdot x}.$$
Since $\Phi\in\mathcal{S}({\mathbb {R}}^n)$ we get that
$$\sum_{m\in{\mathbb Z}^n}|{\Phi}(\epsilon m)|\leq\sum_{m\in{\mathbb Z}^n}\frac{C'_n}{(1+\epsilon|m|)^{n+1}}\leq \frac{C_n}{\epsilon^{n}}.$$
Recall that the linear Bochner-Riesz means of order $n-\frac{1}{2}$ acting on Dirac mass is given by
\begin{eqnarray*}
B^{n-\frac{1}{2}}_R(\delta_0)(x)&=&\sum_{|m|\leq R}\left(1-\frac{|m|^2}{R^2}\right)^{n-\frac{1}{2}}e^{2\pi ix\cdot m}\\
&=&c_n \sum_{m\in{\mathbb Z}^n}R^n\frac{J_{\frac{3n-1}{2}}(2\pi R|x-m|)}{(R|x-m|)^{\frac{3n-1}{2}}}
\end{eqnarray*}
Observe that if $R\leq 10$ we have $|B^{n-\frac{1}{2}}_R(\delta_0)(x)|\leq C_1$ for all $x\in Q_n$. When $R>10$ and $|x|\geq\frac{1}{10}$, we have $R|x-m|\geq1$ for all $m\in{\mathbb Z}^n$. Therefore, using asymptotics of Bessel function for $x\in E=[\frac{1}{10},\frac{1}{2})\cup [-\frac{1}{2},\frac{1}{10}]$ we get that
\begin{eqnarray*}
\sup_{x\in E}\sup_{R>10}|B^{n-\frac{1}{2}}_R(\delta_0)(x)|&\leq&c_n\sup_{x\in E}\sup_{R>10}\sum_{m\in{\mathbb Z}^n}\frac{R^n\cos(2\pi R|x-m|-\frac{3n\pi}{2})}{(R|x-m|)^{\frac{3n}{2}}}
\leq C_2
\end{eqnarray*}
Let $C=\max\{C_1,C_2\}$.
Next, we use an inductive argument to construct measurable subsets $E_j\subset E$ with $|E_j|\geq\frac{4}{5}-\frac{1}{j}$, an increasing sequence ${R_j}$ and two positive null sequences $\epsilon_j\leq\delta_j,~j\geq1$ such that
\begin{equation}\label{toprove}
\sup_{R\leq R_j}|\mathcal B^{{n-\frac{1}{2}}}_R(f,\delta_0)(x)|\geq j \text{\hspace{10mm} for all $x\in E_j$},
\end{equation}
where $f=\sum\limits_{s=1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s})\in L^1({\mathbb {R}}^n)$.
Observe that the desired property holds trivially with the initial choice of $E_1=\emptyset$, $R_1=1,$ and $\epsilon_1=\delta_1=1$. Next, suppose that we have chosen $E_j,R_j,\epsilon_j,\delta_j$ satisfying \eqref{toprove} for all $1\leq j\leq k-1$. We need to construct $E_k,R_k,\epsilon_k$ and $\delta_k$ so that \eqref{toprove} holds for $j=k$. We will choose $\delta_k$ first. Let $B$ be a constant such that
$$|\Phi(x)-\Phi(y)|\leq B|x-y|,~x,y\in{\mathbb {R}}^n.$$
Choose $\delta_k>0$ such that
\begin{equation*}\label{lipschitz}
B\delta_k \sum_{|(m_1,m_2)|\leq R_{k-1}}|m_1|\leq 1.
\end{equation*}
Write $A_k=CC_n\left(2^{-k}\delta_k^{-n}+\sum\limits_{s=1}^{k-1}2^{-s}(\epsilon_s^{-n}+\delta_s^{-n})\right)$. Consider
\begin{eqnarray*}
&&\mathcal B^{{n-\frac{1}{2}}}_R\left(-2^{-k}\phi_{\delta_k}+\sum_{j=1}^{k-1}2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)\\
&=&\nonumber\sum_{|(m_1,m_2)|\leq R}\left(1-\frac{|m_1|^2+|m_2|^2}{R^2}\right)^{{n-\frac{1}{2}}}\left(-2^{-k}\phi_{\delta_k}+\sum_{s=1}^{k-1}2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s})\right)^{\widehat{}}(m_1)e^{2\pi ix\cdot(m_1+m_2)}\\
&=&\nonumber\sum_{|m_1|\leq R}\left(1-\frac{|m_1|^2}{R^2}\right)^{{n-\frac{1}{2}}}\left(-2^{-k}\Phi(\delta_k m_1)+\sum_{s=1}^{k-1}2^{-s}(\Phi(\epsilon_s m_1)-\Phi(\delta_s m_1))\right)e^{2\pi ix\cdot m_1}\\
&&\nonumber\sum_{|m_2|\leq\sqrt{R^2-|m_1|^2}}\left(1-\frac{|m_2|^2}{R^2-|m_1|^2}\right)^{{n-\frac{1}{2}}}e^{2\pi ix\cdot m_2}.
\end{eqnarray*}
We make a crude estimate for the terms above in the following way.
\begin{eqnarray}\label{smaller}
&&\sup_{x\in E}\sup_{R>0}|\mathcal B^{{n-\frac{1}{2}}}_R\left(-2^{-k}\phi_{\delta_k}+\sum_{s=1}^{k-1}2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)|\\
&\leq& C \nonumber\sup_{R>0}\sum_{|m_1|\leq R}\left(2^{-k}|\Phi(\delta_k m_1)|+\sum_{s=1}^{k-1}2^{-s}(|\Phi(\epsilon_s m_1)|+|\Phi(\delta_s m_1)|)\right)\\
&\leq&\nonumber C\sum_{m_1\in{\mathbb Z}^n}\left(2^{-k}|\Phi(\delta_k m_1)|+\sum_{s=1}^{k-1}2^{-s}(|\Phi(\epsilon_s m_1)|+|\Phi(\delta_s m_1)|)\right)\\
&\leq&\nonumber CC_n\left(2^{-k}\delta_k^{-n}+\sum_{s=1}^{k-1}2^{-s}(\epsilon_s^{-n}+\delta_s^{-n})\right)= A_k
\end{eqnarray}
Using Fatou's lemma and the estimate proved in Lemma \ref{Lemma:Dirac}, we get that
$$\liminf_{N\to\infty}\left|\left\{x\in E:\sup_{0<R\leq N} |\mathcal B^\alpha_R(\delta_0,\delta_0)(x)|>2^k(A_k+k+2)\right\}\right|=\frac{4}{5},$$
Thus, there exists an $R_k>R_{k-1}$ such that the set
$$E_k=\left\{x\in E:\sup_{0<R\leq R_k} |\mathcal B^\alpha_R(2^{-k}\delta_0,\delta_0)(x)|>A_k+k+2\right\}$$
has measure at least $\frac{4}{5}-\frac{1}{k}$.
Next, we choose $0<\epsilon_k\leq \delta_k$ so that
\begin{eqnarray*}
&&\sup_{x\in Q_n}\sup_{R\leq R_k}|\mathcal B^{{n-\frac{1}{2}}}_R(2^{-k}\delta_0,\delta_0)(x)-\mathcal B^{{n-\frac{1}{2}}}_R(2^{-k}\phi_{\epsilon_k},\delta_0)(x)|\\
&\leq& \sum_{|(m_1,m_2)|\leq R_k}2^{-k}\left(1-\frac{|m_1|^2+|m_2|^2}{R_k^2}\right)^{{n-\frac{1}{2}}}|1-\Phi(\epsilon_k m_1)|\leq 1
\end{eqnarray*}
Note that such a choice of $\epsilon_k$ is possible for a fixed $R_k$ because $|1-\Phi(\epsilon m_1)|\to 0$ as $\epsilon\to 0$. Therefore, we have
\begin{equation}\label{equal}
\inf_{x\in E_k}\sup_{R\leq R_k} |\mathcal B^{n-\frac{1}{2}}_R(2^{-k}\phi_{\epsilon_k},\delta_0)(x)|\geq A_k+k+1
\end{equation}
The choice of $\delta_k$ allows us to deduce the following estimate
\begin{eqnarray}\label{greater}
&& \sup_{x\in Q_n}\sup_{R\leq R_k} \left|\mathcal B^{n-\frac{1}{2}}_R\left(\sum_{s=k+1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)\right|\\
&\leq&\nonumber \sum_{|(m_1,m_2)|\leq R_k}\left[\sum_{s=k+1}^\infty2^{-s}|\Phi(\epsilon_s m_1)-\Phi(\delta_s m_1)|\right]\\
&\leq& \nonumber \sum_{|(m_1,m_2)|\leq R_k}\left[\sum_{s=k+1}^\infty2^{-s}B|\delta_s-\epsilon_s||m_1|\right]\\\nonumber
&\leq& B\delta_{k+1} \sum_{|(m_1,m_2)|\leq R_k}|m_1|\leq 1.
\end{eqnarray}
Now for $j=k$ we have
\begin{eqnarray*}
\mathcal B^{{n-\frac{1}{2}}}_R\left(\sum_{s=1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)&=&\mathcal B^{{n-\frac{1}{2}}}_R\left(-2^{-k}\phi_{\delta_k}+\sum_{s=1}^{k-1}2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)\\
&&+~ \mathcal B^{n-\frac{1}{2}}_R(2^{-k}\phi_{\epsilon_k},\delta_0)(x)+\mathcal B^{n-\frac{1}{2}}_R\left(\sum_{s=k+1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)
\end{eqnarray*}
Using the estimates \eqref{smaller}, \eqref{equal} and \eqref{greater} for $x\in E_k$, we get that
\begin{eqnarray*}
\sup_{R\leq R_k}|\mathcal B^{{n-\frac{1}{2}}}_R\left(\sum_{s=1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)|&\geq&\sup_{R\leq R_k}|\mathcal B^{{n-\frac{1}{2}}}_R(2^{-k}\phi_{\epsilon_k},\delta_0)(x)|\\
&&-~\sup_{R\leq R_k}|\mathcal B^{{n-\frac{1}{2}}}_R\left(-2^{-k}\phi_{\delta_k}+\sum_{s=1}^{k-1}2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)|\\
&&-~\sup_{R\leq R_k}|\mathcal B^{n-\frac{1}{2}}_R\left(\sum_{s=k+1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s}),\delta_0\right)(x)|\\
&\geq&k
\end{eqnarray*}
Denote $f=\sum_{s=1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s})$ and observe that we have $\sup\limits_{R>0}|\mathcal B^{{n-\frac{1}{2}}}_R(f,\delta_0)(x)|\geq k$ for all $x\in\cup_{r\geq k}E_k$. Therefore,
\begin{equation}\label{one1}
\sup_{R>0}|\mathcal B^{n-\frac{1}{2}}_R(f,\delta_0)(x)|=\infty
\end{equation}
for all $x\in E=\cap_{r\geq1}\cup_{r\geq k}E_k$. Note that $|E|=\frac{4}{5}$.
\subsection*{Step II:}
In this step we replace the Dirac mass in the second place by $f$ as constructed in the previous step. We need to make minor modifications to the arguments used in the previous step. We provide essential details here for a self contained proof. We will use the same notation as in the previous step. However, the parameters may differ from the previous step.
Let $M$ denote the classical Hardy-Littlewood maximal function defined by
$$Mf(x):=\sup_{t>0}\frac{1}{|B(x,t)|}\int_{B(x,t)} |f(y)|dy,$$
where $B(x,t)$ is the euclidean ball of radius $t$ and center $x$.
Since $f\in L^1(Q_n)$, we know that $M(f)(x)$ is finite a.e. $x\in Q_n$ and there holds weak-type $(1,1)$ estimate
$$|\{x\in E:|M(f)(x)|>N\}|\leq \frac{c_n}{N}\|f\|_1. $$
Choose $N$ large enough that $|\{x\in E:|M(f)(x)|>N\}|<\frac{1}{5}$. Let $F=\{x\in E:|M(f)(x)|\leq N\}$. Then $|F|\geq \frac{3}{5}$.
For $j\geq1$, we will construct measurable subsets $F_j\subset F$ such that $|F_j|\geq\frac{3}{5}-\frac{1}{j}$, an increasing sequence ${R_j}$ and two positive null sequences $\epsilon_j\leq\delta_j$ such that
\begin{equation}\label{toprove1}
\sup_{R\leq R_j}|\mathcal B^{{n-\frac{1}{2}}}_R\left(f,\sum_{s=1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s})\right)(x)|\geq j \text{\hspace{10mm} for all $x\in F_j$}.
\end{equation}
As previously, we begin with $F_1=\emptyset, R_1=1,$ and $\epsilon_1=\delta_1=1$. Suppose we have chosen $F_j,R_j,\epsilon_s,\delta_s$ for all $1\leq j\leq k-1$ satisfying \eqref{toprove1}. Choose $\delta_k>0$ small enough so that
\begin{equation*}\label{lipschitz1}
B\|f\|_1\delta_k \sum_{|(m_1,m_2)|\leq R_{k-1}}|m_2|\leq 1.
\end{equation*}
Denote $B_k=NC_n\left(2^{-k}\delta_k^{-n}+\sum_{s=1}^{k-1}2^{-s}(\epsilon_s^{-n}+\delta_s^{-n})\right)$ and as in the previous step we get that
\begin{eqnarray}\label{smaller1}
&& \nonumber \sup_{x\in F}\sup_{R>0}|\mathcal B^{{n-\frac{1}{2}}}_R\left(f,-2^{-k}\phi_{\delta_k}+\sum_{s=1}^{k-1}2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s})\right)(x)|\\
&\leq&\nonumber N~\sup_{R>0}\sum_{|m_2|\leq R}\left(2^{-k}|\Phi(\delta_k m_2)|+\sum_{s=1}^{k-1}2^{-s}(|\Phi(\epsilon_s m_2)|+|\Phi(\delta_s m_2)|)\right)\\
&\leq&\nonumber N\sum_{m_2\in{\mathbb Z}^n}\left(2^{-k}|\Phi(\delta_k m_2)|+\sum_{s=1}^{k-1}2^{-s}(|\Phi(\epsilon_s m_2)|+|\Phi(\delta_s m_2)|)\right)\\
&\leq&\nonumber NC_n\left(2^{-k}\delta_k^{-n}+\sum_{s=1}^{k-1}2^{-s}(\epsilon_s^{-n}+\delta_s^{-n})\right)\\
&=& B_k.
\end{eqnarray}
Using Fatou's lemma and the estimate \eqref{one1}, we have
$$\liminf_{N\to\infty}\left|\left\{x\in F:\sup_{0<R\leq N} |\mathcal B^\alpha_R(f,\delta_0)(x)|>2^k(B_k+k+2)\right\}\right|=\frac{3}{5},$$
Choose $R_k>R_{k-1}$ such that the set
$F_k=\left\{x\in F:\sup\limits_{0<R\leq R_k} |\mathcal B^\alpha_R(f,2^{-k}\delta_0)(x)|>B_k+k+2\right\}$
has measure at least $\frac{3}{5}-\frac{1}{k}$. Next, we choose $\epsilon_k\leq \delta_k$ so that
\begin{eqnarray*}
&&\sup_{x\in F}\sup_{R\leq R_k}|\mathcal B^{{n-\frac{1}{2}}}_R(f,2^{-k}\delta_0)(x)-\mathcal B^{{n-\frac{1}{2}}}_R(f,2^{-k}\phi_{\epsilon_k})(x)|\\
&\leq& \sum_{|(m_1,m_2)|\leq R_k}2^{-k}\left(1-\frac{|m_1|^2+|m_2|^2}{R_k^2}\right)^{{n-\frac{1}{2}}}|\hat{f}(m_1)||1-\Phi(\epsilon_k m_2)|\\
&\leq& \sum_{|(m_1,m_2)|\leq R_k}2^{-k}\left(1-\frac{|m_1|^2+|m_2|^2}{R_k^2}\right)^{{n-\frac{1}{2}}}\|f\|_1|1-\Phi(\epsilon_k m_2)|\leq 1.
\end{eqnarray*}
Therefore, we get that
\begin{equation}\label{equal1}
\inf_{x\in F_k}\sup_{R\leq R_k} |\mathcal B^{n-\frac{1}{2}}_R(2^{-k}\phi_{\epsilon_k},\delta_0)(x)|\geq B_k+k+1.
\end{equation}
Also, we have
\begin{eqnarray}\label{greater1}
&& \sup_{0<R\leq R_k} \left|\mathcal B^{n-\frac{1}{2}}_R\left(f,\sum_{s=k+1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s})\right)(x)\right|\\
&\leq&\nonumber \sum_{|(m_1,m_2)|\leq R_k}|\hat{f}(m_1)|\left[\sum_{s=k+1}^\infty2^{-s}|\Phi(\epsilon_s m_2)-\Phi(\delta_s m_2)|\right]\\\nonumber
&\leq& \sum_{|(m_1,m_2)|\leq R_k}\|f\|_1\left[\sum_{s=k+1}^\infty2^{-s}B|\delta_s-\epsilon_s||m_2|\right]\\\nonumber
&\leq& B\|f\|_1\delta_{k+1} \sum_{|(m_1,m_2)|\leq R_k}|m_2| \leq 1.
\end{eqnarray}
When $j=k$, using the estimates \eqref{smaller1}, \eqref{equal1} and \eqref{greater1} for $x\in F$ we can get the following estimate (as in the previous step)
\begin{eqnarray*}
\sup_{R\leq R_k}|\mathcal B^{{n-\frac{1}{2}}}_R\left(f,\sum_{s=1}^\infty2^{-s}(\phi_{\epsilon_s}-\phi_{\delta_s})\right)(x)|
&\geq&k
\end{eqnarray*}
This implies that $\sup\limits_{R>0}|\mathcal B^{{n-\frac{1}{2}}}_R(f,f)(x)|\geq k$ for all $x\in\cup_{r\geq k}F_k$. Therefore,
\begin{equation*}\label{one}
\sup_{R>0}|\mathcal B^{n-\frac{1}{2}}_R(f,f)(x)|=\infty
\end{equation*}
for all $x\in A=\cap_{r\geq1}\cup_{r\geq k}F_k$. Clearly, the set $A$ has positive measure. This completes the proof of Theorem~\ref{div}. \qed
\section{Weighted estimates for the maximal function \texorpdfstring{$\mathcal B^{n-\frac{1}{2}}_*$}{B1}}\label{max:sec}
The $L^p$ estimates for the maximal function $\mathcal B^{\alpha}_*$ were studied by~Grafakos, He and Honzik~\cite{GHH} and Jeong and Lee~\cite{JL}, which were later improved by Jotsaroop and Shrivastava~\cite{JS}. The problem of weighted boundedness of the bilinear Bochner-Riesz means $\mathcal B_R^{n-\frac{1}{2}}$ and the maximal function $\mathcal B^{n-\frac{1}{2}}_*$ was addressed in~\cite{JSK} for $n\geq 2$. The case of $n=1$ does not follow from their method. We complete the picture by giving a different proof of the weighted $L^p$ boundedness of $\mathcal B^{n-\frac{1}{2}}_*$. This proof works uniformly in all dimensions. We make use of the idea developed in~\cite{JS} to decompose the bilinear Bochner-Riesz multiplier $(1-\frac{|\xi|^2+|\eta|^2}{R^2})^{\alpha}_{+}$ in a specific manner. This idea along with the Stein's complex interpolation for analytic family of bilinear operators is used to deduce the desired weighted estimates. This approach naturally requires us to consider the operator $\mathcal B^{\alpha}_*$ for complex parameter $\alpha$ which can be defined in a similar fashion by simply taking the multiplier $(1-\frac{|\xi|^2+|\eta|^2}{R^2})^{\alpha}_{+}$ for $\alpha\in {\mathbb C}$ with $\text{Re}(\alpha)>0$.
The following lemma play a key role in proving Theorem~\ref{mainthm} for $\mathcal B^{n-\frac{1}{2}}_*$.
\begin{lemma}\label{keylem1}
Let $n\geq 1$ and $z$ be a complex number such that $0< Re(z)< n-\frac{1}{2}$. Then we have the following estimate
\begin{eqnarray*} \int_{\mathbb{R}^n} |\partial_z \mathcal {B}^{z}_*(f,g)(x)| dx &\leq & C_{n+Re(z)} e^{\mathfrak{C} | Im(z)|^{2}}\| f\|_{L^{2}}\| g\|_{L^{2}},
\end{eqnarray*}
where $\mathfrak{C}>0$ is a constant.
\end{lemma}
\begin{remark}
Along with Lemma~\ref{keylem1} we will also require $L^2\times L^2\rightarrow L^1$ boundedness of the maximal bilinear Bochner-Riesz function $\mathcal {B}^{z}_*(f,g)$ from ~\cite{JS}. We will make use of the ideas developed in~\cite{JS} to prove Lemma~\ref{keylem1}.
\end{remark}
We postpone the proof of Lemma~\ref{keylem1} to the next section and complete the proof of Theorem~\ref{mainthm} first. The following auxiliary results will be used in the proof of Theorem~\ref{mainthm}.
\begin{lemma}\cite{JSK} \label{weighted} Let $n\geq 1$ and $z\in \mathbb C$ be such that $Re(z)>n-\frac{1}{2}.$ Then the operator $\mathcal {B}^{z}_*$ is bounded from $L^{p_{1}}(\omega_{1})\times L^{p_{2}}(\omega_{2})\rightarrow L^{p}(v_{\omega})$ for all $\vec{\omega}\in A_{\vec{P}}$ with $1<p_{1}, p_{2}<\infty$ and $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}$.
\end{lemma}
\begin{lemma}\label{wei1}\cite{Ler1}
Let $\vec{\omega}=(\omega_{1},\omega_{2})\in A_{\vec{P}}$, where $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ with $1<p_1,p_2 <\infty$, then there exists a $\delta>0$, such that $\vec{\omega}_{\delta}=(\omega^{1+\delta}_{1},\omega^{1+\delta}_{2})\in A_{\vec{P}}$.
\end{lemma}
\subsection{Proof of Theorem~\ref{mainthm}}
First, note that in view of the multilinear extrapolation theorem, see~\cite{KJS} for details, it is enough to prove Theorem~\ref{mainthm} for $\vec{P}=(2,2).$ More precisely, we need to prove
\begin{eqnarray}\label{mainthm1}
\|\mathcal {B}_*^{n-\frac{1}{2}}(f,g)\|_{L^1(v_{\omega})}\lesssim \|f\|_{L^{2}(\omega_{1})} \|g\|_{L^{2}(\omega_{2})} ~\text{for~all~}\vec{\omega}\in A_{\vec{P}}, ~\vec{P}=(2,2).
\end{eqnarray}
We linearize the maximal function using a standard trick. Let $R(x)$ be an arbitrary positive measurable function on ${\mathbb {R}}^n$ such that both $R(x)^{-1}$ and $R(x)$ are bounded. It is enough to prove the estimate ~\eqref{mainthm1} for $\mathcal {B}_{R(x)}^{n-\frac{1}{2}}(f,g)$ with bounds independent of the function $R(x)$.
Fix such a function $R(x)$. Let $\epsilon_1,\epsilon_2>0$ (to be chosen later), $N\in \mathbb N$ and $A>\mathfrak{C}$, where $\mathfrak{C}$ is the constant appearing in Lemma~\ref{keylem1}. Consider the operator
$$\tilde{\mathcal {B}}^{z,\epsilon_{1},\epsilon_{2},N}_{R(x)}(f,g)(x)=\mathcal {B}^{(1-z)\epsilon_{1}+z(n-\frac{1}{2}+\epsilon_{2})}_{R(x)}(f,g)(x)(v_{N}(x))^{z}e^{A z^{2}},$$
where
$$ v_{N}(x)= \begin{cases} v_{\omega}(x), & \mbox{if } v_{\omega}(x)\leq N \\ N, & \mbox{if }v_{\omega}(x)>N. \end{cases}$$
Note that $v_{N}(x)\leq v_{\omega}(x)$ a.e. $x$.
Let $f$ and $g$ be compactly supported positive smooth functions. Given $\delta_{0}>0$ define $$f^{z}_{\delta_{0}}(x)=f(x)(\omega_{1}(x)+\delta_{0})^{-\frac{z}{2}}~~~\text{and}~~~g^{z}_{\delta_{0}}(x)=g(x)(\omega_{2}(x)+\delta_{0})^{-\frac{z}{2}}.$$
For $h\in L^\infty({\mathbb {R}}^n)$ consider
\begin{eqnarray*}
\psi(z)&=&\int_{\mathbb{R}^{n}}\tilde{\mathcal {B}}_{R(x)}^{z,\epsilon_{1},\epsilon_{2},N}(f^{z}_{\delta_{0}},g^{z}_{\delta_{0}})(x)h(x)dx \\ &=&\int_{\mathbb{R}^{n}}\mathcal {B}^{(1-z)\epsilon_{1}+z(n-\frac{1}{2}+\epsilon_{2})}_{R(x)}(f^{z}_{\delta_{0}},g^{z}_{\delta_{0}})(x)(v_{N}(x))^{z}h(x)e^{A z^{2}}dx,
\end{eqnarray*}
where $0\leq \text{Re}(z)\leq 1$.
Use Lemma \ref{keylem1} to conclude that $\psi$ is analytic in the strip $S=\lbrace z\in \mathbb{C} : 0<Re(z)<1\rbrace$, bounded and continuous on the closure $\bar{S}=\lbrace z\in \mathbb{C} : 0\leq Re(z)\leq 1\rbrace$. Moreover, we have the following estimates at the boundary.
\begin{eqnarray*}
\sup_{t\in\mathbb{R}}| \psi(i t)|
&\leq & \| h\|_{\infty}\sup_{t\in\mathbb{R}}e^{-At^{2}}\int_{{\mathbb {R}}^n} | \mathcal {B}_{R(x)}^{(1-i t)\epsilon_{1}+i t(n-\frac{1}{2}+\epsilon_{2})}\left(f(\omega_{1}+\delta_{0})^{\frac{-i t}{2}}, g(\omega_{2}+\delta_{0})^{\frac{-i t}{2}} \right)(x)| dx\\
&& (\text{Since}~~ \text{Re}[(1-i t)\epsilon_{1}+i t(n-\frac{1}{2}+\epsilon_{2})]=\epsilon_{1}>0~\text{apply~Lemma}~\ref{keylem1}) \\
&\leq &
C_{\epsilon_{1},\epsilon_{2}}\| h\|_{\infty}\sup_{t\in\mathbb{R}}e^{-(A-\mathfrak{C})t^{2}}\left(\int_{{\mathbb {R}}^n} | f(\omega_{1}+\delta_{0})^{-\frac{i t}{2}}|^{2} dx \right)^{\frac{1}{2}}\left(\int_{{\mathbb {R}}^n} | g(\omega_{2}+\delta_{0})^{-\frac{i t}{2}}|^{2} dx \right)^{\frac{1}{2}}\\
&\leq & C_{\epsilon_{1},\epsilon_{2}}\| h\|_{\infty}\| f\|_{2}\| g\|_{{2}}.
\end{eqnarray*}
Similarly,
\begin{eqnarray*}
& & \sup_{t\in\mathbb{R}} | \psi(1+i t)|\\
& \leq & \| h\|_{\infty}\sup_{t\in \mathbb{R}}e^{A(1-t^{2})}\int_{{\mathbb {R}}^n} | \mathcal {B}_{R(x)}^{(-i t)\epsilon_{1}+(1+it)(n-\frac{1}{2}+\epsilon_{2})}\left(f(\omega_{1}+\delta_{0})^{-\frac{1+i t}{2}}, g(\omega_{2}+\delta_{0})^{-\frac{1+i t}{2}} \right)(x)| v_{N}(x)dx\\
& \leq & \| h\|_{\infty}\sup_{t\in \mathbb{R}}e^{A(1-t^{2})}\int_{{\mathbb {R}}^n} | \mathcal {B}_{R(x)}^{(-i t)\epsilon_{1}+(1+i t)(n-\frac{1}{2}+\epsilon_{2})}\left(f(\omega_{1}+\delta_{0})^{-\frac{1+i t}{2}}, g(\omega_{2}+\delta_{0})^{-\frac{1+i t}{2}} \right)(x)| v_{\omega}(x)dx.
\end{eqnarray*}
Note that
$\text{Re}[(-it)\epsilon_{1}+(1+it)(n-\frac{1}{2}+\epsilon_{2})]=n-\frac{1}{2}+\epsilon_{2}>n-\frac{1}{2}$ and
$(\omega_{j}+\delta_{0})^{-1}\leq \omega^{-1}_{j}, ~j=0,1$. Therefore, applying Lemma~\ref{weighted}, we get that
\begin{eqnarray*}
&& \sup_{t\in\mathbb{R}}| \psi(1+i t)| \\
&\leq & C_{\epsilon_{1},\epsilon_{2}}\| h\|_{\infty}\sup_{t\in\mathbb{R}}e^{-(A-\mathfrak C)t^{2}}\left(\int_{{\mathbb {R}}^n} | f(\omega_{1}+\delta_{0})^{-\frac{1+i t}{2}}|^{2}\omega_{1}(x) dx \right)^{\frac{1}{2}}\left(\int_{{\mathbb {R}}^n} | g(\omega_{2}+\delta_{0})^{-\frac{1+i t}{2}}|^{2}\omega_{2}(x) dx \right)^{\frac{1}{2}}\\
& \leq & C_{\epsilon_{1},\epsilon_{2}}\| h\|_{\infty}\| f\|_{2}\| g\|_{2}.
\end{eqnarray*}
With these estimates on the boundary of the strip $S$ apply `Three lines lemma' from complex analysis to get that
\begin{eqnarray*}
| \psi(\theta)|
&\leq & \nonumber C\left(\sup_{t\in \mathbb{R}}| \psi(\iota t)|\right)^{1-\theta}\left(\sup_{t\in \mathbb{R}}| \psi(1+\iota t)|\right)^{\theta}\\
&\leq & \label{three} C_{\epsilon_{1},\epsilon_{2}}\| h\|_{\infty}\| f\|_{2}\| g\|_{2},~~0<\theta<1.
\end{eqnarray*}
Note that the constant in the estimate above does not depend on $R(x)$. This gives us that
\begin{eqnarray*}
|\psi(\theta)|
&=&\left|\int_{\mathbb{R}^{n}}\mathcal {B}_{R(x)}^{(1-\theta)\epsilon_{1}+\theta(n-\frac{1}{2}+\epsilon_{2})}(f^{\theta}_{\delta_{0}},g^{\theta}_{\delta_{0}})(x)(v_{N}(x))^{\theta}h(x)dx \right| \\
&=&\left|\int_{{\mathbb {R}}^n} \mathcal {B}_{R(x)}^{(1-\theta)\epsilon_{1}+\theta(n-\frac{1}{2}+\epsilon_{2})}\left(f(\omega_{1}+\delta_{0})^{\frac{-\theta}{2}}, g(\omega_{2}+\delta_{0})^{\frac{-\theta}{2}} \right)(x)(v_{N}(x))^{\theta}h(x)dx \right|\\
&\leq & C_{\epsilon_{1},\epsilon_{2}}\| f\|_{2}\| g\|_{2}.
\end{eqnarray*}
Since the constant $C$ in the inequality above is independent of $N$ and $\delta_{0}$, let $N\rightarrow \infty$ and $\delta_{0}\rightarrow0$ and then replace $f$ and $g$ by $f\omega_{1}^{\frac{\theta}{2}}$ and $g\omega_{2}^{\frac{\theta}{2}}$ respectively to get that
\begin{align} \label{estimate}
\int_{{\mathbb {R}}^n} |\mathcal {B}_{R(x)}^{(1-\theta)\epsilon_{1} +\theta(n-\frac{1}{2}+\epsilon_{2})}\left(f, g \right)(x)|(v_{\omega}(x))^{\theta}dx
&\leq & C \left(\int_{{\mathbb {R}}^n} | f(x)|^{2}\omega_{1}^{\theta} dx\right)^{\frac{1}{2}}\left(\int_{{\mathbb {R}}^n} | g(x)|^{2}\omega_{2}^{\theta} dx\right)^{\frac{1}{2}},
\end{align}
where $0<\theta<1$.
At this point invoke the reverse H\"{older} inequality for bilinear weights from Lemma~\ref{wei1}. This tells us that given a bilinear weight $\vec{\omega}\in A_{\vec{P}}$ there exists $\delta>0$ such that $\vec{\omega}_{\delta}=(\omega_{1}^{1+\delta},\omega_{2}^{1+\delta})\in A_{\vec{P}}$. Using the estimate \eqref{estimate} for $\vec{\omega}_{\delta}\in A_{\vec{P}}$ with $\theta=\frac{1}{1+\delta}$ we get the desired result.
\begin{eqnarray*}\label{critical}
\| \mathcal {B}_{R(x)}^{\lambda}(f,g)\|_{L^{1}(v_{\omega})}
&\leq & C \| f\|_{L^{2}(\omega_{1})}\| g\|_{L^{2}(\omega_{2})},
\end{eqnarray*}
where $\lambda=(1-\frac{1}{1+\delta})\epsilon_{1}+\frac{1}{1+\delta}(n-\frac{1}{2}+\epsilon_{2})$.
Finally, observe that we can choose $\epsilon_1$ and $\epsilon_2$ so that $\lambda=n-\frac{1}{2}.$
This completes the proof of Theorem~\ref{mainthm}.
\qed
\section{Proof of Lemma~\ref{keylem1}}\label{sec:keylem1}
Let $z\in {\mathbb C}$ be such that $0<\text{Re}(z)<n-\frac{1}{2}$. Choose functions $\psi\in C^\infty_0[\frac{1}{2},2]$ and $\psi_0\in C_0^{\infty}[-\frac{3}{4},\frac{3}{4}]$ such that
$$1=\sum_{j\geq 2}\psi(2^j(1-t))+\psi_0(t),~ t\in[0,1].$$
This gives us
\begin{eqnarray*}
m_R^{z}(\xi,\eta)
=\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{z}
= \sum\limits_{j\geq 2}{m}^{z}_{j,R}(\xi,\eta)+m^{z}_{0,R}(\xi,\eta),
\end{eqnarray*}
where $$m^{z}_{j,R}(\xi,\eta)=\psi\left(2^j\left(1-\frac{|\xi|^2}{R^2}\right)\right)\left(1-\frac{|\xi|^2}{R^2}\right)_+^{z}\left(1-\frac{|\eta|^2}{R^2}\left(1-\frac{|\xi|^2}{R^2}\right)^{-1}\right)^{z}_+$$
and
$$m^{z}_{0,R}(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{z}.$$
Let $\mathcal{B}_{j,R}^{z}$ denote the bilinear multiplier operator associated with $m^{z}_{j,R}(\xi,\eta)$, i.e.,
\begin{equation*}\mathcal{B}_{j,R}^{z}(f,g)(x)=\int_{{\mathbb {R}}^n}\int_{{\mathbb {R}}^n}m^{z}_{j,R}(\xi,\eta)\hat{f}(\xi)\hat{g}(\eta)e^{2\pi ix\cdot (\xi+\eta)}d\xi d\eta.
\end{equation*}
Let $\beta>\frac{1}{2}$ and note that $\text{Re}(z)-\beta>-\frac{1}{2}$. Using the decomposition of the bilinear Bochner-Riesz multiplier from~Kaur and Shrivastava~[\cite{JS}, Section $3$] we have the following representation
\begin{eqnarray*}\label{decomope}
\mathcal B^{z}_{j,R}(f,g)(x)
&=& c_{z}\int_0^{\sqrt{2^{-j+1}}}S_{j,\beta}^{R,t}f(x)B_{Rt}^{z-\beta}g(x)t^{2(z-\beta)+1}dt,
\end{eqnarray*}
where $c_z=\frac{\Gamma(z+1)}{\Gamma(\beta)\Gamma(z-\beta+1)}$, $B_{Rt}^{z-\beta}$ is the linear Bochner-Riesz mean and
$${S}_{j,\beta}^{R,t}f(x)=\int_{{\mathbb {R}}^n}\psi\left(2^j\left(1-\frac{|\xi|^2}{R^2}\right)\right)\left(1-\frac{|\xi|^2}{R^2}-t^2\right)_+^{\beta-1}\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi.$$
Therefore, we need to prove the desired boundedness results for maximal functions
$$\mathcal{B}_{j,*}^{z}(f,g)(x)=\sup_{R>0}|\mathcal{B}_{j,R}^{z}(f,g)(x)|$$
for $j=0$ and $j\geq2$.
The derivative (with respect to $z$) of $\mathcal{B}_{j,R}^{z}(f,g)(x)$ is given by
\begin{eqnarray*}\label{keylem1:der}
\partial_z \mathcal {B}^{z}_{j,R}(f,g)(x)&=& I_{j,R}+II_{j,R}+III_{j,R}
\end{eqnarray*}
where
\begin{eqnarray*}
I_{j,R} &=& (\partial_zc_{z})\left(\int_0^{\sqrt{2^{-j+1}}}S_{j,\beta}^{R,t}f(x)B_{Rt}^{z-\beta}g(x)t^{2(z-\beta)+1}dt\right)
\end{eqnarray*}
\begin{eqnarray*} II_{j,R}&=& c_{z}\int_0^{\sqrt{2^{-j+1}}}S_{j,\beta}^{R,t}f(x)\tilde{B}_{Rt}^{z-\beta}g(x)t^{2(z-\beta)+1}dt
\end{eqnarray*}
\begin{eqnarray*} III_{j,R}&=&
c_{z}\int_0^{\sqrt{2^{-j+1}}}S_{j,\beta}^{R,t}f(x)B_{Rt}^{z-\beta}g(x)t^{2(z-\beta)+1}\log tdt
\end{eqnarray*}
where $$\tilde{B}_t^{z-\beta}g(x)=\int_{{\mathbb {R}}^n}\hat{g}(\eta)\left(1-\frac{|\eta|^2}{t^2}\right)_+^{z-\beta}\log\left(1-\frac{|\eta|^2}{t^2}\right)_+ e^{2\pi ix.\eta} d\eta.$$
We will prove estimates for maximal functions associated with each of the terms above separately. Let us first record the bounds for constant $c_z$ and its derivative $\partial_zc_{z}$.
Write $z=\alpha+i\tau$. Using estimates of gamma function, see~ [\cite{Grafakosclassical}, pages 569-570], we know that
\begin{eqnarray*}
|\Gamma(\alpha+1+i \tau)|\leq |\Gamma(\alpha+1)|\quad\text{and}\quad \frac{1}{|\Gamma(\alpha-\beta+1+i \tau)|}\leq \frac{e^{C_{\alpha,\beta}|\tau|^{2}}}{|\Gamma(\alpha-\beta+1)|},
\end{eqnarray*}
where $C_{\alpha,\beta}=\max\{(1+\alpha-\beta)^{-2},(1+\alpha-\beta)^{-1}\}$.
Therefore $|c_z|$ increases at most by a constant multiple of $e^{\mathfrak{C}|\tau|^2}$ when $0<\alpha<n-\frac{1}{2}$, where $\mathfrak{C}$ is a fixed constant.
Next we estimate the growth of $|\partial_zc_z|$. We have
$$\partial_zc_z=\frac{1}{\Gamma(\beta)}\left(\frac{\Gamma(z-\beta+1)\Gamma'(z+1)-\Gamma(z+1)\Gamma'(z-\beta+1)}{\Gamma(z-\beta+1)^2}\right).$$
It is easy to see that for $\text{Re}(z)>0$
\begin{eqnarray*}
|\Gamma'(z)|
&\lesssim& |\Gamma(Re(z)-\epsilon)|+|\Gamma(Re(z)+1)|,
\end{eqnarray*}
where $\epsilon$ is small enough so that $\text{Re}(z)-\epsilon>0$. Further, using estimates of gamma function we can show that
\begin{eqnarray*}
|\partial_zc_z|
&\lesssim& \frac{|\Gamma(z-\beta+1)\Gamma'(z+1)|+|\Gamma(z+1)\Gamma'(z-\beta+1)|}{\Gamma(z-\beta+1)^2}\\
&\lesssim& \frac{|\Gamma(z-\beta+1)|(|\Gamma(Re(z)-\epsilon+1)|+|\Gamma(Re(z)+2)|)}{\Gamma(z-\beta+1)^2}\\
&&~+ \frac{|\Gamma(z-\beta+1)|(|\Gamma(Re(z)-\beta+1-\epsilon)||+|\Gamma(Re(z)-\beta+2)|}{\Gamma(z-\beta+1)^2}\\
&\lesssim& C_{Re(z)}e^{2C_{\alpha,\beta}|\tau|^{2}}.
\end{eqnarray*}
Therefore, $|\partial_zc_z|$ also increases at most by a constant multiple of $e^{\mathfrak{C}|\tau|^2}$.
\noindent {\bf Estimate for the term $I_{j,R}$:}~Note that $\Gamma(z+1)$ and $\frac{1}{\Gamma(z-\beta+1)}$ are analytic functions in the region $0<Re(z)<n-\frac{1}{2}$. Applying Cauchy-Schwarz inequality twice and a change of variable argument in the second term we get that
\begin{eqnarray*}
\left\|\sup_{R>0}|I_{j,R}|\right\|_1
&\leq& 2^{-\frac{j}{4}}|\partial_zc_{z}|\left\|\sup_{R>0}\left(\int_0^{\sqrt{2^{-j+1}}}|{S}_{j,\beta}^{R,t}f(x)t^{2({z-\beta})+1}|^2dt\right)^{1/2}\right\|_2\\
&& \left\|\sup_{R>0}\left(R_j^{-1}\int_0^{R_j}|B_t^{z-\beta}g(x)|^2dt\right)^{1/2}\right\|_2.
\end{eqnarray*}
Invoking Theorem 5.1 from~Kaur and Shrivastava~\cite{JS}) for $\beta>\frac{1}{2}$ we have
$$\left\|\sup_{R>0}\left(\int_0^{\sqrt{2^{-j+1}}}|{S}_{j,\beta}^{R,t}f(x)t^{2(z-\beta)+1}|^2dt\right)^{1/2}\right\|_2\lesssim2^{j(\frac{1}{4}-Re(z)+\gamma)}\|f\|_2.$$
Also, the other operator satisfies the following $L^2-$estimate, see~[\cite{JS}, Lemma 4.3].
$$\left\|\sup_{R>0}\left(R_j^{-1}\int_0^{R_j}|B_t^{z-\beta}g(x)|^2dt\right)^{1/2}\right\|_2\lesssim \|f\|_2,~~\text{Re}({z-\beta})>-\frac{1}{2}.$$
Putting these estimates together we get that
\begin{eqnarray*}
\left\|\sup_{R>0}|I_{j,R}|\right\|_1
&\lesssim& 2^{-j(\text{Re}(z)-\gamma)}e^{\mathfrak{C}|\text{Im}(z)|^2}\|f\|_2\|g\|_2.
\end{eqnarray*}
\noindent {\bf Estimate for the term $II_{j,R}$:}~~
As in the previous step the Cauchy-Schwarz inequality gives us
\begin{eqnarray*}
\left\|\sup_{R>0}|II_{j,R}|\right\|_1
&\leq& 2^{-\frac{j}{4}}|c_z|\left\|\sup_{R>0}\left(\int_0^{\sqrt{2^{-j+1}}}|{S}_{j,\beta}^{R,t}f(x)t^{2(z-\beta)+1}|^2dt\right)^{1/2}\right\|_2\\
&&\nonumber \left\|\sup_{R>0}\left(R_j^{-1}\int_0^{R_j}|\tilde{B}_t^{z-\beta}g(x)|^2dt\right)^{1/2}\right\|_2.
\end{eqnarray*}
We already have the required bounds for the constant $c_z$ and the first term involving the operator ${S}_{j,\beta}^{R,t}$.
We claim that the following $L^2$ estimate holds
\begin{eqnarray}\label{L2}
\left\|\sup_{R>0}\left(R_j^{-1}\int_0^{R_j}|\tilde{B}_t^{\delta}g(x)|^2dt\right)^{1/2}\right\|_2 \lesssim \|f\|_2,~~~\text{for~Re}(\delta)>-\frac{1}{2}.
\end{eqnarray}
Consequently, we get that
\begin{eqnarray*}
\left\|\sup_{R>0}|II_{j,R}|\right\|_1\lesssim 2^{-j(Re(z)-\gamma)}e^{\mathfrak{C}|Im(z)|^2}\|f\|_2\|g\|_2,~~~\text{Re}(z)-\beta>-\frac{1}{2}.
\end{eqnarray*}
We can use a trick from~Stein~\cite{SW} involving square function to prove~\eqref{L2}. Let $\text{Re}(\delta)>-\frac{1}{2}$ and choose $d$ such that $\text{Re}(\delta)+d>\frac{n-1}{2}$. Write
$$\tilde{B}_t^{\delta}g=\sum_{k=1}^{d}\left(\tilde{B}_t^{\delta+k-1}g-\tilde{B}_t^{\delta+k}g\right) + \tilde{B}_t^{\delta+d}g.$$
This implies that $$\left(\int_0^{R}|\tilde{B}_t^{\delta}g(x)|^2dt\right)^{1/2}\leq \sum_{k=1}^{d}\left(\int_0^{R}|\tilde{B}_t^{\delta+k}g(x)-\tilde{B}_t^{\delta+k-1}g(x)|^2dt\right)^{1/2} + \left(\int_0^{R}|\tilde{B}_t^{\delta+d}g(x)|^2dt\right)^{1/2}.$$
Observe that $\sup\limits_{R>0}\left(R^{-1}\int_0^{R}|\tilde{B}_t^{\delta+k}g(x)-\tilde{B}_t^{\delta+k-1}g(x)|^2dt\right)^{1/2},~1\leq k\leq d$ is dominated by $\left(\int_0^{\infty}|\tilde{B}_t^{\delta+k}g(x)-\tilde{B}_t^{\delta+k-1}g(x)|^2 t^{-1}dt\right)^{1/2}.$ Using Plancherel's theorem, we get that
\begin{eqnarray*}
&&\left\|\left(\int_0^{\infty}|\tilde{B}_t^{\delta+k}g(x)-\tilde{B}_t^{\delta+k-1}g(x)|^2 \frac{dt}{t}\right)^{1/2}\right\|_2^2\\
&=& \int_0^{\infty}\int_{{\mathbb {R}}^n}\left|\left(1-\frac{|\eta|^2}{t^2}\right)_+^{\delta+k}\frac{|\eta|^2}{t^2}\log\left(1-\frac{|\eta|^2}{t^2}\right)_+\hat{g}(\eta)\right|^2dx \frac{dt}{t}\\
&\lesssim & \int_0^{\infty}\int_{{\mathbb {R}}^n}\left|\left(1-\frac{|\eta|^2}{t^2}\right)_+^{\delta+k-\epsilon}\frac{|\eta|^2}{t^2}\hat{g}(\eta)\right|^2dx \frac{dt}{t}\\
&\lesssim & \|g\|_2^2
\end{eqnarray*}
where we have chosen $\epsilon>0$ such that $\text{Re}(\delta)-\epsilon>-\frac{1}{2}$.
Finally, for the remaining term with $\text{Re}(\delta)+d>\frac{n-1}{2}$, one can easily verify that the kernel of $\tilde{B}_t^{\delta+d}$ is an integrable function. For,
\begin{eqnarray*}
K_t^{\delta+d}(x)&=&\int_{{\mathbb {R}}^n}\left(1-\frac{|\eta|^2}{t^2}\right)_+^{\delta+d}\log\left(1-\frac{|\eta|^2}{t^2}\right)_+e^{2\pi ix\cdot\eta}d\eta\\
&=& t^n\int_0^\infty \int_{\mathbb{S}^{n-1}}(1-r^2)^{\delta+d}_+\log(1-r^2)_+e^{2\pi itx\cdot r\theta}d\theta r^{n-1}dr\\
&=& \frac{2\pi t^n}{|x|^\frac{n-2}{2}} \int_0^1 (1-r^2)^{\delta+d}_+\log(1-r^2)_+J_{\frac{n}{2}-1}(2\pi rt|x|)r^{\frac{n}{2}}dr\\
\end{eqnarray*}
Therefore, we get that
\begin{eqnarray*}
|K_t^{\delta+d}(x)| &\leq& \frac{2\pi t^n}{|tx|^\frac{n-2}{2}}\sup_{r\in[0,1]}\{(1-r^2)^\epsilon_+\log(1-r^2)_+\} \int_0^1 (1-r^2)^{\delta+d-\epsilon}_+J_{\frac{n}{2}-1}(2\pi rt|x|)r^{\frac{n}{2}}dr\\
&=& Ct^n\frac{J_{\frac{n}{2}+\delta+d-\epsilon}(2\pi t|x|)}{|tx|^{\frac{n}{2}+\delta+d-\epsilon}},
\end{eqnarray*}
where $\epsilon$ is small enough so that $\delta+d-\epsilon>\frac{n-1}{2}$. Clearly, we get that
$$\sup_{t>0}|\tilde{B}_t^{\delta+d}g(x)\leq c(\delta+d,n)Mg(x),$$
where $M$ is the classical Hardy-Littlewood maximal function. Consequently, we obtain the desired estimate~\eqref{L2}. \\
\noindent {\bf Estimate for the term $III_{j,R}$:}~~
In this case we have
\begin{eqnarray*}
\left\|\sup_{R>0}|III_{j,R}|\right\|_1
&\leq& 2^{-j/4}|c_z|\left\|\sup_{R>0}\left(\int_0^{\sqrt{2^{-j+1}}}|{S}_{j,\beta}^{R,t}f(x)t^{2(z-\beta)+1}\log t|^2dt\right)^{1/2}\right\|_2\\
&&\nonumber \left\|\sup_{R>0}\left(R_j^{-1}\int_0^{R_j}|B_t^{z-\beta}g(x)|^2dt\right)^{1/2}\right\|_2.
\end{eqnarray*}
Following the discussion in the previous cases observe that we only need to deal with the term involving ${S}_{j,\beta}^{R,t}$ in equation above. This can be done easily in the following manner.
For $\text{Re}(z-\beta)>-\frac{1}{2}$, choose $\epsilon>0$ small enough so that $\text{Re}(2(z-\beta)+1)-\epsilon>0$. Then
$$\sup_{R>0}\left(\int_0^{\sqrt{2^{-j+1}}}|{S}_{j,\beta}^{R,t}f(x)t^{2(z-\beta)+1}\log t|^2dt\right)^{1/2}\lesssim \sup_{R>0}\left(\int_0^{\sqrt{2^{-j+1}}}|{S}_{j,\beta}^{R,t}f(x)t^{2(z-\beta)+1-\epsilon}|^2dt\right)^{1/2}$$
as $t^\epsilon\log t$ is bounded in $[0,\sqrt{2^{-j+1}}]$.
This reduces our job to a known situation as considered in~\cite{JS} and hence we have that
$$\left\|\sup_{R>0}\left(\int_0^{\sqrt{2^{-j+1}}}|{S}_{j,\beta}^{R,t}f(x)t^{2(z-\beta)+1-\epsilon}|^2dt\right)^{1/2}\right\|_2\lesssim 2^{j(\frac{1}{4}-\text{Re}(z)+\epsilon+\gamma)}\|f\|_2,$$
for $\text{Re}(z)-\beta-\epsilon>-\frac{1}{2}$ and $\beta>\frac{1}{2}$.
Putting the estimates above together, we get that
$$\left\|\sup_{R>0}|III_{j,R}|\right\|_1\lesssim 2^{-j(\text{Re}(z)-\epsilon-\gamma)}e^{\mathfrak{C}|\text{Im}(z)|^2}\|f\|_2\|g\|_2.$$
Since $\text{Re}(z)>0$, we can sum over $j\geq 2$ in all the cases obtained as above.
It remains to deal with the operator $\mathcal{B}_{0,*}^z$. \\
\noindent {\bf Boundedness of the operator $\mathcal{B}_{0,*}^z$:}
This part is dealt with similarly. We will make use of the decomposition of the multiplier $m_{0,R}^\alpha(\xi,\eta)$ as carried out Kaur and Shrivastava~[\cite{JS}, Section 5.3]. This time we decompose $m_{0,R}^\alpha(\xi,\eta)$ with respect to $\eta$ variable. Note that previously we did the same with respect to $\xi$ variable. This gives us
\begin{eqnarray*}
m_{0,R}^\alpha(\xi,\eta)&=&\sum_{j\geq0}\Tilde{m}_{j,R}^\alpha(\xi,\eta)
\end{eqnarray*}
where
$$\Tilde{m}_{j,R}^\alpha(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi\left(2^j\left(1-\frac{|\eta|^2}{R^2}\right)\right)\left(1-\frac{|\eta|^2}{R^2}\right)_+^{\alpha}\left(1-\frac{|\xi|^2}{R^2}\left(1-\frac{|\eta|^2}{R^2}\right)^{-1}\right)^{\alpha}_+$$
for $j\geq2$, and for $j=0,1$
$$\Tilde{m}_{j,R}^\alpha(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi_0^{j+1}\left(\frac{|\eta|^2}{R^2}\right)\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{\alpha}.$$
Here $\psi_0, \psi_0^1,$ and $ \psi_0^2$ are smooth functions supported in $[0,3/4], [0,3/16],$ and $[\frac{3}{32},\frac{3}{4}]$ respectively. Also, they satisfy the identity $\psi_0(x)=\psi_0^1(x)+\psi_0^2(x)$.
Let $\tilde{\mathcal B}^{\alpha}_{j,R}$ denote the bilinear multiplier operator associated with $\tilde{m}^{\alpha}_{j,R}(\xi,\eta)$ and let $\tilde{\mathcal B}^{\alpha}_{j,*}$ denote the corresponding maximal function.
We will deal with maximal function $\tilde{\mathcal B}^{z}_{j*}, j\geq 0$ separately.
Consider the case of $j=0$ first. In this case the multiplier is given by
$$\tilde{m}_{0,R}^\alpha=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi_0^1\left(\frac{|\eta|^2}{R^2}\right)\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{\alpha}.$$
Taking the derivative we see that the bilinear multiplier for the operator $\left(\partial_z\right) \tilde{\mathcal B}^{z}_{0,R}$ is given by
$$M(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi_0^1\left(\frac{|\eta|^2}{R^2}\right)\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{z}\log\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+.$$
Observe that $\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi_0^1\left(\frac{|\eta|^2}{R^2}\right)$ is a smooth function with its support in a ball of radius $\sqrt{\frac{15}{16}}R$ and observe that $\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{z}\log\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+$ is a smooth function on this set. Therefore, using standard argument we can show that $\tilde{\mathcal B}^{z}_{0,*}$ is dominated by the bilinear Hardy-Littlewood maximal function which is defined by
$$\mathcal M(f,g)(x):=\sup_{t>0}\frac{1}{|B(x,t)|^2}\int_{B(x,t)} |f(y)|dy\int_{B(x,t)} |g(z)|dz.$$
We refer to~\cite{Ler1} for more details about the maximal function $\mathcal M$.
The $L^p$ boundedness of $\mathcal M$ yields the desired estimates for $\tilde{\mathcal B}^{z}_{0,*}$.
Next, for $j=1$ Stein's identity [\cite{SW}, page 278] allows us to express
$$\tilde{m}_{1,R}^\alpha(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi_0^2\left(\frac{|\eta|^2}{R^2}\right)R^{-2\alpha}\int_0^{uR}\left(R^2\varphi_R(\eta)-t^2\right)_+^{\beta-1}t^{2\delta+1}\left(1-\frac{|\xi|^2}{t^2}\right)^{\delta}_+dt,$$
where $\varphi_R(\eta)=\left(1-\frac{|\eta|^2}{R^2}\right)_+$ and $u=\sqrt{\frac{29}{32}}$. Therefore,
\begin{eqnarray*}
\tilde{\mathcal B}^{z}_{1,R}(f,g)(x)&=&c_{z}\int_0^{u}H^{\beta}_{R,t}g(x)B_{Rt}^{z-\beta}f(x)t^{2(z-\beta)+1}dt,
\end{eqnarray*}
where $H_{R,t}^{\beta}g(x)=\int_{{\mathbb {R}}^n}\psi_0^2\left(\frac{|\eta|^2}{R^2}\right)\left(1-t^2-\frac{|\eta|^2}{R^2}\right)_+^{\beta-1}\hat{g}(\eta)e^{2\pi ix.\eta} d\eta$.
This gives us that
\begin{eqnarray}\label{j=1}
\partial_z \tilde{\mathcal B}^{z}_{1,R}(f,g)(x)&=& \left(\partial_zc_{z}\right)\int_0^{u}H^{\beta}_{R,t}g(x)B_{Rt}^{z-\beta}f(x)t^{2(z-\beta)+1}dt\\
&&\nonumber ~+ c_{z}\int_0^{u}H^{\beta}_{R,t}g(x)\tilde{B}_{Rt}^{z-\beta}f(x)t^{2(z-\beta)+1}dt\\
&&\nonumber~+ c_{z}\int_0^{u}H^{\beta}_{R,t}g(x)B_{Rt}^{z-\beta}f(x)t^{2(z-\beta)+1}\log tdt.
\end{eqnarray}
Note that from this point onward the requires estimate can be deduced by following the corresponding argument (as in the case of $\mathcal B_{j,*}^z$) from the previous section along with $L^2$-estimate for the operator $f\rightarrow \left(\sup_{R>0}\int_0^{u}|H^{\beta}_{R,t}g(x)t^{2\delta+1}|^2 dt\right)^{1/2}$ from [\cite{JS}, Section 5.3]
Finally, when $j\geq2$, notice that $\tilde{m}^{\alpha}_{j,R}(\xi,\eta)$ is similar to $m^{\alpha}_{j,R}(\xi,\eta)$ except that there is an extra factor of $\psi_0\left(\frac{|\xi|^2}{R^2}\right)$ present in $\tilde{m}^{\alpha}_{j,R}(\xi,\eta)$. Let $K\in {\mathbb N}$ be such that $\{\xi: (\xi,\eta)\in \operatorname{supp}(\tilde{m}^{\alpha}_{j,R})\}\subseteq \{\xi:|\xi|\leq \frac{R}{8}\}, j\geq K$. We can assume that $\psi_0\left(\frac{|\xi|^2}{R^2}\right)= 1~\text{for}~|\xi|\leq \frac{R}{8}.$ Therefore, for $j\geq K$ the maximal function $\tilde{\mathcal B}^{\alpha}_{j,*}$ behaves the same way as $\mathcal B^{\alpha}_{j,*}$ and hence the desired results follow in this situation.
For $2\leq j<K$, using Stein's identity once again we can write
\begin{eqnarray*}
\tilde{\mathcal B}^{z}_{j,R}(f,g)(x)
&=& c_{z}\int_0^{\sqrt{2^{-j+1}}}S_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}B_{Rt}^{z-\beta}f(x)t^{2(z-\beta)+1}dt,
\end{eqnarray*}
where $c_z=\frac{\Gamma(z+1)}{\Gamma(\beta)\Gamma(z-\beta+1)}$ and $B^{\psi_0}_{R}f(x)=\int_{{\mathbb {R}}^n}\psi_0\left(\frac{|\xi|^2}{R^2}\right)\hat{f}(\xi)e^{2\pi ix.\xi} d\xi.$
Therefore,
\begin{eqnarray*}
\partial_z \tilde{\mathcal B}^{z}_{j,R}(f,g)(x)&=& \left(\partial_zc_{z}\right)\int_0^{\sqrt{2^{-j+1}}}S_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}B_{Rt}^{z-\beta}f(x)t^{2(z-\beta)+1}dt\\
&&~+ c_{z}\int_0^{\sqrt{2^{-j+1}}}S_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}\tilde{B}_{Rt}^{z-\beta}f(x)t^{2(z-\beta)+1}dt\\
&&~+ c_{z}\int_0^{\sqrt{2^{-j+1}}}S_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}B_{Rt}^{z-\beta}f(x)t^{2(z-\beta)+1}\log tdt
\end{eqnarray*}
The $L^2\times L^2\to L^1$-boundedness of the maximal functions associated with all the three terms can be proved similarly as in the previous case except that in place of estimate ~\eqref{L2} here we will require the following $L^2-$estimate
\begin{eqnarray}\label{L21}
\left\|\sup_{R>0}\left(R^{-1}\int_0^{R}|B^{\psi_0}_{R}\tilde{B}_t^{\delta}f(x)|^2dt\right)^{1/2}\right\|_2\lesssim \|f\|_2~~\text{for}~ ~\text{Re}(\delta)>-\frac{1}{2}.
\end{eqnarray}
This is proved combining the estimate \eqref{L2} along with the estimate $$\sup\limits_{R>0}|B^{\psi_0}_{R}f(x)|\lesssim M(f)(x).$$ Note that the later assertion holds because $\psi_0$ is a compactly supported smooth function.
This completes the proof of Lemma~\ref{keylem1}.
\qed
\section{End-point estimates for the square function \texorpdfstring{$\mathcal G^{n-\frac{1}{2}}$}{G}}\label{sec:sqrexam}
Recall that the kernel (in the sense of vector-valued operator) of square function $\mathcal G^{\alpha}$ is given by
\begin{eqnarray*}
{\mathcal K^{\alpha}_t}(y_1,y_2) &=& c_{n,\alpha}t^{2n-2}\Delta\left(\frac{ J_{\alpha+n} (|t(y_1,y_2)|)} {|t(y_1,y_2)|^{\alpha+n}}\right)\\
&=&c_{n,\alpha}t^{2n}\left(\frac{J_{n+\alpha}(2\pi |t(y_1,y_2)|)}{(|t(y_1,y_2)|)^{n+\alpha}}-\frac{J_{n+\alpha+1}(2\pi |t(y_1,y_2)|)}{(|t(y_1,y_2)|)^{n+\alpha+1}}\right).
\end{eqnarray*}
Using the asymptotics of Bessel functions for large $|x|$ and $t\geq1$, we have
$$J_{n+\alpha}(2\pi t|x|)=\frac{\cos(2\pi t|x|+\frac{\pi}{2}(n+\alpha)+\frac{\pi}{4})}{\pi\sqrt{t|x|}}+O((t|x|)^{-\frac{3}{2}}).$$
Then,
$$\mathcal{K}^{n-\frac{1}{2}}_t(y_1,y_2)=c_{n}\left(\frac{\cos(2\pi |t(y_1,y_2)|+n\pi)}{{|(y_1,y_2)|}^{2n}}+O\left(\frac{1}{|(y_1,y_2)|^{2n+1}}\right)\right).$$
Let $\psi\in \mathcal S({\mathbb {R}}^n)$ be such that $\operatorname{supp}(\hat{\psi})$ is contained in $B(0,2)$ and $\hat{\psi}(\xi)=1$ in $B(0,1)$. Let $\psi_N(x)=N^n\psi(N^nx).$ Consider
\begin{eqnarray*}
\mathcal G^{n-\frac{1}{2}}(\psi_N,\psi_N)(x) &\geq& \left(\int_1^N\left|\int_{{\mathbb {R}}^n\times{\mathbb {R}}^n}\left(1-\frac{|\xi|^2+|\eta|^2}{t^2}\right)^{n-\frac{1}{2}}_+\frac{|\xi|^2+|\eta|^2}{t^2}\hat{\psi}_N(\xi)\hat{\psi}_N(\eta)e^{2\pi ix\cdot(\xi+\eta)}d\xi\eta\right|^2\frac{dt}{t}\right)^{\frac{1}{2}}\\
&=& \left(\int_1^N\left|\mathcal K^{n-\frac{1}{2}}_t(x,x)\right|^2\frac{dt}{t}\right)^{\frac{1}{2}}\\
&\gtrsim & \frac{1}{|(x,x)|^{2n}}\left(\int_1^N|\cos(2\pi t|(x,x)|)|^2\frac{dt}{t}\right)^{\frac{1}{2}}\\
&= & \frac{1}{|(x,x)|^{2n}}\left(\int_{|(x,x)|}^{N|(x,x)|}|\cos(2\pi t)|^2\frac{dt}{t}\right)^{\frac{1}{2}}\\
&\gtrsim & \frac{\log N}{|(x,x)|^{2n}}.
\end{eqnarray*}
Since $\|\psi_N\|_1=1$, we conclude that $\|\mathcal G^{n-\frac{1}{2}}(\psi_N,\psi_N)\|_{\frac{1}{2},\infty}\gtrsim \log N$. Therefore, $\mathcal G^{n-\frac{1}{2}}$ cannot be bounded from $L^1({\mathbb {R}}^n)\times L^1({\mathbb {R}}^n)$ to $L^{\frac{1}{2},\infty}({\mathbb {R}}^n)$. This proves Proposition~\ref{endpoint:sf}.
\qed
\section{Weighted estimates for the square function \texorpdfstring{$\mathcal G^{n-\frac{1}{2}}$}{G1}}\label{sec:sqr}
In this section we prove Theorem~\ref{mainthm} for $T= \mathcal G^{n-\frac{1}{2}}$. The scheme of proof is exactly the same as in the case of maximal function $\mathcal B^{n-\frac{1}{2}}_*$. However, some of the estimates require different arguments. We will point out only the differences to avoid repetition. The $L^p$ boundedness of $\mathcal G^{\alpha}$ for a wide range exponents is proved in~\cite{CKSS}. We will exploit the techniques developed in~\cite{CKSS} to prove our proofs.
First note that in view of the multilinear extrapolation theorem from \cite{KJS}, it is enough to prove the main Theorem~\ref{mainthm} for $\vec{P}=(2,2)$ and all weights in the corresponding class of bilinear weights. More precisely, we need to prove the following.
\begin{theorem} \label{mainthm3} The bilinear Bochner-Riesz operator $\mathcal G^{n-\frac{1}{2}}$ is bounded from $L^{2}(\omega_{1})\times L^{2}(\omega_{2})\rightarrow L^{1}(v_{\omega})$ for all $\vec{\omega}\in A_{\vec{P}},$ where $\vec{P}=(2,2).$
\end{theorem}
Moreover, following the discussion in Section~\ref{max:sec} in order to prove Theorem~\ref{mainthm3} we will require weighted estimates for $\mathcal G^{\alpha}$ when $\alpha>n-\frac{1}{2}$, $L^2\times L^2 \rightarrow L^1$ boundedness of $\mathcal G^{\alpha}$ for $0<\text{Re}(\alpha)<n-\frac{1}{2}$ and an analogue of the key Lemma~\ref{keylem1} in the context of square function. The weighted estimates for $\mathcal G^{\alpha}, \alpha>n-\frac{1}{2}$ are known from \cite{CKSS} as follows
\begin{theorem}\cite{CKSS}\label{weighted1}
Let $n\geq 1$ and $z\in \mathbb C$ be such that $\text{Re}(z)>n-\frac{1}{2}.$ Then the operator $\mathcal G^{z}$ is bounded from $L^{p_{1}}(\omega_{1})\times L^{p_{2}}(\omega_{2})\rightarrow L^{p}(v_{\omega})$ for all $\vec{\omega}\in A_{\vec{P}}$ with $1<p_{1}, p_{2}<\infty$ and $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}$.
\end{theorem}
Also, $L^2\times L^2 \rightarrow L^1$ boundedness of $\mathcal G^{\alpha}$ for $0<\text{Re}(\alpha)<n-\frac{1}{2}$ has been obtained in~\cite{CKSS}. Therefore, we need to establish the following analogue of the key lemma, Lemma~\ref{keylem1}.
\begin{lemma}\label{keylem2}
Let $n\geq 1$ and $z$ be a complex number such that $0< \text{Re}(z)< n-\frac{1}{2}$. Then the following holds
\begin{eqnarray*} \int_{\mathbb{R}^n} |\partial_z \mathcal G^{z}(f,g)(x)| dx &\leq & C_{n+\text{Re}(z)} e^{\mathfrak{C} | \text{Im}(z)|^{2}}\| f\|_{2}\| g\|_{2},
\end{eqnarray*}
where $\mathfrak{C}>0$ is a constant.
\end{lemma}
\noindent
{\bf Proof of Theorem~\ref{mainthm3} :} Assuming Lemma~\ref{keylem2} we follow the method of proof of Theorem~\ref{mainthm} for $T=\mathcal B^{n-\frac{1}{2}}_*$ to deduce the proof of Theorem~\ref{mainthm3}.
Let $z=\alpha+i\tau\in {\mathbb C}$ and consider the linearized version of the square function
\begin{equation*}
\mathcal T^z_b(f,g)(x)=\int_0^\infty\int_{{\mathbb {R}}^n\times{\mathbb {R}}^n}\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)^{z}_{+}\frac{|\xi|^2+|\eta|^2}{R^2}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi ix.(\xi+\eta)}d\xi d\eta b(x,R)\frac{dR}{R},
\end{equation*}
where $b(x,R)\in L^2((0,\infty),\frac{dR}{R})$ with $\int_0^\infty|b(x,R)|^2\frac{dR}{R}\leq 1.$
As in the proof of Theorem~\ref{mainthm} we let $\epsilon_1,\epsilon_2>0$ and $N\in \mathbb N$ and consider the operator
$$\tilde{\mathcal {T}}^{z,\epsilon_{1},\epsilon_{2},N}_{b}(f,g)(x)=\mathcal {T}^{(1-z)\epsilon_{1}+z(n-\frac{1}{2}+\epsilon_{2})}_{b}(f,g)(x)(v_{N}(x))^{z}e^{A z^{2}},$$
such that $A>\mathfrak{C}$ and $v_{N}(x)$ is defined by
$$ v_{N}(x)= \begin{cases} v_{\omega}(x), & \mbox{if } v_{\omega}(x)\leq N \\ N, & \mbox{if }v_{\omega}(x)>N. \end{cases}.$$
The proof from this point onwards may be completed imitating the method of proof of Theorem~\ref{mainthm} without any difficulty. We skip the details to avoid repetition. \qed
\section{Proof of Lemma~\ref{keylem2}}
We begin with the decomposition of the multiplier as previously. Also, see~[\cite{CKSS}, Section~3] for more details. We have
\begin{eqnarray*}
m_R^{\alpha}(\xi,\eta)
&=&\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{\alpha}\frac{|\xi|^2+|\eta|^2}{R^2}
= \sum\limits_{j\geq 2}{m}^{\alpha}_{j,R}(\xi,\eta)+m^{\alpha}_{0,R}(\xi,\eta),
\end{eqnarray*}
where $$m^{\alpha}_{j,R}(\xi,\eta)=\psi\left(2^j\left(1-\frac{|\xi|^2}{R^2}\right)\right)\left(1-\frac{|\xi|^2}{R^2}\right)_+^{\alpha}\left(1-\frac{|\eta|^2}{R^2}\left(1-\frac{|\xi|^2}{R^2}\right)^{-1}\right)^{\alpha}_+\frac{|\xi|^2+|\eta|^2}{R^2}$$
and
$$m^{\alpha}_{0,R}(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{\alpha}\frac{|\xi|^2+|\eta|^2}{R^2}.$$
Let $\mathfrak g_R^{\alpha}$ denote the bilinear operator associated with the multiplier $m^{\alpha}_{j,R}(\xi,\eta)$ and $\mathcal G_j^{\alpha}$ denote the corresponding bilinear square function. Then, we have
\begin{eqnarray*}\label{firstdecom} \mathcal G^{\alpha}(f,g)(x)\leq \mathcal G_0^{\alpha}(f,g)(x)+\sum_{j\geq 2} \mathcal G_j^{\alpha}(f,g)(x).
\end{eqnarray*}
Following the decomposition of the multiplier $m^{\alpha}_{j,R}(\xi,\eta)$ from~[\cite{CKSS}, equation $(7)$] we can write
\begin{eqnarray*}
\mathfrak g^{\alpha}_{j,R}(f,g)(x)
&=&c_{\alpha}\int_0^{\sqrt{2^{-j+1}}}(S_{j,\beta}^{R,t}f(x)A_{Rt}^{\delta}g(x)+\tilde{S}_{j,\beta}^{R,t}f(x)B_{Rt}^{\delta}g(x))t^{2\delta+1}dt,
\end{eqnarray*}
where $\beta>\frac{1}{2}$, $\delta>-\frac{1}{2}$ and $\beta+ \delta=\alpha$ and
\begin{eqnarray*}B_t^{\delta}g(x)=\int_{{\mathbb {R}}^n}\hat{g}(\eta)\left(1-\frac{|\eta|^2}{t^2}\right)^{\delta}_+e^{2\pi ix\cdot\eta} d\eta,
\end{eqnarray*}
\begin{eqnarray*}A_t^{\delta}g(x)=\int_{{\mathbb {R}}^n}\hat{g}(\eta)\frac{|\eta|^2}{t^2}\left(1-\frac{|\eta|^2}{t^2}\right)^{\delta}_+e^{2\pi ix\cdot\eta} d\eta,
\end{eqnarray*}
$${S}_{j,\beta}^{R,t}f(x)=\int_{{\mathbb {R}}^n}\psi\left(2^j\left(1-\frac{|\xi|^2}{R^2}\right)\right)\left(1-\frac{|\xi|^2}{R^2}-t^2\right)_+^{\beta-1}\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi,$$
and
$$\tilde{S}_{j,\beta}^{R,t}f(x)=\int_{{\mathbb {R}}^n}\psi\left(2^j\left(1-\frac{|\xi|^2}{R^2}\right)\right)\frac{|\xi|^2}{R^2}\left(1-\frac{|\xi|^2}{R^2}-t^2\right)_+^{\beta-1}\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi.$$
The same decomposition can be preformed for the multiplier with complex exponent. This gives us the following representation of $g^{z}_{j,R}(f,g)(x)$ for $z\in {\mathbb C}$ with $0<Re(z)<n-\frac{1}{2}$
\begin{eqnarray*}\label{decomope1}
\mathfrak g^{z}_{j,R}(f,g)(x)
&=& c_{z}\int_0^{\sqrt{2^{-j+1}}}[S_{j,\beta}^{R,t}f(x)A_{Rt}^{z-\beta}g(x)+\tilde{S}_{j,\beta}^{R,t}f(x)B_{Rt}^{z-\beta}g(x)]t^{2(z-\beta)+1}dt,
\end{eqnarray*}
where $\beta>\frac{1}{2}, \text{Re}(z)-\beta>-\frac{1}{2}$ and $c_z=\frac{\Gamma(z+1)}{\Gamma(\beta)\Gamma(z-\beta+1)}$, see \cite[page $279$]{SW} for the precise form of the constant $c_z$.
\subsection*{Boundedness of $\mathcal G^{\alpha}_j, j\geq 2$:}
The derivative of $\mathcal \mathfrak g^{z}_{j,R}(f,g)(x)$ is given by
\begin{eqnarray*}
\left(\partial_z\right) \mathcal \mathfrak g^{z}_{j,R}(f,g)(x)
&=& \left(\partial_zc_{z}\right)\int_0^{\sqrt{2^{-j+1}}}[S_{j,\beta}^{R,t}f(x)A_{Rt}^{z-\beta}g(x)+\tilde{S}_{j,\beta}^{R,t}f(x)B_{Rt}^{z-\beta}g(x)]t^{2(z-\beta)+1}dt\\
&&~+ c_{z}\int_0^{\sqrt{2^{-j+1}}}[S_{j,\beta}^{R,t}f(x)\tilde{A}_{Rt}^{z-\beta}g(x)+\tilde{S}_{j,\beta}^{R,t}f(x)\tilde{B}_{Rt}^{z-\beta}g(x)]t^{2(z-\beta)+1}dt\\
&&~+ c_{z}\int_0^{\sqrt{2^{-j+1}}}[S_{j,\beta}^{R,t}f(x)A_{Rt}^{z-\beta}g(x)+\tilde{S}_{j,\beta}^{R,t}f(x)B_{Rt}^{z-\beta}g(x)]t^{2(z-\beta)+1}\log tdt\\
&=& I_{j,R}+II_{j,R}+III_{j,R}
\end{eqnarray*}
where $$\tilde{B}_t^{z-\beta}g(x)=\int_{{\mathbb {R}}^n}\hat{g}(\eta)\left(1-\frac{|\eta|^2}{t^2}\right)_+^{z-\beta}\log\left(1-\frac{|\eta|^2}{t^2}\right)_+ e^{2\pi ix.\eta} d\eta$$ and $$\tilde{A}_t^{z-\beta}g(x)=\int_{{\mathbb {R}}^n}\hat{g}(\eta)\left(1-\frac{|\eta|^2}{t^2}\right)_+^{z-\beta}\frac{|\eta|^2}{t^2}\log\left(1-\frac{|\eta|^2}{t^2}\right)_+ e^{2\pi ix.\eta} d\eta.$$
From the proof of Lemma~\ref{keylem1} we know that constants $|c_z|$ and $|\partial_zc_z|$ increase at most by a constant multiple of $e^{\mathfrak{C}\tau^2}$, where $\mathfrak{C}>0$ is a fixed constant.
Next, we need to prove the desired estimates for each of the three square functions associated with quantities $I_{j,R}, II_{j,R}$ and $III_{j,R}.$ The proof of these estimates may be completed following the scheme of proof for the operator $\partial_z \mathcal {B}^{z}_{j,R}(f,g)$ as in Section~\ref{sec:keylem1}. Of course, we will have to make minor modifications in the arguments, but this part can be completed without much difficulty imitating the proof of its counterpart in Lemma~\ref{keylem1}. In doing so we will require [\cite{JS}, Theorem 5.1], [\cite{CKSS}, Theorem 3.2] and the following proposition.
\begin{proposition}\label{sqf2}
The operator
$$g\to\left(\int_0^\infty\int_{0}^{\sqrt{2^{-j+1}}}|\tilde{A}_{Rt}^{\delta}g(x)|^2dt\frac{dR}{R}\right)^{\frac{1}{2}}$$
is bounded on $L^2({\mathbb {R}}^n)$ for $\text{Re}(\delta)>-\frac{1}{2}$
\end{proposition}
\begin{proof}
Consider
\begin{eqnarray*}
\left\|\left[\int_0^\infty\left(\int_{0}^{\sqrt{2^{-j+1}}}|\tilde{A}_{Rt}^{\delta}g(x)|^2dt\right)\frac{dR}{R}\right]^{\frac{1}{2}}\right\|_2
&=& \left\|\left[\int_{0}^{\sqrt{2^{-j+1}}}\left(\int_0^\infty|\tilde{A}_{R}^{\delta}g(x)|^2\frac{dR}{R}\right)t^4dt\right]^{\frac{1}{2}}\right\|_2\\
&\lesssim& 2^{\frac{5}{4}(-j+1)}\left\|\left[\int_0^\infty|\tilde{A}_{R}^{\delta}g(x)|^2\frac{dR}{R}\right]^{\frac{1}{2}}\right\|_2.
\end{eqnarray*}
Use Plancherel's theorem to deduce that
\begin{eqnarray*}
\left\|\left(\int_0^{\infty}|\tilde{A}_R^{\delta}g(x)|^2 \frac{dR}{R}\right)^{1/2}\right\|_2^2&=&\int_0^{\infty}\int_{{\mathbb {R}}^n}\left|\left(1-\frac{|\eta|^2}{R^2}\right)_+^{\delta}\frac{|\eta|^2}{R^2}\log\left(1-\frac{|\eta|^2}{t^2}\right)_+\hat{g}(\eta)\right|^2dx\frac{dR}{R}\\
&\lesssim & \int_0^{\infty}\int_{{\mathbb {R}}^n}\left|\left(1-\frac{|\eta|^2}{R^2}\right)_+^{\delta+k-\epsilon}\frac{|\eta|^2}{R^2}\hat{g}(\eta)\right|^2dx \frac{dR}{R}\\
&\lesssim& \|g\|_2^2
\end{eqnarray*}
where $\epsilon>0$ is small enough so that $\delta-\epsilon>-\frac{1}{2}$.
\end{proof}
\noindent {\bf Boundedness for $\mathcal G_0^\alpha(f,g)$:} Finally, we need to prove $L^2\times L^2\to L^1$-boundedness of the square function $\mathcal G_0^\alpha(f,g)$.
As earlier we decompose the multiplier in the following manner.
\begin{eqnarray*}
m_{0,R}^\alpha(\xi,\eta)&=&\sum_{j\geq0}\Tilde{m}_{j,R}^\alpha(\xi,\eta)
\end{eqnarray*}
where
$$\Tilde{m}_{j,R}^\alpha(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi\left(2^j\left(1-\frac{|\eta|^2}{R^2}\right)\right)\frac{|\xi|^2+|\eta|^2}{R^2}\left(1-\frac{|\eta|^2}{R^2}\right)_+^{\alpha}\left(1-\frac{|\xi|^2}{R^2}\left(1-\frac{|\eta|^2}{R^2}\right)^{-1}\right)^{\alpha}_+$$
for $j\geq2$, and for $j=0,1$
$$\Tilde{m}_{j,R}^\alpha(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi_0^{j+1}\left(\frac{|\eta|^2}{R^2}\right)\frac{|\xi|^2+|\eta|^2}{R^2}\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{\alpha}.$$
Let $\tilde{\mathfrak g}^{\alpha}_{j,R}$ denote the bilinear multiplier operator associated with $\tilde{m}^{\alpha}_{j,R}(\xi,\eta)$ and let $\tilde{\mathcal G}^{\alpha}_{j}$ be the square function corresponding to $\tilde{\mathfrak g}^{\alpha}_{j,R}$.
Note that for $j=0$ the multiplier
$M(\xi,\eta)=\psi_0\left(\frac{|\xi|^2}{R^2}\right)\psi_0^1\left(\frac{|\eta|^2}{R^2}\right)\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+^{z}\frac{|\xi|^2+|\eta|^2}{R^2}$ $\log\left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)_+$
is a smooth function with its support in a ball of radius $\sqrt{\frac{15}{16}}R$. Consequently, this part can be dominated by the bilinear Hardy-Littlewood maximal function and the desired estimate follows.
When $j=1$ similar to the expression~\eqref{j=1} we get that
\begin{eqnarray*}
\partial_z\left( \tilde{\mathfrak g}^{z}_{1,R}(f,g)(x)\right)&=& \left(\partial_zc_{z}\right)\int_0^{u}[H^{\beta}_{R,t}g(x)A_{Rt}^{z-\beta}f(x)+\tilde{H}^{\beta}_{R,t}g(x)B_{Rt}^{z-\beta}f(x)]t^{2(z-\beta)+1}dt\\
&&~+ c_{z}\int_0^{u}[H^{\beta}_{R,t}g(x)\tilde{A}_{Rt}^{z-\beta}f(x)+\tilde{H}^{\beta}_{R,t}g(x)\tilde{B}_{Rt}^{z-\beta}f(x)t]^{2(z-\beta)+1}dt\\
&&~+ c_{z}\int_0^{u}[H^{\beta}_{R,t}g(x)A_{Rt}^{z-\beta}f(x)+\tilde{H}^{\beta}_{R,t}g(x)B_{Rt}^{z-\beta}f(x)]t^{2(z-\beta)+1}\log tdt
\end{eqnarray*}
where $u=\sqrt{\frac{29}{32}}$, $$H_{R,t}^{\beta}g(x)=\int_{{\mathbb {R}}^n}\psi_0^2\left(\frac{|\eta|^2}{R^2}\right)\left(1-t^2-\frac{|\eta|^2}{R^2}\right)_+^{\beta-1}\hat{g}(\eta)e^{2\pi ix.\eta} d\eta,$$ and
$$\tilde{H}_{R,t}^{\beta}g(x)=\int_{{\mathbb {R}}^n}\psi_0^2\left(\frac{|\eta|^2}{R^2}\right)\left(1-t^2-\frac{|\eta|^2}{R^2}\right)_+^{\beta-1}\frac{|\eta|^2}{R^2}\hat{g}(\eta)e^{2\pi ix.\eta} d\eta.$$
It follows that the bilinear square function associated to each of the terms in the equation above is bounded from $L^2\times L^2$ into $L^1$. Here we need to use $L^2$-boundedness of operators $g\rightarrow \left(\sup_{R>0}\int_0^{u}|H^{\beta}_{R,t}g(x)t^{2\delta+1}|^2 dt\right)^{1/2}$ and $g\rightarrow \left(\int_0^\infty\int_0^{u}|\tilde{H}^{\beta}_{R,t}g(x)t^{2\delta+1}|^2 dt\frac{dR}{R}\right)^{1/2}$ from [\cite{JS}, Section 5.3] and [\cite{CKSS}, page 16] respectively.
Finally, consider the case $j\geq2$. Let $K\in {\mathbb N}$ be such that $\{\xi: (\xi,\eta)\in \operatorname{supp}(\tilde{m}^{\alpha}_{j,R})\}\subseteq \{\xi:|\xi|\leq \frac{R}{8}\}$ for all $j\geq K$. As earlier we may assume that $\psi_0\left(\frac{|\xi|^2}{R^2}\right)= 1$ for all $|\xi|\leq \frac{R}{8}.$ Therefore, for $j\geq K$ the square function $\tilde{\mathcal G}^{\alpha}_{j}$ can be dealt with exactly the same way as $\mathcal G^{\alpha}_{j}$. For the remaining terms, i.e., for $2\leq j<K$ using the approach similar to the expression \eqref{decomope1}, we get that
\begin{eqnarray*}
\tilde{\mathfrak g}^{z}_{j,R}(f,g)(x)
&=& c_{z}\int_0^{\sqrt{2^{-j+1}}}[S_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}A_{Rt}^{z-\beta}f(x)+\tilde{S}_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}B_{Rt}^{z-\beta}f(x)]t^{2(z-\beta)+1}dt,
\end{eqnarray*}
where $c_z=\frac{\Gamma(z+1)}{\Gamma(\beta)\Gamma(z-\beta+1)}$.
Therefore, we see that the derivative is given by
\begin{eqnarray*}
\partial_z \left(\tilde{\mathfrak g}^{z}_{j,R}(f,g)(x)\right)
&=& \left(\partial_zc_{z}\right)\int_0^{\sqrt{2^{-j+1}}}[S_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}A_{Rt}^{z-\beta}f(x)+\tilde{S}_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}B_{Rt}^{z-\beta}f(x)]t^{2(z-\beta)+1}dt\\
&&~+ c_{z}\int_0^{\sqrt{2^{-j+1}}}[S_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}\tilde{A}_{Rt}^{z-\beta}f(x)+\tilde{S}_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}\tilde{B}_{Rt}^{z-\beta}f(x)]t^{2(z-\beta)+1}dt\\
&&~+ c_{z}\int_0^{\sqrt{2^{-j+1}}}[S_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}A_{Rt}^{z-\beta}f(x)+\tilde{S}_{j,\beta}^{R,t}g(x)B^{\psi_0}_{R}B_{Rt}^{z-\beta}f(x)]t^{2(z-\beta)+1}\log tdt\\
&=& \tilde{I}_{j,R}+\tilde{II}_{j,R}+\tilde{III}_{j,R}
\end{eqnarray*}
The $L^2\times L^2\to L^1$ estimates for the square functions corresponding to the expressions $\tilde{I}_{j,R}$ and $\tilde{III}_{j,R}$ can be obtained similarly as for the operator $\mathcal G_{j}^z$. We will need to make use of $L^2$-boundedness of $f\rightarrow \left(\sup_{R>0}R_j^{-1}\int_0^{R_j}|B^{\psi_0}_{R}B_t^{\delta}f(x)|^2 dt\right)^{1/2}$ from [\cite{JS}, page 24] and the operator $f\rightarrow \left(\int_0^{\sqrt{2^{-j+1}}}\int_0^\infty|B^{\psi_0}_{R}A_{Rt}^{\delta}f(x)|^2 dt\right)^{1/2}$ from [\cite{CKSS}, page 15].
Finally, the square function for the second expression $\tilde{II}_{j,R}$ is dealt with as follows.
\begin{eqnarray*}
&& \left\|\left(\int_0^\infty|\tilde{II}_{j,R}|^2\frac{dR}{R}\right)^{\frac{1}{2}}\right\|_1\\
&\lesssim& |c_{z}|\left\|\sup_{R>0}\left(\int_{0}^{\sqrt{2^{-j+1}}}|S_{j,\beta}^{R,t}g(\cdot)t^{2\delta+1}|^2dt\right)^{\frac{1}{2}}\right\|_2 \left\|\left[\int_0^\infty\left(\int_{0}^{\sqrt{2^{-j+1}}}|B^{\psi_0}_{R}\tilde{A}_{Rt}^{\delta}f(x)|^2dt\right)\frac{dR}{R}\right]^{\frac{1}{2}}\right\|_2\\
&&~+ 2^{-\frac{j}{4}}|c_{z}|\left\|\left(\int_0^\infty\int_{0}^{\sqrt{2^{-j+1}}}|\tilde{S}_{j,\beta}^{R,t}g(\cdot)t^{2\delta+1}|^2dt\frac{dR}{R}\right)^{\frac{1}{2}}\right\|_2 \left\|\sup_{R>0}\left(\frac{1}{R_j}\int_0^{R_j}|B^{\psi_0}_{R}\tilde{B}_t^{\delta}f(x)|^2dt\right)^{\frac{1}{2}}\right\|_2
\end{eqnarray*}
Recall that the operator $f\rightarrow \left(\sup_{R>0}R_j^{-1}\int_0^{R_j}|B^{\psi_0}_{R}\tilde{B}_t^{\delta}f(x)|^2 dt\right)^{1/2}$ satisfies the desired $L^2$ estimates, see equation~\eqref{L21}. Also, the $L^2$ estimates for $f\rightarrow \sup_{R>0}\left(\int_{0}^{\sqrt{2^{-j+1}}}|S_{j,\beta}^{R,t}g(\cdot)t^{2\delta+1}|^2dt\right)^{\frac{1}{2}}$ and $f\rightarrow \left(\int_0^\infty\int_{0}^{\sqrt{2^{-j+1}}}|\tilde{S}_{j,\beta}^{R,t}g(\cdot)t^{2\delta+1}|^2dt\frac{dR}{R}\right)^{\frac{1}{2}}$ are known from [\cite{JS}, Theorem 5.1] and [\cite{CKSS}, Theorem 3.2] respectively. Therefore, in order to conclude the $L^2\times L^2\to L^1$ boundedness of the square function we only need to prove the following estimate.
\begin{eqnarray}\label{L22}
\left\|\left(\int_0^\infty\int_{0}^{\sqrt{2^{-j+1}}}|B^{\psi_0}_{R}\tilde{A}_{Rt}^{\delta}f(\cdot)|^2dt\frac{dR}{R}\right)^{\frac{1}{2}}\right\|_2\lesssim 2^{\frac{5}{4}(-j+1)}\|f\|_2~\text{for}~ \text{Re}(\delta)>-\frac{1}{2}\end{eqnarray}
Consider
\begin{eqnarray*}
\left(\int_0^\infty\int_{0}^{\sqrt{2^{-j+1}}}|B^{\psi_0}_{R}\tilde{A}_{Rt}^{\delta}f(x)|^2dt\frac{dR}{R}\right)^{\frac{1}{2}}
&=&\left(\int_{0}^{\sqrt{2^{-j+1}}}\int_0^\infty|B^{\psi_0}_{R}\tilde{A}_{Rt}^{\delta}f(x)|^2\frac{dR}{R}dt\right)^{\frac{1}{2}}\\
&\lesssim& 2^{\frac{5}{4}(-j+1)}\left(\int_0^\infty|B^{\psi_0}_{R}\tilde{A}_{Rt}^{\delta}f(x)|^2\frac{dR}{R}\right)^{\frac{1}{2}}\\
&\lesssim& 2^{\frac{5}{4}(-j+1)} \left(\int_0^{\infty}|M(\tilde{A}_t^{\delta}f(x)|^2 t^{-1}dt\right)^{1/2}
\end{eqnarray*}
Here we have used that $\sup_{R>0}|B^{\psi_0}_{R}f(x)|\lesssim M(f)(x)$. Finally,
invoking vector-valued estimates for the Hardy-Littlewood Maximal function, see~\cite{DK, FS}, and $L^2$ estimate for the square function corresponding to $\tilde{A}_{t}^{\delta}$ we get the desired estimate~\eqref{L22}.
This completes the proof of the key Lemma~\ref{keylem2}.
\qed
\section*{Acknowledgement} The first author is supported by CSIR (NET), file no. 09/1020(0182)/2019-EMR-I. The second author acknowledges the support from Science and Engineering Research Board, Department of Science and Technology, Govt. of India under the scheme Core Research Grant with file no. CRG/2021/000230.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,884
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/**
*/
package guizmo.structure;
import org.eclipse.emf.common.util.EList;
/**
* <!-- begin-user-doc -->
* A representation of the model object '<em><b>Radio Group</b></em>'.
* <!-- end-user-doc -->
*
* <p>
* The following features are supported:
* <ul>
* <li>{@link guizmo.structure.RadioGroup#isMultipleSelection <em>Multiple Selection</em>}</li>
* <li>{@link guizmo.structure.RadioGroup#getSelected <em>Selected</em>}</li>
* </ul>
* </p>
*
* @see guizmo.structure.StructurePackage#getRadioGroup()
* @model
* @generated
*/
public interface RadioGroup extends SingleWidget, Itemizable {
/**
* Returns the value of the '<em><b>Multiple Selection</b></em>' attribute.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Multiple Selection</em>' attribute isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Multiple Selection</em>' attribute.
* @see #setMultipleSelection(boolean)
* @see guizmo.structure.StructurePackage#getRadioGroup_MultipleSelection()
* @model
* @generated
*/
boolean isMultipleSelection();
/**
* Sets the value of the '{@link guizmo.structure.RadioGroup#isMultipleSelection <em>Multiple Selection</em>}' attribute.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Multiple Selection</em>' attribute.
* @see #isMultipleSelection()
* @generated
*/
void setMultipleSelection(boolean value);
/**
* Returns the value of the '<em><b>Selected</b></em>' reference list.
* The list contents are of type {@link guizmo.structure.Item}.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Selected</em>' reference list isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Selected</em>' reference list.
* @see guizmo.structure.StructurePackage#getRadioGroup_Selected()
* @model
* @generated
*/
EList<Item> getSelected();
} // RadioGroup
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,275
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The 2021 FIVB Beach Volleyball World Tour was the final edition of the global elite professional beach volleyball circuit organized by the Fédération Internationale de Volleyball (FIVB) for the 2021 beach volleyball season. When it started in late February 2021, it comprised eight FIVB World Tour 4-star tournaments, three 2-star events, 17 categorized as 1-star and the World Tour Finals, all organized by the FIVB.
The full calendar of events was updated on September 13, 2021.
Schedule
Key
Men
Women
Medal table by country
References
External links
2021 FIVB Beach Volleyball World Tour at FIVB.org
World Tour
2021
FIVB
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,927
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Imran demands FATA's merger with KP as party lawmakers submit resignations
PTI chief says vacuum exists in FATA which will lead to terrorism if not filled
Says police reforms necessary in wake of Kasur rape cases, Naqeebullah's murder
ISLAMABAD: Pakistan Tehreek-e-Insaf (PTI) Chairman Imran Khan on Tuesday demanded the immediate merger of FATA with Khyber Pakhtunkhwa (KP) as he confirmed receiving resignations of party lawmakers.
Addressing a press conference, the PTI chief said that the final decision regarding the resignations will be taken after consultations with senior PTI leaders.
Sources said that in a session of the PTI's central executive committee (CEC) earlier, party members had given the authority to decide on resignations from assemblies to Imran Khan and agreed that party chief's decision in the matter would be final.
During the meeting, committee members also proposed suggestions for nominations on party tickets for the upcoming general elections.
The CEC approved, during the session, the process for concluding the party's membership drive and passed motions of solidarity with Palestine and Kashmir.
Two factions in the party seemed to have emerged – one in favour of resignations from the assembly, and the other strongly opposed to it.
The group opposed to resignations from assemblies and dissolving the Khyber Pakhtunkhwa legislature reasoned that PTI would not be able to claim its share in the caretaker government if it takes the extreme measure, sources said.
The PTI chairman in his presser said that a vacuum exists in FATA, which if not filled, will again lead to a resurgence of terrorism in tribal areas. "There is now a vacuum in the tribal areas as the old system has been done away with. We feel there is a danger that if the vacuum remains than terrorism will again take root in the region."
Elaborating further, Imran said the country, the people of the tribal areas and the armed forces have rendered great sacrifices in the fight against terrorism and a resurgence of terrorism will hurt the entire country.
"Hence, we are demanding the immediate merger of FATA with KP. Extending the jurisdiction of the court by itself is not enough. There is no longer a structure present there. They are hurting the people of the tribal areas and Pakistan," added the PTI chairman.
"We feel Maulana Fazlur Rehman is playing a very dangerous game. The government and Fazlur Rehman are scared as they think if FATA is merged with KP then it will benefit PTI," said Imran.
Corruption in Sindh and Punjab
Speaking in regards to alleged corruption in Punjab and Sindh, the PTI chairman said the party will be forming committees of experts to collect data.
He alleged that there are huge kickbacks involved in construction projects in Punjab and questioned the contract of Orange Line Train Project regarding its confidential nature. The PTI chief further alleged that the incumbent leaders make billions through such large-scale construction projects.
"Why is the contract for LNG confidential? This is mega corruption. They build mega projects and mint money from them," alleged the PTI chairman.
Explaining the role of the committees, Imran said they would be responsible for collecting data regarding corruption in public sector projects.
Police reforms
Giving the reference to child abuse cases in Kasur and the extrajudicial killing of Naqeebullah in Karachi, the Imran said that police reforms are the need of the hour.
"The way Najeebullah was killed extrajudicially in Karachi, the way Kasur cases have come to the forefront, it is evident that police reforms are sorely needed."
"The police has been politicised and is corrupt. When you promote police officers without merit, the police become incompetent," said the PTI chairman. He added that the PTI government in KP has passed a Police Act, which allows only the IG to confirm transfers and postings and there is no interference.
Imran added that the people of KP have confidence in their police force.
In regards to the regular firing by Indian forces on the Line of Control (LoC), Imran claimed that entire villages in the Narowal area have been emptied and significant casualties have been sustained by the civilian population residing along the border with India.
"Our delegates ask us what is the reason behind the firing, as whenever Nawaz Sharif gets into trouble such incidents increase," he said.
Imran, on the occasion, also condemned the US move to move its embassy to Jerusalem.
PTI chief
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,219
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MyPlate for Adults — What's on Your Plate?
This series of tipsheets and a brochure help consumers make healthy food choices for their families. They promote MyPlate released by the United States Departments of Agriculture and Health and Human Services in June of 2011 to promote healthy food choices.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,065
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Q: Export Domino 10 view to Excel without Lotus 1-2-3 An old topic, but a new problem. I have been using Export AS 1-2-3 and saving as .wk4 file, then using the old 1-2-3 app to save as Excel. In Domino 10, that export method has been removed. I have a view with 2500 documents and hundreds of rows. I have tried for years to export as CSV, Tabular, and structured, but the data never exports totally correctly regardless of how I try to format, use the tools in Excel when opening, etc. Without a script to write to an excel file, is there no simple way to export data from a Domino view into a spreadsheet?
A: You can get back the WK4 option by setting in the notes.ini :
IncludeOldFileTypes=1
True to say that "hundreds of columns" cannot be exported as easily, cause of the great size of data. It is also a kind of security... why would you not put a script, it would be so simple ?
Other ways to export data :
*
*Notesmanager freeware https://unplugged.openntf.org/main.nsf/project.xsp?r=project/Notesmanager/summary
*NotesSQL driver (not simple)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,595
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Album of the Day 11/24/10
Every day our 1000 best albums of all time countdown continues all the way to #1. Wednesday's album is #797:
Lefty Frizzell – 16 Biggest Hits
Lefty Frizzell was a legendary Texas honkytonk singer from the 50s, a guy who sounded a lot older than he was. By the 70s, now in his 40s, he sounded close to 70. One of the songs here, an early proto-rockabilly number, is titled Just Can't Live That Fast (Any More), but in real life he didn't seem to have any problem with that. He drank himself to death at 47 in 1975. But he left a rich legacy. This album is missing some of his best-known songs – notably Cigarettes & Coffee Blues – but it's packed with classics. Frizzell's 1950 version of If You've Got The Money I've Got The Time topped the country charts and beat Hank Williams – a frequent tourmate – at his own game. Other 50s hits here include the western swing-tinged Always Late (With Your Kisses), the fast shuffle She's Gone, Gone, Gone and Frizzell's iconic version of Long Black Veil – with its echoey, ghostly vocals and simple acoustic guitar, it's even better than the Johnny Cash version. From the 60s, there's the surprisingly folkie version of Saginaw Michigan, the sad drinking ballad How Far Down Can I Go, the torchy, electric piano-based That's the Way Love Goes and I'm Not the Man I'm Supposed to Be. His later period is best represented by I Never Go Around Mirrors, later covered by both George Jones and Merle Haggard. This is one of those albums that pops up in used vinyl stores from time to time, but isn't easy to find online. There's a popular "500 greatest country songs" torrent with several of these on it out there; if you see one for this particular album, let us know!
November 24, 2010 Posted by theamyb | country music, lists, Music, music, concert | best albums, best albums all time, best albums alltime, best albums ever, best albums list, best albums lucid culture, best country albums all time, best country albums alltime, best country albums ever, best country music, best music, best music ever, best obscure albums, best obscure albums all time, best obscure albums alltime, best obscure albums ever, best underrated albums, classic country, classic country music, country music, greatest albums all time, greatest albums alltime, greatest albums ever, greatest country albums all time, greatest country albums alltime, greatest country albums ever, greatest obscure albums, hank williams, hillbilly music, honkytonk music, johnny cash, lefty frizzell, lefty frizzell 16 biggest hits, most underrated albums, most underrated albums all time, Music, rockabilly, top albums all time, top albums alltime, top albums ever, vintage country | Leave a comment
Uncle Leon and the Alibis Raise the Roof at Rodeo Bar
"I love you, Leon!" a girl hollered from the back of the bar. Uncle Leon, frontman of Uncle Leon and the Alibis is not your typical babe magnet – he could be Joba Chamberlain's wiser, older brother (they have a similar midwestern blue-collar look). But he pulls demographics that your average bunch of Strokes wannabes would kill for. Back in the early-to-mid zeros these guys put on some of the funnest, funniest shows in town…and then they broke up. It didn't really matter that they weren't particularly tight, because Leon's David Allan Coe-style songs were so funny. The first thing that hits you is what a good band this new version of the group is – they don't need to be funny all the time to be interesting. Lead guitarist Charlie Aceto plays the stuff Leon can't, and has a good handle on Bakersfield guitar – and he can do Social Distortion roots-punk and blues too. Maria on the drums is missed – she was always at least half of why the original band was so irresistible – but the guy who replaced her is solid and and can really swing, teaming up with bassist Neil Magnuson.
The thing that separates these guys from the rest of the funny country bands out there is that their jokes are usually pretty smart and edgy: they don't just rely on cornball cornpone humor. Leon's specialty is the battle of the sexes: the good guys always lose, badly. That's how he comes across – that, and his resonant baritone probably explain the presence of all the women at his shows. Sure, he's having fun up there, but the guy can flat-out sing. That this particular set was successful without either of his big hits, I Hate My Job or Drugstore Roses (or his cover of Baby Got Back), speaks for how good the rest of the material was. They opened with a blackly funny faux murder ballad based on a real-life encounter between Leon and a bounty hunter in a Dairy Queen parking lot somewhere in Kansas. My Love Is Like a Monster Truck was what you'd think it was: monster trucks use up a lot of rubber (that might not have actually been one of the lyrics, but it could have been). A slowly swaying, mournful ballad turned into a kiss-off anthem: "When you said 'I love you,' I thought that meant just me," Leon explained. They blasted through a truck-driving number, Blue Sky and Asphalt and then a boisterous version of Hot Rod Mamas, where he skewered "catalog girls" with their perfect everything and their selfcenteredness – he likes a girl with a little junk in the trunk but with brains too.
They did three covers: an understatedly vicious version of Hank Williams' My Love For You Has Turned to Hate, the Merle Haggard classic Swingin' Doors and a practically halfspeed, swinging, straight-up country take of the Stones' Dead Flowers – that song's retirement date may have come and gone a long time ago, but damned if these guys didn't make it sound fresh. They wrapped up their first set with a cowpunk number – Good Time Woman? Two Time Woman? Two Ton Woman? It could have been any of them, maybe more than one.
Uncle Leon is not only a singer, he's a co-founder of Brooklyn Country, who maintain an excellent site dedicated to country and roots music in New York, with a concert calendar, interviews and the occasional album review. Kind of like us, but more specialized. Uncle Leon and the Alibis' next gig is at midnight on 9/11 at Southpaw as part of the excellent three-day Brooklyn Country Music Festival.
July 26, 2010 Posted by delarue | concert, country music, Live Events, Music, music, concert, New York City, review, Reviews | brooklyn country, brooklyn country music festival, brooklyn country music festival 2010, charlie aceto, country and western music, country music, country music parody, cowpunk, david allan coe, funny country music, funny songs, hank williams, honkytonk music, joke band, merle haggard, musical satire, neil magnuson, parody band, rolling stones cover, roots music, satirical music, uncle leon, uncle leon and the alibis, uncle leon and the alibis rodeo bar, uncle leon rodeo bar | 2 Comments
CD Review: Jay Bennett – Whatever Happened, I Apologize
What a harrowing way to start the new year. This cd hits you with a gale force, bitter, brutal and direct. Even if you try to get out of the way, Jay Bennett – the talented multi-instrumentalist who for all intents and purposes was Wilco until he left the band and Jeff Tweedy decided to become Brian Wilson – will still knock the wind out of you. Most of this cd – Bennett's fourth solo album – is just voice and acoustic guitar, occasionally embellished with organ and bass that are so good that you're left wanting more. While the songs on this album scream out for a full band to flesh them out, even if this is as far as they ever get, that's fine: they still pack a wallop. Stylistically, Bennett evokes Matt Keating or Richard Buckner in particularly energetic mode: this is smart, terse, gorgeously melodic Americana rock with equally smart, tersely unwinding lyrics. It's a concept album about a relationship gone awry, spectacularly: this one was doomed right from the start, and if Bennett is to be taken at face value, it's something of a miracle he got out alive.
The cd starts with a road song, just a bit of ominous foreshadowing in the same vein as the Wilco classic Far, Far Away (from the Being There cd), followed by the matter-of-fact, dismissive I Don't Have the Time. Bennett knows there's drama coming down the line and he wants no part of it. "I don't have the good looks, but I know yours won't last," he caustically tells the woman. With the next cut, I'll Decorate My Love, the genie's out of the bottle, Pandora's out of the box and all hell breaks loose, setting the tone for the rest of the cd:
There will be no profit in protection
Even when you're walking miles in the rain
I will curse the day I met you
And you will curse the day I lost control
And there will be no reward for your actions
Even when you're trying to save your lover's soul…
You were down before me
The themes that recur again and again here are missed opportunities and wasted time (go figure), notably on the cd's towering centerpiece, the big, crescendoing 6/8 ballad The Engines Are Idle:
The engines are idle and the trees are all bare
And the issues are clouded and hang in the air
The best part of the story is the part that you missed…
The best part of the record is the part where it skips
And she lost the lyrics and the jacket is ripped…
Cos it's ageless and timeless but beauty must fade
And you looked so much better when the picture was made
The pace picks up and emotions reach a fever pitch on How Dull They Make the Razor: Bennett wants to wait this one out, but he ends up getting dragged in anyway:
It don't matter how dull they make the razor
You won't feel it when you're dead
On the next track, Without the Benefit of Sight, Bennett likens himself to a block of ice on a Chicago rooftop in early spring, loosened just enough to become deadly. Exasperation and despair take over center stage:
If you want to weigh me down there's just one layer left
I've been repainted so many times it's anybody's guess
And that's pretty much where it's left. Bennett muses on how Hank Williams might have written this story, then throws up his hands and lets that work as a smokescreen: he's through with trying to cut through the smoldering underbrush, and the songs follow suit. "I lost my best friend last night, I'm working on number two/Won't you give me a chance cause your chances are through," he warns on the stark, mandolin-spiced ballad Talk and Talk and Talk. The cd ends with a lament for the world as a whole – the relationship seems to be a microcosm of something far worse – and then with the understatement of Little Blue Pills, "that don't make you ill – someday they will."
Intensely personal yet not the least bit self-absorbed, this is the best thing Bennett's ever done. And the best thing about it is that the cd is absolutely free: Bennett is giving it away as a free download at rockproper.com, click here and then hang on, this is not exactly easy listening.
January 5, 2009 Posted by delarue | Music, music, concert, review, Reviews, rock music | acoustic rock, americana, americana music, americana rock, best songwriters, best songwriters all time, best songwriters ever, chicago bands, chicago musicians, death album, folk music, folk singer, free download, hank williams, i'll decorate my love, Jay Bennett, jay bennett download, jay bennett final album, jay bennett free download, jay bennett free mp3, jay bennett i'll decorate my love, jay bennett last album, jay bennett torrent, jay bennett whatever happed i apologize, jeff tweedy, matt keating, multi-instrumentalist, richard buckner, rock proper, rock proper chicago, rockproper, singer-songwriter, songwriter, suicide album, whatever happened I apologize, wilco, wilco band | 3 Comments
Real Live Bluegrass in New York City? Yee Ha!
All you out-of-towners might be shocked to know that there's a vibrant bluegrass scene in New York. The Dixie Bee-Liners, whose new album just hit #1 on the Roots Music Report got their start here. Since they left town, the best band around these parts is Straight Drive, whose gorgeously soulful performance of old-time, old school style bluegrass at Banjo Jim's Saturday night would have made Bill Monroe proud. A lot of new bluegrass bands give off a coldly sterile, fussily technical vibe, but not this crew. Fiddle player Ronnie Feinberg made his marvelously precise runs look effortless. Banjo player Terry McGill was even more impressive when not soloing than when he was. He has great technique and a terrific way of building to a crescendo, but when he plays rhythm, he doesn't just comp chords: he uses the whole fretboard, toying expertly with the melody. He threw everybody for a surprise by ending one song with a couple of high chromatics, and then bent the neck of his banjo ever so slightly to raise the pitch. Their new mandolinist is a vast improvement over the guy he replaced, the bass player pushed the beat along and frontwoman Jen Larson was brilliant as usual. Incongruous as it may seem, the most striking and haunting voice in maybe all of bluegrass belongs not to someone south of the Mason-Dixon line, but to this casually captivating architecture historian originally from Boxford, Massachusetts.
But she didn't do the haunting thing tonight. This was Straight Drive's fun set. This crew knows that a lot of bluegrass is dance music, and while they didn't get the crowd on their feet, everybody except the trio of trendoids in the corner yakking away, oblivious to the music, were swaying back and forth and clapping along. Their version of Bill Monroe's (Why Put Off Til Tomorrow) What You Can Do Today had fire and bounce; their cover of Hank Williams' Blue Love was nothing short of sultry. The best of the vocal numbers, which they interspersed among the instrumentals, was a warmly swaying 6/8 number written by Larson that wouldn't be out of place on a Dolly Parton record from the mid-sixties. Larson can give you chills but tonight's show proved she can also make you smile and keep your head bobbing in time with the melody. Like most of the best New York bands, they don't do a lot of shows here because the money is on the road, where audiences are used to lousy cover bands, and a show by a group like Straight Drive is a special treat that you can't just see any old day.
February 11, 2008 Posted by delarue | concert, Live Events, Music, music, concert, New York City, review, Reviews | americana, americana music, banjo jim's, banjo jim's nyc, best bluegrass singer, best country singer, bill monroe, bluegrass, bluegrass music, bluegrass new york, bluegrass nyc, country music, country music new york, country music nyc, dolly parton, hank williams, hillbilly music, jen larson, jen larson bluegrass, jennifer larson bluegrass, ron feinberg, ron feinberg fiddle, ron feinberg violin, ronnie feinberg, roots music, straight drive, straight drive bluegrass, terry mcgill, terry mcgill banjo, women in bluegrass | 2 Comments
CD Review: The Jack Grace Band – The Martini Cowboy
The Jack Grace Band have been sort of the opening act du jour on the country circuit, opening for Merle and Willie Nelson and Jerry Lee, et al.. If this is an attempt to get some notice from the retro country crowd, it ought to work. Hell, this ought to get them on the Grand Old Opry, if they don't mind songs about cocaine at the Ryman Auditorium.
The Jack Grace Band's last album I Like It Wrong put in some serious overtime on some of the better jukeboxes across the counry. In fact, you could say that it was the party album of the summer of 2004. Suffused in booze and tested live on crowds of drunks in dives all over town, those songs were every smart party animal's alternative to Jimmy Buffett. It may therefore come as some surprise that the new album by the Jack Grace Band is an attempt to – gasp – make a serious record. I say record because the cd is divided into a distinct side 1 and side 2. A concept album, no less, complete with little instrumental fragments separating the songs, and something of a central, unifying theme. The most surprising thing about it is that it actually works. Tight, focused, thoughtfully conceived, in other words, everything Grace's previous work was NOT. Which ironically was always his saving grace – the band may have been a little loose, the whiskey may have run rivers but you always knew that if you went to see these guys live you would have a good time. While it doesn't look like anybody left the bar for very long to make this album, it's a hundred eighty degrees from what you might expect after hearing the last one. Is it possible that Grace has actually matured?
The Martini Cowboy is packed with haunting, gorgeously old-fashioned, 1960s style country songs with tasteful electric guitar, soaring pedal steel, piano and a rhythm section that swings like the dickens. You can dance to this stuff more than you can Grace's older stuff. Because ultimately that's why honkytonks exist: where else can you squeeze your cheatin' lover against the jukebox and sway to the strains of Merle Haggard? Who happens to be exactly who the first song, the album's title track, evokes. Straight up. When he's on top of his game Jack Grace's songs sound like country classics from 40 or 50 years ago. The cd's second song, Broken Man continues in a purist vein, driven by Jon Dryden's beautiful, incisively minimal honkytonk piano "I'm not gonna go out there tonight," swears the Martini Cowboy. He's been burned too many times. Which leads perfectly into the next song, Cry, a sexy bossa beat and groovalicious bass player Daria Grace's bop-bop backing vocals only momentarily distracting from its eerie minor-key drive and bitter lyrics. When after a surprisingly jaunty, jazzy guitar solo the thing stumbles out of its groove and literally falls apart, the effect is nothing short of heartbreaking.
The album's next track Trying to Get Away from Nothing at All zooms in on our protagonist trying to pull himself away from the brink. It's a showcase for Jack Grace's voice, a big, Johnny Cash style baritone that can handle the over-the-top whiskey-drinking anthems and the dark, disturbing ballads with equal aplomb.
After that song, we get Sugarbear, another minor-key Waits-esque number with ambient steel guitar, and Rotary Phone, arguably the album's best song , a haunting, skeletal minor-key blues: "Let me tell a story about the way it used to be/With a rotary phone don't leave a message for me/You're gonna be an old man too…"
The last song of the "A side", What I Drink and Who I Meet at the Track (Is My Business) is completely self-explanatory – it's one of those songs that someone should have written long ago, and that it took this long before someone did is a mystery. It's a good thing that it was this guy who wrote it and not Neil Sedaka. I mean, can you imagine Neil Sedaka at the track? No, you can't. He'd get killed before he got to the stands.
The "B side" begins with Uncle Luther. By now, the Martini Cowboy has fallen in love. His Uncle Luther is moving back to the shack he hasn't lived in for ten years and the Martini Cowboy has to get out. But that's not what's bugging him. It's that he can't stop thinking about her. Yeah, her, and it scares the hell out of him. The following tune, Verge of Happiness is so George Jones it's not funny, in fact it's scary, right down to the vocals. Nobody ever did desperate, lost love songs better than Jones, anyway, so it makes sense. Happy in the Fall continues in the No Show Jones vein "I'm happy in the fall, but I don't like the landing," Grace muses ruefully as the band swings behind him. The album's climactic track, Something to Look Forward To – where the guy finally gets the girl – is a bit of a letdown. Like at the end of Siddhartha when the guy finally gets to India and all he finds is…OMMMMM (hey, this is a serious album, I'm trying to be serious about this).The cd concludes with a real old-timey number called Spike Down, which sounds like an electrified version of some obscure 19th century folk blues.
There's not a weak song on this album – which is more impressive than you think. Hell, even Sergeant Pepper had that stupid phony raga tune that Harrison sang. And Merle Haggard's greatest hits albums all seem to have those horrid pro-Vietnam War ditties he did before he woke up and smelled the coffee. So the Martini Cowboy's in pretty good company. If this doesn't get him the big record deal (memo to the band – WATCH YOUR BACK), Jack Grace can always fall back on his side project Van Hayride, which plays country covers of Van Halen songs. I'm not making this up. Not a word.
May 9, 2007 Posted by delarue | Music, music, concert, review, Reviews | americana music, blues music, classic country, country blues, country music, country rock, daria grace, george jones, hank williams, hard country, i like it wrong, Jack Grace, Jack Grace band, jack grace band martini cowboy, jack grace i like it wrong, jack grace martini cowboy, jimmy buffett, johnny cash, jon dryden, jon dryden piano, martini cowboy, merle haggard, Van Halen cover, van halen cover band, van hayride | 7 Comments
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,384
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Q: Android BLE Gatt connection change statuses I have an android app to connect to a BLE device and write to it. I can successfully connect, read and write to it. As a part of testing, we are trying different disconnection scenarios.
Sometimes, if BLE device disconnects the connection, I get the connection change as disconnect with status value as 19. Also if there is any bond error, status equals 22. If I programmatically disconnect the connection, this status gives me 0. But none of these states except 0 are specified in android documentation.
Posting a sample BluetoothGattCallback
private BluetoothGattCallback bluetoothGattCallback = new BluetoothGattCallback() {
@Override
public void onConnectionStateChange(BluetoothGatt gatt, int status, int newState) {
Log.i(TAG, "onConnectionStateChange status: "+status+", newState: "+newState);
/*i need to know the possible values for this status variable*/
if(newState == BluetoothProfile.STATE_CONNECTED) {
gatt.discoverServices();
} else {
gatt.close();
}
}
@Override
public void onServicesDiscovered(BluetoothGatt gatt, int status) {
Log.i(TAG, "onServicesDiscovered service discovered");
}
};
Does anyone face this same problem and sorted out the list of statuses. I need to know the possible values for status variable in onConnectionStateChange method
A: Sorry to bring up an old question, but here is the solution for many of the problems i've had with Bluetooth (BLE) 4.0. Sorry again for the big classes below but be sure they are needed and no method there is irrelevant or unused.
public abstract class AbstractBluetoothBroadcaster extends BroadcastReceiver {
protected static final String LOG_TAG = BluetoothLowEnergy.LOG_TAG;
protected BluetoothLowEnergy bluetoothLowEnergy;
public AbstractBluetoothBroadcaster(BluetoothLowEnergy bluetoothLowEnergy, String action){
super();
this.bluetoothLowEnergy = bluetoothLowEnergy;
IntentFilter intentFilterStateChange = new IntentFilter(action);
intentFilterStateChange.setPriority(IntentFilter.SYSTEM_HIGH_PRIORITY);
this.bluetoothLowEnergy.getActivity().registerReceiver(this, intentFilterStateChange);
}
public void onDestroy(){
this.bluetoothLowEnergy.getActivity().unregisterReceiver(this);
}
}
public class BluetoothBondStateBroadcaster extends AbstractBluetoothBroadcaster {
private BluetoothLowEnergy bluetoothLowEnergy;
private boolean deviceBonded;
public BluetoothBondStateBroadcaster(BluetoothLowEnergy bluetoothLowEnergy) {
super(bluetoothLowEnergy, BluetoothDevice.ACTION_BOND_STATE_CHANGED);
this.bluetoothLowEnergy = bluetoothLowEnergy;
this.deviceBonded = false;
}
@Override
public void onReceive(Context context, Intent intent) {
String action = intent.getAction();
if (action == null){
return;
}
BluetoothDevice bluetoothDevice = intent.getParcelableExtra(BluetoothDevice.EXTRA_DEVICE);
if (action.equals(BluetoothDevice.ACTION_BOND_STATE_CHANGED) &&
bluetoothDevice != null &&
bluetoothDevice.getAddress().equals(bluetoothLowEnergy.getDeviceUUID())) {
int state = intent.getIntExtra(BluetoothDevice.EXTRA_BOND_STATE, -1);
switch (state) {
case BluetoothDevice.BOND_NONE:
Log.d(LOG_TAG, " NOT BONDED - dev " + bluetoothDevice.getAddress());
this.deviceBonded = false;
break;
case BluetoothDevice.BOND_BONDING:
Log.d(LOG_TAG, " BONDING ... - dev " + bluetoothDevice.getAddress());
break;
case BluetoothDevice.BOND_BONDED:
Log.d(LOG_TAG, " BONDED - dev " + bluetoothDevice.getAddress());
deviceBonded = true;
bluetoothLowEnergy.onBluetoothBonded();
break;
default:
break;
}
}
}
public void resetDeviceBonded(){
this.deviceBonded = false;
}
public boolean isDeviceBonded() {
return deviceBonded;
}
}
public class BluetoothPairingBroadcaster extends AbstractBluetoothBroadcaster {
private String devicePIN;
public BluetoothPairingBroadcaster(BluetoothLowEnergy bluetoothLowEnergy){
super(bluetoothLowEnergy, BluetoothDevice.ACTION_PAIRING_REQUEST);
this.devicePIN = "";
}
@Override
public void onReceive(Context context, Intent intent) {
String action = intent.getAction();
if (action == null){
return;
}
BluetoothDevice bluetoothDevice = intent.getParcelableExtra(BluetoothDevice.EXTRA_DEVICE);
int pairingType = intent.getIntExtra(BluetoothDevice.EXTRA_PAIRING_VARIANT, BluetoothDevice.ERROR);
if (action.equals(BluetoothDevice.ACTION_PAIRING_REQUEST) &&
bluetoothDevice != null &&
bluetoothDevice.getAddress().equals(bluetoothLowEnergy.getDeviceUUID()) &&
!getDevicePIN().isEmpty()) {
if (pairingType == BluetoothDevice.PAIRING_VARIANT_PIN){
bluetoothDevice.setPin(getDevicePIN().getBytes());
Log.d(LOG_TAG," Auto-entering pin - " + getDevicePIN());
bluetoothDevice.createBond();
Log.d(LOG_TAG," pin entered and request sent...");
abortBroadcast();
}
}
}
public void setDevicePIN(String pin){
this.devicePIN = pin;
}
public String getDevicePIN(){
return this.devicePIN ;
}
}
public class BluetoothLowEnergy extends BluetoothGattCallback {
// listener that has the methods that the application (activity)
// will use to send / receive data, or to reflect the system state
// in the UI
public interface BluetoothListener {
/**
* Triggered when the scanning has started successfully
*/
void onBluetoothStartScan();
/**
* Triggered when the scanning stops
* @param scanResults results of the scanning
*/
void onBluetoothStopScan(Collection<BluetoothDevice> scanResults);
/**
* Triggered when the device is ready to send/receive data
*/
void onBluetoothConnectionReady();
/**
* Triggered when a bluetooth msg is received
* @param msg message received
*/
void onBluetoothReceiveMsg(String msg);
/**
* Triggered whenever data is send
* @param success true means data was sent fine to the remote device, false otherwise
*/
void onBluetoothSend(String data, boolean success);
/**
* Triggered if no bluetooth is connected, and we need a connection
* to send / receive / discover services
*/
void onBluetoothNotConnected();
}
// custom exceptions
public class BluetoothNotEnabledException extends Exception { }
public class BluetoothLowEnergyNotSupported extends Exception { }
public class BluetoothDeviceNotFound extends Exception { }
// service and characteristic uuids that are going to be used to
// send / receive data between central and peripheral GATTs
private static final String SERVICE_UUID = "FFE0-";
private static final String CHARACTERISTIC_UUID = "FFE1-";
// timeout for bluetooth scan (in ms)
public static final int SCAN_TIMEOUT = 5000;
// BLE LOG TAG
public static final String LOG_TAG = "BLUETOOTH_BLE";
// model
private boolean bluetoothScanning;
private boolean bluetoothConnected;
private Map<String, BluetoothDevice> bluetoothScanResults;
// gui
private Activity activity;
// bluetooth
private BluetoothAdapter bluetoothAdapter;
private BluetoothLeScanner bluetoothLeScanner;
private ScanCallback bluetoothScanCallback;
private BluetoothGatt bluetoothGatt;
private BluetoothGattCharacteristic characteristic;
public BluetoothLowEnergy(Activity activity, BluetoothListener bluetoothListener){
this.activity = activity;
this.bluetoothListener = bluetoothListener;
// this keeps track of the scanning and connection states
this.bluetoothScanning = this.bluetoothConnected = false;
// keeps track of the scanning results
this.bluetoothScanResults = new HashMap<>();
// set bluetooth pairing request and bonded callback
// these broadcasters will be responsible to detect and validate
// the bonded state of your device
this.pairingRequestBroadcaster = new BluetoothPairingBroadcaster(this);
this.bondedBroadcaster = new BluetoothBondStateBroadcaster(this);
// set the scan callback methods that will add results to
// this.bluetoothScanResults map
this.bluetoothScanCallback = new ScanCallback() {
@Override
public void onScanResult(int callbackType, ScanResult result) {
super.onScanResult(callbackType, result);
addScanResult(result);
}
@Override
public void onBatchScanResults(List<ScanResult> results) {
super.onBatchScanResults(results);
for (ScanResult result: results) {
addScanResult(result);
}
}
@Override
public void onScanFailed(int errorCode) {
super.onScanFailed(errorCode);
Log.e(LOG_TAG, "Scan Failed with code " + errorCode);
}
private void addScanResult(ScanResult result) {
BluetoothDevice device = result.getDevice();
String deviceAddress = device.getAddress();
bluetoothScanResults.put(deviceAddress, device);
Log.d(LOG_TAG, "Found device " + deviceAddress);
}
};
// Use this to determine whether BLE is supported on the device.
if (!this.activity.getPackageManager().hasSystemFeature(PackageManager.FEATURE_BLUETOOTH_LE)) {
throw new BluetoothLowEnergyNotSupported();
}
}
/**
* This method should be called when the activity is destroyed
*/
public void onDestroy(){
this.bondedBroadcaster.onDestroy();
this.pairingRequestBroadcaster.onDestroy();
this.disconnect();
}
/**
* This method is called when we finish pairing/bonding to the device
*/
public void onBluetoothBonded(){
// if we have the services already discovered, then we can
// send/receive data, to do so we call the bluetooth listener below
if (servicesDiscovered){
this.bluetoothListener.onBluetoothConnectionReady();
// if we know we have a connection established, then we can
// discover services
} else if (bluetoothConnected){
bluetoothGatt.discoverServices();
}
}
/**
* This method is called whenever a connection is established or a disconnection happens
*/
@Override
public void onConnectionStateChange(BluetoothGatt gatt, int status, int newState) {
super.onConnectionStateChange(gatt, status, newState);
BluetoothDevice bluetoothDevice = gatt.getDevice();
// if these conditions == true, then we have a disconnect
if ( status == BluetoothGatt.GATT_FAILURE ||
status != BluetoothGatt.GATT_SUCCESS ||
newState == BluetoothProfile.STATE_DISCONNECTED) {
Log.d(LOG_TAG, String.format(Locale.getDefault(),
"Disconnected from %s (%s) - status %d - state %d",
bluetoothDevice.getName(),
bluetoothDevice.getAddress(),
status,
newState
));
this.disconnect();
// if these conditions == true, then we have a successful connection
} else if (newState == BluetoothProfile.STATE_CONNECTED) {
bluetoothConnected = true;
Log.d(LOG_TAG, String.format(Locale.getDefault(),
"Connected to %s (%s) - status %d - state %d",
bluetoothDevice.getName(),
bluetoothDevice.getAddress(),
status,
newState
));
// this sleep is here to avoid TONS of problems in BLE, that occur whenever we start
// service discovery immediately after the connection is established
try {
Thread.sleep(600);
} catch (InterruptedException e) {
e.printStackTrace();
}
gatt.discoverServices();
}
}
@Override
public void onServicesDiscovered(BluetoothGatt gatt, int status) {
super.onServicesDiscovered(gatt, status);
if (status != BluetoothGatt.GATT_SUCCESS) {
return;
}
// BEGIN - find the service and characteristic that we want (defined as a static attribute
// of the BluetoothLowEnergy class)
Log.d(LOG_TAG, "Discovering services ...");
BluetoothGattService service = null;
for (BluetoothGattService serv: gatt.getServices()){
Log.d(LOG_TAG, "Found service " + serv.getUuid().toString());
if (serv.getUuid().toString().toUpperCase().contains(SERVICE_UUID)){
service = serv;
Log.d(LOG_TAG, "---> Selected service " + serv.getUuid().toString());
break;
}
}
if (service == null){
return;
}
for (BluetoothGattCharacteristic charac: service.getCharacteristics()){
Log.d(LOG_TAG, "Found characteristic " + charac.getUuid().toString());
if (charac.getUuid().toString().toUpperCase().contains(CHARACTERISTIC_UUID)){
this.characteristic = charac;
Log.d(LOG_TAG, "---> Selected characteristic " + charac.getUuid().toString());
break;
}
}
if (this.characteristic == null){
return;
}
Log.d(LOG_TAG, "Setting write and notification to the characteristic ...");
bluetoothAdapter.cancelDiscovery();
// END - find the service and characteristic
// set that we want to write to the selected characteristic and be notified if
// it changes (the remote GATT peripheral sends data to the Android's GATT Center)
this.characteristic.setWriteType(BluetoothGattCharacteristic.WRITE_TYPE_DEFAULT);
gatt.setCharacteristicNotification(this.characteristic, true);
// we finished service discovery
this.servicesDiscovered = true;
// if we have paired/bonded then we are ready to send/receive data
if (pairingRequestBroadcaster.getDevicePIN().isEmpty() || bondedBroadcaster.isDeviceBonded()) {
this.bluetoothListener.onBluetoothConnectionReady();
}
}
@Override
public void onCharacteristicRead(BluetoothGatt gatt, BluetoothGattCharacteristic charac, int status) {
super.onCharacteristicRead(gatt, charac, status);
restartDisconnectTimeout();
if (status != BluetoothGatt.GATT_SUCCESS) {
return;
}
try {
String characValue = new String(charac.getValue(), CHARSET)
.replaceAll(DATA_FILTER_REGEX,"");
Log.i(LOG_TAG, String.format(Locale.getDefault(),
"Characteristic Read - %s",
characValue
));
if (charac.getUuid().equals(this.characteristic.getUuid())) {
this.bluetoothListener.onBluetoothReceiveMsg(characValue);
}
} catch (UnsupportedEncodingException e) {
Log.e(LOG_TAG, "Characteristic Read - Failed to convert message string to byte array");
}
}
@Override
public void onCharacteristicWrite(BluetoothGatt gatt, BluetoothGattCharacteristic charac, int status) {
super.onCharacteristicWrite(gatt, charac, status);
restartDisconnectTimeout();
try {
String characValue = new String(charac.getValue(), CHARSET);
Log.i(LOG_TAG, String.format(Locale.getDefault(),
"Characteristic Write - SUCCESS - %s",
characValue
));
bluetoothListener.onBluetoothSend( characValue, (status == BluetoothGatt.GATT_SUCCESS) );
} catch (UnsupportedEncodingException e) {
Log.e(LOG_TAG, "Characteristic Write - Failed to convert message string to byte array");
}
}
@Override
public void onCharacteristicChanged(BluetoothGatt gatt, BluetoothGattCharacteristic charac) {
super.onCharacteristicChanged(gatt, charac);
Log.d(LOG_TAG,"Characteristic Changed");
onCharacteristicRead(gatt, charac, BluetoothGatt.GATT_SUCCESS);
}
/**
* Remove pairing/bonding of the device
* @param device Device to remove bonding
*/
public static void removeBond(BluetoothDevice device){
try {
if (device == null){
throw new Exception();
}
Method method = device.getClass().getMethod("removeBond", (Class[]) null);
method.invoke(device, (Object[]) null);
Log.d(LOG_TAG, "removeBond() called");
Thread.sleep(600);
Log.d(LOG_TAG, "removeBond() - finished method");
} catch (Exception e) {
e.printStackTrace();
}
}
/**
* Clears the GATT services cache, so that new services can be discovered
* @param bluetoothGatt GATT Client to clear service's discovery cache
*/
public static void refresh(BluetoothGatt bluetoothGatt){
try {
Method method = bluetoothGatt.getClass().getMethod("refresh", (Class[]) null);
method.invoke(bluetoothGatt, (Object[]) null);
} catch (Exception e){
e.printStackTrace();
}
}
/**
* Connect to the GATT Peripheral device
* @param uuid GATT Peripheral address / mac / uuid to connect to
* @param pin PIN to authenticate and pair to the device
*/
public void connect(String uuid, String pin) throws BluetoothNotEnabledException, BluetoothDeviceNotFound {
checkBluetooth();
// do not connect twice
if (this.isConnected()){
return;
}
// get device
BluetoothDevice device = this.bluetoothScanResults.get(uuid);
if (device == null){
throw new BluetoothDeviceNotFound();
}
this.deviceUUID = uuid;
pairingRequestBroadcaster.setDevicePIN(pin);
removeBond(device);
// create connection to the bluetooth device
bluetoothGatt = device.connectGatt(activity, false, this);
refresh(bluetoothGatt);
}
/**
* Disconnect from BLE device. This method should be called whenever we want to
* close the APP, or the BLE connection.
*/
public void disconnect() {
Log.d(LOG_TAG, "disconnect() - executed");
if (bluetoothGatt != null) {
if (characteristic != null) {
bluetoothGatt.setCharacteristicNotification(characteristic, false);
}
//remove device authorization/ bond/ pairing
removeBond(bluetoothGatt);
// disconnect now
bluetoothGatt.disconnect();
bluetoothGatt.close();
Log.d(LOG_TAG, "disconnect() - bluetoothGatt disconnect happened");
}
bluetoothGatt = null;
characteristic = null;
bluetoothConnected = false;
servicesDiscovered = false;
// set device as not bonded anymore
bondedBroadcaster.resetDeviceBonded();
}
/**
* bluetooth nearby devices scan is on
* @return true if scanning is on, false otherwise
*/
public boolean isScanning(){
return (this.bluetoothScanning);
}
/**
* Check bluetooth system state (on or off)
* @return true if system is on, false otherwise
*/
public boolean isEnabled(){
try {
checkBluetooth();
return bluetoothAdapter.isEnabled();
} catch (BluetoothNotEnabledException e) {
return false;
}
}
/**
* Check bluetooth connection
* @return true if connected, false otherwise
*/
public boolean isConnected(){
return (this.bluetoothConnected);
}
/**
* Start bluetooth scan for nearby devices
* @param filters Scan filters that define what devices to scan for
*/
public void startScan(List<ScanFilter> filters)
throws BluetoothNotEnabledException{
checkBluetooth();
// dont run two scans simultaneously
if (isScanning()) {
return;
}
// disconnect previously connected devices
if (isConnected()) {
this.disconnect();
return;
}
// setup bluetooth scanning settings
ScanSettings settings = new ScanSettings.Builder()
.setScanMode(ScanSettings.SCAN_MODE_LOW_POWER)
.build();
// start scanning
this.bluetoothScanning = true;
this.bluetoothScanResults.clear();
this.bluetoothLeScanner = bluetoothAdapter.getBluetoothLeScanner();
// Stops scanning after a pre-defined scan period.
Handler bluetoothHandler = new Handler();
bluetoothHandler.postDelayed(new Runnable() {
@Override
public void run() {
stopScan();
}
}, SCAN_TIMEOUT);
// start scan with default scan callback
this.bluetoothLeScanner.startScan(filters, settings, bluetoothScanCallback);
// we have started successfully the BLE scanning
bluetoothListener.onBluetoothStartScan();
}
/**
* Stop bluetooth scan for nearby devices
*/
public void stopScan(){
if (!bluetoothScanning) {
return;
}
// set app scan state to false
bluetoothScanning = false;
if (bluetoothLeScanner != null) {
bluetoothLeScanner.stopScan(bluetoothScanCallback);
bluetoothLeScanner = null;
}
// we have stopped BLE scanning, call the user's callback
bluetoothListener.onBluetoothStopScan(bluetoothScanResults.values());
}
/**
* Send a message via bluetooth
* @param msg message to send
*/
public void send(String msg) {
if (!bluetoothConnected || characteristic == null){
bluetoothListener.onBluetoothNotConnected();
return;
}
try {
msg = msg.replaceAll(DATA_FILTER_REGEX, "") + TERMINATION_CHAR;
Log.d(LOG_TAG, String.format(Locale.getDefault(),
"Sending message: %s",
msg));
characteristic.setValue(msg.getBytes(CHARSET));
bluetoothGatt.writeCharacteristic(characteristic);
} catch (UnsupportedEncodingException e) {
Log.e(LOG_TAG,
"BluetoothLowEnergy.send: Failed to convert message string to byte array");
}
}
public String getDeviceUUID(){
return deviceUUID;
}
public Activity getActivity(){
return activity;
}
/**
* Check if bluetooth is enabled and working properly
*/
private void checkBluetooth() throws BluetoothNotEnabledException{
if (bluetoothAdapter == null) {
final BluetoothManager bluetoothManager =
(BluetoothManager) activity.getSystemService(Context.BLUETOOTH_SERVICE);
if (bluetoothManager == null){
throw new BluetoothNotEnabledException();
}
bluetoothAdapter = bluetoothManager.getAdapter();
}
// Ensures Bluetooth is available on the device and it is enabled. If not,
// displays a dialog requesting user permission to enable Bluetooth.
if (bluetoothAdapter == null || !bluetoothAdapter.isEnabled()) {
throw new BluetoothNotEnabledException();
}
}
}
The key methods and functions to avoid problems used above are:
*
*Thread.sleep(600)
*removeBond(device)
*refresh(gatt)
*gatt.disconnect()
*gatt.close()
A: Here is the list of codes i have
*
*Programmatically disconnected - 0
*Device went out of range - 8
*Disconnected by device - 19
*Issue with bond - 22
*Device not found - 133(some phone it gives 62)
I have tested disconnect scenario's in 5.0.2, 5.1, 6.0 and 6.0.1. But only found this bond issue code in 6.0.1 android version.
A: In my case I got this response from bluetooth stack because the device was already bonded with my phone. I removed it from my settings and the error 22 vanished.
A: in aosp (android source code). you can find any error in bluetooth source code, and know the meaning of status code.
the file path is system/bt/stack/include/gatt_api.h
Here's the link: https://android.googlesource.com/platform/system/bt/+/ea7ab70a711e642653dd5922b83aa04a53af9e0e/stack/include/gatt_api.h but it all display by hex.
for example:
hex
Decimal
reason
0x08
8
connection timeout
0x13
19
connection terminate by peer user
0x16
22
connectionterminated by local host
0x22
34
connection fail for LMP response tout
0x85
133
gatt_error
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,798
|
This Broadway musical-comedy classic features original music by Tony- and 8-time Oscar winner Alan Menken.
The play tells the hilarious story of Deloris Van Cartier, a wannabe diva whose life takes a surprising turn when she witnesses a crime and the cops hide her in the last place anyone would think to look -- a convent! Under the suspicious watch of Mother Superior, Deloris helps her fellow sisters find their voices as she unexpectedly rediscovers her own. A sparkling tribute to the universal power of friendship, Sister Act is reason to rejoice!
I watched the movie before and it was quite good!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,954
|
{"url":"http:\/\/papers.nips.cc\/paper\/5798-learning-with-group-invariant-features-a-kernel-perspective","text":"# NIPS Proceedings\u03b2\n\n## Learning with Group Invariant Features: A Kernel Perspective.\n\nA note about reviews: \"heavy\" review comments were provided by reviewers in the program committee as part of the evaluation process for NIPS 2015, along with posted responses during the author feedback period. Numerical scores from both \"heavy\" and \"light\" reviewers are not provided in the review link below.\n\n[PDF] [BibTeX] [Supplemental] [Reviews]\n\n### Abstract\n\nWe analyze in this paper a random feature map based on a theory of invariance (\\emph{I-theory}) introduced in \\cite{AnselmiLRMTP13}. More specifically, a group invariant signal signature is obtained through cumulative distributions of group-transformed random projections. Our analysis bridges invariant feature learning with kernel methods, as we show that this feature map defines an expected Haar-integration kernel that is invariant to the specified group action. We show how this non-linear random feature map approximates this group invariant kernel uniformly on a set of $N$ points. Moreover, we show that it defines a function space that is dense in the equivalent Invariant Reproducing Kernel Hilbert Space. Finally, we quantify error rates of the convergence of the empirical risk minimization, as well as the reduction in the sample complexity of a learning algorithm using such an invariant representation for signal classification, in a classical supervised learning setting","date":"2018-11-18 00:44:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5833999514579773, \"perplexity\": 989.2537384078569}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-47\/segments\/1542039743913.6\/warc\/CC-MAIN-20181117230600-20181118012600-00113.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/applied-mathematics\/elementary-technical-mathematics\/chapter-8-section-8-4-the-equation-of-a-line-exercises-page-315\/34","text":"# Chapter 8 - Section 8.4 - The Equation of a Line - Exercises - Page 315: 34\n\nDraw a line through points (0,0) and (3,-1).\n\n#### Work Step by Step\n\nTo find a 2nd point when the slope is $-\\frac{1}{3}$ count down 1 and right 3 from the given point.\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2022-05-24 03:36:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5872557163238525, \"perplexity\": 999.1140623126034}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662562410.53\/warc\/CC-MAIN-20220524014636-20220524044636-00375.warc.gz\"}"}
| null | null |
{"url":"https:\/\/en.wikipedia.org\/wiki\/Cofactor","text":"# Cofactor\n\nGiven a factor a of a number ${\\displaystyle x=ab}$, the cofactor of a is ${\\displaystyle b={\\frac {x}{a}}.}$","date":"2017-09-26 18:52:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 2, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7400115132331848, \"perplexity\": 842.5019854811704}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818696677.93\/warc\/CC-MAIN-20170926175208-20170926195208-00387.warc.gz\"}"}
| null | null |
Home GCC Saudi Arabia Al-Mabani wins Saudi airports work
Al-Mabani wins Saudi airports work
21 May 2008 5:11 PM By Meed
The local Al-Mabani has been awarded a contract from the General Authority of Civil Aviation (Gaca) for the renovation and repair of seven domestic airports in the kingdom.
The SR300m ($80m) contract covers runway renovation and maintenance at Najran, Al-Taif, Buraydah, Hail, Al-Wajeh, Ar Ar and Qourayat. The work is expected to take two to three years.
In a separate contract, the contractor secured a further SR300m contract for the construction of new terminal buildings at Najran airport.
The awards are part of a wider effort by Saudi Arabia to update its aviation infrastructure.
Together with its new contracts, Al-Mabani is currently working on a SR1bn contract to upgrade and renovate the airfield of King Abdulaziz International Airport in Jeddah. It is also carrying out works at Tabuk and is due to complete further work at Yanbu airport in the coming weeks.
Several project clients across the GCC are expected to imminently appoint transaction advisers for planned railway PPP schemes
Saudi Railway and Huawei sign 'smart railway' deal
17 July 2019 11:22 AM By Jennifer Aguinaldo
Saudi Railway Company has also invited consultants to bid for a contract to review and update the kingdom's railway masterplan
EXCLUSIVE: Saudi Arabia seeks firms for railway masterplan update
Saudi Arabia's existing railway masterplan proposed the construction of 9,900km of railway between 2010 and 2040
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,154
|
{"url":"https:\/\/astronomy.stackexchange.com\/tags\/particles\/hot","text":"# Tag Info\n\n28\n\nI'm going to give you an intuitive answer. Keep in mind, this is not the \"actual\" answer, as the Hawking radiation is quite a bit more complex than the typical pop-sci explanation with virtual particles. But some intuitive justification is possible nevertheless. I don't see how this event contributes to evaporation of the black hole (, since the ...\n\n9\n\nThese lecture notes address the issues to some degree, especially on slides 33-35. Because in the strongly warped spacetime near the horizon, virtual particles made from vacuum fluctuations turn out to have negative energy density. Energy density = energy per unit volume. These particles indeed have positive mass -- look at the one that ...\n\n9\n\nNeutrinos are typically produced in AGN jets through what we refer to as hadronic processes. Protons are accelerated to relativistic speeds and interact with nearby photons. Depending on the particular type of AGN, there are a number of possible sources of these photons: Ambient light within the jet Emission from the accretion disk Photons from the broad ...\n\n8\n\nYou can not check if a dimensional constant has changed because you can always reverse that change by a smart change of coordinates (system of units). Despite that, since the current Physics assumes the immutability of certain constants, you can verify this assumption by testing the change of an adimensional constant. One of the most common adimensional ...\n\n1\n\nI don't know if the experts will agree with this description, but here is how I understand it: Both space and the event horizon are in constant quantum fluctuation. Essentially, the event horizon has tiny ripples. At points where the event horizon ripples up (above the average radius of the black hole), it has an above average amount of local energy. The ...\n\n1\n\nLet's look at the validity of tachyons first. Travelling faster than light, as said by userLTK, would create time travel, which would create the famous paradox, the tachyonic antitelephone. The tachyonic antitelephone is a violation of causality, and thus most likely, tachyons would not exist. Now, take a look at another assumption. Wikipedia is right ...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2020-05-24 22:52:48","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.828437328338623, \"perplexity\": 595.0412518541724}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347385193.5\/warc\/CC-MAIN-20200524210325-20200525000325-00400.warc.gz\"}"}
| null | null |
package cn.ucai.superwechat.widget;
import android.content.Context;
import android.content.res.TypedArray;
import android.graphics.drawable.Drawable;
import android.util.AttributeSet;
import android.view.LayoutInflater;
import android.view.View;
import android.widget.ImageView;
import android.widget.LinearLayout;
import android.widget.TextView;
import cn.ucai.superwechat.R;
public class ContactItemView extends LinearLayout{
private TextView unreadMsgView;
public ContactItemView(Context context, AttributeSet attrs) {
super(context, attrs);
init(context, attrs);
}
public ContactItemView(Context context) {
super(context);
init(context, null);
}
private void init(Context context, AttributeSet attrs){
TypedArray ta = context.obtainStyledAttributes(attrs, R.styleable.ContactItemView);
String name = ta.getString(R.styleable.ContactItemView_contactItemName);
Drawable image = ta.getDrawable(R.styleable.ContactItemView_contactItemImage);
ta.recycle();
LayoutInflater.from(context).inflate(R.layout.em_widget_contact_item, this);
ImageView avatar = (ImageView) findViewById(R.id.avatar);
unreadMsgView = (TextView) findViewById(R.id.unread_msg_number);
TextView nameView = (TextView) findViewById(R.id.name);
if(image != null){
avatar.setImageDrawable(image);
}
nameView.setText(name);
}
public void setUnreadCount(int unreadCount){
unreadMsgView.setText(String.valueOf(unreadCount));
}
public void showUnreadMsgView(){
unreadMsgView.setVisibility(View.VISIBLE);
}
public void hideUnreadMsgView(){
unreadMsgView.setVisibility(View.INVISIBLE);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,807
|
<html>
<head>
<title>Camunda commons UI library</title>
<base href="/" />
<link rel="icon" href="resources/img/favicon.ico" />
<link href="styles.css" rel="stylesheet" />
<link href="test-styles.css" rel="stylesheet" />
</head>
<body class="cam-widget-loader-test-page">
<!-- gh-pages-menu -->
<header>
<div>
<h1>loader</h1>
</div>
</header>
<section class="widget-description">
<header>
<h2>Description</h2>
</header>
<p>A loader is aimed to wrap the loading of content in a standard way. It shows a loading state at first and then either the what has been loaded (or the results) or a "no records".<br/>
You can customize what comes in the center with <a href="//docs.angularjs.org/api/ng/directive/ngTransclude">ng-transclude</a>.</p>
</section>
<section class="widget-reference">
<header>
<h2>Usage</h2>
</header>
<h3>Options</h3>
<dl>
<dt><span class="badge">@</span> text-loading</dt>
<dd>can be used to provide a custom "loading" text</dd>
<dt><span class="badge">@</span> text-empty</dt>
<dd>can be used to provide a custom "empty response" text (which is blank by default)</dd>
<dt><span class="badge">@</span> text-error</dt>
<dd>can be used to provide a custom "error" text (which is blank by default)</dd>
<dt><span class="badge">@</span> loading-state</dt>
<dd class="possibilities">
<dl>
<dt>INITIAL</dt>
<dd>is the default state, used when the loading process has not been started</dd>
<dt>LOADING</dt>
<dd>is used when the loading process has started and is not complete</dd>
<dt>LOADED</dt>
<dd>is used when the loading process is complete and data were received</dd>
<dt>EMPTY</dt>
<dd>is used when the loading process is complete and no data were received</dd>
<dt>ERROR</dt>
<dd>is used when the loading process is failed</dd>
</dl>
</dd>
</dl>
</section>
<section class="widget-examples">
<header>
<h2>Examples</h2>
</header>
<div class="widget-example"
id="initial-state">
<h3>Initial state</h3>
<pre ng-non-bindable><div cam-widget-loader></div></pre>
<div class="test-container">
<div cam-widget-loader></div>
</div><!-- /.test-container -->
</div><!-- /.widget-example -->
<div class="widget-example"
id="loading-state">
<h3>Loading state</h3>
<pre ng-non-bindable><div cam-widget-loader
loading-state="LOADING"></div></pre>
<div class="test-container">
<div cam-widget-loader
loading-state="LOADING"></div>
</div><!-- /.test-container -->
</div><!-- /.widget-example -->
<div class="widget-example"
id="loading-state-custom">
<h3>Loading state with custom text</h3>
<pre ng-non-bindable><div cam-widget-loader
text-loading="Custom loading text"
loading-state="LOADING"></div></pre>
<div class="test-container">
<div cam-widget-loader
text-loading="Custom loading text"
loading-state="LOADING"></div>
</div><!-- /.test-container -->
</div><!-- /.widget-example -->
<div class="widget-example"
id="loaded-state">
<h3>Loaded state</h3>
<pre ng-non-bindable><div cam-widget-loader
loading-state="LOADED">
<div class="panel panel-default">
<div class="panel-heading">
This has been loaded
</div>
<div class="panel-body">
<p>Bacon ipsum dolor amet shank sirloin meatball, salami jerky short loin corned beef andouille strip steak cow ham kevin pastrami.</p>
</div>
</div><!-- /.panel -->
</div></pre>
<div class="test-container">
<div cam-widget-loader
loading-state="LOADED">
<div class="panel panel-default">
<div class="panel-heading">
This has been loaded
</div>
<div class="panel-body">
<p>Bacon ipsum dolor amet shank sirloin meatball, salami jerky short loin corned beef andouille strip steak cow ham kevin pastrami.</p>
</div>
</div><!-- /.panel -->
</div>
</div><!-- /.test-container -->
</div><!-- /.widget-example -->
<div class="widget-example"
id="empty-state">
<h3>Empty state</h3>
<pre ng-non-bindable><div cam-widget-loader
loading-state="EMPTY">
<div class="expected-hidden">
Should not be visible.
</div>
</div></pre>
<div class="test-container">
<div cam-widget-loader
loading-state="EMPTY">
<div class="expected-hidden">
Should not be visible.
</div>
</div>
</div><!-- /.test-container -->
</div><!-- /.widget-example -->
<div class="widget-example"
id="empty-state-custom">
<h3>Empty state with custom text</h3>
<pre ng-non-bindable><div cam-widget-loader
loading-state="EMPTY"
text-empty="Custom 'empty' text">
<div class="expected-hidden">
Should not be visible.
</div>
</div></pre>
<div class="test-container">
<div cam-widget-loader
loading-state="EMPTY"
text-empty="Custom 'empty' text">
<div class="expected-hidden">
Should not be visible.
</div>
</div>
</div><!-- /.test-container -->
</div><!-- /.widget-example -->
<div class="widget-example"
id="interactive"
ng-controller="testInteractiveController">
<h3>Interactive</h3>
<pre ng-non-bindable><div class="row">
<div class="btn-group col-xs-4">
<button class="btn btn-default reload"
ng-click="reload()"
ng-enabled="ctrlState !== 'LOADING'">
Load
</button>
<button class="btn btn-default reload-empty"
ng-click="reload(true)"
ng-enabled="ctrlState !== 'LOADING'">
Load empty
</button>
<button class="btn btn-danger fail-load"
ng-click="fail()"
ng-disabled="['LOADED', 'EMPTY', 'ERROR'].indexOf(ctrlState) !== -1">
Fail! <span class="glyphicon glyphicon-fire"></span>
</button>
</div>
<div class="col-xs-8">
Current state: <code class="state-display">{{ ctrlState }}</code>
</div>
</div><!-- /.row -->
<hr />
<div cam-widget-loader
loading-state="{{ ctrlState }}"
text-error="Custom error text">
<div class="panel panel-default">
<div class="panel-heading">
This has been loaded
</div>
<div class="panel-body">
<p>testInteractiveController has <code>ctrlVar1</code> with <code>{{ ctrlVar1 }}</code>.</p>
</div>
</div><!-- /.panel -->
</div></pre>
<div class="test-container"
id="interactive-container">
<div class="row">
<div class="btn-group col-xs-4">
<button class="btn btn-default reload"
ng-click="reload()"
ng-enabled="ctrlState !== 'LOADING'">
Load
</button>
<button class="btn btn-default reload-empty"
ng-click="reload(true)"
ng-enabled="ctrlState !== 'LOADING'">
Load empty
</button>
<button class="btn btn-danger fail-load"
ng-click="fail()"
ng-disabled="['LOADED', 'EMPTY', 'ERROR'].indexOf(ctrlState) !== -1">
Fail! <span class="glyphicon glyphicon-fire"></span>
</button>
</div>
<div class="col-xs-8">
Current state: <code class="state-display">{{ ctrlState }}</code>
</div>
</div><!-- /.row -->
<hr />
<div cam-widget-loader
loading-state="{{ ctrlState }}"
text-error="Custom error text">
<div class="panel panel-default">
<div class="panel-heading">
This has been loaded
</div>
<div class="panel-body">
<p>testInteractiveController has <code>ctrlVar1</code> with <code>{{ ctrlVar1 }}</code>.</p>
</div>
</div><!-- /.panel -->
</div>
</div><!-- /.test-container -->
</div><!-- /.widget-example -->
</section>
<!-- gh-pages-footer -->
<script src="node_modules/requirejs/require.js"></script>
<script>
require.config({
paths: {text: 'node_modules/requirejs-text/text'}
});
require(['text!test-conf.json'], function(rConf) {
'use strict';
rConf = JSON.parse(rConf);
require.config(rConf);
require([
'angular',
'lib/widgets/loader/cam-widget-loader',
'angular-bootstrap'
], function (
angular,
loaderDefinition
) {
var loaderModule = angular.module('loaderModule', ['ui.bootstrap']);
loaderModule.directive('camWidgetLoader', loaderDefinition);
var testModule = angular.module('testModule', [loaderModule.name]);
testModule.controller('testInteractiveController', [
'$scope',
'$timeout',
function(
$scope,
$timeout
) {
$scope.ctrlState = 'INITIAL';
$scope.timeoutPromise = null;
$scope.reload = function (simulateEmpty) {
$scope.ctrlState = 'LOADING';
$scope.timeoutPromise = $timeout(function () {
$scope.ctrlVar1 = 'Control variable';
$scope.ctrlState = !!simulateEmpty ? 'EMPTY' : 'LOADED';
}, 1000);
};
$scope.fail = function () {
$scope.ctrlState = 'ERROR';
$scope.ctrlError = 'Something wen really wrong';
if ($scope.timeoutPromise) {
$timeout.cancel($scope.timeoutPromise);
}
};
}]);
angular.element(document).ready(function() {
angular.bootstrap(document.body, [testModule.name]);
});
});
});
</script>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,523
|
If you want to ask a question about the use of this plugin, please avoid submmitting an issue and ask on [Discussions]https://github.com/NiklasMerz/cordova-plugin-fingerprint-aio/discussions) instead.
# Submitting bugs
Create a new issue and please use the template
# Contributing
* Fork
* Create new feature branch (git checkout -b feature-or-fix-something)
* Commit your changes (git commit -am 'Add fix for android ...')
* Push to the branch (git push origin feature-or-fix-something)
* Create new Pull Request with description what you did and why you did it
## Tips
* Please avoid changing the indentation of a complete file in your pull request, because that makes reviewing changes hard
* Use the command `npm test PLATFORM` to run automatic and manual tests first
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,129
|
Пјер-Иг Ербер и Никола Маи су освојили турнир. У финалу су савладали Џејмија Марија и Џона Пирса са 6:4, 6:4.
Носиоци
Жреб
Завршница
Горња половина
Група 1
Група 2
Доња половина
Група 3
Група 4
Извори
Жреб турнира на сајту atptour.com
Отворено првенство САД у тенису 2015.
Отворено првенство Сједињених Америчких Држава у тенису — мушки парови
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,732
|
\section{Introduction}\label{intro}
Detailed knowledge of surface properties is important to understanding a wide
variety of phenomena such as catalysis, surface reactivity, growth, etc. Of
particular importance are such quantities as the step formation energy (SFE),
which determines the equilibrium shape of islands on flat terraces, and the
potential energy surface seen by, e.g., an adatom, which provides information
on the preferred sites for adsorption and the kinetics of diffusion. We
present here a detailed study of these properties for the (111) surface of
platinum within the framework of density-functional theory (DFT).\cite{kohn}
The (111) surface of fcc metals, and in particular Pt, is of interest for (at
least) two reasons. First, as displayed in Fig.\ \ref{steps}, it exhibits two
possible step geometries, named according to the micro-facet that step-edge
atoms form with atoms in the layer underneath, namely a (100)-faceted step,
where an edge atom has a single near neighbor at the base of the step (i.e.,
the micro-facet constitutes a square lattice) and a (111)-faceted step, where
each atom has two neighbors (triangular lattice). The only difference between
the two geometries, as far as nearest-neighbors are concerned, is that atoms
at the base of the step have a coordination of 10 for the (100)-faceted step
and 11 for the (111)-faceted step. The two steps are evidently very similar
and the formation energies are thus {\em expected} to be comparable, i.e.,
the equilibrium island shape should be very nearly hexagonal, with the sides
consisting of, alternately, (100)- and (111)-faceted steps. This has indeed
been observed in the case of Ir\cite{fu} and Ag.\cite{morgenstern} For Pt,
however, scanning-tunneling microscopy (STM) reveals, rather, a strong
preference for (111)-faceted steps --- 0.87$\pm$0.02 as measured by the ratio
of step formation energies per unit length.\cite{michely} On the theory side,
DFT calculations in the local-density approximation\cite{ceperley} (LDA) by
Stumpf and Scheffler predict this behaviour in the case of Al, with a ratio
of SFE of 0.93;\cite{stumpf1} this has not yet been confirmed experimentally.
For Pt, in contrast, corresponding calculations by Feibelman failed to
reproduce the experimental results, leading to essentially equal SFE for the
two types of steps.\cite{feibelman0}
A second feature that makes the fcc(111) surface interesting is that it
possesses two different adsorption sites: the fcc (normal) site, where an
adatom sits in a position appropriate to the stacking of atomic planes in a
perfect fcc crystal, and the hcp (stacking fault) site, corresponding to an
hcp stacking. Both sites have three nearest neighbors, the difference between
the two lying in the second-layer arrangement, as illustrated in Fig.\
\ref{sites}. It has been shown that, for transition metals, the preferred
site for adsorption depends on the filling of the $d$
band.\cite{mortensen,piveteau} For Pt, DFT-LDA calculations predict the fcc
site to be much more favourable,\cite{mortensen,feibelman1} by 0.12--0.18 eV,
depending on the state of relaxation of the substrate. This is the largest
difference observed (so far) for late transition metals and noble
metals.\cite{mortensen,boisvert1,wang} Experimentally, also, there is
evidence that the fcc site is preferred over the hcp site in
Pt;\cite{feibelman1,golzhauser} while a precise numerical value cannot be
inferred from the measurements, it is estimated that the difference should be
{\em at least} 0.06 eV.\cite{golzhauser} However, despite this apparent
agreement, the DFT calculations for Pt(111)\cite{mortensen,feibelman1} have
been unable to reproduce correctly the diffusion barrier --- 0.38--0.41
versus $\sim$0.26 eV from experiment.\cite{feibelman1,bott} Since it is of
primary importance to have reliable and accurate energy barriers in order to
predict growth (see for example Ref. \onlinecite{ruggerone}), it is essential
that this discrepancy be resolved.
Clearly, a quantitative picture of the surface properties of Pt(111) is still
missing. In order to address this problem, we present here the results of
extensive {\em ab-initio} total-energy calculations of the formation energies
of the two kinds of steps, as well as of the energetics of adatom adsorption
and diffusion. The calculations have been carried out within the LDA, but we
have also carried out a series of calculations in the generalized-gradient
approximation (GGA)\cite{perdew} so as to assess the applicability of the LDA
to this system. We find the LDA to offer a better description (compared to
experiment) of the lattice constant of bulk Pt than the GGA, in agreement
with previous calculations,\cite{ozolins,khein} as well as some properties of
the clean (111) surface (surface relaxation and work function). Within
numerical accuracy, however, we observe no sizeable effect on the ratio of
SFE and on the energetics of adatom adsorption and diffusion.
As discussed in more detail below, the electronic wave-functions were
expanded in plane waves, which enables us to deal easily, and completely,
with the effect of relaxation on adsorption and adatom diffusion. In the
calculations of Refs.\ \onlinecite{mortensen} and \onlinecite{feibelman1},
only nearest-neighbor relaxation was included, at best. We observe that
proper account of atomic relaxation leads to a significant decrease, by
$\sim$20$\%$, of the value of the diffusion barrier on the flat (111)
surface. Our estimate for this quantity, while closer to
experiment\cite{feibelman1,bott} than previous
calculations,\cite{mortensen,feibelman1} however remains high --- 0.33 versus
$\sim0.26$ eV; possible reasons for this discrepancy will be discussed. In
contrast, the difference in energy between the two adsorption sites increases
upon relaxing, from 0.10 to 0.17 eV.
One other important advantage of plane waves is that, contrary to the
Gaussian orbitals used in Ref.\ \onlinecite{feibelman0}, they are independent
of the atomic positions and should therefore provide a more adequate
description of the subtle difference in SFE expected here. In agreement with
experiment, our highly-converged calculations indicate a clear preference for
(111)-faceted steps over (100)-faceted steps, the SFE being in a ratio of
0.88 (versus about 0.87 experimentally).\cite{michely} Our calculations,
further, provide a simple explanation for the origin of the energy difference
between the two steps in terms of the release of surface stress through
relaxation. Before discussing our results in detail, we give a brief
description of our computational approach.
\section{Computational details}\label{model}
As already noted above, the calculations reported here were performed within
the framework of density-functional theory,\cite{kohn} using both the
LDA\cite{ceperley} and the GGA\cite{perdew} for the exchange-and-correlation
energy. The ion cores were approximated by pseudopotentials with $5d$
electrons treated as valence states. The pseudopotentials were generated
using the semi-relativistic scheme of Troullier and Martins\cite{troullier}
and expressed in the Kleinman-Bylander form using the $s$ component as the
local one.\cite{kleinman,gonze,fuchs} The electronic wave-functions were
represented using a plane-wave basis set with kinetic energy up to 40 Ry in
the LDA and 45 Ry in the GGA. To improve convergence, the electronic states
were occupied according to a Fermi distribution with $k_BT_{\rm el} = 0.1$ eV
and the total energies obtained by extrapolating to zero electronic
temperature. For similar reasons, the calculations were initiated using
wave-functions obtained from the self-consistent solution of the Kohn-Sham
Hamiltonian in a mixed basis set composed of pseudo-atomic orbitals and plane
waves cut off at 4 Ry.\cite{kley} The minimization of the energy with respect
to the electronic degrees of freedom was done using an iterative
procedure.\cite{stumpf} After achieving electronic convergence, the atoms
were moved according to a damped Newton dynamics until forces became less
than $0.01$ eV/\AA. All the calculations were performed using the supercell
approach. Details of the cell shape and size, as well as {\bf k}-point
sampling, for the different geometries considered, are given along with the
results in the following sections.
\section{Results}\label{res}
\subsection{Bulk and Surface Properties}\label{res_clean}
In order to assess the validity of our approach, and for completeness, we
first determined the lattice constant of the bulk material as well as the
properties of the clean Pt(111) surface --- surface energy, relaxation,
excess surface stress, and work function. As a reminder, the surface stress
tensor, $g_{\alpha\beta}$, is given by
\begin{equation}
g_{\alpha \beta} = \gamma \delta_{\alpha \beta} +
d \gamma / d \varepsilon_{\alpha \beta}
\end{equation}
where $\gamma$ is the surface energy per unit area,
$\varepsilon_{\alpha\beta}$ is the strain tensor and $\delta_{\alpha\beta}$
is the Kronecker delta function. The second term in this equation represents
the excess surface stress. For the clean surface, we used a ($1\times1$)
supercell consisting of 5 or 7 (111) atomic planes plus $\sim$10 \AA\ of
vacuum. Integration of the first Brillouin zone was done over a uniform grid
of 100 {\bf k} points in the $x-y$ plane, which was found to yield
well-converged results, e.g., within 2 meV for the surface energy. The latter
was calculated by comparing to a bulk-like ($1\times1$) supercell containing
3 layers (and of course no vacuum). The same {\bf k}-point density was used
for the $x-y$ plane; to compensate for the smaller size of the cell in the
$z$ direction, the two-dimensional grid was replicated 4 times along $z$ so
as to get a density of points similar to that in the $x-y$ plane. For the
excess surface stress, we varied the in-plane lattice constant for both the
clean surface and the bulk, while keeping the atoms' $z$ coordinates fixed to
their bulk-like values, and examined the concomitant variations in the total
energy.
The results are listed in Table \ref{tests}, along with those from other {\em
ab-initio} calculations and experimental values when available. Evidently,
our LDA lattice constant is consistent with previous calculations and with
experiment. The GGA value, in contrast, while in agreement with other
calculations, overestimates somewhat the lattice constant, by more than
$2\%$; it is a well-known fact that the GGA yields larger lattice constant
than the LDA\cite{khein} --- actually overcompensates in the present case.
For the surface energy and excess surface stress, our LDA results agree well
with previous calculations. We note that the excess surface stress is large
and positive, meaning that the Pt(111) surface is under significant tensile
stress, i.e., would prefer a smaller lattice constant. Unfortunately, to our
knowledge, there exists no experimental determination of these quantities. In
the GGA, both the surface energy and the excess surface stress decrease. This
effect of the GGA on the surface energy was actually predicted from jellium
calculations.\cite{perdew}
We have also calculated the top and second layer relaxation, $\Delta d_{12}$
and $\Delta d_{23}$, i.e., the change in interlayer spacing relative to the
bulk value. We find a small (0.4$\%$) outward relaxation for the top layer.
This is quite a bit smaller than the value reported in Ref.\
\onlinecite{feibelman1}, 1.25$\%$. The origin of the discrepancy between the
two LDA calculations is not clear; it may be due to different choices of
basis sets (LCAO in Ref.\ \onlinecite{feibelman1} versus plane waves here).
The GGA, interestingly, leads to a small {\em inward} relaxation. Currently
available experimental data vary widely, in the range 0--2.5$\%$, and are
therefore not of much help in resolving the issue. For the second layer,
theory and experiment agree that it should be insignificant, i.e., $\Delta
d_{23}$ is a small fraction of a percent.
For the work function, finally, the experimental values also vary quite a
bit, in the range 5.77--6.10 eV. According to Kaack and Fick,\cite{kaack}
however, the work function has a small temperature dependence, decreasing
slightly with temperature. Since our calculations are performed at 0~K, we
expect that they should compare well with the largest experimental values. We
find, indeed, that the LDA result, 6.07 eV, is in excellent agreement with
the largest experimental number, 6.10 eV. The GGA value, in contrast, is
significantly smaller --- 5.70 eV --- indicating, once more, that the LDA
provides a better description of Pt than the LDA. In spite of this, the two
approximations will carefully be examined in the context of SFE and diffusion
barriers.
\subsection{Step Formation Energy}\label{res_vic}
We come now to the heart of the matter, namely the energetics of step
formation. In order to determine the SFE, we constructed vicinal surfaces
(using rectangular surface cells) appropriate to each type of steps. For the
(100)-faceted step, we examined both a (211) and a (332) surface; the former
has 3 atoms per terrace while the latter has 5. For the (111)-faceted step,
only the (221) surface, which contains 4 atoms per terrace, is considered.
Again, here, a vacuum region of approximately 10 \AA\ was included in all
cases.
The energies of the vicinal surfaces were determined using the same procedure
as in the case of the clean surface, i.e., by comparing to an appropriate
bulk model. In order to minimize the error arising from the use of different
geometries, the bulk reference system for a given vicinal surface was always
taken to have the same in-plane geometry as the surface. Thus, the same {\bf
k} points were used in the $x-y$ plane, while for the $z$ coordinate, the
grid was adjusted to yield a comparable density. In total, the bulk
supercells corresponding to the (211), (322), and (221) surfaces contained 6,
34, and 18 atoms, respectively.
The SFE is given, simply, by the difference in energy between a surface with
a step and one without. Since we are dealing with vicinal surfaces here, this
is equivalent to subtracting from the vicinal-surface energy (per terrace),
$\sigma_{\rm vic}$, that portion of the clean-surface energy corresponding to
the exposed (111) area.\cite{feibelman0} If we call $\sigma_{(111)}$ the
clean-(111)-surface energy per atom and neglect step-step interactions we
find, for the (100)-faceted step:
\begin{equation}
E^{\rm SF}_{(100)} = \sigma_{\rm vic} - ( N - \frac{1}{3} ) \sigma_{(111)},
\label{step-100}
\end{equation}
and for the (111)-faceted step:
\begin{equation}
E^{\rm SF}_{(111)} = \sigma_{\rm vic} - ( N - \frac{2}{3} ) \sigma_{(111)},
\label{step-111}
\end{equation}
where $N$ is the number of atoms per terrace, as defined earlier.
It is evident from Eqs.\ \ref{step-100} and \ref{step-111} that, in order to
determine reliably the SFE ratio, {\em very} accurate surface energies are
required for both vicinal and clean surfaces. It is our purpose here to
assess carefully the accuracy of our calculations through a detailed
convergence study. As explained earlier, the error arising from the supercell
geometry is minimized by always comparing surface and bulk energies obtained
using the same in-plane periodicity and {\bf k}-point density. Of course, it
is essential that the energies be converged with respect to Brillouin-zone
integration; for the clean (111) surface, the {\bf k}-point sampling scheme
used here leads to values converged within 2 meV, as discussed in Sec.\
\ref{res_clean}. For the vicinal surfaces, we used a similar sampling scheme
and, as we will see below, the error is of the order of a few meV, so that
differences in energy (e.g., between steps) of a few hundredths of an eV {\em
are} significant.
The results for the (100)-faceted step under a variety of theoretical
conditions are listed in Table \ref{step100}. First, we examine the effect of
relaxation and size within the LDA. With all atoms in the bulk-like
configuration (referred to as ``rigid'' in Table \ref{step100}), we find the
SFE to be rather insensitive to the size of the supercell: adding two layers
to the (211) slab increases the SFE by a mere 0.02 eV/(step atom), while
using a (322) surface instead of a (211) leads to a small decrease of 0.01
eV/(step atom). (The number of layers refers to the number of (111)-like
layers in the slab before a rotation is applied to make the surface vicinal.)
However, upon relaxing {\em all} atoms, except those in the central (111)
layer (configurations referred to as ``relaxed'' in Table \ref{step100}), the
SFE is found to decrease strongly, from 0.62 to 0.43 eV/(step atom) for the
(322) surface. For the relaxed configurations, we have also examined the
convergence with respect to the {\bf k}-point sampling. In all cases, the SFE
changes by at most 0.01 eV/(step atom) upon increasing the number of {\bf k}
points. Thus, we estimate the SFE for the (100)-faceted step to be
0.43$\pm$0.02 eV/(step atom) within the LDA.
For the (111)-faceted step, in view of the above results, we have studied a
single vicinal surface, namely the (221), at fixed and converged {\bf
k}-point density, as indicated in Table \ref{step111}. Again, here,
relaxation affects strongly the SFE, which decreases very markedly --- from
0.64 to 0.38 eV for a 5-layer slab in the LDA. However, increasing the
thickness from 5 to 7 layers brings about no significant changes in the SFE.
Thus, our best LDA-SFE value for the (111)-faceted step is 0.38$\pm$0.02
eV/(step atom).
Our calculations indicate, therefore, that the (111)-faceted step has a lower
formation energy than the (100)-faceted step --- 0.38 versus 0.43 eV (in the
LDA), leading to a ratio $E^{\rm SF}_{(111)} / E^{\rm SF}_{(100)}$ of
0.88$\pm$0.07. This is in excellent agreement with the experimental value of
0.87$\pm$0.02,\cite{michely} but at variance with a previous LDA calculation
by Feibelman,\cite{feibelman0} who found that the two steps are nearly
equivalent, that is, 0.46 eV for the (111)-faceted step versus 0.47 eV for
the (100), i.e., a ratio of 0.98. The SFE values differ from Feibelman's not
only in a relative sense, but also in an absolute sense: the values we find
are significantly smaller, by 0.08 eV/(step atom) for the (111)-faceted step
and 0.04 eV/(step atom) for the (100)-faceted step. Though the reasons for
these differences are not clear, they may originate in the choice of basis
sets: while we use plane waves, Feibelman employs Gaussian orbitals which are
more sensitive to the details of the atomic configuration, as discussed in
Ref.\ \onlinecite{feibelman0}. In view of this, it might perhaps be the case
that a Gaussian basis set lacks the accuracy needed to resolve such small
energy differences as those involved here.
A value of 0.37 eV/(step atom) for the (111)-faceted SFE has also been
estimated from the experimental surface free-energy anisotropy between the
(110) and (111) surfaces.\cite{bonzel} While this corresponds quite closely
to our value of 0.38 eV/(step atom), the agreement is fortuitous since the
above result was obtained assuming a value of the surface energy of 0.097
eV/\AA $^2$, much lower than that calculated here, 0.124 eV/\AA $^2$. [Using
the latter value for the surface energy would lead to a (111)-faceted SFE
0.45 eV/(step atom) in the approach of Ref.\ \onlinecite{bonzel}.]
Recently, some concerns have been expressed regarding the procedure used here
to determine the surface energy, which should diverge as the thickness of the
slab increases.\cite{boettger,fiorentini} We reinterpreted our results using
the approach suggested in Ref.\ \onlinecite{fiorentini} and found only small
changes in the (111) surface energy, now 0.121 eV/\AA$^2$ rather than 0.124
eV/\AA$^2$; for the SFE, we obtain now 0.41 and 0.46 eV/(step atom) for the
(111)- and (100)-faceted step, respectively, compared to 0.38 and 0.43
eV/(step atom) using the usual approach. The SFE ratio remains approximately
unchanged, 0.89 versus 0.88. We are thus led to conclude that, while the
uncertainty on the SFE might be of the order a 0.03 eV, the value we find for
the ratio is accurate to a few percent, and is not affected by the numerical
procedure used.
As mentioned in the previous section, the LDA seems to provide a better
description of bulk Pt, as well as of the (111) surface, than the GGA. In
view of this, it is expected that the vicinal surfaces are also better
represented within the LDA. The question remains open, however, because there
exists no firm experimental data to compare our results to, and it is
therefore of interest to calculate the SFE also within the GGA. The results
are given in Tables \ref{step100} and \ref{step111}. We observe the GGA-SFE
to be systematically lower than the corresponding LDA values, as is also true
of the (111) surface energy. Further, convergence with respect to both size
and {\bf k}-point density is similar in the two approximations. We therefore
conclude to GGA values of 0.29$\pm$0.02 and 0.25$\pm$0.02 eV/(step atom) for
the (100)- and (111)-faceted step, respectively. The resulting SFE ratio is
0.86$\pm$0.10, essentially unchanged from the LDA value, namely
0.88$\pm$0.07.
In order to understand why the two steps have different formation energies,
it is of interest to consider, first, the (111)- to (100)-faceted SFE ratio
in the {\em unrelaxed} (bulk-like) configuration. We find, from Tables
\ref{step100} and \ref{step111}, this ratio to be equal to 1.03$\pm$0.07
(using the LDA), compared to about 0.88 for the relaxed configurations, as we
have seen above. Thus, before the atoms relax, the two steps are nearly
equivalent [with perhaps a slight preference for the (100)-faceted step], as
could be expected from a simple nearest-neighbor model as explained in the
Introduction. Evidently, therefore, the observed step anisotropy is closely
related to relaxation, and this can be understood in the following way: As we
have seen in Sec.\ \ref{res_clean}, the Pt(111) surface is under large
tensile stress, which can be locally relieved at steps. However, because the
atomic configurations are different (albeit slightly) for the two kinds of
steps, the relaxation patterns also differ, and lead to different energetics.
This can in fact be seen very clearly in Fig.\ \ref{rel_geo}, where we plot
the displacement patterns for the two types of steps: Some atoms suffer very
large displacements --- by as much as a few percent (relative to the bulk
nearest-neighbor distance) for those that sit closest to the steps. More
important, it is also clear from this figure that the displacements
associated to the (111)-faceted step are larger than for the (100)-faceted
step, i.e., the former can relieve stress more efficiently than the latter,
and is thus energetically more favorable. It should be mentioned that the
displacements we find here differ from Feibelman's\cite{feibelman0} by as
much as 1$\%$ in some cases, and might possibly explained the discrepancy
between the two sets of results; this is likely related, again, to different
choices of basis functions.
The relation between relaxation and stress can be understood in a more
quantitative manner by considering the change in energy resulting from the
displacement inwards of an edge atom, i.e., in the direction normal to the
step and parallel to the terrace. Starting with both step models in their
bulk-like geometry and moving an edge atom by the same amount for the two
steps, we find that the (111)-faceted step, because of its triangular
geometry, releases more energy than the (100)-faceted step: for a
displacement of 0.14 \AA, corresponding approximately to the observed
relaxation, we find the (111)-faceted step to be already 22 meV lower in
energy than the (100)-faceted step.
Clearly, therefore, the difference in SFE arises from the large excess
surface stress of the Pt(111) surface --- 0.25 eV/\AA$^2$ (cf.\ Table
\ref{tests}), about twice as large as the surface energy, 0.124 eV/\AA$^2$.
We may compare this with the corresponding situation for Ir/Ir(111), where
the equilibrium island shape is nearly hexagonal.\cite{fu} In this case, the
excess surface stress is 0.128 eV/\AA$^2$,\cite{needs} significantly smaller
than the surface energy, 0.204 eV/\AA$^2$. Evidently, large surface energies
lead to large SFE, and large excess surface stresses to large differences
between the two steps. These observations therefore suggest that, as a ``rule
of thumb'', the SFE ratio should differ from one (i.e., non-hexagonal
equilibrium island shape) when the excess surface stress is sizeably larger
than the surface energy. This is in fact the case of Al(111)\cite{needs2} and
Au(111),\cite{needs} while the opposite is true of Rh\cite{filippetti} and
Cu.\cite{boisvert2} To our knowledge, no information on the equilibrium
island shape is available for Rh and Cu, but we would predict it to be
hexagonal in both cases. For Au, reconstruction\cite{barth} is likely to be
important in determining the island shape. For Al, finally, {\em ab initio}
calculations of the kind presented here have been performed by Stumpf and
Scheffler,\cite{stumpf1} and a SFE ratio of 0.93 is indeed found. In this
case, the ratio does not seem to be affected by relaxation, contrary to our
results for Pt, but different electronic orbitals are involved --- $sp$ for
Al and $d$ for Pt.
There have been other calculations of the equilibrium island shape on
Pt(111). Using a tight-binding model, Papadia {\em et al.\ } found
essentially no difference between the two steps in their bulk-like
configuration, as is the case here, but did not consider
relaxation.\cite{papadia} Within equivalent-crystal theory, Khare and
Einstein found a ratio close to unity (0.968) for the SFE, without allowing
in-plane relaxation;\cite{khare} also, in this approach, the surface energy
is predicted to be 0.076 eV/\AA $^2$, quite a bit smaller than our 0.124
eV/\AA $^2$. Fully-relaxed calculations using the semi-empirical
embedded-atom method (EAM) have also been performed and lead to a SFE ratio
very close to one;\cite{nelson} it is however doubtful that the EAM potential
is robust enough to account for the small energy differences involved here.
Other approaches have been proposed, based on coordination- or
orientation-dependent bonds,\cite{fallis,barkema} which do not take
relaxation effects into account.
We also list, in Tables \ref{step100} and \ref{step111}, the work function
for the different surfaces examined. As expected (see for instance Ref.\
\onlinecite{meth92}), and already observed by Feibelman,\cite{feibelman0} the
work function is smaller for the vicinal surfaces than for the (111) surface.
We also observe, in the case of the (100)-faceted step, a small dependence on
the the terrace length. This agrees with Feibelman's calculations, while a
stronger dependence is reported from experiment.\cite{besocke} This might be
due to the fact that terraces studied in experiment are much wider than ours
and/or surface contamination.\cite{feibelman0}
\subsection{Atom Adsorption and Diffusion}\label{res_ad}
We now discuss adsorption and diffusion of a Pt atom on the Pt(111) surface.
For these calculations, an adatom is added on one surface of the Pt slab
while the other surface is constrained to its bulk-like configuration. In
order to determine the energies at the two adsorption sites as well as the
barrier for diffusion, we considered both a ($2\times2$) and a ($3\times3$)
cell with, again, approximately 10 \AA\ of vacuum. Unless otherwise noted,
the integration over reciprocal space was performed using a mesh of 16
equidistant {\bf k} points for the ($2\times2$) cell and 9 {\bf k} points for
the ($3\times3$) cell.
First, starting with the ($2\times2$) system, we examined convergence with
respect to the number of layers, which we varied from 3 to 6. The results are
given in Table \ref{barrier}. We observe significant changes upon going from
3 to 4 (for a given state of relaxation), while increasing this number
further brings about changes of at most 0.01 eV; thus, 4 layers seem to be
sufficient for reliable estimates of both the diffusion barrier, $E_{\rm d}$,
and the difference in adsorption energies, $\Delta E_{ads}$.
The barrier we obtain for the {\em unrelaxed} substrate, 0.41 eV, and the
difference in adsorption energies, 0.10 eV, agree well with previous LDA
calculations.\cite{mortensen,feibelman1} It is however clear from Table
\ref{barrier} that relaxation effects are again here important: If we allow
the topmost layer to relax, $E_{\rm d}$ decreases to 0.34 eV and $\Delta
E_{\rm ads}$ increases to 0.17 eV; including second-layer relaxation as well
results in relatively minor changes to the energies (less than 0.02 eV).
It should be noted that, because of the asymmetry between the two adsorption
sites, there are in fact two barriers for diffusion. However, the difference
between the two barriers is such that, for temperatures of interest, it is
the highest-energy barrier that limits diffusion, i.e., the adatom will
escape rapidly from the low-energy adsorption state but get trapped in the
high-energy site.
We have also examined convergence with respect to {\bf k}-point sampling,
energy cutoff, and lateral size. As indicated in Table \ref{barrier}, we find
in all cases very modest changes of at most 0.01 eV. Likewise, using the GGA
does not lead to appreciable changes to the energy barrier, while $\Delta
E_{\rm ads}$ decreases by about 0.03 eV. For the reasons discussed in Sec.\
\ref{res_clean}, we suspect that the LDA values are more accurate; our best,
highly-converged estimates of $E_{\rm d}$ and $\Delta E_{\rm ads}$ are thus
0.33$\pm$0.03 and 0.17$\pm$0.03 eV, respectively. As discussed in Ref.\
\onlinecite{mortensen}, the large difference between the fcc and the hcp site
is related to the angular character of the $d$ orbitals, whose bonding
strength depends on the filling and radial quantum number of the bands. Due
to this difference, the transition site for jump diffusion does not lie {\em
exactly} midway between the two equilibrium sites but, rather, about 0.07
\AA\ towards the hcp site. The potential energy surface is however very flat
in this region, changing by no more than 0.01 eV upon going from the
transition to the midpoint site.
There is very little experimental information available for $\Delta E_{\rm
ads}$; as mentioned in the Introduction, only a lower limit of 0.06 eV has
been determined,\cite{golzhauser} and this is consistent with our results.
For the diffusion barrier, a value of 0.25$\pm$0.02 eV has been inferred from
field-ion microscopy (FIM) measurements in the temperature range 92--100
K.\cite{feibelman1} Also, based on a comparison between STM measurements of
the island density and kinetic Monte Carlo simulations between 110 K and 160
K, an estimate of 0.26$\pm$0.01 eV is obtained.\cite{bott} Both values agree,
but disagree with our result of 0.33$\pm$0.03 eV, which is surprising in view
of the high-level of accuracy and convergence of our calculations.
One possible explanation for this disagreement would be the neglect, in our
calculations, of dynamical effects. These cannot be assessed directly from
first-principles but have been shown to be insignificant at low temperatures,
either from the calculation of dynamical corrections to the transition state
theory\cite{cohen} or from empirical molecular-dynamics
simulations.\cite{boisvert3,ferrando,kallinteris} Experimentally, it is clear
that the FIM value\cite{feibelman1} suffers from poor statistics --- only two
points are used to determine the Arrhenius parameters --- and therefore the
error bar is large. Concerning the STM experiment, it has been pointed out by
the authors that small-cluster mobility could affect the Monte-Carlo estimate
of the barrier if its energy is close to that for adatom diffusion, in which
case the quoted value would be a lower bound to the actual barrier. As a
final point, it should be mentioned that many different empirical potentials
have been used to determine the diffusion barrier of Pt adatoms on
Pt(111),\cite{feibelman1,majerus,li,liu,stoltze,basset} yielding to values in
the range 0.01--0.18 eV, i.e., much lower than the experimental value, which
we argue, is a lower bound to the actual barrier. Thus, such models are
clearly too crude to provide a proper description of the energetics of
diffusion for the present system.
\section{Summary}\label{conc}
We have used highly-accurate {\em ab-initio} methods to calculate the ratio
of (111)- to (100)-faceted step formation energies on the (111) surface of
Pt. We find, in excellent agreement with experiment, (111)-faceted steps to
be favoured over (100) in a ratio of about 0.88; as a consequence, the island
on this surface should be clearly non-hexagonal. The difference between the
two steps is related to the large tensile stress of the Pt(111) surface,
which is released in a different manner because of differences in the local
topology. Our calculations underline the importance of relaxation in such
cases: while the two steps are about equivalent for the unrelaxed substrate,
relaxation does bring about large changes in the formation energies.
Likewise, relaxation is important to a proper determination of
equilibrium-site energies on the flat (111) surface. We find the fcc site to
be preferred over the hcp by a sizeable 0.17 eV (after full relaxation),
consistent with experiment which provides a lower bound of 0.06 eV for the
difference between the two sites.
We have also calculated the energy barrier for adatom diffusion and found a
fully-relaxed value of 0.33 eV. While constituting an improvement over
previous calculations, this value remains larger than experiment, by about
0.07 eV. The discrepancy might be due to limitations of our theoretical
approach (e.g., finite size), but it might also due to errors in the
interpretation of the experimental data --- poor statistics, neglect of
small-cluster contributions (e.g., dimers) to mass transport, i.e., incorrect
assumption regarding the critical nucleus size. More experiments are needed
to clarify this point. Likewise, it would be of interest that measurements of
the SFE and difference in energy between fcc and hcp sites be carried out so
as to assess the validity of LDA (versus GGA) in the present context.
\acknowledgements
G.B.\ acknowledges the warm hospitality of the Theory Department of the
Fritz-Haber-Institut where a large part of this work was performed. We are
grateful to Martin Fuchs for help with generating the pseudopotentials and to
Alexander Kley, Christian Ratsch, Paolo Ruggerone, Ari P Seitsonen, and Byung
Deok Yu for stimulating discussions. This work was supported by grants from
the Natural Sciences and Engineering Research Council (NSERC) of Canada and
the ``Fonds pour la formation de chercheurs et l'aide {\`a} la recherche''
(FCAR) of the Province of Qu{\'e}bec. One of us (G.B.) is thankful to NSERC and
FCAR for financial support. We are grateful to the ``Services informatiques
de l'Universit{\'e} de Montr{\'e}al'' for generous allocations of computer
resources. Part of the calculations reported here were carried out on the
IBM/SP-2 at the CACPUS (``Centre d'applications du calcul parall{\`e}le de
l'Universit{\'e} de Sherbrooke'').
|
{
"redpajama_set_name": "RedPajamaArXiv"
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O Efebo de Anticítera é uma escultura da Grécia Antiga, parte dos destroços de Anticítera, um naufrágio encontrado com várias preciosidades, incluindo o Mecanismo de Anticítera. O efebo é uma das raras obras em bronze que sobreviveram da Antiguidade Clássica.
A estátua pertence ao Museu Arqueológico Nacional de Atenas. A autoria da obra é desconhecida, bem como sua localização original. Alguns estudos sugerem que estava sendo levada da região grega da Ásia Menor, talvez de Éfeso, em direção à Itália. Representa um efebo (jovem) de pé, todo nu, na clássica postura do contrapposto, com o braço direito elevado e a mão em atitude de segurar um objeto, que se perdeu. Os olhos são incrustações de pedra colorida. Mede 1,94 m de altura, e foi fundido em várias partes juntas posteriormente.
O Efebo foi encontrado em 1900 junto aos destroços do naufrágio, ao largo da costa de Anticítera, por pescadores de esponjas, estando partido em muitos fragmentos e bastante corroído pela longa ação da água do mar. Um restauro preliminar foi realizado pelo escultor francês Alfred André logo após o achamento, quando foram recriados os fragmentos ausentes, especialmente nas áreas do abdômen, da pelve, dos flancos e dos ombros. As lacunas criaram problemas sérios para a recomposição da postura da figura e o resultado não foi muito satisfatório. Adicionalmente, o tratamento que aplicou no metal para livrá-lo da corrosão causou danos generalizados na superfície. O seu trabalho também sofreu críticas pelas liberdades que tomou na recriação das partes faltantes, por ter coberto toda a superfície com uma película de massa para ocultar as múltiplas emendas e rugosidades, e por tê-la pintado inteira, tornando impossível estudá-la cientificamente e determinar seu real estilo.
Assim, na década de 1950 foi decidido um outro restauro, empreendido por Christos Karouzos. O trabalho envolveu a remoção de todos os acréscimos de massa e tinta e a desmontagem total da obra, a fim de recompô-la com um posicionamento mais adequado. Ao longo deste processo foi descoberto que muitos fragmentos haviam sido reintegrados por André fora de sua posição correta. Este restauro produziu uma estátua à primeira vista similar ao resultado de André, mas uma análise mais detida mostra que muitos detalhes são diferentes, e o conjunto evidencia muito mais de perto sua aparência como deve ter sido originalmente, especialmente na sua postura. Um artigo descrevendo o restauro foi publicado em 1969.
O Efebo é uma obra altamente apreciada pela crítica especializada, considerado uma obra-prima da tradição grega. Opiniões mais antigas chegaram a atribuí-lo a algum dos grandes mestres do Classicismo Tardio, nomeadamente Praxíteles, Escopas ou Lísipo, ou a algum dos seus seguidores diretos, mas também se cogitou que fosse uma criação helenística inspirada em modelos clássicos, o que trouxe incerteza para sua datação. Em meados do século XX formou-se um consenso em sua datação como sendo 340–330 a.C., e foi enfatizada uma derivação da escola de Policleto, do Alto Classicismo, mas a situação não se consolidou. A partir da década de 1960 a polêmica aumentou em torno de sua datação e da caracterização do seu estilo, com alguns pesquisadores assinalando traços típicos do Alto Classicismo, como o modelado da musculatura do abdômen, outros indicando elementos que seriam típicos de um período um pouco posterior, como a estrutura do dorso, e outros ainda sugerindo que esse ecletismo de traços aponta para um artista helenístico, corrente que ressuscitou estilos anteriores produzindo combinações estilísticas tipicamente ecléticas. A polêmica continua.
A identificação do sujeito também tem se prestado a muito debate. A ausência de qualquer atributo distintivo torna impossível uma identificação precisa, surgindo sugestões de que tenha representado um deus, um heroi ou um mortal. A principal pista para uma definição é a posição do braço e mão direitos. Os dedos da mão estão em posição de parecerem ter segurado um objeto arredondado, e daí vários críticos sugeriram que tenha sido a maçã que Páris ofereceu à vencedora do concurso de beleza entre as deusas Afrodite, Atena e Hera. Esta teoria é criticada porque na tradição iconográfica de Páris o heroi é usualmente representado mais jovem e menos musculoso. Alternativamente, pode ter sido uma maçã do Jardim das Hespérides, que Hércules em seus Doze Trabalhos foi incumbido de coletar, e a postura do Efebo é consistente com a tradição iconográfica de Hércules no Jardim das Hespérides em ato de colher uma das maçãs de ouro. Remanescentes de um objeto também perdido em sua mão esquerda poderiam ser de uma pele de leão, elemento regularmente representado com o semideus. O posicionamento dos dedos, porém, não é uma evidência forte, pois podem também ter sustentado uma variedade de outros objetos, como uma coroa de louros, uma bola, uma grinalda de flores ou ainda outros.
Ver também
Escultura do Classicismo grego
Escultura helenística
Estátuas
Esculturas da Grécia Antiga
Esculturas no Museu Arqueológico Nacional de Atenas
Esculturas de bronze
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: ui-router - ui view and link not working I'm trying out ui-router for angular. I added this line of code to my app module in my app.js:
angular
.module("ngClassifieds", ['ngMaterial', 'ui.router'])
.config(function($mdThemingProvider, $stateProvider){
$mdThemingProvider.theme('default')
.primaryPalette('teal')
.accentPalette('orange');
$stateProvider
.state('stateone', {
url:'/stateone',
template: '<h1>State one</h1>'
})
.state('statetwo', {
url: '/statetwo',
template: '<h1>State two</h1>'
});
});
On my html I just put an empty ui-view to test whether or not i'm getting those two headers but with no luck:
<ui-view></ui-view>
Testing it out on my localhost link by putting in:
localhost:8080/#/stateone
localhost:8080/#/statetwo
But for some reasons it's not loading/showing any of the headers on any of those links but just showing my page without the headers.
I have already included angular-ui-router.js into my index.html
If anyone could point out what I did wrong, that would great.
In case anyone is wondering : this is the order of my scripts:
<script src="node_modules/angular/angular.js"></script>
<script src="node_modules/angular-animate/angular-animate.js"></script>
<script src="node_modules/angular-aria/angular-aria.js"></script>
<script src="node_modules/angular-material/angular-material.js"></script>
<script src="node_modules/angular-ui-router/release/angular-ui-router.js"></script>
<script src="scripts/app.js"></script>
<script src="components/classifieds.ctr.js"></script>
<script src="components/classifieds.fac.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.4/lodash.min.js"></script>
A: Since you did not get any error it is difficult to understand what went wrong but you can have a look at my plunkr and figure it out. I removed angular-material.
https://plnkr.co/edit/WGjzY6flSx5Ces1moSPk?p=preview
<script>
angular
.module("ngClassifieds", ['ui.router'])
.config(function( $stateProvider){
$stateProvider
.state('stateone', {
url:'/stateone',
template: '<h1>State one</h1>'
})
.state('statetwo', {
url: '/statetwo',
template: '<h1>State two</h1>'
});
});
</script>
</head>
<body>
<h1>Hello Plunker!</h1>
<a href="#/stateone">Link1</a>
<a href="#/statetwo">Link2</a>
<ui-view></ui-view>
</body>
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"redpajama_set_name": "RedPajamaStackExchange"
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'use strict';
angular.module('crossfitApp')
.controller('LogoutController', function (Auth) {
Auth.logout();
});
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"redpajama_set_name": "RedPajamaGithub"
}
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\section{Introduction}
We consider a cubic microlattice scaffold constructed of connected particles of micrometer scale, within a nematic liquid crystal. In this article, we treat the particles of the cubic microlattice as being inclusions from the mathematical point of view, while they might be interpreted as colloids from the physical point of view, even though they do not possess all of their properties. The cubic microlattice scaffold is also called a bicontinuous porous solid matrix (BPSM) in the physics literature (for example, see \cite{BPSM1}, \cite{nematiccage} or \cite{BPSM2}). By cubic microlattice scaffold we understand a connected family of parallelipipeds or cubes of different sizes, placed in a periodic fashion, as in Figure \ref{fig:nematiccage1}, where only the embedded particles have been shown. For simplicity, we might refer to this object as being a scaffold or a cubic microlattice. This type of scaffold is usually obtained using the \emph{two-photon polymerization} (TPP or 2PP) process, which represents a technique of 3D-manufacturing structures and which can generate stand-alone objects. An overview of the field of TPP processes can be found in \cite{2PP}. There are numerous experiments, theory and computer simulations regarding embedding microparticles into nematic liquid crystals (for example, see \cite{Ravnik1}, \cite{Ravnik2} and \cite{Ravnik3}).
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{limit0,25-eps0,05.png}
\caption{Example of a cubic microlattice.}
\label{fig:nematiccage1}
\end{figure}
The system bears mathematical similarities to that of colloids embedded into nematic liquid crystals. The mathematical studies of nematic colloids (the mixture of colloidal particles embedded into nematic liquid crystals) are split into two broad categories:
\begin{itemize}
\item[•] one is dealing with the effect produced by a small number of particles in this mixture, with a focus on the defect patterns that arise in the alignment of the nematic particles induced by the interaction at the boundary of the colloid between the two combined materials (see, for example, \cite{Alama1, Alama2, Canevari1, Canevari2, Canevari3, Canevari4, Wang});
\item[•] the other one treats the study of the collective effects, that is the homogenisation process (see, for example, \cite{Bennett, Berlyland, Calderer, CanevariZarnescu1} and \cite{CanevariZarnescu2}).
\end{itemize}
This work continues within the second direction, that is studying the homogenised material, and it is built on the work from \cite{CanevariZarnescu1} and \cite{CanevariZarnescu2}, which was also based on \cite{Bennett, Berlyland, Calderer}. In \cite{CanevariZarnescu1} and \cite{CanevariZarnescu2}, the \textit{inclusion} is considered to be the union of some disconnected particles, obtained from different or identical model particles, in such a way that the distance between the particles is considerable larger than the size of them, which is called the dilute regime. Also, in this regime, the volume fraction of colloids tends to zero.
In this article, we are going to consider the case of a cubic microlattice scaffold, as shown in Figure \ref{fig:nematiccage1}. The idea of using such a particular geometry for the scaffold comes from the work done in \cite{nematiccage}. At the same time, this geometric configuration is more relevant from the physical point of view, since in \cite{CanevariZarnescu1} and in \cite{CanevariZarnescu2} one cannot position \textit{a priori} the colloidal particles in a periodic fashion. Here the periodicity is automatically generated by the structure of the cubic microlattice. We construct two types of scaffolds: one with identical cubes centered in a periodic 3D lattice of points, cubes which are inter-connected by parallelipipeds, and one where we replace the cube with a parallelipiped with three different length sides. If by cubic symmetry we understand the family of rotations that leave a cube invariant, then the first case is when the scaffold particles have cubic symmetry and the second one is with the loss of this type of symmetry. The main new aspects of this work are:
\begin{itemize}
\item[•] the set of all the inclusions is now connected;
\item[•] the model particle that we use (that is, a parallelipiped or a cube) grants us the possibility to compute the surface contribution for arbitrarily high order terms in the surface energy density - hence, a generalisation has been done for higher order polynomials in the bulk energy potential that admit at least one local minimiser (see \cref{th:gen});
\item[•] in the case where the cubic symmetry is lost, we obtain a new term into the homogenised limit that can be seen as a change in the preferred alignment of the liquid crystal particles inside the domain (see \cref{th:asym});
\item[•] we obtain a rate of convergence for how fast the surface energies converge to the homogenised one (more details in \cref{prop:rate_of_conv}); in \cref{remark:rate_of_convergence}, we also obtain a rate of convergence for how fast the sequence of minimisers of the free energies tend to a minimiser of the homogenised free energy;
\end{itemize}
Liquid crystal materials, which typically consist of either rod-like or disc-like molecules, can achieve a state of matter which has properties between those of conventional liquids and those of solid crystals. The liquid crystal state of matter is one where there exists a long range orientational order for the molecules. In order to quantify the local preferred alignment of the rod-like molecules, we use the theory of Q-tensors (for more details, see \cite{Mottram}). A background of the field of liquid crystal materials can be found in \cite{deGennes}.
Let $\Omega\subset\mathbb{R}^3$ be an open and bounded domain from $\mathbb{R}^3$. For every $\varepsilon>0$, we construct a cubic microlattice $\mathcal{N}_\varepsilon$ inside of $\Omega$, such that, as $\varepsilon\rightarrow 0$, the volume of the scaffold tends to 0. More details regarding the construction of the cubic microlattice can be found in \cref{section:assumptions} and in \cref{section:constructing_lattice}.
Let $\Omega_{\varepsilon}=\Omega\setminus\mathcal{N}_{\varepsilon}$, which represents the space where only liquid crystal particles can be found. We use functions $Q:\Omega_{\varepsilon}\rightarrow\mathcal{S}_0$ to describe the orientation of the liquid crystal particles, where: $$\mathcal{S}_{0}=\{Q\in\mathbb{R}^{3\times 3}:\;Q=Q^T,\;\text{tr}(Q)=0\},$$ is denoted as the set of $Q$-tensors. In the space $\mathcal{S}_0$, if we define $|Q|=(\text{tr}(Q^2)\big)^{1/2}$, for any $Q\in\mathcal{S}_0$, we can see that $\mathcal{S}_0$ is a normed linear space and the so-called Frobenius norm is induced by the scalar product $Q\cdot P=\text{tr}(Q\cdot P)$.
We consider the following Landau-De Gennes free energy functional:
\begin{equation}\label{eq:variat1}
\mathcal{F}_{\varepsilon}[Q]:=\int_{\Omega_{\varepsilon}} \big(f_e(\nabla Q)+f_b(Q)\big)\text{d}x+\dfrac{\varepsilon^{3}}{\varepsilon^{\alpha}(\varepsilon-\varepsilon^{\alpha})}\int_{\partial\mathcal{N}_{\varepsilon}}f_s(Q,\nu)\text{d}\sigma,
\end{equation}
where $f_e$ represents the \textit{elastic energy}, $f_b$ the \textit{bulk energy}, $f_s$ the \textit{surface density energy}, $\alpha$ is a real parameter and $\partial\mathcal{N}_{\varepsilon}$ the \textit{surface of the scaffold}. The parameter $\alpha$ is chosen such that the term $\varepsilon^{\alpha}(\varepsilon-\varepsilon^{\alpha})$ is proportional with the surface terms from $\partial\mathcal{N}_{\varepsilon}$ in order that, in the limit $\varepsilon\rightarrow 0$, it balances the effect given by the areas of the surface terms. All of these quantities are described and detailed more in \cref{section:assumptions}.
The \textit{elastic energy}, also called the distortion energy, penalises the distortion of $Q$ in the space and, in the Landau-de Gennes theory, it is usually considered to be a positive definite quadratic form in $\nabla Q$.
The \textit{bulk energy} in our case consists only of the thermotropic energy, which is a potential function that describes the preferred state of the liquid crystal, that is either uniaxial, biaxial or isotropic\footnote{The isotropic case corresponds to the case in which $Q=0$. The uniaxial case corresponds to the one in which two of the eigenvalues of $Q$ are equal and the third one has a different value. The biaxial case corresponds to the case in which all the eigenvalues have different values.}. For large values of the temperature, the minimum of this energy is obtained in the isotropic case, that is $Q=0$, and for small values, the minimum set is a connected set of the form $s(\nu\otimes \nu-\mathbb{I}_3/3)$, with $\nu\in \mathbb{S}^2$ and $\mathbb{I}_3$ the identity $3\times 3$ matrix, and this is a connected set diffeomorphic with the real projective plane. The simplest form that we can take for the \textit{bulk energy} in our case is the quartic expansion:
\begin{align*}
f_b(Q)=a\;\text{tr}(Q^2)-b\;\text{tr}(Q^3)+c\;\text{tr}(Q^2)^2,
\end{align*}
where the coefficient $a$ depends on the temperature of the liquid crystal and $b$ and $c$ depend on the properties of the liquid crystal material, with $b,c>0$.
The \textit{surface energy} describes the interaction between the liquid crystal material and the boundary of the scaffold. We assume, for simplicity, that it depends only on $Q$ and on $\nu$, where $\nu$ is the outward normal at the boundary of the cubic microlattice. One of the most common forms for the \textit{surface energy} is the Rapini-Papoular energy:
\begin{align}\label{eq:intro_Rapini-Papoular}
f_s(Q,\nu)=W\;\text{tr}\big(Q-s_+\big(\nu\otimes\nu-\mathbb{I}_3/3\big)\big)^2,
\end{align}
where $W$ is a coefficient measuring the strength of the anchoring, $s_+$ is measuring the deviation from the homeotropic (perpendicular) anchoring to the boundary and $\mathbb{I}_3$ is the $3\times 3$ identity matrix.
We are interested in studying the behaviour of the whole material when $\varepsilon\rightarrow 0$. We will show that in our dilute regime we obtain for the homogenised material an energy functional of the following form
\begin{equation*}
\mathcal{F}_0[Q]:=\int_{\Omega}\big(f_e(\nabla Q)+f_b(Q)+f_{hom}(Q)\big)\text{d}x,
\end{equation*}
where $f_{hom}$ is defined in \eqref{defn:f_hom} and in \eqref{defn:f_hom_sym}, depending on the choice of $f_b$. Unlike in the standard homogenisation studies, our focus will be on \textit{a priori} designing the $f_{hom}$, in terms of the available parameters of the system.
The article is organised in the following manner:
\begin{itemize}
\item[•] in \cref{section:assumptions_and_main_results} we present the technical assumptions that we have chosen and the main results of this article;
\item[•] in \cref{section:prop_F_eps} we present the study of the properties of the functional $\mathcal{F}_{\varepsilon}$ for a fixed value for $\varepsilon>0$;
\item[•] in \cref{section:conv_local_min} we glue together the properties studied in the previous section and analyse the $\Gamma$-limit of $\mathcal{F}_{\varepsilon}$ as $\varepsilon\rightarrow 0$ and we prove the main theorems stated in \cref{section:assumptions_and_main_results};
\item[•] in \cref{section:rate_of_conv} we analyse the rate of convergence of the sequence of surface energies to the homogenised surface functional, where the main result is \cref{prop:rate_of_conv}, but we also analyse the rate of convergence of the sequence of minimisers of the free energies to a minimiser of the homogenised free energy (see \cref{remark:rate_of_convergence})
\end{itemize}
\noindent and
\begin{itemize}
\item[•] in \cref{section:appendix} we prove various results, the most important of which is the proposition regarding the explicit extension function that we use in \cref{subsection:extension}.
\end{itemize}
\section{Tehnical assumptions and main results}\label{section:assumptions_and_main_results}
\subsection{Assumptions, notations and main result}\label{section:assumptions}
Let $\Omega\subset\mathbb{R}^3$ be a bounded, Lipschitz domain, that models the ambient liquid crystal, and let $\mathcal{C}\subset\mathbb{R}^3$ be the model particle for the cubic microlattice. Since $\Omega$ is bounded in $\mathbb{R}^3$, then:
\begin{align}\label{defn:L_0_l_0_h_0}
\exists\; L_0,\;l_0,\;h_0\in[0,+\infty)\;\text{such that}\;\overline{\Omega}\subseteq[-L_0,L_0]\times[-l_0,l_0]\times[-h_0,h_0].
\end{align}
In Figure \ref{fig:nematiccage}, we illustrate some examples of cubic microlattices, where the ``connecting" boxes (which can be seen better in Figure \ref{fig:nematiccage1} as being the black cubes) are cubes of size $\varepsilon^{\alpha}$, with $\alpha=1.4999$ \footnote{We choose $\alpha$ close to the value 3/2 in order to make the difference between the lengths of the sides of the black cubes and the gray parallelipipeds from Figure \ref{fig:nematiccage1} more visible, for relatively ``large" values of $\varepsilon$ ($0.01$, $0.05$ or $0.001$).} and $\varepsilon$ has a positive value close to 0, since we desire to work in the dilute regime. The distance between two closest black cubes is equal to $\varepsilon$, therefore the length of the black cubes is significantly smaller than the distance between them, by using the exponent $\alpha$. \footnote{The reason why we represent the lattice only in the box $[0,l]^3$, with $l=5\varepsilon$, is that if we keep the same $l$ and shrink $\varepsilon$, then the number of boxes appearing in the image would be significantly larger, hence, as we make $\varepsilon$ smaller, we also zoom in to have a better picture of what is happening for small values of $\varepsilon$.} For Figure \ref{fig:nematiccage1}, $\varepsilon=0.05$, $\alpha=1.4999$ and $l=0.25$, so we have the same ratio between $\varepsilon$ and $l$.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{limit0,05-eps0,01.png}
\caption{$\varepsilon=0.01$ and $l=0.05$;}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{limit0,005-eps0,001.png}
\caption{$\varepsilon=0.001$ and $l=0.005$.}
\end{subfigure}
\caption{Cubic microlattices constructed in the box $[0,l]^3$ with $\alpha=1.4999$.}
\label{fig:nematiccage}
\end{figure}
In order to construct such a scaffold, we use as a model particle the cube:
\begin{align}\label{defn:initial_cube}
\mathcal{C}=\bigg[-\dfrac{1}{2},\dfrac{1}{2}\bigg]^3.
\end{align}
We denote by $\partial\mathcal{C}$ the surface of the cube $\mathcal{C}$, which we also write it as:
\begin{align}\label{defn:C_x_C_y_C_z}
\partial\mathcal{C}=\mathcal{C}^x\;\cup\;\mathcal{C}^y\;\cup\;\mathcal{C}^z,
\end{align}
where $\mathcal{C}^x$ is the union of the two faces of the cube that are perpendicular to the $x$ direction and in the same way are defined $\mathcal{C}^y$ and $\mathcal{C}^z$.
Then, for a fixed value of $\varepsilon>0$ and an $\varepsilon$-independent positive constant $\alpha$, we define
\begin{equation}\label{defn:initial_cube_alpha}
\mathcal{C}^{\alpha}=\bigg[-\dfrac{\varepsilon^{\alpha}}{2p},+\dfrac{\varepsilon^{\alpha}}{2p}\bigg]\times\bigg[-\dfrac{\varepsilon^{\alpha}}{2q},+\dfrac{\varepsilon^{\alpha}}{2q}\bigg]\times\bigg[-\dfrac{\varepsilon^{\alpha}}{2r},+\dfrac{\varepsilon^{\alpha}}{2r}\bigg],
\end{equation}
with $p,\;q,\;r\in[1,+\infty)$. We call the scaffold symmetric whenever $p=q=r$. In Figures \ref{fig:nematiccage1} and \ref{fig:nematiccage}, we have $p=q=r=1$.
We construct now the lattice
\begin{equation}\label{defn:points1}
\mathcal{X}_{\varepsilon}=\{x\in\Omega\;:\;x=(x_1,x_2,x_3),\;\text{dist}(x,\partial\Omega)\geq\varepsilon\;\text{and}\;x_k/\varepsilon\in\mathbb{Z}\;\text{for}\;k\in\overline{1,3}\},
\end{equation}
which we rewrite it as:
\begin{align}\label{defn:points2}
\mathcal{X}_{\varepsilon}=\{x_{\varepsilon}^i\;:\;i\in\overline{1,N_{\varepsilon}}\},\;\text{where}\;N_{\varepsilon}=\text{card}\big(\mathcal{X}_{\varepsilon}\big).
\end{align}
Hence, the first part of the scaffold is the family of parallelipipeds
\begin{equation}\label{defn:ceps}
\mathcal{C}_{\varepsilon}=\bigcup_{i=1}^{N_{\varepsilon}}\mathcal{C}_{\varepsilon}^i,\;\text{where}\;\mathcal{C}_{\varepsilon}^i=x^i_{\varepsilon}+\mathcal{C}^{\alpha},\;\text{for every}\;i\in\overline{1,N_{\varepsilon}},
\end{equation}
which represents the union of all black parallelipipeds from Figure \ref{fig:nematiccage1}.
We add now the lattice
\begin{align}\label{defn:Y_eps}
\mathcal{Y}_{\varepsilon}&:=\bigg\{y_{\varepsilon}\in\Omega\;:\;\exists i,j\in\overline{1,N_{\varepsilon}}\;\text{such that}\;\big|x_{\varepsilon}^i-x_{\varepsilon}^j\big|=\varepsilon\;\text{and}\;y_{\varepsilon}=\dfrac{1}{2}\big(x_{\varepsilon}^i+x_{\varepsilon}^j\big)\bigg\}\notag\\
&\hspace{-16mm}\text{and let}\;M_{\varepsilon}=\text{card}\big(\mathcal{Y}_{\varepsilon}\big).
\end{align}
The lattice $\mathcal{Y}_{\varepsilon}$ helps us construct the gray parallelipipeds from Figure \ref{fig:nematiccage1}, that is, the ``connecting boxes". We split this lattice into three parts, since the gray parallelipipeds are elongated into three different directions, granted by the axes of the Cartesian coordinate system in $\mathbb{R}^3$. We denote by $\mathcal{P}_{\varepsilon}$ the union of all of these parallelipipeds. More details regarding the construction of these objects can be found in \cref{section:constructing_lattice}.
Let $\mathcal{N}_{\varepsilon}=\mathcal{C}_{\varepsilon}\cup\mathcal{P}_{\varepsilon}$ be the entire scaffold and $\partial\mathcal{N}_{\varepsilon}$ its surface.
Regarding the \textit{elastic energy} that we use, we observe that having a fixed distortion of $Q$, the liquid crystal material should remain unchanged under translations and rotations. We consider the following form for the \textit{elastic energy}:
\begin{align*}
f_e(\nabla Q)&:=\sum_{i,j,k\in\{1,2,3\}}\bigg[\dfrac{L_1}{2}\bigg(\dfrac{\partial Q_{ij}}{\partial x_k}\bigg)^2+\dfrac{L_2}{2}\dfrac{\partial Q_{ij}}{\partial x_j}\dfrac{\partial Q_{ik}}{\partial x_k}+\dfrac{L_3}{2}\dfrac{\partial Q_{ik}}{\partial x_j}\dfrac{\partial Q_{ij}}{\partial x_k}\bigg],
\end{align*}
where $Q_{ij}$ is the $(ij)^{th}$ component of $Q$, $(x_1,x_2,x_3)$ represents the usual cartesian coordinates and $e_{ijk}$ represents the Levi-Civita symbol.
Regarding the \textit{surface energy densities}, we use different forms. One is the Rapini-Papoular \textit{surface energy density}, presented in \eqref{eq:intro_Rapini-Papoular}, which has a single minimum at the point in which the dependent variables take value granted by the surface treatment, which in this case represents the perpendicular \textit{alignment}.
Another form for the \textit{surface energy density} is represented by the planar degenerate anchoring case:
\begin{align*}
f_s(Q,\nu)=k_a\;(\nu\cdot Q^2\nu)+k_b\;(\nu\cdot Q\nu)(\nu\cdot Q^2\nu)+k_c\;(\nu\cdot Q^2\nu)^2+a'\;\text{tr}(Q^2)+\dfrac{2b'}{3}\;\text{tr}(Q^3)+\dfrac{c'}{2}\;\text{tr}(Q^2)^2,
\end{align*}
where $k_a$, $k_b$, $k_c$, $a'$, $b'$ and $c'$ are constants, in which the preferred alignment for the liquid crystal material is to lie parallel to the boundary of the scaffold.
In order to describe the \textit{surface energy}, we need a better description of $\partial\mathcal{N}_{\varepsilon}$, therefore we analyse what faces from every parallelipiped constructed are in contact with the liquid crystal. More precisely, the liquid crystal is in contact with the scaffold:
\begin{itemize}
\item[•] on only four of the six faces of the parallelipipeds centered in points from $\mathcal{Y}_{\varepsilon}$, that is, on every $\mathcal{T}_x^k$, $\mathcal{T}_y^l$ and $\mathcal{T}_z^m$, defined in \eqref{defn:T_x_k}, \eqref{defn:T_y_l} and \eqref{defn:T_z_m};
\item[•] only on the edges of the parallelipipeds centered in some of the points from $\mathcal{X}_{\varepsilon}$, parallelipipeds which are in the ``interior" of the scaffold, meaning that they are not ``visible" - in this case, the interaction is neglected (the black parallelipipeds from Figure \ref{fig:nematiccage1} which have the only role of connecting the six adjacent gray parallelipipeds and they are not visible from that point of view, since they are ``inside" of the scaffold); let
\begin{align}\label{defn:N_eps_1}
N_{\varepsilon,1}\;=\text{ the total number of parallelipipeds from this case;}
\end{align}
\item[•] on at most five of the six faces of the parallelipipeds centered in some of the points from $\mathcal{X}_{\varepsilon}$, parallelipipeds which are at the ``outer" boundary of the scaffold (which are the closest to $\partial\Omega$) - in this case, we prove that the interaction is neglectable (the black parallelipipeds from Figure \ref{fig:nematiccage1} that are visible); let
\begin{align}\label{defn:N_eps_2}
N_{\varepsilon,2}\;=\text{ the total number of parallelipipeds from this case;}
\end{align}
and let
\begin{align}\label{defn:S_i}
\mathcal{S}^i\;=&\text{ the union of all the rectangles (at most five in this case) that}\notag\\
&\text{\hspace{4mm} are in contact with the liquid crystal material,}
\end{align}
for any $i\in\overline{1,N_{\varepsilon,2}}$.
\end{itemize}
From relations \eqref{defn:N_eps_1} and \eqref{defn:N_eps_2}, we have $N_{\varepsilon}=N_{\varepsilon,1}+N_{\varepsilon,2}$. Using \eqref{defn:T_x_k}, \eqref{defn:T_y_l}, \eqref{defn:T_z_m} and \eqref{defn:S_i}, we can write $\partial\mathcal{N}_{\varepsilon}=\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}\cup\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}$, where:
\begin{equation}\label{defn:surface}
\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}=\bigg(\bigcup_{i=1}^{N_{\varepsilon,2}}\mathcal{S}^i\bigg)\;\;\text{and}\;\;\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}=\bigg(\bigcup_{k=1}^{X_{\varepsilon}}\mathcal{T}_x^k\bigg)\cup\bigg(\bigcup_{l=1}^{Y_{\varepsilon}}\mathcal{T}_y^l\bigg)\cup\bigg(\bigcup_{m=1}^{Z_{\varepsilon}}\mathcal{T}_z^m\bigg).
\end{equation}
Let $J_{\varepsilon}[Q]$ be the \textit{surface energy} term from \eqref{eq:variat1} and let us split this term into two parts:
\begin{align}\label{defn:Jeps}
J_{\varepsilon}[Q]&=J_{\varepsilon}^{\mathcal{S}}[Q]+J_{\varepsilon}^{\mathcal{T}}[Q],
\end{align}
where
\begin{align}\label{defn:Jeps_S}
J_{\varepsilon}^{\mathcal{S}}[Q]&=\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\displaystyle{\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}}f_s(Q,\nu)\text{d}\sigma}=\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{i=1}^{N_{\varepsilon,2}}\int_{\mathcal{S}^i}f_s(Q,\nu)\text{d}\sigma,
\end{align}
using \eqref{defn:surface}, and
\begin{align}
J_{\varepsilon}^{\mathcal{T}}[Q]&=\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\displaystyle{\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}f_s(Q,\nu)\text{d}\sigma},
\end{align}
which can be also expressed using \eqref{defn:surface} as
\begin{align}\label{defn:Jeps_T}
J_{\varepsilon}^{\mathcal{T}}[Q]&=J_{\varepsilon}^{X}[Q]+J_{\varepsilon}^{Y}[Q]+J_{\varepsilon}^{Z}[Q],
\end{align}
where:
\begin{align}\label{defn:Jeps_T_X_Y_Z}
\begin{cases}
J_{\varepsilon}^{X}[Q]&=\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\displaystyle{\sum_{k=1}^{X_{\varepsilon}}\int_{\mathcal{T}_x^k}f_s(Q,\nu)\text{d}\sigma};\\
\vspace{-4mm}&\\
J_{\varepsilon}^{Y}[Q]&=\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\displaystyle{\sum_{l=1}^{Y_{\varepsilon}}\int_{\mathcal{T}_y^l}f_s(Q,\nu)\text{d}\sigma};\\
\vspace{-4mm}&\\
J_{\varepsilon}^{Z}[Q]&=\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\displaystyle{\sum_{m=1}^{Z_{\varepsilon}}\int_{\mathcal{T}_z^m}f_s(Q,\nu)\text{d}\sigma}.
\end{cases}
\end{align}
The main reason why we are using relation \eqref{defn:Jeps} is because we can prove that $J_{\varepsilon}^{\mathcal{S}}[Q]$ has no influence over the homogenised material, since this surface term describes only the interaction between the liquid crystal and some parts of the ``outer" boundary of the scaffold (close to $\partial\Omega$), which are small and fewer in number compared to the interactions with the other parts of the scaffold (see \cref{subsection:nocontribution}).
\begin{remark}
In this paper, we use the notation $A\lesssim B$ for two real numbers $A$ and $B$ whenever there exists an $\varepsilon$-independent constant $C$ such that $A\leq C\cdot B$.
\end{remark}
We assume furthermore that:
\begin{itemize}
\item[($A_1$)] $\Omega\subset\mathbb{R}^3$ is a smooth and bounded domain
\item[($A_2$)] \textit{$1<\alpha<\dfrac{3}{2}$;}
\end{itemize}
As we have seen already, the condition $1<\alpha$ grants the existence of the parallelipipeds generated in equations \eqref{defn:P_x}, \eqref{defn:P_y} and \eqref{defn:P_z}, but it also ensures the dilute regime. For a detailed proof of the last statement, consult \cref{subsection:appendix1}.
\begin{itemize}
\item[($A_3$)] \textit{There exists a constant $\lambda_{\Omega}>0$ such that} $$\text{dist}(z^i_{\varepsilon},\partial\Omega)+\dfrac{1}{2}\inf_{j\neq i}|z^j_{\varepsilon}-z^i_{\varepsilon}|\geq \lambda_{\Omega}\varepsilon$$ \textit{for any $\varepsilon>0$ and any center $z^i_{\varepsilon}$ of an object (either a black cube or a gray parallelipiped) that is contained the cubic microlattice, where $i\in\overline{1,(N_{\varepsilon}+M_{\varepsilon})}$.}
\end{itemize}
\begin{itemize}
\item[($A_4$)] \textit{As $\varepsilon\rightarrow 0$, the measures
\begin{equation}\label{defn:measures}
\mu_{\varepsilon}^X:=\varepsilon^3\sum_{k=1}^{X_{\varepsilon}}\delta_{y^{x,k}_{\varepsilon}},\;\mu_{\varepsilon}^Y:=\varepsilon^3\sum_{l=1}^{Y_{\varepsilon}}\delta_{y^{y,l}_{\varepsilon}}\;\text{and}\;\mu_{\varepsilon}^Z:=\varepsilon^3\sum_{m=1}^{Z_{\varepsilon}}\delta_{y^{z,m}_{\varepsilon}}
\end{equation}
converge weakly* (as measures in $\mathbb{R}^3$) to the Lebesgue measure restricted on $\Omega$, denoted $\emph{dx}\mres\Omega$.}
\end{itemize}
We say that a function $f:\mathcal{S}_0\otimes\mathbb{R}^{3}\rightarrow\mathbb{R}$ is strongly convex if there exists $\theta>0$ such that the function $\tilde{f}:\mathbb{S}_0\otimes\mathbb{R}^{3}\rightarrow\mathbb{R}$ defined by $\tilde{f}(D)=f(D)-\theta|D|^2$ is convex.
\begin{itemize}
\item[($A_5$)] \textit{$f_e:\mathcal{S}_0\otimes\mathbb{R}^3\rightarrow[0,+\infty)$ is differentiable, strongly convex and there exists a constant $\lambda_e>0$ such that} $$\lambda_e^{-1}|D|^2\leq f_e(D)\leq \lambda_e|D|^2,\;\;\;\;|(\nabla f_e)(D)|\leq\lambda_e(|D|+1),$$ \textit{for any $D\in\mathcal{S}_0\times\mathbb{R}^3$. }
\item[($A_6$)] \textit{$f_b:\mathcal{S}_0\rightarrow\mathbb{R}$ is continuous, bounded from below and there exists a constant $\lambda_b>0$ such that \linebreak $|f_b(Q)|\leq\lambda_b(|Q|^6+1)$ for any $Q\in\mathcal{S}_0$.}
\item[($A_7$)] \textit{$f_s:\mathcal{S}_0\times\mathbb{S}^2\rightarrow\mathbb{R}$ is continuous and there exists a strictly positive constant $\lambda_s$ such that, for any $Q_1,Q_2\in\mathcal{S}_0$ and any $\nu\in\mathbb{S}^2$, we have $$|f_s(Q_1,\nu)-f_s(Q_2,\nu)|\leq\lambda_s|Q_1-Q_2|\big(|Q_1|^3+|Q_2|^3+1\big).$$}
\end{itemize}
It is easy to see from here that $f_s$ has a quartic growth in $Q$.
\vspace{3mm}
\textbf{The homogenised functional.} Let $f_{hom}:\mathcal{S}_0\rightarrow\mathbb{R}$ be the function defined as:
\begin{equation}\label{defn:f_hom}
f_{hom}(Q):=\dfrac{q+r}{qr}\int_{\mathcal{C}^x}f_s(Q,\nu)\text{d}\sigma+\dfrac{p+r}{pr}\int_{\mathcal{C}^y}f_s(Q,\nu)\text{d}\sigma+\dfrac{p+q}{pq}\int_{\mathcal{C}^z}f_s(Q,\nu)\text{d}\sigma,
\end{equation}
\noindent for any $Q\in\mathcal{S}_0$, where $\mathcal{C}^x$, $\mathcal{C}^y$ and $\mathcal{C}^z$ are defined in \eqref{defn:C_x_C_y_C_z}. From ($A_7$), we can deduce that $f_{hom}$ is also continuous and that it has a quartic growth. If we work in the symmetric case, that is $p=q=r$, then relation \eqref{defn:f_hom} becomes:
\begin{equation}\label{defn:f_hom_sym}
f_{hom}(Q):=\dfrac{2}{p}\int_{\partial\mathcal{C}}f_s(Q,\nu)\text{d}\sigma.
\end{equation}
The main results of these notes concerns the asymptotic behaviour of local minimisers of the functional $\mathcal{F}_{\varepsilon}$, as $\varepsilon\rightarrow 0$.
Let $g\in H^{1/2}(\partial\Omega,\mathcal{S}_0)$ be a boundary datum. We denote by $H^1_g(\Omega,\mathcal{S}_0)$ the set of maps $Q$ from $H^1(\Omega,\mathcal{S}_0)$ such that $Q=g$ on $\partial\Omega$ in the trace sense. Similarly, we define $H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$ to be $H^1(\Omega_{\varepsilon})$ with $Q=g$ on $\partial\Omega$ in the trace sense.
We use the harmonic extension operator, $E_{\varepsilon}:H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)\rightarrow H^1_g(\Omega,\mathcal{S}_0)$, defined in the following way: $E_{\varepsilon}Q:=Q$ on $\Omega_{\varepsilon}$ and inside the scaffold, $E_{\varepsilon}Q$ is the unique solution of the following problem:
\begin{equation*}
\left\{
\begin{array}{ll}
\Delta E_{\varepsilon}Q=0 & \text{in}\;\mathcal{N}_{\varepsilon}\\
E_{\varepsilon}Q\equiv Q & \text{on}\; \partial\mathcal{N}_{\varepsilon}.
\end{array}
\right.
\end{equation*}
Using this framework, we can produce the main result of this work:
\begin{theorem}\label{th:local_min}
Suppose that the assumptions ($A_1$)-($A_7$) are satisfied. Let $Q_0\in H^1_g(\Omega,\mathcal{S}_0)$ be an isolated $H^1$-local minimiser for $\mathcal{F}_0$, that is, there exists $\delta_0>0$ such that $\mathcal{F}_0[Q_0]<\mathcal{F}_0[Q]$ for any $Q\in H^1_g(\Omega,\mathcal{S}_0)$ such that $\big\|Q-Q_0\big\|_{H^1_g(\Omega,\mathcal{S}_0)}\leq \delta_0$ and $Q\neq Q_0$. Then for any $\varepsilon$ sufficiently small enough, there exists a sequence of $H^1$-local minimisers $Q_{\varepsilon}$ of $\mathcal{F}_{\varepsilon}$ such that $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q_0$ strongly in $H^1_g(\Omega,\mathcal{S}_0)$.
\end{theorem}
\subsection{Applications to the Landau-de Gennes model}
We use the classical terms from the Landau-de Gennes model for nematic liquid crystals.
For the \textit{elastic energy density}, we take:
\begin{align*}
f_e(\nabla Q):=L_1\partial_k Q_{ij}\partial_k Q_{ij}+L_2\partial_j Q_{ij}\partial_k Q_{ik}+L_3 \partial_j Q_{ik}\partial_k Q_{ij},
\end{align*}
where the Einstein's summation convention is assumed.
In order to fulfill assumption ($A_5$), we take as in \cite{Longa}:
\begin{align}\label{eq:ineq_for_elastic}
L_1>0, \hspace{5mm}-L_1<L_3<2 L_1, \hspace{5mm}-\dfrac{3}{5}L_1-\dfrac{1}{10}L_3<L_2.
\end{align}
For the \textit{bulk energy density}, we use several versions of it. The first one is the classical quartic polynomial in the scalar invariants of $Q$, that is:
\begin{align}\label{defn:f_b_LDG}
f_b^{LDG}(Q):=a\,\text{tr}(Q^2)-b\,\text{tr}(Q^3)+c\,\big(\text{tr}(Q^2)\big)^2,
\end{align}
which verifies the conditions of \cref{th:local_min}.
We also prove similar results for a general polynomial in the scalar invariants of $Q$, that is:
\begin{align}\label{defn:f_b_gen}
f_b^{gen}[Q]=\sum_{k=2}^{N} a_k\,\text{tr}(Q^k),
\end{align}
where $N\in\mathbb{N}$, $N\geq 4$ is fixed, with the coefficients $a_k\in\mathbb{R}$ chosen such that the polynomial $h:\mathbb{R}\rightarrow\mathbb{R}$, defined by $h(x)=\sum_{k=2}^{N}a_k x^k$, for any $x\in\mathbb{R}$, admits at least one local minimum over $\mathbb{R}$.
In all the cases, the coefficient of $\text{tr}(Q^2)$ depends on the temperature at which the phase transition occurs. More specifically, $a$ from \eqref{defn:f_b_LDG} is of the form $a:=a_{*}(T-T_{*})$, in which $a_{*}$ is a material parameter and $T_{*}$ is the characteristic temperature of the nematic liquid crystal material (the temperature where the isotropic state starts losing \textit{local} stability).
For each of the versions of the \textit{bulk energy densities}, we choose suitable \textit{surface energy densities}, such that, in the homogenised functional, the surface terms have a similar form with the effective bulk energy, which is now in the same form as in the non-homogenised situation, but with different coefficients, most important of which the coefficient of $\text{tr}(Q^2)$ is now different.
\cref{th:local_min} holds for any values of $p$, $q$ and $r$, that is, for any type of parallelipiped chosen for the construction of the scaffold. In reality, 2PP (two-photon polymerization) materials with cubic symmetry properties have been obtained (for example, \cite{nematiccage}) and they do represent an object of interest in the construction of nematic scaffolds.
If the scaffold presents symmetries, then the physical invariances require
\begin{equation*}
f_s(UQU^T,Uu)=f_s(Q,u),\;\forall (Q,u)\in\mathcal{S}_0\times\mathbb{R}^3,\;U\in\mathcal{O}(3)
\end{equation*}
\noindent and this leads, according to Proposition 2.6 from \cite{CanevariZarnescu1}, to a \textit{surface energy} of the form
\begin{equation*}
f_s(Q,\nu)=\tilde{f}_s(\text{tr}(Q^2),\text{tr}(Q^3),\nu\cdot Q\nu,\nu\cdot Q^2\nu),\;\forall (Q,\nu)\in\mathcal{S}_0\times\mathbb{R}^3.
\end{equation*}
Even though we have this result, we still use a more general form for $f_s(Q,\nu)$, in which we include terms of the form $\nu\cdot Q^k\nu$, since they grant an easier way to compute the homogenised functional, in this case with this type of scaffold. Still, according to \cref{prop:int_en_dens_LDG}, we obtain in the homogenised functional terms of the form $\text{tr}(Q^k)$, with $k\geq 4$, which depend only on $\text{tr}(Q^2)$ and $\text{tr}(Q^3)$, since $\text{tr}(Q)=0$. In order to prove this statement, let $\lambda_1$, $\lambda_2$ and $\lambda_3$ the eigenvalues of $Q$. Then they satisfy the system:
\begin{align*}
\begin{cases}
\lambda_1+\lambda_2+\lambda_3=0\\
\lambda_1^2+\lambda_2^2+\lambda_3^2=\text{tr}(Q^2)\\
\lambda_1^3+\lambda_2^3+\lambda_3^3=\text{tr}(Q^3)
\end{cases}
\end{align*}
and, by solving the system, we can see that $\lambda_1$, $\lambda_2$ and $\lambda_3$ can be viewed as functions of $\text{tr}(Q^2)$ and $\text{tr}(Q^3)$. Since $\text{tr}(Q^k)=\lambda_1^k+\lambda_2^k+\lambda_3^k$, for any $k\in\mathbb{N}$, $k\geq 1$, then it is easy to see from here that $\text{tr}(Q^k)$, for $k\geq 4$, is depending only on $\text{tr}(Q^2)$ and $\text{tr}(Q^3)$. Indeed, by Cayley-Hamilton theorem, the identity:
\begin{align*}
Q^3-\dfrac{1}{2}\text{tr}(Q^2)Q-\dfrac{1}{3}\text{tr}(Q^3)\mathbb{I}_3=0
\end{align*}
becomes valid for any Q-tensor $Q$, where $\mathbb{I}_3$ is the $3\times 3$ identity matrix. Multiplying this identity succcessively by $Q$, $Q^2$, $Q^3$ and so on and taking the trace we obtain the claim.
\subsubsection{The case $p=q=r$}
Assuming $p=q=r$ implies that the parallelipipeds constructed in \eqref{defn:ceps} are actually cubes and that the ``cells" of the nematic scaffold are also cubes.
For each of the \textit{bulk energies} presented before, we take different \textit{surface energy densities}. One of the corresponding choices of the \textit{surface energy} $f_s$ in the case of \eqref{defn:f_b_LDG} is:
\begin{align}\label{defn:f_s_LDG}
f_s^{LDG}(Q,\nu)=\dfrac{p}{4}\bigg((a'-a)(\nu\cdot Q^2\nu)-(b'-b)(\nu\cdot Q^3\nu)+2(c'-c)(\nu\cdot Q^4\nu)\bigg)
\end{align}
where $a'$, $b'$ and $c'$ are the desired coefficients in the homogenised bulk potential, such that in the homogenised material, we have:
\begin{align}\label{defn:f_hom_LDG}
f_{hom}^{LDG}(Q)=(a'-a)\,\text{tr}(Q^2)-(b'-b)\,\text{tr}(Q^3)+(c'-c)\,\big(\text{tr}(Q^2)\big)^2.
\end{align}
We are interested in studying the behaviour of the whole material when $\varepsilon\rightarrow 0$, that is, studying the following functionals:
\begin{align}\label{defn:F_eps_LDG}
\mathcal{F}_{\varepsilon}^{LDG}[Q_{\varepsilon}]&:=\int_{\Omega_{\varepsilon}}\big(f_e(\nabla Q_{\varepsilon})+a\,\text{tr}(Q_{\varepsilon}^2)-b\,\text{tr}(Q_{\varepsilon}^3)+c\,\big(\text{tr}(Q_{\varepsilon}^2)\big)^2\big)\text{d}x+\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}}f_s^{LDG}(Q_{\varepsilon},\nu)\text{d}\sigma
\end{align}
and
\begin{align}\label{defn:F_0_LDG}
\mathcal{F}_0^{LDG}[Q]&:=\int_{\Omega}\big(f_e(\nabla Q)+a'\,\text{tr}(Q^2)-b'\,\text{tr}(Q^3)+c'\,\big(\text{tr}(Q^2)\big)^2\big)\text{d}x,
\end{align}
with $a'$, $b'$ and $c'$ being the desired parameters.
\begin{theorem}\label{th:LDG}
Let $(a,b,c)$ and $(a',b',c')$ be two set of parameters with $c>0$ and $c'>0$. Suppose that the assumptions ($A_1$)-($A_7$) are satisfied and also the inequalities from \eqref{eq:ineq_for_elastic}. Then, for any isolated $H^1$-local minimiser $Q_0$ of the functional $\mathcal{F}_0^{LDG}$ defined by \eqref{defn:F_0_LDG}, and for $\varepsilon>0$ sufficiently small enough, there exists a sequence of local minimisers $Q_{\varepsilon}$ of the functionals $\mathcal{F}_{\varepsilon}^{LDG}$, defined by \eqref{defn:F_eps_LDG}, such that $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q_0$ strongly in $H^1_g(\Omega,\mathcal{S}_0)$.
\end{theorem}
\begin{proof}
This theorem is a particular case of \cref{th:local_min}. It is sufficient to prove that relation \eqref{defn:f_hom_LDG} can be obtained via \eqref{defn:f_hom_sym}, that is:
\begin{align*}
f_{hom}^{LDG}(Q)=\dfrac{2}{p}\int_{\partial\mathcal{C}}f_s^{LDG}(Q,\nu)\text{d}\sigma=(a'-a)\;\text{tr}(Q^2)-(b'-b)\;\text{tr}(Q^3)+(c'-c)\;\big(\text{tr}(Q^2)\big)^2.
\end{align*}
Using \cref{prop:int_en_dens_LDG}, we have:
\begin{align*}
\int_{\partial\mathcal{C}}\nu\cdot Q^2\nu\text{d}\sigma =2\text{tr}(Q^2),\hspace{3mm}
\int_{\partial\mathcal{C}}\nu\cdot Q^3\nu\text{d}\sigma =2\text{tr}(Q^3)\hspace{3mm}\text{and}\hspace{3mm}\int_{\partial\mathcal{C}}\nu\cdot Q^4\nu\text{d}\sigma =2\text{tr}(Q^4),
\end{align*}
from which we get
\begin{align*}
\dfrac{2}{p}\int_{\partial\mathcal{C}}f_s^{LDG}(Q,\nu)\text{d}\sigma &=\dfrac{2}{p}\cdot\dfrac{p}{4}\big((a'-a)\cdot 2\text{tr}(Q^2)-(b'-b)\cdot 2\text{tr}(Q^3)+2(c'-c)\cdot 2\text{tr}(Q^4)\big)\Rightarrow\\
\Rightarrow f_{hom}^{LDG}(Q)&=(a'-a)\text{tr}(Q^2)-(b'-b)\text{tr}(Q^3)+(c'-c)\cdot 2\text{tr}(Q^4).
\end{align*}
Since $Q\in\mathcal{S}_0$, then, by Cayley-Hamilton theorem, if $\lambda_1$, $\lambda_2$ and $\lambda_3$ are the eigenvalues of $Q$, we have:
\begin{align*}
\begin{cases}
\lambda_1+\lambda_2+\lambda_3=\text{tr}(Q)=0\\
\lambda_1\lambda_2+\lambda_2\lambda_3+\lambda_3\lambda_1=\dfrac{1}{2}\big(\big(\text{tr}(Q)\big)^2-\text{tr}(Q^2)\big)=-\dfrac{1}{2}\text{tr}(Q^2)
\end{cases}
\end{align*}
and
\begin{align*}
\text{tr}(Q^4)=\lambda_1^4+\lambda_2^4+\lambda_3^4&=(\lambda_1^2+\lambda_2^2+\lambda_3^2)^2-2(\lambda_1^2\lambda_2^2+\lambda_2^2\lambda_3^2+\lambda_3^2\lambda_1^2)\\
&=(\lambda_1^2+\lambda_2^2+\lambda_3^2)^2-2\big((\lambda_1\lambda_2+\lambda_2\lambda_3+\lambda_3\lambda_1)^2-2\lambda_1\lambda_2\lambda_3(\lambda_1+\lambda_2+\lambda_3)\big)\\
&=\big(\text{tr}(Q^2)\big)^2-2\bigg(-\dfrac{1}{2}\text{tr}(Q^2)\bigg)^2\\
&=\dfrac{1}{2}\big(\text{tr}(Q^2)\big)^2
\end{align*}
from which we get the relation $2\text{tr}(Q^4)=\big(\text{tr}(Q^2)\big)^2$.
Hence, we conclude that:
\begin{align*}
f_{hom}^{LDG}(Q)&=(a'-a)\text{tr}(Q^2)-(b'-b)\text{tr}(Q^3)+(c'-c)\big(\text{tr}(Q^2)\big)^2.
\end{align*}
\end{proof}
If we assume that in the case of \eqref{defn:f_b_LDG} we have $b=c=0$ ($f_b^{RP}=a\;\text{tr}(Q^2)$) and we desire $b'=c'=0$, that is, the only nonzero coefficients are $a$ and $a'$, then another suitable choice for $f_s$ is given by the Rapini-Papoular form \eqref{eq:intro_Rapini-Papoular}:
\begin{align}\label{defn:f_s_RP}
f_s^{RP}(Q,\nu)=\dfrac{p}{12}(a'-a)\;\text{tr}(Q-Q_{\nu})^2,
\end{align}
where $Q_{\nu}=\nu\otimes\nu-\mathbb{I}_3/3$ and $\mathbb{I}_3$ is the $3\times 3$ identity matrix.
In this case, we have:
\begin{align}\label{defn:F_eps_RP}
\mathcal{F}_{\varepsilon}^{RP}[Q_{\varepsilon}]&:=\int_{\Omega_{\varepsilon}}\big(f_e(\nabla Q_{\varepsilon})+a\,\text{tr}(Q_{\varepsilon}^2)\big)\text{d}x+\dfrac{p}{2}\cdot(a'-a)\cdot\bigg(\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}}\;\text{tr}(Q_{\varepsilon}-Q_{\nu})^2\text{d}\sigma\bigg)
\end{align}
and we prove that
\begin{align}\label{defn:f_hom_RP}
f_{hom}^{RP}(Q)=(a'-a)\;\text{tr}(Q^2),
\end{align}
and
\begin{align}\label{defn:F_0_RP}
\mathcal{F}_0^{RP}[Q]&:=\int_{\Omega}\big(f_e(\nabla Q)+a'\,\text{tr}(Q^2)\big)\text{d}x.
\end{align}
\begin{theorem}\label{th:RP}
Let $a$ and $a'$ be two parameters. Suppose that the assumptions ($A_1$)-($A_7$) are satisfied and also the inequalities from \eqref{eq:ineq_for_elastic}. Then, for any isolated $H^1$-local minimiser $Q_0$ of the functional $\mathcal{F}_0^{RP}$ defined by \eqref{defn:F_0_RP}, and for $\varepsilon>0$ sufficiently small enough, there exists a sequence of local minimisers $Q_{\varepsilon}$ of the functionals $\mathcal{F}_{\varepsilon}^{RP}$, defined by \eqref{defn:F_eps_RP}, such that $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q_0$ strongly in $H^1_g(\Omega,\mathcal{S}_0)$.
\end{theorem}
\begin{proof}
The proof follows the same steps as in the proof of \cref{th:LDG}, using \cref{prop:int_en_dens_RP}. We only have to prove that relation \eqref{defn:f_hom_RP} can be obtained using \eqref{defn:f_hom_sym}, knowing that \eqref{defn:f_s_RP} holds.
From \eqref{defn:f_hom_sym} and \cref{prop:int_en_dens_RP}, we have:
\begin{align*}
f_{hom}^{RP}(Q)&=\dfrac{2}{p}\int_{\partial\mathcal{C}}f_s^{RP}(Q,\nu)\text{d}\sigma=\dfrac{2}{p}\cdot\dfrac{p}{12}(a'-a)\int_{\partial\mathcal{C}}\text{tr}(Q-Q_{\nu})\text{d}\sigma \\
&=\dfrac{(a'-a)}{6}\big(6\text{tr}(Q^2)+4\big)=(a'-a)\text{tr}(Q^2)+\dfrac{2}{3}(a'-a).
\end{align*}
We can eliminate the constant $\dfrac{2}{3}(a'-a)$ from $f_{hom}^{RP}$, since it does not influence the minimisers of the functional $\mathcal{F}_{\varepsilon}^{RP}$, so we obtain: $f_{hom}^{RP}(Q)=(a'-a)\text{tr}(Q^2)$.
\end{proof}
For the more general case described by \eqref{defn:f_b_gen}, we can choose:
\begin{align}\label{defn:f_s_gen}
f_s^{gen}(Q,\nu)=\dfrac{p}{4}\sum_{k=2}^{M}b_k(\nu\cdot Q^k\nu),
\end{align}
where $(b_k)_{k\in\overline{2,M}}$ are the coefficients of the polynomial $i:\mathbb{R}\rightarrow\mathbb{R}$ of degree $M\in\mathbb{N}$, $M\geq 4$, defined by $i(x)=\sum_{k=2}^{M}b_k x^k$, for any $x\in\mathbb{R}$, with the property that $i$ admits at least one local minimum over $\mathbb{R}$.
In the same manner, we have
\begin{align*}
f_{hom}^{gen}(Q)=\sum_{k=2}^{\text{max}\{M,N\}}c_k\;\text{tr}(Q^k),
\end{align*}
where, for any $k\in\overline{2,\text{max}\{M,N\}}$, we have
\begin{align*}
c_k=\begin{cases}
a_k+b_k,\;\text{if}\;2\leq k\leq\text{min}\{M,N\}\\
a_k,\;\text{if}\;\text{min}\{M,N\}<k\leq\text{max}\{M,N\}\;\text{and}\;M\leq N\\
b_k,\;\text{if}\;\text{min}\{M,N\}<k\leq\text{max}\{M,N\}\;\text{and}\;M\geq N.
\end{cases}
\end{align*}
In this case, $\mathcal{F}_{\varepsilon}$ and $\mathcal{F}_0$ become:
\begin{align}\label{defn:F_eps_gen}
\mathcal{F}_{\varepsilon}^{gen}[Q_{\varepsilon}]:=\int_{\Omega_{\varepsilon}}\bigg(f_e(\nabla Q_{\varepsilon})+\sum_{k=2}^{N}a_k\;\text{tr}(Q_{\varepsilon}^k)\bigg)\text{d}x+\dfrac{p}{4}\cdot\sum_{k=2}^{M}b_k\cdot\bigg(\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}}(\nu\cdot Q_{\varepsilon}^k\nu)\text{d}\sigma\bigg)
\end{align}
and
\begin{align}\label{defn:F_0_gen}
\mathcal{F}_{0}^{gen}[Q]=\int_{\Omega}\bigg(f_e(\nabla Q)+\sum_{k=2}^{\text{max}\{M,N\}}c_k\;\text{tr}(Q^k)\bigg)\text{d}x.
\end{align}
\begin{theorem}\label{th:gen}
Let $(a_k)_{k\in\overline{2,N}}$ and $(b_k)_{k\in\overline{2,M}}$ be such that the polynomials $h$ and $i$ defined earlier admit at least one local minimum over $\mathbb{R}$. Suppose that the assumptions ($A_1$)-($A_7$) are satisfied and also the inequalities from \eqref{eq:ineq_for_elastic}. Then, for any isolated $H^1$-local minimiser $Q_0$ of the functional $\mathcal{F}_0^{gen}$ defined by \eqref{defn:F_0_gen}, and for $\varepsilon>0$ sufficiently small enough, there exists a sequence of local minimisers $Q_{\varepsilon}$ of the functionals $\mathcal{F}_{\varepsilon}^{gen}$, defined by \eqref{defn:F_eps_gen}, such that $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q_0$ strongly in $H^1_g(\Omega,\mathcal{S}_0)$.
\end{theorem}
\begin{proof}
This theorem is a particular case of \cref{th:local_min}. Using once again \cref{prop:int_en_dens_LDG}, the proof is finished.
\end{proof}
\subsubsection{The case $p\neq q\neq r\neq p$}
Assume now that $p$, $q$ and $r$ are three different real values, each greater than or equal to 1. For this case, we only present one of the theorems stated in the last subsection.
Relation \eqref{defn:f_hom} states that:
\begin{equation*}
f_{hom}(Q)=\dfrac{q+r}{qr}\int_{\mathcal{C}^x}f_s(Q,\nu)\text{d}\sigma+\dfrac{p+r}{pr}\int_{\mathcal{C}^y}f_s(Q,\nu)\text{d}\sigma+\dfrac{p+q}{pq}\int_{\mathcal{C}^z}f_s(Q,\nu)\text{d}\sigma.
\end{equation*}
We only illustrate how to proceed for the case in which we have
\begin{align*}
f_b(Q)=a\;\text{tr}(Q^2)-b\;\text{tr}(Q^3)+c\;\text{tr}(Q^4)=a\;\text{tr}(Q^2)-b\;\text{tr}(Q^3)+\dfrac{c}{2}\;\big(\text{tr}(Q^2)\big)^2,
\end{align*}
with $c>0$, and similar results can be obtained for the other cases in which we modify the form of $f_b$.
Let
\begin{align*}
A=\dfrac{1}{3}\begin{pmatrix}
-\dfrac{2}{p}+\dfrac{1}{q}+\dfrac{1}{r} & 0 & 0\\
0 & \dfrac{1}{p}-\dfrac{2}{q}+\dfrac{1}{r} & 0\\
0 & 0 & \dfrac{1}{p}+\dfrac{1}{q}-\dfrac{2}{r}
\end{pmatrix}\hspace{3mm}\text{and}\hspace{3mm}B=\begin{pmatrix}
\dfrac{1}{q}+\dfrac{1}{r} & 0 & 0\\
0 & \dfrac{1}{p}+\dfrac{1}{r} & 0\\
0 & 0 & \dfrac{1}{p}+\dfrac{1}{q}
\end{pmatrix}.
\end{align*}
and $\omega=\dfrac{2}{3}\bigg(\dfrac{1}{p}+\dfrac{1}{q}+\dfrac{1}{r}\bigg)$. Note that $A$, $B$ and $\omega$ are constants depending only on the choice of $p$, $q$ and $r$. Moreover, we have $\text{tr}(A)=0$ and $B=\omega\mathbb{I}_3+A$, where $\mathbb{I}_3$ is the $3\times 3$ identity matrix.
Consider now
\begin{align}\label{defn:f_s_asym}
f_s^{asym}(Q,\nu)=\dfrac{1}{2\omega}\big((a'-a)(\nu\cdot Q^2\nu)-(b'-b)(\nu\cdot Q^3\nu)+(c'-c)(\nu\cdot Q^4\nu)\big),
\end{align}
with $a'$, $b'$ and $c'$ real parameters such that $c'>0$ and the associated free energy functional:
\begin{align}\label{defn:F_eps_asym}
\mathcal{F}_{\varepsilon}^{asym}[Q_{\varepsilon}]:=\int_{\Omega}\big(f_e(\nabla Q_{\varepsilon})+a\;\text{tr}(Q_{\varepsilon}^2)-b\;\text{tr}(Q_{\varepsilon}^3)+c\;\text{tr}(Q_{\varepsilon}^4)\big)\text{d}x+\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}}f_s^{asym}(Q_{\varepsilon},\nu).
\end{align}
We prove in the next theorem that the homogenised functional is:
\begin{align}\label{defn:f_hom_asym}
f_{hom}^{asym}(Q)&=\big((a'-a)\text{tr}(Q^2)-(b'-b)\text{tr}(Q^3)+(c'-c)\text{tr}(Q^4)\big)+\notag\\
&\hspace{2mm}+\dfrac{1}{\omega}\big((a'-a)\text{tr}(A\cdot Q^2)-(b'-b)\text{tr}(A\cdot Q^3)+(c'-c)\text{tr}(A\cdot Q^4)\big).
\end{align}
\begin{theorem}\label{th:asym}
Let $(a,b,c)$ and $(a',b',c')$ be two set of parameters with $c>0$ and $c'>0$. Suppose that the assumptions ($A_1$)-($A_7$) are satisfied and also the inequalities from \eqref{eq:ineq_for_elastic}. Then, for $\varepsilon>0$ sufficiently small enough and for any isolated $H^1$-local minimiser $Q_0$ of the functional:
\begin{align*}
\mathcal{F}_0^{asym}[Q]&:=\int_{\Omega}\big(f_e(\nabla Q)+a'\emph{tr}(Q^2)-b'\emph{tr}(Q^3)+c'\big(\emph{tr}(Q^2)\big)^2\big)\emph{d}x+\notag\\
&\hspace{2mm}+\dfrac{1}{\omega}\int_{\Omega}\big((a'-a)\emph{tr}(A\cdot Q^2(x))-(b'-b)\emph{tr}(A\cdot Q^3(x))+(c'-c)\emph{tr}(A\cdot Q^4(x))\big)\emph{d}x
\end{align*}
there exists a sequence of local minimisers $Q_{\varepsilon}$ of the functionals $\mathcal{F}_{\varepsilon}^{asym}$, defined by \eqref{defn:F_eps_asym}, such that $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q_0$ strongly in $H^1_g(\Omega,\mathcal{S}_0)$.
\end{theorem}
\begin{proof}
We follow the same steps as in \cref{th:LDG} and in \cref{th:RP}, that is, we prove that relation \eqref{defn:f_hom_asym} can be obtained using \eqref{defn:f_hom} and \eqref{defn:f_s_asym}.
In the proof of \cref{prop:int_en_dens_LDG}, we obtain that:
\begin{align*}
\int_{\mathcal{C}^x}\nu\cdot Q^k\nu\text{d}\sigma =2q_{11,k},\hspace{3mm}
\int_{\mathcal{C}^y}\nu\cdot Q^k\nu\text{d}\sigma =2q_{22,k}\hspace{3mm}\text{and}\hspace{3mm}\int_{\mathcal{C}^z}\nu\cdot Q^k\nu\text{d}\sigma =2q_{33,k},
\end{align*}
for any $k\in\mathbb{N}$, $k\neq 0$, where $q_{ij,k}$ is the $ij$-th component of $Q^k$, from which we get:
\begin{align*}
\int_{\mathcal{C}^x}f_s^{asym}(Q,\nu)\text{d}\sigma=\dfrac{1}{\omega}((a'-a)q_{11,2}-(b'-b)q_{11,3}+(c'-c)q_{11,4})\\
\int_{\mathcal{C}^y}f_s^{asym}(Q,\nu)\text{d}\sigma=\dfrac{1}{\omega}((a'-a)q_{22,2}-(b'-b)q_{22,3}+(c'-c)q_{22,4})\\
\int_{\mathcal{C}^z}f_s^{asym}(Q,\nu)\text{d}\sigma=\dfrac{1}{\omega}((a'-a)q_{33,2}-(b'-b)q_{33,3}+(c'-c)q_{33,4}).
\end{align*}
Using now \eqref{defn:f_hom}, we obtain:
\begin{align*}
f_{hom}^{asym}(Q)&=\dfrac{1}{\omega}(a'-a)\bigg(q_{11,2}\bigg(\dfrac{1}{q}+\dfrac{1}{r}\bigg)+q_{22,2}\bigg(\dfrac{1}{p}+\dfrac{1}{r}\bigg)+q_{33,2}\bigg(\dfrac{1}{p}+\dfrac{1}{q}\bigg)\bigg)-\\
&-\dfrac{1}{\omega}(b'-b)\bigg(q_{11,3}\bigg(\dfrac{1}{q}+\dfrac{1}{r}\bigg)+q_{22,3}\bigg(\dfrac{1}{p}+\dfrac{1}{r}\bigg)+q_{33,3}\bigg(\dfrac{1}{p}+\dfrac{1}{q}\bigg)\bigg)+\\
&+\dfrac{1}{\omega}(c'-c)\bigg(q_{11,4}\bigg(\dfrac{1}{q}+\dfrac{1}{r}\bigg)+q_{22,4}\bigg(\dfrac{1}{p}+\dfrac{1}{r}\bigg)+q_{33,4}\bigg(\dfrac{1}{p}+\dfrac{1}{q}\bigg)\bigg)\\
\end{align*}
which we can see as:
\begin{align*}
f_{hom}^{asym}(Q)&=\dfrac{1}{\omega}\big((a'-a)\text{tr}(B\cdot Q^2)-(b'-b)\text{tr}(B\cdot Q^3)+(c'-c)\text{tr}(B\cdot Q^4)\big)\\
\end{align*}
and since $B=\omega\mathbb{I}_3+A$, we obtain:
\begin{align*}
f_{hom}^{asym}(Q)&=\big((a'-a)\text{tr}(Q^2)-(b'-b)\text{tr}(Q^3)+(c'-c)\text{tr}(Q^4)\big)+\\
&+\dfrac{1}{\omega}\big((a'-a)\text{tr}(A\cdot Q^2)-(b'-b)\text{tr}(A\cdot Q^3)+(c'-c)\text{tr}(A\cdot Q^4)\big),
\end{align*}
from which we conclude.
\end{proof}
\begin{remark}
We have obtained in this case a part which is exactly the same as in the case in which we have cubic symmetry, but also three terms of the form $\emph{tr}(A\cdot Q^k)$ which describe a new preferred alignment of the liquid crystal particles inside of the domain, given by the loss of the cubic symmetry of the scaffold.
\end{remark}
\section{Properties of the functional \texorpdfstring{$\mathcal{F}_{\varepsilon}$}{Fe}}\label{section:prop_F_eps}
\subsection{Analytical tools: trace and extension}\label{subsection:extension}
The main result of this subsection consists on a $L^p$ inequality, which is adapted from lemma 3.1. from \cite{CanevariZarnescu1}, because our scaffold now consists on inter-connected particles and the interaction between the liquid crystal and the cubic microlattice happens only up to five faces of the particles of the scaffold.
In the following, given a set $\mathcal{P}\subset\mathbb{R}^{2}$ and a real number $a>0$, we define $a\mathcal{P}=\{ax:x\in\mathcal{P}\}$.
\begin{lemma}\label{lemma:bP-aP}
Let $\mathcal{P}\subseteq\mathbb{R}^2$ be a compact, convex set whose interior contains the origin. Let $a$ and $b$ be positive numbers such that $a<b$. Then there exists a bijective, Lipschitz map $\phi:b\mathcal{P}\setminus a\mathcal{P}\rightarrow \overline{B}_b\setminus\overline{B}_a$ that has a Lipschitz inverse and satisfies $$\|\nabla\phi\|_{L^{\infty}(b\mathcal{P}\setminus a\mathcal{P})}+\|\nabla(\phi^{-1})\|_{L^{\infty}(\overline{B}_b\setminus\overline{B}_a)}\leq C(\mathcal{P}),$$ where $C(\mathcal{P})$ is a positive constant that depends only on $\mathcal{P}$ and neither on $a$ nor $b$.
\end{lemma}
The proof of \cref{lemma:bP-aP} follows the same steps as Lemma 3.2. from \cite{CanevariZarnescu1}, the only difference being that now we are in the case of $\mathbb{R}^2$ instead of $\mathbb{R}^3$.
\begin{lemma}\label{lemma:bP-aP-ineq}
Let $\mathcal{P}\subseteq\mathbb{R}^2$ be a compact, convex set whose interior contains the origin and $n\in[2,4]$. Then, there exists $C=C(\mathcal{P},\phi)>0$, such that for any $0<a\leq b$ and any $u\in H^1(b\mathcal{P}\setminus a\mathcal{P})$, there holds $$\oint_{\partial\big(a\mathcal{P}\big)}|u|^n\text{d}s\lesssim \dfrac{2aC}{b^2-a^2}\int_{b\mathcal{P}\setminus a\mathcal{P}}|u|^n\text{d}x+\dfrac{nC}{2}\int_{b\mathcal{P}\setminus a\mathcal{P}}\big(|u|^{2n-2}+|\nabla u|^2\big)\text{d}x,$$
where $\displaystyle{\oint}$ represents the curvilinear integral in $\mathbb{R}^2$.
\end{lemma}
\begin{proof}
Using \cref{lemma:bP-aP}, we can restrict without loss of generality to the case in which $\mathcal{P}=\overline{B}_1$, which is the two dimensional unit disk, centered in origin. Then $\partial B_{\tau}=\{x\in\mathbb{R}^2:|x|=\tau\}=\{(\rho,\theta):\rho=\tau,\;\theta\in[0,2\pi]\}$, for any $\tau>0$, and we can write, for any $\rho\in[a,b]$ and any $\theta\in[0,2\pi]$:
\begin{align*}
|u|^n(a,\theta)&=|u|^n(\rho,\theta)-\int_{a}^{\rho}\partial_{\tau}\big(|u|^n\big)(\tau,\theta)\text{d}\tau\\
&\leq |u|^n(\rho,\theta)+n\int_a^{\rho}\big(|u|^{n-1}\cdot |\partial_{\tau}u|\big)(\tau,\theta)\text{d}\tau\\
&\leq |u|^n(\rho,\theta)+\dfrac{n}{2}\int_a^{\rho}\big(|u|^{2n-2}+|\partial_{\tau}u|^2\big)(\tau,\theta)\text{d}\tau\\
|u|^n(a,\theta)&\leq |u|^n(\rho,\theta)+\dfrac{n}{2}\int_a^b\big(|u|^{2n-2}+|\nabla u|^2\big)(\tau,\theta)\text{d}\tau
\end{align*}
If we multiply both sides by $\rho$ and integrate over $[a,b]$ with respect to $\rho$, we get:
\begin{align*}
|u|^n(a,\theta)\int_a^b\rho\;\text{d}\rho&\leq \int_a^b|u|^n(\rho,\theta)\cdot\rho\;\text{d}\rho+\dfrac{n}{2}\int_a^b\rho\;\text{d}\rho\int_a^b\big(|u|^{2n-2}+|\nabla u|^2\big)(\tau,\theta)\text{d}\tau\\
\dfrac{b^2-a^2}{2}|u|^n(a,\theta)&\leq \int_a^b|u|^n(\rho,\theta)\cdot\rho\;\text{d}\rho+\dfrac{n(b^2-a^2)}{4}\int_a^b\big(|u|^{2n-2}+|\nabla u|^2\big)(\tau,\theta)\text{d}\tau.
\end{align*}
Since for any $\tau\in[a,b]$ we have $\tau>a$, then:
\begin{align*}
\dfrac{b^2-a^2}{2a}|u|^n(a,\theta)\cdot a&\leq \int_a^b|u|^n(\rho,\theta)\cdot\rho\;\text{d}\rho+\dfrac{n(b^2-a^2)}{4a}\int_a^b\big(|u|^{2n-2}+|\nabla u|^2\big)(\tau,\theta)\cdot \tau\;\text{d}\tau.
\end{align*}
Now we integrate with respect to $\theta$ over $[0,2\pi]$ and we get:
\begin{align*}
\dfrac{b^2-a^2}{2a}\int_0^{2\pi}|u|^n(a,\theta)\cdot a\;\text{d}\theta&\leq \int_0^{2\pi}\int_a^b|u|^n(\rho,\theta)\cdot\rho\;\text{d}\rho\text{d}\theta+\dfrac{n(b^2-a^2)}{4a}\int_0^{2\pi}\int_a^b\big(|u|^{2n-2}+|\nabla u|^2\big)(\tau,\theta)\cdot \tau\;\text{d}\tau\text{d}\theta\\
\dfrac{b^2-a^2}{2a}\oint_{\partial B_a}|u|^n\text{d}s &\leq \int_{B_b\setminus B_a}|u|^n\text{d}x+\dfrac{n(b^2-a^2)}{4a}\int_{B_b\setminus B_a}\big(|u|^{2n-2}+|\nabla u|^2\big)\text{d}x,
\end{align*}
therefore
\begin{align*}
\oint_{\partial B_a}|u|^n\text{d}s\leq \dfrac{2a}{b^2-a^2}\int_{B_b\setminus B_a}|u|^n\text{d}x+\dfrac{n}{2}\int_{B_b\setminus B_a}\big(|u|^{2n-2}+|\nabla u|^2\big)\text{d}x.
\end{align*}
If we apply now the Lipschitz homeomorphism $\phi$ defined by \cref{lemma:bP-aP}, the conclusion follows.
\end{proof}
\begin{lemma}\label{lemma:surface}
For any $Q\in H^1(\Omega_{\varepsilon},\mathcal{S}_0)$ and any $n\in[2,4]$, there holds:
\begin{equation*}
\dfrac{\varepsilon^3}{\varepsilon^{\alpha}(\varepsilon-\varepsilon^{\alpha})}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q|^n\emph{d}\sigma \lesssim \dfrac{n}{2}\cdot \dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\int_{\Omega_{\varepsilon}}\big(|Q|^{2n-2}+|\nabla Q|^2\big)\emph{d}x + \dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\int_{\Omega_{\varepsilon}}|Q|^n \emph{d}x.
\end{equation*}
\end{lemma}
\begin{proof}
Let $I_{\varepsilon}[Q]=\dfrac{\varepsilon^3}{\varepsilon^{\alpha}(\varepsilon-\varepsilon^{\alpha})}\displaystyle{\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q|^n\text{d}\sigma}$ and $$I_{\varepsilon}^{X}[Q]=\dfrac{\varepsilon^{3}}{\varepsilon^{\alpha}(\varepsilon-\varepsilon^{\alpha})}\sum_{k=1}^{X_{\varepsilon}}\int_{\mathcal{T}_x^k} |Q|^n \text{d}\sigma.$$
Let $\overline{e_1}=(1,0,0)^T$, $\overline{e_2}=(0,1,0)^T$, $\overline{e_3}=(0,0,1)^T$ and $k\in\overline{1,X_{\varepsilon}}$. Then, according to previous definitions, $y_{\varepsilon}^{x,k}$ is the center of the parallelipiped $\mathcal{P}_{\varepsilon}^{x,k}$ with the ``contact" faces $\mathcal{T}_x^k$. If this is an ``inner" parallelipiped of the scaffold, then Figure \eqref{fig:crosss} shows a cross section of a neighbourhood of $\mathcal{P}_{\varepsilon}^{x,k}$, surrounding $\mathcal{T}_x^k$, a section which is parallel to the $yOz$ plane and which is passing through $y_{\varepsilon}^{x,k}+\delta\overline{e_1}$, where $\delta\in I_p:=\bigg[-\dfrac{p\varepsilon-\varepsilon^{\alpha}}{2p},\dfrac{p\varepsilon-\varepsilon^{\alpha}}{2p}\bigg]$. If the parallelipiped is next to $\partial\Omega$, then the same argument works, since we have relations \eqref{defn:points1} and \eqref{defn:Y_eps}.
Let $\mathcal{T}_x^k(\delta)$ be
\begin{equation*}
\mathcal{T}_x^k(\delta)=\bigg\{y_{\varepsilon}^{x,k}+\delta\overline{e_1}+y\overline{e_2}+z\overline{e_3}\;\bigg|\; -\dfrac{\varepsilon^{\alpha}}{2q}\leq y\leq\dfrac{\varepsilon^{\alpha}}{2q};\;-\dfrac{\varepsilon^{\alpha}}{2r}\leq z\leq\dfrac{\varepsilon^{\alpha}}{2r}\bigg\},
\end{equation*}
which represents the centered white rectangle from Figure \eqref{fig:crosss}.
Let $\mathcal{V}_x^k(\delta)$ be
\begin{equation*}
\mathcal{V}_x^k(\delta)=\bigg\{y_{\varepsilon}^{x,k}+\delta\overline{e_1}+y\overline{e_2}+z\overline{e_3}\;\bigg|\; -\varepsilon+\dfrac{\varepsilon^{\alpha}}{2q}\leq y\leq\varepsilon-\dfrac{\varepsilon^{\alpha}}{2q};\;-\varepsilon+\dfrac{\varepsilon^{\alpha}}{2r}\leq z\leq\varepsilon-\dfrac{\varepsilon^{\alpha}}{2r}\bigg\}\setminus\mathcal{T}_x^k(\delta),
\end{equation*}
which represents the darker shaded area from Figure \eqref{fig:crosss}, containing only liquid crystal particles, that is $\mathcal{V}_x^k(\delta)\subset\Omega_{\varepsilon}$, for any $\delta\in I_p$.
In our case, $\mathcal{V}_x^k(\delta)$ plays the role of $b\mathcal{P}\setminus a\mathcal{P}$ from \cref{lemma:surface}.
If for every $\delta\in I_p$, we apply the translation $y_{\varepsilon}^k+\delta\overline{e_1}$ to the origin of the system, then for
\begin{align*}
\mathcal{P}=\{0\}\times\bigg[-\dfrac{1}{2q},\dfrac{1}{2q}\bigg]\times\bigg[-\dfrac{1}{2r},\dfrac{1}{2r}\bigg],
\end{align*}
we can choose $a=\varepsilon^{\alpha}$, therefore $\varepsilon^{\alpha}\mathcal{P}=\mathcal{T}_x^k(\delta)$. In order to choose $b$, we assume: $\dfrac{b}{2r}\leq \varepsilon-\dfrac{\varepsilon^{\alpha}}{2r}$ and $\dfrac{b}{2q}\leq \varepsilon-\dfrac{\varepsilon^{\alpha}}{2q}$, that is: $b\leq 2q\varepsilon-\varepsilon^{\alpha}$ and $b\leq 2r\varepsilon-\varepsilon^{\alpha}$. Since $p,q,r\geq 1$, we can choose $b=2\varepsilon-\varepsilon^{\alpha}$. In this way, we have $b\mathcal{P}\setminus a\mathcal{P}\subset\mathcal{V}_x^k(\delta)$ and we also have $b\geq a\Leftrightarrow 2\varepsilon-\varepsilon^{\alpha}\geq \varepsilon^{\alpha}\Leftrightarrow \alpha\geq 1$.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.35]{crosssectiondelta2.png}
\caption{Cross section of the scaffold, parallel to $yOz$ plane, passing through $y_{\varepsilon}^{x,k}+\delta\overline{e_1}$. The gray shaded areas represent the liquid crystal and the white rectangles represent the sections of the parts of the scaffold nearby.}
\label{fig:crosss}
\end{figure}
Therefore, we can apply \cref{lemma:bP-aP-ineq} for $Q$ with $a=\varepsilon^{\alpha}$, $b=2\varepsilon-\varepsilon^{\alpha}$ and $\mathcal{P}$ defined as before, hence:
\begin{align*}
\oint_{\partial\mathcal{T}_x^k(\delta)}|Q|^n\text{d}s &\lesssim \dfrac{2\varepsilon^{\alpha}}{(2\varepsilon-\varepsilon^{\alpha})^2-\varepsilon^{2\alpha}}\int_{b\mathcal{P}\setminus\mathcal{T}_x^k(\delta)}|Q|^n\text{d}x+\dfrac{n}{2}\int_{b\mathcal{P}\setminus\mathcal{T}_x^k(\delta)}\big(|Q|^{2n-2}+|\nabla Q|^2\big)\text{d}x
\end{align*}
and since $b\mathcal{P}\subset\mathcal{V}_x^k(\delta)$, we have:
\begin{align*}
\oint_{\partial\mathcal{T}_x^k(\delta)}|Q|^n\text{d}s &\lesssim \dfrac{\varepsilon^{\alpha}}{2\varepsilon(\varepsilon-\varepsilon^{\alpha})}\int_{\mathcal{V}_x^k(\delta)}|Q|^n\text{d}x+\dfrac{n}{2}\int_{\mathcal{V}_x^k(\delta)}\big(|Q|^{2n-2}+|\nabla Q|^2\big)\text{d}x,
\end{align*}
for every $\delta\in I_p$. Integrating now with respect to $\delta$ over $I_p$, we get:
\begin{align*}
\int_{I_p}\bigg(\oint_{\partial\mathcal{T}_x^k(\delta)}|Q|^n\text{d}s\bigg)\text{d}\delta &\lesssim \dfrac{\varepsilon^{\alpha}}{2\varepsilon(\varepsilon-\varepsilon^{\alpha})}\int_{I_p}\bigg(\int_{\mathcal{V}_x^k(\delta)}|Q|^n\text{d}x\bigg)\text{d}\delta+\dfrac{n}{2}\int_{I_p}\bigg(\int_{\mathcal{V}_x^k(\delta)}\big(|Q|^{2n-2}+|\nabla Q|^2\big)\text{d}x\bigg)\text{d}\delta\\
\int_{\mathcal{T}_x^k}|Q|^n\text{d}\sigma &\lesssim \dfrac{\varepsilon^{\alpha}}{2\varepsilon(\varepsilon-\varepsilon^{\alpha})}\int_{\mathcal{U}_x^k}|Q|^n\text{d}x+\dfrac{n}{2}\int_{\mathcal{U}_x^k}\big(|Q|^{2n-2}+|\nabla Q|^2\big)\text{d}x,
\end{align*}
where $\mathcal{U}_x^k:=\bigcup_{\delta\in I_p}\mathcal{V}_x^k(\delta)\subset\Omega_{\varepsilon}$ is now a three dimensional object. Hence:
\begin{align*}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\mathcal{T}_x^k}|Q|^n\text{d}\sigma &\lesssim \dfrac{n}{2}\cdot \dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\int_{\mathcal{U}_x^k}\big(|Q|^{2n-2}+|\nabla Q|^2\big)\text{d}x+\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\int_{\mathcal{U}_x^k}|Q|^n\text{d}x.
\end{align*}
Repeating the same argument for all the other parallelipipeds of type $\mathcal{T}$ and considering the fact that parts of $\mathcal{U}_x^k$ are added only up to four times (by constructing the same sets for the nearby particles from the scaffold), then the conclusion follows.
\end{proof}
Since we are interested in the homogenised material, it is useful to consider maps defined on the entire $\Omega$ and for this we use the harmonic extension operator $E_{\varepsilon}:H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)\rightarrow H^1_g(\Omega,\mathcal{S}_0)$, defined as follows: for $Q\in H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$, we take $E_{\varepsilon}Q\equiv Q$ in $\Omega_{\varepsilon}$ and inside $\mathcal{N}_{\varepsilon}$, $E_{\varepsilon}Q$ solves the following PDE:
\begin{equation}\label{eq:extensionpb}
\left\{
\begin{array}{ll}
\Delta E_{\varepsilon}Q=0 & \text{in}\;\mathcal{N}_{\varepsilon}\\
E_{\varepsilon}Q\equiv Q & \text{on}\; \partial\mathcal{N}_{\varepsilon}
\end{array}
\right.
\end{equation}
Since $\mathcal{N}_{\varepsilon}$ has a Lipschitz boundary, we can apply Theorem 4.19 from \cite{CioranescuDonato} and see that there exists a unique solution $E_{\varepsilon}Q\in H^{1}(\mathcal{N}_{\varepsilon})$ to the problem \eqref{eq:extensionpb}. Hence the operator $E_{\varepsilon}$ is well defined. Moreover, from \eqref{eq:extensionpb}, we can see that $E_{\varepsilon}Q$ verifies:
\begin{equation}\label{eq:extensionmin}
\|\nabla E_{\varepsilon}Q\|_{L^2(\mathcal{N}_{\varepsilon})}=\text{min}\big\{\|\nabla u\|_{L^2(\mathcal{N}_{\varepsilon})}\;\big|\;u\in H^1(\mathcal{N}_{\varepsilon}),\;u=Q\;\text{on}\;\partial\mathcal{N}_{\varepsilon}\big\}.
\end{equation}
Our aim is now to prove that the extension operator $E_{\varepsilon}$ is uniformly bounded with respect to $\varepsilon>0$. More specifically, we prove that the following lemma holds.
\begin{lemma}\label{lemma:extensionineq}
There exists a constant $C>0$ such that $\|\nabla E_{\varepsilon}Q\|_{L^2(\Omega)}\leq C\|\nabla Q\|_{L^2(\Omega_{\varepsilon})}$ for any $\varepsilon\in(0,\varepsilon_0)$, where $\varepsilon_0$ is suitably small enough, and for any $Q\in H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$.
\end{lemma}
\begin{proof}
By \cref{subsection:existence_of_extension}, we know that there exists $v\in H^1(\Omega)$ such that:
\begin{align*}
\begin{cases}
v\equiv Q\;\text{in}\;\Omega_{\varepsilon}\\
v=Q\;\text{on}\;\partial\mathcal{N}_{\varepsilon}\\
\big\|\nabla v\big\|_{L^2(\Omega)}\lesssim \big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}.
\end{cases}
\end{align*}
Using relation \eqref{eq:extensionmin}, we see that
\begin{align*}
\big\|\nabla E_{\varepsilon}Q\big\|_{L^2(\mathcal{N}_{\varepsilon})}\leq \big\|\nabla v\big\|_{L^2(\mathcal{N}_{\varepsilon})}
\end{align*}
and because $E_{\varepsilon}Q\equiv Q$ in $\Omega_{\varepsilon}$, we have $E_{\varepsilon}Q\equiv v\equiv Q$ in $\Omega_{\varepsilon}$ and therefore:
\begin{align*}
\big\|\nabla E_{\varepsilon}Q\big\|_{L^2(\Omega)}\leq \big\|\nabla v\big\|_{L^2(\Omega)}\lesssim \big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}.
\end{align*}
\end{proof}
\subsection{Zero contribution from the surface terms depending on $\mathcal{S}^i$}\label{subsection:nocontribution}
In this section, we prove that the surface term $J_{\varepsilon}^{\mathcal{S}}$ has a neglectable contribution to the homogenised material, that is $J_{\varepsilon}^{\mathcal{S}}[Q]\rightarrow 0$ as $\varepsilon\rightarrow 0$, for any $Q\in H^{1}_{g}(\Omega,\mathcal{S}_0)$, since we can use the extension operator $E_{\varepsilon}$ defined in the previous subsection.
We start by proving if $Q:\overline{\Omega}\rightarrow\mathcal{S}_0$ is a bounded, Lipschitz map, then $J_{\varepsilon}^{\mathcal{S}}[Q]\rightarrow 0$ as $\varepsilon\rightarrow 0$ and then, by a density argument, for all $Q\in H^1_g(\Omega,\mathcal{S}_0)$.
\begin{lemma}\label{lemma:zerocontrbounded}
Let $Q:\overline{\Omega}\rightarrow\mathcal{S}_0$ be a bounded, Lipschitz map. Then $J_{\varepsilon}^{\mathcal{S}}[Q]\rightarrow 0$, as $\varepsilon\rightarrow 0$, where $J_{\varepsilon}^{\mathcal{S}}$ is defined in \eqref{defn:Jeps} and in \eqref{defn:Jeps_S}.
\end{lemma}
\begin{proof}
By \eqref{defn:Jeps_S}, we have:
\begin{align*}
\bigg|\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{i=1}^{N_{\varepsilon,2}}\int_{\mathcal{S}^i}f_s(Q(t),\nu)d\sigma(t)\bigg| & \leq \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{i=1}^{N_{\varepsilon,2}}\int_{\mathcal{S}^i}\big|f_s(Q(t),\nu)\big|d\sigma(t)\\
&\leq \dfrac{C\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{i=1}^{N_{\varepsilon,2}}\int_{\mathcal{S}^i}\big(|Q|^4(t)+1\big)d\sigma(t) \\
&\leq \dfrac{C\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{i=1}^{N_{\varepsilon,2}}\int_{\partial\mathcal{C}^i_{\varepsilon}}\big(|Q|^4(t)+1\big)d\sigma(t) \\
&\leq \dfrac{\varepsilon^{3+\alpha}}{\varepsilon-\varepsilon^{\alpha}}\cdot\dfrac{2C(p+q+r)}{pqr}\cdot\big(\|Q\|_{L^{\infty}(\overline{\Omega})}^4+1\big)\cdot\sum_{i=1}^{N_{\varepsilon,2}}\int_{\partial\mathcal{C}}d\sigma(t)\\
&\leq \dfrac{\varepsilon^{3+\alpha}}{\varepsilon-\varepsilon^{\alpha}}\cdot\big(\|Q\|^4_{L^{\infty}(\overline{\Omega})}+1\big)\cdot\dfrac{2C(p+q+r)}{pqr}\cdot\sigma(\partial\mathcal{C})\cdot N_{\varepsilon,2},\\
\end{align*}
where $\partial\mathcal{C}$ represents the surface of the model particle $\mathcal{C}$ defined in \eqref{defn:initial_cube}, $\mathcal{C}^{i}_{\varepsilon}$ represents the parallelipipeds constructed in relation \eqref{defn:ceps}, $N_{\varepsilon,2}$ is defined in \eqref{defn:N_eps_2} and $C$ is the $\varepsilon$-independent constant given from the inequality that states that $f_s$ has a quartic growth in $Q$, which can be obtained from assumption ($A_7$). We have also used that $Q$ is bounded on $\overline{\Omega}$. In the proof of \cref{prop:outer_surfaces_go_to_0}, we obtain $N_{\varepsilon,2}\leq\dfrac{L_0l_0+l_0h_0+h_0L_0}{\varepsilon^2}$, hence:
\begin{align*}
\bigg|\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{i=1}^{N_{\varepsilon,2}}\int_{\mathcal{S}^i}f_s(Q(t),\nu)d\sigma(t)\bigg| &< C'\cdot\dfrac{\varepsilon^{3+\alpha}}{\varepsilon-\varepsilon^{\alpha}}\cdot\dfrac{L_0l_0+L_0h_0+l_0h_0}{\varepsilon^{2}}\Rightarrow\\
\Rightarrow\bigg|\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{i=1}^{N_{\varepsilon,2}}\int_{\mathcal{S}^i}f_s(Q(t),\nu)d\sigma(t)\bigg| &< C''\cdot\dfrac{\varepsilon^{\alpha}}{1-\varepsilon^{\alpha-1}}\rightarrow 0\;\text{as}\;\varepsilon\rightarrow 0,
\end{align*}
since $\alpha\in\bigg(1,\dfrac{3}{2}\bigg)$, where $L_0$, $l_0$ and $h_0$ are defined in \eqref{defn:L_0_l_0_h_0} and $C'$ and $C''$ are $\varepsilon$-independent constants.
\end{proof}
\begin{lemma}
For any $Q\in H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$, we have $J_{\varepsilon}^{\mathcal{S}}[Q]\rightarrow 0$ as $\varepsilon\rightarrow 0$.
\end{lemma}
\begin{proof}
Let $(Q_j)_{j\geq 1}$ be a sequence of smooth maps that converge strongly in $H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$ to $Q$. By \cref{lemma:zerocontrbounded}, we have $J_{\varepsilon}^{\mathcal{S}}[Q_j]\rightarrow 0$ as $\varepsilon\rightarrow 0$, for any $j\geq 1$. By assumption ($A_7$), we have on $\mathcal{N}_{\varepsilon}^{\mathcal{S}}$:
\begin{align*}
|f_s(Q_j,\nu)-f_s(Q,\nu)|&\leq |Q_j-Q|\big(|Q_j|^3+|Q|^3+1\big)\\
&\lesssim |Q_j-Q|\big(|Q_j-Q|^3+|Q|^3+1\big)\\
&\lesssim |Q_j-Q|^4+|Q_j-Q|\big(|Q|^3+1\big)
\end{align*}
Thanks to the continuity of the trace operator from $H^1(\Omega_{\varepsilon})$ to $H^{1/2}(\partial\Omega_{\varepsilon})$, the Sobolev embedding \linebreak $H^{1/2}(\partial\Omega_{\varepsilon})\hookrightarrow L^4(\partial\Omega_{\varepsilon})$ and the strong convergence $Q_j\rightarrow Q$ in $H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$, we get that $Q_j\rightarrow Q$ a.e. on $\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}$, since $\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}\subset\partial\Omega_{\varepsilon}$. Therefore, there exists $\psi\in L^4(\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}})$ such that $|Q_j-Q|\leq \psi$ a.e. in $\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}$ and we can write:
\begin{align}\label{eq:noname}
|f_s(Q_j,\nu)-f_s(Q,\nu)|&\lesssim \psi^4+\psi\big(|Q|^3+1)
\end{align}
on $\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}$, for every $j\geq 1$.
At the same time, we have the compact Sobolev embedding $H^{1/2}(\partial\Omega_{\varepsilon})\hookrightarrow L^3(\partial\Omega_{\varepsilon})$, therefore $|Q|^3$ is in $L^1(\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}})$. Hence, the right hand side from \eqref{eq:noname} is in $L^1(\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}})$ and we can apply the Lebesgue dominated convergence theorem and get:
\begin{align*}
\lim_{j\rightarrow +\infty}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}}|f_s(Q_j,\nu)-f_s(Q,\nu)|\text{d}\sigma&=0,
\end{align*}
for any $\varepsilon>0$ fixed.
Now, because for $\varepsilon\rightarrow 0$ we get $|\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}|\rightarrow 0$ (according to \cref{prop:outer_surfaces_go_to_0}) and $\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\rightarrow 0$, the conclusion follows.
\end{proof}
Therefore, from now on we omit the term $J_{\varepsilon}^{\mathcal{S}}$ from the free energy functional and we only study the behaviour of:
\begin{align*}
\mathcal{F}_{\varepsilon}^{\mathcal{T}}[Q]:=\int_{\Omega_{\varepsilon}}\big(f_e(\nabla Q)+f_b(Q)\big)\text{d}x+\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}f_s(Q,\nu)\text{d}\sigma,
\end{align*}
which we denote simply by $\mathcal{F}_{\varepsilon}[Q]$, but we keep the same notation for surfaces generated by the scaffold.
\subsection{Equicoercivity of \texorpdfstring{$\mathcal{F}_{\varepsilon}$}{Fe}}
\begin{prop}\label{prop:equicoercivity}
Suppose that the assumptions ($A_1$)-($A_7$) hold and also that there exists $\mu>0$ such that $f_b(Q)\geq\mu|Q|^6-C$, for any $Q\in\mathcal{S}_0$. Let $Q\in H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$ satisfy $\mathcal{F}_{\varepsilon}[Q]\leq M$, for some $\varepsilon$-independent constant. Then there holds $$\int_{\Omega_{\varepsilon}}|\nabla Q|^2\leq C_M$$ for $\varepsilon>0$ small enough and for some $C_M>0$ depending only on $M$, $f_e$, $f_b$, $f_s$ and $\Omega$.
\end{prop}
\begin{proof}
Assumption $(A_6)$ ensures that $|f_s(Q,\nu)|\lesssim |Q|^4+1$, therefore:
\begin{equation*}
J_{\varepsilon}^{\mathcal{T}}[Q]\geq -C_1\cdot \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}(|Q|^4+1)\text{d}\sigma\geq -C_1\cdot \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q|^4\text{d}\sigma-C_1\cdot C_s,
\end{equation*}
\noindent according to \cref{prop:C_s}. Using \cref{lemma:surface} with $n=4$, we have:
\begin{equation*}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q|^4\text{d}\sigma \lesssim \dfrac{2\varepsilon^{2-\alpha}}{(1-\varepsilon^{\alpha-1})}\int_{\Omega_{\varepsilon}}\big(|Q|^6+|\nabla Q|^2\big)\text{d}x+\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\int_{\Omega_{\varepsilon}}|Q|^4\text{d}x,
\end{equation*}
hence
\begin{equation*}
J_{\varepsilon}^{\mathcal{T}}[Q]\geq -C_1\cdot C_2\cdot \dfrac{2\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\int_{\Omega_{\varepsilon}}\big(|Q|^6+|\nabla Q|^2\big)\text{d}x-C_1\cdot C_2\cdot\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\int_{\Omega_{\varepsilon}}|Q|^4\text{d}x-C_1\cdot C_s.
\end{equation*}
At the same time, from the generalised version of the Hölder's inequality and from the fact that $\Omega$ is bounded, we have
\begin{equation*}
\bigg(\int_{\Omega_{\varepsilon}}|Q|^4\text{d}x\bigg)^{1/4}\leq|\Omega_{\varepsilon}|^{1/12}\cdot\bigg(\int_{\Omega_{\varepsilon}}|Q|^6\text{d}x\bigg)^{1/6}\Rightarrow \int_{\Omega_{\varepsilon}}|Q|^4\text{d}x<|\Omega|^{1/3}\bigg(\int_{\Omega_{\varepsilon}} |Q|^6\text{d}x\bigg)^{2/3}
\end{equation*}
and so
\begin{equation}\label{eq:equi1}
J_{\varepsilon}^{\mathcal{T}}[Q]\geq -C_3\cdot\dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\int_{\Omega_{\varepsilon}}\big(|Q|^6+|\nabla Q|^2)\text{d}x-C_3\cdot\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\cdot |\Omega|^{1/3}\bigg(\int_{\Omega_{\varepsilon}}|Q|^6\text{d}x\bigg)^{2/3}-C_3,
\end{equation}
where $C_3=\text{max}\{C_1\cdot C_2,\;C_1\cdot C_s\}$.
\vspace{2mm}
Since $f_b(Q)\geq\mu |Q|^6-C$, $f_e(\nabla Q)\geq \lambda_e^{-1}|\nabla Q|^2$ (according to ($A_5$) and ($A_6$)) and $|\Omega_{\varepsilon}|\leq|\Omega|$, we have:
\begin{equation}\label{eq:equi2}
\int_{\Omega_{\varepsilon}}\big(f_b(Q)+f_e(\nabla Q)\big)\text{d}x\geq \mu\int_{\Omega_{\varepsilon}}|Q|^6\text{d}x+\lambda_e^{-1}\int_{\Omega_{\varepsilon}}|\nabla Q|^2\text{d}x-C|\Omega|
\end{equation}
and because $\mathcal{F}_{\varepsilon}[Q]\leq M$, combining \eqref{eq:equi1} and \eqref{eq:equi2}, we obtain
\begin{equation}\label{eq:equi3}
\bigg(\lambda_e^{-1}-C_3\cdot \dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\bigg)\int_{\Omega_{\varepsilon}}|\nabla Q|^2\text{d}x \leq h_{\varepsilon}\bigg(\bigg(\int_{\Omega_{\varepsilon}}|Q|^6\text{d}x\bigg)^{1/3}\bigg),
\end{equation}
where
\begin{equation*}
h_{\varepsilon}(t)=t^2\cdot\big(C_4(\varepsilon)+t\cdot C_5(\varepsilon)\big)+C_6,
\end{equation*}
for any $t\geq 0$, with $C_4(\varepsilon)=\dfrac{C_3\cdot|\Omega|^{1/3}}{2(1-\varepsilon^{\alpha-1})^2}$, $C_5(\varepsilon)=C_3\cdot \dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}-\mu$ and $C_6=\big(M+C_3+C|\Omega|\big)$. As $\varepsilon\rightarrow 0$, we have $C_4(\varepsilon)\searrow \dfrac{C_3\cdot|\Omega|^{1/3}}{2}>0$ and $C_5(\varepsilon)\searrow (-\mu)<0$. Hence, for $\varepsilon>0$ small enough, we have:
\begin{align}\label{eq:ineq_C4_C5}
\dfrac{C_3\cdot|\Omega|^{1/3}}{2}<C_4(\varepsilon)<C_3\cdot|\Omega|^{1/3}\;\text{and}\;-\mu<C_5(\varepsilon)<-\dfrac{\mu}{2}<0.
\end{align}
Let $t_0(\varepsilon)$ be the solution of the equation $C_4(\varepsilon)+t\cdot C_5(\varepsilon)=0$. We prove that $h_{\varepsilon}(t)$ is bounded from above on $[0,+\infty)$. Computing the critical points of $h_{\varepsilon}$, it is easy to check that $2t_0(\varepsilon)/3$ is the point in which the function attains its maximum over $[0,+\infty)$, which is:
\begin{align*}
\text{max}\big\{h_{\varepsilon}(t):t\in[0,+\infty)\big\}&=\dfrac{4C_4^3(\varepsilon)}{27C_5^2(\varepsilon)}+C_6<\dfrac{4}{27}\cdot C_3^3\cdot |\Omega|\cdot \dfrac{4}{\mu^2}+C_6,
\end{align*}
using \eqref{eq:ineq_C4_C5}. Therefore, the function $h_{\varepsilon}$ is bounded from above on $[0,+\infty)$.
Using the same arguments we can see that $\lambda_e^{-1}-C_3\cdot\dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}$ is also bounded from below, away from $0$, for $\varepsilon>0$ small enough, and from here the conclusion follows, based on relation \eqref{eq:equi3}.
\end{proof}
\subsection{Lower semi-continuity of \texorpdfstring{$\mathcal{F}_{\varepsilon}$}{Fe}}
\begin{prop}\label{prop:lsc}
Suppose that the assumptions ($A_1$)-($A_7$) are satisfied. Then, the following statement holds: for any positive $M>0$, there exists $\varepsilon_0(M)>0$ such that for any $\varepsilon\in\big(0,\varepsilon_0(M)\big)$ and for any sequence $(Q_j)_{j\in\mathbb{N}}$ from $H^1(\Omega_{\varepsilon},\mathcal{S}_0)$ that converges $H^1$-weakly to a function $Q\in H^1(\Omega_{\varepsilon},\mathcal{S}_0)$ and which satisfies $\|\nabla Q_j\|_{L^{2}(\Omega_{\varepsilon})}\leq M$ for any $j\in\mathbb{N}$, then $$\mathcal{F}_{\varepsilon}[Q]\leq\liminf_{j\rightarrow +\infty} \mathcal{F}_{\varepsilon}[Q_j].$$
\end{prop}
\begin{proof}
The proof of \cref{prop:lsc} follows the same steps as in \cite{CanevariZarnescu1}. We prove this proposition on each component of $\mathcal{F}_{\varepsilon}$. Before that, let $$\omega=\liminf_{j\rightarrow +\infty}\int_{\Omega_{\varepsilon}}|\nabla Q_j|^2 \text{d}x-\int_{\Omega_{\varepsilon}}|\nabla Q|^2\text{d}x.$$
Since $Q_j\rightharpoonup Q$ in $H^1$, then $\nabla Q_j\rightharpoonup \nabla Q$ in $L^2$, therefore $\omega\geq 0$. Moreover, up to extracting a subsequence, we can assume that
\begin{equation}\label{eq:Q_j_goes_to_Q_(nabla)}
\int_{\Omega_{\varepsilon}}|\nabla Q_j|^2\text{d}x\rightarrow\int_{\Omega_{\varepsilon}}|\nabla Q|^2\text{d}x+\omega
\end{equation}
as $j\rightarrow +\infty$.
From the assumption ($A_5$), we have that $f_e$ is strongly convex, that is for $\theta>0$ small enough, $\tilde{f_e}(D):=f_e(D)-\theta|D|^2$ is a convex function from $\mathcal{S}_0\otimes \mathbb{R}^3$ to $[0,+\infty)$. In this case, the functional $\int_{\Omega_{\varepsilon}}\tilde{f_e}(\cdot)\text{d}x$ is lower semicontinuous. Therefore $$\liminf_{j\rightarrow +\infty
}\int_{\Omega_{\varepsilon}}\tilde{f_e}(\nabla Q_j)\text{d}x\geq\int_{\Omega_{\varepsilon}}\tilde{f_e}(\nabla Q)\text{d}x,$$ from which we get
\begin{equation}\label{eq:liminffe}
\liminf_{j\rightarrow +\infty}\int_{\Omega_{\varepsilon}}f_e(\nabla Q_j)\text{d}x-\int_{\Omega_{\varepsilon}}f_e(\nabla Q)\text{d}x\geq \bigg(\liminf_{j\rightarrow +\infty}\int_{\Omega_{\varepsilon}}\tilde{f_e}(\nabla Q_j)\text{d}x-\int_{\Omega_{\varepsilon}}\tilde{f_e}(\nabla Q)\text{d}x\bigg)+\theta\omega\geq 0.
\end{equation}
Since $Q_j\rightharpoonup Q$ in $H^1(\Omega_{\varepsilon})$ and the injection $H^1(\Omega_{\varepsilon})\subset L^2(\Omega_{\varepsilon})$ is compact, then we can assume, up to extracting a subsequence, that $Q_j\rightarrow Q$ a.e. in $\Omega_{\varepsilon}$. Then, from the assumption ($A_6$), we can see that the sequence $\big(f_b(Q_j)\big)_{j\in\mathbb{N}}$ satisfies all the conditions from Fatou's lemma, therefore:
\begin{equation}\label{eq:liminffb}
\liminf_{j\rightarrow +\infty}\int_{\Omega_{\varepsilon}}f_b(Q_j)\text{d}x\geq\int_{\Omega_{\varepsilon}}\liminf_{j\rightarrow +\infty}f_b(Q_j)\text{d}x=\int_{\Omega_{\varepsilon}}f_b(Q)\text{d}x.
\end{equation}
Regarding the \textit{surface energy}, we split $\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}$ into:
\begin{align*}
A_j&=\{x\in\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}\;:\;|Q_j(x)-Q(x)|\leq |Q(x)|+1\}\\
B_j=\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}\setminus A_j&=\{x\in\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}\;:\;|Q_j(x)-Q(x)|>|Q(x)|+1\},
\end{align*}
for any $j\in\mathbb{N}$.
Using ($A_7$), we have
\begin{align*}
\int_{A_j}|f_s(Q_j,\nu)-f_s(Q,\nu)|\text{d}\sigma&\leq \int_{A_j}\big(|Q_j|^3+|Q|^3+1\big)\cdot |Q_j-Q|\text{d}\sigma\\
&\leq \int_{A_j}\big((|Q_j-Q|+|Q|)^3+|Q|^3+1\big)\cdot\big(|Q|+1\big)\text{d}\sigma\\
&\lesssim \int_{A_j}(|Q|^3+1)(|Q|+1)\text{d}\sigma\lesssim \int_{A_j}(|Q|^4+1)\text{d}\sigma.
\end{align*}
Then due to the continuous embedding of $H^{1/2}(\partial\mathcal{N}_{\varepsilon})$ into $L^{4}(\partial\mathcal{N}_{\varepsilon})$:
\begin{align*}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{A_j}|f_s(Q_j,\nu)-f_s(Q,\nu)|\text{d}\sigma\lesssim \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{A_j}\big(|Q|^4+1\big)\text{d}\sigma<+\infty,
\end{align*}
according also to \cref{prop:C_s}. At the same time, the compact embedding $H^{1/2}(\partial\mathcal{N}_{\varepsilon})\hookrightarrow L^{2}(\partial\mathcal{N}_{\varepsilon})$ and the continuity of the trace operator from $H^{1}(\Omega_{\varepsilon})$ into $H^{1/2}(\partial\mathcal{N}_{\varepsilon})$ grants that $Q_j\rightarrow Q$ a.e. on $\partial\mathcal{N}_{\varepsilon}$, up to extracting a subsequence. We can now apply the dominated convergence theorem and get:
\begin{equation}\label{eq:liminfaj}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{A_j}|f_s(Q_j,\nu)-f_s(Q,\nu)|\text{d}\sigma\rightarrow 0\;\text{as}\;j\rightarrow +\infty.
\end{equation}
Regarding the $B_j$ sets, we have, according to ($A_7$):
\begin{align*}
|f_s(Q_j,\nu)-f_s(Q,\nu)|&\leq \lambda_s |Q_j-Q|\big(|Q_j|^3+|Q|^3+1\big)\\
& \lesssim |Q_j-Q|\big(|Q_j|^3+|Q_j-Q|^3+1\big)\;\text{(using}\;|Q|+1<|Q_j-Q|)\\
& \lesssim |Q_j-Q|\big(|Q_j-Q|^3+|Q|^3+|Q_j-Q|^3+1\big)\\
& \lesssim |Q_j-Q|\big(|Q_j-Q|^3+1) \lesssim |Q_j-Q|^4.
\end{align*}
Using \cref{lemma:surface} for $(Q_j-Q)$ with $n=4$, we have:
\begin{align*}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{B_j}|f_s(Q_j,\nu)-f_s(Q,\nu)|\text{d}\sigma &\lesssim \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{B_j}|Q_j-Q|^4\text{d}\sigma\lesssim \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q_j-Q|^4\text{d}\sigma
\end{align*}
\begin{equation*}
\lesssim \dfrac{2\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\int_{\Omega_{\varepsilon}}|Q_j-Q|^6+|\nabla Q_j-\nabla Q|^2\text{d}x+\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\int_{\Omega_{\varepsilon}}|Q_j-Q|^4\text{d}x.
\end{equation*}
Since $\text{H}^1(\Omega_{\varepsilon})$ is compactly embedded into $\text{L}^{4}(\Omega_{\varepsilon})$ and $Q_j\rightharpoonup Q$ in $\text{H}^1(\Omega_{\varepsilon})$, then $Q_j\rightarrow Q$ in $\text{L}^4(\Omega_{\varepsilon})$ and so
\begin{equation*}
\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\int_{\Omega_{\varepsilon}}|Q_j-Q|^4\text{d}x\rightarrow 0,\;\text{as}\;j\rightarrow +\infty.
\end{equation*}
For the term containing $|Q_j-Q|^6$, we proceed in the following way:
\begin{align*}
\int_{\Omega_{\varepsilon}}|Q_j-Q|^6\text{d}x &=\int_{\Omega_{\varepsilon}}|E_{\varepsilon}(Q_j-Q)|^6\text{d}x \leq \int_{\Omega}|E_{\varepsilon}(Q_j-Q)|^6\text{d}x=\big\|E_{\varepsilon}(Q_j-Q)\big\|^6_{L^{6}(\Omega)}\\
&\lesssim \big\|E_{\varepsilon}(Q_j-Q)\big\|^6_{\text{H}^1(\Omega)}\;\text{(by the continuous injection}\;\text{H}^1(\Omega)\subset \text{L}^6(\Omega))\\
&\lesssim \big\|E_{\varepsilon}(Q_j-Q)\big\|^6_{\text{H}^1_0(\Omega)}\;\text{(because}\;Q_j\equiv Q\;\text{on}\;\partial\Omega)\\
&\lesssim \big\|\nabla E_{\varepsilon}(Q_j-Q)\big\|^6_{\text{L}^{2}(\Omega)} = \bigg(\int_{\Omega}|\nabla E_{\varepsilon}(Q_j-Q)|^2\text{d}x\bigg)^3\\
&\lesssim \bigg(\int_{\Omega_{\varepsilon}}|\nabla Q_j-\nabla Q|^2\text{d}x\bigg)^3\;\text{(using}\;\text{\cref{lemma:extensionineq}}).
\end{align*}
Now, because $\|\nabla Q_j\|_{\text{L}^2(\Omega_{\varepsilon})}\leq M$, then:
\begin{align*}
\int_{\Omega_{\varepsilon}}|\nabla Q_j-\nabla Q|^2\text{d}x &\leq \int_{\Omega_{\varepsilon}} \big(|\nabla Q_j|^2+|\nabla Q|^2\big)\text{d}x \lesssim M^2
\end{align*}
and therefore
\begin{equation}\label{eq:liminfbj}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{B_j}|f_s(Q_j,\nu)-f_s(Q,\nu)|\text{d}\sigma \lesssim \dfrac{2\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}(1+M^4)\int_{\Omega_{\varepsilon}}|\nabla Q_j-\nabla Q|^2\text{d}x+o(1).
\end{equation}
Using that $Q_j\rightharpoonup Q$ in $\text{H}^1(\Omega_{\varepsilon})$ and \eqref{eq:Q_j_goes_to_Q_(nabla)}, we obtain that $\displaystyle{\int_{\Omega_{\varepsilon}}|\nabla Q_j-\nabla Q|^2\text{d}x\rightarrow \omega}$ as $j\rightarrow +\infty$ and combining this with \eqref{eq:liminfaj} and \eqref{eq:liminfbj}, we get:
\begin{equation}\label{eq:liminffs}
\liminf_{j\rightarrow +\infty}J_{\varepsilon}^{\mathcal{T}}[Q_j]-J_{\varepsilon}^{\mathcal{T}}[Q]\geq -C_M\cdot\omega\cdot\dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}},
\end{equation}
where $C_M$ is a constant dependent of $M$ and independent of $\varepsilon$.
According to \eqref{eq:liminffe}, \eqref{eq:liminffb} and \eqref{eq:liminffs}, we finally obtain that
\begin{equation*}
\liminf_{j\rightarrow +\infty}\mathcal{F}_{\varepsilon}[Q_j]-\mathcal{F}_{\varepsilon}[Q]\geq \bigg(\theta -C_M\dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\bigg)\omega
\end{equation*}
and since $\dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\rightarrow 0$ as $\varepsilon\rightarrow 0$, the conclusion follows.
\end{proof}
\section{Convergence of local minimisers}\label{section:conv_local_min}
\subsection{Pointwise convergence of the surface integral}
The aim of this section is to prove the following statement:
\begin{theorem}\label{th:Je_goes_to_J0}
Suppose that the assumptions ($A_1$)-($A_7$) are satisfied. Then, for any bounded, Lipschitz map $Q:\overline{\Omega}\rightarrow\mathcal{S}_0$, there holds $J_{\varepsilon}^{\mathcal{T}}[Q]\rightarrow J_0[Q]$ as $\varepsilon\rightarrow 0$, where
\begin{align}\label{defn:J_0}
J_0[Q]=\displaystyle{\int_{\Omega}f_{hom}(Q)\emph{d}x}.
\end{align}
\end{theorem}
\begin{proof}
Let us fix a bounded, Lipschitz map $Q:\overline{\Omega}\rightarrow\mathcal{S}_0$ and let $\tilde{J}_{\varepsilon}$ be the following functional:
\begin{equation}\label{defn:J_tilde_eps}
\tilde{J}_{\varepsilon}[Q]=\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\bigg(\sum_{k=1}^{X_{\varepsilon}}\int_{\mathcal{T}_x^k}f_s(Q(y^{x,k}_{\varepsilon}),\nu)\text{d}\sigma+\sum_{l=1}^{Y_{\varepsilon}}\int_{\mathcal{T}_y^l}f_s(Q(y^{y,l}_{\varepsilon}),\nu)\text{d}\sigma+\sum_{m=1}^{Z_{\varepsilon}}\int_{\mathcal{T}_z^m}f_s(Q(y^{z,m}_{\varepsilon}),\nu)\text{d}\sigma\bigg),
\end{equation}
where $y_{\varepsilon}^{x,k}$, $y_{\varepsilon}^{y,l}$ and $y_{\varepsilon}^{z,m}$ are defined in \eqref{defn:Y_eps_x}, \eqref{defn:Y_eps_y} and \eqref{defn:Y_eps_z}, $\mathcal{T}_x^k$, $\mathcal{T}_y^l$ and $\mathcal{T}_z^m$ are defined in \eqref{defn:T_x_k}, \eqref{defn:T_y_l} and \eqref{defn:T_z_m} and $X_{\varepsilon}$, $Y_{\varepsilon}$ and $Z_{\varepsilon}$ are defined in \eqref{defn:Y_eps_x_cardinal}, \eqref{defn:Y_eps_y_cardinal} and \eqref{defn:Y_eps_z_cardinal}.
We prove that $\tilde{J}_{\varepsilon}[Q]\rightarrow J_0[Q]$ and that $\big|J_{\varepsilon}^{\mathcal{T}}[Q]-\tilde{J}_{\varepsilon}[Q]\big|\rightarrow 0$ as $\varepsilon\rightarrow 0$, for any $Q$ with the properties set earlier.
Let
\begin{align}\label{defn:PSI}
\begin{cases}
\displaystyle{\Psi^{X}(Q(\tau_0))=\int_{\mathcal{C}^x}f_s(Q(\tau_0),\nu(\tau))\text{d}\sigma(\tau)}\\
\vspace{-4mm}\\
\displaystyle{\Psi^Y(Q(\tau_0))=\int_{\mathcal{C}^y}f_s(Q(\tau_0),\nu(\tau))\text{d}\sigma(\tau)}\\
\vspace{-4mm}\\
\displaystyle{\Psi^Z(Q(\tau_0))=\int_{\mathcal{C}^z}f_s(Q(\tau_0),\nu(\tau))\text{d}\sigma(\tau)}
\end{cases}
\end{align}
for any $\tau_0\in\Omega$, where $\mathcal{C}^x$, $\mathcal{C}^y$ and $\mathcal{C}^z$ are defined in \eqref{defn:C_x_C_y_C_z}. Because $f_s$ is continuous on $\mathcal{S}_0\times\mathbb{S}^2$, then $\Psi^X$, $\Psi^Y$ and $\Psi^Z$ are also continuous. In this case, for example, the first sum from \eqref{defn:J_tilde_eps}, denoted as $\tilde{J}_{\varepsilon}^{X}$, becomes:
\vspace{-3mm}
\begin{align}\label{defn:J_tilde_eps_nice_form}
\tilde{J}_{\varepsilon}^{X}[Q]&=\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{k=1}^{X_{\varepsilon}}\int_{\mathcal{T}^k_x}f_s(Q(y^{x,k}_{\varepsilon}),\nu)\text{d}\sigma\notag\\
&=\dfrac{(p-\varepsilon^{\alpha-1})}{pr(1-\varepsilon^{\alpha-1})}\cdot\varepsilon^{3}\sum_{k=1}^{X_{\varepsilon}}\int_{\mathcal{C}^{y}}f_s(Q(y_{\varepsilon}^{x,k}),\nu)\text{d}\sigma+\dfrac{(p-\varepsilon^{\alpha-1})}{pq(1-\varepsilon^{\alpha-1})}\cdot\varepsilon^{3}\sum_{k=1}^{X_{\varepsilon}}\int_{\mathcal{C}^{z}}f_s(Q(y_{\varepsilon}^{x,k}),\nu)\text{d}\sigma\notag\\
&=\dfrac{(p-\varepsilon^{\alpha-1})}{pr(1-\varepsilon^{\alpha-1})}\cdot\int_{\Omega}\Psi^Y(Q(\tau))\text{d}\mu_{\varepsilon}^{X}(\tau)+\dfrac{(p-\varepsilon^{\alpha-1})}{pq(1-\varepsilon^{\alpha-1})}\cdot\int_{\Omega}\Psi^Z(Q(\tau))\text{d}\mu_{\varepsilon}^{X}(\tau),
\end{align}
where $\mu_{\varepsilon}^{X}$ is defined in \eqref{defn:measures}, that is, assumption ($A_4$).
According to ($A_4$), as $\varepsilon\rightarrow 0$, $\mu_{\varepsilon}^{X}$ converges weakly* to the Lebesgue measure restricted to $\Omega$ and because $\Psi^Y$ and $\Psi^Z$ are continuous, then:
\begin{equation*}
\tilde{J}_{\varepsilon}^{X}[Q]\rightarrow\dfrac{1}{r}\int_{\Omega}\Psi^Y(Q(\tau))\text{d}\tau+\dfrac{1}{q}\int_{\Omega}\Psi^Z(Q(\tau))\text{d}\tau.
\end{equation*}
Computing in a similar way $\tilde{J}_{\varepsilon}^y$ and $\tilde{J}_{\varepsilon}^z$, we get:
\begin{equation*}
\tilde{J}_{\varepsilon}[Q]\rightarrow \int_{\Omega}\bigg(\dfrac{q+r}{qr}\Psi^X(Q(\tau))+\dfrac{p+r}{pr}\Psi^Y(Q(\tau))+\dfrac{p+q}{pq}\Psi^Z(Q(\tau))\bigg)\text{d}\tau=\int_{\Omega}f_{hom}(Q(\tau))\text{d}\tau=J_0[Q]
\end{equation*}
which implies that $\tilde{J}_{\varepsilon}[Q]\rightarrow J_0[Q]$. For $J_{\varepsilon}^{X}[Q]$ and $\tilde{J}_{\varepsilon}^{X}[Q]$, we have:
\begin{align}\label{eq:J_eps_J_eps_tilde_control1}
\big|J_{\varepsilon}^{X}[Q]-\tilde{J}_{\varepsilon}^{X}[Q]\big|&\leq \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{k=1}^{X_{\varepsilon}}\int_{\mathcal{T}_x^k}\big|f_s(Q(\tau),\nu(\tau))-f_s(Q(y^{x,k}_{\varepsilon}),\nu(\tau))\big|\text{d}\sigma(\tau)\notag\\
&\lesssim \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\sum_{k=1}^{X_{\varepsilon}}\int_{\mathcal{T}_x^k}\big(|Q(\tau)|^3+|Q(y^{x,k}_{\varepsilon})|^3+1\big)|Q(\tau)-Q(y^{x,k}_{\varepsilon})|\text{d}\sigma(\tau)\notag\\
&\lesssim \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\cdot\big(\|Q\|^3_{L^{\infty}(\overline{\Omega})}+1\big)\cdot \text{Lip}(Q)\cdot \sum_{k=1}^{X_{\varepsilon}} \sigma(\mathcal{T}_x^k)\cdot \text{diam}(\mathcal{T}_x^k),
\end{align}
using that $Q$ is bounded on $\overline{\Omega}$ and where $\text{Lip}(Q)$ is the Lipschitz constant of $Q$, $\sigma(\mathcal{T}_x^k)$ is the total area of $\mathcal{T}_x^k$ and $\text{diam}(\mathcal{T}_x^k)$ is the diameter of $\mathcal{T}_x^k$, which coincides with the diameter of the parallelipiped $\mathcal{P}_{\varepsilon}^{x,k}$, defined in \eqref{defn:P_x}. Hence
\begin{align}\label{eq:J_eps_J_eps_tilde_control2}
\big|J_{\varepsilon}^{X}[Q]-\tilde{J}_{\varepsilon}^{X}[Q]\big|&\lesssim\dfrac{2(r+q)}{pqr}\cdot \dfrac{p\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}\cdot X_{\varepsilon}\cdot\varepsilon^{3}\cdot\sqrt{\bigg(\varepsilon-\dfrac{\varepsilon^{\alpha}}{p}\bigg)^2+\bigg(\dfrac{\varepsilon^{\alpha}}{q}\bigg)^2+\bigg(\dfrac{\varepsilon^{\alpha}}{r}\bigg)^2}
\end{align}
Now, as $\varepsilon\rightarrow 0$, we have: $\sqrt{\bigg(\varepsilon-\dfrac{\varepsilon^{\alpha}}{p}\bigg)^2+\bigg(\dfrac{\varepsilon^{\alpha}}{q}\bigg)^2+\bigg(\dfrac{\varepsilon^{\alpha}}{r}\bigg)^2}\rightarrow 0$; $\dfrac{p\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}=\dfrac{p-\varepsilon^{\alpha-1}}{1-\varepsilon^{1-\alpha}}\rightarrow p$ because $1<\alpha$; $X_{\varepsilon}\cdot\varepsilon^{3}<\bigg(\dfrac{L_0}{\varepsilon}-1\bigg)\cdot\dfrac{l_0}{\varepsilon}\cdot\dfrac{h_0}{\varepsilon}\cdot\varepsilon^{3}=L_0l_0h_0-\varepsilon l_0h_0$, according to \cref{prop:volumegoesto0}, where $L_0$, $l_0$ and $h_0$ are defined in \eqref{defn:L_0_l_0_h_0}. Since $X_{\varepsilon}$ is positive, we see that:
\begin{align}\label{eq:control_over_X_eps}
0\leq \lim_{\varepsilon\rightarrow 0} \big(X_{\varepsilon}\cdot \varepsilon^{3}\big)\leq\lim_{\varepsilon\rightarrow 0}\big(L_0l_0h_0-\varepsilon l_0h_0)=L_0l_0h_0<+\infty.
\end{align}.
Therefore $J_{\varepsilon}^{X}[Q]\rightarrow\tilde{J}_{\varepsilon}^{X}[Q]$ as $\varepsilon\rightarrow 0$. We get the same result for the other two components, from which we conclude.
\end{proof}
\begin{remark}
It is easy to see that if we replace the coefficient $\dfrac{\varepsilon^{3}}{\varepsilon^{\alpha}(\varepsilon-\varepsilon^{\alpha})}$ of the \textit{surface energy} term $J_{\varepsilon}$ with:
\begin{itemize}
\item[•] $\dfrac{\varepsilon^3}{(\varepsilon-\varepsilon^{\alpha})^2}\rightarrow 0$, then $J_{\varepsilon}^{X}[Q]\rightarrow 0$ as $\varepsilon\rightarrow 0$;
\item[•] $\dfrac{\varepsilon^3}{\varepsilon^{2\alpha}}=\varepsilon^{3-2\alpha}\rightarrow 0$, then $J_{\varepsilon}^{X}[Q]\rightarrow +\infty$ as $\varepsilon\rightarrow 0$.
\end{itemize}
In both cases, we lose the convergence $J_{\varepsilon}^{\mathcal{T}}[Q]\rightarrow J_0[Q]$.
\end{remark}
\subsection{$\Gamma$-convergence of the approximating free energies}
\begin{lemma}\label{lemma:boundednabla}
Suppose that the assumption ($A_7$) is satisfied. Let $Q_1$ and $Q_2$ from $H^1_g(\Omega,\mathcal{S}_0)$ be such that
\begin{equation}\label{eq:controlovernabla}
max\{\|\nabla Q_1\|_{\text{L}^2(\Omega)},\|\nabla Q_2\|_{\text{L}^{2}(\Omega)}\}\leq M
\end{equation}
for some $\varepsilon$-independent constant $M$. Then, for $\varepsilon$ sufficiently small, we have:
\begin{equation}\label{eq:alpha/4}
\big|J_{\varepsilon}^{\mathcal{T}}[Q_2]-J_{\varepsilon}^{\mathcal{T}}[Q_1]\big|\leq C_M\big(\varepsilon^{1/2-\alpha/4}+\|Q_2-Q_1\|_{\text{L}^4(\Omega)}\big)
\end{equation}
for some $C_M>0$ depending only on $M$, $f_s$, $\Omega$, $\mathcal{C}$ and $g$.
\end{lemma}
\begin{proof}
According to ($A_7$) and H\"{o}lder inequality, we have:
\begin{align*}
\big|J_{\varepsilon}^{\mathcal{T}}[Q_2]-J_{\varepsilon}^{\mathcal{T}}[Q_1]\big|&\lesssim \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}} \big(|Q_1|^3+|Q_2|^3+1\big)|Q_2-Q_1|\text{d}\sigma
\end{align*}
\begin{equation*}
\lesssim \dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\bigg(\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q_2-Q_1|^4\text{d}\sigma\bigg)^{1/4}\bigg(\bigg(\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q_1|^4\text{d}\sigma\bigg)^{3/4}+\bigg(\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q_2|^4\text{d}\sigma\bigg)^{3/4}+|\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}|^{3/4}\bigg).
\end{equation*}
If we make use of \cref{lemma:surface}, then:
\begin{align*}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q_i|^4\text{d}\sigma &\lesssim \dfrac{2\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\int_{\Omega_{\varepsilon}}\big(|Q_i|^6+|\nabla Q_i|^2\big)\text{d}x+\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\int_{\Omega_{\varepsilon}}|Q_i|^4\text{d}x,
\end{align*}
for any $i\in\{1,2\}$. By the continuous injection $\text{H}^1(\Omega_{\varepsilon})$ into $\text{L}^6(\Omega_{\varepsilon})$, we have
\begin{align}\label{eq:control_Q_i}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q_i|^4\text{d}\sigma &\lesssim \dfrac{2\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\bigg(\big\|\nabla Q_i\big\|^2_{\text{L}^2(\Omega_{\varepsilon})} + \big\|Q_i\big\|^6_{\text{H}^{1}(\Omega_{\varepsilon})}\bigg)+\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\big\|Q_i\big\|^4_{\text{L}^4(\Omega_{\varepsilon})}.
\end{align}
Using the Poincar\' e inequality as in Theorem 4.4.7, page 193, from \cite{Ziemer}, the compact embedding \linebreak $H^1(\Omega_{\varepsilon})\hookrightarrow L^4(\Omega_{\varepsilon})$ and the fact that $\Omega_{\varepsilon}\subset\Omega$, we get for $\varepsilon$ small enough:
\begin{align*}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q_i|^4\text{d}\sigma &\lesssim \dfrac{2\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\bigg(\big\|\nabla Q_i\big\|^2_{L^2(\Omega)}+\big\|\nabla Q_i\big\|^6_{L^2(\Omega)}\bigg)+\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}\big\|\nabla Q_i\big\|^4_{L^2(\Omega)}\\
&\lesssim \dfrac{2\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\big(M^2+M^6\big)+\dfrac{1}{2(1-\varepsilon^{\alpha-1})^2}M^4,
\end{align*}
where we used \eqref{eq:controlovernabla} and we can see that the right-hand side from the last inequality can be bounded in terms of $M$, since $\dfrac{\varepsilon^{2-\alpha}}{1-\varepsilon^{\alpha-1}}\searrow 0$ and $\dfrac{1}{(1-\varepsilon^{\alpha-1})^2}\searrow 1$ as $\varepsilon\rightarrow 0$. But since $\dfrac{1}{(1-\varepsilon^{\alpha-1})^2}\searrow 1$ as $\varepsilon\rightarrow 0$, we can choose $\varepsilon>0$ such that $\dfrac{1}{(1-\varepsilon^{\alpha-1})^2}<2$ and we can move the constant $2$ under the ``$\lesssim$" sign. Hence, the last relation can be written as:
\begin{align*}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}|Q_i|^4\text{d}\sigma &\lesssim \varepsilon^{2-\alpha}\big(M^2+M^6\big)+M^4.
\end{align*}
In a similar fashion, using the same arguments as before for \eqref{eq:control_Q_i}, we get in the case of $\big(Q_2-Q_1\big)$:
\begin{align*}
\dfrac{\varepsilon^{3-\alpha}}{\varepsilon-\varepsilon^{\alpha}}\int_{\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}}\big|Q_2-Q_1\big|^4\text{d}\sigma &\lesssim \varepsilon^{2-\alpha}\big(M^2+M^6\big)+\big\|Q_2-Q_1\big\|^4_{L^4(\Omega)}.
\end{align*}
Using the same bounds as in \eqref{eq:control_Q_i}, we conclude by observing that there exists a constant $C_M>0$ such that:
\begin{equation*}
\big|J_{\varepsilon}^{\mathcal{T}}[Q_2]-J_{\varepsilon}^{\mathcal{T}}[Q_1]\big|\leq C_M\cdot\big(\big(\varepsilon^{2-\alpha}\big)^{1/4}+\|Q_2-Q_1\|_{\text{L}^4(\Omega)}\big)
\end{equation*}
\end{proof}
\begin{lemma}\label{lemma:J_eps_goes_to_J_0_for_any_Q}
For any $Q\in H^1_g(\Omega,\mathcal{S}_0)$, there holds $J_{\varepsilon}^{\mathcal{T}}[Q]\rightarrow J_0[Q]$ as $\varepsilon\rightarrow 0$.
\end{lemma}
\begin{proof}
Let $(Q_j)_{j\geq 1}$ be a sequence of smooth functions that converge strongly to $Q$ in $H^1_g(\Omega,\mathcal{S}_0)$. Then there holds:
\begin{equation*}
\big|J_{\varepsilon}^{\mathcal{T}}[Q]-J_0[Q]\big|\leq\big|J_{\varepsilon}^{\mathcal{T}}[Q]-J_{\varepsilon}^{\mathcal{T}}[Q_j]\big|+\big|J_{\varepsilon}^{\mathcal{T}}[Q_j]-J_0[Q_j]\big|+\big|J_0[Q_j]-J_0[Q]\big|.
\end{equation*}
From \cref{lemma:boundednabla}, we have that
\begin{equation*}
\big|J_{\varepsilon}^{\mathcal{T}}[Q]-J_{\varepsilon}^{\mathcal{T}}[Q_j]\big|\lesssim \varepsilon^{1/2-\alpha/4}+\|Q-Q_j\|_{\text{L}^4(\Omega)}.
\end{equation*}
and we recall that for $\varepsilon\rightarrow 0$ we have $\varepsilon^{1/2-\alpha/4}\rightarrow 0$ because $\alpha\in(1,3/2)$.
Since the $\big(Q_j)_{j\geq 1}$ converge strongly in $H^1_g(\Omega)$, from the compact Sobolev embedding, we get that $Q_j\rightarrow Q$ in $L^4(\Omega)$ as $j\rightarrow +\infty$, therefore $Q_j\rightarrow Q$ a.e. in $\Omega$.
From \cref{th:Je_goes_to_J0}, we obtain that $J_{\varepsilon}^{\mathcal{T}}[Q_j]\rightarrow J_0[Q_j]$ as $\varepsilon\rightarrow 0$, for any $j\geq 1$.
For the last term, we can write $\big|J_0[Q_j]-J_0[Q]\big|\leq\int_{\Omega}\big|f_{hom}[Q_j]-f_{hom}[Q]\big|\text{d}x$. In here, we have: $f_{hom}$ is continuous, $Q_j\rightarrow Q$ a.e. in $\Omega$ and $f_{hom}$ has a quartic growth in $Q$ (because $f_s$ has the same growth), which implies that: $\big|f_{hom}[Q_j]\big|\lesssim|Q_j|^4+1$. At the same time, we can assume that there exists $\psi \in L^1(\Omega)$ such that $|Q_j|^4\leq \psi$, for any $j\geq 1$, a.e. in $\Omega$. Therefore, we can apply the Lebesgue dominated convergence theorem and get that $J_0[Q_j]\rightarrow J_0[Q]$ as $j\rightarrow +\infty$.
Combining the results from above, we obtain:
\begin{align*}
\limsup_{\varepsilon\rightarrow 0}\big|J_{\varepsilon}^{\mathcal{T}}[Q]-J_{\varepsilon}^{\mathcal{T}}[Q_j]\big|\lesssim \|Q-Q_j\|_{L^4(\Omega)}+\big|J_0[Q_j]-J_0[Q]\big|\rightarrow 0,\hspace{5mm}\text{as }j\rightarrow +\infty,
\end{align*}
from which we conclude.
\end{proof}
We now prove that $\mathcal{F}_{\varepsilon}$ $\Gamma$-converges to $\mathcal{F}_0$ as $\varepsilon\rightarrow 0$, with respect to the weak $H^1$-topology.
\begin{prop}\label{prop:gamma_conv_inf}
Suppose that the assumptions ($A_1$)-($A_7$) are satisfied. Let $Q_{\varepsilon}\in H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$ be such that $E_{\varepsilon}Q_{\varepsilon}\rightharpoonup Q$ weakly in $H^1(\Omega)$ as $\varepsilon\rightarrow 0$. Then:
\begin{equation*}
\liminf_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}[Q_{\varepsilon}]\geq \mathcal{F}_0[Q],\hspace{10mm} \lim_{\varepsilon\rightarrow 0}J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]=J_0[Q].
\end{equation*}
\end{prop}
\begin{proof}
The proof follows the same steps as in Proposition 4.2. from \cite{CanevariZarnescu1}.
Since $E_{\varepsilon}Q_{\varepsilon}\rightharpoonup Q$ in $H^1(\Omega)$, then $\big(E_{\varepsilon}Q_{\varepsilon}\big)_{\varepsilon >0}$ is a bounded sequence in $H^1(\Omega)$. Therefore, we can choose a subsequence $\big(E_{\varepsilon_j}Q_{\varepsilon_j}\big)_{j\geq 1}\subset\big(E_{\varepsilon}Q_{\varepsilon}\big)_{\varepsilon >0}$ such that
\begin{align*}
\liminf_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}[Q_{\varepsilon}]&=\lim_{j\rightarrow +\infty}\mathcal{F}_{\varepsilon_j}[Q_{\varepsilon_j}].
\end{align*}
Furthermore, by the compact embeddings $H^1(\Omega)\hookrightarrow L^s(\Omega)$, with $s\in[1,6)$, we have that $E_{\varepsilon_j}Q_{\varepsilon_j}\rightarrow Q$ strongly in $L^s(\Omega)$, for any $s\in[1,6)$. As a result, we also obtain that $E_{\varepsilon_j}Q_{\varepsilon_j}\rightarrow Q$ a.e. in $\Omega$. We denote the subsequence $E_{\varepsilon_j}Q_{\varepsilon_j}$ as $E_{\varepsilon}Q_{\varepsilon}$ for the ease of notation.
Now, according to ($A_5$), we have:
\begin{align}\label{eq:nabla_f_e}
\int_{\Omega_{\varepsilon}}\big(f_e(\nabla Q_{\varepsilon})-f_e(\nabla Q)\big)\text{d}x &\geq \int_{\Omega_{\varepsilon}}\nabla f_e(\nabla Q):(\nabla Q_{\varepsilon}-\nabla Q)\text{d}x=\notag\\
&=\int_{\Omega}\nabla f_e(\nabla Q):(\nabla Q_{\varepsilon}-\nabla Q)\text{d}x-\int_{\mathcal{N}_{\varepsilon}}\nabla f_e(\nabla Q):(\nabla Q_{\varepsilon}-\nabla Q)\text{d}x \geq\notag\\
&\geq \int_{\Omega}\nabla f_e(\nabla Q):(\nabla Q_{\varepsilon}-\nabla Q)\text{d}x-\big\|\nabla f_e(\nabla Q)\big\|_{L^2(\mathcal{N}_{\varepsilon})}\cdot\big\|\nabla Q_{\varepsilon}-\nabla Q\big\|_{L^2(\mathcal{N}_{\varepsilon})}
\end{align}
Because $Q\in H^1(\Omega)$, then $\nabla Q\in L^2(\Omega)$ and, according to ($A_5$), the relation $\big|\nabla f_e(\nabla Q)\big|\lesssim |\nabla Q|+1$ implies that $\nabla f_e(\nabla Q)\in L^2(\Omega)$. Therefore, by the weak convergence $E_{\varepsilon}Q_{\varepsilon}\rightharpoonup Q$ in $H^1(\Omega)$, the first term from the right hand side in \eqref{eq:nabla_f_e} goes to $0$ as $\varepsilon\rightarrow 0$. The second term goes to $0$ as well thanks additionally to the fact that the volume of the scaffold $\mathcal{N}_{\varepsilon}$ tends to $0$ as $\varepsilon\rightarrow 0$, according to \cref{prop:volumegoesto0}. Hence:
\begin{align}\label{eq:liminf_f_e}
\liminf_{\varepsilon\rightarrow 0}\int_{\Omega_{\varepsilon}}f_e(\nabla Q_{\varepsilon})\text{d}x &\geq \lim_{\varepsilon\rightarrow 0}\int_{\Omega_{\varepsilon}}f_e(\nabla Q)\text{d}x=\int_{\Omega}f_e(\nabla Q)\text{d}x.
\end{align}
For the bulk potential we apply Fatou's lemma, since $f_b(Q_{\varepsilon})\chi_{\Omega_{\varepsilon}}\rightarrow f_b(Q)$ a.e. in $\Omega$ (because $f_b$ is continuous, $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q$ a.e. in $\Omega$ and $|\mathcal{N}_{\varepsilon}|\rightarrow 0$, according to \cref{prop:volumegoesto0}) and $f_b$ is bounded from below (according to ($A_6$)), in order to obtain:
\begin{align}\label{eq:liminf_f_b}
\liminf_{\varepsilon\rightarrow 0}\int_{\Omega_{\varepsilon}}f_b(Q_{\varepsilon})\text{d}x &\geq \int_{\Omega}f_b(Q)\text{d}x.
\end{align}
For the \textit{surface energy}, we first use \cref{lemma:boundednabla} in the following inequality:
\begin{align*}
\big|J_{\varepsilon}^{\mathcal{T}}[E_{\varepsilon}Q_{\varepsilon}]-J_0[Q]\big| &\leq \big|J_{\varepsilon}^{\mathcal{T}}[E_{\varepsilon}Q_{\varepsilon}]-J_{\varepsilon}^{\mathcal{T}}[Q]\big|+\big|J_{\varepsilon}^{\mathcal{T}}[Q]-J_0[Q]\big|\\
& \lesssim \varepsilon^{1/2-\alpha/4}+\big\|E_{\varepsilon}Q_{\varepsilon}-Q\big\|_{L^4(\Omega)}+\big|J_{\varepsilon}^{\mathcal{T}}[Q]-J_0[Q]\big|.
\end{align*}
Since we have $\varepsilon^{1/2-\alpha/4}\rightarrow 0$ for $\varepsilon\rightarrow 0$ (because $\alpha\in(1,3/2)$), then combining the result from \cref{lemma:J_eps_goes_to_J_0_for_any_Q} with the fact that $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q$ strongly in $L^4(\Omega)$, we obtain
\begin{align}\label{eq:liminf_f_s}
\lim_{\varepsilon\rightarrow 0} J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]=J_0[Q].
\end{align}
The proof is now complete, considering \eqref{eq:liminf_f_e}, \eqref{eq:liminf_f_b} and \eqref{eq:liminf_f_s}.
\end{proof}
\begin{prop}\label{prop:gamma_conv_sup}
Suppose that the assumptions ($A_1$)-($A_7$) are verified. Then, for any $Q\in H^1_g(\Omega,\mathcal{S}_0)$, there exists a sequence $\big(Q_{\varepsilon}\big)_{\varepsilon>0}$ such that $Q_{\varepsilon}\in H^1(\Omega_{\varepsilon})$, for any $\varepsilon>0$, $E_{\varepsilon}Q_{\varepsilon}\rightharpoonup Q$ in $H^1(\Omega)$ and:
\begin{align*}
\limsup_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}[Q_{\varepsilon}]&\leq \mathcal{F}_0[Q].
\end{align*}
The sequence $\big(Q_{\varepsilon}\big)_{\varepsilon>0}$ is called a recovery sequence.
\end{prop}
\begin{proof}
Let us define in this case $Q_{\varepsilon}=Q\cdot\chi_{\Omega_{\varepsilon}}$. Since $|\mathcal{N}_{\varepsilon}|\rightarrow 0$ as $\varepsilon\rightarrow 0$ (according to \cref{prop:volumegoesto0}), then $\chi_{\Omega_{\varepsilon}}\rightarrow 1$ strongly in $L^1(\Omega)$ and we can apply Lebesgue's dominated converge theorem in order to obtain that:
\begin{align*}
\lim_{\varepsilon\rightarrow 0}\int_{\Omega_{\varepsilon}}f_e(\nabla Q_{\varepsilon})+f_b(Q_{\varepsilon})\text{d}x &=\int_{\Omega}f_e(\nabla Q)+f_b(Q)\text{d}x\\
\lim_{\varepsilon\rightarrow 0}\big(\mathcal{F}_{\varepsilon}[Q]-J_{\varepsilon}^{\mathcal{T}}[Q]\big)&=\mathcal{F}_0[Q]-J_0[Q].
\end{align*}
By \cref{prop:gamma_conv_inf}, we have that $\displaystyle{\lim_{\varepsilon\rightarrow 0}J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]=J_0[Q]}$, hence the conclusion follows.
\end{proof}
\cref{prop:gamma_conv_inf} and \cref{prop:gamma_conv_sup} show that $\mathcal{F}_{\varepsilon}$ $\Gamma$-converges to $\mathcal{F}_0$, as $\varepsilon\rightarrow 0$, with respect to the weak $H^1$ topology.
\subsection{Proof of main theorems}
\begin{proof}[Proof of \cref{th:local_min}]
Let $Q_0$ from $H^1_g(\Omega,\mathcal{S}_0)$ be an isolated $H^1$-local minimiser for $\mathcal{F}_0$, that is, there exists $\delta_0 >0$ such that $\mathcal{F}_0[Q_0]<\mathcal{F}_0[Q]$, for any $Q\in H^1_g(\Omega,\mathcal{S}_0)$, such that $0<\big\|Q-Q_0\big\|_{H^1(\Omega)}\leq \delta_0$.
We would like to prove that for any $\varepsilon>0$, there exists $Q_{\varepsilon}\in H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)$, which is a $H^1$-local minimiser for $\mathcal{F}_{\varepsilon}$, such that $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q_0$ strongly in $H^1_g(\Omega,\mathcal{S}_0)$ as $\varepsilon\rightarrow 0$.
For this, let
\begin{align*}
\mathcal{B}_{\varepsilon}:=\big\{Q\in H^1_g(\Omega_{\varepsilon},\mathcal{S}_0)\;:\:\big\|E_{\varepsilon}Q-Q_0\big\|_{H^1(\Omega)}\leq \delta_0\big\}.
\end{align*}
Using Mazur's lemma, we can show that the set $\mathcal{B}_{\varepsilon}$ is sequentially weakly closed in $H^1(\Omega_{\varepsilon})$. Then, by \cref{prop:lsc}, we can see that, for $\varepsilon$ small enough, $\mathcal{F}_{\varepsilon}$ is lower semicontinuous on $\mathcal{B}_{\varepsilon}$ and, by \cref{prop:equicoercivity}, is also coercive on $\mathcal{B}_{\varepsilon}$, since any $Q\in\mathcal{B}_{\varepsilon}$ has $\big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}<\big\|\nabla Q_0\big\|_{L^2(\Omega)}+\delta_0$. Hence, for any $\varepsilon$ sufficiently small, the functional $\mathcal{F}_{\varepsilon}$ admits at least one minimiser $Q_{\varepsilon}$ from $\mathcal{B}_{\varepsilon}$.
Firstly, we prove that $E_{\varepsilon}Q_{\varepsilon}\rightharpoonup Q_0$ weakly in $H^1(\Omega)$, as $\varepsilon\rightarrow 0$.
Let $\mathcal{B}_0:=\big\{Q\in H^1_g(\Omega,\mathcal{S}_0)\;:\;\big\|Q-Q_0\big\|_{H^1(\Omega)}\leq\delta_0\big\}$. Because $Q_{\varepsilon}\in\mathcal{B}_{\varepsilon}$, then $(E_{\varepsilon}Q_{\varepsilon})_{\varepsilon>0}$ represents a bounded sequence in $H^1(\Omega)$, hence there exists a subsequence, which we still denote $(E_{\varepsilon}Q_{\varepsilon})_{\varepsilon>0}$ for the ease of notation, that converges weakly to a $\tilde{Q}\in \mathcal{B}_0$. We show that $\tilde{Q}=Q_0$.
Since $E_{\varepsilon}Q_{\varepsilon}\rightharpoonup\tilde{Q}$ in $H^1_g(\Omega,\mathcal{S}_0)$, we can apply \cref{prop:gamma_conv_inf} and get:
\begin{align*}
\mathcal{F}_0[\tilde{Q}]&\leq\liminf_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}[Q_{\varepsilon}]\leq\limsup_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}[Q_{\varepsilon}].
\end{align*}
But $Q_{\varepsilon}$ is a minimiser of $\mathcal{F}_{\varepsilon}$ on $\mathcal{B}_{\varepsilon}$, therefore, since $Q_0\big|_{\Omega_{\varepsilon}}\in\mathcal{B}_{\varepsilon}$, we get that
\begin{align*}
\limsup_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}[Q_{\varepsilon}]&\leq \lim_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}[Q_0]=\lim_{\varepsilon\rightarrow 0}\bigg(\int_{\Omega_{\varepsilon}}f_e(\nabla Q_0)+f_b(Q)\text{d}x+\mathcal{J}_{\varepsilon}^{\mathcal{T}}[Q_0]\bigg)=\mathcal{F}_0[Q_0].
\end{align*}
Hence, we have $\mathcal{F}_0[\tilde{Q}]\leq\mathcal{F}_0[Q_0]$. Because $\tilde{Q}$ is in $\mathcal{B}_0$, that is $\|\tilde{Q}-Q_0\|_{H^1(\Omega)}\leq\delta_0$, then by the definition of $Q_0$, we get that $\tilde{Q}=Q_0$.
We now prove that $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q_0$ strongly in $H^1(\Omega)$, as $\varepsilon\rightarrow 0$.
By ($A_5$), there exists $\theta >0$ such that the function $\tilde{f_e}(D)=f_e(D)-\theta|D|^2$ is convex. We can repeat the same arguments from \cref{prop:gamma_conv_inf}, more specifically, steps \eqref{eq:liminf_f_e} and \eqref{eq:liminf_f_b}, to get:
\begin{align*}
\liminf_{\varepsilon\rightarrow 0}\int_{\Omega_{\varepsilon}}\tilde{f_e}(\nabla Q_{\varepsilon})\text{d}x &\geq \int_{\Omega}\tilde{f_e}(\nabla Q_0)\text{d}x,\\
\theta\liminf_{\varepsilon\rightarrow 0}\int_{\Omega_{\varepsilon}}|\nabla Q_{\varepsilon}|^2\text{d}x &\geq \theta\int_{\Omega}|\nabla Q_0|^2\text{d}x,\\
\liminf_{\varepsilon\rightarrow 0}\int_{\Omega_{\varepsilon}}f_b(Q_{\varepsilon})\text{d}x &\geq \int_{\Omega}f_b(Q_0)\text{d}x.
\end{align*}
From \cref{prop:gamma_conv_inf}, we have that $J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]\rightarrow J_0[Q_0]$ as $\varepsilon\rightarrow 0$. Also, from the proof that $\tilde{Q}=Q_0$, we can see that
\begin{align*}
\lim_{\varepsilon\rightarrow 0}\mathcal{F}_{\varepsilon}[Q_{\varepsilon}]=\mathcal{F}_0[Q_0],
\end{align*}
which implies that
\begin{align*}
\lim_{\varepsilon\rightarrow 0}\int_{\Omega_{\varepsilon}}|\nabla Q_{\varepsilon}|^2\text{d}x=\int_{\Omega}|\nabla Q_0|^2\text{d}x.
\end{align*}
This shows us that $\nabla (E_{\varepsilon}Q_{\varepsilon})\chi_{\Omega_{\varepsilon}}$ converges strongly to $\nabla Q_0$ in $L^2(\Omega)$, where $\chi_{\Omega_{\varepsilon}}$ is the characteristic function of $\Omega_{\varepsilon}$. We now show that $\nabla (E_{\varepsilon}Q_{\varepsilon})\chi_{\mathcal{N}_{\varepsilon}}$ converges strongly to $0$ in $L^2(\Omega)$. We have:
\begin{align*}
\big\|\nabla(E_{\varepsilon}Q_{\varepsilon})\chi_{\mathcal{N}_{\varepsilon}}\big\|_{L^2(\Omega)} &=\bigg(\int_{\Omega}|\nabla E_{\varepsilon}Q_{\varepsilon}|^2\cdot |\chi_{\mathcal{N}_{\varepsilon}}|^2\text{d}x\bigg)^{1/2}\leq \bigg(\int_{\Omega}|\nabla E_{\varepsilon}Q_{\varepsilon}|^2\text{d}x\bigg)^{1/2}\cdot \big\|\chi_{\mathcal{N}_{\varepsilon}}\big\|_{L^{\infty}(\Omega)}\\
&\leq \big\|\nabla E_{\varepsilon}Q_{\varepsilon}\big\|_{L^2(\Omega)}\cdot\big\|\chi_{\mathcal{N}_{\varepsilon}}\big\|_{L^{\infty}(\Omega)}\leq C\cdot \big\|\nabla Q_{\varepsilon}\big\|_{L^2(\Omega_{\varepsilon})}\cdot\big\|\chi_{\mathcal{N}_{\varepsilon}}\big\|_{L^{\infty}(\Omega)}\\
&\leq C\cdot\big(\big\|\nabla Q_0\big\|_{L^2(\Omega)}+\delta_0\big)\cdot\big\|\chi_{\mathcal{N}_{\varepsilon}}\big\|_{L^{\infty}(\Omega)},
\end{align*}
where we have used \cref{lemma:extensionineq} and the fact that $Q_{\varepsilon}\in\mathcal{B}_{\varepsilon}$. According to \cref{prop:volumegoesto0}, we have $|\mathcal{N}_{\varepsilon}|\rightarrow 0$, therefore we have $\nabla (E_{\varepsilon}Q_{\varepsilon})\chi_{\mathcal{N}_{\varepsilon}}\rightarrow 0$ strongly in $L^2(\Omega)$.
Combining all the results, we obtain that $\nabla E_{\varepsilon}Q_{\varepsilon}\rightarrow \nabla Q_0$ strongly in $L^2(\Omega)$, hence $E_{\varepsilon}Q_{\varepsilon}$ converges strongly to $Q_0$ in $H^1(\Omega)$, since the weak convergence $E_{\varepsilon}Q_{\varepsilon}\rightharpoonup Q_0$ in $H^1(\Omega)$ automatically implies the strong convergence $E_{\varepsilon}Q_{\varepsilon}\rightarrow Q_0$ in $L^2(\Omega)$.
\end{proof}
\section{Rate of convergence}\label{section:rate_of_conv}
The aim of this section is the study the rate of convergence of the sequence $J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]$ to $J_0[Q_0]$, where $J_{\varepsilon}^{\mathcal{T}}$ is defined in \eqref{defn:Jeps_T} and in \eqref{defn:Jeps_T_X_Y_Z}, $J_0$ is defined in \eqref{defn:J_0} and $(Q_{\varepsilon})_{\varepsilon>0}$ is a sequence from $H^1_g(\Omega,\mathcal{S}_0)$ that converges $H^1$-strongly to $Q\in H^1_g(\Omega,\mathcal{S}_0)$. We omit the term $J_{\varepsilon}^{\mathcal{S}}$ because in \cref{subsection:nocontribution} we proved that this term has no contribution to the homogenised functional.
First, we recall some notations used in the previous sections. For a $Q\in H^1_g(\Omega,\mathcal{S}_0)$, we write $J_0[Q]$ in the following form:
\begin{align}\label{defn:J_0_new_form}
J_0[Q]&=\int_{\Omega}\bigg(\dfrac{1}{r}\Psi^{Y}(Q)+\dfrac{1}{q}\Psi^{Z}(Q)\bigg)\text{d}x+\notag\\
&+\int_{\Omega}\bigg(\dfrac{1}{r}\Psi^{X}(Q)+\dfrac{1}{p}\Psi^{Z}(Q)\bigg)\text{d}x+\notag\\
&+\int_{\Omega}\bigg(\dfrac{1}{q}\Psi^{X}(Q)+\dfrac{1}{p}\Psi^{Y}(Q)\bigg)\text{d}x,
\end{align}
where $\Psi^{X}$, $\Psi^{Y}$ and $\Psi^{Z}$ are defined in \eqref{defn:PSI}. We also write $\tilde{J}_{\varepsilon}[Q]$, defined in \eqref{defn:J_tilde_eps}, as:
\begin{align}\label{defn:J_tilde_eps_new_form}
\tilde{J}_{\varepsilon}[Q]&=\int_{\Omega}\bigg(\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\Psi^{Y}(Q)+\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pq(\varepsilon-\varepsilon^{\alpha})}\Psi^{Z}(Q)\bigg)\text{d}\mu_{\varepsilon}^{X}+\notag\\
&+\int_{\Omega}\bigg(\dfrac{q\varepsilon-\varepsilon^{\alpha}}{qr(\varepsilon-\varepsilon^{\alpha})}\Psi^{X}(Q)+\dfrac{q\varepsilon-\varepsilon^{\alpha}}{pq(\varepsilon-\varepsilon^{\alpha})}\Psi^{Z}(Q)\bigg)\text{d}\mu_{\varepsilon}^{Y}+\notag\\
&+\int_{\Omega}\bigg(\dfrac{r\varepsilon-\varepsilon^{\alpha}}{qr(\varepsilon-\varepsilon^{\alpha})}\Psi^{X}(Q)+\dfrac{r\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\Psi^{Y}(Q)\bigg)\text{d}\mu_{\varepsilon}^{Z},
\end{align}
using \eqref{defn:J_tilde_eps_nice_form} and the analogous formulae.
We suppose now that:
\begin{itemize}
\item[($H_1$)] the \textit{surface energy density} $f_s$ is locally Lipschitz continuous.
\end{itemize}
Using the assumption ($A_7$), from \cref{section:assumptions}, we have:
\begin{align}\label{eq:properties_f_s_1}
|f_s(Q_1,\nu)-f_s(Q_2,\nu)|&\lesssim |Q_2-Q_1|(|Q_1|^3+|Q_2|^3+1),
\end{align}
for any $Q_1,Q_2\in\mathcal{S}_0$ and any $\nu\in\mathbb{S}^2$, and
\begin{align}\label{eq:properties_f_s_2}
|f_s(Q,\nu)|&\lesssim |Q|^4+1,
\end{align}
for any $Q\in\mathcal{S}_0$ and any $\nu\in\mathbb{S}^2$.
We now have the following lemma:
\begin{lemma}\label{lemma:control_Psi}
For any $K\in\{X,Y,Z\}$, the function $\Psi^{K}$ is locally Lipschitz continuous and there holds:
\begin{align*}
|\Psi^{K}(Q)|\lesssim |Q|^4+1\hspace{10mm} \big|\nabla\Psi^{K}(Q)\big|\lesssim |Q|^3+1,
\end{align*}
for any $Q\in\mathcal{S}_0$. Moreover, the function $\Psi^K$ satisfies:
\begin{align*}
|\Psi^K(Q_1)-\Psi^K(Q_2)|\lesssim |Q_2-Q_1|(|Q_1|^3+|Q_2|^3+1),
\end{align*}
for any $Q_1,Q_2\in\mathcal{S}_0$.
\end{lemma}
\begin{proof}
The proof of this lemma follows immediatly, using the definitions of the functions $\Psi^X$, $\Psi^Y$ and $\Psi^Z$ from \eqref{defn:PSI}, the assumption ($H_1$) and the properties of the function $f_s$ from \eqref{eq:properties_f_s_1} and \eqref{eq:properties_f_s_2}.
\end{proof}
We recall now that the measures $\mu_{\varepsilon}^{X}$, $\mu_{\varepsilon}^Y$ and $\mu_{\varepsilon}^Z$, which are defined in \eqref{defn:measures}, converge weakly*, as measures in $\mathbb{R}^3$, to the Lebesgue measure restricted to $\Omega$, according to ($A_4$) from \cref{section:assumptions}. We need to prescribe a rate of convergence and for this we use the $W^{-1,1}$-norm (that is, the dual Lipschitz norm, also known as flat norm in some contexts):
\begin{align*}
\mathbb{F}_{\varepsilon}:=\max_{K\in\{X,Y,Z\}}\sup\bigg\{\int_{\Omega}\varphi\text{d}\mu_{\varepsilon}^{K}-\int_{\Omega}\varphi\text{d}x\;:\;\varphi\in W^{1,\infty}(\Omega), \|\nabla\varphi\|_{L^{\infty}(\Omega)}+\|\varphi\|_{L^{\infty}(\Omega)}\leq 1\bigg\}.
\end{align*}
\begin{lemma}\label{lemma:Fbb_eps}
There exists a constant $\lambda_{\text{flat}}>0$ such that $\mathbb{F}_{\varepsilon}\leq \lambda_{\text{flat}}\varepsilon$ for any $\varepsilon>0$.
\end{lemma}
\begin{proof}
Let $\varphi\in W^{1,\infty}(\Omega)$. Then, according to the definition of $\mu_{\varepsilon}^{X}$ from \eqref{defn:measures}, we have:
\begin{align*}
\int_{\Omega}\varphi\text{d}\mu_{\varepsilon}^X=\varepsilon^{3}\sum_{k=1}^{X_{\varepsilon}}\varphi(y_{\varepsilon}^{x,k})=\sum_{k=1}^{X_{\varepsilon}}\int_{y_{\varepsilon}^{x,k}+[-\varepsilon/2,\varepsilon/2]^3}\varphi(y_{\varepsilon}^{x,k})\text{d}x,
\end{align*}
where in the last equality we integrate over the cube with length $\varepsilon$ centered in $y_{\varepsilon}^{x,k}$. Let $\Omega_{\varepsilon}^X$ be the following set:
\begin{align*}
\Omega_{\varepsilon}^{X}:=\bigcup_{k=1}^{X_{\varepsilon}}\big(y_{\varepsilon}^{x,k}+[-\varepsilon/2,\varepsilon/2]^3\big).
\end{align*}
Hence, we can write:
\begin{align*}
\int_{\Omega}\varphi\text{d}\mu_{\varepsilon}^{X}=\int_{\Omega_{\varepsilon}^X}\varphi(y_{\varepsilon}^{x,k})\text{d}x.
\end{align*}
Then:
\begin{align}
\bigg|\int_{\Omega}\varphi\text{d}\mu_{\varepsilon}^X-\int_{\Omega}\varphi\text{d}x\bigg|&\leq \int_{\Omega_{\varepsilon}^X}\big|\varphi-\varphi(y_{\varepsilon}^{x,k})\big|\text{d}x+\int_{\Omega\setminus\Omega_{\varepsilon}^X}\big|\varphi\big|\text{d}x\notag\\
&\leq \dfrac{\varepsilon\sqrt{3}}{2}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\cdot |\Omega_{\varepsilon}^X|+\|\varphi\|_{L^{\infty}(\Omega)}\cdot |\Omega\setminus\Omega_{\varepsilon}^{X}|\label{eq:diff_of_varphi},
\end{align}
where $\dfrac{\varepsilon\sqrt{3}}{2}$ comes from the largest possible value for $|x-y_{\varepsilon}^{x,k}|$, with $x\in\big(y_{\varepsilon}^{x,k}+[-\varepsilon/2,\varepsilon/2]^3\big)$.
If we look now at the definition of the points $y_{\varepsilon}^{x,k}$ in \eqref{defn:y_eps_x_k}, hence also at the definition of the points $x_{\varepsilon}^{i}$ in \eqref{defn:points1} and \eqref{defn:points2}, we observe that $\Omega\setminus\Omega_{\varepsilon}^{X}\subset\{x\in\Omega\;:\;\text{dist}(x,\partial\Omega)<\varepsilon\}$, therefore, we have $|\Omega\setminus\Omega_{\varepsilon}^X|\leq C\cdot\varepsilon$, where $C$ is an $\varepsilon$-independent constant. At the same time, we have $\Omega_{\varepsilon}^X\subset\Omega\Rightarrow |\Omega_{\varepsilon}^X|\leq |\Omega|$, so \eqref{eq:diff_of_varphi} becomes:
\begin{align*}
\bigg|\int_{\Omega}\varphi\text{d}\mu_{\varepsilon}^X-\int_{\Omega}\varphi\text{d}x\bigg|&\leq \dfrac{\varepsilon\sqrt{3}}{2}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\cdot|\Omega|+C\cdot\varepsilon\|\varphi\|_{L^{\infty}(\Omega)}\lesssim \varepsilon\cdot\|\varphi\|_{W^{1,\infty}(\Omega)}.
\end{align*}
Computing in the same fashion for $\mu_{\varepsilon}^Y$ and $\mu_{\varepsilon}^Z$, we obtain the conclusion.
\end{proof}
We also suppose that:
\begin{itemize}
\item[($H_2$)] $g$ is bounded and Lipschitz, where $g$ represents the prescribed boundary data.
\end{itemize}
Since $\Omega$ is bounded and smooth (by assumption ($A_1$) from \cref{section:assumptions}), we can extend the function $g$ to a bounded and Lipschitz map from $\mathbb{R}^3$ to $\mathcal{S}_0$, denoted still as $g$.
We present an auxiliary result proved in \cite{CanevariZarnescu2}:
\begin{lemma}\label{lemma:Q_eps_beta}
Let $\Omega\subseteq\mathbb{R}^3$ a bounded, smooth domain, and let $g:\Omega\rightarrow\mathcal{S}_0$ be a bounded, Lipschitz map. For any $Q\in H^1_g(\Omega,\mathcal{S}_0)$ and $\sigma\in(0,1)$, there exists a bounded, Lipschitz map $Q_{\sigma}:\overline{\Omega}\rightarrow\mathcal{S}_0$ that satisfies the following properties:
\begin{equation*}
Q_{\sigma}=g\hspace{3mm}\text{on}\;\partial\Omega
\end{equation*}
\begin{equation}\label{eq:Q_sigma}
\|Q_{\sigma}\|_{L^{\infty}(\Omega)}\lesssim \sigma^{-1/2}\big(\|Q\|_{H^1(\Omega)}+\|g\|_{L^{\infty}(\Omega)}\big)
\end{equation}
\begin{equation}\label{eq:nabla_Q_sigma}
\|\nabla Q_{\sigma}\|_{L^{\infty}(\Omega)}\lesssim \sigma^{-3/2}\big(\|Q\|_{H^1(\Omega)}+\|g\|_{W^{1,\infty}(\Omega)}\big)
\end{equation}
\begin{equation}\label{eq:Q-Q_sigma_L^2}
\|Q-Q_{\sigma}\|_{L^2(\Omega)}\lesssim \sigma\|Q\|_{H^1(\Omega)}
\end{equation}
\begin{equation}\label{eq:nabla_Q-Q_sigma_L^2}
\|\nabla Q-\nabla Q_{\sigma}\|_{L^2(\Omega)}\rightarrow 0\hspace{3mm}\text{as}\;\sigma\rightarrow 0.
\end{equation}
\end{lemma}
The main result from this section is the following:
\begin{prop}\label{prop:rate_of_conv}
Suppose that assumptions ($A_1$)-($A_7$) (from \cref{section:assumptions}) and ($H_1$)-($H_2$) (from this section) hold. Then, for any $Q\in H^1_g(\Omega,\mathcal{S}_0)$, there exists a sequence $(Q_{\varepsilon})_{\varepsilon>0}$ in $H^1_g(\Omega,\mathcal{S}_0)$ that converges $H^1(\Omega)$-strongly to $Q$ and satisfies
\begin{equation*}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-J_{0}[Q]|\lesssim \varepsilon^{(\alpha-1)/3}\big(\|Q\|^4_{H^1(\Omega)}+1\big),
\end{equation*}
for $\varepsilon$ small enough, where $J_{\varepsilon}^{\mathcal{T}}$ is defined in \eqref{defn:Jeps_T} and $J_0$ is defined in \eqref{defn:J_0_new_form}. The constant that merged with the sign ``$\lesssim$" depends only on the $L^{\infty}$-norms of $g$ and $\nabla g$, but also on $\Omega$, $f_s$ and the initial cube $\mathcal{C}$.
\end{prop}
\begin{remark}\label{remark:rate_of_convergence}
The previous proposition allows us to obtain, as claimed, a rate of convergence for the minimisers $\bar Q_\varepsilon$ of $\mathcal{F}_\varepsilon$ to a minimiser $Q$ of $\mathcal{F}_0$ in terms of $\|\bar Q_\varepsilon-Q\|_{H^1(\Omega)}=o(1)$ as $\varepsilon\to 0$ (i.e. relation \eqref{eq:rate_of_convergence}).
Indeed, this is obtained in the following way. First, let us fix a value for $0<\varepsilon<1$ such that equation \eqref{eq:alpha/4} holds. Then we use the inequality
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[\bar Q_{\varepsilon}]-J_0[Q]|&\leq |J_{\varepsilon}^{\mathcal{T}}[\bar Q_{\varepsilon}]-J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]|+|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-J_0[Q]|,
\end{align*}
where $Q_{\varepsilon}$ is the function from $H^1_g(\Omega,\mathcal{S}_0)$ granted by \cref{lemma:Q_eps_beta}, with $\sigma=\varepsilon^{(\alpha-1)/3}$.
For the first term from the right-hand side from the last inequality, we use relation \eqref{eq:alpha/4} and we obtain, for a fixed $\varepsilon$ sufficiently small:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[\bar Q_{\varepsilon}]-J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]|&\leq C\cdot\big(\varepsilon^{1/2-\alpha/4}+\|\bar Q_{\varepsilon}-Q_{\varepsilon}\|_{L^4(\Omega)}\big),
\end{align*}
where $C$ is $\varepsilon$-independent.
From the compact Sobolev embedding $H^1(\Omega)\hookrightarrow L^4(\Omega)$, we obtain:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[\bar Q_{\varepsilon}]-J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]|&\leq C\cdot\big(\varepsilon^{1/2-\alpha/4}+\|\bar Q_{\varepsilon}-Q_{\varepsilon}\|_{H^1(\Omega)}\big).
\end{align*}
Now, we observe that:
\begin{align*}
\|\bar Q_{\varepsilon}-Q_{\varepsilon}\|_{H^1(\Omega)}&\leq \|\bar Q_{\varepsilon}-Q\|_{H^1(\Omega)}+\|Q_{\varepsilon}-Q\|_{H^1(\Omega)}\\
&\leq \|\bar Q_{\varepsilon}-Q\|_{H^1(\Omega)}+\|Q_{\varepsilon}-Q\|_{L^2(\Omega)}+\|\nabla Q_{\varepsilon}-\nabla Q\|_{L^2(\Omega)}\\
&\leq \|\bar Q_{\varepsilon}-Q\|_{H^1(\Omega)}+\varepsilon^{(\alpha-1)/3}\|Q\|_{H^1(\Omega)}+\|\nabla Q_{\varepsilon}-\nabla Q\|_{L^2(\Omega)},
\end{align*}
where we have used relation \eqref{eq:Q-Q_sigma_L^2} in the last row. Relation \eqref{eq:nabla_Q-Q_sigma_L^2} tells us that $\|\nabla Q_{\varepsilon}-\nabla Q\|_{L^2(\Omega)}\rightarrow 0$ as $\varepsilon\rightarrow 0$, hence, by the choice of $\varepsilon$, we can control it with a constant. Since $Q$ is fixed, we can also control $\|Q\|_{H^1(\Omega)}$ with an $\varepsilon$-independent constant. Therefore, we can write:
\begin{align*}
\|\bar Q_{\varepsilon}-Q_{\varepsilon}\|_{H^1(\Omega)}&\lesssim \|\bar Q_{\varepsilon}-Q\|_{H^1(\Omega)}+\varepsilon^{(\alpha-1)/3}.
\end{align*}
Hence, we have:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[\bar Q_{\varepsilon}]-J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]|&\lesssim \varepsilon^{1/2-\alpha/4}+\varepsilon^{(\alpha-1)/3}+\|\bar Q_{\varepsilon}-Q\|_{H^1(\Omega)}.
\end{align*}
and if we denote by $m_{\alpha}=\text{min}\{1/2-\alpha/4,(\alpha-1)/3\}$ (which is defined depending whether $\alpha$ is bigger or smaller than $10/7$), we can rewrite:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[\bar Q_{\varepsilon}]-J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]|&\lesssim \varepsilon^{m_{\alpha}}+\|\bar Q_{\varepsilon}-Q\|_{H^1(\Omega)},
\end{align*}
since $\varepsilon$ is chosen from $(0,1)$.
For the term $|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-J_0[Q]|$, we apply \cref{prop:rate_of_conv}:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-J_0[Q]|&\lesssim \varepsilon^{(\alpha-1)/3}(\|Q\|^4_{H^1(\Omega)}+1)
\end{align*}
and since $Q$ is fixed, we obtain:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-J_0[Q]|&\lesssim \varepsilon^{(\alpha-1)/3}\lesssim \varepsilon^{m_{\alpha}},
\end{align*}
using the definition of $m_{\alpha}$.
If we go back to our initial inequality, we obtain:
\begin{align}\label{eq:rate_of_convergence}
|J_{\varepsilon}^{\mathcal{T}}[\bar Q_{\varepsilon}]-J_0[Q]|&\lesssim \varepsilon^{\text{min}\{1/2-\alpha/4,(\alpha-1)/3\}}+\|\bar Q_{\varepsilon}-Q\|_{H^1(\Omega)}.
\end{align}
\end{remark}
\begin{proof}[Proof of \cref{prop:rate_of_conv}]
Let us fix a small $\varepsilon\in(0,1)$ such that:
\begin{align}\label{eq:choice_of_eps_2p}
\dfrac{p\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}<2p,\;\dfrac{q\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}<2q\;\text{and}\;\dfrac{r\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}<2r.
\end{align}
This is possible since $\dfrac{p\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}\searrow p$, $\dfrac{q\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}\searrow q$ and $\dfrac{r\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}\searrow r$ as $\varepsilon\rightarrow 0$ and $p,q,r\geq 1$.
Let now $\beta$ be a positive parameter, to be chosen later, and let $Q_{\varepsilon}:=Q_{\varepsilon^{\beta}}\in H^1_g(\Omega,\mathcal{S}_0)$ be the Lipschitz map given by \cref{lemma:Q_eps_beta}. Then, we have:
\begin{align}\label{eq:control_J_eps_Q_eps_J_0_Q}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-J_0[Q]|\leq |J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-\tilde{J}_{\varepsilon}[Q_{\varepsilon}]|+|\tilde{J}_{\varepsilon}[Q_{\varepsilon}]-J_0[Q_{\varepsilon}]|+|J_0[Q_{\varepsilon}]-J_0[Q]|,
\end{align}
where $\tilde{J}_{\varepsilon}$ is defined in \eqref{defn:J_tilde_eps}.
We analyse the first term from the right-hand side from \eqref{eq:control_J_eps_Q_eps_J_0_Q}. Using the same notations as in \cref{th:Je_goes_to_J0}, replacing $\text{Lip}(Q_{\varepsilon})$ (the Lipschitz constant) with $\|\nabla Q_{\varepsilon}\|_{L^{\infty}(\Omega)}$ and combining relations \eqref{eq:J_eps_J_eps_tilde_control1} and \eqref{eq:J_eps_J_eps_tilde_control2}, we obtain:
\begin{align*}
|J_{\varepsilon}^X[Q_{\varepsilon}]-\tilde{J}^X_{\varepsilon}[Q_{\varepsilon}]|&\lesssim \big(\|Q_{\varepsilon}\|^3_{L^{\infty}(\Omega)}+1\big)\cdot\|\nabla Q_{\varepsilon}\|_{L^{\infty}(\Omega)}\cdot\dfrac{2(r+q)}{pqr}\cdot\dfrac{p\varepsilon-\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}\cdot\\
&\hspace{2mm}\cdot X_{\varepsilon}\cdot \varepsilon^{3}\cdot \sqrt{\bigg(\varepsilon-\dfrac{\varepsilon^{\alpha}}{p}\bigg)^2+\bigg(\dfrac{\varepsilon^{\alpha}}{q}\bigg)^2+\bigg(\dfrac{\varepsilon^{\alpha}}{r}\bigg)^2}.
\end{align*}
Using \eqref{eq:control_over_X_eps} and \eqref{eq:choice_of_eps_2p}, we can rewrite the last inequality as follows:
\begin{align}\label{eq:control_J_eps_X_J_tilde_eps_X_sqrt}
|J_{\varepsilon}^X[Q_{\varepsilon}]-\tilde{J}^X_{\varepsilon}[Q_{\varepsilon}]|&\lesssim \big(\|Q_{\varepsilon}\|^3_{L^{\infty}(\Omega)}+1\big)\cdot\|\nabla Q_{\varepsilon}\|_{L^{\infty}(\Omega)}\cdot\sqrt{\bigg(\varepsilon-\dfrac{\varepsilon^{\alpha}}{p}\bigg)^2+\bigg(\dfrac{\varepsilon^{\alpha}}{q}\bigg)^2+\bigg(\dfrac{\varepsilon^{\alpha}}{r}\bigg)^2},
\end{align}
since the term $\dfrac{2(r+q)}{pqr}$ can be bounded with an $\varepsilon$-independent constant. Now, because $p,q,r\geq 1$, we have:
\begin{align*}
\dfrac{\varepsilon^{2\alpha}}{p^2},\dfrac{\varepsilon^{2\alpha}}{q^2},\dfrac{\varepsilon^{2\alpha}}{r^2}\leq \varepsilon^{2\alpha}
\end{align*}
and, because $\varepsilon>0$ and $\alpha\in(1,3/2)$, we also have:
\begin{align*}
0<\varepsilon-\varepsilon^{\alpha}\leq \varepsilon-\dfrac{\varepsilon^{\alpha}}{k}\leq \varepsilon,\;\text{for}\;k\in\{p,q,r\}.
\end{align*}
Therefore, \eqref{eq:control_J_eps_X_J_tilde_eps_X_sqrt} becomes:
\begin{align*}
|J_{\varepsilon}^X[Q_{\varepsilon}]-\tilde{J}^X_{\varepsilon}[Q_{\varepsilon}]|&\lesssim \big(\|Q_{\varepsilon}\|^3_{L^{\infty}(\Omega)}+1\big)\cdot\|\nabla Q_{\varepsilon}\|_{L^{\infty}(\Omega)}\cdot\sqrt{\varepsilon^2+2\varepsilon^{2\alpha}},
\end{align*}
and using the same arguments for $J_{\varepsilon}^Y[Q_{\varepsilon}]$ and $J_{\varepsilon}^Z[Q_{\varepsilon}]$, we obtain:
\begin{align}\label{eq:control_J_eps_J_tilde_no_sqrt}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-\tilde{J}_{\varepsilon}[Q_{\varepsilon}]|&\lesssim \big(\|Q_{\varepsilon}\|^3_{L^{\infty}(\Omega)}+1\big)\cdot\|\nabla Q_{\varepsilon}\|_{L^{\infty}(\Omega)}\cdot\sqrt{\varepsilon^2+2\varepsilon^{2\alpha}},
\end{align}
where the constant 3, which comes from adding the three relations obtained, has merged into the ``$\lesssim$" sign.
Using \cref{lemma:Q_eps_beta}, we have:
\begin{align*}
\|Q_{\varepsilon}\|^3_{L^{\infty}(\Omega)}&\lesssim \varepsilon^{-3\beta/2}\big(\|Q\|_{H^1(\Omega)}+\|g\|_{L^{\infty}(\Omega)}\big)^3\\
\|\nabla Q_{\varepsilon}\|_{L^{\infty}(\Omega)}&\lesssim \varepsilon^{-3\beta/2}\big(\|Q\|_{H^1(\Omega)}+\|g\|_{W^{1,\infty}(\Omega)}\big).
\end{align*}
Now, the constant involved by using the sign ``$\lesssim$" is going to depend also on the $L^{\infty}$-norms of $g$ and $\nabla g$, hence, relation \eqref{eq:control_J_eps_J_tilde_no_sqrt} becomes:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-\tilde{J}_{\varepsilon}[Q_{\varepsilon}]|&\lesssim \dfrac{\sqrt{\varepsilon^2+\varepsilon^{2\alpha}}}{\varepsilon^{3\beta}}\big(\|Q\|^4_{H^1(\Omega)}+1\big)\lesssim \sqrt{\varepsilon^{2(1-3\beta)}+\varepsilon^{2(\alpha-3\beta)}}\big(\|Q\|^4_{H^1(\Omega)}+1\big).
\end{align*}
Since $\alpha\in(1,3/2)$, we have $1-3\beta<\alpha-3\beta$. Therefore, we can write the last inequality as follows:
\begin{align}\label{eq:first_control_J_eps_J_0_Q_eps}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-\tilde{J}_{\varepsilon}[Q_{\varepsilon}]|&\lesssim\varepsilon^{1-3\beta}\big(\|Q\|^4_{H^1(\Omega)}+1\big),
\end{align}
since $\varepsilon\in(0,1)$.
In order to analyse better the second term from \eqref{eq:control_J_eps_Q_eps_J_0_Q}, which contains $\tilde{J}_{\varepsilon}[Q_{\varepsilon}]$ and $J_0[Q_{\varepsilon}]$, we analyse the first terms from \eqref{defn:J_0_new_form} and \eqref{defn:J_tilde_eps_new_form}:
\begin{align*}
\bigg|\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}\mu_{\varepsilon}^{X}-\dfrac{1}{r}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|&\leq \dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\bigg|\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}\mu_{\varepsilon}^{X}-\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|+\\
&\hspace{2mm}+\bigg|\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}-\dfrac{1}{r}\bigg|\cdot\bigg|\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|.
\end{align*}
As we have seen before, we have $\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\searrow\dfrac{1}{r}$ and we have chosen $\varepsilon>0$ such that $\dfrac{1}{r}\leq \dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}<\dfrac{2}{r}$. Moreover, we have $\bigg|\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}-\dfrac{1}{r}\bigg|=\dfrac{\varepsilon^{\alpha}(p-1)}{pr(\varepsilon-\varepsilon^{\alpha})}$ and we can impose further conditions regarding the choice of $\varepsilon$, such that $\dfrac{\varepsilon^{\alpha}(p-1)}{pr(\varepsilon-\varepsilon^{\alpha})}<\varepsilon^{\alpha-1}$, which is equivalent to choosing $\varepsilon$ such that $\varepsilon^{\alpha-1}< 1-\dfrac{1}{r}+\dfrac{1}{pr}$. Hence, we have:
\begin{align*}
\bigg|\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}\mu_{\varepsilon}^{X}-\dfrac{1}{r}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|&\leq \dfrac{2}{r}\bigg|\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}\mu_{\varepsilon}^{X}-\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|+\varepsilon^{\alpha-1}\bigg|\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|.
\end{align*}
Using the definition of $\mathbb{F}_{\varepsilon}$, we have:
\begin{align*}
\bigg|\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}\mu_{\varepsilon}^{X}-\dfrac{1}{r}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|\leq \dfrac{2}{r}\cdot\mathbb{F}_{\varepsilon}\cdot\|\Psi^{Y}(Q_{\varepsilon})\|_{W^{1,\infty}(\Omega)}+\varepsilon^{\alpha-1}\|\Psi^{Y}(Q_{\varepsilon})\|_{L^{\infty}(\Omega)}.
\end{align*}
Using now the fact that now $Q_{\varepsilon}\in H^1_g(\Omega,\mathcal{S}_0)$, \cref{lemma:control_Psi} and \cref{lemma:Fbb_eps}, we obtain (also by moving the constant $\dfrac{2}{r}$ under the ``$\lesssim$" sign):
\begin{align*}
&\bigg|\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}\mu_{\varepsilon}^{X}-\dfrac{1}{r}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|\lesssim \varepsilon\big(\|Q_{\varepsilon}\|^4_{L^{\infty}(\Omega)}+1\big)+\\
&\hspace{20mm}+\varepsilon\big(\|Q_{\varepsilon}\|^3_{L^{\infty}(\Omega)}+1\big)\cdot\|\nabla Q_{\varepsilon}\|_{L^{\infty}(\Omega)} + \varepsilon^{\alpha-1}\big(\|Q_{\varepsilon}\|^4_{L^{\infty}(\Omega)}+1\big),
\end{align*}
Applying \cref{lemma:Q_eps_beta}, we get:
\begin{align*}
&\bigg|\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}\mu_{\varepsilon}^{X}-\dfrac{1}{r}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg|\lesssim \varepsilon\bigg(\varepsilon^{-2\beta}\big(\|Q\|_{H^1(\Omega)}+\|g\|_{L^{\infty}(\Omega)}\big)^4+1\bigg)+\\
&\hspace{10mm}+\varepsilon\bigg(\varepsilon^{-3\beta/2}\big(\|Q\|_{H^1(\Omega)}+\|g\|_{L^{\infty}(\Omega)}\big)^3+1\bigg)\cdot\varepsilon^{-3\beta/2}\big(\|Q\|_{H^1(\Omega)}+\|g\|_{W^{1,\infty}(\Omega)}\big) +\\
&\hspace{10mm}+\varepsilon^{\alpha-1}\bigg(\varepsilon^{-2\beta}\big(\|Q\|_{H^1(\Omega)}+\|g\|_{L^{\infty}(\Omega)}\big)^4+1\bigg),
\end{align*}
Moving the terms $\|g\|_{L^{\infty}(\Omega)}$ and $\|g\|_{W^{1,\infty}(\Omega)}$ under the ``$\lesssim$" sign and using the fact that $\beta>0$ and $\varepsilon\in(0,1)$, we have:
\begin{align*}
&\bigg|\dfrac{p\varepsilon-\varepsilon^{\alpha}}{pr(\varepsilon-\varepsilon^{\alpha})}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}\mu_{\varepsilon}^{X}-\dfrac{1}{r}\int_{\Omega}\Psi^{Y}(Q_{\varepsilon})\text{d}x\bigg| \lesssim \big(\varepsilon^{1-2\beta}+\varepsilon^{1-3\beta}+\varepsilon^{\alpha-2\beta-1}\big)\big(\|Q\|^4_{H^1(\Omega)}+1\big).
\end{align*}
Applying the same technique for the other five terms from $J_0$ and $\tilde{J}_{\varepsilon}$, which are in \eqref{defn:J_0_new_form} and \eqref{defn:J_tilde_eps_new_form}, we obtain:
\begin{align*}
|\tilde{J}_{\varepsilon}[Q_{\varepsilon}]-J_0[Q_{\varepsilon}]|\lesssim\big(\varepsilon^{1-2\beta}+\varepsilon^{1-3\beta}+\varepsilon^{\alpha-2\beta-1}\big)\big(\|Q\|^4_{H^1(\Omega)}+1\big)
\end{align*}
and using once again that $\beta>0$ and $\varepsilon\in(0,1)$, we can write:
\begin{align}\label{eq:second_control_J_eps_J_0_Q_eps}
|\tilde{J}_{\varepsilon}[Q_{\varepsilon}]-J_0[Q_{\varepsilon}]|\lesssim\big(\varepsilon^{1-3\beta}+\varepsilon^{\alpha-2\beta-1}\big)\big(\|Q\|^4_{H^1(\Omega)}+1\big).
\end{align}
Moving now to the last term from \eqref{eq:control_J_eps_Q_eps_J_0_Q}, which is $|J_0[Q_{\varepsilon}]-J_0[Q]|$, we once again analyse every difference that can be formed with the six terms from the definition of \eqref{defn:J_0_new_form}. Hence:
\begin{align*}
\bigg|\int_{\Omega}\dfrac{1}{r}\Psi^{Y}(Q_{\varepsilon})\text{d}x-\int_{\Omega}\dfrac{1}{r}\Psi^{Y}(Q)\text{d}x\bigg|\leq \dfrac{1}{r}\int_{\Omega}\big|\Psi^{Y}(Q_{\varepsilon})-\Psi^{Y}(Q)\big|\text{d}x.
\end{align*}
Using \cref{lemma:control_Psi} and moving the constant $\dfrac{1}{r}$ under the ``$\lesssim$" sign, we have:
\begin{align*}
\bigg|\int_{\Omega}\dfrac{1}{r}\Psi^{Y}(Q_{\varepsilon})\text{d}x-\int_{\Omega}\dfrac{1}{r}\Psi^{Y}(Q)\text{d}x\bigg|&\lesssim \int_{\Omega}\big(|Q|^3+|Q_{\varepsilon}|^3+1)|Q-Q_{\varepsilon}|\text{d}x\\
&\lesssim \bigg(\int_{\Omega}\big(|Q|^3+|Q_{\varepsilon}|^3+1\big)^2\text{d}x\bigg)^{1/2}\cdot\bigg(\int_{\Omega}|Q-Q_{\varepsilon}|^2\text{d}x\bigg)^{1/2}\\
&\lesssim \bigg(\int_{\Omega}\big(|Q|^6+|Q_{\varepsilon}|^6+1\big)\text{d}x\bigg)^{1/2}\cdot\|Q-Q_{\varepsilon}\|_{L^2(\Omega)}\\
&\lesssim \big(\|Q\|^3_{L^6(\Omega)}+\|Q_{\varepsilon}\|^3_{L^6(\Omega)}+1\big)\cdot\|Q-Q_{\varepsilon}\|_{L^2(\Omega)}.
\end{align*}
The sequence $(Q_{\varepsilon})_{\varepsilon>0}$ is bounded in $L^6(\Omega)$, due to the continuous Sobolev embedding $H^1(\Omega)\hookrightarrow L^6(\Omega)$ and to \cref{lemma:Q_eps_beta}. Using once again \cref{lemma:Q_eps_beta} to control $\|Q-Q_{\varepsilon}\|_{L^2(\Omega)}$, we obtain:
\begin{align*}
\bigg|\int_{\Omega}\dfrac{1}{r}\Psi^{Y}(Q_{\varepsilon})\text{d}x-\int_{\Omega}\dfrac{1}{r}\Psi^{Y}(Q)\text{d}x\bigg|&\lesssim \varepsilon^{\beta}\big(\|Q\|^4_{H^1(\Omega)}+1\big),
\end{align*}
hence
\begin{align}\label{eq:third_control_J_eps_J_0_Q_eps}
|J_0[Q_{\varepsilon}-J_0[Q]|\lesssim \varepsilon^{\beta}\big(\|Q\|^4_{H^1(\Omega)}+1\big).
\end{align}
Combining now relations \eqref{eq:first_control_J_eps_J_0_Q_eps}, \eqref{eq:second_control_J_eps_J_0_Q_eps} and \eqref{eq:third_control_J_eps_J_0_Q_eps}, we obtain:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-J_0[Q]|\lesssim \big(\varepsilon^{1-3\beta}+\varepsilon^{\alpha-2\beta-1}+\varepsilon^{\beta}\big)\big(\|Q\|^4_{H^1(\Omega)}+1\big).
\end{align*}
Now we need to find a suitable value for $\beta>0$ such that we can put the minimum positive value between the exponents $1-3\beta$, $\alpha-2\beta-1$ and $\beta$, in order to obtain the best rate of convergence. Since we desire that all exponents are positive, $\alpha-2\beta-1>0\Rightarrow \beta<\dfrac{\alpha-1}{2}$. Since $\dfrac{\alpha-1}{2}<2-\alpha\Leftrightarrow \alpha<\dfrac{5}{3}$, which is true because $\alpha\in(1,3/2)$, then we have: $\alpha-2\beta-1<1-3\beta\Leftrightarrow \beta<2-\alpha$, which is true, because $\beta<\dfrac{\alpha-1}{2}<2-\alpha$. Hence we only consider the exponents $\alpha-2\beta-1$ and $\beta$ and we can see that the optimal choice for $\beta$ is $\dfrac{\alpha-1}{3}$, which is positive because $\alpha\in(1,3/2)$. Hence, we obtain:
\begin{align*}
|J_{\varepsilon}^{\mathcal{T}}[Q_{\varepsilon}]-J_0[Q]|\lesssim \varepsilon^{(\alpha-1)/3}\big(\|Q\|^4_{H^1(\Omega)}+1\big).
\end{align*}
\end{proof}
\begin{appendix}
\section{Appendix}\label{section:appendix}
\subsection{Constructing the cubic microlattice}\label{section:constructing_lattice}
In this subsection, we provide more details regarding the construction of the gray parallelipipeds from Figure \ref{fig:nematiccage1}.
In each of the points from $\mathcal{Y}_{\varepsilon}$ we construct a parallelipiped that connects the parallelipipeds $\mathcal{C}^i_{\varepsilon}$ and $\mathcal{C}^j_{\varepsilon}$, where $i,j\in\overline{1,N_{\varepsilon}}$ such that $|x_{\varepsilon}^i-x_{\varepsilon}^j|=\varepsilon$.
If $x^i_{\varepsilon}-x^j_{\varepsilon}=\pm(\varepsilon,0,0)^T$, then let:
\begin{align}
& \bullet X_{\varepsilon}=\text{card}\big(\big\{(i,j)\in\overline{1,N_{\varepsilon}}^2\;\big|\;x_{\varepsilon}^i-x_{\varepsilon}^j=(\varepsilon,0,0)^T,\;i<j\big\}\big)\label{defn:Y_eps_x_cardinal}\\
& \bullet \Upsilon_{\varepsilon}^x:\big\{(i,j)\in\overline{1,N_{\varepsilon}}^2\;\big|\;x_{\varepsilon}^i-x_{\varepsilon}^j=(\varepsilon,0,0)^T,\;i<j\big\}\rightarrow\overline{1,X_{\varepsilon}}\text{ a bijection;}\notag\\
& \bullet y_{\varepsilon}^{x,k}=\dfrac{1}{2}(x_{\varepsilon}^i+x_{\varepsilon}^j),\text{ where }k=\Upsilon_{\varepsilon}^x(i,j);\label{defn:y_eps_x_k}\\
& \bullet \mathcal{Y}_{\varepsilon}^x=\bigg\{y_{\varepsilon}^{x,k}\in\mathcal{Y}_{\varepsilon}\;\bigg|\;y_{\varepsilon}^{x,k}=\dfrac{1}{2}(x_{\varepsilon}^i+x_{\varepsilon}^j),\;k=\Upsilon_{\varepsilon}^x(i,j)\bigg\};\label{defn:Y_eps_x}\\
& \bullet \mathcal{P}_{\varepsilon}^{x,k}\;\text{the parallelipiped centered in }y_{\varepsilon}^{x,k}\text{, defined by }\mathcal{P}_{\varepsilon}^{x,k}=y_{\varepsilon}^{x,k}+T_x\mathcal{C}^{\alpha}\text{, where}\label{defn:P_x}\\
&\hspace{14mm} T_x\mathcal{C}^{\alpha}=\bigg[-\dfrac{p\varepsilon-\varepsilon^{\alpha}}{2p},\dfrac{p\varepsilon-\varepsilon^{\alpha}}{2p}\bigg]\times\bigg[-\dfrac{\varepsilon^{\alpha}}{2q},\dfrac{\varepsilon^{\alpha}}{2q}\bigg]\times\bigg[-\dfrac{\varepsilon^{\alpha}}{2r},\dfrac{\varepsilon^{\alpha}}{2r}\bigg];\\
& \bullet \mathcal{T}_x^k\text{ be the union of the four transparent faces of }\mathcal{P}_{\varepsilon}^{x,k}\text{ that have the length equal to }\dfrac{p\varepsilon-\varepsilon^{\alpha}}{p},\notag\\
&\text{which are represented in Figure \ref{fig:T_x}.\label{defn:T_x_k}}
\end{align}
If $x^i_{\varepsilon}-x^j_{\varepsilon}=\pm(0,\varepsilon,0)^T$, then let:
\begin{align}
& \bullet Y_{\varepsilon}=\text{card}\big(\big\{(i,j)\in\overline{1,N_{\varepsilon}}^2\;\big|\;x_{\varepsilon}^i-x_{\varepsilon}^j=(0,\varepsilon,0)^T,\;i<j\big\}\big)\label{defn:Y_eps_y_cardinal}\\
& \bullet \Upsilon_{\varepsilon}^y:\big\{(i,j)\in\overline{1,N_{\varepsilon}}^2\;\big|\;x_{\varepsilon}^i-x_{\varepsilon}^j=(0,\varepsilon,0)^T,\;i<j\big\}\rightarrow\overline{1,Y_{\varepsilon}}\text{ a bijection;}\notag\\
& \bullet y_{\varepsilon}^{y,l}=\dfrac{1}{2}(x_{\varepsilon}^i+x_{\varepsilon}^j),\text{ where }l=\Upsilon_{\varepsilon}^y(i,j);\notag\\
& \bullet \mathcal{Y}_{\varepsilon}^y=\bigg\{y_{\varepsilon}^{y,l}\in\mathcal{Y}_{\varepsilon}\;\bigg|\;y_{\varepsilon}^{y,l}=\dfrac{1}{2}(x_{\varepsilon}^i+x_{\varepsilon}^j),\;l=\Upsilon_{\varepsilon}^y(i,j)\bigg\};\label{defn:Y_eps_y}\\
& \bullet \mathcal{P}_{\varepsilon}^{y,l}\;\text{the parallelipiped centered in }y_{\varepsilon}^{y,l}\text{, defined by }\mathcal{P}_{\varepsilon}^{y,l}=y_{\varepsilon}^{y,l}+T_y\mathcal{C}^{\alpha}\text{, where}\label{defn:P_y}\\
&\hspace{14mm} T_y\mathcal{C}^{\alpha}=\bigg[-\dfrac{\varepsilon^{\alpha}}{2p},\dfrac{\varepsilon^{\alpha}}{2p}\bigg]\times\bigg[-\dfrac{q\varepsilon-\varepsilon^{\alpha}}{2q},\dfrac{q\varepsilon-\varepsilon^{\alpha}}{2q}\bigg]\times\bigg[-\dfrac{\varepsilon^{\alpha}}{2r},\dfrac{\varepsilon^{\alpha}}{2r}\bigg];\\
& \bullet \mathcal{T}_y^l\text{ be the union of the four transparent faces of }\mathcal{P}_{\varepsilon}^{y,l}\text{ that have the length equal to }\dfrac{q\varepsilon-\varepsilon^{\alpha}}{q},\notag\\
&\text{which are represented in Figure \ref{fig:T_y}.\label{defn:T_y_l}}
\end{align}
If $x^i_{\varepsilon}-x^j_{\varepsilon}=\pm(0,0,\varepsilon)^T$, then let:
\begin{align}
& \bullet Z_{\varepsilon}=\text{card}\big(\big\{(i,j)\in\overline{1,N_{\varepsilon}}^2\;\big|\;x_{\varepsilon}^i-x_{\varepsilon}^j=(0,0,\varepsilon)^T,\;i<j\big\}\big)\label{defn:Y_eps_z_cardinal}\\
& \bullet \Upsilon_{\varepsilon}^y:\big\{(i,j)\in\overline{1,N_{\varepsilon}}^2\;\big|\;x_{\varepsilon}^i-x_{\varepsilon}^j=(0,0,\varepsilon)^T,\;i<j\big\}\rightarrow\overline{1,Z_{\varepsilon}}\text{ a bijection;}\notag\\
& \bullet y_{\varepsilon}^{z,m}=\dfrac{1}{2}(x_{\varepsilon}^i+x_{\varepsilon}^j),\text{ where }m=\Upsilon_{\varepsilon}^z(i,j);\notag\\
& \bullet \mathcal{Y}_{\varepsilon}^z=\bigg\{y_{\varepsilon}^{z,m}\in\mathcal{Y}_{\varepsilon}\;\bigg|\;y_{\varepsilon}^{z,m}=\dfrac{1}{2}(x_{\varepsilon}^i+x_{\varepsilon}^j),\;m=\Upsilon_{\varepsilon}^z(i,j)\bigg\};\label{defn:Y_eps_z}\\
& \bullet \mathcal{P}_{\varepsilon}^{z,m}\;\text{the parallelipiped centered in }y_{\varepsilon}^{z,m}\text{, defined by }\mathcal{P}_{\varepsilon}^{z,m}=y_{\varepsilon}^{z,m}+T_z\mathcal{C}^{\alpha}\text{, where}\label{defn:P_z}\\
&\hspace{14mm} T_z\mathcal{C}^{\alpha}=\bigg[-\dfrac{\varepsilon^{\alpha}}{2p},\dfrac{\varepsilon^{\alpha}}{2p}\bigg]\times\bigg[-\dfrac{\varepsilon^{\alpha}}{2q},\dfrac{\varepsilon^{\alpha}}{2q}\bigg]\times\bigg[-\dfrac{r\varepsilon-\varepsilon^{\alpha}}{2r},\dfrac{r\varepsilon-\varepsilon^{\alpha}}{2r}\bigg];\\
& \bullet \mathcal{T}_z^m\text{ be the union of the four transparent faces of }\mathcal{P}_{\varepsilon}^{z,m}\text{ that have the length equal to }\dfrac{r\varepsilon-\varepsilon^{\alpha}}{r},\notag\\
&\text{which are represented in Figure \ref{fig:T_z}.\label{defn:T_z_m}}
\end{align}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{T_x_5.png}
\caption{The parallelipiped $\mathcal{P}_{\varepsilon}^{x,k}$, with the center in $y_{\varepsilon}^{x,k}$, with lateral tran\-spa\-rent faces $\mathcal{T}_x^k$.}
\label{fig:T_x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{T_y_2.png}
\caption{The parallelipiped $\mathcal{P}_{\varepsilon}^{y,l}$, with the center in $y_{\varepsilon}^{y,l}$, with lateral tran\-spa\-rent faces $\mathcal{T}_y^l$.}
\label{fig:T_y}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{T_z_2.png}
\caption{The parallelipiped $\mathcal{P}_{\varepsilon}^{z,m}$, with the center in $y_{\varepsilon}^{z,m}$, with lateral tran\-spa\-rent faces $\mathcal{T}_z^m$.}
\label{fig:T_z}
\end{subfigure}
\caption{The three types parallelipipeds with centers from $\mathcal{Y}_{\varepsilon}$.}
\end{figure}
\begin{remark} We can already see that we need to set $\varepsilon^{\alpha-1}<\text{min}\{p,q,r\}$, otherwise we have $\dfrac{\varepsilon^{\alpha-1}}{p}\geq 1\Rightarrow \dfrac{\varepsilon^{\alpha}}{2p}\geq\dfrac{\varepsilon}{2}$ and, in the same way, $\dfrac{\varepsilon^{\alpha}}{2q}\geq\dfrac{\varepsilon}{2}$ and $\dfrac{\varepsilon^{\alpha}}{2r}\geq\dfrac{\varepsilon}{2}$, hence the inclusions from the family $\mathcal{C}_{\varepsilon}$ are not disjoint anymore and they overlap. More specifically, the gray parallelipipeds from Figure \ref{fig:nematiccage1} cannot be constructed anymore. Since the parameters $p$, $q$ and $r$ are fixed and we are interested what happens when $\varepsilon\rightarrow 0$, then the condition $\varepsilon^{\alpha-1}<\text{min}\{p,q,r\}$ implies that $\alpha\geq1$. If $\alpha=1$, then it is easy to see that the volume of the scaffold does not tend to zero as $\varepsilon\rightarrow 0$, so we are not in the dilute regime anymore.
\end{remark}
\subsection{Volume and surface area of the scaffold}\label{subsection:appendix1}
\begin{prop}\label{prop:volumegoesto0}
The volume of the scaffold $\mathcal{N}_{\varepsilon}$ tends to $0$ as $\varepsilon\rightarrow 0$.
\end{prop}
\begin{proof}
According to \eqref{defn:L_0_l_0_h_0}, \eqref{defn:points1}, \eqref{defn:points2}, \eqref{defn:Y_eps}, \eqref{defn:Y_eps_x_cardinal}, \eqref{defn:Y_eps_y_cardinal} and \eqref{defn:Y_eps_z_cardinal}, we have:
\vspace{2mm}
\begin{tabular}{llll}
$\bullet\; N_{\varepsilon}<\dfrac{L_0 l_0 h_0}{\varepsilon^3}$; & $\bullet\; X_{\varepsilon}<\bigg(\dfrac{L_0}{\varepsilon}-1\bigg)\cdot\dfrac{l_0h_0}{\varepsilon^2}$; & $\bullet\; Y_{\varepsilon}<\bigg(\dfrac{l_0}{\varepsilon}-1\bigg)\dfrac{L_0h_0}{\varepsilon^2}$; & $\bullet\; Z_{\varepsilon}<\bigg(\dfrac{h_0}{\varepsilon}-1\bigg)\dfrac{L_0l_0}{\varepsilon^2}$.
\end{tabular}
Furthermore, we have:
\begin{align*}
|\mathcal{N}_{\varepsilon}|&=N_{\varepsilon}\cdot\dfrac{\varepsilon^{3\alpha}}{pqr}+X_{\varepsilon}\cdot\dfrac{\varepsilon^{2\alpha}}{pqr}\big(p\varepsilon-\varepsilon^{\alpha}\big)+Y_{\varepsilon}\cdot\dfrac{\varepsilon^{2\alpha}}{pqr}\big(q\varepsilon-\varepsilon^{\alpha}\big)+Z_{\varepsilon}\cdot\dfrac{\varepsilon^{2\alpha}}{pqr}\big(r\varepsilon-\varepsilon^{\alpha}\big),
\end{align*}
where $\dfrac{\varepsilon^{3\alpha}}{pqr}$ represents the volume of a parallelipiped defined in \eqref{defn:ceps} and $\dfrac{\varepsilon^{2\alpha}}{pqr}\big(p\varepsilon-\varepsilon^{\alpha}\big)$, $\dfrac{\varepsilon^{2\alpha}}{pqr}\big(q\varepsilon-\varepsilon^{\alpha}\big)$ and \linebreak $\dfrac{\varepsilon^{2\alpha}}{pqr}\big(r\varepsilon-\varepsilon^{\alpha}\big)$ represent the volume of a parallelipiped $\mathcal{P}_{\varepsilon}^{x,k}$, $\mathcal{P}_{\varepsilon}^{y,l}$ and, respectively, $\mathcal{P}_{\varepsilon}^{z,m}$, which are defined in \eqref{defn:P_x}, \eqref{defn:P_y} and \eqref{defn:P_z}. Hence:
\begin{align*}
|\mathcal{N}_{\varepsilon}|&<\dfrac{L_0l_0h_0(p+q+r)}{pqr}\varepsilon^{2(\alpha-1)}-2\dfrac{L_0l_0h_0}{pqr}\varepsilon^{3(\alpha-1)}-\bigg(\dfrac{L_0l_0}{pq}+\dfrac{L_0h_0}{pr}+\dfrac{l_0h_0}{qr}\bigg)\varepsilon^{2\alpha-1}+\\
&\;\;\;\;+\dfrac{L_0l_0+L_0h_0+l_0h_0}{pqr}\varepsilon^{3\alpha-2}.
\end{align*}
Because $\alpha>1$, according to ($A_2$), then $2(\alpha-1)>0$, $3(\alpha-1)>0$, $2\alpha-1>0$ and $3\alpha-2>0$, therefore $|\mathcal{N}_{\varepsilon}|\rightarrow 0$ as $\varepsilon\rightarrow 0$.
\end{proof}
\begin{prop}\label{prop:C_s}
There exists an $\varepsilon$-independent constant $C_s=C_s(p,q,r,\Omega)>0$ such that: $$\lim_{\varepsilon\rightarrow 0}\dfrac{\varepsilon^{3}}{\varepsilon^{\alpha}(\varepsilon-\varepsilon^{\alpha})}|\partial\mathcal{N}_{\varepsilon}|<C_s.$$
\end{prop}
\begin{proof}
Using the same tehnique as in \cref{prop:volumegoesto0}, we have, considering relations \eqref{defn:T_x_k}, \eqref{defn:T_y_l}, \eqref{defn:T_z_m} and \eqref{defn:surface}: \begin{align*}
|\partial\mathcal{N}_{\varepsilon}|&<N_{\varepsilon}\cdot\varepsilon^{2\alpha}\cdot\dfrac{2(p+q+r)}{pqr}+X_{\varepsilon}\cdot\varepsilon^{\alpha}(p\varepsilon-\varepsilon^{\alpha})\cdot\dfrac{2(q+r)}{pqr}+\\
&\hspace{5mm}+Y_{\varepsilon}\cdot\varepsilon^{\alpha}(q\varepsilon-\varepsilon^{\alpha})\cdot\dfrac{2(p+r)}{pqr}+Z_{\varepsilon}\cdot\varepsilon^{\alpha}(r\varepsilon-\varepsilon^{\alpha})\cdot\dfrac{2(p+q)}{pqr}\\
&<C(p,q,r,L_0,l_0,h_0)\cdot \varepsilon^{\alpha-3}\cdot \big((p+q+r)\varepsilon-2\varepsilon^{\alpha}\big).
\end{align*}
Hence
$$\lim_{\varepsilon\rightarrow 0}\dfrac{\varepsilon^{3}}{\varepsilon^{\alpha}(\varepsilon-\varepsilon^{\alpha})}|\partial\mathcal{N}_{\varepsilon}|<C(p,q,r,L_0,l_0,h_0)\cdot \lim_{\varepsilon\rightarrow 0}\dfrac{(p+q+r)\varepsilon-2\varepsilon^{\alpha}}{\varepsilon-\varepsilon^{\alpha}}<+\infty.$$ We denote $C_{s}$ the constant obtained in the last inequality.
\end{proof}
\begin{prop}\label{prop:outer_surfaces_go_to_0}
Let $\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}$ be the set defined in \eqref{defn:surface}. Then, for $\varepsilon\rightarrow 0$, we have $|\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}|\rightarrow 0$.
\end{prop}
\begin{proof}
According to \eqref{defn:surface}, we have:
\begin{align*}
\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}=\bigg(\bigcup_{i=1}^{N_{\varepsilon,2}}\mathcal{S}^i\bigg),
\end{align*}
therefore, we can write:
\begin{align*}
\big|\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}\big|&\leq \sum_{i=1}^{N_{\varepsilon,2}}\big|\mathcal{S}^i\big|\leq\sum_{i=1}^{N_{\varepsilon,2}}\big|\mathcal{C}_{\varepsilon}^{i}\big|\leq \sum_{i=1}^{N_{\varepsilon,2}}\dfrac{\varepsilon^{2\alpha}(p+q+r)}{pqr}\leq \dfrac{\varepsilon^{2\alpha}(p+q+r)}{pqr}\cdot N_{\varepsilon,2},
\end{align*}
where we have used \eqref{defn:S_i} and \eqref{defn:ceps}. Since $N_{\varepsilon,2}$ counts only the parallelipipeds that are ``close" to the boundary of $\Omega$ (see \eqref{defn:N_eps_2}), then we can write:
\begin{align*}
N_{\varepsilon,2}<\dfrac{L_0\cdot l_0}{\varepsilon^2}+\dfrac{L_0\cdot h_0}{\varepsilon^2}+\dfrac{l_0\cdot h_0}{\varepsilon^2},
\end{align*}
where $L_0$, $l_0$ and $h_0$ are defined in \eqref{defn:L_0_l_0_h_0} and they describe the parallelipiped that contains the entire domain $\Omega$. From here, we obtain:
\begin{align*}
\big|\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}\big|&\leq \dfrac{\varepsilon^{2\alpha}(p+q+r)}{pqr}\cdot\dfrac{L_0l_0+l_0h_0+h_0L_0}{\varepsilon^2}\lesssim \varepsilon^{2(\alpha-1)}\rightarrow 0,\;\text{ as}\;\varepsilon\rightarrow 0,
\end{align*}
since $\alpha>1$.
\end{proof}
\subsection{Constructing an explicit extension of $Q$ inside the scaffold}\label{subsection:existence_of_extension}
The aim of this subsection is to prove that there exists a function $v\in H^1(\Omega)$ such that $v=Q$ on $\partial\mathcal{N}_{\varepsilon}$, $v=Q$ in $\Omega_{\varepsilon}$ and $\big\|\nabla v\big\|_{L^2(\Omega)}\lesssim \big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}$.
In order to prove it, we first construct an explicit extension $u:\Omega_{\varepsilon}\cup\mathcal{N}_{\varepsilon}^{\mathcal{T}}\rightarrow\mathcal{S}_0$ such that $u\in H^1(\Omega_{\varepsilon}\cup\mathcal{N}_{\varepsilon}^{\mathcal{T}},\mathcal{S}_0)$ and there exists a constant $C$, independent of $\varepsilon$, for which we have
\begin{align*}
\big\|\nabla u\big\|_{L^2(\mathcal{N}_{\varepsilon}^{\mathcal{T}})}\leq C\big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})},
\end{align*}
which implies $\big\|\nabla u\big\|_{L^2(\Omega_{\varepsilon}\cup\mathcal{N}_{\varepsilon}^{\mathcal{T}})}\leq C\big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}$.
Then we construct $v:\Omega\rightarrow\mathcal{S}_0$ such that $v\in H^1(\Omega)$, $v\equiv u$ on $\Omega_{\varepsilon}\cup\mathcal{N}_{\varepsilon}^{\mathcal{T}}$, $v=u$ on $\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}$ and there exists a constant $c$ such that:
\begin{align*}
\big\|\nabla v\big\|_{L^2(\Omega)}\leq c\big\|\nabla u\big\|_{L^2(\Omega_{\varepsilon}\cup\mathcal{N}_{\varepsilon}^{\mathcal{T}})},
\end{align*}
which implies that $\big\|\nabla v\big\|_{L^2(\Omega)}\lesssim\big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}$, using the properties mentioned for $u$.
We prove first the following result.
\begin{lemma}\label{lemma:extension_u}
Let $z_0,a,b\in\mathbb{R}$, $a,b,z_0>0$ and let $A_{a,b}=\big\{(\rho\cos\theta,\rho\sin\theta)\;:\; 0\leq\theta< 2\pi,\;a<\rho<b\big\}$ be a two dimensional annulus, $B_a=\big\{(\rho\cos\theta,\rho\sin\theta)\;:\;0\leq \theta<2\pi,\;0\leq\rho<a\big\}$ be a two dimensional ball with radius $a$, $\mathcal{A}_{a,b}^{z_0}=A_{a,b}\times(-z_0,z_0)\subset\mathbb{R}^3$ and $\mathcal{B}_a^{z_0}=B_a\times(-z_0,z_0)\subset\mathbb{R}^3$ be a three dimensional cylinder.
Let $Q\in H^1\big(\mathcal{A}_{1,2}^{z_0},\mathcal{S}_0\big)$. Then the function $u:\mathcal{B}_1^{z_0}\rightarrow\mathcal{S}_0$ defined for any $z\in(-z_0,z_0)$ as
\begin{equation}\label{eq:ext_def_u}
u(x,y,z)=\begin{cases}\varphi(\sqrt{x^2+y^2})\; Q\bigg(\bigg(\dfrac{2}{\sqrt{x^2+y^2}}-1\bigg)x,\bigg(\dfrac{2}{\sqrt{x^2+y^2}}-1\bigg)y,z\bigg)+\\
\hspace{3mm}+(1-\varphi(\sqrt{x^2+y^2}))\displaystyle{\fint_{A_{1,3/2}}Q(s,t,z)\emph{d}(s,t)},\;\emph{for}\;\dfrac{1}{2}\leq\sqrt{x^2+y^2}<1\\
\displaystyle{\fint_{A_{1,3/2}}Q(s,t,z)\emph{d}(s,t)},\;\emph{for}\;0\leq\sqrt{x^2+y^2}\leq\dfrac{1}{2}\end{cases}
\end{equation}
is from $H^1(\mathcal{B}_1^{z_0},\mathcal{S}_0)$, where $\varphi\in C_c^{\infty}\bigg(\bigg(\dfrac{1}{2},\dfrac{3}{2}\bigg)\bigg)$ is the following bump function defined as
\begin{equation*}
\varphi(\rho)=\begin{cases}\exp\bigg\{4-\dfrac{4}{(2\rho-1)(3-2\rho)}\bigg\},\;\forall\rho\in\bigg(\dfrac{1}{2},\dfrac{3}{2}\bigg)\\
0,\;\forall\rho\in\mathbb{R}\setminus\bigg(\dfrac{1}{2},\dfrac{3}{2}\bigg)\end{cases},
\end{equation*}
the product $\varphi(\rho)\;Q$ represents product between a scalar and a Q-tensor and $\displaystyle{\fint}$ represents the average integral sign. Moreover, there exists a constant $c>0$, independent of $z_0$, such that: $\big\|u_{t}\big\|_{L^2(\mathcal{B}_1^{z_0})}\leq c\big\|Q_{t}\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}$ for any $t\in\{x,y,z\}$, where $u_t$ represents the partial derivative of $u$ with respect to $t$, and:
\begin{align*}
\|\nabla u\|_{L^2(\mathcal{B}_1^{z_0})}\leq c\|\nabla Q\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
\end{lemma}
\begin{proof}
First of all, we can assume without loss of generality that $Q$ and $u$ are scalar functions, instead of $Q$-tensors. Hence, we prove this lemma for each of the component of $Q$ and $u$.
Let $T:A_{1/2,1}\rightarrow A_{1,3/2}$ be the reflection defined as:
\begin{equation*}
T(x,y)=\bigg(\bigg(\dfrac{2}{\sqrt{x^2+y^2}}-1\bigg)x,\bigg(\dfrac{2}{\sqrt{x^2+y^2}}-1\bigg)y\bigg):=(x',y'),\;\forall(x,y)\in A_{1/2,1}.
\end{equation*}
Then $T$ is invertible and also bi-Lipschitz.
Let $Q\in H^1(\mathcal{A}_{1,3/2}^{z_0})$ and $u$ defined by \eqref{eq:ext_def_u}. By Theorem 3.17 from \cite{Adams}, we can approximate the function $Q\in H^1(\mathcal{A}_{1,3/2}^{z_0})$ with smooth functions from $C^{\infty}\big(\overline{\mathcal{A}_{1,3/2}^{z_0}}\big)$.
Let $(Q_k)_{k\geq 1}\subset C^{\infty}\big(\overline{\mathcal{A}_{1,3/2}^{z_0}}\big)$ such that $Q_k\rightarrow Q$ strongly in $H^1(\mathcal{A}_{1,3/2}^{z_0})$ and, for any $k\geq 1$, let $u_k:\overline{\mathcal{B}_{1}^{z_0}}\rightarrow\mathbb{R}$ defined for all $z\in(-z_0,z_0)$ as:
\begin{equation*}
u_k(x,y,z)=\begin{cases}\varphi(\sqrt{x^2+y^2})\; Q_k(T(x,y),z)+\\
\hspace{3mm}+(1-\varphi(\sqrt{x^2+y^2}))\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\text{d}(s,t)},\;\text{for}\;\dfrac{1}{2}\leq\sqrt{x^2+y^2}\leq 1\\
\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\emph{d}(s,t)},\;\text{for}\;0\leq\sqrt{x^2+y^2}\leq\dfrac{1}{2}\end{cases}
\end{equation*}
By the above definition, we have that $u_k\in C^0\big(\overline{\mathcal{B}_1^{z_0}}\big)$ and that:
\begin{align*}
\dfrac{\partial u_k}{\partial x}(x,y,z)&=\begin{cases}\dfrac{x}{\sqrt{x^2+y^2}}\cdot \varphi'(\sqrt{x^2+y^2})\cdot\bigg(Q_k(x',y',z)-\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\text{d}(s,t)}\bigg)+\\
\hspace{3mm}+\bigg(\dfrac{2y^2}{\big(\sqrt{x^2+y^2)}\big)^3}-1\bigg)\cdot \varphi(\sqrt{x^2+y^2})\cdot\bigg(\dfrac{\partial Q_k}{\partial x'}(x',y',z)\bigg)-\\
\hspace{3mm}-\dfrac{2xy}{\big(\sqrt{x^2+y^2}\big)^3}\cdot\varphi(\sqrt{x^2+y^2})\cdot \bigg(\dfrac{\partial Q_k}{\partial y'}(x',y',z)\bigg),\;\text{for}\;\dfrac{1}{2}\leq\sqrt{x^2+y^2}<1\\
0,\;\text{for}\;0\leq\sqrt{x^2+y^2}\leq\dfrac{1}{2}.
\end{cases}
\end{align*}
Since $\varphi\bigg(\dfrac{1}{2}\bigg)=\varphi'\bigg(\dfrac{1}{2}\bigg)=0$ and $\varphi\in C^{\infty}\bigg(\bigg(\dfrac{1}{2},\dfrac{3}{2}\bigg)\bigg)$, then we have $\dfrac{\partial u_k}{\partial x}\in C^0\big(\overline{\mathcal{B}_1^{z_0}}\big)$. Moreover, we obtain:
\begin{align*}
\dfrac{1}{3}\int_{A_{1/2,1}}\bigg|\dfrac{\partial u_k}{\partial x}\bigg|^2&(x,y,z)\text{d}(x,y) \leq \int_{A_{1/2,1}}\bigg|\dfrac{2xy}{\big(\sqrt{x^2+y^2}\big)^3}\bigg|^2 \big|\varphi(\sqrt{x^2+y^2})\big|^2\bigg|\dfrac{\partial Q_k}{\partial y'}\bigg|^2(x',y',z)\text{d}(x,y)+\\
&\hspace{2mm}+\int_{A_{1/2,1}}\bigg|\dfrac{2y^2}{\big(\sqrt{x^2+y^2}\big)^3}-1\bigg|^2\big|\varphi(\sqrt{x^2+y^2})\big|^2\bigg|\dfrac{\partial Q_k}{\partial x'}\bigg|^2(x',y',z)\text{d}(x,y)+\\
&\hspace{2mm}+\int_{A_{1/2,1}}\bigg|\dfrac{x}{\sqrt{x^2+y^2}}\bigg|^2\big|\varphi'(\sqrt{x^2+y^2})\big|^2\bigg|Q_k(x',y',z)-\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\text{d}(s,t)}\bigg|^2\text{d}(x,y).
\end{align*}
By the definition of $\varphi$, we have $\|\varphi\|_{L^{\infty}(\mathbb{R})}=1$ and $\|\varphi'\|_{L^{\infty}(\mathbb{R})}=2\sqrt{9+6\sqrt{3}}\cdot e^{1-\sqrt{3}}\approx 4.23<5$ (the maximum is obtained for $\rho=1-\dfrac{1}{6}\sqrt{6\sqrt{3}-9}$).
For any $(x,y)$ such that $\dfrac{1}{2}\leq\sqrt{x^2+y^2}\leq 1$, we have:
\begin{itemize}
\item[•] $\bigg|\dfrac{x}{\sqrt{x^2+y^2}}\bigg|^2=\dfrac{x^2}{x^2+y^2}\leq 1$;
\item[•] $\bigg|\dfrac{2xy}{x^2+y^2}\bigg|\leq 1\Rightarrow \bigg|\dfrac{2xy}{\big(\sqrt{x^2+y^2}\big)^3}\bigg|^2\leq\dfrac{1}{x^2+y^2}\leq 2$;
\item[•] $0\leq\dfrac{2y^2}{\big(\sqrt{x^2+y^2}\big)^3}\leq\dfrac{2(x^2+y^2)}{(\sqrt{x^2+y^2}\big)^3}\leq 4\Rightarrow -1\leq\dfrac{2y^2}{\big(\sqrt{x^2+y^2}\big)^3}-1\leq 3\Rightarrow \bigg|\dfrac{2y^2}{\big(\sqrt{x^2+y^2}\big)^3}\bigg|^2\leq 9.$
\end{itemize}
Therefore
\begin{align*}
\dfrac{1}{3}\int_{A_{1/2,1}}\bigg|\dfrac{\partial u_k}{\partial x}\bigg|^2(x,y,z)\text{d}(x,y)&\leq 25\int_{A_{1/2,1}}\bigg|Q_k(x',y',z)-\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\text{d}(s,t)}\bigg|^2\text{d}(x,y)+\\
&\hspace{2mm}+9\int_{A_{1/2,1}}\bigg|\dfrac{\partial Q_k}{\partial x'}\bigg|^2(x',y',z)\text{d}(x,y)+4\int_{A_{1/2,1}}\bigg|\dfrac{\partial Q_k}{\partial y'}\bigg|^2(x',y',z)\text{d}(x,y).
\end{align*}
Using now the change of variables $(x',y')=T(x,y)$, we obtain
\begin{align*}
\text{d}(x,y)=\bigg(\dfrac{2}{\sqrt{(x')^2+(y')2}}-1\bigg)\text{d}(x',y')
\end{align*}
and since $(x',y')\in A_{1,3/2}$, we get $1\geq\dfrac{2}{\sqrt{(x')^2+(y')^2}}-1\geq\dfrac{1}{3}$, which implies
\begin{align*}
\int_{A_{1/2,1}}\bigg|\dfrac{\partial u_k}{\partial x}\bigg|^2(x,y,z)\text{d}(x,y) &\leq 75\int_{A_{1,3/2}}\bigg|Q_k(x',y',z)-\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\text{d}(s,t)}\bigg|^2\text{d}(x',y')+\\
&\hspace{2mm}+27\int_{A_{1,3/2}}\bigg|\dfrac{\partial Q_k}{\partial x'}\bigg|^2(x',y',z)\text{d}(x',y')+12\int_{A_{1,3/2}}\bigg|\dfrac{\partial Q_k}{\partial y'}\bigg|^2(x',y',z)\text{d}(x',y').
\end{align*}
For the first term from the right hand side from the last inequality we can apply the Poincar\'{e} inequality, since $Q_k(\cdot,\cdot,z)\in H^1(A_{1,3/2})$, for any $z\in(-z_0,z_0)$. Therefore
\begin{align*}
\int_{A_{1/2,1}}\bigg|\dfrac{\partial u_k}{\partial x}\bigg|^2(x,y,z)\text{d}(x,y)&\leq 75\cdot C_P(A_{1,3/2})\int_{A_{1,3/2}}\bigg(\bigg|\dfrac{\partial Q_k}{\partial x'}\bigg|^2(x',y',z)+\bigg|\dfrac{\partial Q_k}{\partial y'}\bigg|^2(x',y',z)\bigg)\text{d}(x',y')+\\
&\hspace{2mm}+27\int_{A_{1,3/2}}\bigg|\dfrac{\partial Q_k}{\partial x'}\bigg|^2(x',y',z)\text{d}(x,y)+12\int_{A_{1,3/2}}\bigg|\dfrac{\partial Q_k}{\partial y'}\bigg|^2(x',y',z)\text{d}(x,y),
\end{align*}
where $C_P(A_{1,3/2})$ is the Poincar\'{e} constant for the two dimmensional domain $A_{1,3/2}$. Hence, there exists $c_1>0$, independent of $z_0$, such that:
\begin{align*}
\int_{A_{1/2,1}}\bigg|\dfrac{\partial u_k}{\partial x}\bigg|^2(x,y,z)\text{d}(x,y,z)&\leq c_1\int_{A_{1,3/2}}\bigg(\bigg|\dfrac{\partial Q_k}{\partial x'}\bigg|^2(x',y',z)+\bigg|\dfrac{\partial Q_k}{\partial y'}\bigg|^2(x',y',z)\bigg)\text{d}(x',y',z).
\end{align*}
Integrating now with respect to $z\in(-z_0,z_0)$, we get:
\begin{align*}
\int_{\mathcal{A}_{1/2,1}^{z_0}}\bigg|\dfrac{\partial u_k}{\partial x}\bigg|^2(x,y,z)\text{d}(x,y,z)&\leq c_1\int_{\mathcal{A}_{1,3/2}^{z_0}}\bigg(\bigg|\dfrac{\partial Q_k}{\partial x'}\bigg|^2(x',y',z)+\bigg|\dfrac{\partial Q_k}{\partial y'}\bigg|^2(x',y',z)\bigg)\text{d}(x',y',z)\Rightarrow\\
\Rightarrow\bigg\|\dfrac{\partial u_k}{\partial x}\bigg\|^2_{L^2(\mathcal{A}_{1/2,1}^{z_0})}&\leq c_1\big\|\nabla Q_k\|^2_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
Using now the fact that $\dfrac{\partial u_k}{\partial x}\in C^0\big(\overline{\mathcal{B}_1^{z_0}}\big)$ and that $\dfrac{\partial u_k}{\partial x}(x,y,z)=0$ if $0\leq\sqrt{x^2+y^2}\leq\dfrac{1}{2}$, then we can write:
\begin{align*}
\bigg\|\dfrac{\partial u_k}{\partial x}\bigg\|^2_{L^2(\mathcal{B}_1^{z_0})}&\leq c_1\big\|\nabla Q_k\big\|^2_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
In a similar fashion, $\dfrac{\partial u_k}{\partial y},\dfrac{\partial u_k}{\partial z}\in C^0\big(\overline{\mathcal{B}_1^{z_0}}\big)$, where
\begin{align*}
\dfrac{\partial u_k}{\partial y}(x,y,z)&=\begin{cases}\dfrac{y}{\sqrt{x^2+y^2}}\cdot \varphi'(\sqrt{x^2+y^2})\cdot\bigg(Q_k(x',y',z)-\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\text{d}(s,t)}\bigg)-\\
\hspace{3mm}-\dfrac{2xy}{\big(\sqrt{x^2+y^2}\big)^3}\cdot\varphi(\sqrt{x^2+y^2})\cdot \bigg(\dfrac{\partial Q_k}{\partial x'}(x',y',z)\bigg)+\\
\hspace{3mm}+\bigg(\dfrac{2x^2}{\big(\sqrt{x^2+y^2)}\big)^3}-1\bigg)\cdot \varphi(\sqrt{x^2+y^2})\cdot\bigg(\dfrac{\partial Q_k}{\partial y'}(x',y',z)\bigg),\;\text{for}\;\dfrac{1}{2}\leq\sqrt{x^2+y^2}\leq 1\\
0,\;\text{for}\;0\leq\sqrt{x^2+y^2}\leq\dfrac{1}{2}
\end{cases}
\end{align*}
with
\begin{align*}
\bigg\|\dfrac{\partial u_k}{\partial y}\bigg\|^2_{L^2(\mathcal{B}_1^{z_0})}&\leq c_1\big\|\nabla Q_k\big\|^2_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
and
\begin{align*}
\dfrac{\partial u_k}{\partial z}(x,y,z)=\begin{cases}\varphi(\sqrt{x^2+y^2})\; \dfrac{\partial Q_k}{\partial z}(x',y',z)+\\
\hspace{3mm}+(1-\varphi(\sqrt{x^2+y^2}))\displaystyle{\fint_{A_{1,3/2}}\dfrac{\partial Q_k}{\partial z}(s,t,z)\text{d}(s,t)},\;\text{for}\;\dfrac{1}{2}\leq\sqrt{x^2+y^2}\leq 1\\
\displaystyle{\fint_{A_{1,3/2}}\dfrac{\partial Q_k}{\partial z}(s,t,z)\emph{d}(s,t)},\;\text{for}\;0\leq\sqrt{x^2+y^2}\leq\dfrac{1}{2}\end{cases}
\end{align*}
since $Q\in C^{\infty}\big(\overline{\mathcal{A}_{1,3/2}^{z_0}}\big)$ and $A_{1,3/2}$ is independent of $z$, so we can move the derivative under the integral.
Then for any $(x,y,z)\in\mathcal{A}_{1/2,1}^{z_0}$, we have:
\begin{align*}
\dfrac{1}{2}\int_{A_{1/2,1}}\bigg|\dfrac{\partial u_k}{\partial z}\bigg|^2(x,y,z)\text{d}(x,y)&\leq \big\|\varphi\big\|^2_{L^{\infty}(\mathbb{R})}\int_{A_{1/2,1}}\bigg|\dfrac{\partial Q_k}{\partial z}\bigg|^2(x',y',z)\text{d}(x,y)+\\
&\hspace{2mm}+\big\|1-\varphi\big\|^2_{L^{\infty}(\mathbb{R})}\int_{A_{1/2,1}}\bigg|\displaystyle{\fint_{A_{1,3/2}}\dfrac{\partial Q_k}{\partial z}(s,t,z)\text{d}(s,t)}\bigg|^2\text{d}(x,y)\\
&\leq\int_{A_{1,3/2}}\bigg|\dfrac{\partial Q_k}{\partial z}\bigg|^2(x',y',z)\text{d}(x',y')+\dfrac{3\pi}{4}\cdot\dfrac{16}{25\pi^2}\int_{A_{1,3/2}}\bigg|\dfrac{\partial Q_k}{\partial z}\bigg|^2(x',y',z)\text{d}(x',y')
\end{align*}
and integrating with respect to $z\in(-z_0,z_0)$, we obtain
\begin{align*}
\int_{\mathcal{A}_{1/2,1}^{z_0}}\bigg|\dfrac{\partial u_k}{\partial z}\bigg|^2(x,y,z)\text{d}(x,y,z)&\leq\bigg(2+\dfrac{24}{25\pi}\bigg)\int_{\mathcal{A}_{1,3/2}^{z_0}}\bigg|\dfrac{\partial Q_k}{\partial z}\bigg|^2(x',y',z)\text{d}(x',y',z).
\end{align*}
For any $(x,y,z)\in\mathcal{B}_1^{z_0}$ with $0\leq\sqrt{x^2+y^2}\leq\dfrac{1}{2}$ we have:
\begin{align*}
\bigg|\dfrac{\partial u_k}{\partial z}\bigg|^2(x,y,z)&\leq\dfrac{16}{25\pi^2}\int_{A_{1,3/2}}\bigg|\dfrac{\partial Q_k}{\partial z}\bigg|^2(x',y',z)\text{d}(x',y')
\end{align*}
which implies
\begin{align*}
\int_{\overline{B_{1/2}}}\bigg|\dfrac{\partial u_k}{\partial z}\bigg|^2(x,y,z)\text{d}(x,y)&\leq\dfrac{4}{25\pi}\int_{A_{1,3/2}}\bigg|\dfrac{\partial Q_k}{\partial z}\bigg|^2(x',y',z)\text{d}(x',y',z)
\end{align*}
and from here we obtain that
\begin{align*}
\bigg\|\dfrac{\partial u_k}{\partial z}\bigg\|^2_{L^2(\mathcal{B}_1^{z_0})}&\leq\bigg(3+\dfrac{3}{25\pi}\bigg)\big\|\nabla Q_k\big\|^2_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
Now we prove that we can control $\|u_k\|_{L^2(\mathcal{B}_1^{z_0})}$ with $\|Q_k\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}$. For any $(x,y)\in A_{1/2,1}$, we have:
\begin{align*}
|u_k|^2(x,y,z)&\leq |\varphi(\sqrt{x^2+y^2}|^2\big|Q_k(x',y',z)|^2+(1-\varphi(\sqrt{x^2+y^2}))^2\bigg|\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\text{d}(s,t)}\bigg|^2
\end{align*}
which implies
\begin{align*}
\dfrac{1}{2}\int_{A_{1/2,1}}|u_k|^2(x,y,z)\text{d}(x,y)&\leq\big\|1-\varphi\|^2_{L^{\infty}(\mathbb{R})}\cdot\dfrac{3}{4\pi}\cdot\dfrac{16}{25\pi^2}\int_{A_{1,3/2}}|Q_k|^2(x',y',z)\text{d}(x,y)+\\
&\hspace{2mm}+\big\|\varphi\|^2_{L^{\infty}(\mathbb{R})}\int_{A_{1/2,1}}|Q_k|^2(x',y',z)\text{d}(x,y).
\end{align*}
Using the same change of variables, the same bounds for $\varphi$ and for $1-\varphi$ and integrating with respect to $z\in(-z_0,z_0)$, we get:
\begin{align*}
\big\|u_k\big\|^2_{L^2(\mathcal{A}_{1/2,1}^{z_0})}\leq\bigg(2+\dfrac{24}{25\pi}\bigg)\big\|Q_k\big\|^2_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
For any $(x,y)\in\overline{B_{1/2}}$ we have:
\begin{align*}
|u_k|^2(x,y,z)=\bigg|\displaystyle{\fint_{A_{1,3/2}}Q_k(s,t,z)\text{d}(s,t)}\bigg|^2
\end{align*}
which implies
\begin{align*}
\big\|u_k\big\|^2_{L^2(\overline{B_{1/2}}\times(-z_0,z_0))}\leq \dfrac{4}{25\pi}\big\|Q_k\big\|^2_{L^2(\mathcal{A}_{1,3/2}^{z_0})},
\end{align*}
hence
\begin{align*}
\big\|u_k\big\|^2_{L^2(\mathcal{B}_1^{z_0})}\leq\bigg(3+\dfrac{3}{25\pi}\bigg)\big\|Q_k\big\|^2_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
Combining all the relations that we have obtained, we see that for any $k\geq 1$ we have:
\begin{itemize}
\item[•] $u_k\in H^1(\mathcal{B}_1^{z_0})$;
\item[•] $\big\|u_k\big\|_{L^2(\mathcal{B}_1^{z_0})}\leq c_2\big\|Q_k\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}$, where $c_2=\sqrt{3+\dfrac{3}{25\pi}}$;
\item[•] $\big\|\nabla u_k\big\|_{L^2(\mathcal{B}_1^{z_0})}\leq c_3\big\|\nabla Q_k\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}$, where $c_3=\max\big\{\sqrt{c_1},c_2\big\}$.
\end{itemize}
Now, if we repeat the same argument (as the one used in order to achieve the $L^2$ control between $u_k$ and $Q_k$) for the functions $(u_k-u)$, for any $k\geq 1$, we get:
\begin{align*}
\big\|u_k-u\big\|_{L^2(\mathcal{B}_1^{z_0})}\leq c_2\big\|Q_k-Q\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}
\end{align*}
and since $Q_k\rightarrow Q$ strongly in $H^1(\mathcal{A}_{1,3/2}^{z_0})$, hence in $L^2(\mathcal{A}_{1,3/2}^{z_0})$, we obtain that $u_k\rightarrow u$ strongly in $L^2(\mathcal{B}_1^{z_0})$.
Because $Q_k\rightarrow Q$ strongly in $H^1(\mathcal{A}_{1,3/2}^{z_0})$, then $(Q_k)_{k\geq 1}$ is a bounded sequence in $H^1(\mathcal{A}_{1,3/2}^{z_0})$ and using the inequalities proved before, we get that $(u_k)_{k\geq 1}$ is a bounded sequence in $H^1(\mathcal{B}_1^{z_0})$, therefore there exists a subsequence $(u_{k_j})_{j\geq 1}$ which has the property that $u_{k_j}\rightharpoonup u_0$, with $u_0\in H^1(\mathcal{B}_1^{z_0})$. From here, we have the following convergences in $L^2(\mathcal{B}_1^{z_0})$: $u_{k_j}\rightharpoonup u_0$ and $u_{k_j}\rightarrow u$, so $u=u_0$ a.e. in $\mathcal{B}_1^{z_0}$. However, since $u_0\in H^1(\mathcal{B}_1^{z_0})$, we obtain that $u\in H^1(\mathcal{B}_1^{z_0})$ with $\nabla u=\nabla u_0$ a.e. in $\mathcal{B}_1^{z_0}$.
Let $\tilde{u}_x:\mathcal{B}_1^{z_0}\rightarrow\mathbb{R}$ be the function defined as:
\begin{align*}
\tilde{u}_x(x,y,z)=\begin{cases}\dfrac{x}{\sqrt{x^2+y^2}}\cdot \varphi'(\sqrt{x^2+y^2})\cdot\bigg(Q(x',y',z)-\displaystyle{\fint_{A_{1,3/2}}Q(s,t,z)\text{d}(s,t)}\bigg)+\\
\hspace{3mm}+\bigg(\dfrac{2y^2}{\big(\sqrt{x^2+y^2)}\big)^3}-1\bigg)\cdot \varphi(\sqrt{x^2+y^2})\cdot\bigg(\dfrac{\partial Q}{\partial x'}(x',y',z)\bigg)-\\
\hspace{3mm}-\dfrac{2xy}{\big(\sqrt{x^2+y^2}\big)^3}\cdot\varphi(\sqrt{x^2+y^2})\cdot \bigg(\dfrac{\partial Q}{\partial y'}(x',y',z)\bigg),\;\text{for}\;\dfrac{1}{2}\leq\sqrt{x^2+y^2}<1\\
0,\;\text{for}\;0\leq\sqrt{x^2+y^2}\leq\dfrac{1}{2}.
\end{cases}
\end{align*}
for every $z\in(-z_0,z_0)$.
Using the same argument as before (we only control the $L^2$ norm), we can see that:
\begin{align*}
\bigg\|\dfrac{\partial u_k}{\partial x}-\tilde{u}_x\bigg\|_{L^2(\mathcal{B}_1^{z_0})}\leq c_3\big\|\nabla Q_k-\nabla Q\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}
\end{align*}
and since $\nabla Q_k\rightarrow \nabla Q$ strongly in $L^2(\mathcal{A}_{1,3/2}^{z_0})$, we obtain that $\dfrac{\partial u_k}{\partial x}\rightarrow\tilde{u}_x$ strongly in $L^2(\mathcal{B}_1^{z_0})$. But at the same time, we have $\dfrac{\partial u_k}{\partial x}\rightharpoonup\dfrac{\partial u}{\partial x}$ weakly in $L^2(\mathcal{B}_1^{z_0})$, hence $\dfrac{\partial u}{\partial x}=\tilde{u}_x$ a.e. in $L^2(\mathcal{B}_1^{z_0})$ and $\dfrac{\partial u_k}{\partial x}\rightarrow\dfrac{\partial u}{\partial x}$ strongly in $L^2(\mathcal{B}_1^{z_0})$. Applying the same argument, we finally prove that $\nabla u_k\rightarrow\nabla u$ strongly in $L^2(\mathcal{B}_1^{z_0})$.
In the end, we see that:
\begin{align*}
\big\|\nabla u\big\|_{L^2(\mathcal{B}_1^{z_0})}&\leq\big\|\nabla u-\nabla u_k\big\|_{L^2(\mathcal{B}_1^{z_0})}+\big\|\nabla u_k\big\|_{L^2(\mathcal{B}_1^{z_0})}\\
&\leq \big\|\nabla u-\nabla u_k\big\|_{L^2(\mathcal{B}_1^{z_0})}+c_3\big\|\nabla Q_k\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}\\
&\leq \big\|\nabla u-\nabla u_k\big\|_{L^2(\mathcal{B}_1^{z_0})}+c_3\big\|\nabla Q_k-\nabla Q\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}+c_3\big\|\nabla Q\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
Because $\nabla u_k\rightarrow \nabla u$ strongly in $L^2(\mathcal{B}_1^{z_0})$ and because $\nabla Q_k\rightarrow \nabla Q$ strongly in $L^2(\mathcal{A}_{1,3/2}^{z_0})$, we conclude that
\begin{align*}
\big\|\nabla u\big\|_{L^2(\mathcal{B}_1^{z_0})}\leq c_3\big\|\nabla Q\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}.
\end{align*}
\end{proof}
Now we transform in several steps the sets $\mathcal{B}_1^{z_0}$ and $\mathcal{A}_{1,3/2}^{z_0}$ from the previous lemma into the corresponding regions related to $\Omega_{\varepsilon}$ and $\mathcal{N}_{\varepsilon}^{\mathcal{T}}$, that is, $\mathcal{B}_1^{z_0}$ into $\mathcal{P}_{\varepsilon}^{z,m}$, which is included in $\mathcal{N}_{\varepsilon}^{\mathcal{T}}$, and $\mathcal{A}_{1,3/2}^{z_0}$ into a parallelipiped with an interior hole, surrounding $\mathcal{P}_{\varepsilon}^{z,m}$, which is included in $\Omega_{\varepsilon}$ (the hole is exactly the parallelipiped $\mathcal{P}_{\varepsilon}^{z,m}$).
Let $T_2:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be the transformation defined as:
\begin{align*}
T_2(x,y,z)=\begin{cases}
(0,0,z) &\text{if}\;x=y=0,\\
\bigg(\sqrt{x^2+y^2},\dfrac{4}{\pi}\sqrt{x^2+y^2}\arctan{\dfrac{y}{x}},z\bigg)&\text{if}\;|y|\leq x,\;x>0,\\
\bigg(-\sqrt{x^2+y^2},-\dfrac{4}{\pi}\sqrt{x^2+y^2}\arctan{\dfrac{y}{x}},z\bigg)&\text{if}\;|y|\leq -x,\;x<0,\\
\bigg(\dfrac{4}{\pi}\sqrt{x^2+y^2}\arctan{\dfrac{x}{y}},\sqrt{x^2+y^2},z\bigg)&\text{if}\;|x|\leq y,\;y>0,\\
\bigg(-\dfrac{4}{\pi}\sqrt{x^2+y^2}\arctan{\dfrac{x}{y}},-\sqrt{x^2+y^2},z\bigg)&\text{if}\;|x|\leq -y,\;y<0,
\end{cases}
\end{align*}
with the inverse
\begin{align*}
T_2^{-1}(\xi,\eta,z)=\begin{cases}
(0,0,z)&\text{if}\;\eta=\xi=0,\\
\bigg(\xi\cos{\dfrac{\pi}{4}\dfrac{\eta}{\xi}},\xi\sin{\dfrac{\pi}{4}\dfrac{\eta}{\xi}},z\bigg)&\text{if}\;|\eta|\leq|\xi|,\;\xi\neq 0,\\
\bigg(\eta\sin{\dfrac{\pi}{4}\dfrac{\xi}{\eta}},\eta\cos{\dfrac{\pi}{4}\dfrac{\xi}{\eta}},z\bigg)&\text{if}\;|\xi|\leq|\eta|,\;\eta\neq 0.
\end{cases}
\end{align*}
More specifically, $T_2(x,y,z)=(\Lambda_2(x,y),z)$, where $\Lambda_2$ is, according to \cite{Rehberg}, a bi-Lipschitz continuous map that maps, in $\mathbb{R}^2$, the unit ball into the unit cube and the Jacobian of $\Lambda_2$ is constant almost everywhere in $\mathbb{R}^2$. Hence, the transformation $T_2$ is bi-Lipschitz and the Jacobian of $T_2$ is constant almost everywhere in $\mathbb{R}^3$.
In our case, we have: $T_2(\mathcal{B}_1^{z_0})=(-1,1)^2\times(-z_0,z_0)$ and $T_2(\mathcal{A}_{1,3/2}^{z_0})=\big((-3/2,3/2)^2\setminus(-1,1)^2\big)\times(-z_0,z_0)$.
Let $u\in H^1(\mathcal{B}_1^{z_0})$, $Q\in H^1(\mathcal{A}_{1,3/2}^{z_0})$ and the constant $c>0$ such that $\big\|\nabla u\big\|_{L^2(\mathcal{B}_1^{z_0})}\leq c\big\|\nabla Q\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}$ be given the previous lemma, constant which is independent of $z_0$. Then we obtain that the functions $\tilde{u}:=u\circ T_2^{-1}\in H^1((-1,1)^2\times(-z_0,z_0))$ and $\tilde{Q}:=Q\circ T_2^{-1}\in H^1\big(\big((-3/2,3/2)^2\setminus(-1,1)^2\big)\times(-z_0,z_0)\big)$ and that there exists constants $c_{j}$ and $c_{J}$, which are also independent of $z_0$, but dependent on the constants given by the Jacobians of $T_2$ and $T_2^{-1}$, such that:
\begin{align*}
c_j\big\|\nabla \tilde{u}\big\|^2_{L^2(T_2(\mathcal{B}_1^{z_0}))}\leq \big\|\nabla u\big\|^2_{L^2(\mathcal{B}_1^{z_0})}\leq c_J\big\|\nabla\tilde{u}\big\|^2_{L^2(T_2(\mathcal{B}_1^{z_0}))}
\end{align*}
and
\begin{align*}
c_j\big\|\nabla \tilde{Q}\big\|^2_{L^2(T_2(\mathcal{A}_{1,3/2}^{z_0}))}\leq \big\|\nabla Q\big\|^2_{L^2(\mathcal{A}_{1,3/2}^{z_0})}\leq c_J\big\|\nabla\tilde{Q}\big\|^2_{L^2(T_2(\mathcal{A}_{1,3/2}^{z_0}))}.
\end{align*}
Hence, the inequality $\big\|\nabla u\big\|_{L^2(\mathcal{B}_1^{z_0})}\leq c\big\|\nabla Q\big\|_{L^2(\mathcal{A}_{1,3/2}^{z_0})}$ implies that there exists a constant $c_0$, also independent of $z_0$, such that:
\begin{align*}
\big\|\nabla\tilde{u}\big\|_{L^2(T_2(\mathcal{B}_1^{z_0}))}\leq c_0\big\|\nabla\tilde{Q}\big\|_{L^2(T_2(\mathcal{A}_{1,3/2}^{z_0}))}.
\end{align*}
Now if we use the transformation $T_3(x,y,z)=\varepsilon^{\alpha}(x,y,z)$ and denote $\overline{u}:=\tilde{u}\circ T_3^{-1}$ and $\overline{Q}:=\tilde{Q}\circ T_3^{-1}$, we get:
\begin{align*}
\varepsilon^{-\alpha}\big\|\nabla\overline{u}\big\|^2_{L^2((T_3\circ T_2)(\mathcal{B}_1^{z_0}))}=\big\|\nabla\tilde{u}\big\|^2_{L^2(T_2(\mathcal{B}_1^{z_0}))}\leq c_0^2\big\|\nabla\tilde{Q}\big\|^2_{L^2(T_2(\mathcal{A}_{1,3/2}^{z_0}))}=c_0^2\varepsilon^{-\alpha}\big\|\nabla\overline{Q}\big\|^2_{L^2((T_3\circ T_2)(\mathcal{B}_1^{z_0}))}
\end{align*}
which implies that
\begin{align*}
\big\|\nabla\overline{u}\big\|_{L^2((T_3\circ T_2)(\mathcal{B}_1^{z_0}))}\leq c_0\big\|\nabla\overline{Q}\big\|_{L^2((T_3\circ T_2)(\mathcal{A}_{1,3/2}^{z_0}))}.
\end{align*}
Since the constant $c_0$ is independent of the choice of $z_0$, we can have $z_0=\dfrac{r\varepsilon-\varepsilon^{\alpha}}{\varepsilon^{\alpha}}$.
The final change of variables is based on the mapping $T_4:\mathbb{R}^3\rightarrow \mathbb{R}^3$ defined as: $T_4(x,y,z)=\bigg(\dfrac{x}{2p},\dfrac{y}{2q},\dfrac{z}{2r}\bigg)$, where $p$, $q$ and $r$ are from relation \eqref{defn:initial_cube_alpha}. In this way, if we translate the origin into the center of the parallelipiped $\mathcal{P}_{\varepsilon}^{z,m}$, we obtain that $(T_4\circ T_3\circ T_2)(\mathcal{B}_1^{z_0})=\mathcal{P}_{\varepsilon}^{z,m}$ and we denote by $\mathcal{R}_{\varepsilon}^{z,m}$ the set $(T_4\circ T_3\circ T_2)(\mathcal{A}_{1,3/2}^{z_0})$, which is the box contained in $\Omega_{\varepsilon}$ (for $\varepsilon$ small enough) that ``surrounds" $\mathcal{P}_{\varepsilon}^{z,m}$.
The transformation $T_4$ is bi-Lipschitz and applying the same arguments as before, we obtain that there exists a function $u\in H^1(\mathcal{P}_{\varepsilon}^{z,m})$ ($u$ can be seen as $\overline{u}\circ(T_4^{-1})$) such that $u=Q$ on the ``contact" faces $\mathcal{T}_z^m$ of $\mathcal{P}_{\varepsilon}^{z,m}$ and an $\varepsilon$-independent constant $c>0$ such that
\begin{align*}
\big\|\nabla u\big\|_{L^2(\mathcal{P}_{\varepsilon}^{z,m})}\leq c\big\|\nabla Q\big\|_{L^2(\mathcal{R}_{\varepsilon}^{z,m})}.
\end{align*}
Since the objects $\mathcal{R}_{\varepsilon}^{z,m}$ are pairwise disjoint (if we look only at the boxes surrounding the parallelipipeds with centers in $\mathcal{Y}_{\varepsilon}^{z}$), repeating the same argument for every parallelipiped of this type (with centers in $\mathcal{Y}_{\varepsilon}^z$) and then repeating the same argument for any parallelipiped from $\mathcal{N}_{\varepsilon}^{\mathcal{T}}$ (that is, with centers in $\mathcal{Y}_{\varepsilon}$), we obtain $u\in H^1(\Omega_{\varepsilon}\cup\mathcal{N}_{\varepsilon}^{\mathcal{T}},\mathcal{S}_0)$ an extension of $Q\in H^1(\Omega_{\varepsilon},\mathcal{S}_0)$ such that:
\begin{align*}
\begin{cases}
u=Q\;\text{in}\;\Omega_{\varepsilon}\\
u=Q\;\text{on}\;\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}\\
\big\|\nabla u\big\|_{L^2(\Omega_{\varepsilon}\cup\mathcal{N}_{\varepsilon}^{\mathcal{T}})}\leq c\big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}.
\end{cases}
\end{align*}
Let $\Omega'_{\varepsilon}=\Omega_{\varepsilon}\cup\mathcal{N}_{\varepsilon}^{\mathcal{T}}$. We want now to construct a function $v:\mathcal{N}_{\varepsilon}^{\mathcal{S}}\rightarrow\mathcal{S}_0$ such that $v=u$ on $\partial\Omega'_{\varepsilon}$ and that there exists a constant $c>0$, independent of $\varepsilon$ such that $\big\|\nabla v\big\|_{L^2(\mathcal{N}_{\varepsilon}^{\mathcal{S}})}\leq c\big\|\nabla u\big\|_{L^2(\Omega'_{\varepsilon})}$.
But in the case of the family $\mathcal{N}_{\varepsilon}^{\mathcal{S}}$, the parallelipipeds are pairwise disjoint for $\varepsilon$ small enough, therefore we can construct $v$ in each $\mathcal{C}_{\varepsilon}^i$, for every $i\in\overline{1,N_{\varepsilon}}$ and control, independent of $\varepsilon$, $\big\|\nabla v\big\|_{L^2(\mathcal{C}_{\varepsilon}^i)}$ with $\big\|\nabla u\big\|_{L^2(\mathcal{P}_{\varepsilon}^i)}$, where $\mathcal{R}_{\varepsilon}^i$ is the ``surrounding" box for $\mathcal{C}_{\varepsilon}^i$, constructed in the same way as $\mathcal{R}_{\varepsilon}^{z,m}$.
\begin{lemma}\label{lemma:extension_v}
Let $a,b\in\mathbb{R}^*_+$ with $a<b$, let $\mathcal{B}_{a}=\big\{x\in\mathbb{R}^3\;\big|\;|x|<a\big\}$ and let $\mathcal{A}_{a,b}=\mathcal{B}_{b}\setminus\overline{\mathcal{B}_a}$.
Let $u\in H^1\big(\mathcal{A}_{1,2},\mathcal{S}_0\big)$. Then the function $v:\mathcal{B}_1\rightarrow\mathcal{S}_0$ defined as
\begin{equation*}
v(x,y,z)=\begin{cases}\varphi(\sqrt{x^2+y^2+z^2})\; u\bigg(\bigg(\dfrac{2}{\sqrt{x^2+y^2+z^2}}-1\bigg)(x,y,z)\bigg)+\\
\hspace{3mm}+(1-\varphi(\sqrt{x^2+y^2+z^2}))\displaystyle{\fint_{\mathcal{A}_{1,3/2}}u(\xi,\eta,\tau)\emph{d}(\xi,\eta,\tau)},\;\emph{for}\;\dfrac{1}{2}\leq\sqrt{x^2+y^2+z^2}<1\\
\displaystyle{\fint_{\mathcal{A}_{1,3/2}}u(\xi,\eta,\tau)\emph{d}(\xi,\eta,\tau)},\;\emph{for}\;0\leq\sqrt{x^2+y^2+z^2}\leq\dfrac{1}{2}\end{cases}
\end{equation*}
is from $H^1(\mathcal{B}_1,\mathcal{S}_0)$, where $\varphi\in C_c^{\infty}\bigg(\bigg(\dfrac{1}{2},\dfrac{3}{2}\bigg)\bigg)$ is the following bump function defined as
\begin{equation*}
\varphi(\rho)=\begin{cases}\exp\bigg\{4-\dfrac{4}{(2\rho-1)(3-2\rho)}\bigg\},\;\forall\rho\in\bigg(\dfrac{1}{2},\dfrac{3}{2}\bigg)\\
0,\;\forall\rho\in\mathbb{R}\setminus\bigg(\dfrac{1}{2},\dfrac{3}{2}\bigg)\end{cases},
\end{equation*}
the product $\varphi(\rho)\;u$ represents product between a scalar and a Q-tensor and $\displaystyle{\fint}$ represents the average integral sign. Moreover, there exists a constant $c>0$ such that: $\big\|v_{t}\big\|_{L^2(\mathcal{B}_1)}\leq c\big\|u_{t}\big\|_{L^2(\mathcal{A}_{1,3/2})}$ for any $t\in\{x,y,z\}$, where $v_t$ represents the partial derivative of $v$ with respect to $t$, and:
\begin{align*}
\|\nabla v\|_{L^2(\mathcal{B}_1)}\leq c\|\nabla u\|_{L^2(\mathcal{A}_{1,3/2})}.
\end{align*}
\end{lemma}
\begin{remark}
\cref{lemma:extension_v} is just a different version of \cref{lemma:extension_u}. The proof follows the same steps as in \cref{lemma:extension_u}.
\end{remark}
Now if we use instead of $T_2$ the transformation $\Lambda_3$, from \cite{Rehberg}, which is a bi-Lipschitz mapping that transforms the unit ball into the unit cube, and then the transformations $T_3$ and $T_4$ as before, we end up with the function $v$ being an extension of $u$ that satisfies:
\begin{align*}
\begin{cases}
v\in H^1(\mathcal{C}^i_{\varepsilon})\\
v=u\;\text{on}\;\partial\mathcal{C}^i_{\varepsilon}\\
\big\|\nabla v\big\|_{L^2(\mathcal{C}^i_{\varepsilon})}\leq c\big\|\nabla u\big\|_{L^2(\mathcal{R}_{\varepsilon}^i)}.
\end{cases}
\end{align*}
Because the objects $\mathcal{R}_{\varepsilon}^i$ are pairwise disjoint for $\varepsilon$ small enough, we construct therefore an extension $v\in H^1(\Omega,\mathcal{S}_0)$ of $u\in H^1(\Omega'_{\varepsilon},\mathcal{S}_0)$ such that:
\begin{align*}
\begin{cases}
v=u\;\text{in}\;\Omega'_{\varepsilon}\Rightarrow v=Q\;\text{in}\;\Omega_{\varepsilon}\;\text{and}\;v=Q\;\text{on}\;\partial\mathcal{N}_{\varepsilon}^{\mathcal{T}}\\
v=u\;\text{on}\;\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}\Rightarrow v=Q\;\text{on}\;\partial\mathcal{N}_{\varepsilon}^{\mathcal{S}}\\
\big\|\nabla v\big\|_{L^2(\Omega)}\leq c\big\|\nabla u\big\|_{L^2(\Omega'_{\varepsilon})}\leq \tilde{c}\big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}.
\end{cases}
\end{align*}
So we have $v\in H^1(\Omega)$, $v=Q$ in $\Omega_{\varepsilon}$, $v=Q$ on $\partial\mathcal{N}_{\varepsilon}$ and there exists an $\varepsilon$-independent constant such that:
\begin{align*}
\big\|\nabla v\big\|_{L^2(\Omega)}\leq c\big\|\nabla Q\big\|_{L^2(\Omega_{\varepsilon})}.
\end{align*}
\subsection{Integrated energy densities}
In this subsection, we present two propositions that are used in order to prove that using relation \eqref{defn:f_hom_sym}, that is:
\begin{align*}
f_{hom}(Q)=\dfrac{2}{p}\int_{\partial\mathcal{C}}f_s(Q,\nu)\text{d}\sigma,
\end{align*}
then by using, for example, the choice of the \textit{surface energy density} defined in \eqref{defn:f_s_LDG}, which is:
\begin{align*}
f_s^{LDG}(Q,\nu)=\dfrac{p}{4}\bigg((a'-a)(\nu\cdot Q^2\nu)-(b'-b)(\nu\cdot Q^3\nu)+2(c'-c)(\nu\cdot Q^4\nu)\bigg),
\end{align*}
we can obtain the corresponding homogenised functional defined in \eqref{defn:f_hom_LDG}, that is:
\begin{align*}
f_{hom}^{LDG}(Q)=(a'-a)\,\text{tr}(Q^2)-(b'-b)\,\text{tr}(Q^3)+(c'-c)\,\big(\text{tr}(Q^2)\big)^2.
\end{align*}
More specifically, \cref{prop:int_en_dens_LDG} treats the case of the classical quartic polynomial in the scalar invariants of $Q$ for the \textit{bulk energy}, defined in \eqref{defn:f_b_LDG}, where the choice of the \textit{surface energy density} is in \eqref{defn:f_s_LDG}, and the more general version of it, defined in \eqref{defn:f_b_gen}, with the \textit{surface energy density} defined in \eqref{defn:f_s_gen}. Both cases have all of the terms from the picked \textit{surface energy densities} of the form $\nu\cdot Q^k\nu$, with $k\geq 2$. \cref{prop:int_en_dens_RP} treats only the Rapini-Papoular case, where the \textit{surface energy density} is defined in \eqref{defn:f_s_RP}.
\begin{prop}\label{prop:int_en_dens_LDG}
For any $k\in\mathbb{N}$, $k\geq 2$ and for a fixed matrix $Q\in\mathcal{S}_0$, we have:
\begin{align*}
\emph{tr}(Q^k)=\dfrac{1}{2}\int_{\partial\mathcal{C}}\big(\nu\cdot Q^k\nu\big)\emph{d}\sigma,
\end{align*}
where $\partial\mathcal{C}$ is defined in \eqref{defn:C_x_C_y_C_z} and $\nu$ is the exterior unit normal to $\partial\mathcal{C}$.
\end{prop}
\begin{proof}
Let $Q^k=\begin{pmatrix}
q_{11,k} & q_{12,k} & q_{13,k}\\
q_{21,k} & q_{22,k} & q_{23,k}\\
q_{13,k} & q_{32,k} & q_{33,k}
\end{pmatrix}$. According to \eqref{defn:C_x_C_y_C_z}, we have $\partial\mathcal{C}=\mathcal{C}^x\cup\mathcal{C}^y\cup\mathcal{C}^z$. We compute first the intergral for $\mathcal{C}^x$, on which $\nu=(\pm 1,0,0)^T$:
\begin{align*}
\int_{\mathcal{C}^x}\big(\nu\cdot Q^k\nu\big)\text{d}\sigma=\int_{\mathcal{C}^x}\big((\pm 1,0,0)^T\cdot(\pm q_{11,k}, \pm q_{21,k}, \pm q_{31,k})^T\big)\text{d}\sigma=\int_{\mathcal{C}^x} q_{11,k}\text{d}\sigma =2q_{11,k},
\end{align*}
since $\mathcal{C}$ has length 1.
In the same way, we obtain:
\begin{align*}
\int_{\mathcal{C}^y}\big(\nu\cdot Q^k\nu\big)\text{d}\sigma=2q_{22,k}\hspace{3mm}\text{and}\hspace{3mm}\int_{\mathcal{C}^z}\big(\nu\cdot Q^k(x_0)\nu\big)\text{d}\sigma=2q_{33,k},
\end{align*}
from which we obtain
\begin{align*}
\int_{\partial\mathcal{C}}\big(\nu\cdot Q^k\nu\big)\text{d}\sigma=2\text{tr}(Q^k).
\end{align*}
\end{proof}
For the Rapini-Papoular case, we prove that:
\begin{prop}\label{prop:int_en_dens_RP}
For a fixed matrix $Q\in\mathcal{S}_0$, we have:
\begin{align*}
6\emph{tr}(Q^2)+4=\int_{\partial\mathcal{C}}\emph{tr}(Q-Q_{\nu})^2\emph{d}\sigma,
\end{align*}
where $\partial\mathcal{C}$ is defined in \eqref{defn:C_x_C_y_C_z}, $Q_{\nu}=\nu\otimes\nu-\mathbb{I}_3/3$, $\nu$ represents the exterior unit normal to $\partial\mathcal{C}$ and $\mathbb{I}_3$ is the $3\times 3$ identity matrix.
\end{prop}
\begin{proof}
First of all, we can see that $\text{tr}(Q-Q_{\nu})^2=\text{tr}(Q^2)-\text{tr}(QQ_{\nu})-\text{tr}(Q_{\nu}Q)+\text{tr}(Q^2_{\nu})$.
According to \eqref{defn:C_x_C_y_C_z}, we have $\partial\mathcal{C}=\mathcal{C}_x\cup\mathcal{C}_y\cup\mathcal{C}_z$. On $\mathcal{C}^x$, we have $\nu=\begin{pmatrix}
\pm 1\\
0\\
0
\end{pmatrix}$ and $Q_{\nu}=\begin{pmatrix}
2/3 & 0 & 0\\
0 & -1/3 & 0\\
0 & 0 & -1/3
\end{pmatrix}$. Then $\text{tr}(Q_{\nu}^2)=\bigg(\dfrac{2}{3}\bigg)^2+\bigg(-\dfrac{1}{3}\bigg)^2+\bigg(-\dfrac{1}{3}\bigg)^2=\dfrac{2}{3}$. We also obtain $\text{tr}(Q_{\nu}^2)=\dfrac{2}{3}$ on $\mathcal{C}^y$ and $\mathcal{C}^z$. Therefore we obtain:
\begin{align*}
\int_{\partial\mathcal{C}}\text{tr}(Q^2_{\nu})\text{d}\sigma=6\cdot\dfrac{2}{3}=4,
\end{align*}
where the constant 6 comes from the total surface of the cube $\mathcal{C}$.
Let $Q=\begin{pmatrix}
q_{11} & q_{12} & q_{13}\\
q_{12} & q_{22} & q_{23}\\
q_{13} & q_{23} & -q_{11}-q_{22}
\end{pmatrix}$. Using the computations done earlier for $Q_{\nu}$ on $\mathcal{C}^x$, $\mathcal{C}^y$ and $\mathcal{C}^z$, we get:
\begin{align*}
\int_{\mathcal{C}^x}\big(\text{tr}(Q_{\nu}Q)+\text{tr}(QQ_{\nu})\big)\text{d}\sigma &=2\bigg(\dfrac{2q_{11}}{3}-\dfrac{q_{22}}{3}+\dfrac{q_{11}+q_{22}}{3}\bigg)=2q_{11}\\
\int_{\mathcal{C}^y}\big(\text{tr}(Q_{\nu}Q)+\text{tr}(QQ_{\nu})\big)\text{d}\sigma &=2\bigg(-\dfrac{q_{11}}{3}+\dfrac{2q_{22}}{3}+\dfrac{q_{11}+q_{22}}{3}\bigg)=2q_{22}\\
\int_{\mathcal{C}^z}\big(\text{tr}(Q_{\nu}Q)+\text{tr}(QQ_{\nu})\big)\text{d}\sigma &=2\bigg(-\dfrac{q_{11}}{3}-\dfrac{q_{22}}{3}-\dfrac{2q_{11}+2q_{22}}{3}\bigg)=-2q_{11}-2q_{22}.
\end{align*}
Combining the last three relations, we get that
\begin{align*}
\int_{\partial\mathcal{C}}\big(\text{tr}(Q_{\nu}Q)+\text{tr}(QQ_{\nu})\big)\text{d}\sigma &=0,
\end{align*}
from which the conclusion follows, with the observation that the constant 6 in front of $\text{tr}(Q^2)$ appears from the total surface of the cube $\mathcal{C}$, which has the length equal to 1.
\end{proof}
\begin{remark}
The constant 4 from \cref{prop:int_en_dens_RP} is neglected when we are studying the asymptotic behaviour of the minimisers of the functional \eqref{defn:F_eps_RP}, since adding constants do not influence the form and the existence of the possible minimisers.
\end{remark}
\end{appendix}
\section*{Acknowledgements}
The work of Razvan-Dumitru Ceuca is supported by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323-17-1 (BES-2017-080630) and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym "DESFLU". The author would like to thank Jamie Taylor and Giacomo Canevari for insightful discussions that have benefited this work and for their support granted during the making of it. The author would also like to thank his Ph.D. supervisor, Arghir-Dani Zarnescu, for the constant mathematical and moral support offered during the proccess of generating this work.
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{
"redpajama_set_name": "RedPajamaArXiv"
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\section{Introduction}
Despite over 80 years of predictive success (reviewed in \cite{schloss06}), the physical interpretation of quantum states, and hence of quantum theory itself remains mysterious (for recent reviews see \cite{schloss07, landsman07, wallace08}). Informally speaking, this mysteriousness results from the apparent dependence of the physical dynamics on the act of observation. Consider Schr\"odinger's cat: the situation is paradoxical because the observer's act of opening the box and looking inside appears to \textit{cause} the quantum state of the cat to ``collapse'' from the distinctly non-classical superposition $|cat\rangle = \frac{1}{\sqrt{2}}(|alive\rangle + |dead\rangle)$ to one of the two classical eigenstates $|alive\rangle$ or $|dead\rangle$. The introduction of decoherence theory in the 1970s and 80s \cite{zeh70, zeh73, zurek81, zurek82, joos-zeh85} transferred this mysterious apparently-causal effect on quantum states from what the observer looks at - the system of interest - to what the observer ignores: the system's environment (reviewed by \cite{joos-zeh03, zurek03rev, schloss04}; see also \cite{schloss07, landsman07, wallace08} for treatments of decoherence in a more general context and \cite{bacci07} for a less formal, more philosophical perspective). Schr\"odinger's poor cat, for example, interacts constantly with the environment within the box - stray photons, bits of dust, etc. - and via the walls of the box with the thermal environment outside. Components of $|cat\rangle$ thereby become entangled with components of the environmental state $|env\rangle$, a state that spreads at the speed of light to encompass all the degrees of freedom of the entire universe (other than the cat's) as the elapsed time $t \rightarrow \infty$. To an observer who does not look at the environment, this entanglement is invisible; the components of the environment can therefore be ``traced out'' of the joint quantum state $|cat \otimes env\rangle$ to produce an ensemble of non-interfering, effectively classical states of just the cat, each with a well-defined probability. Such reasoning about what observers do not look at is employed to derive effectively classical states of systems of interest throughout the applied quantum mechanics literature. For example, Martineau introduces decoherence calculations intended to explain why the Cosmic Background Radiation displays only classical fluctuations with the remarks: ``Decoherence is, after all, an observer dependent effect - an observer who could monitor every degree of freedom in the universe wouldn't expect to see any decoherence. However, our goal is to determine a lower bound on the amount of decoherence as measured by any observer ... we trace out only those modes which we must ... and take our system to be composed of the rest'' (\cite{martineau06} p. 5821). Noting that the setting for these calculations is the inflationary period immediately following the Big Bang, one might ask, ``\textit{Observer}? \textit{What} observer? Looking at \textit{what}?''
Ordinary observers in ordinary laboratories interact with ordinary, macroscopic apparatus in order to gain classical information in the form of macroscopically and stably recordable experimental outcomes. The reconceptualization of physics as an information science that developed in the last quarter of the $20^{\mathrm{th}}$ century, motivated by Feynman's speculation that all of physics could be simulated with a quantum computer \cite{feynman82}, Wheeler's ``it from bit'' proposal that ``all things physical ... must in the end submit to an information-theoretic description'' (\cite{wheeler92} p. 349), Deutsch's proof of the universality of the quantum Turing machine (QTM \cite{deutsch85}) and Rovelli's explicitly information-theoretic derivation of relational quantum mechanics \cite{rovelli96}, reformulated the problem of describing measurement as the problem of describing how observers could obtain classical information in a world correctly described by the quantum mechanical formalism. Theoretical responses to this reconceptualization can be divided into two broad categories by whether they maintain the standard Dirac - von Neumann Hilbert-space formalism as fundamental to quantum mechanics and adopt information-theoretic language to its interpretation, or adopt information-theoretic postulates as fundamental and attempt to derive the Hilbert-space formalism from them. Responses in the first category treat decoherence as a fundamental physical process and derive an account of measurement from it; examples include traditional relative-state (i.e. many-worlds or many-minds) interpretations \cite{joos-zeh03, tegmark98, zeh00, tegmark10, wallace10, susskind11}, the consistent histories formulation \cite{griffiths02, hartle08, griffiths11} and quantum Darwinism \cite{zurek03rev, zurek04, zurek05, zurek06, zurek07grand, zurek09rev}. Those in the second treat measurement as a fundamental physical process; they are distinguished by whether they treat information and hence probabilities as objective \cite{clifton03, bub04, lee11} or subjective \cite{rovelli96, fuchs02, fuchs10, chiribella11, rau11, spekkens11}.
While observers appear as nominal recipients of information in all interpretative approaches to quantum theory, the \textit{physical structure} of an observer is rarely addressed. Zurek \cite{zurek03rev}, for example, remarks that observers differ from apparatus in their ability to ``readily consult the content of their memory'' (p. 759), but nowhere specifies either what memory contents are consulted or what memory contents might be required, stating that ``the observer's mind (that verifies, finds out, etc.) constitutes a primitive notion which is prior to that of scientific reality'' (p. 363-364). Hartle \cite{hartle08} characterizes observers as ``information gathering and utilizing systems (IGUSes)'' but places no formal constraints on the structure of an IGUS and emphasizes that the information gathered by IGUSes is ``a feature of the universe independent of human cognition or decision'' (p. 983). Rovelli \cite{rovelli96} insists that ``The observer can be any physical system having a definite state of motion'' (p. 1641). Schlosshauer \cite{schloss07} adopts the assumption that appears most commonly throughout the literature: ``We simply treat the observer as a quantum system interacting with the observed system'' (p. 361). Fuchs \cite{fuchs10} treats observers as Bayesian agents, and not only rejects but lampoons the idea that the physical implementation of the observer could be theoretically important: ``would one ever imagine that the notion of an agent, the user of the theory, could be derived out of its conceptual apparatus?'' (p. 8). While such neglect (or dismissal) of the structure of the observer is both traditional and \textit{prima facie} consistent with the goal of building a fully-general, observer-independent physics, it seems surprising in a theoretical context motivated by ``it from bit'' and the conceptualization of physical dynamics as quantum computing.
It is the contention of the present paper that the physical structure of the observer is important to quantum theory, and in particular that the information employed by the observer to \textit{identify} the system of interest as an information source must be taken into account in the description of measurement. This contention is motivated by the intuition expressed by Rovelli, that ``the unease (in the interpretation of quantum theory) may derive from the use of a concept which is inappropriate to describe the world at the quantum level'' (\cite{rovelli96} p. 1638). On the basis of this intuition, Rovelli rejects the assumption of observer-independent quantum states, an assumption also rejected by quantum Bayesians \cite{fuchs02, fuchs10, rau11, spekkens11}. The present paper rejects an equally-deep assumption: the assumption of a ``Galilean'' observer, an observer that is simply ``a quantum system interacting with the observed system'' without further information-theoretic constraints. As the analysis of Rovelli \cite{rovelli96} demonstrates, measurement interactions between a Galilean observer and a physical system can be described in terms of Shannon information, but this can only be done from the perspective of a second observer or a theorist who stipulates what is to count as ``observer'' and ``system.'' The use of Galilean observers in an information-theoretic formulation of physical theory thus requires that the identities of ``systems'' be given in advance. That this requirement is problematic has been noted by Zurek, who states that ``a compelling explanation of what the systems are - how to define them given, say, the overall Hamiltonian in some suitably large Hilbert space - would undoubtedly be most useful'' (\cite{zurek98rough} p. 1818), and requires as ``axiom(o)'' of quantum mechanics that ``(quantum) systems exist'' (\cite{zurek03rev} p. 746; \cite{zurek07grand} p. 3; \cite{zurek05env} p. 2) as objective entities. Zurek adopts Wheeler's \cite{wheeler75} view that the universe itself can be considered to be the ``second observer,'' and proposes from this ``environment as witness'' perspective that decoherence provides the physical mechanism by which systems ``emerge'' into objectivity \cite{zurek03rev, zurek04, zurek05, zurek06, zurek07grand, zurek09rev}. Decoherence is similarly proposed to be the mechanism by which quantum information becomes classical \cite{griffiths07} and by which both Everett branches \cite{tegmark10, wallace10} and the frameworks defining consistent histories \cite{griffiths02, hartle08, griffiths11} are distinguished. By rejecting the assumption of Galilean observers, the present paper also rejects the idea that the objective existence of systems can be taken as given \textit{a priori}, either by an axiom or by a physical process of emergence. Instead, it proposes that not just quantum states but systems themselves are definable only relative to observers, and in particular, that quantum systems are defined only relative to classical information encoded by observers. An alternative approach to understanding quantum theory in informational terms is proposed, one that explicitly recognizes the requirement that observers encode sufficient information to enable the identification and hence the definition of the systems being observed.
That ordinary observers in ordinary laboratories must be in possession of information sufficient to identify systems of interest as classical information sources, not just instantaneously but over extended time, is uncontroversial in practice. It follows immediately, moreover, from Moore's 1956 proof that no finite sequence of observations of the outputs generated by a finite automaton in response to given inputs could identify the automaton being observed (\cite{moore56} Theorem 2; cf. \cite{ashby56} Ch. 6). Hence ordinary observers are not Galilean. The information employed by an ordinary, non-Galilean observer to identify a system being observed is ``pragmatic'' information in the sense defined by Roederer \cite{roederer05, roederer11}, although as will be seen below, without Roederer's restriction of such information to living (i.e. evolved self-reproducing) systems. That observers must encode such pragmatic information in their physical structures follows from the physicalist assumption - the complement of ``it from bit'' - that all information is physically encoded \cite{landauer99}. The notion of an ``observer'' as a physical device encoding input-string parsers or more general input-pattern recognizers that fully specify its observational capabilities underlies not only the design and implementation of programming languages and other formal-language manipulation tools (e.g. \cite{kleene67, tan76, hopcroft79}), but also computational linguistics and the cognitive neuroscience of perception (e.g. \cite{marr82, dretske83, rock83, kosslyn94, ullman96, pinker97}).
It is shown in what follows that when the pragmatic information encoded by ordinary observers is explicitly taken into account, distinctive features of the quantum world including the contexuality of observations, the violation of Bell's inequality and the requirement for complex amplitudes to describe quantum states follow naturally from simple physical assumptions. The next section, ``Interaction and System Identification'' contrasts the description of measurement as physical interaction with its description as a process of information transfer, and shows how the problem of system identification arises in the latter context. The third section, ``Informational Requirements for System Identification'' formalizes the minimal information that an observer must encode in order to identify a macroscopic system - a canonical measurement apparatus - that reports the pointer values of two non-commuting observables. It then defines a \textit{minimal observer} in information-theoretic terms as a virtual machine encoding this minimal required information within a control structure capable of making observations and recording their results. The following section, ``Physical Interpretation of Non-commutative POVMs'' considers the physical implementation of a minimal observer in interaction with a physical channel. It shows that if the physical dynamics of the information channel are time-symmetric, deterministic, and satisfy assumptions of decompositional equivalence and counterfactual definiteness, any minimal observer encoding POVMs that jointly measure physical action will observe operator non-commutativity independently of any further assumptions about the observed system. The fifth section, ``Physical Interpretation of Bell's Theorem, the Born Rule and Decoherence'' shows that the familiar phenomenology of quantum measurement follows from the assumptions of minimal observers and channel dynamics that are time-symmetric, deterministic, and satisfy decompositional equivalence and counterfactual definiteness. It shows, in particular, that decoherence can be understood as a consequence of hysteresis in quantum information channels, and that the use of complex Hilbert spaces to represent observable states of quantum systems is required by this hysteresis. The sixth section, ``Adding Minimal Observers to the Interpretation of Quantum Theory'' reviews the ontology that naturally follows from the assumption of minimal observers, an ontology that is realist about the physical world but virtualist about ``systems'' smaller than the universe as a whole. It shows that any interpretative framework that treats ``systems'' as objective implicitly assumes that information is free, i.e. implicitly assumes that the world is classical. The paper concludes by suggesting that the interpretative problem of interest is that of understanding the conditions under which a given physical dynamics implements a given virtual machine, i.e. the problem of understanding the ``emergence'' not of ``classicality'' but of observers.
\section{Interaction and System Identification}
The extraordinary empirical success of quantum theory suggests strongly that quantum theory is the correct description of the physical world, and that classical physics is an approximation that, at best, describes the appearance of the physical world under certain circumstances. Landsman \cite{landsman07} calls the straightforward acceptance of this suggestion ``stance 1'' and contrasts it with the competing view (``stance 2'') that quantum theory is itself an approximation of some deeper theory in which the world remains classical after all. This paper assumes the correctness of quantum theory; Landsman's ``stance 1'' is thus adopted. In particular, it assumes \textit{minimal} quantum theory, in which the universe as a whole undergoes deterministic, unitary time evolution described by a Schr{\"o}dinger equation. The question that is addressed is how the formal structure of minimal quantum theory can be understood physically, as a description of the conditions under which observers can obtain classical information about the evolving states of quantum systems.
As emphasized by Rovelli \cite{rovelli96}, minimal quantum theory treats all systems, including observers, in a single uniform way. The interaction between an observer and a system being observed can, therefore, be represented as in Fig. 1a: both observer and observed system are collections of physical degrees of freedom that are embedded in and interact with the much larger collection of physical degrees of freedom - the ``environment'' - that composes the rest of the universe. The present paper adopts a realist stance about these physical degrees of freedom; they can be considered to be the quantum degrees of freedom of the most elementary objects with which the theory is concerned. The observer - system interaction is described by a Hamiltonian $\mathcal{H}_{\mathbf{O-S}}$; this Hamiltonian is well-defined to the extent that the boundaries separating the observer and the system from the rest of the universe are well-defined. In practice, however, neither the system - environment nor the observer - environment boundaries are determined experimentally. The degrees of freedom composing the system $\mathbf{S}$ are typically specified by specifying a set $\lbrace |s_{i}\rangle \rbrace$ of orthonormal basis vectors, e.g. by saying ``let $|S\rangle = \sum_{i} \lambda_{i} |s_{i}\rangle$.'' The set $\lbrace |s_{i}\rangle \rbrace$ is a subset of a set of basis vectors spanning the Hilbert space $\mathbf{H}_{\mathbf{U}}$ of the universe as a whole; it defines a subspace of $\mathbf{H}_{\mathbf{U}}$ with finite dimension $d$ that represents $\mathbf{S}$. The state of $\mathbf{O}$, on the other hand, is typically left unspecified, and the $\mathbf{O - S}$ interaction is represented not as a Hamiltonian but as a measurement that yields classical information. Traditionally, measurements are represented as orthonormal projections along allowed basis vectors of the system (e.g. \cite{vonNeumann32}); distinct real ``pointer values'' representing distinct observable outcomes are associated with each of these projections. In current practice, the requirement of orthogonality is generally dropped and measurements are represented as positive operator-valued measures (POVMs), sets of positive semi-definite operators $\lbrace \mathcal{E}_{\mathit{j}} \rbrace$ that sum to the identity operator on the Hilbert space of $\mathbf{S}$ (e.g. \cite{nielsen-chaung00} Ch. 2). As shown by Fuchs \cite{fuchs02}, a ``maximally informative'' POVM can be constructed from a set of $d^{2}$ projections $\lbrace \Pi_{j} \rbrace$ on the Hilbert space spanned by $\lbrace |s_{i}\rangle \rbrace$. The first $d$ components of such a POVM are the orthogonal projections $|s_{i}\rangle\langle s_{i}|$; ``pointer values'' can be associated with these $d$ orthogonal components in the usual way.
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Replacing ``physical interaction'' with ``informative measurement'' and hence $\mathcal{H}_{\mathbf{O-S}}$ with $\lbrace \mathcal{E}_{\mathit{j}} \rbrace$ effectively replaces Fig. 1a with Fig. 1b, in which a well-defined observer obtains information from a well-defined system. The surrounding physical environment of Fig. 1a is abstracted into the information channel of Fig. 1b. This idea that information is transferred from system to observer via the environment is made explicit in quantum Darwinism \cite{zurek06, zurek07grand, zurek09rev}. However, it is implicit in the assumption of standard decoherence theory that the observer ``ignores'' the surrounding environment and obtains information only from the system; an observer will receive information from the system alone only if the observer - environment interaction transfers no information, i.e. only if the information content of the environment is viewed as transferred entirely through the system - observer channel.
In the case of human observers of macroscopic systems, the information channel is in many cases physically implemented by the ambient photon field. If the system of interest is stipulated to be microscopic - the electrons traversing a double-slit apparatus, for example, or a pair of photons in an anti-symmetric Bell state - the information channel is often taken to be the macroscopic measurement apparatus that is employed to conduct the observations. For the present purposes, the system will be assumed to be macroscopic, and to comprise both the apparatus employed and any additional microscopic degrees of freedom that may be under investigation. As Fuchs has emphasized \cite{fuchs02, fuchs10}, some intervention in the time-evolution of the system is always required to extract information; hence the channel is two-way as depicted in Fig. 1b. The fact that the channel delivers \textit{classical} information - real values of pointer variables computed by the component operators of POVMs - imposes on the observer an implicit requirement of classical states into which these classical values may be recorded. Viewing observation as POVM-mediated information transfer thus requires observers also to be effectively macroscopic. Consistent with the above characterization of both system and observer as embedded in a ``much larger'' physical environment, the number of states available to either system or observer will be assumed to be much smaller than the number of states within $\mathbf{H}_{\mathbf{U}}$.
Considering the channel through which information flows to be a physical and hence quantum system forcefully raises the question of how the observer identifies as ``$\mathbf{S}$'' the source of the signals that are received. This is the question that was addressed by Moore \cite{moore56} in the general case of interacting automata. Moore's answer, that no finite sequence of observations is sufficient to uniquely identify even a classical finite-state machine, calls into question the standard assumption that the observed system can be identified, either by the observer or by a third party, as a collection of physical degrees of freedom represented by a specified set $\lbrace |s_{i}\rangle \rbrace$ of basis vectors. \textit{Stipulating} that the system can be so represented does not resolve the issue; it merely reformulates the question from one of identifying the system being observed to one of identifying and employing a POVM that acts on the stipulated system and not on something else. This latter question is eminently practical: it must be addressed in the design of every apparatus and every experimental arrangement.
By allowing both the degrees of freedom composing the system of interest and the operators composing the POVM employed to perform observations to be arbitrarily stipulated, the standard quantum-mechanical formalism systematically obscures the question of system identification by observers. While it facilitates computations, placing the ``Heisenberg cut'' delimiting the domain that is to be treated by quantum-mechanical methods around a microscopic collection degrees of freedom further obscures the issue, as it introduces an intermediary - the apparatus - that must also be identified. It has been shown, moreover, that decoherence considerations alone cannot resolve the question of system identification, as decoherence calculations require the assumption of a boundary that must itself be identified: a boundary in Hilbert space that specifies a collection of degrees of freedom, or a boundary in the space of all possible frameworks or Everett branches that distinguishes the framework or branch under consideration from all others \cite{fields10, fields11a}. Absent a metaphysical assumption not just of Zurek's axiom(o), but of the specific \textit{a priori} existence of all and only the systems that observers actually observe, the only available sources of such boundary specifications are observers themselves. The next section examines the question of what such specifications look like in practice.
\section{Informational Requirements for System \\ Identification}
A primary distinction between quantum mechanics and classical mechanics is the failure, in the former but not the latter, of commutativity between physical observables. Implicit in this statement is the phrase, ``for any given system.'' For example, $[\hat{x}, \hat{p}~] = (\hat{x} \hat{p} - \hat{p} \hat{x}) \neq 0$ says that the position and momentum observables $\hat{x}$ and $\hat{p}$ do not commute for states of any particular, identified system $\mathbf{S}$. An observation that $\hat{x}$ and $\hat{p}$ do not commute for states of two spatially separated and apparently distinct systems $\mathbf{S}^{\mathrm{1}}$ and $\mathbf{S}^{\mathrm{2}}$ is \textit{prima facie} evidence that $\mathbf{S}^{\mathrm{1}}$ and $\mathbf{S}^{\mathrm{2}}$ are not distinct systems after all. If $\mathbf{S}^{\mathrm{1}}$ and $\mathbf{S}^{\mathrm{2}}$ are truly distinct, commutativity is not a problem: $[\hat{x}^{\mathrm{1}}, \hat{p}^{\mathrm{2}}] = [\hat{x}^{\mathrm{2}}, \hat{p}^{\mathrm{1}}] = 0$ for all states $|\mathbf{S}^{\mathrm{1}}\rangle$ and $|\mathbf{S}^{\mathrm{2}}\rangle$ operationally defines separability of $\mathbf{S}^{\mathrm{1}}$ from $\mathbf{S}^{\mathrm{2}}$, and warrants the formal representation $|\mathbf{S}^{\mathrm{1}} \otimes \mathbf{S}^{\mathrm{2}}\rangle = |\mathbf{S}^{\mathrm{1}}\rangle \otimes |\mathbf{S}^{\mathrm{1}}\rangle$ of the state of the combined system as separable. Hence quantum mechanics can only be distinguished from classical mechanics by observers that know when they are observing the same system $\mathbf{S}$ twice, as opposed to observing distinct systems $\mathbf{S}^{\mathrm{1}}$ and $\mathbf{S}^{\mathrm{2}}$, when they test operators for commutativity.
The assumption that a single system $\mathbf{S}$ is being observed is indicated in the standard quantum-mechanical formalism by simply writing down ``$\mathbf{S}$'' and saying: ``Let $\mathbf{S}$ be a physical system ...'' In foundational discussions, however, such a facile and implicit indication of sameness can introduce deep circularity. Ollivier, Poulin and Zurek, for example, define ``objectivity'' as follows:
\begin{quotation}
``A property of a physical system is \textit{objective} when it is:
\begin{list}{\leftmargin=2em}
\item
1. simultaneously accessible to many observers,
\item
2. who are able to find out what it is without prior knowledge about the system of interest, and
\item
3. who can arrive at a consensus about it without prior agreement.''
\end{list}
\begin{flushright}
(p. 1 of \cite{zurek04}; p. 3 of \cite{zurek05})
\end{flushright}
\end{quotation}
On the very reasonable assumption that knowing how to identify the system of interest counts as having knowledge about it - exactly what kind of knowledge is discussed in detail below - this definition is clearly circular: each observer must have ``prior knowledge'' to even begin her observations, and the observers must have a ``prior agreement'' that they are observing the same thing to arrive at a consensus about its properties \cite{fields10, fields11a}. Hence while the assumption that observers \textit{can} know that they are observing one single system over time is natural and even essential to experimentation and practical calculations, both its role as a foundational assumption and its relationship to other assumptions that are explicitly written down as axioms of quantum theory bear examination.
Let us fully specify, therefore, the information that an observer $\mathbf{O}$ must have in order to confirm that $[\mathcal{A}_{\mathrm{1}},\mathcal{A}_{\mathrm{2}}] \neq \mathrm{0}$ for two observables $\mathcal{A}_{\mathrm{1}}$ and $\mathcal{A}_{\mathrm{2}}$ and some physical system $\mathbf{S}$. The situation can be represented as in Fig. 2: $\mathbf{O}$ is faced with a macroscopic system $\mathbf{S}$, and at any given time $t$ can measure a value for either $\mathcal{A}_{\mathrm{1}}$ or $\mathcal{A}_{\mathrm{2}}$ but not both. For example, $\mathbf{S}$ could be a Stern-Gerlach apparatus, including ion source, vacuum pump, magnet and power supply, and particle detectors. In this case, $\mathcal{A}_{\mathrm{1}}$ and $\mathcal{A}_{\mathrm{2}}$ are the spin directions $\hat{s}_{x}$ and $\hat{s}_{z}$, the meters are event counters, and the selector switch sets the position of a mask at either of two fixed angles. Let us explicitly assume that $\mathbf{O}$ is herself a finite physical system, that $\mathbf{O}$ can make any finite number of measurements in any order, and that $\mathbf{O}$ has been tasked with recording the values for $\mathcal{A}_{\mathrm{1}}$ or $\mathcal{A}_{\mathrm{2}}$ along with the time $t_{\mathit{k}}$ of each observation. Let us, moreover, explicitly assume that information is physical: that obtaining it requires finite time and recording it requires finite physical memory. For simplicity, assume also that the information channel $\mathbf{C}$ from $\mathbf{S}$ to $\mathbf{O}$ has sufficient capacity to be regarded as effectively infinite; as this channel is implemented by the environment surrounding the experimental set-up, this assumption is realistic.
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Common sense as well as Moore's theorem entail that in order to carry out observations of $\mathbf{S}$, $\mathbf{O}$ must encode information sufficient to (1) distinguish signals from $\mathbf{S}$ from other signals that may flow from the channel; (2) distinguish signals from $\mathbf{S}$ that encode information about the positions of the $\mathcal{A}_{\mathrm{1}} - \mathcal{A}_{\mathrm{2}}$ selector switch and the pointers $\mathbf{P}_{\mathrm{1}}$ and $\mathbf{P}_{\mathrm{2}}$ from signals from $\mathbf{S}$ that do not encode this kind of information; and (3) distinguish between signals that encode different positions of the selector switch and different pointer values for $\mathbf{P}_{\mathrm{1}}$ and $\mathbf{P}_{\mathrm{2}}$. For example, if $\mathbf{S}$ is a Stern-Gerlach apparatus, $\mathbf{O}$ must encode information sufficient to distinguish $\mathbf{S}$ from other systems of similar size, shape and composition, such as leak detectors or general-purpose mass spectrometers. Once $\mathbf{O}$ has identified $\mathbf{S}$, she must be capable of identifying the mask selector and the event counters, and determining both the position of the mask and the numbers displayed on the counters. As $\mathbf{O}$ is finite, all of the information that $\mathbf{O}$ can obtain about $\mathbf{S}$, the selector switch, the pointers, and the values that the pointers indicate can be considered, without loss of generality, to be encoded by finite-precision representations of real numbers. Assuming that one can talk about a well-defined physical state $|\mathbf{C}\rangle$ of the channel $\mathbf{C}$, the information that $\mathbf{O}$ must encode in order to identify and characterize $\mathbf{S}$ and its components can, therefore, be taken to be encoded by four operators that assign (indicated by ``$\mapsto$'') fine-precision real numbers to states $|\mathbf{C}\rangle$ of $\mathbf{C}$:
\begin{equation*}
\mathcal{S}^{\mathbf{O}}(|\mathbf{C}\rangle) \mapsto \left\{
\begin{array}{rl}
(s_{\mathrm{1}}, ..., s_{\mathit{k}}) & \text{if } |\mathbf{C}\rangle \text{ encodes } |\mathbf{S}\rangle \\
\text{NULL} \quad & \text{otherwise}
\end{array} \right.
\end{equation*}
where the $s_{\mathrm{1}}, ..., s_{\mathit{k}}$ are finite real values of a set of \textit{control variables} of $\mathbf{S}$;
\begin{equation*}
\mathcal{P}^{\mathbf{O}}(|\mathbf{C}\rangle) \mapsto \left\{
\begin{array}{rl}
(p_{\mathrm{1}}, p_{\mathrm{2}}) & \text{if } |\mathbf{C}\rangle \text{ encodes } |\mathbf{S}\rangle \\
\text{NULL} & \text{otherwise}
\end{array} \right.
\end{equation*}
where $(p_{\mathrm{1}}, p_{\mathrm{2}}) = (1, 0)$ if the selector switch points to ``$\mathcal{A}_{\mathrm{1}}$'' and $(p_{\mathrm{1}}, p_{\mathrm{2}}) = (0, 1)$ if the selector switch points to ``$\mathcal{A}_{\mathrm{2}}$'';
\begin{equation*}
\mathcal{A}^{\mathbf{O}}_{\mathrm{1}}(|\mathbf{C}\rangle) \mapsto \left\{
\begin{array}{rl}
(a_{\mathrm{1 1}} ... a_{\mathrm{1}\mathit{n}}) & \text{if } |\mathbf{C}\rangle \text{ encodes } |\mathbf{P}_{\mathrm{1}}\rangle \\
\quad & \quad \text{AND } p_{\mathrm{1}} = 1 \\
\text{NULL} & \text{otherwise}
\end{array} \right.
\end{equation*}
where $a_{\mathrm{1 1}} ... a_{\mathrm{1}\mathit{n}}$ are finite real values, and;
\begin{equation*}
\mathcal{A}^{\mathbf{O}}_{\mathrm{2}}(|\mathbf{C}\rangle) \mapsto \left\{
\begin{array}{rl}
(a_{\mathrm{2 1}} ... a_{\mathrm{2}\mathit{m}}) & \text{if } |\mathbf{C}\rangle \text{ encodes } |\mathbf{P}_{\mathrm{2}}\rangle \\
\quad & \quad \text{AND } p_{\mathrm{2}} = 1 \\
\text{NULL} & \text{otherwise}
\end{array} \right.
\end{equation*}
where $a_{\mathrm{2 1}} ... a_{\mathrm{2}\mathit{m}}$ are finite real values. In these expressions, ``NULL'' indicates that the relevant operator returns no value under the indicated conditions. The allowed values of $a_{\mathrm{1}\mathit{k}}$ and $a_{\mathrm{2}\mathit{k}}$ are the $\mathbf{O}$-distinguishable ``pointer values'' for $\mathcal{A}_{\mathrm{1}}$ and $\mathcal{A}_{\mathrm{2}}$ respectively; they are guaranteed to be both individually finite and finite in number, irrespective of the size of the physical state space of $\mathbf{S}$, by the requirement that a finite observer $\mathbf{O}$ records them with finite precision in a finite memory. Figure 3 illustrates the action of these operators on $|\mathbf{C}\rangle$, assuming that $\mathbf{S}$ is in the state shown in Fig. 2.
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As illustrated in Fig. 3, the values of the control variables $s_{\mathrm{1}}, ..., s_{\mathit{k}}$ are what indicate to $\mathbf{O}$ that she is in fact observing $\mathbf{S}$ and not something else. In the case of the Stern-Gerlach apparatus, these may include details of its size, shape and components, as well as conventional symbols such as brand names or read-out labels. In order for $\mathbf{O}$ to recognize these values, they clearly must be real and finite. The control variables must, moreover, take on ``acceptable'' values at $t$ indicating to $\mathbf{O}$ that $\mathbf{S}$ is in a state suitable for making observations. A Stern-Gerlach apparatus, for example, must have an acceptable value for the chamber vacuum and the magnets and particle detectors must be turned on. The entire apparatus must not be disassembled, under repair, or on fire. The existence, recognition by the observer, and acceptable values of such control variables are being assumed whenever ``$\mathbf{S}$'' is written down as the name of a quantum system that is being observed. It is commonplace in the literature (e.g. \cite{tegmark00} where this is explicit) to treat quantum systems as represented during the measurement process by their pointer states alone, but as Figs. 3c and 3d illustrate, such a ``bare pointer'' provides no information by which the system for which it indicates a pointer value can be identified, much less be determined to be in an acceptable state for making observations.
The operators $\mathcal{S}^{\mathbf{O}}$, $\mathcal{P}^{\mathbf{O}}$, $\mathcal{A}^{\mathbf{O}}_{\mathrm{1}}$ and $\mathcal{A}^{\mathbf{O}}_{\mathrm{2}}$ defined above assign finite values, i.e. do not assign ``NULL,'' only for subsets of the complete set of states of $\mathbf{C}$. As discussed above, the information channel $\mathbf{C}$ is physically implemented by the environment in which $\mathbf{S}$ and $\mathbf{O}$ are embedded. Let $\mathbf{H}_{\mathbf{C}}$ be the Hilbert space of this environment. As the environment of any experiment is contiguous with the universe as a whole, with increasing elapsed time the dimension $dim(\mathbf{H}_{\mathbf{C}}) \sim \mathit{dim}(\mathbf{H}_{\mathbf{U}})$; $\mathbf{H}_{\mathbf{C}}$ can therefore be considered to be much larger than the state spaces of either $\mathbf{S}$ or $\mathbf{O}$, and in particular much larger than the memory available to $\mathbf{O}$. Let $\mathcal{S}^{\mathbf{O}}_{\mathit{NULL}}$, $\mathcal{P}^{\mathbf{O}}_{\mathit{NULL}}$, $\mathcal{A}^{\mathbf{O}}_{\mathrm{1} \mathit{NULL}}$ and $\mathcal{A}^{\mathbf{O}}_{\mathrm{2} \mathit{NULL}}$, respectively, be operators defined on $\mathbf{H}_{\mathbf{C}}$ that assign a value of zero to all states within $\mathbf{H}_{\mathbf{C}}$ that do not encode information about the states of $\mathbf{S}$, the selector switch of $\mathbf{S}$, $\mathbf{P}_{\mathrm{1}}$ and $\mathbf{P}_{\mathrm{2}}$ respectively, and ``NULL'' for states within $\mathbf{H}_{\mathbf{C}}$ that do encode such information. A POVM $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$ acting on $\mathbf{H}_{\mathbf{C}}$ can then be defined as follows: let $\mathcal{S}^{\mathbf{O}}_{\mathit{0}} = \mathcal{S}^{\mathbf{O}}_{\mathit{NULL}}$, and for $k \neq 0$ let $\mathcal{S}^{\mathbf{O}}_{\mathit{k}}$ be the component of $\mathcal{S}^{\mathbf{O}}$ that assigns the value $s_{k}$, normalized so that $\mathcal{S}^{\mathbf{O}}_{\mathit{0}} + \sum_{\mathit{k \neq 0}} \mathcal{S}^{\mathbf{O}}_{\mathit{k}} = \mathit{Id}$ where $Id$ is the identity operator for $\mathbf{H}_{\mathbf{C}}$. The component $\mathcal{S}^{\mathbf{O}}_{\mathit{0}}$ of $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$ is by definition orthogonal to the $\mathcal{S}^{\mathbf{O}}_{\mathit{k}}$ with $k \neq 0$; however, these latter components are not, in general, required to be orthogonal to each other. The component of $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$ that assigns the value ``ready'' to $\mathbf{S}$, for example, will not in general be orthogonal to components that establish the identity of $\mathbf{S}$; many parts of $\mathbf{S}$ must be examined to determine that it is ready for use. Practical experimental apparatus are, nonetheless, generally designed to assure that many non-NULL components of $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$ are orthogonal and hence distinguishable and informationally independent. The vacuum gauge on a Stern-Gerlach apparatus, for example, is designed to be distinguishable from and independent of the ammeter on the magnet power supply or the readout on the event counter. In general, the distinguishability and informational independence of components is an operational definition of their separability and hence of the appearance of classicality. The practical requirement that observer-identifiable systems have distinguishable and informationally-independent control and pointer variables is analogous to Bohr's requirement \cite{bohr28} that measurement apparatus be regarded as classical.
Additional POVMs $\lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{2}\mathit{k}} \rbrace$ can be defined by including $\mathcal{P}^{\mathbf{O}}_{\mathit{NULL}}$, $\mathcal{A}^{\mathbf{O}}_{\mathrm{1} \mathit{NULL}}$ and $\mathcal{A}^{\mathbf{O}}_{\mathrm{2} \mathit{NULL}}$ as $0^{th}$ components. As in the case of $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$, these $0^{th}$ components are by definition orthogonal to the others. If $\mathbf{S}$ is assumed to be designed so as to allow only a single kind of measurement to be performed at any given time, and if all observations are assumed to be carried out at maximum resolution, then the non-NULL components of $\lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{2}\mathit{k}} \rbrace$ can also be taken to be orthogonal. For simplicity, orthogonality of these components will be assumed in what follows; the general case can be accomodated by assuming that the components of $\lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$ that indicate incompatible measurements are orthogonal, that components of $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{2}\mathit{k}} \rbrace$ that assign values at maximum resolution are orthogonal, and by considering only these orthogonal components when defining inverse images as described below.
Regarding $\mathcal{S}^{\mathbf{O}}$, $\mathcal{P}^{\mathbf{O}}$, $\mathcal{A}^{\mathbf{O}}_{\mathrm{1}}$ and $\mathcal{A}^{\mathbf{O}}_{\mathrm{2}}$ respectively as POVMs $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{2}\mathit{k}} \rbrace$ acting on $\mathbf{H}_{\mathbf{C}}$ is useful because it removes any dependence on an explicit specification of the boundaries between $\mathbf{C}$ and $\mathbf{S}$ or between the selector switch, $\mathbf{P}_{\mathrm{1}}$, $\mathbf{P}_{\mathrm{2}}$ and the non-switch and non-pointer components of $\mathbf{S}$. These boundaries are replaced, from $\mathbf{O}$'s perspective, by the boundaries of the $\mathbf{O}$-detectable encodings of $\mathbf{S}$ and its components in $\mathbf{H}_{\mathbf{C}}$. Let $\epsilon$ be $\mathbf{O}$'s detection threshold for encodings in $\mathbf{C}$; $\mathbf{O}$ is able to record a value $s_{k}$, for example, only if $\langle \mathbf{C}|\mathcal{S}^{\mathbf{O}}_{\mathit{k}}| \mathbf{C}\rangle \geq \epsilon$. Because $\mathbf{O}$ is a finite observer, $\epsilon > 0$; arbitrarily weak encodings are not detectable. Given this threshold, the encoding of $\mathbf{S}$ can be defined, from $\mathbf{O}$'s perspective, as $\cup_{k} (Im^{-1}(s_{k}))$, where $Im^{-1}(s_{k})$ is the inverse image in $\mathbf{H}_{\mathbf{C}}$ of the detectable value $s_{k}$. Because $\mathcal{S}^{\mathbf{O}}_{\mathit{0}}$ is orthogonal to all of the $\mathcal{S}^{\mathbf{O}}_{\mathit{k}}$ with $k \neq 0$, the intersection $Im^{-1} (\mathcal{S}^{\mathbf{O}}_{\mathit{0}}) \mathit{\cap (\cup_{k} (Im^{-1}(s_{k}))) = \emptyset}$; indeed these inverse images are separated by states for which $0 \leq \langle \mathbf{C}|\mathcal{S}^{\mathbf{O}}_{\mathit{k}}| \mathbf{C}\rangle \leq \epsilon$ for all $\mathcal{S}^{\mathbf{O}}_{\mathit{k}}$ with $k \neq 0$. Let ``$Im^{-1} \lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$'' denote $\cup_{k} (Im^{-1}(s_{k}))$; $Im^{-1} \lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$ is then the proper subspace of $\mathbf{H}_{\mathbf{C}}$ containing vectors to which the POVM $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$ assigns finite real values with probabilities greater than $\epsilon$. The proper subspaces $Im^{-1} \lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $Im^{-1} \lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $Im^{-1} \lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ can be defined in an analogous fashion. As any state $|C\rangle$ that encodes an acceptable value of either the pointer position or the pointer value for either $\mathcal{A}^{\mathbf{O}}_{\mathrm{1}}$ or $\mathcal{A}^{\mathbf{O}}_{\mathrm{2}}$ also encodes acceptable values of the $s_{k}$, it is clear that $Im^{-1} \lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $Im^{-1} \lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $Im^{-1} \lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ are properly contained within $Im^{-1} \lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$.
Specifying $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{2}\mathit{k}} \rbrace$ in terms of the values that they assign for each state $|\mathbf{C}\rangle$ of $\mathbf{C}$ completely specifies $\mathbf{O}$'s observational capabilities regarding $\mathbf{S}$; no further specification of $\mathbf{S}$ or its states is necessary. The notion that ``systems exist'' can, therefore, be dropped; all that is necessary for the description of measurement, other than observers equipped with POVMs, is that \textit{channels} exist. By regarding all POVMs that identify systems or their components as observer-specific (hence dropping the superscript ``$\mathbf{O}$''), the minimal capabilities required by any observer can be defined in purely information-theoretic terms. Given an information channel $\mathbf{C}$, a \textit{minimal observer} on $\mathbf{C}$ is a finite system $\mathbf{O}$ that encodes collections of POVMs $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ within a control structure such that, for each $i$:
\begin{enumerate}
\item The inverse images $Im^{-1} \lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $Im^{-1} \lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$ and $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ for all $j$ are non-empty proper subspaces of $\mathbf{H}_{\mathbf{C}}$ such that $Im^{-1} \lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ properly contains $Im^{-1} \lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$ and the $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ for all $j$.
\item The $s^{\mathit{i}}_{\mathrm{1}}, ..., s^{\mathit{i}}_{n^{\mathit{i}}}$ are accepted by the control structure of $\mathbf{O}$ as triggering the action of the POVM $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ for which $p^{\mathit{i}}_{j} = 1$.
\item The control structure of $\mathbf{O}$ is such that the action on $|\mathbf{C}\rangle$ with $\mathcal{A}^{\mathit{i}}_{\mathit{jk}}$ is followed by recording of the single non-zero value $a^{\mathit{i}}_{\mathit{jk}}$ to memory.
\end{enumerate}
The control structure required by this definition consists of one ``if - then - else'' block for each POVM component, organized as shown in Fig. 4 for a minimal observer with $N$ POVMs $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$. Together with the specified POVMs and a memory allocation process, this control structure specifies a classical virtual machine (e.g. \cite{tan76}), i.e. a consistent semantic interpretation of some subset of the possible behaviors of a computing device. Such a virtual machine may be implemented as software on any Turing-equivalent functional architecture, and hence may be physically implemented by any quantum system that provides a Turing-equivalent functional architecture, such as a QTM \cite{deutsch85} or any of the alternative quantum computing architectures provably equivalent to a QTM \cite{nielsen-chaung00, farhi96, galindo01, perdrix-jorrand06}. \textit{Constructing} such an implementation using a programming language provided by a quantum computing architecture is equivalent to constructing a semantic interpretation of the behavior of the quantum computing architecture that defines the virtual machine using the pre-defined semantics of the programming language. As in the classical case, programming languages for quantum computing architectures provide the required semantic mappings from formal computational constructs (e.g. logical operations or arithmetic) to the operations of the underlying architecture (e.g. unitary dynamics for a QTM or a Hamiltonian oracle) \cite{gay06, rudiger07}; for any universal programming language, however, higher-level interpretations that define specific programs are independent of these lower-level semantic mappings. Hence from an ontological perspective, a minimal observer is a \textit{classical virtual machine} that is \textit{physically implemented} by a quantum system $\mathbf{O}$ that, if not universal, nonetheless provides a sufficient quantum computing architecture to realize all the functions of the minimal observer. A physically-implemented minimal observer interacts with and obtains physically-encoded information from a physically implemented information channel $\mathbf{C}$. Laboratory data acquisition systems that incorporate signal-source identification criteria and stably record measurement results are minimal observers under this definition. As is the case for all physical implementations of classical virtual machines, and for all operations involving classically-characterized inputs to or outputs from quantum computers, the semantic interpretation of a physical (i.e. quantum) system as an implementation of a minimal observer requires, at least implicitly via the semantics of the relevant programming language, an interpretative approach to the quantum measurement problem. The consequences of replacing Galilean observers with minimal observers as defined here for interpretative approaches to the measurement problem are discussed in Sect. 6 below.
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A minimal observer as defined above, and as illustrated in Fig. 4, is clearly not Galilean; it is rather a richly-structured information-encoding entity. The information encoded by a minimal observer is \textit{relative to} a specified control structure, and is therefore \textit{pragmatic}, i.e. used for doing something \cite{roederer05, roederer11}. Hence a minimal observer is not just a ``physical system having a definite state of motion'' or ``a quantum system interacting with the observed system.'' Indeed, if considered apart from its physical implementation, a minimal observer as defined above is not a \textit{quantum system} at all; it is a classical virtual machine, an entity defined purely informationally. One cannot, therefore, talk about the ``quantum state'' of a minimal observer. The traditional von Neumann chain representation (\cite{vonNeumann32}, reviewed e.g. by \cite{schloss07}), in which the observer becomes entangled with the system of interest, after which the observer's quantum state must ``collapse'' to a definite outcome, cannot be defined for a minimal observer, and the information encoded by a minimal observer cannot be characterized by a von Neumann entropy. The \textit{physical implementation} of a minimal observer can be characterized by a quantum state, and hence does have a von Neumann entropy; however, \textit{any} physical implementation that provides a Turing-equivalent architecture and sufficient coding capacity will do. The history of compilers, interpreters, programming languages, and distributed architecures demonstrates that the emulation mapping from a virtual machine to its physical implementation can be arbitrarily complex, indirect, and de-localized in space and time; any straightforward interpretation of von Neumann's principle of ``psychophysical parallelism'' as a constraint on the implementation of minimal observers is, therefore, undone by the architecture that von Neumann himself helped devise two decades after the publication of \textit{Mathematische Grundlagen der Quantenmechanische}.
In consequence of their finite supplies of executable POVMs and finite memories, minimal observers display \textit{objective ignorance} of two distinct kinds. First, a minimal observer cannot, by any finite sequence of observations, fully specify the set of states of $\mathbf{C}$ that encode states of any system $\mathbf{S}$, regardless of the size of the state space of $\mathbf{S}$. This form of objective ignorance follows solely from the large size of $\mathbf{H}_{\mathbf{C}}$ compared to memory available to $\mathbf{O}$. A minimal observer cannot, therefore, determine with certainty that any specification of the states of $\mathbf{S}$ derived from observations is complete. If the observational data characterizing $\mathbf{S}$ obtained by $\mathbf{O}$ are viewed as outputs from an oracle, this failure of completeness can be viewed as an instance of the Halting Problem \cite{tan76, hopcroft79}: $\mathbf{O}$ cannot, in principle, determine whether any oracle that produces a specification of the states of $\mathbf{S}$ will halt in finite time. This first form of objective ignorance blocks for minimal observers the standard assumption of particle physics that the states of elementary particles are specified completely by their observable quantum numbers, downgrading this to a ``for all practical purposes'' specification; it then extends this restriction to all systems, elementary or not. The second form of objective ignorance is that required by Moore's theorem: any system $\mathbf{S}^{\prime}$ that interacts with $\mathbf{C}$ in a way that is indistinguishable using $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{2}\mathit{k}} \rbrace$ from $\mathbf{S}$ will be identified by $\mathbf{O}$ as $\mathbf{S}$. The information provided to $\mathbf{O}$ by $\lbrace \mathcal{S}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathbf{O}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{1}\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathbf{O}}_{\mathrm{2}\mathit{k}} \rbrace$ is, therefore, objectively ambiguous concerning the physical degrees of freedom that generate the encodings in $\mathbf{C}$ on which these operators act. This second form of objective ignorance extends to all systems the indistinguishability within types familiar from particle physics. Neither of these forms of objective ignorance can be remedied by further data acquisition by $\mathbf{O}$; they thus differ fundamentally from subjective or classical ignorance. As will be shown in the two sections that follow, these two forms of objective ignorance together assure that the observational results recorded by a minimal observer will display the typical characteristics predicted by quantum theory, independently of any specific assumptions about the observer's physical implementation.
\section{Physical Interpretation of Non-commutative POVMs}
The definition of a minimal observer given above relies only on the classical concept of information and the system-identification requirements placed on observers by classical automata theory, the assumption that the channel $\mathbf{C}$ is physically implemented by the environment, the idea that information is physical, and the formal notion of a POVM. It provides, however, a robust formal framework with which arbitrary measurement interactions can be characterized. This formal framework makes no mention of ``systems'' other than $\mathbf{O}$ and $\mathbf{C}$, requires no strict specification of the boundary between the physical degrees of freedom that implement $\mathbf{O}$ and those that implement $\mathbf{C}$, and makes no assumption that $\mathbf{O}$ and $\mathbf{C}$ are separable. The physical interpretation of information transfer by POVMs within this framework thus provides a ``systems-free'' interpretation of quantum mechanics with no \textit{a priori} assumptions about the nature of quantum states. This interpretation does not violate the axiomatic assumptions of minimal quantum mechanics in any way; hence it requires no changes in the standard quantum-mechanical formalism or its application in practice to specific cases.
Let us drop temporarily the assumption of minimal quantum mechanics adopted in Sect. 2, and assume only that the physical degrees of freedom composing the coupled system $\mathbf{O \otimes C}$, where ``$\mathbf{O}$'' here refers to the physical implementation of a minimal observer, evolve under some dynamics $\mathcal{H}$ that is time-symmetric and fully deterministic. A natural, classical ``arrow of time'' is imposed on this dynamics, from the perspective of $\mathbf{O}$, by the sequence of memory allocations executed by $\mathbf{O}$'s functional architecture. From a perspective exterior to $\mathbf{O}$ (e.g. the perspective of $\mathbf{C}$), the minimal observer $\mathbf{O}$ is only one of an arbitrarily large number of virtual machines that could describe the physical dynamics of its hardware implementation; hence this $\mathbf{O}$-specific arrow of time is unavailable from such an exterior perspective. Any alternative minimal observer $\mathbf{O}^{\prime}$ will, however, have its own arrow of time determined by its own memory-allocation process.
The large size of $\mathbf{H}_{\mathbf{C}}$ renders the physical degrees of freedom implementing $\mathbf{C}$ \textit{fine-grained} compared to both the detection resolution $\epsilon$ and the memory capacity of any minimal observer $\mathbf{O}$; in particular, these degrees of freedom are fine-grained compared to the inverse images of the POVMs $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ with which $\mathbf{O}$ obtains information about an external system $\mathbf{S}$. As illustrated in Fig. 1a, $\mathbf{O}$ is implemented by the same kinds of physical degrees of freedom that implement $\mathbf{C}$; the degrees of freedom implementing $\mathbf{O}$ are, therefore, also fine-grained compared to $\mathbf{O}$'s memory. Let us assume a weak version of counterfactual definiteness: that the fine-grained degrees of freedom within the inverse images of the $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ implemented by any $\mathbf{O}$ are well-defined at all times; this assumption is a natural correlate, if not a consequence, of the realist stance toward physical degrees of freedom adopted in Sect. 2. Note that this assumption of counterfactual definiteness does not apply to the states of any ``system'' other than $\mathbf{C}$, and that it applies to states of $\mathbf{C}$ without assuming that $\mathbf{C}$ is separable from $\mathbf{O}$. This assumption renders any physical interpretation based on it a ``hidden variables'' theory. However, it does not violate the Kochen-Specker contextuality theorem \cite{kochen67}; indeed it provides a mechanism for satisfying it. The ``hidden'' fine-grained state variables of $\mathbf{C}$ are inaccessible in principle to $\mathbf{O}$, although they fully determine the course-grained measurement results that $\mathbf{O}$ obtains. As discussed above, no two instances of the execution of a $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ triple at times $t$ and $t^{\prime}$ can be assumed by $\mathbf{O}$ to act on the same fine-grained state $|\mathbf{C}\rangle$, nor is any measure of similarity or dissimilarity of channel states $|\mathbf{C}\rangle$ and $|\mathbf{C}\rangle^{\prime}$ other than a $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ triple available to $\mathbf{O}$. All executions by $\mathbf{O}$ of a single measurement $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ are thus contextualized by prior executions of $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ and $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$; executions of pairs $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ and $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{lm}} \rbrace$, commutative or otherwise, are contextualized by two executions of $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ and $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$.
Finally, let us assume that the dynamic evolution of $\mathbf{C}$ does not depend in any way on the POVMs or the control structure implemented by $\mathbf{O}$. Given that $\mathbf{O}$ is by definition a virtual machine, this is an assumption that the physical dynamics $\mathcal{H}$ is independent of its semantic interpretation by any observer. This assumption of \textit{decompositional equivalence} assures that the allocation by $\mathcal{H}$ of fine-grained degrees of freedom to the inverse images of the $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ are independent of the information $\mathbf{O}$ encodes, and hence of $\mathbf{O}$'s ``expectations'' about $\mathbf{C}$ or $\mathcal{H}$. This assumption renders the interpretative framework free of ``subjective'' dependence on the observer. By ruling out any dependence of $\mathcal{H}$ on system - environment boundaries drawn by observers, it also renders the interpretative framework consistent with the common scientific practice of stipulating systems of interest \textit{ad hoc} either demonstratively by pointing and saying ``that'' or formally by specifying lists of degrees of freedom to be included within the boundaries of the stipulated systems.
With these assumptions, the interpretation of $\mathbf{O \otimes C}$ is both realist and objectivist about the fine-grained degrees of freedom implementing $\mathbf{O \otimes C}$, and free, via decompositional equivalence, of any dependence on what observables and hence what descriptions of $\mathbf{C}$ or $\mathcal{H}$ are available to $\mathbf{O}$. The physical interpretation of $[\mathcal{A}^{\mathit{i}}_{\mathit{jk}}, \mathcal{A}^{\mathit{i}}_{\mathit{lm}}] \neq \mathrm{0}$ for POVM components $\mathcal{A}^{\mathit{i}}_{\mathit{jk}}$ and $\mathcal{A}^{\mathit{i}}_{\mathit{lm}}$ must, therefore, also be realist, objectivist, and independent of the descriptions available to $\mathbf{O}$. Suppose that at $t$, $\mathbf{C}$ is in a fine-grained state $|\mathbf{C}\rangle$ such the action $\mathcal{A}^{\mathit{i}}_{\mathit{jk}} |\mathbf{C}\rangle$ would cause $\mathbf{O}$ to record a value $a^{\mathit{i}}_{\mathit{jk}}$ and the action $\mathcal{A}^{\mathit{i}}_{\mathit{lm}} |\mathbf{C}\rangle$ would cause $\mathbf{O}$ to record a value $a^{\mathit{i}}_{\mathit{lm}}$; $|\mathbf{C}\rangle$ at $t$ is thus in the intersection $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}}) \cap Im^{-1}(a^{\mathit{i}}_{\mathit{lm}})$ of the inverse images of $a^{\mathit{i}}_{\mathit{jk}}$ and $a^{\mathit{i}}_{\mathit{lm}}$. In this case, the failure of commutativity can be expressed intuitively (e.g. \cite{schloss07} Ch. 2) in terms of the physical dynamics $\mathcal{H}$ by a pair of counterfactual conditionals:
\begin{quote}
If $|\mathbf{C}\rangle \in \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{jk}}) \cap \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{lm}})$ at $t$ and $\mathbf{O}$ does nothing at $t$, then at a subsequent $t + \Delta t$, $\mathcal{H} |\mathbf{C}\rangle \in \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{jk}}) \cap \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{lm}})$; however, if $|\mathbf{C}\rangle \in \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{jk}}) \cap \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{lm}})$ at $t$ and $\mathbf{O}$ measures either $\mathcal{A}^{\mathit{i}}_{\mathit{jk}}$ or $\mathcal{A}^{\mathit{i}}_{\mathit{lm}}$ at $t$, then at a subsequent $t+ \Delta t$, $\mathcal{H} |\mathbf{C}\rangle \notin \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{jk}}) \cap \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{lm}})$.
\end{quote}
Figure 5 illustrates this situation, the familiar ``dependence of the physical dynamics on the act of observation'' mentioned in the Introduction.
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Implicit in this intuitive formulation of non-commutativity as a counterfactual conditional, and in Fig. 5a, is the idea that the observer could ``do nothing'' at $t$, thus avoiding the ``perturbation'' of $|\mathbf{C}\rangle$ with either $\mathcal{A}^{\mathit{i}}_{\mathit{jk}}$ or $\mathcal{A}^{\mathit{i}}_{\mathit{lm}}$. The definition of a minimal observer, however, permits $\mathbf{O}$ to ``do nothing'' only if the control values $s^{\mathit{i}}_{\mathrm{1}}, ..., s^{\mathit{i}}_{n^{\mathit{i}}}$ are not accepted, i.e. only if (to use the usual language of external systems momentarily) the ``system'' $\mathbf{S}^{\mathit{i}}$ is not identified as ``ready'' by the POVM $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$. If $\mathbf{S}^{\mathit{i}}$ is identified by the action of $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ as ready, $\mathbf{O}$ deterministically makes an observation and records a value. The dynamics depicted in Fig. 5a is thus inconsistent with the condition that $|\mathbf{C}\rangle \in \mathit{Im^{-1}} \lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ at $t$. Consistency with $|\mathbf{C}\rangle \in \mathit{Im^{-1}} \lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ at $t$ requires that if $|\mathbf{C}\rangle \in \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{jk}}) \cap \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{lm}})$ at $t$, $|\mathbf{C}\rangle \notin \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{jk}}) \cap \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{lm}})$ at an immediately-previous $t - \Delta t$. This consistent situation is illustrated in Fig. 6, in which the uncertainties about the state of $\mathbf{C}$ before and after $t$ are symmetric.
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Realism and objectivism demand that the forward and reverse dynamics of $\mathcal{H}$ depicted in Fig. 6 receive the same physical interpretation. Viewing the dynamics symmetrically and considering $\mathbf{O}$'s control structure as shown in Fig. 4 makes the causal structure of the sequence from $t - \Delta t$ to $t$ clear: \textit{if} the physical evolution of $\mathbf{O \otimes C}$ under the action of $\mathcal{H}$ results in $|\mathbf{C}\rangle \in \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{jk}}) \cap \mathit{Im}^{-1}(a^{\mathit{i}}_{\mathit{lm}})$ at $t$, \textit{either} $\mathcal{A}^{\mathit{i}}_{\mathit{jk}}$ or $\mathcal{A}^{\mathit{i}}_{\mathit{lm}}$ will be executed by $\mathbf{O}$ at $t$, with precedence determined by $\mathbf{O}$'s control structure. The control structure of $\mathbf{O}$, however, is a \textit{virtual machine} implemented by the collection of physical degrees of freedom $\mathbf{O}$, the time evolution of which are driven by $\mathcal{H}$. Every action of $\mathbf{O}$, therefore, is fully determined by $\mathcal{H}$ via the emulation mapping that defines $\mathbf{O}$ as a physically-implemented virtual machine. Far from ``dependence of the physical dynamics on the act of observation,'' the transition from $t - \Delta t$ to $t$ illustrates the \textit{deterministic} dependence of the act of observation on the physical dynamics. If the dynamics determines the observation from $t - \Delta t$ to $t$, however, it must determine the observation from $t$ to $t + \Delta t$ as well. There is nothing particular to quantum mechanics in this claim: once information is viewed as physical, the conclusion that an interaction that transfers information from $\mathbf{C}$ to $\mathbf{O}$ also transfers information from $\mathbf{O}$ back to $\mathbf{C}$ follows straightforwardly from Newton's Third Law.
Given this physical interpretation of non-commutativity as a consequence of the reaction of $\mathbf{O}$ on $\mathbf{C}$ that is required by a time-symmetric, deterministic $\mathcal{H}$, $\mathbf{O}$ will observe non-commutativity between any pair of POVMs $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ and $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{lm}} \rbrace$ with $j \neq l$ for which the action of $\mathcal{H}$ on $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})$ alters the subsequent distribution of degrees of freedom into $Im^{-1}(a^{\mathit{i}}_{\mathit{lm}})$ for some $m$ or vice versa. Commutativity of $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ and $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{lm}} \rbrace$ thus requires that $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})$ and $Im^{-1}(a^{\mathit{i}}_{\mathit{lm}})$ are separable under the dynamics $\mathcal{H}$ for all $k$ and $m$. Operators that jointly measure the action of $\mathcal{H}$, in particular, will never satisfy this condition; hence such operators cannot commute. It is impossible, moreover, for any minimal observer to predict the effect of $\mathcal{H}$ on a given $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})$ and alter the choice of subsequent measurement to avoid the appearance of non-commutativity, as doing so would require an ability to represent the state of $\mathbf{O \otimes C}$, a state about which minimal observers are objectively ignorant.
The present framework offers, therefore, a straightforward answer to van Fraassen's \cite{vanFraassen91} question ``How could the world possibly be the way a quantum theory says it is?'' The world is a physically-implemented information channel, it evolves through the action of a time-symmetric, deterministic dynamics that satisfies decompositional equivalence and counterfactual definiteness, and it contains minimal observers implementing pairs of POVMs with non-separable inverse images, in particular pairs of POVMs that jointly measure action. Within the present framework, the more interesting question is the reverse of van Fraassen's: what would the world have to be like for \textit{classical} mechanics to be true, i.e. for dynamics to be time-symmetric, deterministic, satisfy decompositional equivalence and counterfactual definiteness, and for all possible physical observables to commute? There are two answers. First, the world would be classical if information transfer required zero time. If information could be transferred instantaneously, multiple POVMs could act on a single channel state $|\mathbf{C}\rangle$ without intervening reactions of $\mathbf{O}$ on $\mathbf{C}$. Second, the world would be classical if observers had effectively infinite coding capacity. With infinite coding capacity, observers could in principle realize the Laplacian dream of completely modeling $\mathcal{H}$, and hence designing time-dependent POVMs with inverse images that accurately predicted the trajectory from any $|\mathbf{C}\rangle$ to the unique subsequent $\mathcal{H} |\mathbf{C}\rangle$. These conditions could both be true if information was not physical. Hence the operator commutativity required by classical mechanics could be true if information were not physical, and can be derived given a fundamental assumption that information is not physical, that information processing in principle costs nothing, is free (c.f. \cite{landauer99} where free information is identified with classicality). What the empirical success of quantum mechanics tells us is that information \textit{is} physical: that information processing is \textit{not} free.
\section{Physical Interpretation of Bell's Theorem, the Born Rule and Decoherence}
The previous section showed that, given reasonable, traditional, and not explicitly quantum-mechanical assumptions about the dynamics driving the evolution of a physical information channel, any physically-implemented minimal observer equipped with sufficiently high-resolution POVMs will discover one of the primary features of the quantum world: pairs of POVMs with mutually non-separable inverse images, including pairs of POVMs that jointly measure action, will not commute. This section will show that minimal observers equipped with sufficiently high-resolution POVMs will also discover several other canonically ``quantum'' phenomena. Before proceeding, however, it is useful to summarize, in Table 1, the meanings given to the fundamental terms of the standard quantum-mechanical formalism by the formal framework for describing the $\mathbf{O - C}$ interaction developed in the last two sections.
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As shown in Table 1, the fundamental difference between the current framework and the standard quantum formalism is the meaning assigned to the notion of a quantum system. In the standard quantum formalism, a quantum system is a collection of physical degrees of freedom, and any quantum system is observable in principle. In the current framework, an observable quantum system is the non-NULL inverse image, in a physical channel $\mathbf{C}$, of a physically-implemented POVM with a finite number of finite, real output values. The current framework thus \textit{limits} quantum theory by placing an observer-relative, information-theoretic restriction on what ``counts'' as an observable quantum system: the POVM $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ must be physically implemented by an observer $\mathbf{O}$ in order for the ``quantum system'' it detects to exist for $\mathbf{O}$. Thus in the current framework, to paraphrase Fuchs' \cite{fuchs10} paraphrase of de Finetti, ``quantum systems do not exist'' as objective, ``given'' entities. This does not, clearly, mean that the \textit{stuff} composing quantum systems does not exist; both $\mathbf{C}$ and $\mathbf{O}$ are implemented by \textit{physical} degrees of freedom. What it means is that their \textit{boundaries} do not exist. Systems are defined only by observer-imposed decompositions, and physical dynamics do not respect decompositional boundaries.
Quantum states are, in the current framework, equivalence classes under the components of a POVM $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ of states of $\mathbf{C}$ that are indistinguishable, in principle, by an observer implementing $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$. As discussed in Sect. 3, other than whether $|\mathbf{C}\rangle$ is identified by an available POVM $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ and the values $a_{jk}$ assigned by the $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$-selected $j^{th}$ available observable $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ that are obtained in the course of a finite sequence of measurements, observers in the current framework are \textit{objectively ignorant} about quantum states. No physical state $|\mathbf{C}\rangle$ of the channel, and therefore no physical state of any ``system'' $\mathbf{S}$, can be either fully characterized or demonstrated to be replicated by any minimal observer, regardless of the amount of data that observer collects. A world in which no observer is able, in principle, to identify any quantum state as a replicate of any other quantum state is, however, equivalent from the perspective of such an observer to a world in which quantum states cannot be replicated. The observational consequences of objective ignorance regarding the replication of quantum states are, therefore, equivalent to the observational consequences of the no-cloning theorem \cite{wooters82}, which forbids the replication of unknown quantum states. These consequences are realized objectively in the current framework for \textit{all} quantum states, since all are ``unknown'' to all observers. In the current framework, the effective inability to clone quantum states is a consequence of the physicality of information and the boundarylessness of quantum systems defined as inverse images of POVMs.
In the current framework, no-cloning renders all observational results observer-specific. Any two observers $\mathbf{O}$ and $\mathbf{O}^{\prime}$ are objectively ignorant about whether the inverse images of any two POVMs $\lbrace \mathcal{A}^{\mathbf{O}\mathit{i}}_{\mathit{jk}} \rbrace$ and $\lbrace \mathcal{A}^{\mathbf{O}^{\prime} \mathit{i}}_{\mathit{lm}} \rbrace$ are the same subsets of $\mathbf{C}$, whether these POVMs commute or not. Whether two observers share observables can, therefore, at best be established ``for all practical purposes'' by comparing the results of multiple observations. Hence it cannot be assumed, without qualifications, that two distinct observers have both measured a single observable such as $\hat{x}$ for a single system $\mathbf{S}$. This reflects laboratory reality: whether an observation has been successfully replicated in all details is always subject to question.
With these understandings of the familiar terms, the physical meaning of Bell's theorem \cite{bell64} for a minimal observer becomes clear. Consider an observer who measures the same observable on two different ``systems'' $\mathbf{S}^{\mathrm{1}}$ and $\mathbf{S}^{\mathrm{2}}$ employing triples $(\lbrace \mathcal{S}^{\mathrm{1}}_{\mathit{k}} \rbrace, \lbrace \mathcal{P}^{\mathrm{1}}_{\mathit{k}} \rbrace, \lbrace \mathcal{A}^{\mathrm{1}}_{\mathit{jk}} \rbrace)$ and $(\lbrace \mathcal{S}^{\mathrm{2}}_{\mathit{k}} \rbrace, \lbrace \mathcal{P}^{\mathrm{2}}_{\mathit{k}} \rbrace, \lbrace \mathcal{A}^{\mathrm{2}}_{\mathit{jk}} \rbrace)$ of POVMs at times $t$ and $t + \Delta t$ respectively. Between $t$ and $t + \Delta t$, the state of $\mathbf{C}$ evolves from $|\mathbf{C}\rangle$ to $\mathcal{H} |\mathbf{C}\rangle$. Clearly $|\mathbf{C}\rangle \in \mathit{Im}^{-1}(\lbrace \mathcal{S}^{\mathrm{1}}_{\mathit{k}} \rbrace)$ at $t$ and $\mathcal{H} |\mathbf{C}\rangle \in \mathit{Im}^{-1}(\lbrace \mathcal{S}^{\mathrm{2}}_{\mathit{k}} \rbrace)$ at $t + \Delta t$; otherwise the measurements could not be performed. What is relevant to Bell's theorem is whether these inverse images overlap, and in particular, whether $Im^{-1}(\lbrace \mathcal{A}^{\mathrm{1}}_{\mathit{jk}} \rbrace)$ evaluated at $t$ intersects $\mathit{Im}^{-1}(\lbrace \mathcal{A}^{\mathrm{2}}_{\mathit{lm}})$ evaluated at $t + \Delta t$ for any $j$ and $l$. If this intersection is empty, the measured ``states'' $|\mathbf{S}^{\mathrm{1}}\rangle$ and $|\mathbf{S}^{\mathrm{2}}\rangle$ are separable. However, the intersection of the inverse image $Im^{-1}(\lbrace \mathcal{A}^{\mathrm{1}}_{\mathit{jk}} \rbrace)$ at $t$ and the inverse image $\mathit{Im}^{-1}(\lbrace \mathcal{A}^{\mathrm{2}}_{\mathit{jk}} \rbrace)$ at $t + \Delta t$ is only guaranteed to be empty if $\mathcal{H}$ respects the $\mathbf{S}^{\mathrm{1}}$ - $\mathbf{S}^{\mathrm{2}}$ boundary, and \textit{assuming} that $\mathcal{H}$ respects the $\mathbf{S}^{\mathrm{1}}$ - $\mathbf{S}^{\mathrm{2}}$ boundary violates decompositional equivalence. Therefore, the default assumption must be that $Im^{-1}(\lbrace \mathcal{A}^{\mathrm{1}}_{\mathit{jk}} \rbrace)$ at $t$ may overlap $\mathit{Im}^{-1}(\lbrace \mathcal{A}^{\mathrm{2}}_{\mathit{jk}} \rbrace)$ at $t + \Delta t$, and hence that $|\mathbf{S}^{\mathrm{1}}\rangle$ and $|\mathbf{S}^{\mathrm{2}}\rangle$ cannot be regarded as separable. That separability between apparently-distinct systems cannot be assumed by default is the operational content of Bell's theorem, accepting the horn of the dilemma on which counterfactual definiteness and hence the ability to talk about the inverse images of POVMs is assumed.
The problem with the classical reasoning that produces Bell's inequality, on the current framework, is that it assumes that observers can have perfect information about distant systems. If $\mathbf{O}$ is making a local measurement of $\mathbf{S}^{\mathrm{1}}$ at $t$, and $\mathbf{S}^{\mathrm{1}}$ has a spacelike separation from $\mathbf{S}^{\mathrm{2}}$ at $t$, then $\mathbf{O}$ cannot be making a local measurement of $\mathbf{S}^{\mathrm{2}}$ at $t$. If at some later time $t + \Delta t$ $\mathbf{O}$ writes down a joint probability distribution for particular states $|\mathbf{S}^{\mathrm{1}}\rangle$ and $|\mathbf{S}^{\mathrm{2}}\rangle$ at $t$, $\mathbf{O}$ must be in possession at $t + \Delta t$ of data obtained about $|\mathbf{S}^{\mathrm{2}}\rangle$ at $t$, such as a report of the state of $\mathbf{S}^{\mathrm{2}}$ at $t$ from some other observer, e.g. Alice, that is was local to $\mathbf{S}^{\mathrm{2}}$ at $t$. The delivery of this report from Alice to $\mathbf{O}$ requires a physical channel, with which $\mathbf{O}$ must interact, using an appropriate POVM, in order to extract the information contained in the report. Writing down the joint probability distribution for $|\mathbf{S}^{\mathrm{1}}\rangle$ and $|\mathbf{S}^{\mathrm{2}}\rangle$ at $t$ therefore requires that $\mathbf{O}$ make two local measurements, one of $|\mathbf{S}^{\mathrm{1}}\rangle$ at $t$, and one of the report from Alice at the later time $t + \Delta t$. Only if the inverse images of the two POVMs required to make these two measurements are separable is the classical assumption of perfect information transfer from Alice to $\mathbf{O}$ warranted. In standard quantum-mechanical practice, $\mathbf{O}$'s interactions with a macroscopic Alice at $t + \Delta t$ are assumed to separable from Alice's interactions with $\mathbf{S}^{\mathrm{2}}$ at $t$ due to decoherence; any entanglement between Alice and $\mathbf{S}^{\mathrm{2}}$ is assumed to be lost to the environment in a way that renders it inaccessible to $\mathbf{O}$. This assumption, however, rests on an implicit assumption that $\mathbf{O}$ can distinguish Alice from the background of the environment without making a measurement of Alice's state \cite{fields11a}, e.g. before asking for her report. If Alice is microscopic - for example, if Alice is a single photon - this latter assumption is unwarranted, as is the assumption that Alice is no longer entangled with $\mathbf{S}^{\mathrm{2}}$ at $t + \Delta t$. A minimal observer, however, cannot identify any system other than by making a measurement of that system's state. A minimal observer cannot, therefore, assume that decoherence has dissipated any previous entanglement into the environment; as will be described below, for a minimal observer decoherence is a property of information channels, not an observer-independent property of system-environment interactions. Hence as discussed above, a minimal observer cannot assume that the inverse images of any two POVMs are separable; for a minimal observer, the default assumption must be that any two systems are entangled. A minimal observer cannot, therefore, assume perfect information transfer from a distant source of data, and hence cannot derive Bell's inequality for spacelike separated systems using classical conditional probabilities that assume perfect information transfer. For a minimal observer, therefore, the failure of Bell's inequality is expected, and the prediction of its failure by minimal quantum mechanics is positive evidence for the theory's correctness.
Viewing both quantum systems and quantum states as inverse images of POVMs also enables a straightforward physical interpretation of the Born rule. Observers are objectively ignorant, at all times, of both the state $|\mathbf{C}\rangle$ of the information channel and the dynamics $\mathcal{H}$ driving its evolution. By assuming decompositional equivalence, however, an observer can be confident that the future evolution of $|\mathbf{C}\rangle$ will not depend on the locations or boundaries within the state space of $\mathbf{C}$ of the inverse images of the POVMs $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$ or $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$. Such an observer can, therefore, be confident that the probability of obtaining an outcome $a^{\mathit{i}}_{\mathit{jk}}$ following a successful application of $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$, $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$ and $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ to $|\mathbf{C}\rangle$ at some future time $t$ will depend only on the number of physical states within $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})$ relative to the total number of states within of $Im^{-1}(\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace)$ at $t$. The Born rule expresses this confidence that $\mathcal{H}$ respects decompositional equivalence.
Let $P(a^{\mathit{i}}_{\mathit{jk}} | ij, t)$ be the probability that $\mathbf{O}$ records the value $a^{\mathit{i}}_{\mathit{jk}}$ at some future time $t$ given that $\mathbf{O}$ has, immediately prior to $t$, identified a ``system'' $\mathbf{S}^{\mathit{i}}$ by successful application of $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ and selected an observable $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ by successful application of $\lbrace \mathcal{P}^{\mathit{i}}_{\mathit{k}} \rbrace$. Given these conditions, $\mathbf{O}$ deterministically records some value $a^{i}_{jk}$, so $\sum_k P(a^{\mathit{i}}_{\mathit{jk}} | ij, t) = 1$. If the POVM $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ is restricted to only the components with $k \neq 0$ and hence considered to act only on the subspace $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ of $\mathbf{H}_{\mathbf{C}}$, it can be renormalized so that $\sum_{k} \mathcal{A}^{\mathit{i}}_{\mathit{jk}}$ is the Identity on $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$. Following the notation used by Zurek in his proof of the Born rule from envariance \cite{zurek05env}, let $m_{\mathit{k}}$ be the number of states in $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})$ and $M = \sum_{\mathit{k}} m_{\mathit{k}}$ be the number of states in $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$; $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ then corresponds to the ``counter'' ancilla $C$ in Zurek's proof, each of the $k$ components of which contains $m_{\mathit{k}}$ fine-grained states. What Zurek shows is that (in his notation \cite{zurek05env} but suppressing phases) if a joint system-environment state $|\psi_{SE}\rangle$ has a Schmidt decomposition $\sum_{k=1}^{N} a_{\mathit{k}} |s_{\mathit{k}}\rangle |e_{\mathit{k}}\rangle$ with $a_{\mathit{k}} \propto \sqrt{m_{\mathit{k}}}$, an ancilla $C$ of $M$ fine-grained states can be chosen with $k$ mutually-orthogonal components $C_{\mathit{k}}$ such that $C = \cup_{\mathit{k}} C_{\mathit{k}}$ and each $C_{\mathit{k}}$ contains $m_{\mathit{k}}$ fine-grained states. Using the $C_{\mathit{k}}$ to count the number of fine-grained states available for entanglement with any given joint state $|s_{\mathit{k}}\rangle |e_{\mathit{k}}\rangle$, Zurek then shows that the probability $p_{\mathit{k}}$ of observing $|s_{\mathit{k}}\rangle |e_{\mathit{k}}\rangle$ is $m_{\mathit{k}}/M$, which equals $|a_{\mathit{k}}|^{\mathrm{2}}$ by the definition of $C$, giving the Born rule.
In the present context, the formalism of Zurek's proof provides a constructive definition of the unknown future quantum state on which a POVM $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ can act to produce $a^{k}_{\mathit{jk}}$ as a recorded outcome. The inverse image $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ is the subset of $\mathbf{C}$ that ``encodes'' the ''quantum state'' of the ``system'' $\mathbf{S}^{\mathit{i}}$ picked out by the POVM $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$; the rest of $\mathbf{C}$ (i.e. $\mathbf{C} \setminus \mathit{Im^{-1}} \lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$) is the ``environment'' of $\mathbf{S}^{\mathit{i}}$. Hence Zurek's ``$|\psi_{SE}\rangle$'' is a coarse-grained representation of $|\mathbf{C}\rangle$, where the coarse-grained basis vectors ``$|s_{\mathit{k}}\rangle$'' and ``$|e_{\mathit{k}}\rangle$'' span the subpaces $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ and $\mathbf{C} \setminus \mathit{Im^{-1}} \lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ respectively. Given Zurek's assumption that all system states are measureable, the $|s_{\mathit{k}}\rangle$ can be readily identified as the $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})$ for the POVM $\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$; the $|e_{\mathit{k}}\rangle$ are notional, as they are for Zurek. Hence the physical content of the Born rule is that, given decompositional equivalence, the inverse images $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})$ can be regarded as coarse-grained basis vectors for $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ that together provide a complete specification of the state of $Im^{-1} \lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace$ as measurable by $\mathbf{O}$. This is in fact the role of the Born rule in standard quantum theory: it assures that the probabilities $P(a^{\mathit{i}}_{\mathit{jk}} | ij, t)$ are exhausted by the amplitudes (squared) of the measureable basis vectors $|s_{\mathit{k}}\rangle$ of the identified system of interest.
Interpreting the Born rule in this way provides, in turn, a natural physical interpretation of decoherence. Observers, as noted in Sect. 3, are virtual machines implemented by physical degrees of freedom. Any ``system'' identified by a POVM $\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace$ implemented by an observer is, therefore, itself a virtual entity: ``quantum systems do not exist'' as objective entities. Decoherence must, therefore, be a virtual process acting on the information available to an observer, not a physical process acting on the degrees of freedom that implement $\mathbf{C}$. Representing decoherence in this way requires re-interpretating it as an intrinsic property of a (quantum) information channel. Such a re-interpretation can be motivated by noting that the usual physical interpretation of decoherence relies on the identification of quantum systems over time and is therefore deeply circular \cite{fields10, fields11a}.
In standard quantum theory, decoherence occurs when a quantum system $\mathbf{S}$ is suddenly exposed to a surrounding environment $\mathbf{E}$. The $\mathbf{S - E}$ interaction $\mathcal{H}_{\mathbf{S - E}}$ rapidly couples degrees of freedom of $\mathbf{S}$ to degrees of freedom of $\mathbf{E}$, creating an entangled joint state in which degrees of freedom ``of $\mathbf{S}$'' can no longer be distinguished from degrees of freedom ``of $\mathbf{E}$.'' The phase coherence of the previous pure state $|\mathbf{S}\rangle$ is dispersed into the entangled joint system $\mathbf{S - E}$. Under ordinary circumstances decoherence is very fast; Schlosshauer (\cite{schloss07} Ch. 3) estimates decoherence times for macroscopic objects exposed to ambient photons and air pressure to be many orders of magnitude less than the light-transit times for such objects (e.g. $10^{-31}$ s to spatially decohere a $10^{-3}$ cm dust particle at normal air pressure versus a light-transit time of $10^{-14}$ s). It is, therefore, safe to regard all ordinary macroscopic objects exposed to the ordinary macroscopic environment as fully decohered.
It is worth asking, however, what is meant physically by the supposition that $\mathbf{S}$ is ``suddenly exposed'' to $\mathbf{E}$. If $\mathbf{S}$ is ``suddenly exposed'' to $\mathbf{E}$ at some time $t$, it must have been isolated from $\mathbf{E}$ before $t$. Call ``$\mathbf{F}$'' whatever imposes the force required to isolate $\mathbf{S}$ from $\mathbf{E}$. On pain of infinite regress, $\mathbf{F}$ must be in contact with $\mathbf{E}$, in which case decoherence theory tells us that $\mathbf{F}$ and $\mathbf{E}$ are almost instantaneously entangled. The interaction of $\mathbf{F}$ with $\mathbf{S}$ that imposes the force that keeps $\mathbf{S}$ isolated will, however, also entangle $\mathbf{S}$ with $\mathbf{F}$. Unless $\mathbf{F}$ can be partitioned into separable components $\mathbf{F1}$ and $\mathbf{F2}$ that separately interact with $\mathbf{S}$ and $\mathbf{E}$ respectively, however, neither $|\mathbf{F} \otimes \mathbf{S}\rangle$ nor $|\mathbf{F} \otimes \mathbf{E}\rangle$ can be considered to be pure states, and nothing prevents the spread of entanglement from $\mathbf{S}$ to $\mathbf{E}$. Hence unless $\mathbf{F}$ can be partitioned into separable components, $\mathbf{S}$ has never been isolated, and can never be ``suddenly exposed.'' In practice, $\mathbf{F}$ is often a piece of laboratory apparatus such as an ion trap, that interacts with an ``isolated'' system on one surface and the environment on another. The assumption that $\mathbf{F}$ can be partitioned into separable systems is, effectively, the assumption of an internal boundary within $\mathbf{F}$ that is not crossed by any entangling interactions. Such an internal boundary would, however, ``isolate'' everything inside it, and hence require another internal boundary to enforce this isolation. Such an infinite regress of boundaries is impossible; hence no such boundary can exist.
That this reasoning applies across the dynamical domains defined by the relation between the self and interaction Hamiltonians of $\mathbf{S}$ (e.g. \cite{schloss07, landsman07}) can be seen by considering a high-energy cosmic ray that collides with the Earth. During its transit of interplanetary space and the upper atmosphere, the interaction of the cosmic ray with its immediate environment is small; it can be considered ``isolated'' as long as no measurements of its state are made. Its sudden collision with dense matter (e.g. a scintillation counter) ``exposes'' it to the local environment defined by that matter, a local environment that is contiguous with the larger environment of the universe as a whole. This ``sudden exposure'' is, however, an artifact of the limited view of the cosmic ray's history just described. The cosmic ray was produced by a nuclear reaction, e.g. in the Sun. Prior to that reaction, its future components were fully exposed to the local environment of the Sun, a local environment that, like the dense matter on Earth, was contiguous with the larger environment of the universe as a whole. The pre-reaction entanglement between components of the future cosmic ray and other components of the Sun, and hence with other components of the universe as a whole, is not physically destroyed by the formation and flight of the cosmic ray; it is merely inaccessible to observers on Earth, who are only able to experimentally take note of the later, local entanglement between the cosmic ray and the Earth-bound matter with which it collides. It is widely acknowledged that the notion of an ``isolated system'' is a holdover from classical physics; Schlosshauer, for example, notes that ``the idealized and ubiquitous notion of isolated systems remained a guiding principle of physics and was adopted in quantum mechanics without much further scrutiny'' (\cite{schloss07} p. 1). Yet if quantum systems are never isolated, if all physical degrees of freedom are entangled at all times with all other physical degrees of freedom, what is the physical meaning of decoherence?
Standard quantum theory resolves this paradox formally. The formalism distinguishes $\mathbf{S}$ from $\mathbf{E}$ by giving them different names. The representation $|\mathbf{S} \otimes \mathbf{E}\rangle = \mathit{\sum_{ij} \lambda_{ij} |s_{i}\rangle |e_{j}\rangle}$ of the entangled joint state preserves this distinction, as does the joint density $\rho = \frac{1}{2} \sum_{ij} |s_{i}\rangle \langle s_{j}||e_{i}\rangle \langle e_{j}|$ and its partial trace over $\mathbf{E}$, $\rho_{\mathbf{S}} = \frac{1}{N} \sum_{ij=1}^{N} |s_{i}\rangle \langle s_{j}| \langle e_{i}|e_{j}\rangle$. These representations all assume, implicitly, that $\mathbf{S}$ can be identified against the background of $\mathbf{E}$; the partial trace additionally assumes, usually explicitly, that $\mathbf{O}$ is employing an observable $\mathcal{A \otimes I}$ that measures states of $\mathbf{S}$ in some basis but acts as the identity operator on states of $\mathbf{E}$. It is this latter assumption that is expressed by the standard proviso that $\mathbf{O}$ cannot or does not observe the states of $\mathbf{E}$. Given these assumptions, however, the claim that decoherence \textit{explains} $\mathbf{O}$'s ability to distinguish $\mathbf{S}$ from $\mathbf{E}$ by providing a physical mechanism for the ``emergence of classicality'' is clearly circular: the ``emergence'' is built in from the beginning by assigning the distinct \textit{names} $\mathbf{S}$ and $\mathbf{E}$ and assuming that they refer to different things. Indeed, the role of decoherence in standard quantum theory appears to be that of an axiom, somewhat more subtle that von Neumann's axiom of wave-function collapse, stating that observers can distinguish quantum systems from their environments even though the two are always and inevitably entangled. The statement ``decoherence is a physical process'' thus appears entirely equivalent to Zurek's ``axiom(o).''
To see how ``axiom(o)'' is employed in practice, consider the now-classic cavity-QED experiments of Brune \textit{et al.} \cite{brune96} (reviewed in \cite{schloss07} Ch. 6), in which decoherence of a mesoscopic ``Schr\"odinger cat'' created by coupling a well-defined excited state of a single Rb atom to a weak photon field inside a superconducting cavity is monitored as a function of time and experimental conditions. In the standard language of quantum systems and states, the system $\mathbf{S}$ in this case provides two observables, the state $e$ (excited) or $g$ (ground) of an Rb atom after it has traversed the cavity, and the correlation $P_{ij}(\Delta t)$ between the states of successive atoms $i$ and $j$ arriving at the detector with a time difference of $\Delta t$. The experimental outcomes are: (1) varying the coupling between the atomic state and the photon field varies the amount of information about the traversing atom's state that was stored in the field (\cite{brune96} Fig. 3); and (2) varying the time interval $\Delta t$ varies the amount of information about the $i^{th}$ atom's state that could be extracted from the $j^{th}$ atom's state (\cite{brune96} Fig. 5). The first result demonstrates that increasing the local interaction between two \textit{identified} degrees of freedom (by increasing the coupling) increases the entanglement between \textit{those} degrees of freedom. The second result demonstrates that after the local interaction between the two identified degrees of freedom (after the $i^{th}$ atom leaves the cavity), the entanglement between those degrees of freedom dissipates; the field in the cavity is also entangled with the atoms in the walls of the cavity, and this latter entanglement decoheres the ``information'' about the $i^{th}$ atom's state that ``the atom leaves in (the cavity) $C$'' (\cite{brune96} p. 4889). Critical to this explanation is the tacit assumption that the states of the atoms in the walls of the cavity are not themselves observed, or equivalently, that the atoms in the walls of the cavity are themselves entangled with the general environment in which the apparatus is embedded. But, this assumption comes with the implicit proviso that this prior system - environment entanglement \textit{does not prevent the identification of quantum states} of the individual Rb atoms traversing the cavity. This assumption that the individual Rb atoms can be regarded ``objectively'' even in the presence of system - environment entanglement is an instance of ``axiom(o).''
The current framework alters this standard account of the physics by re-casting it in informational terms and rejecting the tacit assumption that the $i^{th}$ and $j^{th}$ Rb atoms are distinguishable quantum systems. The ``system'' $\mathbf{S}^{\mathit{B}}$ in this framework (``$B$'' for Brune \textit{et al.}) is the inverse image of a POVM $\lbrace \mathcal{S}^{\mathit{B}}_{\mathit{k}} \rbrace$ with control variables $s^{B}_{\mathrm{1}}, ..., s^{B}_{n^{B}}$. Distinct acceptable sets of values of these variables describe distinct preparation conditions for the system. This system can be considered an information channel from $Im^{-1}(\lbrace \mathcal{A}^{\mathit{B}}_{\mathit{g}}, \mathcal{A}^{\mathit{B}}_{\mathit{e}} \rbrace)$ to $\mathbf{O}$, where the components of $\mathcal{A}^{\mathit{B}}$ report the outcomes $g$ and $e$ respectively. In this representation, long-lived entanglement between the atom traversing the cavity and the photon field within it causes delocalization in time of the outcome: the values of the control variables $s^{B}_{\mathrm{1}}, ..., s^{B}_{n^{B}}$ - specifically, those indicating the mirror separation and hence tuning of the cavity - can be adjusted in a way that smears an outcome $g$ (for example) out over pairs of applications of $\mathcal{A}^{\mathit{B}}$. Figure 7 illustrates this smearing in time using a simple circuit model, in which the (approximately) fixed ``resistance'' $R$ represents information loss from the channel (e.g. the approximately fixed coupling of the photon field to the cavity) and the variable ``capacitance'' $C$ represents the intrinsic memory of the channel (e.g. the manipulable coupling of the atomic beam to the photon field). An instantaneous input impulse $\delta (t - t_{0})$ at $t = t_{0}$ results in an output $\propto e^{-t/RC}$ for $t > t_{0}$ at $\mathbf{O}$. The time constant $RC$ is the decoherence time; it is a measure of the channel's memory of each outcome.
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The ``capacitance'' $C$ in Fig. 7 is clearly a measure of the ``quantum-ness'' of the channel; as $C \rightarrow 0$, the channel appears classical. The condition $C = 0$ corresponds to infinite temporal resolution for measurement events; hence it corresponds to the ``free information'' (i.e. $\hbar \rightarrow 0$) assumption of classical physics discussed at the end of Sect. 4. If $C = 0$, the channel stores no information about previous outcomes, so all pairs of POVMs, including those that jointly measure action commute. The ``resistance'' $R$ measures the leakiness of the channel in either direction; as $R \rightarrow 0$, the channel approaches infinite decoherence time, i.e. perfect isolation, in the quantum ($C > 0$) case, and the ideal of noise-free communication in the classical ($C = 0$) case.
Given the representation of an information channel as an $RC$ circuit, consider a random sequence of measurements with the POVM $\lbrace \mathcal{A}^{\mathit{B}}_{\mathit{g}}, \mathcal{A}^{\mathit{B}}_{\mathit{e}} \rbrace$. These measurements correspond to a random sequence of ``states'' of $Im^{-1}(\lbrace \mathcal{A}^{\mathit{B}}_{\mathit{g}}, \mathcal{A}^{\mathit{B}}_{\mathit{e}} \rbrace)$. The no-cloning theorem requires that these ``states'' be non-identical, and hence that the collections of fine grained states $|\mathbf{C}(\mathit{t})\rangle$ that physically implement them be non-identical. The individual measurement outcomes cannot, therefore, be ``remembered'' at $C$ as identical; the ``memory traces'' of distinct $|\mathbf{C}(\mathit{t_{\mathrm{1}}})\rangle$ and $|\mathbf{C}(\mathit{t_{\mathrm{2}}})\rangle$ stored at $C$ must interfere. From $\mathbf{O}$'s perspective, this interference can be represented formally by adding a random phase factor $e^{-i \phi}$ to each transmission through the channel. Without such interference, the signal at $\mathbf{O}$ would increase monotonically with time if measurements were made with a time separation less that $RC$, since $C$ would never fully discharge. Such arbitrarily temporally-delocalized outcomes are never observed in practical experiments. Adding the random phase term assures that, for $t \gg RC$, interference between measurements drives the time-averaged signal at $\mathbf{O}$ toward zero. In this purely informational $RC$-circuit model of decoherence, therefore, no-cloning is what requires the use of a complex Hilbert space to represent ``states'' in the inverse image $Im^{-1}(\lbrace \mathcal{A}^{\mathit{i}}_{\mathit{jk}} \rbrace)$ of any observable associated with an identified system. Treating the $Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})$ as names of coarse-grained basis vectors for the ``system'' $Im^{-1}(\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace)$ as discussed above, an unknown quantum state of $Im^{-1}(\lbrace \mathcal{S}^{\mathit{i}}_{\mathit{k}} \rbrace)$ as measured at a future time $t$ using the $j^{th}$ available POVM $\mathcal{A}^{\mathit{i}}_{\mathit{jk}}$ can be written $|\psi^{\mathit{i}}_{j} (t)\rangle = \sum_{\mathit{k}} \alpha_{\mathit{k}} e^{-i \phi_{\mathit{k}}} |Im^{-1}(a^{\mathit{i}}_{\mathit{jk}})\rangle$ with $\alpha_{\mathit{k}}$ real, exactly as expected within standard quantum theory.
A ``quantum channel'' defined solely by non-commutativity between observables jointly measuring action is, therefore, a quantum channel as defined by standard quantum theory, provided that information is physical and the observer is a minimal observer as defined in Sect. 3. If determinism, time-symmetry, counterfactual definiteness and decompositional equivalence are assumed, observations made through such channels satisfy the Kochen-Specker, Bell, and no-cloning theorems. The Born rule emerges as a consequence of decompositional equivalence. Complex phases are required by objective ignorance of the physical states implementing the channel, i.e. by no cloning. Decoherence is understandable not as a physical process acting on quantum states, but as an intrinsic hysteresis in quantum information channels. Measurement, in this framework, is unproblematic; \textit{if} minimal observers exist, the determinate, ``classical'' nature of their observations follows straightforwardly from their structure as classical virtual machines and the physics of a quantum channel. The fundamental interpretative assumptions that must be added to quantum theory appear, then, to be that information is physical and that minimal observers exist.
\section{Adding Minimal Observers to the Interpretation of Quantum Theory}
If Galilean observers are replaced by minimal observers as defined in Sect. 3, the interpretation of quantum theory is radically simplified. The traditional problems of why some measurement bases, such as position, are ``preferred'' and how superpositions can ``collapse'' onto determinate eigenstates of those bases are immediately resolved: a minimal observer ``prefers'' the bases in which she encodes POVMs, and is only capable of recording eigenvalues in these bases. The problem of the ``emergence'' of the classical world also vanishes: the classical world is the world of recorded observations made by minimal observers. Minimal observers are virtual machines implemented by physical degrees of freedom; hence the classical world is a virtual world. What the current framework adds to previous proposals along these lines (e.g. \cite{whitworth10}) is a precisely formulated model theory: the model theory expressed by the POVMs implemented by the minimal observer.
From an ontological perspective, the current framework can be viewed as an interpolation between two interpretative approaches generally regarded as diametrical opposites: a ``pure'' relative-state interpretation such as that of Tegmark \cite{tegmark10} and the quantum Bayesianism (``QBism'') of Fuchs \cite{fuchs10}. Like QBism, the current framework views quantum states as observer-specific virtual entities. However, instead of ``beliefs'' as they are in QBism, these virtual entities are inverse images of observer-specific POVMs in the space of possible states of the real physical world. Like a pure relative-state interpretation, the current framework postulates a deterministic, time-symmetric Hamiltonian satisfying counterfactual definiteness and decompositional equivalence. However, ``branching'' into arbitrarily many dynamically-decoupled simultaneous actualities is replaced by the classical notion that a sufficiently complex physical system can be interpreted as implementing arbitrarily many semantically-independent virtual machines. Like QBism, the current framework rejects the interpretation of decoherence as a physical mechanism that generates actuality; unlike QBism, it views the ``classical world'' as entirely virtual and rejects the observer-independent ``real existence'' of bounded, separable macroscopic objects. Like a pure relative-state interpretation, the current framework embraces non-locality as an intrinsic feature of the universe; unlike a pure relative-state interpretation, it views non-locality as a temporal relationship between instances of observation, not as a spatial relationship between objects. The current framework is, therefore, ontologically very spare. It postulates as ``real'' only the in-principle individually unobservable physical degrees of freedom that implement both channel and observer. The virtual machines that are postulated are not in any sense physical; unlike Everett branches \cite{tegmark10}, there is no sense in which virtual machines constitute parallel physical actualities. This strongly Kantian ontology is similar to that of the recent ``possibilist'' extension \cite{kastner10a} of the transactional interpretation \cite{cramer86, cramer88}, but without the notion that transactions ``actualize'' quantum phenomena in an observer-independent way.
What the current interpretative framework emphatically rejects is the notion that the ``environment'' is a \textit{witness} that monitors quantum states and defines systems \textit{for} observers. The idea that the environment \textit{preferentially} encodes certain ``objective'' quantum states and makes information about these states and not others available to observers is the foundation of quantum Darwinism \cite{zurek03rev, zurek04, zurek05, zurek06, zurek07grand, zurek09rev}. It is implicit, however, in all interpretative approaches in which the classical world ``emerges'' from the dynamics in an observer-independent way. The bounded and separable ``real existences'' postulated by QBism \cite{fuchs10}, for example, are effectively the observations of the ``rest of the universe'' viewed as an observing agent \cite{fields11b}. The ``witness'' assumption can be found in interpretative approaches as distant in terms of fundamental assumptions from both QBism and quantum Darwinism as the possibilist transactional interpretation, where an ``experimental apparatus seems persistent in virtue of the highly probable and frequent transactions comprising it'' (\cite{kastner10b} p. 8) not from the perspective of an observer, but from the perspective of an observer-free universe. It is this assumption of emergence via environmental witnessing that enables, explicitly or otherwise, the traditional and ubiquitous assumption of information-free Galilean observers, mere points of view or (as ``preparers'' of physical systems) points of manipulation of a pre-defined objective reality.
As pointed out in the Introduction, the logical coherence of Galilean observers must be rejected on the basis of classical automata theory alone \cite{moore56, ashby56}. It is useful, however, to examine the Galilean observer from the perspective of the ``environment as witness.'' Consider the classic Wigner's friend scenario \cite{wigner61}, but with an omniscient ``friend'' who monitors not just an atomic decay but the states of all possible ``systems'' in the universe. An observer can then obtain information about the state of any system by asking his friend, i.e. by interacting with the local environment as envisaged by quantum Darwinism. A minimal observer asks his friend \textit{in language}, by executing a POVM. The information that such an observer can obtain from the environment, whether viewed as a communication channel or as an omniscient oracle, is limited by the observer's repertoire of POVMs; a minimal observer can obtain no information about a system he cannot describe, and cannot ``observe'' that a system is in a state he cannot represent and record. A Galilean observer, in contrast, stores no prior information and hence has no language. Having an omniscient friend does not help a Galilean observer; they have no way to communicate. The assumption that a Galilean observer can nonetheless obtain any information encoded by the environment is, effectively, the assumption that the observer has the same encoding capacity as the environment: what is ``given'' to the omniscient environment is also ``given'' to the Galilean observer. This assumption was encountered at the end of Sect. 3; it is the familiar, classical assumption that information is free.
Replacing Galilean observers with minimal observers replaces the intractable philosophical problem of why observers never observe superpositions - a pseudoproblem that results from the informationally-impoverished and hence unconstrained nature of the Galilean observer - with two straightforwardly scientific problems. The first is a problem in quantum computer science: what \textit{classical} virtual machines can be implemented by a given \textit{quantum} computer, e.g. by a given Hamiltonian oracle \cite{farhi96}? One answer to this question is known: a quantum Turing machine \cite{deutsch85} can implement any classical virtual machine. A second, more practical, answer is partially known: the quantum systems, whatever they are, that implement our everyday classical computers are Turing equivalent. What we do not know is how to describe these familiar systems quantum mechanically, or how to approach the analysis of an arbitrary quantum system capable of implementing some limited set of classical virtual machines. The second problem straddles the border between machine intelligence and biopsychology. It is the question of what physically-realized virtual machines share POVMs, and of how these systems came to share them. If we are to understand how multiple observers can reach an agreement that they are observing the same properties of the same thing, it is this question that we must be able to answer.
\section{Conclusion}
This paper has investigated the consequences of replacing the Galilean observer traditionally employed in interpretations of quantum theory with an observer that fully satisfies the requirements of classical automata theory. It has shown that if both the observer and the information channel with which it interacts are implemented by physical degrees of freedom, the state space of which admits a linear measure enabling the definition of POVMs, and if the temporal dynamics of these physical degrees of freedom are deterministic, time symmetric, and satisfy decompositional equivalence and counterfactual definiteness, then the observations made by the observer are correctly described by standard quantum theory. Quantum theory does not, therefore, require more than these assumptions. The unmotivated and \textit{ad hoc} nature of the formal postulates that have been employed to axiomitize quantum theory, both traditionally \cite{vonNeumann32} and more recently (e.g. \cite{bub04, rau11}) can be seen as a side-effect of the assumption of Galilean observers and the compensatory, generally tacit assumption of ``axiom(o).''
The introduction of information-rich minimal observers into quantum theory brings to the fore the distinction between Shannon or von Neumann information defined solely by the dynamics and pragmatic information defined relative to an emulation mapping that specifies a control structure and hence a virtual machine. A deterministic, time-symmetric Hamiltonian conserves fine-grained dynamic information; the von Neumann entropy of the channel $\mathbf{C}$ is zero. Nonetheless, the pragmatic information - the list of observational outcomes - recorded by a minimal observer with an approximately ideal memory increases monotonically with time. Pragmatic information appears, therefore, not to be conserved; ``history'' appears actual, objective and given. This apparent asymmetry is, however, illusory. Pragmatic information is only definable relative to an emulation mapping, a semantic interpretation of $\mathbf{C}$. Every classical bit encoded by a minimal observer must be computed when such an emulation mapping is specified. Hence pragmatic information is not free; it is balanced by the computational effort required to specify emulation mappings. This effort is ``expended'' by $\mathcal{H}$ as dynamic evolution unfolds; minimal observers and the outcomes that they record are the result. ``It from bit'' is thus balanced by ``bit from it.''
\section*{Acknowledgement}
Thanks to Eric Dietrich, Ruth Kastner and Juan Roederer for stimulating discussions of some of the ideas presented here. Three anonymous referees provided helpful comments on the manuscript.
\bibliographystyle{mdpi}
\makeatletter
\renewcommand\@biblabel[1]{#1. }
\makeatother
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Android - How to make the recyclerview ListAdpater stop blinking? I have a recycler view whose adapter uses ListAdapter (version 1.1.0) :
class InnerEpisodeFragmentAdapter(
private val actCtx: Context,
) : ListAdapter<Episode, InnerEpisodeFragmentAdapter.MViewHolder>(COMPARATOR) {
...
The recycler view is fed by a kotlin flow coming from Room database Episode table :
vm.episodesFlow().asLiveData().observe(viewLifecycleOwner) { episodes ->
episodes.let { adapter.submitList(it) }
}
@Query("SELECT * FROM Episode ORDER BY pubDate DESC")
fun episodesFlow(): Flow<List<Episode>>
As excepted, each time a tuple changes in Episode table, a new list of episodes is emitted and the recycler view is update.
It works fine, but with an horrible BLINK at each update. It gives a bad user experience.
How can I avoid this blinking when the flow emit new values ?
In the previous version of my app, I used functions like notifyDataSetChanged() or notifyItemChanged() which never blink.I know I could still try to use these functions but I would be very disappointed if I could not avoid the blinks when using the kotlin flow as shown above. Thanks.
A: you comparators
areItemsThesame() is wrongly implemented. What you are comparing is reference where is old object and newObject might not same objects. Instead you some use some kind of uuid to compare the two like primary key. If objects are different it will return false and hence areContentsTheSame will never be called otherwise if objects are same their is no point of comparing contents as both oldObject and newObject are pointing to same reference.
Sorry for errors as I am writing from Mobile
A: ======== REQUESTED ADDITIONAL INFORMATION ======
My comparator is fine and does exactly what I want.
For example, I want isPlayed to be updated in real time in each recycler view item. And it is. But with a blink of the whole list. The recycler view disappears for a fraction of a second then reappear with the updated info.
companion object {
private val COMPARATOR = object : DiffUtil.ItemCallback<Episode>() {
override fun areItemsTheSame(oldItem: Episode, newItem: Episode): Boolean {
return oldItem === newItem
}
override fun areContentsTheSame(oldItem: Episode, newItem: Episode): Boolean {
return (oldItem.id == newItem.id)
&& (oldItem.isOnDisk == newItem.isOnDisk)
&& (oldItem.downloadId == newItem.downloadId)
&& (oldItem.isPlayed == newItem.isPlayed)
&& (oldItem.isDeleted == newItem.isDeleted)
}
}
}
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"
"http://www.w3.org/TR/html4/strict.dtd">
<META HTTP-EQUIV="CACHE-CONTROL" CONTENT="NO-CACHE">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>{% block title %} OpenMDAO Image Viewer {% end %}</title>
<style type="text/css">
html, body {
margin: 0;
padding: 0;
height: 100%;
}
#images {
height: 100%;
width: 100%;
}
</style>
<!-- ben alman's debug script -->
<script type="text/javascript" src="/static/js/ba-debug.min.js"></script>
<!-- jQuery -->
<script src="/static/js/jquery-1.8.2.min.js"></script>
<!-- galleria -->
<link type="text/css" rel="stylesheet" href="/static/js/galleria/themes/classic/galleria.classic.css">
<script type="text/javascript" src="/static/js/galleria/galleria-1.2.9.min.js"></script>
<script type="text/javascript" src="/static/js/galleria/themes/classic/galleria.classic.min.js"></script>
<!-- openmdao -->
<script src="/static/js/openmdao/Project.js"></script>
<script src="/static/js/openmdao/Util.js"></script>
<script>
jQuery(function() {
// define openmdao namespace & create interface to openmdao in global scope
openmdao = (typeof openmdao === 'undefined' || !openmdao ) ? {} : openmdao ;
if (opener && opener.openmdao && opener.openmdao.project ) {
openmdao.project = opener.openmdao.project;
}
else {
openmdao.project = new openmdao.Project();
}
openmdao.project.getFiles()
.done(function(files) {
var images = [],
index = 0,
show_index = 0;
function getImages(files) {
for (var file in files) {
if (typeof files[file] === 'object') {
getImages(files[file]);
}
else if (openmdao.Util.hasImageExtension(file)) {
file = file.replace(/\\/g,'/');
{% if 'filename' in globals() %}
if (file === "{{ filename }}") {
show_index = index;
}
{% end %}
images.push({
title: file,
image: '/workspace/file'+file
});
index++;
}
}
};
getImages(files);
jQuery('#images').galleria({
dataSource: images,
width: '100%',
height: '100%',
responsive: true,
show: show_index
});
})
.fail(function(jqXHR, textStatus, errorThrown) {
debug.error('Error getting files',
jqXHR, textStatus, errorThrown);
});
});
</script>
</head>
<body>
<div id="images"></div>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 5,963
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Martin Ulf Pringle (born 18 November 1970) is a Swedish former professional footballer who played as a centre forward, and the current manager of Varbergs BoIS.
As a player, he played from 1991 until his career was cut short by injury in 2002. Having started his career with Stenungsund he soon moved on to Helsingborgs IF before a switch to Portugal in 1996 with Benfica. In 1999, he moved to England with Charlton Athletic. Having suffered a horrific injury at Athletic that sidelined him for up to a year he returned in 2002 in a loan spell with Grimsby Town but was forced to retire in his second game for the club following a leg breaking tackle that ended his professional football career.
Pringle has since subsequently forged a career in coaching, notably in women's football.
Club career
Early career and Benfica
Born in Gothenburg of Jamaican descent, Pringle did not play top flight football until well into his 20s, when he joined Helsingborgs IF. After consistent performances, he caught the eye of Portugal's S.L. Benfica, which signed him in August 1996.
However, Pringle's chances at the Lisbon club were very limited, and he amassed just over 40 league appearances in nearly three full seasons. His best individual campaign was his first, as he started in 11 of his 15 matches and scored three times with Benfica eventually ranking third.
Charlton Athletic
In January 1999, Pringle signed for Premier League outfit Charlton Athletic on a two-month loan from Benfica. Under the stewardship of Alan Curbishley, the club was embroiled in a relegation battle and had added Pringle to bolster the club's attacking ranks which included the likes of Clive Mendonca, Andy Hunt, Kevin Lisbie and Bradley Allen. He was handed his debut on 9 January 1999 in the club's 1–3 away defeat against Southampton when he replaced Steve Jones in the 71st minute.
In his second game for Charlton, Pringle scored a dramatic 90th minute equaliser against Newcastle United in a 2–2 draw at The Valley. In March 1999, following his two-month loan at the club, he signed permanently for a fee of £800,000 but, ultimately was unable to help the Addicks avoid relegation, finishing the season with three goals in 18 games.
During the 1999–2000 season, Pringle and Charlton were eventual winners of the First Division and earned an automatic return to the top flight. He remained a favoured forward in the team's strike force, scoring another Premier League goal against Chelsea in a 2–0 win on 18 November 2000, but spent the entire following campaign on the sidelines due to injury.
Loan to Grimsby Town
On his return to fitness, Pringle was loaned out to First Division outfit Grimsby Town, along with Charlton teammate Andy Todd, in February 2002. He made his debut for his new team in a 0–0 away draw with Nottingham Forest on 23 February 2002. In his second game for the Mariners, with the score 3–1 in their favour against Stockport County, opposition defender Dave Challinor went in for a challenge on Pringle, which would break one of his legs in two places. He was rushed immediately to the Diana, Princess of Wales Hospital in Grimsby.
Retirement
Despite his injury, Pringle was subsequently given the opportunity to stay at Charlton and was given a squad number and registered for the club for the 2002–03 season. On 4 November 2002, however, he was forced to retire from professional football.
International career
Pringle was capped twice by Sweden during one year, his debut coming in 1995.
Manager career
Pringle then took up coaching, his first head coach spell being with Kopparbergs/Göteborg FC in the Swedish Women's League. In 2008, he returned to the men's game, assisting at Örgryte IS.
In December 2009, Pringle was named Västra Frölunda IF manager. On 11 July 2011, he and Johan Lange joined Roland Nilsson's coaching staff at FC Copenhagen. In 2014 he returned to Sweden, first to assist Ängelholms FF and then to manage Eskilsminne IF. In December 2019 he started as director of sports in then-newly promoted Allsvenskan club Varbergs BoIS.
Career statistics
International
Scores and results list Sweden's goal tally first, score column indicates score after each Pringle goal.
References
External links
1970 births
Living people
Swedish people of Jamaican descent
Sportspeople of Jamaican descent
Swedish footballers
Association football forwards
Allsvenskan players
Helsingborgs IF players
Primeira Liga players
S.L. Benfica footballers
Premier League players
English Football League players
Charlton Athletic F.C. players
Grimsby Town F.C. players
Sweden international footballers
Swedish expatriate footballers
Expatriate footballers in England
Expatriate footballers in Portugal
Swedish expatriate sportspeople in England
Swedish expatriate sportspeople in Portugal
Swedish football managers
F.C. Copenhagen non-playing staff
Swedish expatriate sportspeople in Denmark
Footballers from Gothenburg
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section{Introduction}
\label{sec:intro}
Water is one of the most abundant molecules detected in interstellar space, {found throughout all stages of the stellar
evolution, from prestellar cores till protoplanetary disks}. The ultimate role of liquid water for the origin of life on Earth is
well recognized in biochemistry and astronomy \citep[e.g., ][]{1993Icar..101..108K, 2006BGD.....3...23H, 2009..BioChem..284,
2010A&ARv..18..383J, 2013ApJ...774L...3L}. One of the main difficulties in tracing the origin of the Earth water is strong
evidence that it came from exogenous sources, since the solar nebula was too hot in the terrestrial planet-formation zone to form
water-rich planetesimals from smaller icy precursors
\cite[][]{1998AREPS..26...53B,2000M&PS...35.1309M,2011Icar..212..416M,2012Icar..221..859E}. In addition, no evidence for hydrous
silicates in the inner regions of protoplanetary disks has been found \citep{2010ApJ...721..431J}.
The most promising mechanism of water delivery to Earth is bombardment by carbonaceous asteroids and/or comets that have formed in
the ice-rich, outer region of the young Solar System \citep[e.g.][]{1961Natur.190..389O, 1992Natur.358...43O, 2000M&PS...35.1309M,
2011NatGe...4...74R, 2013ApJ...767...54I}. This idea is supported by modern sophisticated dynamical models of the early Solar
System, which show that complex gravitational interactions between Jupiter and Saturn had likely scattered other small bodies into
the inner and far outer regions \citep[see e.g.][]{2005Natur.435..462M, 2005Natur.435..466G, 2005Natur.435..459T,
2012M&PS...47.1941W}. {More evidence of the Solar System's dynamical past comes from} the presence of crystalline silicates
annealed at temperatures above 800~K in cometary samples collected by the {\it Stardust} mission \citep{2012M&PS...47..660W} and
revealed by infrared observations of comets \citep[e.g.][]{Crovisier1997, 2006ApJ...651.1256K} suggests that transport or highly
energetic local processes occurred in the solar nebula. The importance of dynamical activity of the young solar nebula and
protoplanetary disks on its chemical and isotopic evolution was also demonstrated in numerous theoretical studies
\citep[e.g.][]{1984ApJ...287..371M, Cyr_ea98,1999ApJ...526..314A, G01, 2004A&A...415..643I, 2004ApJ...616.1265B,
2006ApJ...644.1202W, 2007A&A...463..369T, 2007ApJ...660..441W, 2009ApJ...703..479W, 2011ApJS..196...25S, Heinzeller_ea11,
2012M&PS...47..660W}.
The individual contributions from particular exogenous sources is still heavily debated. Water has likely formed on surfaces of
dust grains even before the formation of the solar nebula in the protosolar molecular cloud and inherited specific isotopic
composition and ortho/para ratios from that epoch \citep[e.g.,][]{2010ApJ...725L.172J, 2011ApJ...741L..34C,2013A&A...550A.127T},
which may have been preserved in primitive Solar System bodies such as comets. It is plausible to assume though that the water
isotopic composition (and may be even the ortho/para ratio) could have been partly or fully reset during the violent physical
conditions in the inner, dynamically active solar nebula \citep[e.g.,][]{2000SSRv...92..201R, 2004ApJ...616.1265B}. Other
processes such as grain growth and sedimentation have shown to bear a profound effect on the water abundance and deuterium
fractionation \citep{2011ApJ...726...29F,2011ApJ...727...76V,ANDES}. The importance of the ortho- and para-states of H$_{2}$ for
deuterium
fractionation is well established and also has to be taken into account \citep[e.g.,][]{2006A&A...449..621F}.
{The water D/H ratio remains the most essential probe for disentangling the individual contributions from the various
exogenous sources.} Since comets are reservoirs of pristine material in the solar nebula, a comparison of the water D/H ratios
measured in comets and asteroids to that measured in protoplanetary disks is vital. These observations are challenging because
most of the water is frozen out onto dust grains in disks around cool Sun-like stars \citep{Dominik_ea05a, 2010A&A...521L..33B,
2011Sci...334..338H}. Furthermore, disks have relatively compact sizes ($\la 100-1000$~AU) and are not massive ($\la
0.01M_\odot$). Therefore, detailed high-resolution studies remain a challenge. Ground-based observations are further compromised
by the absorption and veiling of water vapor lines in the Earth atmosphere. {In order} to better understand the origin of
water on Earth, other Solar System bodies, and possibly in other exoplanetary systems, one has to study the chemical evolution of
water prior
and during the onset of planet formation.
{The water evolution begins in prestellar cores, where the low temperatures cause many molecules to freeze-out onto grains,
and the gas remains largely void of molecules \citep{2010A&A...521L..29C}. This drives a rich surface-chemistry along with an
efficient deuterium fractionation, resulting in high water abundances on the grains ($\sim 10^{-4}$ relative to H) and D/H ratios
($\sim 10^{-2}$). As the cores contracts, and begins to heat up, volatiles are released into the gas. Water has been detected
towards several class 0 protostellar objects \citep[e.g.][]{1999A&A...342L..21C, 2010A&A...521L..30K, 2013ApJ...769...19V,
2013arXiv1308.5119M}, but only a few detections of HDO (and D$_{2}$O) have allowed determination of D/H ratios, with values$\sim
10^{-4}$ in the hot cores, and retaining $\sim 10^{-2}$ in the cold outer regions \citep{2010ApJ...725L.172J, 2011A&A...527A..19L,
2012A&A...539A.132C, 2013A&A...549L...3P, 2013ApJ...768L..29T, 2013A&A...553A..75C}. }
Several detections {of water} have been done in protoplanetary disks \citep[e.g.][]{2005ApJ...622..463P, 2010A&A...518L.124M,
2010A&A...521L..33B, 2012A&A...544L...9F}, {including also several using \textit{PACS/Herschel} \citep{2011Sci...334..338H,
2012A&A...538A..20T, 2012A&A...544L...9F, 2012A&A...540A..84H, 2012A&A...538L...3R}} within the WISH program \citep[Water In
Star-forming Regions with Herschel][]{2011PASP..123..138V}. {In particular,} hot water vapor ($\ga 500$~K) has been discovered
in the inner, planet-forming regions of protoplanetary disks around young solar-mass T~Tauri stars, mainly, by infrared
spectroscopy with the \textit{Spitzer} satellite
\citep{2008Sci...319.1504C,Pontoppidan_ea08,2010ApJ...720..887P,2011ApJ...743..147N,2011ApJ...734...27T,2012ApJ...745...90B}.
The first detection of the ground transition of HDO in absorption in the disk around DM~Tau was reported by
\citet{2005ApJ...631L..81C}. Although they have not observed H$_2$O, \citet{2005ApJ...631L..81C} predicted water D/H ratios of $>
0.01$, based on H$_{2}$O measurements in molecular clouds and protostellar envelopes. This discovery was later disputed by
\citet{2006A&A...448L...5G}. \citet{2008ApJ...681.1396Q} have derived upper limits for the water column densities in the disk
around TW~Hya, $\la 9.0\times10^{14}$~cm$^{-2}$, but were not able to determine its D/H ratio.
The discovery of the ground-state rotational emission of both spin isomers of H$_2$O from the outer, $\ga 50-100$~AU disk around
TW~Hya with \textit{Herschel} allowed to derive a water abundance of $\approx 10^{-7}$ relative to H$_2$
\citep{2011Sci...334..338H}. The authors also derived a water ortho/para ratio of $0.77 \pm 0.07$, which indicates that the water
formation took place on cold dust grains with temperatures $\sim 10-20$~K. In contrast, the water ortho/para ratios measured in
comets are indicative of higher temperatures of $\sim 25-50$~K, more typical for the inner solar nebula at $\la 20-30$~AU
\citep{Mumma_ea87, Woodward_ea07, BM_ea09, Shinnaka_ea12, Bonev_ea13a}.
Studies of carbonaceous chondrites have revealed that their phyllosilicates have D/H ratios that are very similar to that in the
Earth's ocean water, $\sim~1.56~\times$~10$^{-4}$, with as large overall water content as $\la 20\%$, thus favoring asteroids as
the main water source \citep[e.g.,][]{1992Natur.358...43O,2006BGD.....3...23H,2012E&PSL.313...56M,2012Sci...337..721A}. Since
observations of Oort-family comets have yielded water D/H ratios in their comae of about 2-3 times higher that in the Earth'
oceans \citep[][and references therein]{2000SSRv...92..201R}, and because of the low amount of volatile noble gases on Earth
\citep[e.g., ][]{Marty01012013}, it was concluded that comets contributed at most up to $\sim 10\%$ to the water delivery on
Earth. However, recent HDO observation of the short-period, Jupiter-family comet Hartley-2 with {\it Herschel} by
\citet{2011Natur.478..218H} revealed a water D/H ratio of $1.6\times 10^{-4}$, which is very close to the Earth value.
Additionally, \citet{2013ApJ...774L...3L} studied the Jupiter-family comet 45P/Honda-Mrkos-Pajdusakova but could not detect HDO
with an upper water D/H ratio $2 \times 10^{-4}$. Their upper limit is still consistent with the value found by
\citet{2011Natur.478..218H} and aids in confirming the diversity of D/H ratios between different comet populations. These
observations have reignited again debates about the relative role of asteroids and comets in the origin of the Earth water. One
has to bear in mind, however, that a major problem could be that the gases in cometary comae may have lower D/H ratios compared to
the D/H ratios of the nucleus, as indicated by experiments of \citet{2012P&SS...60..166B}.
{While we have made many advances in our understanding in regards of the origin of Earth's oceans, we still do not fully
understand the reprocessing of water during the protoplanetary disk stage, nor is there any consensus for contribution from the
various exogenous sources. }
In this paper, we focus on the history and evolution of the early solar nebula and investigate the chemistry of frozen and gaseous
water at the stage when planetesimals have not yet formed, and the micron-sizes dust grains were the dominant population of
solids. We use an extended gas-grain chemical model that includes multiply-deuterated species, high-temperature reactions, surface
reactions, and nuclear spin-states processes. This network is combined with a 1+1D steady-state $\alpha$-viscosity nebula model to
calculate molecular abundances and D/H ratios within the first 1~Myr of its lifetime. We consider both the laminar case with
chemistry not affected by dynamics and a model with 2D turbulent mixing transport. The organization of the paper is the following.
In Section~\ref{sec:model} we describe our physical and chemical model of the solar nebula and introduce the considered deuterium
processes. The results are presented in Sect.~\ref{sec:results}, followed by Discussion and Conclusions.
\section{Model}
\label{sec:model}
\subsection{Physical model}
\label{sec:model:physics}
We used a similar disk structure as in \citet{2011ApJS..196...25S}, but adopted parameters characteristic of the young Sun and
early solar nebula. The physical model is based on a 1+1D steady-state $\alpha$-model of a flaring protoplanetary disk described
by \citet{1999ApJ...527..893D}, see Fig.~\ref{fig:disk_struc}. We used the parametrization of \citet{1973A&A....24..337S} of the
turbulent viscosity $\nu$ in terms of the characteristic scale height $H(r)$, the sound speed $c_{\rm s}(r,z)$, and the
dimensionless parameter $\alpha$:
\begin{equation}
\nu(r,z) = \alpha\,c_{\rm s}(r,z)\,H(r).
\end{equation}
From observations and detailed MHD modeling the $\alpha$ parameter has values $\sim 0.001-0.1$
\citep[][]{Andrews_Williams07,Guilloteau_ea11a,2011ApJ...735..122F}. A constant value of $\alpha=0.01$ was used in our
simulations. Equal gas and dust temperatures were assumed.
The central star has a mass of $1M_\odot$, a radius of $1.2R_\odot$, and an effective temperature of 6\,000~K. The non-thermal FUV
radiation from the young Sun is represented by the interstellar radiation field of \citet{G}, with the un-attenuated intensity at
100~AU of $\chi_*(100)=10\,000$ $\chi_{\rm draine}$. For the X-ray luminosity of the young Sun we adopted a typical value for
T~Tauri stars, $10^{30}$~erg\,s$^{-1}$. The model disk has an inner radius of $0.03$~AU (dust sublimation front, $T\approx
1\,600$~K), an outer radius of $800$~AU, and an accretion rate of $\dot{M}= 10^{-8}\,M_\odot$\,yr$^{-1}$. The mass of the solar
nebula disk is not well-known but studies suggest it to be $\gtrsim$ 0.1 M$_{\odot}$ \citep[see e.g.][]{1993LPI....24.1225R,
2007ApJ...671..878D}, and hence we adopt a disk model with a total mass of $M=0.11\,M_\odot$.
We assume that the dust grains are uniform spherical particles, with a radius of $a_{\rm d}=0.1\,\mu$m, made of amorphous
silicates with olivine stoichiometry, with a density $\rho_d=3$~g\,cm$^{-3}$ and a dust-to-gas mass ratio $m_{d/g}=0.01$. The
surface density of sites is $N_s=1.5\times 10^{15}$~sites\,cm$^{-2}$, and the total number of sites per each grain is
$S=1.885\times 10^6$. No substantial grain growth is assumed in this early disk phase, an assumption which may be challenged in
other studies.
In our simulations we assume that gas-phase species and dust grains are well mixed and coupled, and transported with the same
diffusion coefficient
\begin{equation}
D_{\rm turb}(r,z) = \nu(r,z)/Sc,
\end{equation}
where $Sc=1$ is the Schmidt number that describes the efficiency of turbulent diffusivity \citep[see e.g.][]{1973A&A....24..337S,
2004ApJ...614..960S}. We treat diffusion of mantle materials similarly to gas-phase molecules, without relating it to individual
grain dynamics.
As boundary conditions for mixing, we assume that there is no inward and outward diffusion across boundaries of the solar nebula,
and that there is no flux through its central plane. All the equations are solved on a non-uniform staggered grid consisting of 65
radial points (from 0.5 to 800~AU) and 91 vertical points.
The physical structure of the solar nebula and the chemical model with and without turbulent mixing are used to solve chemical
kinetics equations \citep[see Eq.~3 in][]{2011ApJS..196...25S}. The equations of chemical kinetics are integrated together with
the diffusion terms in the Eulerian description, using a fully implicit 2D integration scheme, and a sparse matrix formalism for
inversion of the Jacobi matrices \citep{Semenov_ea10}.
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.32\textwidth,angle=90]{fig1.eps}
\caption{(Left to right) Distributions of temperature in K,
density in cm$^{-3}$ ($\log_{10}$ scale), and diffusion coefficient in cm$^2$\,s$^{-1}$
($\log_{10}$ scale) in the solar nebula model. }
\label{fig:disk_struc}
\end{figure*}
\subsection{Chemical network and model}
\label{sec:model:chemistry}
To calculate photoreaction rates through the various disk environments, we adopt precomputed fits of \citet{2006FaDi..133..231V}
for a 1D plane-parallel slab, using the Draine far-UV interstellar radiation field. The self-shielding of H$_{2}$ from
photodissociation is calculated by Equation (37) from \citet{1996ApJ...468..269D} and the shielding of CO by dust grains, H$_{2}$,
and its self-shielding are calculated using the precomputed table of \citet{1996A&A...311..690L}. We do not take the Ly$_\alpha$
radiation into account.
We used the gas-grain chemical network developed by \citet{2013ApJS..207...27A}, which includes an extended list of fractionation
reactions for up to triply-deuterated species. In addition, the high-temperature reaction network from \citet{2010ApJ...721.1570H,
2012ApJ...756..104H} has been added. Recently, it was realized that the ortho/para ratio of H$_{2}$ and other species can lower
the pace of deuterium fractionation \citep{2006A&A...449..621F, 2009A&A...494..623P}. The internal energy of ortho-H$_2$ is higher
than that of para-H$_2$, which helps to overcome the barrier of the backward reaction of deuterium enrichment. Consequently, it
results in a lower degree of deuterium fractionation in a medium having a sufficient amount of ortho-H$_{2}$
\citep{2006A&A...449..621F}. Given the importance of the ortho/para ratios of H$_2$ and H$_3^+$ for efficiency of deuterium
fractionation, particularly, in the solar nebula regions with $T\ga 15-30$~K, the nuclear spin-state processes and ortho-para
states of H$_{2}
$, H$_{2}^{+}$ and H$_{3}^{+}$ were added to our chemical network.
{Reaction rates for a small number of reactions have already been measured or theoretically predicted. For this, we have added
rates from several sources \citep{1990JChPh..92.2377G, 2004A&A...418.1035W, 2004A&A...427..887F, 2009A&A...494..623P,
2011PhRvL.107b3201H}, including reaction rates for the H$_{3}^{+}$ + H$_{2}$ system by \citet{2009JChPh.130p4302H}. In order to
include the nuclear spin states of any reactions including H$_{2}$, H$_{2}^{+}$, H$_{3}^{+}$, or any of their isotopologues, we
employed a separation scheme similar to that described in \citet{2013A&A...554A..92S}. This routine will be adopted and evaluated
in Albertsson et al. 2014, Submitted.} Contrary to \citet{2013A&A...554A..92S}, we allow reactions without H$_3^+$ or H$_2^+$ as
reactants to form both ortho- and para-H$_2$, since we are interested in modeling chemistry in the inner warm nebula region. The
size of the original network was reduced to facilitate the performance of our 2D chemo-dynamical model by allowing
fractionation only for species with $<4$ hydrogen atoms, $<4$ carbon atoms, and species not larger than $7$ atoms.
We find that the inclusion of the high-temperature reactions can increase the abundance of HDO and H$_2$O by up to a factor of 2
inside the snow line ($\lesssim 2-3$ AU). In contrast, the reduction of the original network to $\sim 39\,000$ reactions and $\sim
1\,300$ species bears only insignificant effect on the calculated time-dependent abundances of the H$_2$O isotopologues.
\begin{deluxetable}{lclc}
\centering
\tabletypesize{\small}
\tablecaption{Initial abundances for the models (fractional abundances). \label{tab:IA}}
\tablehead{
\colhead{Species} & \colhead{Abundances} & \colhead{Species} & \colhead{Abundances}
}
\startdata
p-H$_{2}$ & 4.783 $\times 10^{-1}$ & CH$_{4}$ (ice) & 5.832
$\times 10^{-6}$ \\
He & 9.750 $\times 10^{-2}$ & N$_{2}$ & 5.135
$\times 10^{-6}$ \\
o-H$_{2}$ & 2.146 $\times 10^{-2}$ & O$_{2}$ (ice) & 4.619
$\times 10^{-6}$ \\
H & 2.681 $\times 10^{-4}$ & O &
2.661 $\times 10^{-6}$ \\
H$_{2}$O (ice) & 7.658 $\times 10^{-5}$ & N$_{2}$ (ice) & 2.464 $\times
10^{-6}$ \\
CO (ice) & 5.219 $\times 10^{-5}$ & C$_{3}$H$_{2}$ (ice) & 6.493 $\times
10^{-6}$ \\
CO & 1.816 $\times 10^{-5}$ & HD &
1.479 $\times 10^{-5}$ \\
HNO (ice) & 3.541 $\times 10^{-6}$ & O$_{2}$ & 9.645
$\times 10^{-6}$ \\
HDO (ice) & 2.887 $\times 10^{-6}$ & NH$_{3}$ (ice) & 8.663
$\times 10^{-6}$ \\
D & 2.731 $\times 10^{-6}$ & OH &
2.295 $\times 10^{-6}$
\enddata
\end{deluxetable}
We also added neutral-neutral reactions. The pre-exponential rate factors were calculated using a simple collisional theory and
the reactions barriers are adopted from \citet{1994GeCoA..58.2927L}. The added neutral-neutral reactions are listed below:
\begin{eqnarray}
\small
\begin{tabular}{lclc}
HDO + $p$-H$_{2}$ &$\Rightarrow$& H$_{2}$O + HD; & 2.0$\times 10^{-10}$ cm$^{-3}$ s$^{-1}$, 5170 K
\nonumber\\
HDO + $o$-H$_{2}$ &$\Rightarrow$& H$_{2}$O + HD; & 2.0$\times 10^{-10}$ cm$^{-3}$ s$^{-1}$, 5000 K
\nonumber\\
H$_{2}$O + HD &$\Rightarrow$& HDO + $p$-H$_{2}$; & 5.0$\times 10^{-11}$ cm$^{-3}$ s$^{-1}$, 4840 K
\nonumber\\
H$_{2}$O + HD &$\Rightarrow$& HDO + $o$-H$_{2}$; & 1.5$\times 10^{-10}$ cm$^{-3}$ s$^{-1}$, 4840 K
\nonumber
\end{tabular}
\label{eq:neutneut}
\end{eqnarray}
{However, we find that the addition of these reactions has no significant effect on our results.} For initial abundances we
model the chemical evolution of a cold and dark TMC 1-like prestellar core, {temperature 10 K, density $10^{4}$ cm$^{-3}$ and
extinction $A_{V} = 10$ mag}, for 1~Myr using the ``low metals'' initial abundance set from
\citet[][Table~11]{1996A&A...311..690L}, {with an initial H$_{2}$ ortho:para ratio 1:100 (as H$_{2}$ exist in para-form in
cold environments)}. We note that the initial H$_{2}$ ortho:para plays an essential role here, as it affects the deuterium
chemistry, and adopting the statistical 3:1 ortho:para for H$_{2}$ as initial abundance for our TMC 1 model, ice HDO abundances
drop by a factor of 10, and gas HDO abundances by a factor 20. Gas and ice H$_{2}$O are not significantly affected. The most
abundant species in the initial abundances are listed in Table~\ref{tab:IA}.
\section{Results}
\label{sec:results}
Our main set of chemical simulations consists of the two runs: (1) the laminar nebular model without transport processes, and (2)
the 2D-mixing model with $Sc=1$. The water abundance is limited by the initial oxygen abundance in the models and change
throughout the disk. Given the rapid, $\sim 2-3$~Myr, {dynamical} evolution of solids in the nebula, we modeled chemistry only
within 1~Myr.
\begin{figure*}[!htb]
\centering
\includegraphics[width=1.00\textwidth]{./fig2a.eps} \\
\includegraphics[width=1.00\textwidth]{./fig2b.eps}
\caption{The distributions of gaseous (upper row) and solid (bottom row) water abundances (wrt total H) in the solar nebula
between 0.8 and 30~AU at 1~Myr. The laminar model is shown on the left panel, the 2D-mixing model is shown in the middle panel.
The vertically integrated column densities are compared in the right panel, with the laminar model depicted by solid line and the
2D-mixing model depicted by dashed lines. The thickness of these lines correspond to intrinsic uncertainties in the calculated
abundances and thus column densities. }
\label{fig:H2O}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=1.00\textwidth]{./fig3a.eps} \\
\includegraphics[width=1.00\textwidth]{./fig3b.eps}
\caption{The same as Fig.~\ref{fig:H2O} but for HDO.}
\label{fig:HDO}
\end{figure*}
In Figure~\ref{fig:H2O} we show the gas- and solid-phase water abundances between 0.8 and 30 AU for the laminar (left panel) and
the 2D-mixing model (middle panel). {D/H ratios from column densities are shown in the right panel}. Column densities vary
strongly with scale height. Both the laminar and 2D-mixing show similar relative abundances of water (with respect to H) vapor
going from $10^{-5}$ close to the midplane and quickly dropping to $< 10^{-12}$ at around 0.15 scale heights in the inner regions,
while in the outer regions relative abundances in the cold midplane are $\sim 10^{-12}$, increasing to $\sim 10^{-6}$ at 0.15
scale heights and then dropping again to $< 10^{-12}$ at $\sim$ 0.4 scale heights. Also for water ice the two models show similar
the relative abundances, with only smaller differences arising in the outer region $\sim$ 10 AU. In the warm inner regions
relative abundances are $\sim 10^{-8} - 10^{-6}$ in the midplane, dropping $< 10^{-12}$ at $\sim$ 0.15 scale heights. Further
out in the disk, beyond the snow line at $\sim 2-3$ AU, a thick layer of abundant ices with relative abundances $\sim 10^{-4}$
stretches between 0.15 up to 0.3 scale heights, after which we see again a quick drop with relative abundances going below
$10^{-12}$. The calculated vertical column densities for both models are compared in the right panel.
The same panels are shown in Figure~\ref{fig:HDO} for HDO. Here layers of different relative abundances are more clear, with
relative abundances of HDO vapor in the inner disk of $\sim 10^{-10}$, dropping quickly $<10^{-12}$ at 0.15 scale heights. In the
outer regions we have a low vapor abundance in the midplane of $\sim 10^{-14} - 10^{-12}$, increasing up to $10^{-10}$ in the
molecular layer and then at around 0.2 scale heights dropping $< 10^{-12}$. For HDO ice both models show relative abundances $\sim
10^{-12}$ in the inner region and a thick layer of abundances $\sim 10^{-8}$ between 0.15 - 0.3 scale heights in the outer
regions. While the 2D-mixing model show a smooth decline in abundances towards the inner disk, the laminar model show larger
variations, especially evident at 5$-$30 AU. As can be clearly seen, the snow line is not affected by slow diffusive transport and
is located at $\sim 2-3$ AU in both models, thus leaving dust grains in the Earth-formation zone barren of ices. Hence, the water
delivery to Earth could occur only at a later stage, when dust grains were assembled to larger planetesimals that were able to
reach 1~AU without complete loss of volatile materials.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.48\textwidth]{fig4.eps}
\caption{The radial distributions of the D/H ratios of the total water budget in the solar nebula between $1-30$~AU at 1~Myr are
shown, both for the the laminar (solid line) and the 2D-mixing model (thick dashed line). The thickness of these lines reflects
the uncertainties in the calculated water abundances, a factor of $\sim 2$~\citep{Vasyunin_ea08}. The elemental D/H ratio of
$1.5\times10^{-5}$ is indicated by the thin straight solid line in the bottom \citep{1998ApJ...509....1S, 2003SSRv..106...49L}.
The D/H ratio for water in the cold ISM, $3\times10^{-2}$, is depicted by the thick solid line on the top (our model). The Earth
ocean's water D/H ratio, $1.59\times10^{-4}$ \citep{Lecuyer1998249}, is marked by the straight blue dashed line. The water D/H
ratios in the Oort-family comets, which are a few times higher than the Earth value, are shown with the red rectangle filled with
lines and denoted by ``OFC'' on the plot (see Table~\ref{tab:obs}).}
\label{fig:DH}
\end{figure}
The resulting distribution of the D/H ratio for the total water budget (ice + vapor) is shown in Figure~\ref{fig:DH}. The water
D/H ratio of the Earth' oceans and the measured value in carbonaceous meteorites (``Earth") are shown for comparison, also similar
to what observed in Jupiter-family comets. The D/H ratios for the Oort-family comets (``OFC'') are also indicated.
As can be clearly seen, the laminar model shows the Earth water D/H ratio at $0.8-2.5$~AU, while for the 2D chemo-dynamical model
such a low D/H value extends towards larger radii, $\la 9$~AU. Similarly, the elevated water D/H ratios representative of the
Oort-family comets, $\sim 2.5-10\times 10^{-4}$, are achieved within $\sim 2-6$~AU and $\sim 2-20$~AU in the laminar and the 2D
model, respectively. The reason for this behavior is a shallower radial gradient of the water D/H ratio in the 2D mixing model.
Turbulent mixing slowly transports some of the water ice into warmer or irradiated regions where it desorbs and is quickly
de-fractionated by ion-molecule and dissociative recombination processes in the gas phase. As a consequence, lower water D/H
ratios can be retained further out in the nebula, which seems to be more consistent with the delivery of water on Earth by comets.
Both the laminar and the mixing nebular models have the Earth' oceans and the Jupiter-family comets' water D/H ratios at radii
$\sim
2-3$~AU, where carbonaceous chondrites are believed to have formed. Thus both the laminar and dynamical nebular model can
reproduce the water D/H ratios observed in carbonaceous asteroids. With regard to the water isotopic composition and the origin of
the Jupiter-family and Oort-family comets, the mixing model seems to be favored over the laminar model. It allows for a larger
region for their formation with the appropriate water D/H ratios, extending to the distance of $\sim 10-30$~AU.
We have also studied how other processes may affect the results of our chemical modeling. First, given observational evidence for
the presence of bigger grains in protoplanetary disks \citep[e.g.][]{Testi_ea03,2005A&A...437..189V, 2006A&A...446..211R,
Bouwman_ea08,Perez_ea12}, for reviews see \citet{2008PhST..130a4014N, 2011ppcd.book..114H}, we have increased the uniform grain
sizes to 1 and $10\mu$m and calculated the nebular water D/H ratios. The average $1\mu$m and $10\mu$m grain sizes imply 10 and 100
times smaller surface area (per unit volume of gas), respectively, making gas-grain interactions less prominent, and allowing the
high-energy radiation to penetrate deeper into the disk. The thermal desorption rates are independent of the grain size, while the
cosmic-ray particle (CRP)-desorption rate depends on the grain size only weakly \citep{Leger_ea85}. The overall photodesorption
rate increases because the nebula becomes more UV-transparent.
In contrast, the accretion rate on to the grains decreases when the total grain surface area per unit gas volume decreases.
Consequently, in the inner warm nebular region at $r\la 10-20$~AU the amount of gaseous water increases by a factor of $\sim 10$
for $a_{\rm dust} = 1\mu$m and $\sim 100$ for $a_{\rm dust} = 10\mu$m, respectively, but it still constitutes only a tiny fraction
of the total water budget. As a result, the water D/H ratio is not significantly affected and the three modeled D/H radial
distributions are within the uncertainties of the calculated D/H values \citep[see][]{Vasyunin_ea08, 2013ApJS..207...27A}. The
same is found for the nebular model with the standard grains but 100 times lower UV-desorption yield of $10^{-5}$.
Next, we studied how the calculated water D/H ratios are affected if only a limited set of the nuclear spin-state processes is
included in our chemical network. For that test we calculated a model in which only the para states of H$_2$, H$_2^+$, and H$_3^+$
isotopologues were considered, implying more efficient deuterium isotope exchange. The water D/H ratios calculated with this model
coincide with the ratios calculated with our standard model with ortho/para-species (within the factor of $\sim 3$ in intrinsic
chemical uncertainties).
This result is a combination of two extreme situations. First, the initial abundances are calculated using the physical conditions
of a dark, dense, 10~K cloud core. At such conditions almost the entire H$_2$ population exists in the lowest energy para-state,
and the isotope exchange reactions involving ortho-species are still out of equilibrium. This leads to similar values of the
initial water D/H ratio. Second, in the water de-fractionation zone in the nebula, at $r \la 10-30$~AU, temperatures are higher
than 30~K. These temperatures are high enough to enable rapid gas-phase de-fractionation regardless of the dominant nuclear
spin-state form of H$_2$, H$_2^+$, and H$_3^+$, mainly due to X-ray- and CRP-driven reprocessing of CO by He$^+$, followed by
rapid gas-phase production of H$_2$O. This combination of effects leads to the somewhat surprising result that a reduced network
produce roughly the same abundances as the complete network.
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.99\textwidth]{./fig5.eps}
\caption{{The distributions of solid HDO D/H ratios in the solar nebula between 0.8 and 30~AU at 1~Myr. The `cold'' TMC-1
initial abundance model is shown on the left panel and the ``warm'' model is shown in the middle panel. The vertically integrated
column densities are compared in the right panel, with the ``cold'' TMC-1 initial abundance model depicted by solid line and
``cold'' model depicted by dashed lines. The thickness of these lines correspond to intrinsic uncertainties in the calculated
abundances and thus column densities.} }
\label{fig:gHDOopr}
\end{figure*}
{
The initial H$_{2}$ ortho:para ratio in cold dark clouds is unknown but predicted to be low \citep[$<0.1$][]{2009A&A...494..623P}.
However, later, during the warmer protostellar phase this ratio is likely modified, such that there are more ortho$-$H$_2$ and
less para$-$H$_2$ molecules. As such the ortho:para ratio can be closer to the statistical ortho:para ratio of 3:1 in the initial
phase of the protoplanetary disk. We take the statistical value 3:1 as the upper extreme of the H$_{2}$ ortho:para ratio and
initiate another run of the laminar model with initial abundances resulting from a TMC-1 model with an initial 3:1 ortho:para
ratio for H$_{2}$. }
{In Figure~\ref{fig:gHDOopr} we show the D/H ratios of solid-phase HDO between 0.8 and 30 AU for the laminar model using our
previous ``cold'' TMC-1 initial abundance (left panel) and ``warm'' TMC-1 initial abundance (middle panel), and in the right panel
the D/H ratio determined from column densities is shown. In the disk midplane there are small but significant differences where
D/H ratios decrease from $\sim 10^{-2}$ down to $\sim 10^{-4} - 10^{-3}$ around 10 AU. This cause the column density D/H ratios in
the right panel to drop approximately an order of magnitude while the minimum and maximum D/H ratios remain unchanged at $\sim 1$
AU and $\gtrsim 30$ AU, respectively. Thus our conclusion that carbonaceous asteroids formed at 2-3 AU will inherit Earth
water-like D/H ratios remains valid. The fact that the H$_{2}$O D/H ratio gets lower at intermediate radii ($\sim 10$ AU) will
only help us to achieve a larger zone where Hartley-2-like comets with the Earth water D/H ratio could form, while still
leaving a zone for the Oort comets to form at $\sim 10$ AU.
}
Finally, we discuss the effects of the underlying chemistry on the D/H ratios throughout the disk.{In the midplane,
temperatures are low enough to drive a rich surface chemistry, where O atoms are adsorbed on the grain surfaces and become
hydrogenated, forming H$_{2}$O, and/or reacting with deuterium atoms to form HDO or D$_{2}$O. There is a small chance for the
water molecules to be released into the gas-phase directly through chemisorption, but more likely they remain on the grains and
desorb as material is transported into warmer regions, i.e. the inner regions or the atmosphere, where water, and its D/H ratio,
will be reprocessed.
}
The reprocessing of water occurs in the gas phase, so the relative adsorption and desorption plays an important role. These
processes are heavily effected by temperature and density. In the warmer regions, the water abundances steadily increase by $\la
30\%$ during the first 10\, 000 - 100\, 000 years. As soon as the water reaches the gas-phase, it is reprocessed. After that, the
abundances of water ice and water vapor are at quasi steady-state. However, constant evaporation and re-accretion result in
cycling of water between solid and gaseous phases. One important fact is that the D/H ratio never drops to the cosmic ratio, even
in the warm, inner regions within a few AU of the Sun. In the inner solar nebula there are several important processes at work
that sets the D/H ratio. First of all these regions experience strong X-ray or CRP-ionization and water is either ionized or
protonated by reacting with various abundant ions, such as H$_{3}^{+}$, HCO$^{+}$, and their isotopologues:
\begin{eqnarray}
\small
\begin{tabular}{lclcc}
H$_{2}$O + $o-$H$_{3}^{+}$ $\Rightarrow$ H$_{3}$O$^{+}$ + $o-$H$_{2}$; & $4.5
\times 10^{-9}$ cm$^{-3}$ s$^{-1}$ & $\beta = -0.5$ \\
H$_{2}$O + $p-$H$_{3}^{+}$ $\Rightarrow$ H$_{3}$O$^{+}$ + $o-$H$_{2}$; & $2.3
\times 10^{-9}$ cm$^{-3}$ s$^{-1}$ & $\beta = -0.5$ \\
H$_{2}$O + $p-$H$_{3}^{+}$ $\Rightarrow$ H$_{3}$O$^{+}$ + $p-$H$_{2}$; & $2.3
\times 10^{-9}$ cm$^{-3}$ s$^{-1}$ & $\beta = -0.5$ \\
H$_{2}$O + HCO$^{+}$ $\Rightarrow$ H$_{3}$O$^{+}$ + CO; &
$2.1 \times 10^{-9}$ cm$^{-3}$ s$^{-1}$ & $\beta = -0.5$ \nonumber
\end{tabular}
\label{eq:waterform}
\end{eqnarray}
{The same set of reactions applies for HDO being protonated into H$_2$DO$^+$ (as well as heavier isotopologues). The
H$_{3}$O$^{+}$ isotopologues can then reform water through dissociative recombination, but the D/H ratios will be reset,
characteristic of the temperature in the environment at which they are reforming:}
\begin{eqnarray}
\small
\begin{tabular}{lclcc}
H$_{3}$O + e$^{-}$ $\Rightarrow$ H$_{2}$O + H; & $1.1 \times 10^{-7}$ cm$^{-3}$
s$^{-1}$ & $\beta = -0.5$ \\
H$_{2}$DO + e$^{-}$ $\Rightarrow$ H$_{2}$O + D; & $7.3 \times 10^{-8}$ cm$^{-3}$ s$^{-1}$
& $\beta = -0.5$ \\
H$_{2}$DO + e$^{-}$ $\Rightarrow$ HDO + H; & $7.3 \times 10^{-8}$ cm$^{-3}$
s$^{-1}$ & $\beta = -0.5$ \nonumber
\end{tabular}
\label{eq:waterform2}
\end{eqnarray}
The reprocessing of water involves several {additional} reactions that compete in both lowering and increasing the D/H ratio.
In the warm inner nebular region the normal H$_{3}^{+}$~fractionation pathway is not active and instead fractionation is driven
mostly via the ``warm'' fractionation route that includes light hydrocarbons such as CH$_2$D$^+$ and C$_2$HD$^+$ and is active at
$T\la 80$~K \citep[e.g.,][]{1989ApJ...340..906M, 2005A&A...438..585R, 2009A&A...508..737P}. Consequently, the initially high D/H
ratios of light hydrocarbons, $\sim 10^{-3}$, is not fully equilibrated with the cosmic D/H ratio of $\sim 10^{-5}$.
Neutral-neutral reactions of these hydrocarbons with atomic or molecular oxygen are able to produce normal or heavy water, e.g.:
\begin{eqnarray}
\small
\begin{tabular}{lc}
CH$_{3}$ + O$_{2} \Rightarrow$ HCO + H$_{2}$O; & $1.66 \times 10^{-12}$ cm$^{-3}$ s$^{-1}$
\end{tabular}
\label{eq:warmfrac}
\end{eqnarray}
Another important channel of the water formation in the gas-phase is through the neutral-neutral reaction with OH or OD of
H/D{ or $ortho/para-$H$_{2}$/HD}:
\begin{eqnarray}
\small
\begin{tabular}{lclcc}
OH + H $\Rightarrow$ H$_{2}$O; & $4.0 \times 10^{-18}$ cm$^{-3}$ s$^{-1}$
& \\
OH + D $\Rightarrow$ HDO; & $4.0 \times 10^{-18}$ cm$^{-3}$
s$^{-1}$ & \\
OD + H $\Rightarrow$ HDO; & $4.0 \times 10^{-18}$ cm$^{-3}$
s$^{-1}$ & \\
OH + $o/p-$H$_{2}$ $\Rightarrow$ H$_{2}$O + H; & $8.4 \times 10^{-13}$ cm$^{-3}$ s$^{-1}$
& 1040 K \\
OH + HD $\Rightarrow$ HDO + H; & $4.2 \times 10^{-13}$
cm$^{-3}$ s$^{-1}$ & 1040 K \\
OH + HD $\Rightarrow$ H$_{2}$O + D; & $4.2 \times 10^{-13}$ cm$^{-3}$
s$^{-1}$ & 1040 K \nonumber
\end{tabular}
\label{eq:waterform3}
\end{eqnarray}
While OH/OD, $ortho/para-$H$_{2}$/HD and H/D are formed through various processes, including the dissociation of water, the D/H
ratio of OH is dominantly controlled by the equilibrium of the following forward and backward processes:
\begin{eqnarray}
\small
\begin{tabular}{lclcc}
OH + D $\Rightarrow$ OD + H; & $1.3 \times 10^{-10}$ cm$^{-3}$ s$^{-1}$ &
\label{eq:warmfrac1}\\
OD + H $\Rightarrow$ OH + H; & $1.3 \times 10^{-10}$ cm$^{-3}$ s$^{-1}$ & 810 K
\label{eq:warmfrac2}
\end{tabular}
\end{eqnarray}
At around 100~K, the equilibrium state of this reaction will lead to an enhancement by a factor of approximately 100 in OD, which
will transfer into the water D/H ratio, and competes with other de-fractionation processes, eventually leading to the enhanced D/H
ratios of $\sim$ 10$^{-4}$. As we move further our in the disk the efficiency of reaction~\ref{eq:warmfrac2} gradually decreases,
whereas the pace of water ice de-fractionation decreases due to slower cycling of water through gas and solid phases, and the
water D/H ratio increases gradually until we reach the colder outer regions of the solar nebula.
{Compared to \citet{2010MNRAS.407..232T}, we do not find that reactions formation of water through reactions between OH/OD and
H$_{2}$/HD are the dominant neutral-neutral reactions, instead we find that OH/OD and H/D reactions are more dominant. This
discrepancy is likely arising due to the energy barrier of reactions with H$_{2}$ or HD, while reactions with H or D are
exothermic. Other differences between our models may also contribute to the discrepancies. \citet{2010MNRAS.407..232T} used a
steady-state model to determine their key reactions, which are not exactly like ours. Bear in mind that time-dependent disk
chemical models usually do not reach a chemical steady-state even after 1 Myr of evolution. If we would allow our calculations to
run over much longer time scales, we could get a more similar set of key reactions as found by \citet{2010MNRAS.407..232T}.
Furthermore, \citet{2010MNRAS.407..232T} adopt reaction rates from \citet{2007A&A...466.1197W} (H$_{2}$O) and
\citet{1996CPL...253..177T} (
HDO), may also account for some of the discrepancy. This also result in different OH and OD column densities, where OH is
approximately an order of magnitude more abundant in our model, decreasing by two order of magnitudes at $\sim 30$ AU, while OD is
approximately an order of magnitude more abundant for \citet{2010MNRAS.407..232T}. Finally, the inclusion of ortho-para chemistry
in our network, and differences in physical structures may contribute further to these differences. Most notably, the stellar
parameters are signficantly different, whereas \citet{2010MNRAS.407..232T} adopts values for a less massive (0.5 M$_\odot$), but
larger (2 R$_\odot$ and cooler (4000 K) central protostar compared to our study. }
\section{Discussion}
We have shown the importance of 2D-mixing on the water D/H gradient in the young solar nebula, which pushes the (lower) water D/H
ratios from the inner disk regions toward larger heliocentric distances as water is defractionated after being transported into
the inner regions. Before discussing the implications of our results, it is important to discuss the limitations of our model.
Foremost, protoplanetary disks are not static environments, but are ever dynamically evolving systems. Dust grains grow into
planetesimals and eventually planets, with an associated evolution of physical conditions affecting the chemistry. As we
previously discussed, a moderate increase of grain sizes had no significant effect on our results (see Section~\ref{sec:results}),
but this might change with even larger particles. Variations in temperature and density will likely play a significant role.
\citet{2011ApJ...741L..34C} predicted that significant variations in HDO/H$_{2}$O ratios can come from even small variations in
dust temperatures at the time of ice formation. This would also affect initial abundances of our model, as the warm-up phase
following the collapse of a dark pre-stellar core will experience a gradual increase in both temperature and density. In addition,
exact masses and structural parameters of the solar nebula remain highly uncertain.
Another large uncertainty is the post-processing of Earth's ocean water. Such processes include biological (life and its
evolution) as well as chemical processes. Atmospheric processes can affect the fractionation if the young Earth had a massive
hydrogen atmosphere, which would experience a slow hydrodynamic escape, which will be slower for deuterium compared to the lighter
hydrogen isotope \citep{2008Icar..194...42G}. Furthermore, as was first proposed by \citet{2005...Campins}, processes involved in
planetary accretion, such as outgassing and the evolution of hydrosphere and atmosphere, are complex and may have affected the
fractionation.
Finally, the giant impact theory which explains the formation of the moon and the similarities of its bulk composition to Earth
\citep[see e.g.][]{2012Sci...338.1052C}, requires a collision between early Earth and a large impactor. This scenario implies a
heavy loss of the primordial Earth atmosphere and would have likely greatly affected the chemical composition, more specifically
the D/H ratio. However, results by \citet{Saal14062013} revealed a similar isotopic composition of the water in the interiors of
Earth and the Moon, as well as the bulk water in carbonaceous chondrites, suggesting that this reprocessing is of minor
importance.
\begin{deluxetable}{lccc}
\centering
\tabletypesize{\small}
\tablecaption{Compilation of observed deuterated water in disks and comets.\label{tab:obs}}
\tablehead{
\colhead{Source} & \colhead{D/H ratio $[\times$ $10^{-4}]$} &
\colhead{References} \\}
\startdata
solar nebula & $1.7-2.5$ & 1
\\
Standard Mean Ocean Water & $1.46 -1.52$ & 2
\\
Carbonaceous chondrites & $1.30-1.50$ & 3
\\[0.07cm]
\multicolumn{3}{l}{Protoplanetary disks}\\ \hline\\[-1.5ex]
LkCa 15 & 640 & 4
\\
DM Tau & $\gtrsim 100$ & 5
\\[0.07cm]
\multicolumn{3}{l}{Oort-family comets}\\ \hline\\[-1.5ex]
Halley & 2.55$-$3.46 & 6
\\
Halley & 2.72$-$3.40 & 7
\\
Hyakutake & 1.90$-$3.90 & 8
\\
Hale-Bopp & 2.50$-$4.10 & 9
\\
Ikeya-Zhang & $<$ 3.1 & 10
\\
C/2002 T7 & 1.80$-$3.20 & 11
\\
C/2001 Q4 & 3.2$-$6.0 & 12
\\
8p/Tuttle & 2.64$-$5.54 & 13
\\
Machholz & 2.71$-$3.69 & 14
\\
C/2007 W1 (Boattini) & $< 12.9$ & 15
\\
C/2007 N3 (Lulin) & $<$ 5.6 & 16
\\
C/2009 P1 (Garradd) & 1.84$-$2.28 & 17
\\[0.07cm]
\multicolumn{3}{l}{Jupiter-family comets}\\ \hline\\[-1.5ex]
103P/Hartley 2 & 1.37$-$1.85 & 18
\\H
45P/Honda-Mrkos-Pajdusakova & $<$ 2 & 19
\enddata
\tablerefs{(1) \citet{2001A&A...370..610L};
(2) \citet{Lecuyer1998249};
(3) Compiled from \citet{Boato1954209, Robert198281, Kerridge19851707};
(4) \citet{2003cdsf.conf..188K};
(5) \citet{2005ApJ...631L..81C};
(6) \citet{1995JGR...100.5827B};
(7) \citet{1995A&A...302..301E};
(8) \citet{1998Icar..133..147B};
(9) \citet{1998Sci...279..842M};
(10) \citet{2006A&A...449.1255B};
(11) \citet{2008A&A...490L..31H};
(12) \citet{2008LPICo1405.8216W};
(13) \citet{2009ApJ...690L...5V};
(14) \citet{2009ApJ...693..388K};
(15) \citet{2011Icar..216..227V};
(16) \citet{2012ApJ...750..102G};
(17) \citet{2012A&A...544L..15B};
(18) \citet{2011Natur.478..218H};
(19) \citet{2013ApJ...774L...3L}
}
\end{deluxetable}
\subsection{Observations in other protoplanetary disks}
\citet{2013ApJ...766..134N} found a clear relation between the mid-infrared HCN/H$_{2}$O flux ratio and submillimeter disk mass
among T Tauri stars in Taurus. They argue that this interesting trend arises as a consequence of a more efficient formation of
large non-migrating bodies in more massive disks that lock up oxygen and water beyond the water line. This depletion of oxygen
enhances the gas C/O ratio which results in an increase of HCN and decrease of H$_{2}$O abundances. Thus, we see this dramatic
trend in the HCN/H$_{2}$O ratios with disk mass. \citet{2011ApJ...733..102C} found that only modest enhancements in the C/O ratio
in inner disk atmospheres are needed for a significant increase of warm HCN/H$_{2}$O. This efficient lock-up of water in massive
disks is necessary to bear in mind when comparing to observations of other protoplanetary disks, as their masses may differ from
that of the young solar nebula ($\sim 0.11$ M$_{\odot}$).
{TW Hydrae has been observed on multiple occasions, and its expected} disk mass is comparable to the solar nebula and believed
to be able to form a planetary system like our own \citep{2013Natur.493..644B}, hence one may expect a similar chemistry.
\citet{2008ApJ...681.1396Q} estimated water column densities of $<~9.0~\times~10^{14}$ cm$^{-2}$ in the outer region of TW Hydrae,
which is lower compared to our models ($\sim 10^{15}$ cm$^{-2}$). {A small difference in disk mass may, partly, explain the
discrepancy, as has been suggested by \citet{2013ApJ...766..134N} from studies of HCN/H$_{2}$O flux ratios, where more massive
disks have a more efficient formation of large non-migrating bodies, which more efficiently locks up oxygen and water beyond the
water line.} \citet{2011Sci...334..338H} observed a cold water vapor reservoir in the disk of TW Hydrae, predicting instead a
relative abundance $\sim 10^{-7}$ at intermediate heights, slightly higher than we predict in the upper layers of the solar
nebula ($\sim 10^{-8}$). It is possible that they probe regions further in than \citet{2008ApJ...681.1396Q}, due to the optical
depth-effect, corresponding to ``warmer'' water at $\sim 50$ AU.
\citet{2005ApJ...622..463P} observed water ice features in an edge-on circumstellar disk using the \textit{Spitzer} telescope. The
orientation means that their observations probe the outermost regions. The high inclination complicates comparisons and the cutoff
radius of the disk will affect results. Using models and adopting the best-fitting parameters to their observations,
\citet{2005ApJ...622..463P} estimated an abundance of $\sim 10^{-4}$ with respect to H$_{2}$, which is in agreement with our
modeled relative abundances in the outer disk, $10^{-5} - 10^{-4}$. \citet{2012A&A...538A..57A} measured with the AKARI satellite
several ice features in five edge-on Class II disks with column densities on the order of $10^{17}$ cm$^{-2}$ as well as a faint
HDO feature in one of the disks (HV Tau, D/H $\sim 19\%$). The observed disk has a high inclination ($i \sim 84^{\circ}$) and
probe the outermost regions where our models estimate a water D/H ratio of $\sim 0.03-0.05$, a D/H ratio lower by a factor $\sim
5$
compared to the observed ice features. Our column densities are vertically integrated, and due to the higher density in the disk
midplane compared to the surface, horizontally integrated column densities would be expected to be larger by approximately one
order of magnitude.
\subsection{Previous theoretical studies}
There has been a vast number of studies on the chemistry in protoplanetary disks, many of these are not targeting the problem in a
similar fashion as us, making any comparison complicated. Therefore, we concentrate on studies with similar approaches to model
chemistry and physics in a protoplanetary disk or solar nebula.
\citet{1999ApJ...526..314A} studied the deuterium chemistry in the outer regions of protoplanetary disks with an 1D accretion
flow, using a collapse model to set up the initial molecular concentrations. They have found that the molecular D/H ratios are
enhanced with respect to the protosolar values, and that the ratios at $\sim 30$~AU agree reasonably well with the D/H ratios
observed in comets, meaning that comets may not necessarily be composed of primordial, unprocessed interstellar matter, {as we
also find in our models. While our two models predict similar abundances, within an order of magnitude, we find different behavior
of D/H ratios with radii, such that our models attain higher D/H ratios in the outer regions. }
\cite{2007ApJ...660..441W} have investigated deuterium chemistry in outer disk regions, using the UMIST RATE'95 database extended
with a set of reactions for multiply-deuterated species, a 1+1D disk model of \cite{2001ApJ...553..321D}, and initial molecular
abundances obtained by the chemical modeling of a cold prestellar core. Furthermore, they implement a lower stellar and disk mass
compared to our model. Similarly to \cite{1999ApJ...526..314A}, they found that the D/H ratios observed in comets may partly
originate from the parental molecular cloud and partly be produced in the disk. They concluded that the D/H ratios of gaseous
species are more sensitive to deuterium fractionation processes in disks due to rapid ion-molecule chemistry compared to the
deuterated ices, whose D/H values are regulated by slow surface chemistry and are imprints of the cold conditions of the
prestellar cloud. {We find signs of the same sensitivities to ion-molecule chemistry, which is, in our models, also further
aided
by high-temperature reactions.}
In their later study, \cite{2009ApJ...703..479W} investigated deuterium chemistry in the inner 30~AU, accounting for gas and dust
thermal balance. While a good agreement between the model predictions and observations of several non-deuterated gaseous species
in a number of protoplanetary disks was obtained, the calculated D/H ratios for ices were higher than measured in the Solar System
Oort-family comets, {while our models are in better agreement to observations of Solar System bodies (see
Section~\ref{sec:SS}). Furthermore, we note that their D/H distributions do not show a smooth increase with radii, as our models
predict, but which show a spike and swinging variations at radii $\gtrsim 10$ AU.} \cite{2009ApJ...703..479W}, however, consider a
small stellar mass and less turbulent mixing ($\alpha = 0.025$). Their results point to the importance of dynamical processes
(shocks, turbulent or advective mixing, non-steady accretion) for deuterium chemistry in the inner regions of disks.
\citet{2010MNRAS.407..232T} focused on understanding deuterium fractionation in the inner warm disk regions and investigated the
origin of the high H$_2$O D/H ratios in dense ($\ga 10^6$~cm$^{-3}$) and warm gas ($\sim 100-1000$~K) by gas-phase photochemistry
(dominated by photoprocesses and neutral-neutral reactions). Using the time-dependent chemical model based the UMIST RATE'06
database~\citep{2007A&A...466.1197W} and the T~Tau disk structure calculated with ``ProDiMo'' \citep{2009A&A...501..383W}, they
predicted that in the terrestrial planet-forming region at $\la 3$~AU the water D/H ratios may remain high, $\ga 1\%$, which is
significantly higher than the value of $\approx 1.5\times 10^{-4}$ measured in the Earth ocean water as well as predictions from
our models. While \citet{2010MNRAS.407..232T} a dynamical code, they only included a simple chemical network and did not account
for the grain chemistry, which is essential for the calculation of water chemistry.
\citet{Yang2013} coupled a classical dynamic $\alpha$-viscosity model of material transport, which calculates the evolution of the
disk's surface density profile, and mixing with a kinetic study of D-H isotopic exchange amongst gas-phase molecules. They found
that the water D/H ratio is low in the hot inner disk due to rapid exchange reactions with molecular hydrogen and increases
outwards where the exchange becomes less efficient. Contrary to previous studies, they found that further out in the disk, the
water D/H ratio decreases again as water exchanged at high temperatures near the young star is transported outwards in the early
evolutionary stages of the disk. However, their chemical approach is simplistic and only includes neutral-neutral reactions as
these would be dominant in the early stages of disk evolution because effects of ion-molecule reactions and photochemistry are
significantly diluted. With the two-dimensional mixing in effect, we however find that these processes play a role in the
chemistry and deuterium fractionation.
{Recently, \citet{2013arXiv1310.3342F} investigated the water chemistry in turbulent protoplanetary disks. They found that
transport by turbulence in to the atmosphere allows a reprocessing of water, which is destroyed by photoreactions and then
transported back to the midplane where water is (re)formed, and this cycle is most effective at radii $\lesssim 30$ AU. While
\citet{2013arXiv1310.3342F} also discussed the effects of radial transport without including it in their models, we have done so
and found that radial transport can further decrease both water abundances and D/H ratios through reprocessing in the warmer,
inner midplane regions. They also studied the effects from different desorption energies for atomic hydrogen and deuterium,
comparing $E_{des, H/D} = 400$ K and $E_{des, H/D} = 600$ K \citep[as found from molecular dynamics simulations for crystalline
and amorphous water ice, respectively
]
{2007MNRAS.382.1648A}. The higher energy increases the residence time on grains for H and D, and results in a higher water reformation rate in the midplane, and hence the resulting water abundances are higher by up to an order of magnitude.. The most significant difference to our model is in the calculated D/H ratios throughout the disk. For \citet{2013arXiv1310.3342F}, in the inner region, D/H drops to the cosmic ratio $\sim 1.5\times 10^{-5}$ at radii $\lesssim 10$ AU, and even lower at radii $\lesssim 2$ AU, while our 2D-mixing model retain an enhanced D/H ratio of $\sim 10^{-4}$ between $\sim 1-7$ AU. We find that the enhanced ratio at radii $\lesssim 7$ AU is largely because of ``warm'' fractionation pathways (see Section~\ref{sec:results}), which is included in both models. It is therefore likely that this discrepancy is a result of different physical parameters, such as stellar parameters adopted, leading to different thermal structure of the nebula.
}
\subsection{Solar System bodies}\label{sec:SS}
There is a great variation in chemical composition in the different bodies of the Solar System. Due to the temperature gradient throughout the solar nebula, \citep[see figures in][]{2011Natur.478..218H,2013ApJ...774L...3L}, the D/H ratio varies radially through the disk, as we have also seen in our results (see Figure~\ref{fig:DH}). The same behavior has been observed in the solar nebula from observations of small Solar System bodies \citep{2000SSRv...92..201R}: the D/H ratios increasing with radial distance from the Sun. Due to sedimentation in the planetary cores and reprocessing in their atmospheres, the D/H ratios in planetary bodies of the Solar System are no direct measure of the pristine D/H ratio.
While we can understand the rough composition of most of the planets in the Solar System, the more important discussion is the composition of small Solar-System bodies, i.e. asteroids and comets. Their origin is less constrained, but we can derive boundary conditions from the Grand Tack scenario \citep{2012AREPS..40..251M}, the most successful theory of the formation and evolution of the Solar System. In this scenario, Jupiter migrated inwards after its formation, only to be stopped at $\sim 1.5$ AU by a mean motion resonance with Saturn after which both giant planets move back outwards to their present positions, and before Jupiter had managed to diminish the available material in the inner disk for build-up of the terrestrial planets. In this theory, many of the observed features of the Solar System today can be reproduced, such as the smaller mass of Mars \citep{2011Natur.475..206W}. As Jupiter moved through the asteroid belt at $\sim$2-3 AU twice, it scattered much of the belt material. This might
explain the mix of C- and S-type asteroids in the inner and outer asteroid belts \citep{2012M&PS...47.1941W}.
Carbonaceous chondrites have measured water D/H ratios very similar to that of Earth's oceans \citep{2003SSRv..106...87R}. Comets, or more specifically Oort-family comets, on the other hand, have been observed with enhanced D/H ratios relative to the Earth's oceans. Observations reveal values $\sim $1.8-6.0~$\times$~10$^{-4}$ (see Table~\ref{tab:obs}), an enhancement to the Earth ocean value by a factor of few. These values would put their formation origin in our models around $10-20$ AU. However, new experimental studies of ice sublimation suggest that the D/H measured in the evaporated vapor of comets might be depleted by 70\% or more with respect to the bulk D/H ratio in the nucleus \citep{2012P&SS...60..166B}. With this in mind, the measurements of D/H ratios in Oort-family comets would be even more enhanced relative to Earth's oceans, by more than an order of magnitude. Regarding the origin of these comets, it means that they would be expected to have originated further out in the disk, $\sim 30-40$ AU,
and is in agreement that these long-period comets are believed to have formed much further in compared to where they are found today in the Oort cloud \citep[see e.g.][]{1950BAN....11...91O, 1987AJ.....94.1330D}.
While Oort-family comets have too high D/H ratios to be considered as a major source in the delivery of Earth's ocean, the observation by \citet{2011Natur.478..218H} has revealed D/H ratios similar to Earth's ocean water in the Jupiter-family comet 103P/Hartley-2. The existence of diversity in D/H ratios between the Jupiter- and Oort-family comets has been further confirmed by the observations of \citet{2013ApJ...774L...3L} who observed a D/H ratio $< 2 \times 10^{-4}$ in the Jupiter-family comet 45P/Honda-Mrkos-Pajdusakova. Therefore, we may consider the D/H ratio of Hartley-2 as representative of Jupiter-family comets, which puts their possible formation location between $\sim$ 1-10 AU. Most likely they originate in the region between Jupiter and Saturn where the gravitational pull of Jupiter and Saturn have scattered them, giving them their current high-eccentricity orbits past the snow line $\sim 3$ AU. If they would have formed in the Kuiper belt located beyond the orbit of Neptune, which is the current
theory \citep[see e.g.][]{1999SSRv...90..301W, 2007MNRAS.381..779E, 0004-637X-687-1-714}, their D/H ratios would have been much higher in our model as the chemical timescale for de-fractionation of HDO ice is too long. It still means that the origins of the Jupiter- and Oort-family comets are not as distinct as previously thought.
\section{Conclusions}
In this paper the isotopic and chemical evolution of water in the early history of the solar nebula before the onset of planetesimal formation is studied. An extended gas-grain chemical model that includes multiply-deuterated species, high-temperature reactions, and nuclear spin-state processes is combined with a 1+1D steady-state $\alpha$-viscosity nebula model. To calculate initial abundances, we simulated the 1~Myr of the evolution of a cold and dark TMC1-like prestellar core, resulting in initially high D/H ratios for water and other molecules of $\sim 1\%$. Two time-dependent chemical models of the solar nebula are calculated over 1~Myr and for radii $0.8-800$~AU: (1) a laminar model and (2) a model with 2D-turbulent mixing transport. We find that both models are able to reproduce the Earth ocean's water D/H ratio of $\approx 1.5\times 10^{-4}$ at the location of the asteroid belt, $\la 2.5-3$~AU, where a transition from predominantly solid to gaseous water occurs.
The water ices there can be incorporated in growing solids, melt, and eventually produce phyllosilicates. At $\la 2$~AU, nebular temperatures are too high for the water ice to exist and the dust grains are water ice-free. Thus the planetesimals, from which Earth would later form, remain water-poor. We find that the radial increase of the D/H ratio in water outward is shallower in the chemo-dynamical nebular model. This is related to more efficient de-fractionation of HDO via rapid gas-phase processes, as the 2D mixing model allows the water ice to be transported either inward and thermally evaporated or upward and photodesorbed. Taking the water D/H abundance uncertainties of the factor of 2 into account, the laminar model shows the Earth water D/H ratio at $r \la 2.5$~AU, while for the 2D chemo-dynamical model this zone is larger, $r \la 9$~AU. Similarly, the enhanced water D/H ratios representative of the Oort-family comets, $\sim 2.5-10\times 10^{-4}$, are achieved within $\sim 2-6$~AU and $\sim 6-30$~AU
in the laminar and the 2D model, respectively. The characteristically, slightly lower, water D/H ratio that has been found for Jupiter-family comets are found further in and we find their possible formation location $\sim 1 - 10$ AU in both models. This means that, in our models, we find an overlap in the possible formation location for Oort- and Jupiter-family comets. However, with regard to the water isotopic composition and the origin of the comets, the mixing model seems to be favored over the laminar model as the former allows Oort-family comets to have formed in the region of Jupiter's and Saturn's present location.
\acknowledgments
This research made use of NASA's Astrophysics Data System. The research leading to these results has received funding from the European Community's Seventh Framework Programme [FP7/2007-2013] under grant agreement no. 238258. DS acknowledges support by the {\it Deutsche Forschungsgemeinschaft} through SPP~1385: ``The first ten million years of the Solar System - a planetary materials approach'' (SE 1962/1-2).
\bibliographystyle{apj}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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ONCE AROUND THE WORLD
It Bites
Tonny Larz
Anyone who doesnt acknowledge the talent of IT BITES..is a serious contender for this years: "Get your ears checked" !! As Mr.Dunnery (he of IT BITES leadguitar and compository fame) headmaster and grand leader of the guitar and lyrically serpent of UK band IT BITES...this their second outing.. is extremely fabulous and another notch to their belt in the history of UK progmusic!! Dunnery excels himself as a writer and a composer....never have i heard such progmusic in the UK vein extremely welldone and exuberantly talented music....i love these guys....they are the new world of progmusic hailing from UK!! Talented song writing...breaks...timesignatures...songs de Luxe....i need this music....!! Five stars....i cant rate otherwise....if you dont agree??... Tuff [&*!#]!! (sorry!)
Report this review (#25310)
Posted Friday, January 9, 2004 | Review Permalink
Sean Trane
SPECIAL COLLABORATOR
This band reminds me of some of the worst moments in prog history (look at the dates) and also reminds a bit of Saga (not in sound but in the approach to music). Just not my cup of tea. On the whole I don't appreciate this band so I won't review their other albums ( i've heard the majority of it) so as not to hurt too many feelings too much.
I am not sure one can consider fully It Bites as prog, but once again because of the dire times, the least group that had a slight bit of longer tracks, did draw some reactions from starved progheads (I include myself in this pack, since I have heard the majority of the group's discography). Just not my cup of tea!!
Posted Tuesday, March 2, 2004 | Review Permalink
This is by far a gem,inthe world of prog music!! Mr.Dunnery are really an example of great guitarplaying and prog composition. Now i havent heard their other works,but if this album is anything to go by! Id like to hear everything by them. Dunnery are a supreme guitarist and composer! And the band supreme progmusicians. GO GET THIS !!
Posted Tuesday, June 8, 2004 | Review Permalink
I agree with the other reviewers that Francis Dunnery is a great guitarist and there is some interesting use of the guitar but on the whole I find this album disappointing. Yes it does have odd signatures and lots of different moods but only in the longer tracks the rest is just very dated sounding 80's pop rock. It Bites were really Marillions only serious opposition (other contempories IQ not really bruising the singles chart) in the genre and while Marillion did pop singles they had a timeless quality that none of these songs posess as the production reeks of the late eighties. The more overtly proggy tracks are good and worth a listen but still give a feeling of "prog for beginners".
Posted Wednesday, June 9, 2004 | Review Permalink
Blacksword
Their best album, without a doubt, and certainly their most progressive. With Steve Hillage co-producing a few of the tracks, it's clear again that IB had prog in mind for this, their second album. The album opens with the explosive, albeit rather poppy 'Midnight' and is followed up 'Kiss like Judas'; the hit single from OATW. With the pop out of the way the band plough into 'Yellow Christian' the first of three conceptual songs on the album. 'YC' is awash with Tony Banks style keyboards, and some quirky Yes-ish vocals, not to mention a few odd time signatures for good prog measure. This is not to say that this album is a rip off of other artists. IB always had their own style and sound, even though their three studio albums were all very different. 'Old man and the angel' was the second single from this album. As a single it failed miserably, but thats hardly surprising as it's a great song!! The CD unlike the vinyl has the full 9 minute version of OMATA, which in my view is the one of the best tracks on the album. The epic is the title track. A thirteen minute collage of English prog eccentricity, woven together by excellent musicianship. The album is well produced, well played and drips proffesionalism and confidence. It Bites were a sadly underated prog act in the 80's. Once described by Mark Lammar as 'Like Marillion with sex. A far more pallatable concept than sex WITH Marillion'
Posted Wednesday, August 25, 2004 | Review Permalink
hdfisch
Most fellow reviewers call this one their best and most prog-ish album and maybe they're right, since their other releases are even worse than this one. First two songs are nothing special in fact, "Midnight" and "Kiss Like Judas" are simple pop-rock songs played in typical 80's style. "Yellow Christian" is the first track showing a bit more prog-ish approach at least for the poor relations of these "darkest times of prog". "Rose Marie" is a rather short rock song with quite funny lyrics and staccato riffing. Next one "Black December" is for sure the weakest one of the album. "Old Man And The Angel" sounds slightly better to me, quite ok showing at least some versatility. The beginning of "Hunting The Whale" appears quite prog-ish with some laughter and sounds of whales but what follows is quite disappointing I have to say. "Plastic Dreamer" is in a similar very typical for the 80's, maybe good for the time being, but since I use to hate 80's at all, it can't offer me anything. The title song is the longest one with a quite unusual length for this era of 15 minutes and for sure the highlight of the album. It's the first one of this album I can call a real prog song and the reason for giving this album 3 stars although I've to admit only by showing much good will not to offense too much fans of this band.
Posted Sunday, March 6, 2005 | Review Permalink
Prog rock didn't get much better than this in the 80s. The opening track "Midnight" is lightweight funk-pop, but don't be put off. "Kiss Like Judas" (performed live on "Wogan" as I recall) is proof that prog bands can do pop singles. "Yellow Christian" is a delightful exploration of time signatures, yet still manages to sound commercial. "Rose Marie" is pop metal with a stunningly fast guitar break from Francis Dunnery. "Black December" is strong but less interesting than their earlier recording on the b-side of the "Whole New World" single. The best is yet to come. "Old Man and the Angel" is a magnificent tour-de-force with John Beck's keyboards to the fore. Despite lyrics about a toy shop coming to life during the night(!), "Plastic Dreamer" sports a Holdsworth-esque liquid guitar solo and fine interplay between Beck and Dunnery. The title track rounds things off in style. "Once Around The World" is a "Supper's Ready" for the 80s, full of changes and surprises, with all four band members contributing ideas and playing up-front. This album is evidence that It Bites were the most original prog band of their generation. These guys were no mere Genesis- soundalikes. Here they combine prog with pop, funk and metal in a seamlessly uncontrived way. Only "Midnight" lets this album down, a rather too obvious shot at reaching the pop charts. Buy it.
Posted Wednesday, March 30, 2005 | Review Permalink
Teaflax
A perfect marriage of Pop and Prog that puts all other Neo-Prog of the 80's to shame. Catchy without being simplistic and hooky without being too repetitive or obvious, It Bites managed to unify two genres that are rarely seen side by side: actual British-style Pop and Metal-tinged Prog.
Yes, the production reeks of the 80's, but that doesn't detract from som great performances and sterling song writing. Dunnery's guitar playing has some truly unique touches and his singing a youthful freshness that is rarely heard in this kind of context.
Posted Tuesday, May 23, 2006 | Review Permalink
Paul Stump
Some great comments here - nobody gets it wrong! "Prog for beginners" - really like it! This is a real minefield - a mate with impeccable taste hates this. I love it. Quite right, the first two songs suck. But the whole point with IB was that they were unashamedly after the big time, and weren't afraid to be proggy, for which a few stars. They had balls, energy and just completely blew any other 80s prog band out of the water (I don't count the Enid in that) because they didn't pretend to be something they weren't. Old Man and the Angel and, particularly, the title track, are awesome. Tinny production, horrid 80s sounds, but we can't blame them for the kit at their disposal. The Live at Montreux stuff shows what they could do live. Dunnery was a great guitarist, but I think it was John Beck on keys who provided the supple and often cinematic harmonies.
Posted Sunday, October 15, 2006 | Review Permalink
Easy Livin
Honorary Collaborator / Retired Admin
Calling all the Bravehearts!
Despite the presence of a couple of longer tracks, "Once around the world" is still very much based in 80's pop rock. Of the nine tracks, six are of a similar simplistic structure. The performances are exemplary, and the songs certainly well above average, but prog it ain't.
The melodies can sometime sound very similar to those of LEVEL 42, this being particularly apparent on "Hunting the whale" which closely resembles "Lessons in love". The message of the song is along the lines of Yes' "Don't kill the whale" the lyrics asking "why can't you leave him alone..".
The opening couple of tracks, "Midnight" and "Kiss like Judas" have strong melodies and harmonic vocals. Even when we get to "Yellow Christian", a slower 6 minute song, the overriding impression is of an enhanced rock ballad. Certainly the instrumentation has a slightly neo-prog feel, with hints of the type of music made by IQ and JADIS. The 9½ minute "Old man and the angel" is really in two distinct parts, the pop element being followed by a lengthy development of the theme.
The final track, which gives the album its title, runs to almost 15 minutes. The song is by far the bands most progressive, although it remains completely accessible. Various instrumental sequences, including frequent synthesiser flourishes, punctuate the melodic storytelling of A day in the life. The lyrics include a section which goes "The Scots have invaded our land again. . . take a piece of turf home to show their wives", an apparent reference to a great football victory at Wembley(!). The track closes with a fine building instrumental refrain.
The three longer more elaborately structured pieces here cannot disguise that fact that this is primarily a sophisticated pop prog album. It is nevertheless a hugely enjoyable piece of work, enhanced by those three standout tracks.
Posted Monday, March 26, 2007 | Review Permalink
First of all I want to say a couple of things: this is the FIRST time I give a 5 stars rating and most important of all, you MUST have at least some appreciation for pop music to enjoy bands like It Bites.
I think it's a mistake to judge It Bites as a progresive band, I'd like to define their music as pop/rock with some notable progressive influences. All songs from "Once around the World" scream "pop" and I don't think the band were proggers trying "desperately" to appeal the masses. All compositions sound very honest, incredibly inspired and mature for such a young band which also shows great musicianship and impressive vocal arrangements. Probably the 80's approach of the production of this record let many people down as well as the approach of the songwriting: there is almost no hint of any 70's prog influence. I can barely imagine how fresh this album might have sounded back in 1988.
There's no weak track here: "Kiss like Judas" I think was a minor hit for the band, next come three fantastic tracks in a row "Yellow Christian", "Rose Marie" and "Black December", great prog-influenced pop songs with complex vocal arrangements. The title track is what I call a more conventional "prog" song because its lenght, instrumentation and structure but in my humble opinion doesn't represent the real approach of the band, the rest of the album is a better example of that.
It's sad this very talented band didn't make it big, perhaps they were too good and sophisticated for mainstream popularity and too "poppy" for most progheads who, two or three decades later, are still willing for some mellotron-fulled 20+-minute epics.
Fortunately It Bites are together again, minus bandleader, main songwriter, guitarist and great vocalist Francis Dunnery, this time replaced by John Mitchell (of Arena fame). Hope this reunion bring some interest for this great band.
Posted Wednesday, October 24, 2007 | Review Permalink
johnnythelowery
5 stars. Nothing else like it. Well, then. When it was released. There is tons now copying this band! For me-this album was the harder side of YES i'd always secretly wished they'd been. or Genesis for that matter. Though I do like those guys-but gave up on the idiom until It Bites came along. It's rock, but with an artful, educated, knowing mastery of their respective instruments and because of it, their confidence enfuses every twist and turn. Fabulous singing, harmony singing, intricate keyboards, great bass and inventive drumming all around a mesmeric guitarist. Lyrics are rather deeper than most prog. bands. It has an amazingly bright/clean sound and doesn't try to hide the intricute instrumentation. THe title track 'once around the world' is one of the finest prog. tracks ever made unveiling vistas of music never heard before. For me it's a total encapsulation of everything that is great in Prog. Rock. One of the highest peaks of rock ever climbed. For that, I am very very greatful to you It Bite's guys indeed!!
SouthSideoftheSky
Symphonic Team
"I may look like some animal, but I am what I am, a man"
According to many Once Around The World is It Bites' best album and it is surely an improvement over the promising but somewhat pre-mature debut. Once again we get very accomplished Crossover Prog with both strongly commercial elements and many progressive elements. As on the debut, the sound of the band is mostly wholly Pop oriented and the songs all have strong hooks, but the arrangements are often unconventional and the music takes some surprising twists and turns that will certainly please the Crossover fan. Like all It Bites albums, this album too is very well produced and it sounds very professional, even more so than the already polished debut.
It Bites were hardly a groundbreaking group but they certainly had their own characteristic sound and approach which is further developed here. Compared to the debut, there is more energy and "punch" in these songs and a slightly harder edge is tried out here for the first time. The voice (which often evokes Peter Gabriel) of singer and guitarist Francis Dunnery is very strong and distinctive and the whole band oozes with musical and instrumental talent. There is a bit more lead guitar work on this album which is also very accomplished. John Beck provides some very nice keyboard work as well.
The album opens with Midnight which is more or less a pure Pop song that probably leaves many Prog fans wondering what the fuss is about, but it gets better. On Kiss Like Judas the progressive influences start to come through within a still basic Pop approach. Yellow Christian is a very good song with some great Queen-like bombast. Here, like in many other places, Dunnery's guitar sound reminds me of that of Brian May. Rose Marie is again a rather straightforward song, whatever progressive elements present are found in the finer details.
The second half of the album is generally more progressive in nature. The first genuinely progressive number is the nine plus minute The Old Man And The Angel which allows the band to stretch out a bit more instrumentally. Overall, this song reminds me quite a bit of the Neo-Prog band IQ. On some passages the guitar sound is strongly Allan Holdsworth-like which brings UK to mind. Some vocal arrangements are quite Yes-like.
The other Prog song here is the closing 15 minute title track which is strongly Genesis-like. In many ways this is It Bites' answer to Supper's Ready. The band are clearly outside of their comfort zone here and I often feel this "epic" is a bit disjointed and it does not reach the same highs as The Old Man And The Angel for me.
Whatever potential there was on the debut is certainly fulfilled here with this second effort and this is surely a good album that will make a nice addition to any Crossover Prog collection. But I hesitate to call this an excellent addition to any Prog collection. The classic Prog purists will not be easily converted to this.
All in all, Once Around The World is a fine album with one foot in Pop and the other in Prog. This might be the most progressive It Bites album, but my favourite is actually their third album, Eat Me In St Louis.
Posted Saturday, December 26, 2009 | Review Permalink
b_olariu
Once around the world is the second release of It Bites from 1988. This much better then the predecesor, the sound is crystal clear, the compositions are tighter, more choesive and has that special atmosphere of late' 80's. Definetly their most mature work and their best album for sure, It Bites release a great album with pleasent music, beutiful voice, and some very good keyboards arrangements, I realy like this album. Bordering between AOR, more rock edged then previous album with some polished pop ventures combined very well with some neo prog elelents It Bites release a solid album in my opinion. Maybe is not very complex like other releases from neo prog, but is very well done. Every pieces is great, specially the longest track from here Once Around The World, nearly 15 min of great music and superb kyboards anf gutar elements. This might be one of the best pop rock albums of late '80' s for sure, everything is well done, very subtile pop, maybe sometimes they remind me of The Box, art pop of the highest calibre. 4 stars for sure for this album, still not very well regarded as their best work in music circles, and still considerated an easy band for traditional progressive listners, to mellow for the progresive metal ones, but in the end a real surprise for me , in agood way. Almost excellent, great album, I like it a lot.
Posted Saturday, January 9, 2010 | Review Permalink
Drake/Sinister
Probably my favorite prog-crossover act of all time, It Bites are one of those bands that I wished I had the opportunity to see when I was overseas in England back in the late 80's. They had a pompy Smiths/Van Halen hybrid approach to proggy songwriting that made them rather fun to listen to in a big stage setting, Kind of like cock rock that girls could swing with you to.
Anyway, Once Around The World is their 1988 sophomore release (produced by Steve Hillage of GONG fame) and arguably the best set of songs they'd ever write until John MItchell reformed them in the mid 2000's. Francis Dunnery in particular, whom I never had the pleasure of meeting personally, brings one hell of a vocal and guitar presence to all the songs present here. Tracks like 'Kiss Like Judas' and "Midnight' were clever subversions of 80's popular music whilst "Old Man and the Angel" was probably the best fusion of pop and proggy pretension the decade ever saw, and easily on par with any of Marillion's best material.
I know this band isn't the most popular here on the Archives, but as someone who recognizes a good original sound when I hear it, I think this band deserves more appreciation from prog. fans than they get. After all, who else sounds like It Bites? NOBODY!
Posted Friday, November 19, 2010 | Review Permalink
Rune2000
Prog Metal Team
This is actually a weird album to revisit for me since I vividly remember really liking it back in the late '90s and actually considered it to be the band's best release. Looking and listening to it today just doesn't bring out that same enthusiasm.
Unlike The Big Lad In The Windmill which hid its gems towards the end of the album, Once Around The World throws in all of its assets upfront and slowly begins to stagnate towards its second half. Songs like Midnight, Kiss Like Judas, Black December and Old Man And The Angel can only be described as classic It Bites material, even though the latter is a bit too long for its own good. The problems arrive with pretty stale tunes like Hunting The Whale and Plastic Dreamer and even though the mini-opus of a title track does balance out some of the weaker material, it just doesn't make the overall feel of this album any better.
Once Around The World might not be the great album I remembered it being, but it's still not a total waste of time for fans of '80s Art Rock and even if The Big Lad In The Windmill might feel more consistent, this album is definitely the more progressive of the two. The melodies that work, work all the way, while those that lack the punch fail miserably and fall into the dark hole of obscurity for me. Luckily there isn't a single moment here that makes me cringe because of the sound production, which is pretty much the best compliment one can give to an '80s album.
Not good enough for an excellent rating, but far from a fans only release. I'd say that a good, but non-essential rating is in order here!
***** star songs: Midnight (4:06)
**** star songs: Kiss Like Judas (4:10) Yellow Christian (6:30) Rose Marie (3:34) Black December (3:51) Old Man And The Angel (9:21) Once Around The World (14:49)
*** star songs: Hunting The Whale (4:47) Plastic Dreamer (3:54)
Posted Saturday, December 4, 2010 | Review Permalink
Don't forget these boys.
It Bites, probably best known to joe public for their top ten fluke hit "Calling All The Heroes" (1986) were quickly destined to be filed away in the long line of faceless pop bands that dominated the late 1980s and ultimately made the discerning listener yearn for the advent of the 1990s. Instead they became one of England's great lost rock bands that left a void which could never really be filled. the cumbrian quartet, centred around two young musical boffins, Francis Dunnery and John Beck, quickly made its mark as a highly idiosyncratic Pop and Rock band that was hard to pigeonhole but compelling to behold nonetheless. Indeed, they were probably the last of its kind, drawing equally from quirky contemporary Pop and sprawling Art Rock and ? even signing a nine album deal in its wake. Somebody dubbed them a "Marillion with sex" for a short while, however, Francis Dunnery's highly individual vocal and guitar stylings and John Beck's keyboard wizardry quickly set them apart. they never managed another hit single after that but instead concentrated on creating masterful albums. Two of these followed, this one and the follow up "Eat Me In St. Louis" before the band split acrimonously and seemingly for good. They have since reformed in 2007 with long time admirer and New Art Rock journeyman John Mitchell (Arena, The Urbane, Kino, Frost* etc.) replacing Dunnery ? i.e. the brand has been revived ? but it's just not the same any more, evolution and artistic development notwithstanding.
Want to know why? Listen to this one, their second and best effort by a mile. This really is the album that they are being remembered for. It has a few stabs at another glorious hit single with the unnervingly angular yet catchy "Midnight", "Kiss Like Judas", "Rose Marie" and "Black December" but on this one they explored their Art- and Prog-Rock tendencies even further with producer, famed old hippie Steve Hillage (ex-Gong) certainly not standing in their way. Elaborate songs like Yellow Christian", "The Old Man And The Angel" (edited as a single release by a progressingly nervous record company...) and "Hunting The Whale" bristle with oddball ideas, strange twists and turns and a stark contemporary vibe. Rarely has there been a more definite transition from being a singles act to being an outright album band ? and it is the elaborate, 15-minute title track that sealed matters forever and definitely made some folks at Virgin Records frown for a long time afterwards. The term "tour de force" must have been invented for this one alone and it's one of those pieces of music that really do take you on a journey while additionally providing a potted history of the best of 20th century English music, straddling elements of Music Hall sentimentality, Brass Band, Northern Folk, flat-out Hard Rock riffola and pastoral serenity, seamlessly pieced together. A work of art, no more, no less.
"Eat Me In St.Louis", released only a year later, veered of into a more commercial hard rock territory and the band was quickly at a loss with their apparent versatility eventually turning against them. For a Rock audience they were too pop... and they were certainly too cumbersome and unwieldy for a more commercially orientated public. One more live album, the aptly titled "Thank You And Goodnight" was released, well capturing the band's sheer energy and playfulness in concert and they were never to be heard of again, until recently in a different configuration and several tours of their homeland. The name has remained in very high regard since. Don't forget them.
Posted Thursday, November 17, 2011 | Review Permalink
Nothing short of being a pop/rock/prog masterpiece.
It Bites really takes the cake here. Influence by heavy progrock like Rush, Saga and even the pop-approach of Mike and the Mechanics, Francis and co set the standard for years to come.
No longer, prog had to be about long songs and relentless repetition. The eighties were a game changer for progrock. Sure, I love Camel, Yes, Crimson, Floyd et al, but this is something totally different and needs to be listened to with a totally different ear.
New Wave was the answer to Punkrock, and bands like Tears for Fears, Duran Duran, a-ha and Simple Minds made fair use of synths again. So a lot of bands (Rush, Saga, Twelfth Night and others) took a bit of a synthpop/new wave approach.
It Bites did something different, they just cranked it up. heavy distorted guitars, thunderous drums, heavy soloiing. This was a form of progrock nobody had heard before. The songstructures were pop/rock: verse-chorus-verse-chorus and simple lyrics about everyday life.
Some songs are simple hardrock with heavy synths thrown in: Judas, Midnight, Black December, Rose Marie, but some songs are just pure heavy progrock: Christian, Old Man & the Angel and the titletrack. Some odd time signatures, extended soloiing, you name it.
It Bites was (in my opinion) the heaviest progrock band around. I don't know if Queensryche was around yet, but they were more metal than rock.
Posted Friday, April 6, 2012 | Review Permalink
FragileKings
This album was a pleasant surprise for me to find out about and listen to. I first heard about It Bites a year ago when they released "Map of the Past" and it showed up on my Amazon page. I thought they were some new band but somehow heard that they'd released an album or two before. Then while reading Stephen Lambe's book "Citizens of Hope and Glory: The Story of Progressive Rock", I was surprised to read that It Bites was a band from the 80's! Lambe wrote that their "Once Around the World" album was a surprise piece of prog in the prog parched 80's. I felt that I must check it out and with little more than a glimpse at a video on YouTube I ordered it.
The first two tracks are what I had expected from an 80's album. This is 80's rock that is too synthesizer-swamped for the guitars to make it hard rock, but too rockin' to be just 80's pop. Call it 80's pop rock if you like. Not quite my taste and a little embarrassing to have playing on the car stereo. But not bad songs for what they are.
The third track "Yellow Christian" is in the same vein but more synth and less guitars, making it seem closer to bubblegum pop except that in the middle there's a smart section that turns to prog flavour. The first time I heard this my ears pricked right up after having tuned out of the music. Now I knew that this album might have a few surprised before the big 14-minute-plus finale.
"Rose Marie" sounds to me like mid-eighties Uriah Heep or Blue Oyster Cult. The guitar playing is enjoyable but particularly so because in the YouTube video segment I watched, guitarist Francis Dunnery explained about using a guitar where the strings are higher off the fret-board, making the finger work necessarily more precise. My first guitar also had such high strings and it was not easy to learn how to play a lot of hard rock songs at first. Later when I bought a Gibson Epiphone (a Les Paul would have been nice but...) I at last had an easier time of playing. So, I could appreciate Dunnery's skill and the different quality of sound his guitar solos have on the album.
"Black December" is much like most of the album sounds so far. But things are about to get more interesting.
While on the surface "Old Man and the Angel" sounds like another pop rock track, it soon changes and fits in a wonderful prog section in the middle. At first I was thinking that this is what Yes should have been doing on "Big Generator" but then I thought It Bites were pre- saging the prog revival of the 90's, in particular sounding a bit like The Flower Kings. When the song concludes with its pop rock chorus it maintains an odd drum beat. It Bites came to the dinner party in an appropriate jacket but has now taken it off and is showing a prog T-shirt underneath.
"Hunting the Whale" and "Plastic Dreamer" both take us away from the pop rock factory in different ways. At times I felt the vocals sounded a bit theatrical like Peter Gabriel but "Hunting the Whale" really comes off sounding like what Genesis might have been had the classic line-up held together into the late 80's. It's a bit bizarre with a raucous tavern dinner atmosphere at the beginning and the end, whale sounds, some crusty old salt singing from his boat all blended with an 80's synthesizer as the main music. "Plastic Dreamer" tells the story of someone who gets himself locked in the toy store so he can confirm his belief that the toys come alive at night. Darth Vader dressed in drag is one of the many humorous images conjured up in the lyrics. The whimsy of the song sounds like what some otherwise serious pop rock band would have put on their album and have had it criticized as filler or inconsistency. But I find this song and the previous one showing the band's humour and willingness to go out on a limb.
Of course the song that Stephen Lambe praised was the album closer "Once Around the World". Clocking in a just under fifteen minutes, this song begins very smoothly and appropriately where the music of "Plastic Dreamer" ended off, with very beautiful and delicate synthesizer. The song picks up and goes through some interesting changes not unlike "Supper's Ready" by Genesis with odd clips and snippets of what could have been other songs fitted in smartly. As the music graduated through its atmospheres, tempos, and flavours, I felt it could easily have appeared on any Flower Kings album.
My conclusion thus far is that this album introduces itself as a pop rock album but reveals its secret intention to keep symphonic prog alive in the 80's. Considering that the old guard of the 70's were either split up or recording pop music and the neo-prog movement was by 1988 turning towards the mainstream more and more, finding an album like this one is quite a surprise. Once again I must restate my impressions that It Bites sound like a mixture of how classic Genesis might have sounded in the 80's and The Flowers Kings with a hint of what 80's Yes could have been. Pop rock songs aside, I think this was a very bold and intriguing album for the band to make. It is perhaps due to be rated as one of my favourite prog rock albums of the 80's.
Not quite essential to any prog rock collection but certainly essential for an 80's prog collection. For the effort put toward prog on this album I'll give it four stars.
Posted Saturday, October 26, 2013 | Review Permalink
Zoltanxvamos
𝗣𝗲𝘁𝗲𝗿 𝗚𝗮𝗯𝗿𝗶𝗲𝗹 𝗵𝗮𝘀 𝗥𝗲𝘁𝘂𝗿𝗻𝗲𝗱
Is it just me or does Francis Dunnery sound like Peter Gabriel? Anyways, after a bunch of Genesis involvement, Francis Dunnery was originally picked to be the singer for the band. This album was almost a decade before 'Calling All Stations'. Unfortunately for Genesis, this album is much better than 'Calling All Stations'. This album starts with a banger and ends with a banger, Midnight and Once Around The World are both incredible, and staple tracks by the band. Kiss Like Judas is a hit sounding track with a prog spin. Probably the best track on the album, 'Yellow Christian' which has a bunch of odd time signatures, alternating time signatures, and fantastic lyrics. 'Rose Marie' is a Rush type track, fast, catchy, and retro rock influenced. 'Black December' has a great mood, its soft and melodic, less accessible, and much warmer in terms of a writing perspective. 'Old Man and the Angel' is quite long, it's very dynamic, but it's not quite an epic. The song does have its writing changes, and its typical epic structure, but it's more of a Neo-Prog/Crossover Prog typical sound. 'Hunting The Whale' and 'Plastic Dreamer' are both incredible tracks, they are more typical songs by the band but still brilliantly written. However I'm going to go on the record and say that the title track is the best song, it's a brilliant epic. The structure, the lyrics, the songwriting was incredible, the concept, the chord structure, etc, etc. Everything on that song is great, but aside from the songwriting, everyone plays really well and Francis is a great singer.
I'm sorry but this album is just ridiculously amazing, this is a very intriguing, and engaging album. This is It Bites magnum opus, their masterpiece, it's Crossover Prog perfection, and I don't understand why people aren't a fan of this. Either way, It Bites should be more appreciated, this is a very underrated album.
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\section{Introduction}
A large number of observations in the past few decades have established the
general properties of the X-ray spectra from accreting luminous sources. These
are systems in which a compact object, either a neutron star or a black hole, affects
the surrounding material with its vast gravitational forces, which leads to accretion
of large quantities of gas and consequently to the release of large amounts of
radiation, mostly in the X-ray band. The compact object can be a supermassive black hole
as those present in the active galactic nuclei (AGN) of many galaxies, or of
stellar-mass, such as is the case of many galactic black holes (GBHs).
In AGN, the X-ray continuum is usually characterized by a power-law spectrum
with photon index $\Gamma$, which can typically be observed to lie in the
$1.8 \gtrsim \Gamma \gtrsim 2.2$ range, although in some extreme cases it can
be either closer to 1 or larger than 3, extending to high-energies with an
exponential cutoff around $100-300$~keV.
At energies below $0.1$~keV, the power-law often meets a soft excess
that mimics a blackbody radiation. In GBHs, the X-ray spectrum in most cases
is dominated by a thermal blackbody-like component that peaks at high-energies
($0.2 \lesssim kT \lesssim 2$~keV), but can also show a high-energy tail component
of emission. In general, this spectral energy distribution is explained in terms of a
geometrically thin, optically-thick accretion disk around the compact object.
The energy dissipation within the disk would be responsible for the quasi-blackbody
emission observed. The power-law continuum is believed to originate through the Compton
up-scattering of the thermal photons by the electrons in a hot corona or a jet
\citep{haa93,dove97,dau13}.
The presence of this dense ($n_{\mathrm H} \gtrsim 10^{12}$~cm$^{-3}$), warm ($T\sim 10^5-10^7$~K),
and optically-thick ($\tau_{\mathrm T} \gtrsim 1$) medium is also supported by the detection of atomic
features from several ions. These and other features constitute an important component of the
X-ray spectrum observed from accreting sources, resulting from the reprocessing of radiation by
the material in the disk. This component is commonly referred to as {\it reflection}, in the sense that
it is the result of radiation that is returned from the accretion disk by fluorescence or
electron scattering. The current paradigm is that the original power-law radiation irradiates
the surface of the accretion disk. The X-ray photons then interact with the material producing
diverse atomic features. These can be produced both via absorption (mostly in form of edges),
and emission (in form of fluorescence lines and radiative recombination continua, RRC).
Therefore, the reflection component provides direct information about structure, temperature,
ionization stage, and composition of the gas in the accretion disk.
The presence of the Fe K-shell fluorescence emission and the absorption K-edge observed
in the $6-8$~keV energy range are recognized as strong evidence for reflection. X-ray photons
that are photoelectrically absorbed have enough energy to remove a 1s electron from its K-shell,
leaving it in a quasi-bound state above the continuum (autoionizing state). The K-hole is
then filled by an electron, and the energy difference can be released by emitting a second
electron (Auger process), or by the emission of a K-shell photon. These transitions are of the
parity-changing type $np-1s$, where $n=2$ and $n=3$ correspond to the K$\alpha$
and K$\beta$ transitions, respectively. Higher $n$ transitions are allowed but with a much
lower probability. The fluorescence yield,
i.e. the probability of emitting a photon over an Auger electron, is proportional to the nuclear
charge $Z$ to the forth-power (i.e., $\propto Z^4$), making the Fe K-shell emission particularly
strong. This has been shown to be true in a large number of observations from X-ray accreting
sources \citep[e.g.][]{got95,win09,ng10}. The Fe K emission line has also proven to be crucial in
the determination of one of the fundamental quantities that describe black holes, its angular
momentum \citep{lao91,dab97,bre06}. If the reflection occurs near the black hole, line photons will suffer
Doppler effects, light bending, and gravitational redshift, which produces a skewed line profile
with a red wing that can extend to very low energies, particularly in the case of high spin
\citep[e.g.,][]{fab00,fab03,rey03,dov04,mil08,ste11,rey12,dau12}.
Therefore, the proper modeling of the reflected disk
component is of vital importance for determining one of the two fundamental parameters that
define a black hole.
The first theoretical studies of X-ray reflection assumed that irradiation on
the surface of the accretion disk was weak enough so the gas remains neutral, but yet would
reprocess the radiation producing observable spectral features \citep{gui88}. Green's functions
to describe the scattering of photons by cold electrons were first derived by \cite{lig80,lig81},
and their implications for AGN observations discussed in \cite{lig88}. The latter approach
is used for the calculation of cold reflection in the model {\sc pexrav} \citep{mag95}. However,
none of these calculations included line production. \cite{geo91} and \cite{mat91} included
the X-ray fluorescence line emission in their Monte Carlo calculations, providing line strength,
angular distribution, and equivalent widths for the Fe K line. \cite{zyc94} carried out similar
calculations including photoionization equilibrium, yet neglecting the intrinsic emission
inside the gas.
Much more detailed calculations of the radiative transfer of X-rays in an optically-thick
medium were carried out by \cite{ros78} and \cite{ros79}. Their code solves the transfer of the continuum
photons using the Fokker-Planck diffusion equation, including a modified Kompaneets operator
to properly treat the Compton scattering, while the transfer of lines is calculated using the
escape probabilities approximation. This code has been updated over the years leading
to the {\sc reflionx} model \citep{ros93,ros05}, which has been widely used to model the
reflected component in the spectra of accreting X-ray sources.
\cite{dum03} used the {\sc titan} code to examine the accuracy
of the escape probability methods versus the ``exact" solution of the radiative transfer by
implementing accelerated lambda iterations. This code has been extended by \cite{roz02} to treat the cases
of Compton-thick media. All these calculations assume constant density in the material. It
has been argued that a plane-parallel slab under hydrostatic equilibrium could represent the surface
of an accretion disk more accurately \citep{nay00}, and that its reflected spectrum is in fact
different from the one predicted by constant density models \citep[see also][]{roz96,nay01,
bal01,dum02,ros07,roz08}.
Besides the techniques used to treat the transfer of photons through the media, the codes
mentioned above may also differ in terms of the atomic data used, which in most cases
offers a limited perspective of the physics governing the atomic processes for the absorption,
excitation, and emission processes. These limitations have been overcome by our new reflection
model {\sc xillver} \citep{gar10,gar11}.
{\sc xillver} calculates the reflected spectrum emerging from the surface of an X-ray illuminated
accretion disk by simultaneously solving the equations of radiative transfer, energy balance,
and ionization equilibrium in a Compton-thick, plane-parallel medium. The transfer is solved
using the Feautrier method under a lambda iteration procedure \citep{mih78}. {\sc xillver} implements
the well-known photoionization code {\sc xstar} \citep{kal01} to solve the ionization structure of
the atmosphere, therefore making use of the most updated, accurate, and complete atomic database
for the X-ray line emission, photoabsorption, and photoionization for all the astrophysically
relevant ions.
In this paper we present a complete library of reflection spectra using an updated version
of our code {\sc xillver}. This grid of models covers a wide range of input parameters, relevant
to model the spectrum from accreting X-ray sources. Each model is defined by the photon index
$\Gamma$, the ionization parameter $\xi$ (given by the ratio of the X-ray flux over the gas
density), and the abundance of Fe with respect to its solar value. The models are provided in
a single table\footnote{\url{http://hea-www.cfa.harvard.edu/~javier/xillver/}}
suitable to be used in the commonly used fitting packages such as {\sc xspec}
\citep{arn96}, and {\sc isis} \citep{hou00}.
We show a detailed analysis of our models along the space of parameters and a
careful comparison with other similar models. The improvements achieved are stressed, and their
implications on the analysis of X-ray spectra is discussed.
This paper is organized as follows. In Section~\ref{secmod} we describe the basic aspects of
the theory used in our calculations, paying particular attention to the changes and improvements
implemented in the code. An analysis of the new reflection models for the different parameters
is presented in Section~\ref{secres}. We show the effect of the photon index, the
ionization parameter, and the iron abundance on the ionization structure and ultimately on
the emergent spectra. A thorough comparison with other reflection models is also provided.
The main conclusions and future prospects are summarized in Section~\ref{seccon}.
\section{The Reflection Model}\label{secmod}
In order to calculate the reflected spectra from X-ray illuminated
accretion disks we made use of our reflection code {\sc xillver}.
The details of the calculations are fully described in \cite{gar10},
thus we shall review only the main aspects, in particular those where
some changes have been applied. One important modification takes place in the
solution of the radiation transfer equation. This equation describes the interaction of the
radiation field with the gas in the illuminated atmosphere, and it is now
expressed as
\begin{equation}\label{ert}
\mu^2\frac{\partial^2 u(\mu,E,\tau)}{\partial\tau^2}
= u(\mu,E,\tau) - S(E,\tau)
\end{equation}
where $u(\mu,E,\tau)$ is the energy density of the radiation field for
a given cosine of the incident angle with respect to the normal $\mu$, energy $E$,
and position in the slab, which now is specified in terms of the {\it total}
optical depth
\begin{equation}\label{eqdt}
d\tau =\chi(E) dz,
\end{equation}
where
\begin{equation}
\chi(E)=\alpha_{kn}(E) + \alpha_a(E)
\end{equation}
is the total opacity of the gas that includes both scattering and absorption.
Here, $\alpha_{kn}(E)$ and $\alpha_a(E)$ are the scattering and absorption coefficients,
respectively. The former is given by the product of the gas density times the Klein-Nishina
cross section $\sigma_{kn}(E)$ for electron scattering (which includes the relativistic
correction). The latter is defined by the product of the gas density times the absorption cross section
$\sigma_a(E)$ due to bound-bound, bound-free, and free-free processes.
The source function $S(E,\tau)$ in Equation~(\ref{ert}), is given by
the ratio of the total emissivity to the total opacity.
It can written as
\begin{equation}\label{eqsou}
S(E,\tau) = \frac{\alpha_{kn}(E)}{\chi(E,\tau)}J_c(E,\tau) + \frac{j(E,\tau)}{\chi(E,\tau)},
\end{equation}
where $j(E,\tau)$ is
the continuum plus lines emissivity, and $J_c(E,\tau)$ is the Comptonized mean
intensity of the radiation field resulting from the convolution
\begin{equation}\label{eqjc}
J_c(E,\tau) = \frac{1}{\sigma_{kn}(E)}\int dE' J(E',\tau)\sigma_{kn}(E')P(E',E).
\end{equation}
Here, $J(E,\tau)=\int u(\mu,E,\tau)d\mu$ is
the unscattered mean intensity of the radiation field. The quantity $P(E',E)$ is the
probability of a photon with energy $E$ to be Compton scattered to an energy $E'$,
which we approximate by assuming a Gaussian profile centered at
\begin{equation}\label{eqec}
E_c = E \left( 1 + \frac{4kT}{m_ec^2} - \frac{E}{m_ec^2}\right)
\end{equation}
\citep{ros78,ros93}, with energy dispersion
\begin{equation}\label{eqsig}
\sigma = E \left[ \frac{2kT}{m_ec^2} + \frac{2}{5}\left(\frac{E}{m_ec^2}\right)^2\right]^{1/2},
\end{equation}
where $k$ is the Boltzmann's constant, $T$ is the gas temperature, $m_e$ is the rest-mass
of the electron, and $c$ is the speed of light. In Equation~(\ref{eqjc}) we now follow
\cite{nay00} and properly take into account the relativistic (Klein-Nishina)
correction to the cross section for electron scattering, rather than using the classical (Thomson)
approximation \citep{gar10}.
Referring to Section~2 in \cite{gar10}, the radiative
transfer equation written in terms of the Thomson optical depth $\tau_{\mathrm T}$ has
an extra linear term when compared with the usual form shown here (Equation~\ref{ert}).
We found that the radiation transfer equation written in terms of the total
optical depth behaves better when there are large gradients in the opacity, despite
the fact that the integration along the path is performed in a different grid for each
energy bin (i.e., the total optical depth is a function of the photon energy). We use
a logarithmically spaced grid fixed in $\tau_{\mathrm T}$ with 200 spatial bins in the
$10^{-4} \leq \tau_{\mathrm T} \leq 10$ range. At each energy, the
total optical depth is calculated by integrating Equation~(\ref{eqdt}).
The boundary conditions are only slightly different from those in \cite{gar10}. At the top of
the slab ($\tau=0$), we specify the radiation field incident at a given angle
$\mu_0$ by
\begin{equation}\label{eqbco}
\mu \left[ \frac{\partial u(\tau,\mu,E)}{\partial \tau} \right]_0
- u(0,\mu,E) = -\frac{2F_x}{\mu_0}\delta(\mu-\mu_0),
\end{equation}
where $F_x$ is the net flux of the illuminating radiation integrated in the whole energy
band. At the inner boundary ($\tau=\tau_{max}$), we specify the outgoing radiation field
to be equal to a blackbody with the expected temperature for the disk:
\begin{equation}\label{eqbcm}
\mu \left[ \frac{\partial u(\tau,\mu,E)}{\partial \tau} \right]_{\tau_{max}}
+ u(\tau_{max},\mu,E) = B(T_{disk}),
\end{equation}
where $B(T)$ is the Planck function, and $T_{disk}$ can be defined using the
\cite{sak73} formulae. Nevertheless, for the models presented in this paper we have
chosen to neglect any illumination from below the atmosphere, i.e., $B(T)=0$. This
is convenient when comparing our models with previous reflection calculations such
as {\sc reflionx} \citep{ros05}. Also, for the parameters typical
in AGN, the temperature of the disk is sufficiently low that the intrinsic black body
emission is weak compared to the power-law incident at the surface.
Since these models are calculated under the assumption of
constant density, we use the common definition of the ionization parameter \citep{tar69}
to characterize each case, namely,
\begin{equation}\label{eqxi}
\xi = \frac{4\pi F_{\rm x}}{n_e},
\end{equation}
where $F_{\rm x}$ is the net integrated flux in the 1-1000~Ry energy range, and
$n_e=1.2 n_{\mathrm H}$, and $n_e$ and $n_{\mathrm H}$ are the the electron
and hydrogen number densities, respectively. The solution of the system is found by
forward elimination and back substitution. With a solution for the radiation field
$u(\mu,E,\tau)$, a new $J_c(E,\tau)$ and thus a new $S(E,\tau)$ can be calculated.
These are then used to update the solution for $u(\mu,E,\tau)$.
The procedure is repeated until these quantities stop changing within a small fraction.
A full transfer solution must be achieved iteratively in order to self-consistently
treat the scattering process. In general, this procedure
requires $\sim \tau_{max}^2$ iterations for convergence. Since our calculations are
carried out up to $10$ Thomson depths, we perform 100 iterations. This ensures convergence
for energies where scattering is dominant, for which $\tau_{\mathrm T} \approx \tau$. For
energies where the photoelectric opacity is large, the total optical depth can be
much larger than 10. Nevertheless, if absorption dominates over scattering, the first
term in the right-hand side of Equation~(\ref{eqsou}) is reduced. In the limit
$\alpha_{a} \gg \alpha_{kn}$, the source function essentially becomes independent of
$J_c(E,\tau)$, and the solution converges very rapidly.
The structure of the gas is determined by solving the ionization balance equations
for a given gas density and for a particular solution of the radiation field. At each point
within the slab, we use the photoionization code {\sc xstar} \citep{kal01} to calculate
level populations, temperature, the total opacity $\chi(E,\tau)$, and the emissivity
$j(E,\tau)$, assuming that all the physical processes are in steady-state and imposing
radiative equilibrium. We have updated our code {\sc xillver} to work with the current
version of {\sc xstar} (version 2.2.1bn)
\footnote{\url{http://heasarc.gsfc.nasa.gov/xstar/xstar.html}},
and the most recent atomic database. Although the physics is the same, this new version
includes improved routines that considerably reduce the computing time, allowing the
exploration of a wider parameter space for a reasonable allocation of resources.
\subsection{Atomic Data}\label{secatomicd}
As mentioned before, {\sc xillver} implements the {\sc xstar} routines for the calculation
of the ionization structure of the gas, while making use of {\sc xstar}'s atomic data. The core of
the {\sc xstar} atomic database is described in detail in \cite{bau01}. It has been
constructed using information from many sources such as CHIANTI \citep{lan06}, ADAS
\citep{sum04}, NIST \citep{ral08}, TOPbase \citep{cun93} and the IRON project \citep{hum93}.
Over the last several years, we have dedicated a significant effort towards the investigation
of the K-vacancy states in many ions. This has allowed us to systematically improve the modeling
of the K lines and edges relevant for high-quality astronomical X-ray spectra.
Such atomic data sets include energy levels, wavelengths, Einstein $A$-coefficients,
radiative and Auger widths computed for a large number of ions using three
different atomic-structure theoretical approaches, including the relativistic
Hartree--Fock \citep{cow81}, the {\sc autostructure} \citep{bad86,bad97},
and the multi-configuration Dirac--Fock \citep{ber87,sea87} methods.
Extensive calculations of photoabsorption and photoionization cross sections have
been also performed using the Breit--Pauli R-Matrix method including the effects of
radiative and Auger damping by means of an optical potential \citep{gor99,gor00}.
In particular, calculations of the atomic data required for the spectral
modeling of the K-shell photoabsorption of oxygen and nitrogen ions have been carried
out in \cite{gar05} and \cite{gar09}. Computations have also been carried out for the
atomic structure of the isonuclear sequences of Ne, Mg, Si, S, Ar, and Ca \citep{pal08a},
nickel \citep{pal08b}, and aluminum \citep{pal11}. More recently, we have focused our
attention on the iron peak and light odd-Z elements, i.e., F, Na, P, Cl,
K, Sc, Ti, V, Cr, Mn, Co, Cu and Zn producing the atomic parameters for more than 3 million
fine-structure K lines \citep{pal12}. Photoabsorption calculations of the cross sections
across the K-edge of Ne, Mg, Si, S, Ar, and Ca have been performed for ions with less
than 11 electrons by \cite{wit09}, and for ions with more than 10 electron by \cite{wit11}.
In these calculations it was shown that damping processes affect the resonances
converging to the K thresholds
causing them to display symmetric profiles of constant width that smear the
otherwise sharp edge at the photoionization thresholds. The Li-like to Ca-like
ion stages of nickel are discussed in \cite{wit11b}. These new data sets are continually
being incorporated into the {\sc xstar} data base in order to generate improved opacities
in the K-edge regions of the ions considered.
The main differences in the current version of the atomic data with respect to the one used in
\cite{gar10} are as follows: The inclusion of the atomic data for the $n=2\rightarrow 3$
inner-shell transitions of Fe~{\sc vi-xvi} ions, the so-called Fe M-shell unresolved transition
array \citep{gu06}. The inclusion of the K-shell atomic data for the Mg, Si, S, Ar, Ca, Al, and Ni
isonuclear sequences \citep{pal08a,pal08b,wit11,pal11,wit11b}. The implementation of the
updated radiative and dielectronic recombination rates from \cite{bad06} and \cite{bad03}.
Also, we have improved the resolution of the high-energy extrapolation implemented in the Fe
photoionization cross sections for energies well above the K-edge.
Finally, by looking at the third-row Fe ions (Fe~{\sc i-viii}), we found that
for the densities used in our simulations ($n_e\sim 10^{15}$~cm$^{-3}$), many metastable
states are populated. Previously, we only used K-shell photoionization data for the ground
state, therefore missing the contribution from these metastable states. This resulted
in a weak Fe K emission line in models with low ionization parameter. This problem has been
corrected in the new version of the data base by including the photoionization cross
sections for all these states.
\subsection{Illumination spectrum}
Another important change in the {\sc xillver} code is the definition of the radiation field
that illuminates the surface of the accretion disk. In \cite{gar10}, the incident spectrum was
assumed to be a power-law with a given photon index $\Gamma$ for all energies in the
$10^{-1}-2\times 10^5$~eV range. This kind of illumination is appropriate for spectra with
$\Gamma\sim 2$. However, for steeper spectra ($\Gamma > 2$) this choice of illumination creates
unphysical conditions, since the incident power-law will tend to have very many
photons concentrated in the low energy part of the spectrum. Thus, we have now adopted a
power-law that breaks at 100~eV and decays with an exponential tail for lower energies. We have
also extended the energy to 1~MeV, and moved the exponential high-energy cutoff from 200~keV
to 300~keV. Figure~\ref{fpowerlaw} shows the impact of this modification for the resulting reflected
spectrum in a case where $\Gamma=3$. The solid lines show the reflected spectra predicted by {\sc xillver} using the old
(black) and the new (red) definitions of the illumination. The incident power-law is shown with
dashed lines for each case. Both calculations have a similar ionization parameter (log~$\xi\sim 2$)
but the integrated flux is different given the differences in the shape of the power-law. Thus,
the model with the new definition (broken power-law) needed to be scaled down such that both incident
spectra will have the same flux near $100$~eV and energies above, region in which the
photon index is $\Gamma=3$.
The huge impact of the illuminating spectrum on the reflected component is clear, especially
at energies below 5~keV.
This difference becomes more dramatic as the photon index increases. However, for harder
spectra ($\Gamma \lesssim 2$), the break in the incident power-law has little impact on the
reflected spectrum, especially in the high-energy band.
The extension of the spectrum to higher energies has also an important influence on
the reflection calculations. This is particularly important in models on the other extreme of the
photon index parameter space. In the case of a very hard illumination spectrum ($\Gamma < 2$), models
with similar ionization parameters will differ in the number of photons that are concentrated in
the high energy part of the spectrum. Figure~\ref{fhighenergy} shows the resulting {\sc xillver} models
for an incident power-law with $\Gamma=1.4$, log~$\xi=2.8$, and solar abundances. For the black
and red curves, the energy range extends upward to $200$~keV and $1$~MeV, respectively.
In panel~(a), the
dashed lines show the incident power-law spectra, while solid lines are the reflected spectra for
each case. The emission lines in the model with the high energy coverage (red) are considerably weaker
than those in the original model (black). This is most evident for the Fe K complex around $6-7$~keV,
and for the O~{\sc vii} L$\alpha$ near 650~eV. The reason for this difference is that in models with the
high-energy extension, the region of the atmosphere near the surface is more strongly heated
by the high-energy photons, and thus its temperature is higher than in the original
model. This can be seen in panel~(b) of Figure~\ref{fhighenergy}, where the temperature profiles
in the vertical direction of the disk are plotted as function of the Thomson optical depth. Clearly,
the two models converge at large depths ($\tau_{\mathrm T}\sim$few), but the one in which the
illumination extends to higher energies results in a considerably hotter atmosphere near the surface.
Although we expect the Fe features to be produced well inside the disk, line photons that escape
through a hotter gas will tend to suffer a larger energy shift from Compton scattering,
with similar probabilities
for up- or down-scattering. This makes the line profile broader and more symmetric.
\section{Results}\label{secres}
As in the previous version of {\sc xillver}, all the models shown here are calculated with an energy
spectral resolution comparable with the best current observations (${\cal R}=E/\Delta E\sim 350$),
which requires 5000 bin points in the energy range considered here ($10^{-1}-10^{6}$~eV). Each
model covers a large column density of gas ($\tau_{\mathrm T}=10$), using 200 spatial zones and
10 bins that describe the angular dependence of the radiation fields. The first step in the
spatial grid has been reduced to $\tau_{\mathrm T}\sim 10^{-4}$ to ensure an accurate representation of
the illuminated layers in the slab. The hydrogen number density is held constant at
$n_{\mathrm H}=10^{15}$~cm$^{-3}$ for all the models presented in this paper.
\subsection{The space of parameters}
A new library of reflection spectra has been produced using the new version
of the code {\sc xillver}. This set of models covers a wide range
of parameter values that are relevant for fitting the spectra
of accreting sources. Each model is characterized by the photon index $\Gamma$
of the incident power-law radiation, the ionization parameter $\xi$, and the
iron abundance $A_\mathrm{Fe}$ with respect to the solar value \citep{gre98}. In order
to have a library of models suitable for both AGN and GBHs, we have produced models
covering photon indices in the $\Gamma = 1.2 - 3.4$ range, in steps of 0.2.
Observations typically show that the reflection signatures from accretion disks
in AGN are produced in a material at a lower ionization stage than those observed
from GBHs \citep{gar11}, consistent with the picture that the accretion disk in the latter are
much hotter than in the former. Consequently, we vary the ionization parameter over a wide
range to cover both classes of sources. We have produced
models with $\xi=1, 2, 5, 10, 20, 50,...,10^4,2\times 10^4, 5\times 10^4$~erg~cm~s$^{-1}$.
Finally, we also allowed the iron abundance to be treated as a free parameter,
given the importance of the Fe K emission profile in most accreting sources.
For simplicity, the abundance of all the other elements considered, namely H, He, C,
N, O, Ne, Mg, Si, S, Ar, and Ca, is kept fixed to the solar values of \cite{gre98}.
Thus, models with $A_\mathrm{Fe}=0.5, 1, 5$ and $10$ were calculated, taking into account
both sub- and super-solar Fe abundances. Here, $A_\mathrm{Fe}=1$ corresponds to an iron abundance
of $2.5\times 10^{-5}$ with respect to hydrogen. These choices of parameter values resulted
in a library of 720 synthetic
reflection spectra that can be used for the modeling of the reprocessed component
in X-ray observations. The present set of models is provided as a single file
\footnote{\url{http://hea-www.cfa.harvard.edu/~javier/xillver/}}
in FITS (Flexible Image Transport System) format, which can be loaded into the fitting
package {\sc xspec}\footnote{\url{http://heasarc.gsfc.nasa.gov/xanadu/xspec/}}
via the {\tt atable} model.
\subsection{The effect of varying the ionization parameter $\xi$}
Figure~\ref{fspec.xi} shows a sub-group of the resulting reflected spectra for three
values of the photon index $\Gamma$ and a range of ionization parameters. The Fe abundance
is set to the solar value ($A_\mathrm{Fe}=1$) for all the models shown here. Panels (a), (b),
and (c) correspond to $\Gamma=1.4$, $2$ and $2.6$, respectively. Each panel shows
the reflected spectra calculated for a different value of the ionization parameter. From
bottom to top, each curve corresponds to $\xi = 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000,
2000, 5000$~erg~cm~s$^{-1}$, respectively. Because the gas density is held fixed at $n_e=1.2\times 10^{15}$~cm$^{-3}$
for all these calculations, increasing the ionization parameter is equivalent to increase
the flux $F_x$ of the illuminating source, where $F_x = \xi n_e/4\pi$. The spectra are plotted
in units of $E F_E$ (equivalent to $\nu F_{\nu}$ if plotted in frequency), so that a power-law of
$\Gamma=2$ is shown as a horizontal line. Note that no rescaling or shift is applied;
instead, each curve is color-coded according to its corresponding value of log~$\xi$,
to improve clarity. This Figure provides a general overview of the effect that the ionization
parameter has on the spectrum reflected from an optically thick plane-parallel slab.
The original power-law shape of the illuminating continuum (for the lowest value of $\xi$),
shown in black dashed lines, suffers drastic modifications due to both absorption and
emission. The spectra are shown in the entire energy range included in the calculations
($0.1 - 10^6$~eV). In the low-energy part of the spectrum ($0.1-10$~eV), the continuum
is dominated by bremsstrahlung emissivity, despite the fact that this is where the
illumination flux is decreased due to the cut-off imposed at $100$~eV. Emission lines
and absorption edges from H, He, and C are clearly visible for most of the models with
low and intermediate ionization. In the $10-10^4$~eV energy range, the photoelectric opacity
dominates over the electron scattering opacity. Therefore, this
region is where most of the absorption occurs, yielding large departures from the
original power-law continuum. Bremsstrahlung emissivity decreases rapidly for energies
above $\sim 100$~eV; however, many emission lines from all the ions included
in these calculations remain visible over the entire energy range. At higher energies ($>10^4$~eV),
electron scattering is the dominant source of opacity since the cross section for
photoelectric absorption decays as $\sim E^{-3}$, while the Klein-Nishina cross section
for electron scattering remains fairly
constant. Thus, in this spectral region Compton scattering is the only relevant process and
the reflected spectrum depends on the shape of the original illuminating field.
By looking at the overall shape of the reflected continuum shown in Figure~\ref{fspec.xi},
it is possible to distinguish between two main regimes: (1) The {\it high-ionization case},
where the resulting spectra mostly shows very narrow emission features, while the continuum
still resembles the original shape of the illuminating power-law; and (2) the
{\it low-ionization case}, where the emerging spectra
are a combination of a very rich and complex set of emission-line profiles superimposed
on a strongly absorbed and modified continuum, which significantly departs from the original
power-law. The specific value of the ionization parameter that separates these
two regimes depends, to a certain degree, on the photon index $\Gamma$. In fact, for the
$\Gamma=1.4$ case shown in panel (a) this transition is quite obvious, as a drastic
change in the reflected spectra can be seen between models with log~$\xi=2.3$ and $2.7$.
These changes are the result of large differences in the ionization balance solutions,
as can be seen in Figure~\ref{ftemp.xi}, which shows the corresponding temperature profiles
for each one of the models shown in Figure~\ref{fspec.xi}, color-coded in the same way.
Because larger ionization implies a larger illumination flux, models with high $\xi$ values
are systematically hotter than those with low $\xi$ values. The illuminating radiation,
incident at the surface of the slab (at $\tau_{\mathrm T}=10^{-4}$ for our proposes), penetrates the
first layers heating the gas. The amount of heating not only depends on the illuminating
flux, but also on the particular shape of the radiation field; e.g., note that the models
for small values of $\Gamma$ are consistently hotter than those with large values.
We shall discuss these effects further in the next Section. As photons are
scattered or absorbed and re-emitted at different energies, the shape of the continuum is
modified reducing the net heating in the gas. At some point H and He recombine which increases
the cooling very rapidly. The gas then suffers a rather sudden transition to a lower temperature,
as can be seen for most of these models. This transition occurs at deeper regions in
the slab for larger ionization parameters and for softer input spectra (large $\Gamma$).
Regardless of the ionization parameter or the shape of the ionizing
radiation, in all cases the field thermalizes and the gas eventually reaches the same
lower temperature around $(0.5-1)\times 10^5$~K. This is the limit in which the illuminating
radiation no longer contributes to the ionization of the material and its temperature
is simply set by the density of the gas.
To illustrate in more detail the effects of the ionization parameter on the reflected spectra,
we will concentrate on models with one particular value of the photon index.
Further, we will examine the low- and high-ionization models separately. Thus,
Figure~\ref{flowhighxi} shows some of the reflected spectra resulting from models with
$\Gamma=2$ in the $10^2 - 5\times10^5$~eV energy range, which is the spectral
band typically covered in X-ray observations. Panel~(a) shows 4 models with low-ionization,
specifically $\xi=1, 5, 20,$ and $100$, multiplied by factors of $1, 10^2, 10^4,$ and $10^6$,
respectively. Their corresponding temperature profiles are curves
1, 3, 5 and 7, from left to right, in panel~(b) of Figure~\ref{ftemp.xi}. The dashed line
represents the illuminating power-law spectrum for the model with the lowest ionization
parameter ($\xi=1$). As mentioned before, these models display a strong decrease of the
continuum flux due to photoelectric absorption, in particular for energies below $\sim 20$~keV.
Nevertheless, a very rich and complex set of fluorescent emission lines due to K-, L-,
and M-shell transitions from many ions is also present, superimposed on a highly
absorbed continuum. The general spectral shape of the lowest
ionization models ($\xi=1-20$) is very similar, where only small changes are seen. The
temperature in these models is relatively low ($T \lesssim 10^6$~K), and the gas settles
to a low temperature regime at a small optical depth ($\tau_{\mathrm T} \lesssim 0.1$), meaning
that most of the slab remains neutral. The emission due to the K$\alpha$ and K$\beta$ transitions
in Fe is distinctive at $\sim 6.4$ and $\sim 7.1$~keV, respectively. The case for $\xi=100$
is where the changes in the reflected spectra become more evident, in both the continuum
absorption and in the emission features. The higher temperature in this model allows for the
excitation of elements such as Mg, Al, Si, S, Ar and Ca, which produces more fluorescence
lines in the $1-10$~keV region. The Fe K$\alpha$ emission becomes slightly broader and
the K$\beta$ is less intense. Some radiative recombination continua (RRC) can also be
seen near $\sim 0.5-1$~keV.
Panel~(b) of Figure~\ref{flowhighxi} shows the reflected spectra for the next five values
of the ionization parameter considered here ($\xi=200, 500, 1\times 10^3, 2\times 10^3,$ and
$5\times 10^3$). Each curve has been rescaled by a constant factor to improve
clarity. These factors are $1, 10, 10^2, 10^3,$ and $10^5$ from low- to high-ionization,
respectively. As before, the dashed line shows the illuminating power-law
spectrum corresponding to the model with the lower ionization ($\xi=200$). These are the
models considered as {\it high-ionization}, since the continuum is not highly affected by the
photoelectric opacity. This can be seen by comparing, for example, the reflected
flux at $10^2$~eV in the spectrum for $\xi=200$ and in the one for $\xi=1$, shown in panel~(a).
As mentioned before, increasing the ionization raises the temperature of the illuminated
region of the slab. This region also extends to deeper zones as the radiation is able to
ionize the gas at larger optical depths, as can be seen in the corresponding temperature
profiles (see panel~b of Figure~\ref{ftemp.xi}). The increase in the temperature has two
main effects. On the one hand, it affects the ionization of the gas. The ions from low-$Z$ elements
are completely stripped from all their electrons, while the ions of the heavier elements are partially ionized.
This changes the emission lines produced inside the gas, and thus the emerging spectra lacks
emission from low-$Z$ ions, progressively showing more emission from highly ionized O, Ne,
Ar, Ca and Fe, as the illumination increases. On the other hand, the high temperature affects
the way line photons are scattered by electrons, given the dependence of Equations~(\ref{eqec}) and
(\ref{eqsig}) on the kinetic energy of the electrons. When $4kT \sim E$, the probability for
a photon to either gain or lose energy after each scattering becomes
comparable, which effectively produces a broadened and more symmetric line profile.
This effect is particularly evident in the K$\alpha$ emission lines from O and Fe, observed
at $\sim 0.65$ and $\sim 6.9$~keV, respectively. We shall return to a more detailed discussion
of the spectral features in Section~\ref{secfea}.
\subsection{The effect of varying the photon index $\Gamma$}
It is clear from the discussion in the previous section that both the net flux and the
spectral shape of the ionizing radiation incident on the surface of the slab have a great impact on the
ionization balance of the gas, and thus on the reflected spectrum that will
emerge at the surface. For this particular library of models we have adopted a
power-law shape for the illumination spectrum, thus the
general shape is controlled by changing the value of the photon index $\Gamma$.
Figure~\ref{fspec.gamma} shows a sub-group of the reflected spectra calculated for various
conditions. Panels~(a), (b), (c) and (d) show the resulting models for a given ionization
parameter, i.e., $\xi=10, 10^2, 10^3$, and $10^4$, respectively. Each panel contains the
spectra of models for all the values of the photon index $\Gamma$ considered in our
calculations ($\Gamma=1.2 - 3.4$), color-coded accordingly. Since the goal of this Figure
is to show the general trends in the spectra introduced by changing $\Gamma$, all the curves
are plotted with the same normalization (thus, no rescaling was applied). It is quite obvious
how the changes in the ionizing continuum affect the resulting continuum
of the emergent radiation, as expected. However, the ionization structure of the gas
seems to follow different trends on $\Gamma$ for different values of $\xi$.
By comparing the two extreme values of $\xi$ (panels a and d), a completely opposite
behavior is seen. In the low ionization case ($\xi=10$), models with low $\Gamma$ values
are the ones with the most absorption, suggesting a much colder gas than those for high
$\Gamma$ values. This is consistent with the temperature profiles obtained for these models,
which are shown in panel~(a) of Figure~\ref{ftemp.gamma}. Therefore, in the low-ionization regime,
the softer the input spectrum (large $\Gamma$ value), the hotter and more ionized the
gas becomes. Looking at the high-ionization case in panel~(d) ($\xi=10^4$), the opposite
occurs: the models with harder illumination spectra (lowest $\Gamma$) are completely ionized,
owing to the large gas temperature. Thus, in the high-ionization regime, the harder the
input spectra, the hotter and more ionized the gas becomes. The intermediate regimes plotted
in panels~(b) and (c) of Figures~\ref{fspec.gamma} and \ref{ftemp.gamma} show the transition
between these two regimes. For $\xi=10^2$ the gas temperature is low for the harder spectrum
($\Gamma=1.2$), and increases with the photon index up to $\Gamma \sim 2.4$, where it starts
decreasing again. For the case with $\xi=10^3$ (panel~c), the transition occurs at
$\Gamma \sim 1.8$.
The reason for this change of behavior is related to the processes that contribute to the heating
and cooling of the gas. In general terms, there are two main competing mechanisms:
photoionization heating plus recombination cooling, and Compton heating and cooling
due to electron scattering. The low ionization models are those for which the illumination
is relatively low. In this case, photoionization is the dominant process that heats the illuminated
layers of the slab. Because the heating rate due to photoionization is essentially given by
the radiation field flux times the photoelectric opacity of the gas, and since the latter
is dominant at energies below $\sim 10$~keV, a very hard input spectrum will produce much
less heating than a soft spectrum, owing to the lack of low energy photons.
The contrary is true in the high-ionization regime where the dominant process is
Compton heating and cooling.
As discussed in Section~3.1 of \cite{gar10}, in this limit the gas temperature
approaches an asymptotic value, the Compton temperature, given by
\begin{equation}\label{etcomp}
T_C = \frac{<E>}{4k}
\end{equation}
where
\begin{equation}\label{emeane}
<E> = \frac{\int F(E)EdE}{\int F(E)dE}
\end{equation}
is the mean photon energy, which is a quantity that only depends on the spectral shape.
In Figure~\ref{ftcomp} we show the resulting Compton temperature as a function of the
photon index $\Gamma$ in the range of our calculations, which agrees very well with the
temperature of the hot layer in the high-ionization models shown in panel~(d) of
Figure~\ref{ftemp.gamma}.\footnote{Note, however, that {\sc xstar} (and consequently {\sc xillver})
employs a full relativistic treatment of the Compton heating and cooling from \cite{gui82},
rather than using Equation~(\ref{etcomp}).} Physically this makes sense, since for the input spectra with
low $\Gamma$ values most of the photons are concentrated in the high energy part of the
spectrum, where Compton scattering becomes very important. Therefore, in the high-ionization
regime, hard spectra are more efficient in heating the illuminated layers of the slab.
\subsection{The effect of varying the Fe abundance}
The elemental abundances considered in a photoionization calculation
can potentially affect the ionization balance, temperature structure, and ultimately
the observable spectral features in the reprocessed radiation. The total amount of
a particular element changes the continuum opacity, which in turn affects the photoionization
heating rate. At the same time, the abundance of a particular element
influences the strength of the emission and absorption
features due to bound-bound and bound-free transitions. Given the relevance of the Fe
emission in the analysis of the X-ray spectra from accreting sources, we have carried
out calculations in which the Fe abundance, normalized to its solar value, is varied between
sub-solar, solar, and super-solar values. All the other elements considered in these calculations
are set to their solar values. Figure~\ref{fafe} shows a comparison of the reflection
calculations for different values of the iron abundance $A_\mathrm{Fe}$. Left panels show
the temperature profiles, while right panels show the corresponding reflected spectrum
in the $10 - 10^5$~eV energy range. In all the panels, each curve corresponds to one
particular value of $A_\mathrm{Fe} = 0.5, 1, 5$ and $10$, where $A_\mathrm{Fe}=1$ corresponds to
$2.5\times 10^{-5}$ of Fe with respect to H \citep{gre96}. In each one of the right panels, the
plotted spectra have been rescaled for clarity. The scaling factors are, from bottom to top,
$10^{-2}, 1, 10^2,$ and $10^4$. Top, medium, and bottom
panels correspond to ionization parameters $\xi=10, 10^2$ and $10^3$, respectively.
The photon index is set to $\Gamma=2$ in all these models. The general tendency is the
same in all these simulations. The increase of the Fe abundance induces more heating
in the illuminated layers due to the increase in the opacity (and thus a larger photoionization
rate), which raises the gas temperature. However, because continuum absorption is also increased,
the radiation field thermalizes at a smaller depth for the high abundance models,
as can be seen from the temperature curves. Note that the increase in temperature
is more subtle in the high-ionization case (top panel), since in this regime the Compton
heating and cooling is the dominant process that controls the gas temperature.
The effects of the Fe abundance are evident in the reflected spectra as well,
both in the emission and the absorption features. The $10^2 - 10^4$~eV energy range clearly
shows a substantial reduction of the flux due to the increase in the continuum opacity;
meanwhile, the Fe K edge near $8$~keV grows deeper as $A_\mathrm{Fe}$ becomes larger. At the same time,
all the Fe emission features are affected as well, which is mostly evident in the Fe K
emission complex in the $6-8$~keV region. The strength of the whole emission profile
increases when the abundance is high, as expected. In the high ionization models (top panel),
there is a distinctive RRC profile right before the Fe K edge, which becomes substantially
more prominent for $A_\mathrm{Fe}=10$. Also, there are enhanced emission lines near
$100$~eV and just above $1$~keV, which correspond to M- and L-shell transitions in iron,
respectively. However, these emission profiles are noticeable only for the $\xi=100$ model
(middle panel), in particular the L-shell. The reason for this is that these transitions occur
in a rather narrow range of ionization stages. If the gas is very neutral, the photoelectric
absorption reduces considerably the number of ionizing photons at those energies, reducing
the number of excitations from the L-shell. On the contrary, if the gas is hot and ionized,
the fraction of Fe ions with L-shell electrons is very low (i.e., Fe~{\sc i}-{\sc xvi}),
as most of them are stripped already. This is of great relevance, since this feature
can be used to constrain both the ionization of the gas as well as the iron abundance.
One particular example is the Seyfert 1 galaxy 1H~0707-495, which X-ray spectrum shows
evidence of a very intense emission in both the Fe K- and L-shell regions. Fits using
reflection models require a high Fe abundance at an ionization parameter consistent with
the present analysis \citep{fab12b,dau12}.
Although somewhat extreme, 1H~0707-495 is not the only case where Fe is found to be over-abundant
based on predictions from reflection models. In fact, at least for AGN,
it is commonly the case that super-solar iron abundance is required to fit the
observed X-ray spectra \citep{fab06}. A possible explanation for the apparent extreme Fe abundance
in some AGN has been proposed by \cite{rey12}, on the grounds of radiative levitation in the
accretion disk. If the radiative force exerted on a Fe ion exceeds the vertical gravity, this
could cause iron to diffuse towards the photosphere of the disk, enhancing its abundance.
But, in general, there is no particular reason why the
other elements should be considered at their solar values, except the simplicity inherent
in such an approximation. Given the capability of {\sc xillver} to treat any particular
choice of elemental
abundances, it can be used to produce smaller set of models custom made for any
specific situation. This could be of great use in peculiar systems such as ultra-compact
X-ray binaries, where a prominent O~{\sc vii} K$\alpha$ emission line observed in the X-ray
spectra is thought to originate from reflection in an accretion disk over-abundant in oxygen
\citep{mad10,mad11}. Also, emission lines from H-like S, Ar and Ca ions detected in some
low-mass X-ray binaries suggest reprocessing material with compositions different from
solar \citep{dai09,dis09,egr12}.
\subsection{Spectral features}\label{secfea}
Figure~\ref{fspec.xi} shows the great complexity of the reprocessed
spectrum emerging from an illuminated, optically-thick
slab. Meanwhile, the models presented here constitute a
high-resolution representation of only one of the components observed
in the X-ray spectrum of accreting sources, namely the reflected
component. In reality, one observes a composite spectrum that
includes the original power-law (presumably coronal) component plus
possibly a thermal blackbody-like component. Additionally, if the
reflection occurs within a few gravitational radii from the compact
object, Doppler and gravitational redshifts will smear the spectral
profiles. Finally, absorption due to intervening gas such as warm
absorbers and outflows can also be present. We shall discuss some of
the most prominent and representative of these features that one
expects to observe given the capabilities of current detectors.
\subsubsection{The Fe K-shell emission}
Undoubtedly, the emission complex near $6-7$~keV, which is due to
transitions from the inner K-shell of Fe ions, is the most prominent
atomic feature in the X-ray spectrum of accreting sources. It is this
feature that provides the clearest evidence for reflection of
high-energy photons in the relatively cold, optically-thick material
of an accretion disk. The ubiquity of this feature has been
established observationally for a large number of sources
\citep[e.g.,][]{got95,win09,ng10,fuk11} for the following two reasons.
First, the fluorescence yield, i.e. the probability of emission of a
photon rather than an Auger electron, is proportional to $Z^4$, where
$Z$ is the nuclear charge. Second, the Fe K-shell lines are emitted
in a clean region of the X-ray spectrum ($6-8$~keV), where few other
ions emit or absorb radiation. Furthermore, in this energy range
galactic absorption is negligible and most detectors operate quite
effectively.
Figure~\ref{falines} shows a compilation of all the radiative
transitions from Fe ions in the $6-10$~keV energy range that are
included in our
database\footnote{\url{http://heasarc.gsfc.nasa.gov/uadb}}\citep[cf.,
Figure~3 in][]{kal04}. There are a total of 2735 lines, all of which
correspond to K-shell transitions. The open circles show the line
energy plotted against the ionization stage of each ion, as determined
by the relation $Z-N_e+1$, where $Z$ is the nuclear charge and $N_e$
is the number of electrons in the ion. Thus, an ionization stage of 2
corresponds to Fe~{\sc ii} (single ionized), 3 to Fe~{\sc iii} (double
ionized), and so forth. Filled circles show the most intense
transitions, i.e., those with large transition probabilities (here we
have chosen $A_r > 10^{13}$~s$^{-1}$). Ionization increases upward:
The H- and He-like ions are near the top and the neutrals near the
bottom. The big group of points to the left corresponds to the
K$\alpha$ transitions ($n = 1 \rightarrow 2$, with $n$ being the
principal quantum number), and the smaller group to the right (for
which most energies are above $7$~keV) corresponds to the K$\beta$
transitions ($n = 1 \rightarrow 3$). Note that what is commonly
referred to as the neutral Fe K line at $6.4$~keV is in fact a
combination of several transitions that span the energy range
$6.39-6.43$~keV; this feature can be produced by many ions ranging
from Fe~{\sc i} up to Fe~{\sc xvii}. For Fe~{\sc xviii} and more
ionized ions, the transition energy spread is larger and the average
energy moves monotonically towards higher energies as the ionization
stage increases. For the He- and H-like ions (Fe~{\sc xxv-xxvi}),
there are only a few lines with energies around $6.9$~keV. Notice also
that the K$\beta$ transitions are only produced up to Fe~{\sc xvii},
since for higher ionization stages the $n=3$ shell is empty. The line
energy for the K$\beta$ transitions varies between $\sim 7.04$~keV for
Fe~{\sc ii} up to $\sim 7.19$~keV for Fe~{\sc xvii}.
It is these line energies and intensities that we use to analyze the
emission in the Fe K region of the reflected spectra.
Figure~\ref{ffekspec} shows, in the $5 - 8$~keV band, the reflected
spectra for $\Gamma=2$ and $A_\mathrm{Fe}=1$ for different values of
the ionization parameter in the range $\xi=5 - 5\times 10^3$.
Ionization increases downward in the figure. Starting at the lowest
level of ionization ($\xi=5-50$), the first few models show similar
groups of Fe lines, with two distinctive emission lines centered at
$\sim 6.4$~keV and $\sim 7.1$~keV, which correspond respectively to
the K$\alpha$ and K$\beta$ lines. The line energies indicate that the
emission is dominated by Fe ions with ionization stages lower than
Fe~{\sc xvii}. The smaller feature below $6.4$~keV, which is produced
by Compton down-scattering of line photons, is usually referred to as
the {\it Compton shoulder}. For models with $\xi \gtrsim 200$, the
K$\beta$ line is no longer visible, which suggests that the emission
is dominated by ions at higher ionization stages. In particular,
models with $\xi=200$ and $500$ show a rich complex of emission lines
at energies between $6.4-6.7$~keV. At $\xi=10^3$, the emission is
centered at $\sim 6.7$~keV and $\sim 6.9$~keV, which implies that the
gas is highly ionized and most of the Fe ions are hydrogenic or
He-like. In this state the temperature is high, and the overall line
emission profile is thermally broadened into a symmetric profile via
Compton scattering. \cite{kal04} present a more detailed but similar
analysis of the Fe K emission complex based on {\sc xstar}
simulations, which includes atomic data for Fe; however, they do not
consider energy redistribution due to Compton scattering.
\subsubsection{Lower-$Z$ elements}
In addition to the important Fe-K emission complex in the reflected
spectrum, our code includes many emission lines of astrophysically
relevant ions at lower energies, which are due to such elements as C,
N, O, Ne, Mg, Si, S, Ar, and Ca. The intensities of these lines tend
to increase as the illuminating spectrum softens (i.e., as $\Gamma$
increases) and as the flux correspondingly increases at lower energies
where photoelectric absorption dominates. Figures~\ref{flowz1},
\ref{flowz2}, and \ref{flowz3} show the reflected spectra for
$\Gamma=3$ and $A_\mathrm{Fe}=1$ and for all the values of the
ionization parameter considered in our library. The spectra cover the
$0.3 - 4.5$~keV energy range, which is the band that contains most of
the inner-shell transitions from low-$Z$ ions. At the top of each
figure is indicated the energy of the strongest K$\alpha$ emission
line for each ion in the isonuclear sequence of each of the elements
contained in our atomic database (except for the neutral and
single-ionized cases). While these K$\alpha$ lines are not at all the
only lines considered in our models, they nevertheless give an
indication of the energy range one expects most of the features of a
given element to appear.
Figure~\ref{flowz1} shows spectra for the lowest ionization models,
namely $\xi=1, 2, 5, 10,$ and $20$ (with ionization increasing from
bottom to top). At the lowest energies ($0.3 - 1$~keV), the lines are
indistinguishably blended together. At higher energies, emission
lines from low ionization states of Mg, Si and S are evident, and
relatively weak lines due to Ar and Ca K$\alpha$ are also present.
The radiative recombination continuum (RRC) due to H-like Ne at $\sim
1.4$~keV becomes visible in models with $\xi \gtrsim 5$.
Such features are produced by recombining electrons with
energies that exceed the ion binding energy. The excess energy is
radiated as a photon. The energy of a typical RRC photon is $E =
E_{IP} + kT_e$, where $E_{IP}$ is the ionization potential of the ion
and $T_e$ is the electron temperature.
For higher ionization parameters, the reflected spectra are dominated
by RRC features, as shown in Figure~\ref{flowz2}, which displays
spectra ranging from $\xi=50$ at the bottom to $\xi=10^3$ at the top.
Also visible are higher-excitation emission lines of Mg, Si, S and Ar.
The H-like oxygen RRC at $\sim 0.87$~keV is quite strong for $\xi=50$
(bottom curve), and it weakens as $\xi$ increases. Strong emission
features due to K$\alpha$ transitions of Mg, Si, and S appear at $\sim
1.4, 1.8,$ and $2.45$~keV, respectively. The latter feature
corresponds to a blend of the S~{\sc xv} K$\alpha$ line and the
Si~{\sc xiii} RRC \citep{gar11}.
Figure~\ref{flowz3} shows the last five models of the series with the
ionization parameter increasing from $\xi=2\times 10^3$ (bottom) to
$\xi=5\times 10^4$ (top). For these large values of $\xi$, fewer
emission lines are present. However, they are quite intense because
the soft illuminating spectrum is rich in low-energy photons. The
strongest emission lines are produced by H-like ions of O, Mg, Si, and
S with energies of $\sim 0.654, 1.47, 2.01,$ and $2.63$~keV,
respectively. The RRC from H-like Si and S are clearly seen at $\sim
2.67$ and $3.5$~keV, respectively. Generally, K$\alpha$ fluorescence
from the low ionization states of O, Mg, Si, S, Ar, and Ca ions is
most important at low column depths because of the higher K shell
opacity of these elements relative to Fe. Conversely, Fe K$\alpha$ is
more important at relatively high columns, i.e., $\tau_{\mathrm T}\sim
1$.
\subsection{Comparison with previous models}
Considering the several reflection models currently available, {\sc
reflionx} \citep{ros05} is the one that is most similar to the models
presented in this paper. It has been widely used by the scientific
community in analyzing spectral data for many observations of various
black-hole and neutron-star sources. We therefore benchmark our
results by making a detailed comparison of {\sc xillver} and {\sc
reflionx}, while describing the advantages of our model.
In {\sc reflionx}, the radiation field is separated into the direct
component of the illuminating radiation, which is unaffected by either
scattering or absorption (assumed to be $\propto e^{-\tau_{\nu}}$),
and the diffuse component, which results from Compton scattering and
emission within the slab. The Compton processes are treated by solving
the Fokker-Planck diffusion equation, which includes a modified
Kompaneets operator\citep{ros78,ros79}. Incident photons that are
Compton scattered in the slab contribute to the emissivity of the
diffuse field, and it is assumed that their distribution in energy is
described by a Gaussian with central energy and energy dispersion
given by Equations~(\ref{eqec}) and (\ref{eqsig}). Due to the
deficiency of the Fokker-Planck equation in handling steep gradients
in the energy spectrum, resonance lines are treated using the escape
probabilities technique. In addition to fully ionized species, the
\cite{ros05} calculations include: C~{\sc iii-iv}, N~{\sc iii-vii},
O~{\sc iii-viii}, Ne~{\sc iii-x}, Mg~{\sc iii-xii}, Si~{\sc iv-xiv},
S~{\sc iv-xvi}, and Fe~{\sc iv-xxvi}. However, none of the neutral or
near-neutral ions are included in their models. The cross sections
for photoionization are calculated from the fits of \cite{ver95}; in
the case of Fe, transition probabilities and Auger decay rates are
taken from \cite{kas93}. The items we have just highlighted are among
the most important differences between the atomic data employed in the
two models (see Section~\ref{secatomicd}). It is also important to
mention that the Fe K$\alpha$ lines treated in {\sc reflionx} are the
recombination lines of Fe~{\sc xxvi} and Fe~{\sc xxv}, and the
fluorescence lines of Fe~{\sc vi-xvi}, while the K$\alpha$
fluorescence of Fe~{\sc xvii-xxii} is assumed to be suppressed by
autoionization. All K$\beta$ and higher $n$ resonances are also
neglected.
The solar abundances of the elements in {\sc reflionx} are taken from
\cite{mor83}, while in {\sc xstar}, and thus in {\sc xillver}, we
adopt the values of \cite{gre98}. Table~\ref{tabund} shows the values
reported from both sources for each element, and the last column shows
the ratio of the two values. Apart from N and Ar, all the abundances
used in {\sc xillver} are lower than those used in {\sc reflionx}. In
particular, O is lower by $\sim 10\%$ and Fe by $\sim 30\%$, while,
most notably, Ne is $\sim 80\%$ lower. The choice of abundance model
is relatively unimportant, as illustrated in Figure~\ref{fabund},
which shows reflected spectra computed using {\sc xillver} for
$\xi=10$ (left panel) and $\xi=10^3$ (right panel). Results obtained
using the model of \cite{gre98} are shown in black, and those obtained
using the model of \cite{mor83} in red. The differences in each panel
are small. For the higher \cite{mor83} abundances, the continuum is
slightly depressed and some of the strongest lines are somewhat
weaker. These small differences are unlikely to be important in
analyzing real data.
Although the solar abundances assumed for both {\sc reflionx} and {\sc
xillver} are fixed and different, both models allow the Fe abundance
to be varied. This allows us to compensate for the abundance
differences in making direct comparisons of the two models. One such
set of comparisons is shown in Figure~\ref{fcompref}, where the
emergent spectra for different values of both the ionization parameter
and the photon index are plotted for {\sc xillver} (in black), and
{\sc reflionx} (in red). Top, middle and bottom panels are for
$\Gamma=1.4$, $2$, and $2.6$, respectively. Each panel shows 3 pairs
of curves, corresponding to $\xi=10, 10^2,$ and $10^3$, from bottom to
top, respectively. The spectra are from the FITS tables that are
accessible via the {\tt atable} model in XSPEC. To ensure that the
energy resolution is the same for all the spectra, in all cases we
used a logarithmic dummy response of 1000 energy points over the
energy range $0.1 - 1000$~keV. Furthermore, appropriate grid point
were chosen in the parameter space to avoid problems that might be
introduced by the interpolation procedure. The normalization for both
models is the same because we use consistent definitions for the
ionization parameter and the power-law spectrum. Thus, all the
parameters were set to be identical for the two models, with the
exception of the iron abundance: It was set to $A_\mathrm{Fe}=1.32$ in
{\sc xillver} in order to compensate for the difference in the solar
values assumed in each model, as discussed above.
In general, the two models are in better agreement for low ionization
and soft spectra, as in the models for $\Gamma=2.6$ shown in the
bottom panel of Figure~\ref{fcompref}. However, even in these cases
there are important differences: For the bottom spectra with $\xi=10$,
the {\sc xillver} model is more absorbed at low energies than the {\sc
reflionx} model. While the energies and intensities of the strongest
lines are in good agreement, some weaker features at $\sim 2$~keV and
$\sim 3$~keV are only present in the {\sc xillver} spectrum, probably
due to the differences in the atomic data. These particular features
are much stronger for the harder spectra shown in the middle and top
panels. The differences in the atomic data also affect the Fe K
emission profile near $6-8$~keV. For example, Fe K$\alpha$ emission
at $\sim 6.4$~keV is somewhat more intense in our model and K$\beta$
is conspicuously absent in the {\sc reflionx} spectra. Concerning the
Fe continuum, one might expect significant differences because the
models use very different photoionization cross sections. However,
the models agree very well on the depth of the Fe K edge near
$7.2$~keV, especially at low ionization. As mentioned above, the
discrepancies are most pronounced for high ionization and for the hard
spectra shown in the top panel ($\Gamma=1.4$). The difference for the
pair of curves for $\xi=10^3$ is remarkable: The {\sc xillver}
continuum below $\sim1$ keV is at least an order of magnitude fainter
than the {\sc reflionx} continuum; the extremely intense O Ly$\alpha$
emission line at $\sim 0.65$~keV in our spectrum is scarcely present
in the {\sc reflionx} spectrum; and the Fe K line in the latter
spectrum is much broader than in the former.
These large discrepancies for high ionization and hard spectra are
unexpected and there is no ready explanation. The lack of emission
lines from low-$Z$ elements suggests that (for the same input
parameters) the slab is hotter and more ionized for {\sc reflionx}
than for {\sc xillver}. We test this idea by comparing the
performance of the two models for different values of the ionization
parameter: In Figure~\ref{fcompref2} we compare the two $\Gamma=1.4$
models where for {\sc reflionx} we have doubled the normalization and
reduced the ionization parameter to $\xi=500$ (half the value for {\sc
xillver}), in order to match the fluxes at the higher energies. The
shape and intensity of the Fe K line is now in better agreement.
However, the {\sc reflionx} continuum is still considerably higher at
low energies; as a consequence, the reflected spectrum is even softer than the
incident power-law spectrum.
In order to test whether our code is reliably modeling this low-energy
continuum, we compared our results with those computed using the {\sc
apec} model \citep{fos12}. For this comparison, with {\sc xillver}
we computed the reflection spectrum from a thin slab ($\tau_{\mathrm
T}=10^{-2}$) for solar abundances at very low ionization with the
temperature held fixed at $10^6$~K. These parameters closely match
the case of a collisionally-ionized gas whose spectrum is dominated by
bremsstrahlung emission. Figure~\ref{fapec} shows the comparison.
Setting aside the differences in the emission lines (a result of the
differences in the atomic data sets), the spectra are in good
agreement, especially the level of the continuum. This comparison
gives us confidence in {\sc xillver}'s implementation of the free-free
emissivity.
An important quantity to compare these models is the equivalent
width (EW) of the Fe K emission complex. Following \cite{gar11}, we
use the well-known formula,
\begin{equation}\label{eew}
EW= \int_{5.5~\mathrm{keV}}^{7.2~\mathrm{keV}}\frac{F(E) - F_c(E)}{F_c(E)}dE,
\end{equation}
where $F(E)$ is the flux of the reflected spectrum and $F_c(E)$ is the
flux in the continuum. The $5.5 - 7.2$~keV range of integration
covers all Fe emission features, including the K$\beta$ lines. The
continuum is approximated by a straight line that passes through the
endpoints of the energy band. Clearly, this straight-line continuum
and this particular choice of integration limits are somewhat
arbitrary and by no means constitutes an accurate determination of a
physical quantity, but it suffices to compare the two models.
Figure~\ref{fews} shows plots of the EWs vs. the ionization parameter
for both {\sc xillver} and {\sc reflionx}, with $\Gamma$ increasing
from top to bottom. For both models, the EWs tend to decrease as the
ionization increases. At low ionization, the EWs are similar,
although the values for {\sc xillver} are consistently higher, which
is expected because the {\sc reflionx} model lacks the Fe K$\beta$
lines. Meanwhile, at high ionization ($\xi \gtrsim 10^3$), where the
Fe K emission is dominated by the H- and He-like ions, the agreement
in the EWs is also good, but values for the {\sc reflionx} model are
in this case somewhat greater for large $\Gamma$ (bottom panel).
Significant discrepancies appear when one compares the two models at
higher levels of ionization, $10^2 \lesssim \xi \lesssim 10^3$: The
EWs for the {\sc reflionx} models decrease drastically, especially for
the softer spectra (bottom panel). This behavior may occur because
{\sc reflionx} does not include the Fe K$\alpha$ lines for most of the
second-row ions (Fe~{\sc xvii-xxii}) under the assumption that these
lines are suppressed by Auger resonant destruction \citep{ros96}.
{\sc xillver}, on the other hand, predicts strong K$\alpha$ emission
at the energies at which many of these ions are expected to emit
(e.g., see Figures~\ref{falines} and \ref{ffekspec}). {\sc xstar},
and consequently {\sc xillver}, automatically takes into account the
effects of Auger decay by accurately calculating the branching ratios
for Auger versus fluorescence emission; however, it does not include
resonant scattering of the lines. This omission seems reasonable
because it is plausible that velocity gradients or turbulence in an
accretion disk will remove photons from the line core, thereby
reducing resonant scattering. Nevertheless, because we disregard this
mechanism we may be over-predicting the line intensity for some
models. As one additional complication, \cite{lie05b} has shown that
Auger resonance destruction is a selective process that highly
attenuates only a limited subset of Fe K$\alpha$ lines. We conclude
that further analysis is required in order to accurately quantify the
emission from L-shell Fe ions.
We have also compared our results to those obtained using the
reflection models {\sc pexrav} and {\sc pexriv} \citep{mag95}, which
compute the reflected spectrum from a completely neutral disk and an
ionized disk, respectively. These models are an implementation of a
semi-analytic Green's function \citep{lig88} that models Compton
reflection of X-rays by cold electrons while including bound-free
continuum opacity. Fluorescence emission line are however not
included. The left panel of Figure~\ref{fpexrav} compares the
reflected spectrum predicted by {\sc xillver} with that predicted by
{\sc pexrav} for an incident power-law spectrum with $\Gamma=2$ and an
exponential cutoff at $300$~keV. Because {\sc pexrav} models a
completely neutral gas, we use the lowest ionization parameter
considered in {\sc xillver}, namely $\xi=1$. The two models agree
fairly well in the level of the reflected continuum, in particular
near the Fe K line region. The edge energy and EW of the feature at
$\sim 7.2$~keV are similar. Our model shows more absorption than {\sc
pexrav} in the low-energy part of the spectrum. However, the {\sc
xillver} model can not be considered completely neutral; there is some
heating near the surface (see the leftmost curve in the middle panel
of Figure~\ref{ftemp.xi}).
The right panel in Figure~\ref{fpexrav} compares a reflected spectrum
from an ionized disk ($\xi=10^3$) computed using {\sc xillver} with
one computed using {\sc pexriv}. At the outset, we note that {\sc
pexriv}, unlike both {\sc xillver} and {\sc reflionx}, does not solve
for the ionization balance of the slab. Instead, {\sc pexriv} assumes
an isothermal medium with a maximum temperature of $T=10^6$~K. Both
the ionization parameter and the gas temperature are allowed to vary.
This approach results in the large discrepancies that are evident in
both the level of the continuum and the edge features. The level of
the continuum is similar only for energies below $1$~keV. For higher
energies, the {\sc pexriv} spectrum is highly absorbed and its Fe K
edge is much deeper. This comparison demonstrates for the case of an
ionized reflector the importance of accurately modeling the ionization
balance. {\sc pexriv}'s assumption of an isothermal gas is a poor one
for large values of the ionization parameter. Another limitation of
{\sc pexriv} that was pointed out by \cite{fab10} is the lack of blurring
effects of Compton scattering, which smears the K edges.
We now discuss in turn the differences at high energies
($\gtrsim 20$~keV) between our results and those delivered by {\sc
reflionx} and {\sc pexrav/pexriv}. At these energies, photoelectric
absorption is negligible, and the only source of opacity is Compton
scattering. Furthermore, the gas temperature has only a very small
effect on the Comptonized spectrum. Therefore, any differences in the
model spectra can be attributed to how each model treats scattering of
photons by cold electrons. {\sc xillver} implements a Gaussian
convolution for all energies and at all depths (Section~\ref{secmod}).
{\sc reflionx} uses this approximation for the lines only, while using
a modified Kompaneets operator to treat the diffuse radiation.
We tested the accuracy of our approximation by comparing with results
obtained using a Monte Carlo (MC) code, which we treat as the gold
standard. This code, which extends the work of \cite{mat02} and
\cite{mag95}, simulates Compton scattering and radiative transfer
through neutral gas for a semi-infinite slab \citep{eik12}. Included
in this code are photoabsorption for all elements with $Z \le 30$ and
fluorescent line emission using the fluorescent yields from
\cite{kas93}. Compton scattering is modeled using the proper
relativistic scattering dynamics and the differential Klein-Nishina
cross section.
Figure~\ref{fmontecarlo} compares reflected spectra computed using
{\sc xillver} and the MC code for $\Gamma=2$, $\xi=1$ and solar
abundances. The result for {\sc reflionx} is also shown for these same
parameters. Overall, there is a good agreement between the three
models. At energies $\lesssim 10$~keV, {\sc xillver} and the MC
simulation predict very nearly the same level of the continuum and the
same spectrum of intense emission lines. For example, the $\sim
6.4$~keV Fe K$\alpha$ line has a very similar intensity and shape,
although the MC code predicts a $\sim 7.2$~keV K$\beta$ line that is
somewhat more intense. The main differences occur above the $\sim
7.2$~keV Fe~K edge, in the continuum feature referred to as the
Compton hump. The MC simulated spectrum has a somewhat shallower Fe K
edge than the other two spectra, and its $\sim 20 - 60$~keV is
somewhat fainter. Above $\sim100$~keV, the {\sc
reflionx} spectrum is the better match to the MC spectrum.
As a bottom line, the differences between the MC simulation and either
{\sc xillver} or {\sc reflionx} are not larger than $30\%$, which is
reasonable given the simplistic approximation used in those models.
However, this level of performance may not be adequate for the
analysis of such current and future X-ray missions as NuSTAR
\citep{har10} and eROSITA \citep{pre11}. Furthermore, the
Compton-hump spectrum can provide important constraints on disk
inclination. Therefore, our goal is to improve this aspect of the
code in future versions of {\sc xillver}.
\section{Conclusions}\label{seccon}
In this paper we have presented a new and complete library of
synthetic spectra for modeling radiation that is reprocessed in an
accretion disk and emitted as a reflected spectrum, which is generated
in response to illumination by an incident power-law spectrum. This
present version of our code {\sc xillver} is an update of those
presented in \cite{gar10} and \cite{gar11}. We have made several
improvements to both the routines and the atomic data and have
produced a large grid of reflection models covering a wide range of
parameters. Each model is characterized by the photon index $\Gamma$
of the illuminating radiation, the ionization parameter $\xi$ at the
surface of the disk, and the Fe abundance with respect to its solar
value. A total of 720 reflected spectra are provided in a single FITS
file\footnote{\url{http://hea-www.cfa.harvard.edu/~javier/xillver/}}
suitable for the analysis of X-rays observations via the {\tt
atable} model in {\sc xspec}. In order to represent the physical
conditions typically observed in most accreting sources, this library
covers the following range of parameters: $1.2 \leq \Gamma \leq 3.4$,
$1 \leq \xi \leq 10^4$, and $0.5 \leq A_\mathrm{Fe} \leq 10$.
The spectrum that illuminates the surface of the accretion disk is
assumed to be a power-law in the $10^{-1} - 10^6$~eV energy range,
with a sharp low-energy cutoff at $100$~eV and an exponential
high-energy cutoff at $300$~keV. The low-energy cutoff is important
for spectra with steep power laws ($\Gamma > 2$); it prevents the
spectrum from being unphysically over-populated with low-energy
photons. The power-law is extended up to $1$~MeV because in the
low-$\Gamma$ case this portion of the spectrum is rich in high-energy
photons, which importantly effect both the gas temperature and the
reflected spectrum.
The intensity of the illuminating spectrum, which is specified by the
ionization parameter $\xi$, significantly impacts the ionization
structure of the gas. In all models, the gas temperature in the
vertical direction has a similar profile. Near the surface, where the
illumination is intense, there is a hot layer ($T \gtrsim 10^6$~K)
where Compton heating and cooling dominate. At larger depths, a
warm/cold region exists where photoelectric opacity and recombination
dominate. The depth at which the transition between these two regimes
occurs increases as $\xi$ increase and as the incident spectrum
softens (large $\Gamma$). Above the transition layer, the radiation
field thermalizes and the gas temperature remains fairly constant at
around $10^5$~K. For low-ionization ($\xi \lesssim 100-500$), the
reflected spectrum displays a very rich set of emission lines
superimposed on a strongly absorbed continuum. For high-ionization
($\xi \gtrsim 500-1000$), the spectrum consists of very narrow
emission lines from ionized species and a continuum that resembles the
incident power-law.
We have presented detailed comparisons with the reflection models {\sc
reflionx} \citep{ros05}, and {\sc pexrav} \citep{mag95}. The {\sc
reflionx} model is more similar to ours because of its complexity and
the range of physical processes considered. Furthermore, {\sc
reflionx} and {\sc xillver} both characterize the illuminating
spectrum as a power law, cover the same energy range, and use the same
definition for the input parameters. The spectra generated by the two
codes are in good agreement. The spectra agree best for large
$\Gamma$ (soft spectra) and low values of $\xi$. However, at low
energies {\sc reflionx} generates an excess of soft flux compared to
{\sc xillver}. For hard spectra, $\Gamma=1.4$, and high ionization,
$\xi=10^3$, the difference in the continuum at $1$~keV is about an
order of magnitude. This large discrepancy is not well understood and
requires further analysis.
At very high or very low ionization, {\sc xillver} and {\sc reflionx}
predict similar EWs for the Fe K emission feature (including both
K$\alpha$ and K$\beta$), although the values for {\sc xillver} are
generally larger. However, at intermediate levels of ionization,
in the $10^2 \lesssim \xi \lesssim 10^3$ range, the two models disagree
strongly: the Fe K EWs computed in this range computed using {\sc
reflionx} drop drastically. The effect is presumably because
{\sc reflionx} assumes that the Fe K lines from second-row ions are
completely suppressed by Auger resonant destruction.
Comparing {\sc xillver} with the more simplistic models {\sc pexrav}
and {\sc pexriv} highlights the importance of performing detailed
calculations which take into account key physical processes. The
greatest deficiency of these models is that they do not generate
emission lines. The neutral {\sc pexrav} model is in reasonable
agreement with {\sc xillver} concerning the level of the continuum and
the strength of the edges. However, for moderate or high ionization
($\xi \gtrsim 20$), the {\sc pexriv} model is in strong disagreement
because it assumes an isothermal disk, as previously pointed out by
\cite{fab10}.
The library of models presented in this paper is suitable for modeling
the reflection spectra of accreting sources when the thermal disk
component of emission is small compared to the incident power-law
component. That is, the present version of {\sc xillver} is suitable
for analyzing the spectra of AGN, and also GBHs in the hard state,
when the disk component is cool and faint. For GBHs in the thermal or
steep power-law states \citep{rem06}, the strong
thermal component entering the atmosphere from below will
significantly change the ionization structure of the disk. In a
future publication, we will report on an extension of {\sc xillver}
that is appropriate for modeling reflection spectra in the presence of
a strong thermal component.
Regardless of how accurate {\sc xillver} (or any such slab reflection
model) becomes, it is by itself inadequate for modeling the reflection
component in the spectra of accreting black holes. For example,
current black hole spin determinations based on fitting of the Fe K
line naively assume a single ionization state for the reflecting
portion of the disk, which can extend over hundreds of gravitational
radii. We plan to construct more realistic models for particular
illumination geometries that allow for an ionization gradient in the
radial direction. This work will combine the general relativistic
approach implemented in the {\sc relline} code \citep{dau10,dau13}
with the model spectra presented in this paper.
\acknowledgments
\bibliographystyle{apj}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,619
|
using System;
using System.Collections.Generic;
namespace BusinessModel.Core.Models
{
public partial class titleview
{
public string title { get; set; }
public Nullable<byte> au_ord { get; set; }
public string au_lname { get; set; }
public Nullable<decimal> price { get; set; }
public Nullable<int> ytd_sales { get; set; }
public string pub_id { get; set; }
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 592
|
\section{Introduction and motivations}
In the continuous setting, polyharmonic functions are functions which cancel some power of the usual Laplacian. More precisely, a function $v$ on some domain $K$ of $\mathbb R^d$ satisfying
\begin{equation*}
\Delta^p v = 0
\end{equation*}
for some $p\geq1$, where $\Delta$ is the usual Laplacian in $\mathbb R^d$, is said to be \textit{polyharmonic} of order $p$, or \textit{polyharmonic} for short. So polyharmonic functions of order 1 are just harmonic functions.
Obviously, a polyharmonic function $v_p$ of order $p$ satisfies $\Delta v_p=v_{p-1}$, where $v_{p-1}$ is polyharmonic of order $p-1$. For example, polynomials are polyharmonic.
Harmonic functions have been tremendously investigated and pioneer works on polyharmonic functions go back to the work of Almansi \cite{Al-1899}. One can consult for instance the monograph \cite{ArNaCr-83} for an introduction to this topic.
In particular, Almansi \cite{Al-1899} proved that if the domain $K$ is star-like with respect to the origin, then every polyharmonic function of order $p$ admits a unique decomposition
\begin{equation}
\label{eq:Almansi}
f(x)=\sum_{k=0}^{p-1} \vert x\vert^{2k} h_k(x) ,
\end{equation}
where each $h_k$ is harmonic on $K$ and $\vert x\vert$ is the Euclidean length of $x$, hence completely characterising continuous polyharmonic functions on such domains.
In comparison with the continuous case, much less is known in the discrete setting, where the Laplacian has to be replaced by a discrete difference operator. Some progress in understanding discrete polyharmonic functions has been made in the last two decades. For instance, one may cite \cite{CoCoGoSi-02}, where the authors investigated polyharmonic functions for the Laplacian on trees, and proved a similar result as Almansi's theorem \eqref{eq:Almansi} for homogeneous trees. Recent works of Woess and co-authors \cite{HiWo-19, SaWo-19} are generalising this previous work.
Our original motivation to study discrete polyharmonic functions comes from the following framework. Consider a walk in $\mathbb Z^d$ with step set $\mathcal S$ confined in some cone $K\subset\mathbb Z^d$. Denote by
$q(x,y;n)$
the number of $n$-length excursions between $x$ and $y$ staying in the cone $K$.
To simplify, we only consider the case where $y$ is the origin, but all considerations below can be generalised to $y\not = 0$.
In various cases \cite{DeWa-15}, the asymptotics of $q(x,0;n)$ as $n\to\infty$ is known to admit the form
\begin{equation}
\label{eq:asym_equiv}
q(x,0;n) \sim v_0(x) \gamma^n n^{-\alpha_0},
\end{equation}
where $v_0(x)>0$ is a function depending only on $x$, $\gamma\in(0,\vert\mathcal S\vert]$ is the exponential growth, and $\alpha_0$ is the critical exponent.
It is easy to see that the function $v_0(x)$ in \eqref{eq:asym_equiv} defines a discrete harmonic function. Indeed, plugging \eqref{eq:asym_equiv} into the obvious recursive relation
\begin{equation}
\label{eq:discrete_heat_equation}
q(x,0;n+1) = \sum_{s\in\mathcal S} q(x+s,0;n)\mathbf{1}_{\{x+s\in K\}},
\end{equation}
dividing by $\gamma^{n+1} n^{-\alpha_0}$ and letting $n\to\infty$, we obtain
\begin{equation}
\label{eq:C(x)_harmonic}
v_0(x) = \frac{1}{\gamma} \sum_{s\in\mathcal S} v_0(x+s)\mathbf{1}_{\{x+s\in K\}},
\end{equation}
which proves that, with the assumption that $v_0(x)=0$ for $x\notin K$, $v_0(x)$ is discrete harmonic for the Laplacian operator
\begin{equation}
\label{eq:uniform_Laplacian}
Lf(x)=\frac{1}{\gamma} \sum_{s\in\mathcal S} f(x+s)-f(x),
\end{equation}
that is, $Lv_0=0$. Denisov and Wachtel \cite{DeWa-15} go further and show that
\begin{itemize}
\item the exponential growth $\gamma$ is $\min_{\mathbb R_+^d} \sum_{(s_1,\ldots,s_d)\in\mathcal S}x_1^{s_1}\cdots x_d^{s_d}$, it does not depend on $K$;
\item the critical exponent $\alpha_0$ equals $1+\sqrt{\lambda_1+(d/2-1)^2}$, where $d$ is the dimension and $\lambda_1$ is the principal Dirichlet eigenvalue on some spherical domain constructed from $K$.
\end{itemize}
As a leading example, consider the simple random walk in the quarter plane, with step set $\{\leftarrow, \uparrow, \rightarrow, \downarrow\}$. In this case, the number of excursions $q((i,j),0;n)$ is $0$ if $m=\frac{n-i-j}{2}$ is not a non-negative integer, and otherwise takes the value
\begin{equation}
\label{eq:SRW_excursion}
q((i,j),0;n) =\frac{(i+1)(j+1)n!(n+2)!}{m!(m+i+j+2)!(m+i+1)!(m+j+1)!},
\end{equation}
see \cite{bou-02counting} and our Example \ref{ex:SRW}. The equivalence \eqref{eq:asym_equiv} is then
\begin{equation}
\label{eq:asymp_SRW}
q((i,j),0;n) \sim \frac{4}{\pi} 4^n \frac{v_0(i,j)}{n^{3}},
\end{equation} where $v_0(i,j)=(i+1)(j+1)$ is the well-known unique (up to multiplicative constants) harmonic function positive within the quarter plane with Dirichlet boundary conditions. Other examples of such asymptotics may be found for instance in \cite{BaFl-02,BMMi-10,CoMeMiRa-17}.
Our aim in this discrete setting is to study more precise estimates than \eqref{eq:asym_equiv}, by considering complete asymptotic expansions of the following form, as $n\to\infty$,
\begin{equation}
\label{eq:asym_full}
q(x,0;n) \sim \gamma^n \sum_{p\geq0} \frac{v_p(x)}{n^{\alpha_p}}.
\end{equation}
From such an asymptotic expansion and using similar ideas as in \eqref{eq:discrete_heat_equation}, \eqref{eq:C(x)_harmonic} and \eqref{eq:uniform_Laplacian}, it is rather easy to prove that the terms $v_p$ are polyharmonic functions, in the sense that a power $L^kv_p$ of the Laplacian operator vanishes. We will provide examples of such asymptotic expansions (at least for the first terms) and of the set of exponents $\{\alpha_p\}_{p\geq0}$ appearing in~\eqref{eq:asym_full}.
On the other hand, the functional equation approach has proved to be fruitful when studying random walk problems. The reference book on this topic is the monograph \cite{FaIaMa-17} by Fayolle, Iasnogorodski and Malyshev. This method has been used in \cite{Ra-14} to construct harmonic functions, both in the discrete and continuous settings. Basically, the method consists of drawing from the harmonicity condition a functional equation satisfied by the generating function (in the discrete setting) or by the Laplace transform (in the continuous setting) of a harmonic function. Solving some boundary value problem for these quantities leads, via Cauchy or Laplace inversion, to the sought harmonic function. We will provide an implementation of this method to construct bi-harmonic functions, which can be generalised to polyharmonic functions.
The main features of our results are as follows:
\begin{itemize}
\item We shine a light on a new link between discrete polyharmonic functions and complete asymptotic expansions in the enumeration of walks.
\item Our approach provides tools to study complete asymptotics expansions as in \eqref{eq:asym_full}, but does not allow to prove their existence. On the other hand,
the powerful approach of Denisov and Wachtel \cite{DeWa-15} seems restricted to the first term in the asymptotics \eqref{eq:asym_equiv}. Indeed, one of the main tools in \cite{DeWa-15} is a coupling result of random walks by Brownian motion, which only provides an approximation of polynomial order, see \cite[Lem.~17]{DeWa-15}.
\item We introduce a new class of functional equations (see \eqref{functional-eq-BM-2} and \eqref{eq:functional_equation_2-harmo}), for which the method of Tutte's invariants introduced in \cite{Tu-95,BeBM-11,BeBMRa-17} proves to be useful.
\item In the unweighted planar case, it has been shown \cite{BoRaSa-14} that knowing the rationality of the exponent $\alpha_0$ in \eqref{eq:asym_full} was sufficient to decide the non-D-finiteness of the series of excursions. However, for walks with big steps in dimension two or walk models in dimension three, this information is not enough \cite{BoBMMe-18}. As a potential application of our results, we might use arithmetic information on the other exponents $\alpha_p$ to study the algebraic nature, for example the transcendance, of the associated combinatorial series.
\end{itemize}
This paper is organised as follows. We choose to start with the continuous setting since computations are more enlightening and accessible. In Section~\ref{section-BM}, we prove that polyharmonic functions naturally arise when performing an asymptotic expansion of the Dirichlet heat kernel in a cone. We next present the functional equation method to construct polyharmonic functions.
Our main result here is Theorem~\ref{thm:Laplace-2}, where a class of solutions for the Laplace transform of a bi-harmonic function is provided. It shows that the Laplace transform of a bi-harmonic function can be expressed in terms of the Laplace transform of the related harmonic function plus some additional terms. This can be thought of as a Laplace transform version of Almansi's theorem \eqref{eq:Almansi}.
In Section~\ref{section-discrete}, we exhibit the same phenomenon in the random walk setting. Discrete polyharmonic functions appear when considering the asymptotic expansion of coefficients counting walks with fixed endpoints in a domain, and the functional equation approach may be used to construct discrete polyharmonic functions.
These notes are the starting point of a long-term research project on discrete polyharmonic functions in cones. Notice that many ideas and techniques are not specific to cones and would work for many other domains of restriction $K$.
\medskip
{\bf Acknowledgements.}
We would like to thank C\'edric Lecouvey, Steve Melczer, Pierre Tarrago and Wolfgang Woess for very interesting discussions. This project has started in July 2019, when two authors were invited at the Institute of Mathematical Statistics of M\"unster University. The institute, and in particular Gerold Alsmeyer, is greatly acknowledged for hospitality. The first author also acknowledges the Institut Denis Poisson for the warm hospitality during his stay at the Universit\'e de Tours, where part of this work has been pursued. Finally, we thank the three anonymous referees for useful comments.
\section{Classical polyharmonic functions and heat kernel expansions}
\label{section-BM}
As pointed out in \cite[Chap.~VI]{ArNaCr-83}, the connection between the heat kernel and polyharmonic functions is very profound. Here, we deepen this connection by proving an exact asymptotic expansion for the heat kernel in terms of polyharmonic functions. We then implement the functional equation method to construct polyharmonic functions.
\subsection{Exact asymptotic expansion for the Brownian semigroup in a cone}
\label{section:asympt-BM}
Let $K$ be some cone in $\mathbb R^d$ and consider the Brownian motion $(B_t)_{t\geq0}$ killed at the boundary of $K$. Denote by $p(x,y;t)$ its transition density, that is the density probability function of the transition probability kernel
\begin{equation*}
\mathbb P_x ( B_t \in dy, \tau >t ),
\end{equation*}
where $\tau$ is the first exit time of $K$. Recall the well-known fact that $p(x,y;t)$ corresponds to the heat kernel, i.e., the fundamental solution of the heat equation on $K$ with Dirichlet boundary condition, see for instance \cite{BaSm-97}.
Here, we prove that the heat kernel admits a complete asymptotic expansion in terms of polyharmonic functions for the Laplacian.
Denote by $\Delta$ the usual Laplacian on $\mathbb R^d$. In polar coordinates $(r,\theta)$, where $r$ is the radial part and $\theta$ the angular part, it writes:
\begin{equation}
\label{eq:Laplacian_B}
\Delta = \frac{\partial^2}{\partial r^2} + \frac{d-1}{r}\frac{\partial}{\partial r} +\frac{1}{r^2}\Delta_{\mathbb S^{d-1}},
\end{equation}
where $\Delta_{\mathbb S^{d-1}}$ denotes the spherical Laplacian. Let respectively $m_j$ and $\lambda_j$ be the Dirichlet (normalised) eigenfunctions and eigenvalues for the spherical Laplacian on the generating set $K\cap\mathbb S^{d-1}$, that is,
\begin{equation}
\label{eq:eigenfunctions}
\left\{\begin{array}{rcll}
\Delta_{\mathbb S^{d-1}}m_j&=&-\lambda_j m_j & \text{in } K\cap\mathbb S^{d-1},\\
m_j&=&0 & \text{in } \partial (K\cap\mathbb S^{d-1}).
\end{array}\right.
\end{equation}
The eigenvalues satisfy $0<\lambda_1<\lambda_2\leq \lambda_3\leq \ldots$ by \cite[Chap.~VII]{Ch-84}.
We introduce, for $j\geq1$,
\begin{equation}
\label{eq:bj_betaj}
\beta_j=\sqrt{\lambda_j+(d/2-1)^2}\quad \text{and}\quad b_j=1-d/2+\sqrt{\lambda_j+(d/2-1)^2}.
\end{equation}
Lemma~1 in \cite{BaSm-97} gives an explicit expression for the transition density $p(x,y;t)$ of the Brownian motion in $K$. It states that, for $x,y\in \mathbb R^d$ and $t\in \mathbb R_+$,
\begin{equation}
\label{eq:heat_kernel}
p(x,y;t)=\frac{\exp\left(-\frac{\rho^2+r^2}{2t}\right)}{t(\rho r)^{\frac{d}{2}-1}}\sum_{j=1}^\infty I_{\beta_j}\left(\frac{\rho r}{t}\right) m_j(\theta)m_j(\eta),
\end{equation}
where in polar coordinates $x=(\rho, \theta)$ and $y=(r,\eta)$. Here, $I_\beta$ is the modified Bessel function of the first kind of order $\beta$, satisfying the differential equation
$
I''_\beta(z)+\frac{1}{z}I'_\beta(z)=(1+\frac{\beta^2}{z^2})I_\beta(z)
$
and admitting the series expansion
\begin{equation}
\label{eq:modified_Bessel_expansion}
I_\beta(z) = \sum_{m=0}^{\infty}\frac{1}{m!\Gamma(m+\beta+1)}\left(\frac{z}{2}\right)^{2m+\beta}.
\end{equation}
The following easy lemma will allow us to define certain polyharmonic functions.
\begin{lemma}
\label{lem:deg-2}
For any $\mu\geq0$ and $j\geq 1$, let $f_{\mu,j}$ be defined in spherical coordinates by
\begin{equation}
\label{not:f_mu_j}
f_{\mu,j}(r,\theta)=r^{\mu}m_j(\theta).
\end{equation}
Then $f_{\mu,j}$ satisfies
\begin{equation}
\label{eq:f_mu_j}
\Delta f_{\mu,j} =(\mu^2 +(d-2)\mu-\lambda_j) f_{\mu-2,j}.
\end{equation}
\end{lemma}
\begin{proof}
The proof is elementary using \eqref{eq:Laplacian_B} and \eqref{eq:eigenfunctions}.
\end{proof}
\begin{corollary}
\label{cor:poly_continuous}
For any $k\in\mathbb N$, the function $f_{b_j+2k,j}$ defined in~\eqref{not:f_mu_j} is $k$-polyharmonic.
\end{corollary}
\begin{proof}
It is obvious that $\mu=b_j$ satisfies $\mu^2 +(d-2)\mu-\lambda_j=0$, see \eqref{eq:bj_betaj}, so that $f_{b_j,j}$ is harmonic by \eqref{eq:f_mu_j}. An induction based on \eqref{eq:f_mu_j} completes the proof.
\end{proof}
Doing an expansion of the heat kernel \eqref{eq:heat_kernel} as $t\to\infty$ and using series expansions of the exponential function and of the Bessel function \eqref{eq:modified_Bessel_expansion}, one immediately obtains:
\begin{theorem}
The Dirichlet heat kernel $p(x,y;t)$ in $K$ admits the following expansion, as $t\to \infty$, where $f_{b_j+2k,j}$ is defined in~\eqref{not:f_mu_j}, and $b_j$ and $\beta_j$ in~\eqref{eq:bj_betaj}:
\begin{multline*}
p(x,y;t)\sim \\
\sum_{j\geq1}\sum_{k,m\geq0 }\sum_{n=0}^{k}\frac{1}{t^{1+\beta_j+k+2m}} \frac{(-1)^k\binom{k}{n}}{2^kk!m!\Gamma(m+\beta_j+1)}f_{b_j+2(m+n),j}(\rho,\theta)f_{b_j+2(m+k-n),j}(r,\eta).
\end{multline*}
\end{theorem}
As such, the above result shows that the transition density of the Brownian motion in $K$ admits, as $t\to\infty$, an asymptotic expansion in descending powers of $t$ and in terms of polyharmonic functions for the Laplacian (see Corollary \ref{cor:poly_continuous}). Moreover, the set of these exponents is (with $\mathbb N=\{0,1,2,\ldots\}$)
\begin{equation}
\label{eq:set_exponents}
\bigcup_{j=1}^\infty (\beta_j+1+\mathbb N).
\end{equation}
Note that, depending on the cone, there might be an overlap between the sets $\beta_j+1+\mathbb N$. For instance, in the quadrant in dimension $2$, one has $\beta_j=2j$ and the set in \eqref{eq:set_exponents} reduces to $\{3,4,5,\ldots\}$. On the other hand, in dimension $2$ in a cone of opening $\alpha$ such that $\pi/\alpha\notin\mathbb Q$, there is no overlap between the points in \eqref{eq:set_exponents}.
As a last remark, we note that the same phenomenon appears for the survival probability $\mathbb P_x(\tau>t)$. Indeed, thanks to its explicit expression given by \cite[Thm~1]{BaSm-97} (in terms of the confluent hypergeometric function), one can write down an asymptotic expansion of $\mathbb P_x(\tau>t)$ in descending powers of $t$ in terms of polyharmonic functions for the Laplacian.
\subsection{The functional equation approach}
We apply here the functional equation approach in order to construct polyharmonic functions for the $2$-dimensional killed Brownian motion in a convex cone. This approach has been previously introduced in \cite{Ra-14} to compute harmonic functions, and is an adaptation of the functional equation method of the random walk case. Our main result is Theorem~\ref{thm:Laplace-2}, which gives the general form of the Laplace transform of a bi-harmonic function.
Consider the Brownian motion $B$ in the quarter plane $\mathbb R^2_+$ (compared to the last section, we use $(x,y)$ for the coordinates of a 2d point) with covariance matrix
\begin{equation*}
\Sigma = \begin{pmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{12} & \sigma_{22} \end{pmatrix} ,
\end{equation*}
with $\sigma_{11},\sigma_{22}>0$ and $\det{\Sigma} = \sigma_{11}\sigma_{22}- \sigma_{12}^2 \geq0$. Its
infinitesimal generator is the operator
\begin{equation*}
\mathcal G f = \frac12 \left( \sigma_{11} \frac{\partial^2 f}{ \partial x^2} + 2 \sigma_{12} \frac{\partial^2 f}{ \partial x \partial y} + \sigma_{22} \frac{\partial^2 f}{ \partial y^2} \right) .
\end{equation*}
Note that through some linear transformation $\phi$ (see \cite[Eq.~(5.1)]{Ra-14}), one obtains the Brownian motion with identity covariance matrix in the cone $\phi(\mathbb R_+^2)$.
The \textit{kernel} associated to the Brownian motion is defined as the quantity
\begin{equation*}
\gamma (x,y) = \frac12 ( \sigma_{11} x^2 + 2 \sigma_{12} xy + \sigma_{22} y^2),
\end{equation*}
for $(x,y) \in \mathbb C^2$. The Laplace transform of a function $f$, which in the continuous case is the analogous quantity of the notion of generating function, is defined as
\begin{equation*}
L(f) (x,y) = \iint_{[0,\infty)^2} f(u,v) e^{-(xu+yv)} dudv,
\end{equation*}
for $(x,y) \in \mathbb C^2$ with positive real parts.
Now, let $h$ be a harmonic function associated with the Brownian motion with covariance matrix $\Sigma$, that is, $h$ vanishes on the boundary axes of the quadrant and satisfies
$ \mathcal G h = 0$.
The functional equation for $h$
takes the following form (see \cite[Eq.~(A.1)]{Ra-14}):
\begin{equation*}
\gamma(x,y) L(h)(x,y) = \frac12 ( \sigma_{11} L_1 (h)(y) + \sigma_{22} L_2 (h) (x) ) +L(\mathcal G h)(x,y),
\end{equation*}
where we have denoted
\begin{equation*}
\left\{\begin{array}{lclcl}
L_1(h)(y) &:=&\displaystyle L \left( \frac{\partial h}{\partial x} (0,\cdot) \right) (y) &=& \displaystyle\int_0^\infty \frac{\partial h}{\partial x} (0,v) e^{-yv} dv ,\medskip\\
L_2(h)(x) &:=& \displaystyle L \left( \frac{\partial h}{\partial y} (\cdot,0) \right) (x) &= &\displaystyle \int_0^\infty \frac{\partial h}{\partial y} (u,0) e^{-xu} du.
\end{array}\right.
\end{equation*}
Using the harmonicity condition $\mathcal G h =0$, the functional equation for $h$ rewrites as
\begin{equation}
\label{functional-eq-BM-1}
\gamma(x,y) L(h)(x,y) = \frac12 ( \sigma_{11} L_1 (h)(y) + \sigma_{22} L_2 (h) (x) ) .
\end{equation}
We recall below the key argument of the method of \cite{Ra-14} to solve the functional equation~\eqref{functional-eq-BM-1}, which leads to harmonic functions for the Brownian motion via Laplace inversion. We will subsequently apply a related method to obtain polyharmonic functions.
Consider the two solutions of
$
\gamma \left(x,Y(x)\right)=0
$,
which, since $\gamma$ is a homogeneous polynomial of degree two, are explicitly given by
$
Y_\pm(x) = c_\pm x
$,
with
\begin{equation}
\label{eq:c_pm}
c_\pm = \frac{-\sigma_{12} \pm i \sqrt{\det{\Sigma}}}{\sigma_{22}},
\end{equation}
so that $c_+ = \overline{c_-}$. We write $c_\pm = c e^{\pm i\theta} $, with $c = \sqrt{\frac{\sigma_{11}}{\sigma_{22}}}$ and $\theta$ such that $\cos \theta =- \frac{\sigma_{12}}{\sqrt{\sigma_{11}\sigma_{22}}}$.
Denote by $\mathcal G_Y$ the domain delimited by the curve $Y_+([0,\infty])\cup Y_-([0,\infty])=c_+[0,\infty]\cup c_-[0,\infty]$ and containing the positive axis $[0,\infty]$.
Plugging each of the solutions $c_\pm x$ into the functional equation \eqref{functional-eq-BM-1}, one obtains a \textit{boundary value problem} for $L_1(h)$, which states that:
\begin{enumerate}
\item $L_1(h)$ is analytic on $\mathcal G_Y$,
\item $L_1(h)$ is continuous on $\overline{\mathcal G_Y}\setminus \{0\}$,
\item For all $x\in (0,\infty]$, $L_1(h)$ satisfies the boundary equation
$
L_1(h)(c_+ x ) = L_1 (h) (c_- x)
$.
\end{enumerate}
In order to solve this problem, one introduces the conformal mapping $\omega$ from $\mathcal G_Y$ onto $\mathbb C \setminus \mathbb R_-$ defined by
$
\omega ( x) = {x^{-\pi/\theta}}
$.
One eventually obtains that a class of solutions is obtained by letting $L_1(h)$ to be of the form
\begin{equation}
\label{eq:L1-P}
L_1(h)(y)= P\left(\frac{1}{y^{\pi/\theta}}\right),
\end{equation}
for any given polynomial $P$.
The same applies to $L_2(h)$ (by considering the solutions of $\gamma(X(y),y)=0$), and using the functional equation~\eqref{functional-eq-BM-1} and the fact that $(c_\pm)^{\pi/\theta}=-c^{\pi/\theta}$, one must have
\begin{equation*}
L_2(h)(y)= -\frac{\sigma_{11}}{\sigma_{22}} P\left(-\frac{1}{c^{\pi/\theta} x^{\pi/\theta}}\right),
\end{equation*}
with the same $P$ as in \eqref{eq:L1-P}. Hence, using again the functional equation~\eqref{functional-eq-BM-1}, we deduce that the Laplace transform of $h$ writes
\begin{equation}
\label{eq:LFP}
L(h)(x,y) = \frac12 \sigma_{11} \frac{P\left(\frac{1}{y^{\pi/\theta}}\right)-P\left(-\frac{1}{c^{\pi/\theta} x^{\pi/\theta}}\right)}{\gamma(x,y)}.
\end{equation}
In particular, taking $P$ to be a polynomial of degree $1$, one gets
\begin{equation*}
L(h)(x,y) = \frac{ \sigma_{22}\frac{\mu_2}{x^{\pi/\theta}} + \sigma_{11} \frac{\mu_1}{y^{\pi/\theta}}}{\gamma(x,y)} ,
\end{equation*}
where the constants are related by $\mu_2 = \mu_1 ( \frac{\sigma_{22}}{\sigma_{11}})^{1-\pi/2\theta}$.
Taking the inverse Laplace transform, one should recover the unique positive harmonic function (written in polar coordinates $(\rho,\eta)$)
\begin{equation*}
h(x,y) = \rho^{\frac{\pi}{\theta}} \sin \left( \frac{\pi}{\theta} \eta \right).
\end{equation*}
Suppose now that $v$ is bi-harmonic and satisfies
$
\mathcal G v = h
$,
where $h$ is harmonic. The functional equation for $v$ now reads
\begin{equation}
\label{functional-eq-BM-2}
\gamma(x,y) L(v)(x,y) = \frac12 ( \sigma_{11} L_1 (v)(y) + \sigma_{22} L_2 (v) (x) ) + L(h)(x,y).
\end{equation}
By considering the roots of the kernel $\gamma$ and using the same method as above, we obtain
\begin{equation}
\label{eq:BVP-1-harmonic}
\frac12 \sigma_{11} L_1(v)(c_+ x) - \frac12 \sigma_{11} L_1(v) (c_- x) = L(h)(x,c_- x ) - L(h)(x, c_+ x) .
\end{equation}
We now have an \textit{a priori} non-homogeneous boundary value problem for $v$, that we can in fact transform into an homogeneous one, thanks to
the (already known) explicit form of $L(h)$. The key remark to this task is that $(c_+x)^{\pi/\theta} = (c_- x )^{\pi/\theta}=-(cx)^{\pi/\theta}$. Rewriting \eqref{eq:LFP} as
\begin{align*}
L(h)(x,y)
& = \frac{\sigma_{11}}{\sigma_{22}} \frac{P\left(\frac{1}{y^{\pi/\theta}}\right)-P\left(\frac{1}{(c_\pm x)^{\pi/\theta}}\right)}{(y-c_- x)(y-c_+ x)}
\end{align*}
and letting $y\to c_+ x$ and $y\to c_- x$, one finds
\begin{equation*}
L(h)(x,c_\pm x) = \mp\frac{\sigma_{11}}{\sigma_{22}} \frac{\pi}{\theta} \frac{1}{ (c_\pm x - c_\mp x)}
P'\left(\frac{1}{(c_\pm x)^{\pi/\theta}}\right) \frac{1}{(c_\pm x)^{\pi/\theta+1}}.
\end{equation*}
Eventually, we get
\begin{align*}
& L(h)(x,c_- x ) - L(h)(x, c_+ x) \\
& =
\frac{\sigma_{11}}{\sigma_{22}} \frac{\pi}{\theta} \left(
\frac{1}{ (c_+ x - c_- x)} \frac{P'\left(\frac{1}{(c_+ x)^{\pi/\theta}}\right)}{(c_+ x)^{\pi/\theta +1}}
-
\frac{1}{(c_- x- c_+ x)} \frac{P'\left(\frac{1}{(c_- x)^{\pi/\theta}}\right)}{(c_- x)^{\pi/\theta +1}} \right) \\
& =
\frac{\sigma_{11}}{\sigma_{22}} \frac{\pi}{\theta} \left(
\frac{c_+}{c_+-c_-} P'\left(\frac{1}{(c_+ x)^{\pi/\theta}}\right) \frac{1}{(c_+ x)^{\pi/\theta +2} }
-
\frac{c_-}{c_- - c_+} P'\left(\frac{1}{(c_- x)^{\pi/\theta}}\right)\frac{1}{(c_- x)^{\pi/\theta +2} } \right) \\
& =
- \frac{\sigma_{11}}{\sigma_{22}} \frac{\pi}{\theta}
\frac{c_+c_-}{(c_+-c_-)^2} \left( P'\left(\frac{1}{(c_+ x)^{\pi/\theta}}\right) \frac{1}{(c_+ x)^{\pi/\theta +2} }
-
P'\left(\frac{1}{(c_- x)^{\pi/\theta}}\right)
\frac{1}{(c_- x)^{\pi/\theta +2} } \right),
\end{align*}
where the last equality follows from $(c_+x)^{\pi/\theta} = (c_- x )^{\pi/\theta}$. Therefore, the boundary value equation~\eqref{eq:BVP-1-harmonic} is now homogeneous, and of the form
\begin{equation*}
\frac12 \sigma_{11} L_1(v)(c_+ x)-F(c_+ x ) = \frac12 \sigma_{11} L_1(v) (c_- x) -F(c_- x ),
\end{equation*}
where $F$ is equal on $Y_+([0,\infty])\cup Y_-([0,\infty])$ to
\begin{equation}
\label{def:bigF}
F(y) = -\frac{\sigma_{11}}{\sigma_{22}} \frac{\pi}{\theta} \frac{c_+c_-}{(c_+-c_-)^2} P'\left(\frac{1}{y^{\pi/\theta}}\right) \frac{1}{y^{\pi/\theta+2}} .
\end{equation}
We note that the simpler case when $F(c_+x)=F(c_- x)$ occurs exactly when $c_+^2=c_-^2$, i.e., $\theta$ is $0$ or $\pi/2$.
In this way, we obtain a boundary value problem analogous to the harmonic case, which, on the boundary of $\mathcal G_Y$ except at $0$, leads to
\begin{equation*}
\frac12 \sigma_{11} L_1(v)(y)-F(y)=Q\left( \frac{1}{y^{\pi/\theta}} \right),
\end{equation*}
for any given polynomial $Q$.
The same computation applies to $L_2(v)$. As such, using the equation~\eqref{functional-eq-BM-2}, the Laplace transform of the bi-harmonic function $v$ admits the following form:
\begin{theorem}
\label{thm:Laplace-2}
For any polynomials $P$ and $Q$, the formula
\begin{equation*}
L(v)(x,y) = \frac{1}{\gamma(x,y)} \left[ Q\left( \frac{1}{y^{\pi/\theta}} \right) -Q \left( \frac{1}{(c_+x)^{\pi/\theta}} \right) + G(x,y) + L(h)(x,y) \right]
\end{equation*}
is the Laplace transform $L(v)$ of a bi-harmonic function $v$ satisfying $\mathcal G v =h$, where $h$ is a harmonic function with Dirichlet boundary conditions, where the Laplace transform $L(h)$ of $h$ has the form \eqref{eq:LFP} and where
\begin{equation*}
G(x,y)= F(y)-F(c_+x)-L(h)(x,c_+x) ,
\end{equation*}
with $F$ defined in Eq.~\eqref{def:bigF}.
\end{theorem}
The above theorem can be understood as a Laplace transform counterpart of Almansi's theorem~\cite{Al-1899}.
Recursively, if $v_n$ is polyharmonic of order $n$ with $\mathcal G v_n= v_{n-1}$, where $v_{n-1}$ is polyharmonic of order $n-1$, the above method permits to express the Laplace transform of $v_n$ through the one of $v_{n-1}$, allowing to construct polyharmonic functions via Laplace inversion.
Further computations for the Brownian motion with identity covariance matrix are proposed in Appendix \ref{sec:app_BM}.
\section{Discrete polyharmonic functions}
\label{section-discrete}
Similarly to the continuous setting, we first investigate the appearance of polyharmonic functions in the asymptotic expansions of the counting coefficients of lattice paths with prescribed endpoints, starting from an exact expression for these coefficients (such exact expressions may typically be obtained from reflection principles). We then implement the functional equation approach to construct polyharmonic functions.
Our framework is thus the following. We consider random walks in the quarter plane $\mathbb Z^2_+$ with the following assumptions:
\begin{enumerate}
\item The walk is homogeneous with transition probabilities $\{p_{i,j}\}_{-1\leq i,j \leq 1}$ to the eight nearest neighbours and $p_{0,0}=0$ (so we are only considering walks with small steps),
\item In the list $p_{1,1},p_{1,0},p_{1,-1},p_{0,-1},p_{-1,-1},p_{-1,0},p_{-1,1},p_{0,1}$, there are no three consecutive zeros (to avoid degenerate cases),
\item The drifts $\sum_{i,j}ip_{i,j}$ and $\sum_{i,j}jp_{i,j}$ are zero.
\end{enumerate}
The Markov operator $P$ of the walk is defined on discrete functions by
\begin{equation*}
Pf(x,y) = \sum_{-1\leq i,j \leq 1} p_{i,j} f(x+i,y+j),
\end{equation*}
and the Laplacian operator is $L=P-I$. A function $f$ is said to be \textit{harmonic} if $Lf=0$ and \textit{polyharmonic} of order $p$ if $L^pf=0$.
\subsection{Examples of asymptotic expansion in walk enumeration problems}
We start by recalling a few exact expressions for the number of quarter plane walks of length $n$ with prescribed endpoints.
\begin{example}[The diagonal walk]
The step set is $\{\nearrow,\nwarrow,\searrow,\swarrow \}$, with uniform transition probabilities $\frac14$. It is well known (see for instance \cite{bou-02counting}) that
\begin{equation}
\label{counting-diagonal}
q((i,j),(0,0);n) = \frac{(i+1)(j+1)}{\frac{n+i+2}{2} \frac{n+j+2}{2}} \binom{n}{\frac{n+i}{2}} \binom{n}{\frac{n+j}{2}} ,
\end{equation}
with $i$ and $j$ having the same parity as $n$.
Starting from \eqref{counting-diagonal}, one can prove that
\begin{equation}
\label{complete-asympt-diagonal}
q((i,j),(0,0);n) \sim \frac{8}{\pi} 4^n \sum_{p\geq0} \frac{v_p(i,j)}{n^{3+p}},
\end{equation}
where
the first few terms in the above asymptotic expansion are given by
\begin{equation*}
\left\{\begin{array}{rcl}
v_0(i,j)& =& (i+1)(j+1) , \\
v_1(i,j) & =& -\frac12 (i+1)(j+1)(i^2+j^2+2i+2j+9) .
\end{array}\right.
\end{equation*}
The first term $v_0$ is the well-known unique (up to multiplicative constants) positive harmonic function, with Dirichlet conditions; it is the same as for the simple walk, see \eqref{eq:asymp_SRW} and \eqref{eq:V3V4V5}. The next term satisfies $Lv_1=-3v_0$, and therefore is bi-harmonic. Note that in fact, using the explicit expression of the Laplacian $L$, it is obvious that any polynomial of degree at most $2p-1$ is polyharmonic of order $p$, since for any polynomial $f$ of degree $k$, $Lf$ has degree at most $k-2$ (it is a discrete equivalent of Lemma~\ref{lem:deg-2}).
To derive a full asymptotic expansion of \eqref{counting-diagonal}, we shall use the Laplace method applied to the counting coefficients rewritten as an integral, in the spirit of~\cite[p.~75--79]{spitzer} (alternatively one can apply the saddle-point method~\cite[Chap.\ B~VIII]{FlSe-09} in the framework of analytic combinatorics in several variables \cite{CoMeMiRa-17,MeWi-19}). We choose to postpone it to Appendix \ref{sec:app}, since the computations are a bit long, though straightforward.
\end{example}
\begin{example}[The simple random walk]
\label{ex:SRW}
The step set is $\{\leftarrow, \uparrow, \rightarrow, \downarrow\}$, with uniform transition probabilities $\frac14$. We have \eqref{eq:SRW_excursion} by \cite{bou-02counting}.
Again, starting from \eqref{eq:SRW_excursion}, one can prove that
\begin{equation*}
q((i,j),(0,0);n) \sim \frac{4}{\pi}4^n\sum_{p\geq0} \frac{v_p(i,j)}{n^{3+p}},
\end{equation*}
where
the first few terms in the asymptotic expansion are
\begin{equation}
\label{eq:V3V4V5}
\left\{\begin{array}{rcl}
v_0(i,j)&=&(i+1)(j+1),\\
v_1(i,j)&=&-\frac{1}{4}(i+1)(j+1)(2i^2+2j^2+4i+4j+15). \\
\end{array}\right.
\end{equation}
Again, $v_0$ is harmonic, and since $Lv_1=-\frac{3}{2}v_0$, $v_1$ is bi-harmonic.
\end{example}
\begin{example}[The tandem walk]
\label{ex:T}
The step set is $\{\nwarrow,\rightarrow,\downarrow\}$ with uniform transition probabilities $\frac13$.
From \cite[Prop.~9]{BMMi-10}, we know that:
\begin{equation*}
q((i,j),(0,0);n) = \frac{(i+1)(j+1)(i+j+2)(3m+2i+j)!}{m!(m+i+1)!(m+i+j+2)!},
\end{equation*}
with $n=3m+2i+j$. In this case, writing the asymptotic expansion
\begin{equation*}
q((i,j),(0,0);n) \sim \frac{\sqrt 3}{2\pi} 3^n \sum_{p\geq0} \frac{v_p(i,j)}{n^{4+p}} ,
\end{equation*}
one has for the harmonic function $v_0$ and the bi-harmonic function $v_1$,
\begin{equation}
\label{eq:V3V4V5-tandem}
\left\{\begin{array}{rcl}
v_0(i,j) & =& (i+1)(j+1)(i+j+2) , \\
v_1(i,j) & =& -\frac19 (i+1)(j+1)(i+j+2) (3i^2 + 3j^2 +3ij+9i+9j+38) .
\end{array}\right.
\end{equation}
\end{example}
\subsection{Functional equation approach in the discrete case}
We implement here the functional equation method to construct polyharmonic functions. We start by recalling the key arguments in the harmonic case; details may be found in \cite{Ra-14}.
For a harmonic function $h$, we denote by $H$ its generating function, namely,
\begin{equation*}
H(x,y)= \sum_{i,j\geq0} h(i,j) x^i y^j.
\end{equation*}
The \textit{kernel} of the random walk is defined as the polynomial
\begin{equation*}
K(x,y)=xy\left(\sum_{-1\leq k, \ell\leq1 }p_{k,\ell}x^{-k}y^{-\ell}-1\right).
\end{equation*}
The harmonic equation $L h =0$ yields the following \textit{functional equation}
\begin{equation}
\label{eq:functional_equation_1-harmo}
K(x,y)H(x,y)=K(x,0)H(x,0)+K(0,y)H(0,y)-K(0,0)H(0,0).
\end{equation}
To solve \eqref{eq:functional_equation_1-harmo}, one first proves that the function $H(x,0)$ (and similarly $H(0,y)$) satisfies a \textit{boundary value problem} (see \cite{Ra-14}):
\begin{enumerate}
\item $H(x,0)$ is analytic in $\mathcal G_X$,
\item $H(x,0)$ is continuous on $\overline{\mathcal G_X}\setminus \{1\}$,
\item For all $x$ in the boundary of $\mathcal G_X$ except at $1$, $H(x,0)$ satisfies the boundary equation:
\begin{equation*}
K(x,0)H(x,0)-K(\overline x ,0) H(\overline x ,0)=0.
\end{equation*}
\end{enumerate}
Here, $\mathcal G_X$ is a certain domain bounded by the curve $X_+([y_1,1]) \cup X_-([y_1,1])$, where $X_\pm(y)$ are the branches of the algebraic function defined by $K(X(y),y)=0$.
Indeed, writing $K$ as
\begin{equation*}
K(x,y)=\widetilde\alpha(y) x^2 + \widetilde\beta(y) x + \widetilde\gamma(y),
\end{equation*}
where $\widetilde\alpha, \widetilde\beta, \widetilde\gamma$ are polynomials of degree 2 whose coefficients depend on the model, we have
\begin{equation*}
X_\pm (y) = \frac{-\widetilde\beta(y)\pm \sqrt{\widetilde\delta(y)}}{2\widetilde\alpha(y)},
\end{equation*}
where $\widetilde\delta(y) = \widetilde\beta(y)^2-4\widetilde\alpha(y)\widetilde\gamma(y)$. The functions $X_\pm$ are thus meromorphic on a cut plane, determined by the zeros of $\widetilde\delta$.
It follows by \cite{Ra-14} that $K(x,0)H(x,0)$ may be written as a function of a certain conformal mapping $\omega$ (see \cite[Eq.~(3.1)]{Ra-14} for its explicit expression):
\begin{equation*}
K(x,0)H(x,0) = P(\omega(x)),
\end{equation*}
where $P$ is an arbitrary entire function, for example a polynomial. This represents the analogous statement as \eqref{eq:L1-P} in the continuous setting. By the functional equation~\eqref{eq:functional_equation_1-harmo}, one eventually finds that
\begin{equation*}
H(x,y) = \frac{P(\omega(x)) - P(\omega(X_+(x)))}{K(x,y)},
\end{equation*}
which again should be compared with \eqref{eq:LFP} in the continuous case.
For a bi-harmonic function $v$, satisfying $Lv=h$ with $h$ a harmonic function, the functional equation now writes
\begin{equation}
\label{eq:functional_equation_2-harmo}
K(x,y)V(x,y)=K(x,0)V(x,0)+K(0,y)V(0,y)-K(0,0)V(0,0)-xyH(x,y),
\end{equation}
where $V$ is the generating function of $v$, i.e., $V(x,y)=\sum_{i,j\geq0} v(i,j) x^i y^j$; compare with \eqref{functional-eq-BM-2}. Notice that the equation \eqref{eq:functional_equation_2-harmo} is very close to functional equations coming up in walk enumeration problems.
Plugging the roots of the kernel into \eqref{eq:functional_equation_2-harmo}, one has
\begin{equation*}
K(X_\pm(y),0)V(X_\pm(y),0)+K(0,y)V(0,y)-K(0,0)V(0,0)-X_\pm(y)yH(X_\pm(y),y)=0,
\end{equation*}
which leads to the boundary equation
\begin{equation}
\label{eq:functional_equation_2-harmo_BVP}
K(x,0)V(x,0)-K(\overline{x},0)V(\overline{x},0)= y\left( xH(x,y)-\overline x H(\overline x,y) \right) ,
\end{equation}
for $x$ on the boundary of $\mathcal G_X$ (except at $1$).
Note that a general method to solve this kind of boundary value problem \eqref{eq:functional_equation_2-harmo_BVP} exists \cite{FaIaMa-17}, for any quantity in the right-hand side, ending up in some contour integral expression for the unknown function $K(x,0)V(x,0)$. We choose to provide below examples with simpler, integral-free expressions. Indeed, the resolution of \eqref{eq:functional_equation_2-harmo_BVP} is made easier in some peculiar cases, for instance when the right-hand side of \eqref{eq:functional_equation_2-harmo_BVP} is zero (which occurs for the simple random walk, see Example~\ref{ex:SRW-funct-method} below), or when it can be decoupled in the terminology of \cite{BeBMRa-17} (which is analogous to the continuous setting and holds for the tandem walk, see Appendix~\ref{sec:app_tandem}).
\setcounter{example}{1}
\begin{example}[continued]
\label{ex:SRW-funct-method}
We consider here the case of the simple random walk,with kernel
\begin{equation*}
K(x,y)=xy\left(\frac{1}{4}\left(x+\frac{1}{x}+y+\frac{1}{y}\right)-1\right).
\end{equation*}
The domain $\mathcal G_X$ is the open unit disk, and the conformal mapping $\omega$ admits the expression $\omega(x)=\frac{x}{(1-x)^2}$, see \cite{Ra-14}. A computation shows that $\omega(X_+(y))=-\omega(y)$, thus one gets that the generating function of a harmonic function $h$ may be written as
\begin{equation*}
H(x,y) = \frac{P(\omega(x))-P(-\omega(y))}{K(x,y)} .
\end{equation*}
Choosing $P(x)=\frac{x}{4}$ leads to
\begin{equation*}
H(x,y)=\frac{\frac{\frac14x}{(1-x)^2}+\frac{\frac14y}{(1-y)^2}}{xy\left(\frac{1}{4}(x+\frac{1}{x}+y+\frac{1}{y})-1\right)}=\frac{1}{(1-x)^2(1-y)^2}=\sum_{i,j\geq0} (i+1)(j+1)x^{i}y^{j},
\end{equation*}
that is, $H$ is the generating function of the unique positive harmonic function, see \eqref{eq:V3V4V5}.
We now consider bi-harmonic functions. Using the explicit form of $H$, one sees that the right-hand side of Eq.~\eqref{eq:functional_equation_2-harmo_BVP} vanishes. Indeed, we have
\begin{multline*}
X_+(y) H(X_+(y),y ) - X_-(y) H(X_-(y),y ) \\
= X_+(y)\frac{P'(\omega(X_+(y))) \omega'(X_+(y))}{\widetilde\alpha(y) (X_+(y)-X_-(y))}
-X_-(y)\frac{P'(\omega(X_-(y))) \omega'(X_-(y))}{\widetilde\alpha(y) (X_-(y)-X_+(y))} ,
\end{multline*}
which is equal to zero since $\omega(X_+(y))=\omega(X_-(y))$ and
\begin{equation*}
X_+(y)\frac{\omega'(X_+(y))}{X_+(y)-X_-(y)}
-X_-(y)\frac{ \omega'(X_-(y))}{X_-(y)-X_+(y)}=0
\end{equation*}
by straightforward computations.
The boundary equation has thus exactly the same form as the one in the harmonic case, so we get that on the boundary of $\mathcal G_X$,
\begin{equation*}
K(x,0) V(x,0) = Q(\omega(x)),
\end{equation*}
for some polynomial $Q$. Using (twice) the functional equation~\eqref{eq:functional_equation_2-harmo}, the general form for the generating function of a bi-harmonic $v$ satisfying $Lv=h$, with $h$ harmonic, is thus
\begin{equation*}
V(x,y)=\frac{Q(\omega(x))-Q(-\omega(y))+X_+(y)yH(X_+(y),y)-xyH(x,y)}{K(x,y)},
\end{equation*}
with
\begin{equation*}
H(x,y) = \frac{P(\omega(x))-P(-\omega(y))}{K(x,y)} \quad \text{and} \quad H(X_+(y),y)= \frac{P'(\omega(X_+(y))) \omega'(X_+(y))}{\widetilde\alpha(y) (X_+(y)-X_-(y))}.
\end{equation*}
For instance, taking $P(x)=x$ and $Q$ the zero polynomial leads to the bi-harmonic function (non symmetrical in $i$ and $j$)
\begin{equation*}
v(i,j)=(i+1)j(j+1) (j+2).
\end{equation*}
Indeed, one has
\begin{equation*}
X_+(y)H(X_+(y),y)=-\frac{y}{(1-y)^4},
\end{equation*}
so the generating function $V$ writes
\begin{equation*}
V(x,y)=\frac{-4y}{(1-x)^2(1-y)^4},
\end{equation*}
which is easily inverted.
On the other hand, taking $P(x)=x$ and $Q(x)=-2x^2-\frac{5}{2}x$, one obtains the bi-harmonic function
\begin{equation*}
v(i,j)= (i+1)(j+1)(2i^2+2j^2+4i+4j+15),
\end{equation*}
which is (up to a multiplicative constant) the bi-harmonic function $v_1$ appearing in
Eq.~\eqref{eq:V3V4V5}. Another example will be treated in Appendix \ref{sec:app_tandem}.
\end{example}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,812
|
Q: Rails: UJS not executing I have some UJS that's meant to replace a partial in my view. The .js file looks like this:
transition("#content_container", "<%= escape_javascript(render 'users/show') %>");
librarySorter("User", "<%= @user.username %>");
The transition function executes as expected, but librarySorter isn't executing. I check in firebug, and it does appear in my response like so:
librarySorter("User", "tbaron");
And stranger yet, if I copy and paste that line into my firebug console, it runs properly and executes. I've been tinkering with this for hours, and I still have no idea why it's not working. Any ideas?
For reference, here's what librarySorter looks like (it's in application.js):
function librarySorter(library_type, id){
function publication_sort_binder(type, order, name){
$("#overview_table").animate({opacity: 0}, 400);
$("#sort_method").html(type+' '+order);
var url = library_type == "User" ? '/users/'+id+'/owned_publications' : '/groups/'+id+'/overview_sort'
$.ajax({
type: 'GET',
data: {'function':type+' '+order},
url: url,
success: function() {
$("#publication_sort_arrow").remove();
var arrow = (order == 'ASC') ? '↓' : '↑';
$('#sort_'+name).unbind("click").click(function() {
publication_sort_binder(type, (order == 'ASC') ? 'DESC' : 'ASC', name);
}).append("<span id='publication_sort_arrow'> "+arrow+"</span>");
}
});
}
$('#sort_title').click(function() {
publication_sort_binder("publications.title", "DESC", "title");
})
$('#sort_author').click(function() {
publication_sort_binder("contributors.name", "DESC", "author");
})
$('#sort_review_date').click(function() {
publication_sort_binder("reviews.created_at", "DESC", "review_date");
})
$('#sort_review_vote').click(function() {
publication_sort_binder("reviews.votes_sum", "DESC", "review_vote");
})
$('#sort_children_total').click(function() {
publication_sort_binder("updates.replies_count", "DESC", "children_total");
})
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,870
|
Q: flex items don't stretch in cross axis when overflow is set to auto As position:fixed has several disadvantages I'm trying to create a fixed footer by use of flexbox. I have a flex container with direction row and applied overflow-y:auto to it. Now when the content of a flex item exceeds the height of the visible area and the scrollbar appears the flex items do not stretch their height to the height of the container but stick to the height of the visible area. The result is that the text is not fully underlayed with the background color.
HTML:
<body>
<div id="middle">
<nav>
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna
aliquyam erat, sed diam voluptua.
</nav>
<main>
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
dolor sit amet.
Content shortened; more content to make the scrollbar appear
</main>
</div>
<!-- Main Bereich Ende -->
<!--Footer Anfang -->
<footer>
<div id="footerleft">
<h2>Letzte Änderung
<br>20.05.2018</h2>
</div>
<div id="footermiddle">
<h3>Copyright © 2018 xxx - Alle Rechte vorbehalten</h3>
</div>
<div id="footerright">
<a href="#jump-body">
<img src="images/buttonup.png" title="Zum Seitenanfang" alt="Zum Seitenanfang"></img>
</a>
</div>
</footer>
<!--Footer Ende -->
</body>
CSS:
body {
margin: 0;
padding: 0;
height: 100vh;
display: flex;
flex-direction: column;
}
#middle {
flex: 1 1 auto;
display: flex;
flex-direction: row;
align-items: stretch;
overflow-y: auto;
}
nav {
flex: 1;
background-color: blue;
}
main {
flex: 4;
background-color: maroon;
}
footer {
width: 100%;
flex: 0 0 auto;
display: flex;
}
#footerleft {
flex: 2;
}
#footermiddle {
flex: 6;
}
#footerright {
flex: 2;
}
#footerright img {
width: 30px;
height: 30px;
display: block;
margin: 0px auto auto;
}
How can this be fixed?
A: Just remove height: 100vh; from body selector. By default height is auto so the body height will stretch according to its content. So instead of that use min-height: 100vh;.
Note: img tag shouldn't have a closing tag </img>
body {
margin: 0;
padding: 0;
min-height: 100vh;
display: flex;
flex-direction: column;
}
#middle {
flex: 1 1 auto;
display: flex;
flex-direction: row;
align-items: stretch;
overflow-y: auto;
}
nav {
flex: 1;
background-color: blue;
}
main {
flex: 4;
background-color: maroon;
}
footer {
width: 100%;
flex: 1 0 auto;
display: flex;
}
#footerleft {
flex: 2;
}
#footermiddle {
flex: 6;
}
#footerright {
flex: 2;
}
#footerright img {
width: 30px;
height: 30px;
display: block;
margin: 0px auto auto;
}
<div id="middle">
<nav>
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna
aliquyam erat, sed diam voluptua.
</nav>
<main>
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
dolor sit amet.Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
dolor sit amet.Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
dolor sit amet.Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
dolor sit amet.
Content shortened; more content to make the scrollbar appear
</main>
</div>
<footer>
<div id="footerleft">
<h2>Letzte Änderung
<br>20.05.2018</h2>
</div>
<div id="footermiddle">
<h3>Copyright © 2018 xxx - Alle Rechte vorbehalten</h3>
</div>
<div id="footerright">
<a href="#jump-body">
<img src="images/buttonup.png" title="Zum Seitenanfang" alt="Zum Seitenanfang">
</a>
</div>
</footer>
A: In the meantime I researched further and found this:
https://github.com/philipwalton/flexbugs/issues/141
A workaround for this issue is given. Can anyone of the flexbox experts explain? Is this really a bug in flexbox and is it cross browser?
Demo of the fix:
http://webentwicklung.ulrichbangert.de/thread-footer-verdeckt-fixed-footer-3.html
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,535
|
NATWEST GROUP (144A)
Transaction in Own Shares
Nov 26 2021 17:52 GMT Source: RNS
RNS Number : 7920T
NatWest Group plc
NatWest Group plc (the 'Company' or 'NWG') announces today that it has purchased the following number of ordinary shares in the Company with a nominal value of £1 each ('Ordinary Shares') from UBS AG, London Branch ('UBS').
Aggregated information:
Number of Ordinary Shares purchased
Highest price paid: (GBp)
Lowest price paid: (GBp)
Volume weighted average price paid per share (GBp)
Such purchases form part of the Company's existing share buyback programme and were effected pursuant to the instructions issued by the Company to UBS on 30 July 2021, as announced on 2 August 2021.
The Company intends to cancel the repurchased Ordinary Shares.
Following the settlement of the above transactions, NWG will hold 189,712,123 Ordinary Shares in treasury and have 11,317,163,293 Ordinary Shares in issue (excluding treasury shares).
Legal Entity Identifier: 2138005O9XJIJN4JPN90
In accordance with Article 5(1)(b) of Regulation (EU) No.596/2014 as it applies in the UK (Market Abuse Regulation), a full breakdown of the individual trades made by UBS on behalf of the Company as part of the buyback programme is available here:
http://www.rns-pdf.londonstockexchange.com/rns/7920T_1-2021-11-26.pdf
This information is provided by RNS, the news service of the London Stock Exchange. RNS is approved by the Financial Conduct Authority to act as a Primary Information Provider in the United Kingdom. Terms and conditions relating to the use and distribution of this information may apply. For further information, please contact rns@lseg.com or visit www.rns.com.
POSDKABQOBDDFDB
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,977
|
\section{Inroduction}
Our note is very much motivated by the paper \cite{CWY}. The authors of \cite{CWY} consider a weak solution to the Cauchy problem for the Navier-Stokes equations with $L_2$-initial data under the additional assumoption that it is bounded in time with values in the weak Lebesgue space $L^{3,\infty}(\mathbb R^3)$. They show that, at each instance of time, there exists at most a finite number of singular points.
What we would like is to extend this result to the local setting, including considerations near a flat part of the boundary, and to the standard notion of suitable solutions, see \cite{CKN}, \cite{Lin}, and \cite{LadSer1999}. Our proof seems to be shorter and straightforword.
Interior and bounded regularity will be analyised separately.
Let us start with the interrior case.
Consider a suitable weak solution $v$ and $q$ in $Q_T=\Omega\times ]0,T[$, where $\Omega$ is a domain $\mathbb R^3$. The corresponding definition is due to F-H Lin, see \cite{Lin} and Definition \ref{sws} of the this paper.
It differs slightly from the original one, introduced by Caffarelli-Kohn-Nirenberg in \cite{CKN}, just by a more convienient class for the pressure field.
\begin{definition}\label{sws} We say that a pair $v$ and $q$ is a suitable weak solution to the Navier-Stokes equations in $Q_T$ if:
\begin{equation}
\label{class}
v\in L_{2,\infty}(Q_T),\qquad v\in L_{2,\infty}(Q_T),\qquad q\in L_\frac 32(Q_T);
\end{equation}
the pair $v$ and $q$ satisfies the Navier-Stokes equations
\begin{equation}
\label{NSE}
\partial_tv+v\cdot\nabla v-\Delta v=-\nabla q,\qquad {\rm div}\,v=0
\end{equation}
in $Q_T$ in the sense of distributions;
for a.a. $t\in ]0,T[$, the local energy inequality
\begin{equation}
\label{locenineq}
\int\limits_{\Omega}|v(x,t)|^2\varphi(x,t)dx+2\int\limits^t_0\int\limits_\Omega |\nabla v|^2\varphi dx dt'\leq \end{equation}
$$\leq \int\limits^t_0\int\limits_\Omega(|v|^2(\partial_t\varphi+\Delta \varphi)+v\cdot\nabla \varphi(|v|^2+2q))dx dt'
$$
holds for all non-negative test functions $\varphi \in C^\infty_0(\Omega \times ]0,2T[)$.
\end{definition}
Our basic additional assumpton is that
\begin{equation}
\label{addass}
\|v\|_{ L_\infty(0,T;L^{3,\infty}(\Omega))}\leq M<\infty.
\end{equation}
In fact, it implies the following: one can select a representative of the function $t\to v(\cdot,t)$ so that
\begin{equation}
\label{conaddass}
\sup\limits_{0<t\leq T}\|v(\cdot,t)\|_{L^{3,\infty}(\Omega)}\leq M.\end{equation}
Indeed, fix a representative for $v$ such that the set of all singular points has zero 1D parabolic Hausdorff measure. Hence, for each time $0<t_0\leq T$, the set of singular points $(x,t_0)$ has zero 1D Hausdorff measure. As it has been shown in \cite{LadSer1999}, the function $z=(x,t)\to v(z)$ is H\"older continuous in a parabolic vicinity of each regular point $(x,t_0)$. So, the following is true:
$$v(x,t)\to v(x,t_0)$$
for a.a. $x\in \Omega$ as $t\to t_0$ and $t<t_0$. Then, selecting a sequence of times $t_k<t_0$ such that
$$\|v(\cdot,t_k)\|_{L^{3,\infty}(\Omega)}:=\sup\limits_{\alpha>0}\alpha|\{x\in \Omega:\,\,|v(x,t_k)|>\alpha\}|^\frac 13\leq M,
$$
observe that
$$\liminf\limits_{k\to \infty}\|v(\cdot,t_k)\|_{L^{3,\infty}(\Omega)}\geq\|v(\cdot,t_0)\|_{L^{3,\infty}(\Omega)}.
$$
It is important to notice that condition (\ref{addass}) provides the existence of non-trivial limit solutions that are arising from rescaling procedure around a singular point.
A local version of the main result of the paper
\cite{CWY} can be proved with the help of an idea from the paper \cite{Seregin1999}.
\begin{theorem}
\label{final} Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q_T$. Assume that $v$ satisies condition (\ref{addass}). Then,
for any subdomain $\Omega_1\Subset\Omega$, there exists at most a finite number of singular points in the set $\{(x,T):\,\,x \in \overline\Omega_1\}$.
\end{theorem}
To prove Theorem \ref{final}, we need intermediate statements that might be interesting themselves. In order to describe them, let us introduce the following scale invariant quantities:
$$A(v,r;z_0):=\frac 1r\sup\limits_{t_0-r^2<t<t_0}\int\limits_{B(x_0,r)}|v(x,t)dx,\qquad C(v,r;z_0)=\frac 1{r^2}\int\limits_{Q(z_0,r)}|v|^3dz,
$$
$$
E(v,r;z_0)=\frac 1{r}\int\limits_{Q(z_0,r)}|\nabla v|^2dz,\qquad K(v,r;z_0)=\frac 1{r}\int\limits_{Q(z_0,r)}|v|^4dz,$$
$$ D(r)=\frac 1{r^2}\int\limits_{Q(r)}|p|^\frac 32dz, \qquad
D_0(q,r;z_0)=\frac 1{r^2}\int\limits_{Q(z_0,r)}|q-[q]_{B(x_0,r)}|^\frac 32dz,
$$
where $z_0=(x_0,t_0)$, $Q(z_0,r)=B(x_0,r)\times ]t_0-r^2,t_0[$, $B(x_0,r)=\{|x-x_0|<r\}$, $[q]_{B(x_0,r)}(t)$ is the mean value of the function $x\to q(x,t)$ over the ball $B(x_0,r)$.
Also, let us abbreviate: $B(r)=B(0,r)$, $B=B(1)$, $Q(r)=Q(0,r)$, $Q=Q(1)$,
$A(v,r)=A(v,r;0)$, etc.
The following proposition is a local version of the main regularity result of the paper \cite{CWY}.
\begin{pro}\label{mainlemma}
Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q$, satisfying additional assumptions:
\begin{equation}
\label{energy}
D_0(q,1)+E(v,1)\leq N
\end{equation}
and
\begin{equation}
\label{weakl3}
\|v\|_{L_\infty(-1,0;L^{3,\infty}(B))}\leq M.
\end{equation}
There exists a positive number $\varepsilon
<\frac 14$, depending on $N$ and $M$ only, such that if, for some $0<r\leq \frac 12$,
\begin{equation}
\label{maincond}
\frac 1{r^3}|\{x\in B(r): |v(x,0)|>\frac \varepsilon r\}|\leq\varepsilon,
\end{equation}
then
\begin{equation}
\label{boundedness}
v\in L_\infty(Q(\varepsilon r)).\end{equation}
\end{pro}
In our further considerations, a scaled version of Proposition \ref{mainlemma} is going to used.
\begin{pro}\label{mainlemmasc}
Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q(z_0,R)$, satisfying additional assumptions:
\begin{equation}
\label{energysc}
D_0(q,R;z_0)+E(v,R:z_0)\leq N
\end{equation}
and
\begin{equation}
\label{weakl3sc}
\|v\|_{L_\infty(t_0-R^2,t_0;L^{3,\infty}(B(x_0,R)))}\leq M.
\end{equation}
If, for some $0<r\leq \frac 12R$,
inequality
\begin{equation}
\label{maincondscale}
\frac 1{r^3}|\{x\in B(x_0,r): |v(x,t_0)|>\frac \varepsilon r\}|\leq\varepsilon
\end{equation}
holds, then $v\in L_\infty(Q(z_0,\varepsilon r))$
\end{pro}
Next, let us discuss local regularity up to a flat part of the boundary. To formulate the corresponding results, the specific notation is needed:
$$B^+(x_0,r)=B(x_0,r)\cap \{x_3>x_{03}\},\qquad Q^+(z_0,r)=B^+(x_0,r)\times ]t_0-r^2,t_0[.$$
For $x_0=0$, abbreviations
$B^+(r)=B^+(0,r)$, $B^+=B^+(1)$ are exploited.
Now, the definition of suitable week solutiuons to the problem
\begin{equation}
\label{bnse}
\partial_tv+v\cdot\nabla v -\Delta v=-\nabla q,\qquad{\rm div}\,v=0
\end{equation}
in $Q^+$ and
\begin{equation}
\label{dirichlet}
v(x',t)=0
\end{equation}
for all $-1<t<0$ ans for all $|x'|<1$, where
$$x'=(x_1,x_2,0)$$
for $x=(x_1,x_2,x_3)$, is as follows, see \cite{Seregin}.
\begin{definition}
\label{swsbc}
We say that a pair $v$ and $q$ is a suitable weak solution to the Navier-Stokes equations in $Q^+$ if:
\begin{equation}
\label{classbc}
v\in L_{2,\infty}(Q^+),\qquad v\in L_{2,\infty}(Q^+),\qquad q\in L_\frac 32(Q^+);
\end{equation}
the pair $v$ and $q$ satisfies (\ref{bnse}) in the sense of distributions and $v$ satifies boundary condition (\ref{dirichlet});
for a.a. $t\in ]-1,0[$, the local energy inequality
\begin{equation}
\label{locenineqbc}
\int\limits_{B^+}|v(x,t)|^2\varphi(x,t)dx+2\int\limits^t_{-1}\int\limits_{B^+} |\nabla v|^2\varphi dx dt'\leq \end{equation}
$$\leq \int\limits^t_{-1}\int\limits_B^+(|v|^2(\partial_t\varphi+\Delta \varphi)+v\cdot\nabla \varphi(|v|^2+2q))dx dt'
$$
holds for all non-negative test functions $\varphi \in C^\infty_0(B \times ]-1,1[)$.
\end{definition}
Here, our main assumtion remain the same:
\begin{equation}
\label{mabc}
\|v\|_{L_\infty(-1,0;L^{3,\infty}(B^+))}\leq M<\infty.
\end{equation} Arguing as above, one can show that $\|v(\cdot,t)\|_{L^{3,\infty}(B^+)}\leq M$ for all $t\in ]-1,0]$.
A boundary version of our main result reads the folowing.
\begin{theorem}
\label{finalbc}
Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q^+$. Assume that $v$ satisies condition (\ref{mabc}). Then,
for any $r_0\in ]-1,0[$, there exists at most a finite number of singular points in the set $\{(x,0):\,\,x \in \overline{B}^+(r_0)\}$.
\end{theorem}
In order state the auxiliary results, let us define similar scale invariant quantities, for example,
$$A^+(v,r;z_0)=\sup\limits_{t_0-r^2<t<t_0}\frac 1r\int\limits_{B^+(x_0,r)}|v(x,t)|^2dx,\, E^+(v,r;z_0)=\frac 1r\int\limits_{Q^+(z_0,r)}|\nabla v|^2dz,
$$ and so on. In addition, we introduce two other pressure quantities:
$$D_2^+(q,r;z_0)=\frac 1{r^\frac {13}8}\int\limits^{t_0}_{t_0-r^2}\Big(\int\limits_{B^+(x_0,r)}|\nabla q|^\frac {12}{11}dx\Big)^\frac {11}8dt$$
and
$$D_2(q,r;z_0)=\frac 1{r^\frac {13}8}\int\limits^{t_0}_{t_0-r^2}\Big(\int\limits_{B(x_0,r)}|\nabla q|^\frac {12}{11}dx\Big)^\frac {11}8dt.$$
Without loss of generality, one may assume that the suitable weak solution in Definition \ref{swsbc} satisfies the additional condition
$D_2^+(q,1)=D_2^+(q,1;0)<\infty$.
Now, an analog of Proposition \ref{mainlemma} can be stated as follows.
\begin{pro}
\label{mainlemmabc}
Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q^+$ in the sense of Definition \ref{swsbc}. Assume that it satisfies assumption (\ref{mabc}) and
\begin{equation}
\label{energybc}
D^+_2(q,1)+E^+(v,1)\leq N<\infty.
\end{equation}
There exists a positive constant $\varepsilon<\frac 14$, depending only on $N$ and $M$ only, such that, if, for some $0<r\leq 1/2$,
\begin{equation}
\label{maincondscalebc}
\frac 1{r^3}|\{x\in B^+(r):\,|v(x,0)|>\frac \varepsilon r\}|\leq\varepsilon,
\end{equation}
then $v\in L_\infty(Q^+(\varepsilon r))$.
\end{pro}
The scale version of Proposition \ref{mainlemmabc} reads the following.
\begin{pro}
\label{mainlemmabcsc}
Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q^+(R)$ in the sense of Definition \ref{swsbc}. Assume that it satisfies assumption (\ref{mabc}) and
\begin{equation}
\label{energybcsc}
D^+_2(q,R)+E^+(v,R)\leq N<\infty.
\end{equation}
There exists a positive constant $\varepsilon<1/4$, depending only on $N$ and $M$ only, such that, if, for some $0<r\leq 1/2R$, inequality (\ref{maincondscalebc})
holds, then $v\in L_\infty(Q^+(\varepsilon r))$.
\end{pro}
\setcounter{equation}{0}
\section{Proof of Theorem \ref{final}}
Let us fix an arbitrary subdomain $\Omega_1\Subset\Omega$ and let $\delta ={\rm dist}(\Omega_1,\partial\Omega)>0$
and $2R_\star=\min(\delta/2,\sqrt T)$.
It is easy to verify that two inequalities
$$A(v,r;z_0)\leq c\|v\|^2_{L_\infty(t_0-r^2,t_0;L^{3,
\infty}(B(x_0,r))}\leq cM^2$$
and
$$K(v,r;z_0)\leq cM^2(E(v,r;z_0)+A(v,r;z_0)) $$
hold provided $Q(z_0,r)\subset Q_T$. Having those inequalities in hands and estimates for the energy scale invariant quantities proved in \cite{Seregin2006}, see Lemma 1.8, and in \cite{LadSer1999}, see Lemma 5.3, one can state that, for all $z_0=(x_0,T)$ with $x_0\in \Omega_1$, the following is true:
$$
\sup\limits_{0<r<R_\star}A(v,r;z_0)+\sup\limits
_{0<r<R_\star}C^\frac 34(v,r;z_0)+\sup\limits_{0<r<R_\star}E(v,r;z_0)+
$$
\begin{equation}
\label{scaleinv}+\sup\limits_{0<r<R_\star}K(v,r;z_0)+\sup\limits_{0<r<R_\star}D(q,r;z_0)
\end{equation}
$$\leq c(M)(D(q,R_\star;z_0)+E(v,R_\star;z_0)+1)\leq$$
$$\leq c(M,R_\star
\|\nabla v\|_{L_2(Q_T)},\|q\|_{L_\frac 32(Q_T)})=:N.
$$
The number $\varepsilon(M,N)$ of Proposition \ref{mainlemma} can be determined as numbers $M$ and $N$ are known.
Let $S$ be a set of all singular points of $v$ in
$\{(x_0,T):\,\, x_0\in \Omega_1\}$. Assume that it contains more than $M^3\varepsilon^{-4}$ elements. Letting $P=[ M^3\varepsilon^{-4}]+1$, one can find $P$ different singular points $(x_k,T)$, $k=1,2,...,P$, of the set $S$. Then, pick up a positive number $R<R_*$ such that $B(x_k,R)\cap B(x_l,R)=\emptyset$ if $k\neq l$, $k,l=1,2,...,P$. According to Proposition \ref{mainlemmasc},
for all $r\in ]0, 1/2R]$, the following should be true:
$$\varepsilon\leq \frac 1{r^3}|\{x\in B(x_k,r):|v(x,T)|>\frac \varepsilon r\}|$$
for all $k=1,2,...,P$. Now, we let $r=r_0=1/2R$ and, after summation over $k$, we arrive at the following inequality
$$P\varepsilon\leq \sum\limits_{k=1}^P\frac 1{r_0^3}|\{x\in B(x_k,r_0):|v(x,T)|>\frac \varepsilon {r_0}\}|=
$$
$$=\frac 1{r_0^3}|\{x\in\bigcup\limits^P_{k=1}B(x_k,r_0):|v(x,T)|>\frac\varepsilon{r_0}\}|\leq
$$$$
\leq \frac 1{r_0^3}|\{x\in \Omega:|v(x,T)|>\frac \varepsilon{r_0}\}|\leq \frac 1{\varepsilon^3}\|v(\cdot,T)\|^3_{L^{3,\infty}(\Omega)}\leq \frac {M^3}{\varepsilon^3}.
$$
The latter inequality implies that
$P\leq M^3\varepsilon^{-4}<P$. It is a contraduction. The theorem is proved.
\setcounter{equation}{0}
\section{Proof of Theorem \ref{finalbc}}
Let us first prove that the number of singular points of $v$ in the set $b(r_0)\times \{t=0\}$, where
$b(r_0)=\{x\in \mathbb R^3: \,x=x', |x'|\leq r_0\}$, is finite.
We let $2R_*=(1-r_0)/2$. Our further arguments are very similar to ones used in the previous chapter, see \cite{Seregin}. Indeed, for all space-time points $z_0=(x_0,0),$ where $x_0\in b(r_0)$, we have
$$
\sup\limits_{0<r<R_\star}A^+(v,r;z_0)+\sup\limits_{0<r<R_\star}C^+(v,r;z_0)+\sup\limits_{0<r<R_\star}E^+(v,r;z_0)+
$$
\begin{equation}
\label{scaleinvbc}+\sup\limits_{0<r<R_\star}K^+(v,r;z_0)+\sup\limits_{0<r<R_\star}D^+_2(q,r;z_0)
\end{equation}
$$\leq c(M)(D_2^+(q,R_\star;z_0)+E^+(v,R_\star;z_0)+1)\leq$$
$$\leq c(M,R_\star
\|\nabla v\|_{L_2(Q^+)},\|\nabla q\|_{L_{\frac {12}{11},\frac 32(Q^+)}})=:N.
$$
Now, having in hands number $M$ and $N$, we may find the number $\varepsilon$ of Proposition \ref{mainlemmabc}.
Let us denote the set of all singulars points of the form $z_0=(x_0,0)$ with $x_0\in b(r_0)$ by $S_b$. Then, repeating arguments of the proof of Theorem \ref{final} with half balls instead of balls, we show that the number of elements of $S_b$ is bounded by $M^3/\varepsilon^4$.
Now, it remains to establish that any singular point, belonging to a flat part of the boundary, cannot be the limit point of a sequence of singular points from the interior of a half ball. To this end,
we argue ad absurdum. Let $x_0=x'_0$ with $|x_0|\leq r_0$ be a singular point of $v$ and there exists a sequence $x^k$ such that $x^k\to x_0$ as $k\to \infty$ and $x^m_3>0$ for all $m$.
Without loss of generality, we may assume $0<x^m_3\leq R_*/2$ for all $m$. In this case, for all $x_*\in B^+(r_0)$ with $0<x_3\leq R_*/2$, the following is valid:
$$B(x_\star,R_\star)\cap \{x_{3\star}>0 \}\subset B^+(x'_*,2R_\star)\subset B^+.
$$
Denoting $z_*=(x_*,0)$ and $z'_*=(x'_*,0)$, observe that
$$\Theta^+(v,q,r;z'_*):=E^+(v,r;z'_*)+D^+_2(q,r;z'_*)+A^+(v,r;z'_*)+K^+(v,r;z'_*)+$$
$$+C^+(v,r;z'_*)\leq c(M)(E^+(v,2R_*;z'_*)+D^+_2(q,2R_*;z'_*)+$$$$+1)
=:C_1(M,R_*,\|\nabla v\|_{L_2(Q^+)},\|\nabla q\|_{L_{\frac {12}{11},\frac 32}(Q^+)}).$$
for all $0<r\leq R_*$.
It is easy to check that
$$E(v,x_{3*};z_*)+D_2(v,x_{3*};z_*)\leq c(E^+(v,2x_{*3};z'_*)+D^+_2(q,2x_{*3};z'_*))\leq $$
$$\leq cC_1$$
as $2x_{*3}\leq R_*$. Hence, the number $N$ is determined by the following inequality
$$E(v,x_{3*};z_*)+D_0(v,x_{3*};z_*)\leq cC_1=:N
$$
and one can find the number $\varepsilon(M,N)$ of Proposition \ref{mainlemma}.
Let us pick up $P$ different elements $x^{k_1}$, $x^{k_2}$,...,$x^{k_P}$ of the sequence $x^k$ assuming that
$$P>\frac {M^3}{\varepsilon^4}.$$
We let $\gamma=\min\{x^{k_1}_3$, $x^{k_2}_3$,...,$x^{k_P}_3\}>0$
and then select $0<R<\min\{\gamma, R_*/10\}$ so that
$B(x^{k_i},R)\cap B(x^{k_j},R)=\emptyset$ if $i\neq j$. Our further arguments are the same as in the proof of Theorem \ref{final}. Theorem \ref{finalbc} is proved.
\setcounter{equation}{0}
\section{Proof of Proposition \ref{mainlemma}}
We need an auxilary local regularity result, which is in fact a sufficient condition of regularity on one scale, see paper \cite{Seregin2016}.
\begin{pro}\label{mainresultinter} Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q(z_0,R)$.
Given $Z>0$, there exist positive numbers $\varepsilon_\star=\varepsilon_*(Z)$ and $c_*=c_*(Z)$ such that if two conditions
$$\frac 1{R^2}\int\limits_{Q(z_0,R)}|v|^3dxdt< \varepsilon_*(Z)$$
and
$$D_0(q,R;z_0)=\frac 1{R^2}\int\limits_{Q(z_0,R)}|q-[q]_{B(x_0,R)}|^\frac 32dxdt<Z$$
hold, then $v$ and $\nabla v$ are H\"older continuous is the closure of $Q(z_0,R/2)$. Moreover,
$$\sup\limits_{z\in Q(z_0,R/2)}|v(z)|+|\nabla v(z)|\leq \frac {c_*(Z)}R.$$
\end{pro}
As in paper \cite{CWY}, we argue as absurdum. Indeed, if we assume that the statement of Proposition \ref{mainlemma} is false, then, according to Proposition \ref{mainresultinter}, there are positive numbers $M$ and $N$ such that there exist a sequence of suitable weak solutions $v^k$ and $q^k$, sequences of numbers $0<r_k\leq 1/2$ and $\varepsilon_k\to +0$ with the following properties:
\begin{equation}
\label{wl3}
\sup\limits_k\|v^k\|_{L_\infty(-1,0;L^{3,\infty}(B))}\leq M;
\end{equation}
\begin{equation}
\label{startingpoint}
\sup\limits_kD_0(q^k,1)\leq \sup\limits_k(D_0(q^k,1)+E(v^k,1))\leq N;
\end{equation}
\begin{equation}
\label{maincondk}
\frac 1{r_k^3}|\{x\in B(r_k):\,\,|v^k(x,0)|>\frac {\varepsilon_k}{r_k}\}|\leq\varepsilon_k
\end{equation}
for all $k=1,2,...$;
\begin{equation}
\label{sing}
\frac 1{\varrho^2}\int\limits_{Q(\varrho)}|v^k|^3dz>\frac 12 \varepsilon_*(N)
\end{equation}
for all $\varrho\in [2\varepsilon_kr_k,r_k]$.
Moreover, the same arguments as in the proof of the main theorem lead to the inequality for energy scale invariant quantities:
\begin{equation}
\label{scale inv}
\Theta(v^k,q^k,r;z_0):= A(v^k,r;z_0)+C(v^k,r;z_0)+E(v^k,r;z_0)+
\end{equation}
$$+K(v^k,r;z_0)+D_0(q^k,r;z_0)\leq c(M)(N+1)$$
for all $0<r\leq 1/2$ and for all $z_0\in Q(1/2)$.
Now, our functions can be scaled in the following way:
$$u^k(y,s)=r_kv^k(x,t),\qquad p^k(y,s)=r^2_kq^k(x,t),$$
where $x=r_ky$, $t=r_k^2s$ and $e=(y,s)\in Q(1/r_k)$. New functions $u^k$ and $p^k$ satisfy the Navier-Stokes equations in $Q(1/r_k)$,
$$
\sup\limits_k(D_0(p^k,1/r_k
+E(u^k,1/r_k))<N,$$
and
$$\|u^k\|_{L_\infty(-1/r^2_k,0;L^{3,
\infty}(B(1/r_k))}\leq M.
$$
Without loss of generality, one may assume that $r_k\to r_*$ as $k\to \infty$. There are two case: $r_*=0$ and $r_*>0$.
Let us first consider the case $r_*=0$. Here, we can fix an arbitrary space-time point $e_0=(y_0,s_0)$, a number $0<R<\frac 12\frac 1{r_k}$ and make change of variables in (\ref{scale inv}) in order to get
\begin{equation}
\label{scaled scale inv}
\Theta(u^k,p^k,R;e_0)\leq c(M)(N+1).
\end{equation}
for sufficiently large $k$. Moreover, (\ref{maincondk}) and (\ref{sing}) can be transformed into the following:
\begin{equation}
\label{scalemaincond}
|\{y\in B:\,|u^k(y,0)|>\varepsilon_k\}|\leq\varepsilon_k
\end{equation}
and
\begin{equation}
\label{scalesing}
\frac 1{\varrho^2}\int\limits_{Q(\varrho)}|u^k|^3de>\frac 12 \varepsilon_*(N)
\end{equation}
for all $\varrho\in [2\varepsilon_k,1]$ and for all $k$.
Higher derivatives can be evaluated as in \cite{Seregin}. So,
\begin{equation}
\label{Higherder}
\Sigma(u^k,p^k,R):=
\frac 1R\Big[\|\partial_tu^k\|_{L_{\frac 98,\frac 32}(Q(R))}+\|\nabla^2 u^k\|_{L_{\frac 98,\frac 32}(Q(R))}+\end{equation}
$$
+\|\nabla p^k\|_{L_{\frac 98,\frac 32}(Q(R))}\Big]\leq c\Big[A^\frac 13(u^k,2R)E^\frac 23(u^k,2R)+A^\frac 12(u^k,2R)+$$
$$+E^\frac 12(u^k,2R)+D_0^\frac 23(p^k,2R)\Big]$$
for all $0<R<\frac 14\frac 1{r_k}$.
Now, let us pass to the limit as $k\to\infty$, taking into account estimates (\ref{scaled scale inv}) and (\ref{Higherder}). Then,
after using known compactness arguments, we get the so-called local energy ancient solution $u$ and $p$, having the following properties:
$$u^k\to u$$
in $L_3(Q(R))$ and in $C([-R^2,0];L_\frac 98(B(R)))$ for any $R>0$;
$$p^k\rightharpoonup p$$
in $L_\frac 32 (Q(R))$ for any $R>0$;
the pair $u$ and $p$ is a suitable weak solutionto the Navier-Stokes equations in each $Q(R)$;
$$\|u\|_{L_\infty(-\infty,0;L^{3,\infty}(\mathbb R^3))}\leq M;$$
$$\Theta(u,p,R;e_0)\leq c(M)(N+1)$$
for any $R>0$ and $e_0\in Q_-$;
$$\Sigma(u,p,R)\leq c(M,N)$$
for any $R>0$;
$$\frac 1{\varrho^2}\int\limits_{Q(\varrho)}|u|^3de\geq \frac 12 \varepsilon_*(N)$$
for any $\varrho\in ]0,1]$, and finally
\begin{equation}
\label{zeroinball} u(x,0)=0
\end{equation}
for any $x\in B$.
The latter identity follows from the known inequality
$$|\{y\in B: |u(y,0)|>\alpha\}|\leq |\{y\in B: |u^k(y,0)|>\alpha/2\}|+$$$$+|\{y\in B: |u^k(y,0)-u(y,0)|>\alpha/2\}|$$
that is valid for any $\alpha>0$.
Now, let us consider the case $r_*>0$. Our first remark is that (\ref{scaled scale inv}) remains to be true for all $e_0=(y_0,s_0)$ from the unit parabolic ball $ Q$ and for the same $R$. Moreover, relationships (\ref{scalemaincond})-(\ref{Higherder}) are completely the same as well. Repeating the same compactness arguments, we can easily pass to the limit as $k\to\infty$ and conclude that there exist functions $u$ and $p$ with the following properties:
the pair $u$ and $p$ is a suitable weak solutionto the Navier-Stokes equations in each $Q(1/4)$;
$$\|u\|_{L_\infty(-1/4^2,0;L^{3,\infty}(B(1/4))}\leq M;$$
$$E(u,1/4)+D_0(p,1/4)\leq c(M)(N+1);$$
$$\frac 1{\varrho^2}\int\limits_{Q(\varrho)}|u|^3de\geq \frac 12 \varepsilon_*(N)$$
for any $\varrho\in ]0,1/4]$, and finally
$u(y,0)=0
$
for any $y\in B(1/4)$.
Obviously, the restriction of $u$ and $p$ of the first case $r_*=0$ to the parabolic ball $Q(1/4)$ have the properties as above and in what follows we shall work with such a restriction.
The crucial point here is a reduction to backward uniqueness for the heat operator with lower order terms, see \cite{ESS2003}.
To this end, we
select a sequence of positive numbers $\varrho_k$, tending to zero. Then, the new scaling is:
$$U^k(y,s)=\varrho_ku(x,t),\qquad P^k(y,s)=\varrho^2_kp(x,t)$$
where $x=\varrho_ky$, $t=\varrho^2_ks$. Repeating arguments of the first part of the proof, we find the following relationships:
given $e_0\in Q_-$,
$$\Theta(U^k,P^k,R;e_0)+\Sigma(U^k,P^k,R/2)\leq c(M,N)$$
for any $0<R<1/(2\varrho_k)$ and for sufficiently large $k$;
$$\|U^k\|_{L_\infty(-1/\varrho_k^2,0;L^{3,\infty}(B(1/\varrho_k))}\leq M;$$
$$U^k(x,0)=0
$$
for all $x\in B(1/(4\varrho_k))$;
$$\frac 1{(\varrho/\varrho_k)^2}\int\limits_{Q(\varrho/\varrho_k)}|U^k(z)|^3dz\geq \frac 12\varepsilon_*(N)$$
for all $0<\varrho\leq1/4$.
Let $\varrho=\varrho_k$ and $k$ tend to infinity and let us see what happens.
The same arguments as in the first scaling lead to the following: there exists a local energy ancient solution $w$ with the associated pressure $r$ such that:
$$\Theta(w,r,\varrho;z_0)+\Sigma(w,r,\varrho)\leq c(M,N)$$
for any $\varrho>0$ and for any $z_0\in Q_-$;
$$\|w\|_{L_\infty(-\infty,0;L^{3,\infty}(\mathbb R^3))}\leq M;$$
$$w(x,0)=0
$$
for all $x\in\mathbb R^3$;
$
\int\limits_{
}|w|^3dz\geq \frac 12\varepsilon_*(N)>0.$$
In order to apply the approach based on the backward uniqueness, we need to show that solution $w$ has a certain decay at infinity, for example, to prove that $w$ and $\nabla w$ belong to $L_\infty((\mathbb R^3\setminus B(R))\times ]-2,0[)$ for some $R>0$.
Just for completeness, we repeat arguments from the paper \cite{CWY}. Indeed, by the definition of weak Lebesgue spaces, we find that
$$|\{(x,t)\in \mathbb R^3\times ]-3,0[:\,\,|w(x,t)|>\gamma\}|\leq \frac 3{\gamma^3}M^3<\infty.$$
Hence, for any positive number $\eta>0$, there exists $R=R(\gamma)>0$ such that
$$|\{(x,t)\in(\mathbb R^3\setminus B(R(\gamma)))\times ]-3,0[:\,\,|w(x,t)|>\gamma\}|<\eta.
$$
So, if $Q(z_0,1)\in (\mathbb R^3\setminus B(R(\gamma)))\times ]-3,0[$, then we have
$$D_0(r,1;z_0
\leq c(M,N)=Z $$
and
$$C(w,1;z_0)\leq \gamma^3|Q(z_0,1)|+\int\limits_{\{(x,t)\in Q(z_0,1):\,|w(x,t)|>\gamma\}}|w|^3dz\leq
$$
$$\leq c\gamma^3+\Big(\int\limits_{Q(z_0,1)}|w|^4dz\Big)^\frac 34|\{(x,t)\in Q(z_0,1):\,|w(x,t)|>\gamma\}|^\frac 14\leq $$
$$\leq c\gamma^3+K^\frac 34(w,1;z_0)\eta^\frac 14 \leq c\gamma^3+c(M,N)\eta^\frac 14.$$
We select first $\gamma$ and then $\eta$ so that the right hand side of the latter inequality is less that $\varepsilon_\star(Z)$. Then, one can conclude, see Proposition \ref{mainresultinter}, that, for any $z_0\in (\mathbb R^3\setminus B(\eta))\times ]-1,0[$,
$$|u(z_0)|+|\nabla u(z_0)|\leq c_\star(Z). $$
Now, using arguments of the paper \cite{ESS2003}, we show that $w\equiv 0$ in $\mathbb R^3\times ]-1,0[$. This is a contradiction. So, Proposition \ref{mainlemma} is proved.
\setcounter{equation}{0}
\section{Proof of Proposition \ref{mainlemmabc}}
Since our proof of the proposition is similar to the proof of Proposition \ref{mainlemma}, we just outline it. We start with
a certain boundary regularity condition, following the paper \cite{Seregin2016}.
\begin{pro}\label{mainresultbc} Let $v$ and $q$ be a suitable weak solution to the Navier-Stokes equations in $Q^+(z_0,R)$ in the sense of Definition \ref{swsbc}.
Given $Z>0$, there exist positive numbers $\varepsilon_\star=\varepsilon_*(Z)$ and $c_*=c_*(Z)$ such that if two conditions
$$\frac 1{R^2}\int\limits_{Q^+(z_0,R)}|v|^3dxdt< \varepsilon_*(Z)$$
and
$$\frac 1{R^2}\int\limits_{Q^+(z_0,R)}|q-[q]_{B^+(x_0,R)}|^\frac 32dxdt<Z$$
hold, then $v$ is H\"older continuous is the closure of $Q^+(z_0,R/2)$. Moreover,
$$\sup\limits_{z\in Q^+(z_0,R/2)}|v(z)|\leq \frac {c_*(Z)}R.$$
\end{pro}
Assume that Proposition \ref{mainlemmabc} is false. Then, there exist sequeneces $v^k$, $q^k$, $0<r_k\leq 1/2$, and $\varepsilon_k\to+0$ such that
$$ \sup\limits_kD_0^+(q^k,1)\leq c \sup\limits_kD_2^+(q^k,1)\leq cN,
$$
$$
\sup\limits_k\|v^k\|_{L_\infty(-1,0;L^{3,\infty}(B^+))}\leq M,
$$
$$
\frac 1{r_k^3}|\{x\in B^+(r_k):\,|v^k(x,0)|>\frac {\varepsilon_k}{r_k}\}|<\varepsilon_k,
$$
but
$$\frac 1{\varrho^2}\int\limits_{Q^+(\varrho)}|v^k|^3dz>\frac 12\varepsilon_*(cN)
$$
for all $\varrho\in [2\varrho_kr_k,r_k]$.
Then, we have a typical estimate of certain energy scale invariant quantities:
$$A^+(v^k,r;z_0)+C^+(v^k,r;z_0)+E^+(v^k,r;z_0)+K^+(v^k,r;z_0)+ $$
$$+D^+2(q^k,r;z_0)\leq C(M)(N+1)$$
for all $z_0\in Q^+(1/2)$ zuch that $z_0=(x_0,t_0)$ and $x_0=x_0'$.
We let $\omega(x_0,r)=B(x_0,r)\cap \mathbb R^3_+$ and $Q_\omega(z_0,r)=\omega(x_0,r)\times ]t_0-r^2,t_0[$.
Using scaling arguments, we get the main estimate
$$\Theta_\omega(v^k,q^k,r;z_0):=A_\omega(v^k,r;z_0)+C_\omega(v^k,r;z_0)+E_\omega(v^k,r;z_0)+$$$$+K_\omega(v^k,r;z_0)+D_{2\omega}(q^k,r;z_0) \leq c(M)(N+1)$$
for all $0<r\leq \frac 14$ and $z_0\in Q^+(1/4)$.
Next, we do scaling $u^k(y,s)=r_kv^k(x,s)$, $p^k(y,s)=r^2_kq^k(x,s)$, where $x=r_ky$, $t=r^2_ks$ and $e=(y,s)\in Q^+(1/r_k)$. Here, we are going to consider the only case in which $r_k\to0$ as $k\to \infty$, leaving the second case to the reader.
From the previous estimates, one can deduce the following:
$$\sup\limits_k(E^+(u^k,1/r_k)+D_2^+(p^k,1/r_k))\leq N.$$
Moreover, we fix $e_0=(y_0,s_0)$ and, for $0<R<1/(4r_k)$, find
$$\Theta_\omega(u^k,p^k,R;e_0)\leq c(M)(N+1),$$
$$|\{y\in B^+:\,\,|u^k(y,0)|>\varepsilon_k\}|<\varepsilon_k,$$
and
$$\frac 1{\varrho^2}\int\limits_{Q^+(\varrho)}|u^k|^3de>\frac 12\varepsilon_*(cN).$$
In order to provide compactness, higher derivatives are evaluated so that:
$$\Sigma^+(u^k,p^k,R)=\frac 1{R^{\frac {13}{12}}}[\|\partial_tu^k\|_{L_{\frac {12}{11},\frac 32,Q^+(R)}}+\|\nabla^2 u^k\|_{L_{\frac {12}{11},\frac 32,Q^+(R)}}+$$$$+\|\nabla p^k\|_{L_{\frac {12}{11},\frac 32,Q^+(R)}}]\leq $$
$$\leq c[D_0^+(p^k,2R)+ (C^+)^\frac 13(u^k,2R)+(E^+)^\frac 12(u^k,2R)+$$
$$+(A^+)^\frac 14(u^k,2R)(E^+)^\frac 12(u^k,2R) (C^+)^\frac 16(u^k,2R)]
$$
for $0<R<1/(8r_k)$.
Passing to the limit as $k\to\infty$, we have (without loss of generality) the following:
$u^k\to u$ in $L_3(Q^+(R))$ and in $C([-R^2,0];L_\frac {12}{11}(B^+(R)))$ and
$p^k\rightharpoonup p$ in $L_\frac 32(Q^+(R))$ for all $R>0$. Let us list the properies of the limit pair $u$ and $p$. It is a suitable weak solution to the Navier-Stokes equations in each $Q^+(R)$;
$$\|u\|_{L_\infty(-\infty,0;L^{3,\infty}(\mathbb R^3_+))}\leq M;$$
$$\Theta_\omega(u,p,R;e_0)\leq c(M)(N+1)$$
for all $R>0$ and for all $e_0\in Q^+_-:=\mathbb R^3_+\times ]-\infty,0[$;
$$\Sigma^+(u,p,R)\leq c(M,N)$$
for all positive $R$;
$$\frac 1{\varrho^2}\int\limits_{Q^+(\varrho)}|u|^3dz\geq\frac 12\varepsilon_*(cN)$$
for $0<\varrho\leq 1$;
$$u(x,0)=0$$
for $x\in B^+(0)$.
Our further arguments are the same in the proof of Proposition \ref{mainlemma}. We need to adopt the last part of the proof based on the backward uniqueness since the boundary local regularity, in general, does not provide boundedness of $\nabla u$ up to the boundary. To this end, let us fix a positive number $h$ and apply the interior boundary regularity result (as we did in the previous section) in order to show that $\nabla u$ is bounded in $(R^3_++he_3)\setminus B(R_0)$ for a large number $R_0$. Then we can use known arguments, based on the backward uniqueness and the unique continuation through spatial boundaries. This implies $u\equiv 0$ in $R^3_++he_3$ for all $h>0$ and thus we arrive at the contardiction. The proposition is proved.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 6,730
|
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When to Revise Your Life Map
Martha Beck's 3-Step Plan for Navigating the Rocky Road of Life
Going through a rough patch? Here's how to know if it's destiny steering you in another direction.
By Martha Beck Published: Apr 30, 2022
aire images//Getty Images
Martha Beck is the bestselling author of The Way of Integrity and host of the podcast Bewildered. This article first appeared as "Off the Beaten Path" in the February 2013 issue of O, The Oprah Magazine.
"What is happening to my life?" said Dorothy, exhaustedly sipping a triple espresso across the table from me. "Did I do something to deserve this?"
By "this," Dorothy meant a series of crises that had recently hit her like a gang of prizefighters. Her husband had filed for divorce—a week after she lost her job, the same day she was diagnosed with diabetes. Then her best friend moved away. Now Dorothy was caring for both her aging parents while paying a divorce lawyer way more than she (or her retirement account) could afford. "I'm not sure I can go on," she told me. "Why is all this happening at once?"
"Well," I said, "according to probability theory, random events can run in streaks. It's like patterned disorder, and in nature it creates beautiful things."
Dorothy looked as though I'd poured mouse droppings into her coffee. "That's your explanation? My screwed-up life is just beautifully random?"
"It's the most rational explanation," I said. "It's not my explanation."
"What is?"
I shrugged. "I think you've hit a rumble strip."
Then I laid out for Dorothy what I'll now lay out for you, just in case your own current luck makes Job look like a lottery winner. I don't know why catastrophes sometimes come in clusters. But experience and observation have convinced me that these patches of awfulness may be purposeful and, in the end, benevolent. If you've had a run of horrible luck, you can tell yourself you're being tortured or punished. Or you can decide you're being steered.
Penguin Life The Way of Integrity: Finding the Path to Your True Self
Imagine that your true self is your essential consciousness, the part of you that still feels what it was like to be you ten years ago, even though most of the atoms in your physical body have been replaced since then. Suppose you set out to experience the adventure of human life by inhabiting your body. And that this essential you sees your life as an epic road trip. Destination: inner wisdom, love, and joy.
Now let's suppose you forgot this destiny at birth. In its place you created a mental map of the life route you preferred—passing through good health, perfect romance, and professional success on the way to a cheery, painless death (say, being struck by a meteorite while bicycling at the age of 110).
Unfortunately, your essential self very probably has in mind a stranger and more exciting road, featuring spooky tunnels, scary precipices, and sharp curves. Which means your destiny isn't at all what you think you want. Which means that as you drive along the road of life, there will be times when your essential self plans to turn even though you most certainly do not.
More From Oprah Daily
Behold the Rumble Strip
If you're paying attention to your environment, relaxing and following the road, detours from your mental map may be unnerving but not catastrophic. Maybe you planned to become a dentist and marry your high school boyfriend, only to realize that (1) you hate staring into other people's mouths, and (2) you actually prefer women. So you quit dental school, break up with Mr. Wrong, and find work and love that suit your innate preferences.
Or not. This is a best-case scenario, and such scenarios virtually never happen.
What virtually always happens is that when destiny swerves, we proceed straight ahead. We step on the gas, ignoring the fact that we feel trapped in the dead relationship, stifled by the secure job. We go blind to the landscape and the road signs, steering by our assumptions about what life should be, as unaware of those assumptions as a sleeping driver is of her unconsciousness.
Et voilà: rumble strip.
Suddenly, everything's shaking, jolting, falling apart. We have no idea what's happening or why, only that all hell has broken loose. It gets worse and worse—until we wake up, see through our false assumptions to the deeper truth of our situation, and revise our life maps. This isn't punishment. It's enlightenment dressed as chaos.
My Rumble Strip
I hit my first rumble strip while driving hell-for-leather toward my third Harvard degree. In six memorable months, I was almost killed in a car accident, in a high-rise fire, and by a violent autoimmune reaction to an accidental pregnancy. I had incessant nausea. And fibromyalgia. And lice. By the time the baby was diagnosed with Down syndrome, I was pretty much done for.
It took all that to shatter my core assumption: that achievement and intellect gave my life its value. Only after my world seemed to completely fall apart did I learn the lesson my true self needed me to learn: that no brass ring is worth a damn compared with the one thing that makes life worth living—love. Duh. You'd think I'd have figured that out earlier. There were signs absolutely everywhere. But until my first rumble strip shook me awake, I never even noticed them.
I've had other streaks of awful "luck" since, but none has ever caused as much suffering. That's because I've developed a rumble-strip coping strategy. If your own luck seems weirdly cursed, try this:
Navigating Rumble Strips
STEP 1: Hit the brakes.
When Dorothy told me over coffee that she wasn't sure she could go on, I secretly rejoiced—not because I wanted her to suffer, but because I didn't.
"Yup," I said, trying not to sound smug. "The rumble strip is telling you to stop."
"Stop what?"
"Everything," I told her. "Except what's necessary to survive. Eat. Sleep. Go to the bathroom. Make sure your children, pets, and sick parents eat, sleep, and go to the bathroom. If that's beyond you, ask for help. Not forever. Just for now."
This time Dorothy looked as though I'd asked her to stab a baby panda, but she was too exhausted to argue. That was a good thing. When you feel so beaten down that you can't sustain normal activities, it's time to stop trying. Surrender, Dorothy.
STEP 2: Put your mind in reverse.
From a place of minimal functioning, you can back off the rumble strip—by reversing the assumptions that steered you onto it in the first place. These key assumptions are clearly marked with intense negative emotions: fear, anger, sadness. Such feelings are big red WRONG WAY signs. Back away from them.
To help Dorothy do this, I asked her which, of all her tribulations, was causing her the most pain. Topping her very long list was the thought "My marriage has failed." So that's where we began shifting Dorothy's mind into reverse.
"Give me three reasons your marriage actually didn't fail," I said.
"But it did!" Dorothy muffled a sob.
"Well, was any part of it good?"
"Yes. Of course."
"Did you learn from it?"
"I learned so much," said Dorothy.
"And is every learning experience that comes to an end a failure?" I asked. "Like school, or childhood, or life?"
"Well, no."
Dorothy paused, thinking. Then her shoulders relaxed just a little. Ta-da! She'd begun reversing a painful assumption.
To be clear, I wasn't trying to minimize Dorothy's pain or plaster a creepy happy face over her legitimate sorrow. I only wanted her to alter her beliefs enough to catch a glimpse of a different road, where a marriage could succeed as a soul adventure even if it didn't last forever.
Try throwing your mind into reverse right now. Think of the worst, most hurtful thing that's happening in your life. Now think of a way this horrible thing might be good. The more rigidly you hold on to your assumptions, the harder this process will be. But with practice you'll improve—and trust me, it's so worth the effort. When life gets rumbly, being able to reverse an assumption turns out to be the handiest skill imaginable.
STEP 3: Find and follow smooth terrain.
Because rumble strips are one of the few experiences that will make sensible people hire a life coach, I've been privy to hundreds of them. And I've noticed a very consistent pattern: At the point when someone sees through a false assumption, the road of life suddenly turns smooth. Instead of crazy bad luck, bits of strangely good luck start showing up. They're small at first, inconspicuous. Never mind—slather them with attention. Your attention is what steers your life, and it's much more pleasant to steer by focusing on the good stuff.
In Dorothy's case, the moment she reversed her assumption that divorce always means failure, the waitress brought her a cupcake, said, "On the house," and walked away. Later that afternoon, Dorothy found an abandoned New York Times unfolded to an article titled "The Good Divorce," which helped and encouraged her. Then she ran into a former boyfriend she hadn't seen in years. During their brief interaction, he told her how much he still respected her, and how valuable their "failed" relationship still was to him.
Little miracles like this will begin happening to you whenever you turn toward your right life, even if you're in the middle of a rumble strip. If you stop everything you think you should be doing, surrender to what's actually happening, reverse your assumptions, and steer toward the glimmers of light that appear as your old beliefs shatter, the small miracles will turn into big ones. Eventually, your good luck will seem as incredible and mysterious as your bad. Once more you'll be asking, "Did I do something to deserve this?" Only this time, the question will arise from a sense of overwhelming gratitude, not overwhelming pain.
By the way, the answer to that question is yes. You did do something to deserve this. You had the courage to keep traveling the precarious road of life. You deserve to be guided. And rewarded. And, when all else fails, rumbled.
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{"url":"https:\/\/cs.stackexchange.com\/questions\/44203\/are-there-any-non-naive-parallel-sparse-matrix-multiplication-algorithms","text":"# Are there any non-naive parallel sparse matrix multiplication algorithms?\n\nI was wondering about a problem in analyzing a social network (counting friends-in-common between all pairs of members) that requires squaring its adjacency matrix, and started reading up on algorithms for multiplying sparse matrices.\n\nHowever, all I found so far were different ways of arranging the more or less naive \"outer product\" algorithm between processors - the same total number of multiplications\/additions with different amounts of communication and additional algorithmic overhead (which, though, is undoubtedly important).\n\nThe most non-trivial algorithm I found was the Yuster-Zwick algorithm described in Fast sparse matrix multiplication, which is basically a combination of the same old naive algorithm and using a fast dense method for the dense part of the problem.\n\nI looked at how sparse matrix multiplication is implemented in MLLib and it, too, appears to use the simple block-based algorithm.\n\nAre there any parallel algorithms for multiplying sparse matrices that are substantially different from the naive one - as different as Strassen's or Coppersmith-Winograd's algorithms are from the naive algorithm for multiplying dense matrices?\n\nFor concreteness, let us assume that the matrices are sparse enough that number of non-zeros in the arguments and in the result are both $O(N)$.\n\n\u2022 Googling \"sparse matrix multiplication parallel\" reveals many hits. Jul 7, 2015 at 7:12\n\u2022 I googled that a lot, but found only what I described in the post. I may have missed or misunderstood something. Do you have a particular result in mind?\n\u2013\u00a0jkff\nJul 7, 2015 at 13:47\n\nThis recent paper proposes a different approach, based on hypergraph partitioning.\n\nGrey Ballard, Alex Druinsky, Nicholas Knight, and Oded Schwartz. 2015. Brief Announcement: Hypergraph Partitioning for Parallel Sparse Matrix-Matrix Multiplication. In Proceedings of the 27th ACM on Symposium on Parallelism in Algorithms and Architectures (SPAA '15). ACM, New York, NY, USA, 86-88. DOI=10.1145\/2755573.2755613 http:\/\/doi.acm.org\/10.1145\/2755573.2755613","date":"2022-09-27 21:33:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8196874260902405, \"perplexity\": 1106.2868311577295}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030335058.80\/warc\/CC-MAIN-20220927194248-20220927224248-00588.warc.gz\"}"}
| null | null |
Home / Jirair Injeyan
Jirair Injeyan
(April 6, 1926 – August 7, 2006)
Jirair Injeyan was born in Aleppo, Syria, where he attended Haigazian Armenian Elementary School and completed his high school at the French Lycee. In 1946, he moved to Paris to further his studies. He obtained his Ph.D. in Physics at Sorbonne and his electronic engineering degree at L'ecole Superieure d'Electricite. He married Evelyne Ajamian in 1953 and moved to Canada in 1956.
Jirair was one of the founding pioneers and strong supporter of the Toronto Armenian Community. He was especially involved in Hamazkayin. He had a particular passion for literature and theatre and spent many years promoting Hamazkayin theatrical productions in various capacities as actor, producer, director, and translator of French plays into Armenian.
In Canada, while employed in General Electric's research department in the radar communications field, he discovered formulas and codes which bear his name today.
He was a loving father to Nadia and Garo Injeyan and doting grandfather to Gassia and Garni Tatikian and Maria Injeyan. Jirair passed August 7, 2006 unexpectedly of a heart attach but is remembered and missed by all especially for his infectious laughter and sense of humour.
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<?php
namespace MwbExporter\Formatter\Doctrine2;
use MwbExporter\Formatter\Formatter as BaseFormatter;
use MwbExporter\Validator\ChoiceValidator;
use MwbExporter\Validator\Validator;
abstract class Formatter extends BaseFormatter
{
const CFG_BUNDLE_NAMESPACE = 'bundleNamespace';
const CFG_ENTITY_NAMESPACE = 'entityNamespace';
const CFG_REPOSITORY_NAMESPACE = 'repositoryNamespace';
const CFG_AUTOMATIC_REPOSITORY = 'useAutomaticRepository';
const CFG_SKIP_COLUMN_WITH_RELATION = 'skipColumnWithRelation';
const CFG_RELATED_VAR_NAME_FORMAT = 'relatedVarNameFormat';
const CFG_NULLABLE_ATTRIBUTE = 'nullableAttribute';
const CFG_GENERATED_VALUE_STRATEGY = 'generatedValueStrategy';
const CFG_DEFAULT_CASCADE = 'defaultCascade';
const CFG_PREFIX_TABLENAME = 'prefixTablename';
const NULLABLE_AUTO = 'auto';
const NULLABLE_ALWAYS = 'always';
const GENERATED_VALUE_AUTO = 'auto';
const GENERATED_VALUE_IDENTITY = 'identity';
const GENERATED_VALUE_SEQUENCE = 'sequence';
const GENERATED_VALUE_TABLE = 'table';
const GENERATED_VALUE_NONE = 'none';
const CASCADE_OPTION_PERSIST = 'persist';
const CASCADE_OPTION_REMOVE = 'remove';
const CASCADE_OPTION_MERGE = 'merge';
const CASCADE_OPTION_DETACH = 'detach';
const CASCADE_OPTION_ALL = 'all';
const CASCADE_OPTION_REFRESH = 'refresh';
protected function init()
{
parent::init();
$this->addConfigurations(array(
static::CFG_BUNDLE_NAMESPACE => '',
static::CFG_PREFIX_TABLENAME => '',
static::CFG_ENTITY_NAMESPACE => '',
static::CFG_REPOSITORY_NAMESPACE => '',
static::CFG_AUTOMATIC_REPOSITORY => true,
static::CFG_SKIP_COLUMN_WITH_RELATION => false,
static::CFG_RELATED_VAR_NAME_FORMAT => '%name%%related%',
static::CFG_NULLABLE_ATTRIBUTE => static::NULLABLE_AUTO,
static::CFG_GENERATED_VALUE_STRATEGY => static::GENERATED_VALUE_AUTO,
static::CFG_DEFAULT_CASCADE => false,
));
$this->addValidators(array(
static::CFG_NULLABLE_ATTRIBUTE => new ChoiceValidator(array(static::NULLABLE_AUTO, static::NULLABLE_ALWAYS)),
static::CFG_GENERATED_VALUE_STRATEGY => new ChoiceValidator(array(
static::GENERATED_VALUE_AUTO,
static::GENERATED_VALUE_IDENTITY,
static::GENERATED_VALUE_SEQUENCE,
static::GENERATED_VALUE_TABLE,
static::GENERATED_VALUE_NONE,
)),
static::CFG_DEFAULT_CASCADE => new ChoiceValidator(array(
static::CASCADE_OPTION_PERSIST,
static::CASCADE_OPTION_REMOVE,
static::CASCADE_OPTION_DETACH,
static::CASCADE_OPTION_MERGE,
static::CASCADE_OPTION_ALL,
static::CASCADE_OPTION_REFRESH,
false
)),
));
}
/**
* (non-PHPdoc)
* @see \MwbExporter\Formatter\Formatter::createDatatypeConverter()
*/
protected function createDatatypeConverter()
{
return new DatatypeConverter();
}
/**
* Get owning side of relation.
*
* @param array $relation
* @param \MwbExporter\Model\ForeignKey $mappedRelation
* @return boolean
*/
public function isOwningSide($relation, &$mappedRelation)
{
$mappedRelation = $relation['reference']->getOwningTable()->getRelationToTable($relation['refTable']->getRawTableName());
// user can hint which side is the owning side (set d:owningSide on the foreign key)
if ($relation['reference']->parseComment('owningSide') === 'true') {
return true;
}
if ($mappedRelation->parseComment('owningSide') === 'true') {
return false;
}
// if no owning side is defined, use one side randomly as owning side (the one where the column id is lower)
return $relation['reference']->getLocal()->getId() < $mappedRelation->getLocal()->getId();
}
/**
* get the cascade option as array. Only returns values allowed by Doctrine.
*
* @param $cascadeValue string cascade options separated by comma
* @return array array with the values or null, if no cascade values are available
*/
public function getCascadeOption($cascadeValue)
{
$defaultCascade = $this->getRegistry()->config->get(static::CFG_DEFAULT_CASCADE);
if (empty($cascadeValue) && !empty($defaultCascade)) {
return [$defaultCascade];
}
/** @var Validator $validator */
$validator = $this->getRegistry()->validator->get(static::CFG_DEFAULT_CASCADE);
$cascadeValue = array_map('strtolower', array_map('trim', explode(',', $cascadeValue)));
$cascadeValue = array_intersect($cascadeValue, $validator->getChoices());
$cascadeValue = array_filter($cascadeValue);
if(empty($cascadeValue)) {
return null;
}
return $cascadeValue;
}
/**
* Parse order option.
*
* @param string $sortValue
* @return array
*/
public function getOrderOption($sortValue)
{
$orders = array();
if ($sortValue = trim($sortValue)) {
$lines = array_map('trim', explode("\n", $sortValue));
foreach ($lines as $line) {
if (count($values = array_map('trim', explode(',', $line)))) {
$column = $values[0];
$order = (count($values) > 1) ? strtoupper($values[1]) : null;
if (!in_array($order, array('ASC', 'DESC'))) {
$order = 'ASC';
}
$orders[$column] = $order;
}
}
}
return $orders;
}
/**
* get the fetch option for a relation
*
* @param $fetchValue string fetch option as given in comment for foreign key
* @return string valid fetch value or null
*/
public function getFetchOption($fetchValue)
{
if ($fetchValue) {
$fetchValue = strtoupper($fetchValue);
if (in_array($fetchValue, array('EAGER', 'LAZY', 'EXTRA_LAZY'))) {
return $fetchValue;
}
}
}
/**
* get the a boolean option for a relation
*
* @param $booleanValue string boolean option (true or false)
* @return boolean or null, if booleanValue was invalid
*/
public function getBooleanOption($booleanValue)
{
if ($booleanValue) {
switch (strtolower($booleanValue)) {
case 'true':
return true;
case 'false':
return false;
}
}
}
/**
* get the onDelete rule. this will set the database level ON DELETE and can be set
* to CASCADE or SET NULL. Do not confuse this with the Doctrine-level cascade rules.
*/
public function getDeleteRule($deleteRule)
{
if ($deleteRule == 'NO ACTION' || $deleteRule == 'RESTRICT' || empty($deleteRule)) {
// NO ACTION acts the same as RESTRICT,
// RESTRICT is the default
// http://dev.mysql.com/doc/refman/5.5/en/innodb-foreign-key-constraints.html
$deleteRule = null;
}
return $deleteRule;
}
/**
* (non-PHPdoc)
* @see \MwbExporter\Formatter\Formatter::getCommentTagPrefixes()
*/
protected function getCommentTagPrefixes()
{
return array_merge(parent::getCommentTagPrefixes(), array('d', 'doctrine'));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 9,476
|
Q: Capacitor ESR Calculation I'm trying to perform esr calculations of this capacitor(1uF):
https://alconelectronics.com/wp-content/uploads/2020/06/KP-6-JUNE-2019.pdf
to do that I need to know dissipation factor and at the bottom of the first page there is a dissipation factor-frequency graph, I'm going to use cap. at 60kHz and I choose dissipation factor as 10^-4 because the value is not clear.
When I calculated the "esr = dissipation factor / 2pif*C"
esr=0.00026 Ohms in my situation which is too small in my opinion.
and when I calculate esr zero frequency: "1 / 2piR*C"
fesrzero= more than 600MHz which look weird to me.
Am I doing the calculations right, I mean the values are unfamiliar to me and I'm not sure about it
source: https://forum.digikey.com/t/calculating-capacitor-esr-from-tan/2633
A: When I look at the graph in the datasheet i see a dissibation factor value of approximately 2.5e-4 at 60kHz which leads to a ESR of 0.66m Ohms.
This is still less than the max. ESR of 5-8mOhms, but the typical value is not the same as the max value.
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{"url":"http:\/\/mathoverflow.net\/questions\/107616\/chern-number-of-a-sphere","text":"# Chern number of a sphere\n\nHi everybody. I think I get a problem with the definitions of the connections 1-form of a vector bundle.\n\nLet's consider the sphere $S^2$ with its tangent bundle as a vector bundle. Let's take a tangent vector field $A$ regular on the sphere and construct using local patches these connections 1-forms:\n\n$\\omega^{\\alpha}_{\\beta}=$ $\\delta^{{\\alpha},{\\beta}}\\sum_j A_jdx^j$,\n\nwhere $\\delta^{{\\alpha},{\\beta}}$ is the Kronecker delta. I supposed that the vector field is regular and defined in the whole sphere, so the connection 1-forms do vanish in a certain point, because of the hairy ball theorem. Is it a problem? Why? I don't find in the definitions that the connections 1- form can't be zero...\n\nAnyway from these connections we can construct the curvature 2-form and the first Chern number integrating that curvature. But the 2-form to integrate in ordero obtain the first Chern number here is essentially an exact form ($\\Omega$=$dA$) and so the integration through the compact surface is zero.\n\nBut the first Chern number of these vector bundle should not be zero...\n\n-\nConnection 1-forms do not transform in the same way as ordinary 1-forms, so the local expressions you have written do not patch up to a well-defined connection. Otherwise you could just set all the connection 1-forms to zero and get a flat connection on any vector bundle. \u2013\u00a0 Johannes Nordstr\u00f6m Sep 20 '12 at 9:33\n\nThank you very much. I'm still not able to check that my definitions don't patch, but I'll try... Anyway I wanted to ask a related question that is: what do we need to talk about \"Chern numbers\"?\n\nI'm a bit confused. Somewhere I read that I need a principal G-bundle. In that case I could consider (having base manyfold $S^2$) the frame bundle with the group $GL(2,R)$ acting on it?\n\nSomewhere else (this is wikipedia) I found that the object to be considered is an Hermitian complex vector bundle. In this case I could consider the vector bundle of complexified tangent spaces of the sphere?\n\nAre these cases both good and well defined? The Chern number can differ?\n\n-\nThis is not the appropriate forum. Try math.stackexchange.com \u2013\u00a0 Chris Gerig Sep 21 '12 at 17:58\nI'm not sure that I agree that this needs to shunted off to stackexchange. However, it is bad form to ask a question in this way. If you want to ask another question, you should ask a new question, and it should be more precisely formulated than this. \u2013\u00a0 Donu Arapura Sep 21 '12 at 18:25\nok... maybe I'm not in the right forum for asking more clarifications. If somebody wants to help, I've posted where Gerig suggested: math.stackexchange.com\/questions\/200657\/\u2026 \u2013\u00a0 ShortEdge Sep 22 '12 at 10:57","date":"2015-04-19 23:04:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8933709263801575, \"perplexity\": 380.9488537579764}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1429246642012.28\/warc\/CC-MAIN-20150417045722-00173-ip-10-235-10-82.ec2.internal.warc.gz\"}"}
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put up with! 'Ghost' hopes to borrow 'Pulisic', reserve wing 'Sing', use 1 season
Manchester United are still looking for players to strengthen before the transfer window closes. By hoping to borrow Christian Pulisic from Chelsea's reserve. But will have to compete with Atletico Madrid, Juventus and Newcastle who are also interested.
The Athletic , reliable media in the elite English Premier League team Manchester United are reportedly interested in loaning United States winger Christian Pulisic to Chelsea's league rivals. Come dragged to the place of Old Trafford during the opening of the first round of players. After the player himself was unable to secure a starting position at Stamford Bridge.
The 23-year-old has also attracted interest from Atletico Madrid Juventus and Newcastle United ,has failed to prove himself since leaving the Borussia d'Or. Dortmund joined the Blues for a fee of 58 million pounds or about 2,485 million baht 3 years ago. Mostly playing a role as a backup. along with injury problems
While United , under the supervision of a new coach, Erik ten Hag, has not been able to increase the efficiency of attacking opponents satisfactorily. The first two games of the season were defeated in the nest against Brighton 1-2 and defeated in a shambles against Brentford 0-4, which the goal was scored from the opponent's own goal. Indicates a lack of sharpness in the offensive game.
For this reason, the Dutch Grandmaster So hoped to negotiate to borrow. The American midfielder went to work to fix problems in the front and finish off the opponent. While the players themselves are ready to move the team as well. To have the opportunity to regularly play on the field. Before going to serve Uncle Sam's team for the 2022 World Cup final in Qatar at the end of this year.
According to reports. The navy blue lion Welcome to the offer of Pulisic has two years left on his current contract by German coach Thomas Tuchel. Currently improving the efficiency of their team's offensive line. Because in the past. Often used quite a waste of opportunity. But we'll have to wait and see. They will send their players to rival clubs in the league. Or will let the UFABET team in another league on loan
Tagged with: football, Newcastle United, Premier League
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Produced by Chuck Greif, Donald Cummings, Adrian Mastronardi
and the Online Distributed Proofreading Team at
http://www.pgdp.net (This file was produced from images
generously made available by The Internet Archive/American
Libraries.)
SPAIN FROM WITHIN
[Illustration: KING ALFONSO XIII. IN HIS STUDY IN MADRID.
[Frontispiece.]
SPAIN FROM WITHIN
BY
RAFAEL SHAW
NEW YORK
FREDERICK A. STOKES COMPANY
PUBLISHERS
1910
(_All rights reserved._)
"For behold! this monstrous twenty million class,
hitherto the dumb sheep which these others had to
agree about the manner of shearing, is now also
arising with hopes."
--CARLYLE, "French Revolution."
PREFATORY NOTE
"Truth is an exile from our political world. Every faction and
every group tells only that part of the truth which reflects
discredit on its neighbour. Thus our political literature is
interesting only as an archive of monstrosities. The other part of
the truth--that which deals with the good qualities of the
neighbour--is so out of fashion that nobody believes in its
existence. To tell the truth to our politicians would be the
greatest proof of friendship that could be offered to them. But who
has sufficient courage to attempt it? No one in this world would
venture upon so difficult, so disagreeable, and so dangerous a
task. The result is that in Spain the only respect paid to the
truth is to leave it unspoken."--NUEVO MUNDO, Madrid, Feb. 24,
1910.
In the following pages I have endeavoured to show what the people of
Spain believe to be the truth about those who exercise authority over
them, as gathered from conversation with Spaniards of all classes, but
principally working people, in town and country, and from my own reading
and observation. Whether my informants are right or wrong in their
opinions and beliefs I do not pretend to decide; all I can declare is
that I have faithfully reported what I have heard and seen. The
importance of the opinions I have collected lies in the fact that,
whether they are justified or not, _the people believe them to be true_,
and on that belief they will assuredly act as soon as circumstances
allow.
I have to acknowledge with thanks the courteous permission of the
Editors of the _Spectator_ and the _Standard_ to incorporate in this
volume the gist of various articles, notes, &c., which first appeared in
their pages. The author and publisher have also to express their thanks
to the Editors and Proprietors of the _Nuevo Mundo_ of Madrid for
permission to use the illustrations in this book, which are taken from
that periodical.
RAFAEL SHAW.
CONTENTS
CHAP. PAGE
INTRODUCTION 13
I. RACIAL AND CLASS DISTINCTIONS 23
II. THE RELIGION OF THE PEOPLE 39
III. MORALITY AND CEREMONIAL 61
IV. THE CONFESSIONAL, AND CHURCH ABUSES 73
V. THE POOR AND THE RELIGIOUS ORDERS 89
VI. THE MONARCHY AND THE PEOPLE 111
VII. THE REVIVAL OF CARLISM 133
VIII. THE CHURCH MILITANT 159
IX. BARCELONA AND THE LAY SCHOOLS 181
X. THE ARMY, PAST AND PRESENT 199
XI. THE POLICE 215
XII. POLITICS 227
XIII. POLITICAL PARTIES 251
XIV. EDUCATION 263
XV. TAXATION 285
XVI. THE PROCESS OF REGENERATION 303
APPENDIX. NOTES ON POLITICIANS AND PERIODICALS 319
INDEX 325
ILLUSTRATIONS
KING ALFONSO XIII. IN HIS STUDY IN MADRID _Frontispiece_
FACING PAGE
FACTORY GIRLS 14
PEASANT WOMEN 39
NEWSPAPER SELLERS AT THE OFFICES OF THE _NUEVO
MUNDO_ IN MADRID 61
THE QUEEN AND THE QUEEN-MOTHER OF SPAIN 111
SEÑOR MAURA, LEADER OF THE ULTRAMONTANES 149
DON JAIME OF BOURBON IN MOROCCO 153
A DEMONSTRATION OF REJOICING AT THE FALL OF THE
ULTRAMONTANE MINISTRY 174
A CONSCRIPT 199
THE WAR IN MELILLA. A FORT ON MOUNT GURUGU 203
A RESERVIST AT THE FRONT 208
DON SEGISMUNDO MORET, LEADER OF THE LIBERAL-MONARCHISTS 227
SEÑOR CANALEJAS, LEADER OF THE LIBERAL DEMOCRATS,
AND GENERAL MARINA, COMMANDER-IN-CHIEF AT
MELILLA 244
A STREET HAWKER DESCRIBING BATTLE SCENES TO AN
ILLITERATE AUDIENCE 263
SAFFRON PICKERS SORTING THEIR CROP 285
A SELLER OF PALM-LEAF BRUSHES AND FANS 289
INTRODUCTION
While a good deal has been written of late years about Spain from the
point of view put forward by the governing classes, little or nothing
has been said about the people--the mass of the nation--who, unable, the
immense majority of them, to read or write, are more inarticulate than
their fellows in any country of Europe west of Russia, but who have,
nevertheless, very definite aspirations and ideals, entirely distinct
from those of their rulers, at whose hands, disheartened as they are by
long years of misgovernment, they have almost abandoned any hope of
amelioration of their lot.
Circumstances have afforded the writer opportunities of seeing a great
deal of the inner life of the people, and of learning what are the
grievances, the aspirations, and the desires of the Spanish working
classes, gathered from conversation with them, and from years of close
personal observation.
Generalisations about an entire nation are usually of doubtful value;
still, it is safe to say that the Spaniard of the working classes is not
the turbulent rascal he is so often depicted, who in the intervals of
_pronunciamentos_ and civil wars occupies his leisure moments in
"holding up" the wayfarer with a blunderbuss. On the contrary, he is a
quiet, industrious, law-abiding citizen, whose chief desire is to be
left to go about his business and make a living for himself and his
family. If he has to fight he fights well, for he does not lack courage,
and he has often been compelled to fight for causes in which he takes no
interest, as the alternative to losing the employment which stands
between him and starvation. But he does not want to fight, because he is
convinced that all Spain's wars, whatever their ostensible object, are
arranged by his "betters" to put money into their own pockets,
regardless of the true interests of the nation. You may talk as you will
about the wealth, health, and happiness that might be obtained, say, in
Melilla, should it become a well-administered colony of Spain. The
Spanish working man has an invariable reply to all such suggestions. He
says: "That might be so under other Governments, but not under ours.
Look at Cuba!"
Emigration goes on to an extent which causes the gravest apprehension to
those who have
[Illustration: FACTORY GIRLS.
[To face page 14.]
their country's good at heart, and the reason is that owing to the
continual increase in taxation, the Spanish labourer cannot make a
living at home. Of all the taxes which crush him, the most oppressive is
the _consumo_, or octroi. Little is heard of this outside Spain, because
those who profit by it have every reason to keep silence, while those
who suffer have not hitherto dared to raise their voice against the
powerful interests which profit by the system. Any statesman who could
abolish this iniquitous tax would gain thereby an amount of popular
support to which ministers of the Crown in Spain have long been
strangers. But he would have to contend with an organised opposition in
the monied classes which would be hard to overcome, and hitherto,
although the reform is constantly talked of, little or nothing has been
done to bring it about.
Next to bread the chief desire of the Spaniard is education for his
children. He is thoroughly conscious of the disadvantages of his own
ignorance, which he bitterly resents, and the blame for which he lays at
the door of the Church. The Inquisition is not forgotten, and if there
is no priest or "pious" person within sight, an interested listener may
hear strange tales told in explanation of the popular detestation of the
religious Orders. Some of these tales are no doubt traditional, handed
down from the time when the Holy Office was an ever-present terror. It
is not easy for more advanced nations to realise the influence of
tradition among a people necessarily dependent on oral teaching for
everything they know, or the extent to which it colours their thoughts
and affects their actions in every direction. Although the working
classes in Spain are of course aware that the Inquisition no longer
exists, the effects of the nightmare of three hundred years continue,
and the fear and hatred with which that tribunal was regarded are now
transferred to the priests, and especially the Religious Orders. The
Church has ruled in Spain, with one short interval, ever since Isabella
and Torquemada revived the Holy Office, and, like all autocracies, it
has come to look upon the nation over which it rules as a tool to be
used for its own ends, an insentient thing, a mere machine to be driven
hither and thither as the interests of the Church dictate.
And now the inevitable is happening. The machine has become sentient,
and instead of submitting to be driven it is beginning to take its own
course and carry its quondam drivers into regions unknown.
The crucial question to-day in Spain is the religious question. Not the
belief or disbelief of the people in their religion, but the relations
of the Church--_i.e._, that of the priests and, far more, of the
Religious Orders--to the nation.
From tradition and from the circumstances of their lives, the mass of
the people have come to look upon the Religious Orders as their evil
genius, and at every turn one meets with evidences of their distrust of
and hostility to those who should be their spiritual guides. Until July,
1909, this feeling, although for long past there have been clear
indications of it, was not openly expressed by the people in public
places. They not only hated the "good fathers," as they satirically call
them, but dreaded their vengeance upon those who offended them. Since
the rising against the Religious Orders in Cataluña, however, the
attitude of the two parties towards each other has been reversed. It is
now the priests and the Religious Orders who are afraid. So little do
they understand the people whom they are supposed to teach, that they go
in fear of their lives lest the working classes should rise _en masse_
against them; whereas the working classes _en masse_ desire nothing
better than a peaceable solution which shall ensure their daily bread to
them and their children.
On every side the people see the baneful hand of the Church, interfering
or trying to interfere in their domestic life, ordering the conditions
of employment, draining them of their hard-won livelihood by trusts and
monopolies established and maintained in the interests of the Religious
Orders, placing obstacles in the way of their children's education,
hindering them in the exercise of their constitutional rights, and
deliberately ruining those of them who are bold enough to run counter to
priestly dictation. Riots suddenly break out in Barcelona: they are
instigated by the Jesuits. The country goes to war in Morocco: it is
dragged into it solely in defence of the mines owned, actually if not
ostensibly, by the Jesuits. The _consumos_ cannot be abolished, because
the Jesuits are financially interested in their continuance, and so
forth. Rightly or wrongly, the people attribute all the ills under which
they suffer to the influence of the Church, and sooner or later, unless
measures are taken to restrain the interference of the Church in public
and private life, an explosion will come which will sweep the whole
institution away. Moreover, the steady and continuous efforts made by
the Church to upset the existing regime and bring back a reign of
absolutism with the proscribed branch of the House of Bourbon, though
not continually present in the minds of the people, are not unknown to
or ignored by them.
But with all this intensely anti-clerical feeling, the mass of the
people are untouched by modern scepticism, and are deeply and sincerely
religious. Their religion is simple in the extreme: many would call it
gross superstition, but such as it is, it suits their stage of
intellectual development and undoubtedly has a considerable effect on
their conduct. To represent the Spanish working man--as the Church
newspapers always do--as an atheist and an anarchist, only to be
restrained by force from overthrowing the social order, merely proves
how completely ignorant the Clericalists are of his real character.
RACIAL AND CLASS DIVISIONS
CHAPTER I
RACIAL AND CLASS DIVISIONS
The relations between rich and poor, between rulers and ruled, between
employers and employed, in Spain are peculiar and not easy to
understand.
The immediate dependents of a well-to-do family are allowed a freedom of
manner and intercourse which is incomprehensible to English
exclusiveness, and a sense of responsibility for their dependents, and
especially for those who have rendered long domestic service, is almost
universal among employers. Thus there is hardly a family of means that
does not, as a matter of course, support for the rest of their lives one
or more of the wet-nurses who brought up the children; and during the
famine in Andalusia a few years ago, most, if not all, of the landowners
continued to pay, to the limit of their means, the wages of their
permanent labourers, although owing to the drought no field work could
be done for months. But with all this very real generosity towards
those with whom they are brought in contact, the rich have no corporate
or class sense of responsibility for the working classes, and make no
effort to understand or provide for their needs as a whole. Spaniards
are liberal in alms-giving, and every good Catholic gives doles on one
day of the week to his or her regular pensioners; but there is no public
provision for the destitute, and it is not in the least realised that an
organised system of poor relief would be less costly, and certainly far
less demoralising, than the haphazard distribution of pence to all and
sundry. It is true that in some towns benevolent societies are carrying
on good work according to their means, but these, consisting only of
voluntary gifts, are not sufficient to do more than touch the fringe of
the poverty produced by the conditions of the country.
The original causes of this combination of an almost patriarchal
relation between the master and his immediate dependents, and complete
ignorance of and indifference to the lot of those outside of the home or
estate, lie deep, and must be sought in the relations between Christians
and Moslems when the Castilians re-conquered Spain. It must be
remembered that the Arabs had brought agriculture and many industries
to a high state of perfection, and after the conquest they continued to
cultivate the land and work at their manufactures for the benefit of
their conquerors. Thus for some hundreds of years the dominant was
living with the subject race, and the conquerors would feel for the
conquered the contempt of the fighting man for the labourer, of the
Western for the Oriental, of the victor for the vanquished, and of the
Christian for the infidel. It is easy to see that when the mass of the
industrial population was of alien race, any idea of responsibility on
the part of the employers for the employed as a class would be unlikely
to arise, while on the other hand the personal relation between master
and servant would become intimate, as it did in the Southern States of
America in the slave-holding days, and as it is in the East to-day. This
accounts for the relation between rich and poor already remarked on:
liberal protection of immediate dependents, coupled with indifference to
the general welfare of the working classes. The tradition, handed down
from the time when the bulk of the proletariat were aliens, has
persisted for two hundred years after the last of the Moslem inhabitants
was expelled.[1]
A right understanding both of the past history of Spain and of its
social and political condition to-day is made still more difficult by
the claim made by Castile, with Madrid as the capital, to speak for
Spain as a whole. Most histories of Spain are written from the Castilian
point of view, and foreign writers naturally go to the capital in search
of their material. But this procedure leaves out of sight the very
important distinctions between the different parts of Spain, and
especially those between the Castilian and the Aragonese of the centre
and north, and the Andalusian, Valencian, and Murcian of the south.
Setting aside Cataluña and the Basque Provinces, with a population in
round numbers of 2,500,000, the rest of Spain north of the Sierra Morena
has a population of 9,000,000, while the three ancient southern
kingdoms, Valencia, Murcia, and Andalusia, have between them a
population of 6,000,000. The distinctive characteristics of these
provinces, which contain about a third of the total inhabitants of the
country, are left unnoticed by Castilian writers and those who follow
them, or, if the southerners are mentioned at all, it is usually with
some expression of contempt. This applies especially to the Andalusian,
who is always spoken of as lazy and incompetent, without ambition,
content to sit in the sun and smoke a cigarette, a windbag who talks
everlastingly and does nothing, and generally a negligible quantity in
Spanish politics, and a person unworthy of serious consideration in
Madrid.
The ingrained orientalism of the south is at the root of the hostility
with which it is regarded by Castile and the north. Andalusia and
Valencia were under Moslem rule for some 500 years--Granada for nearly
750--and this long occupation and colonisation has left an indelible
impress on the race, language, customs, and modes of thought of the
south. On the other hand, the Arab invasion of the north was soon driven
back beyond the Sierra de Guadarrama, and even in New Castile and
Estremadura, north of the Guadiana, their occupation was more in the
nature of a military tenure than a colonisation, and, such as it was,
came to an end 160 years before the Christians were able to win any
footing in the southern provinces. There is, therefore, comparatively
little Eastern blood in the veins of the Castilian, while in those of
the southerner the Arabic strain is at least as strong as the European.
How little sympathy exists between Castile and Andalusia may be judged
from the following facts: In 1904 the south-west of Spain was afflicted
by ten months of drought, causing the worst famine known for many
years. Men literally died of starvation by the roadside, and the
suffering among women and children was something terrible. No national
or combined effort was attempted for the relief of the distress, which,
indeed, the Clericalist organs of Madrid minimised and almost mocked at,
saying that "every one knew that the Andalusians were all farmers, and
farmers would grumble whatever the weather was." On the other hand, when
comparatively small districts in Castile, Leon, and Galicia suffered
from floods in 1910, over 100,000 pesetas were collected by voluntary
subscription within a week.
It must be remembered that, while the reconquest of the whole of Spain
except the Kingdom of Granada was completed by the middle of the
thirteenth century, there was no large exodus of the Moslem inhabitants
until their expulsion in 1609,[2] and that, until Isabella's religious
fervour made things unpleasant for them, they lived side by side with
their Spanish conquerors, and were, on the whole, not badly treated
until the persecution and expulsion ordered by Philip III. Indeed, all
the evidence goes to show that a steady amalgamation of the races went
on, with so much intermarriage, that in some parts of the country there
is hardly a family without Eastern blood in its veins. But necessarily
and naturally the conquered race gradually fell more and more into the
position of servants and slaves.
Although the great preponderance of the Arabs and Moriscos was in the
south, numbers of them were scattered over other parts of Spain, even so
late as the beginning of the sixteenth century, which accounts for the
position of the working classes elsewhere being much on a par, so far as
their employers' view of them is concerned, with that of their fellows
in the south.
Thus Spain is now divided into two unconsciously hostile camps, with an
ingrained tradition of racial and religious hostility at the root of
their antagonism, which is a fatal obstacle to mutual understanding. The
Spanish labourer has replaced his predecessor of alien race, but the
tradition of contempt and indifference remains, and the employer--and
especially the employer who is "addicted to the priests"--still regards
him, as his predecessor regarded his Moslem servant, as a hewer of wood
and drawer of water, whose duty is to pay his taxes, and to use the
suffrage nominally bestowed on him by the Constitution in the interests
of his master. The working classes, if we are to believe the assertions
of their "superiors," are a godless lot with anarchical leanings, whose
vandalistic tendencies have to be suppressed with a strong hand lest
they break out to the total subversion of society. But ask a peasant
about his politics, and he will say that all he wants is a sufficient
wage to provide for his family and a decent education for his children,
and he will add that he has no hope that any political party will help
him to realise this modest ambition, or do anything whatever for him,
because "all Governments, whatever they call themselves, are of one
kidney, and care for nothing but pocketing the public funds, and
pleasing the Religious Orders; the Conservatives because they love them,
and the Liberals because they fear them, and both because the Jesuits
are the richest people in Spain."
The patient submission of the labourer to conditions which he believes
to be unalterable is partly the result of three hundred years of corrupt
government, during which he has been steadily squeezed to provide money
for the wars, luxuries, and amusements of the governing classes; partly
of the terror of the Inquisition and the tradition of silence that it
has left behind it; partly of Oriental fatalism; but is certainly not
due to the animal indifference and stupidity to which his "betters"
attribute it. The peasant refrains from open complaint, not because he
is contented and has nothing to complain of, but because long experience
has taught him the uselessness and the danger of protest. He may offend
his employer and lose his place, or, still worse, he may offend the
Church and the Jesuits, in which case he will be a marked man, and can
never hope to get permanent employment again.
Here is a paragraph which appeared in a leading Clericalist organ on
December 1, 1909:
"Canovas and Sagasta attracted to the Monarchy the most
aristocratic elements of the Carlist and Republican masses, through
the mediation of Pidal and Castelar. Señor Moret (leader of the
Liberal-Monarchist party) does not act in this way. Instead of
considering the honourable people he considers the masses, the
elements which bring about disturbances of the social order."
This summarises in a few words the attitude which has always been
maintained by the Church, and the aristocracy attached to the Church,
towards the democracy. The people must be restrained from making their
voice heard in the counsels of the nation, although they have nominally
possessed the suffrage for some forty years, because, if the masses are
given the free use of the vote, they will disturb a social order
maintained exclusively in the interests of the classes. Such sentiments
were common in France before 1789, but one hardly expects to find them
so badly expressed in the twentieth century.
The upper classes in Spain are in the majority thoroughly materialised.
Their object in life is simple--wealth and power, with all that they
bring in their train, often without too nice a regard for the means
whereby those ambitions are realised. Their religion consists in a
diligent observance of the ordinances of the Church, and submission to
the dictates of the priesthood.[3] Of any higher ideals--of any
amelioration in the general lot of the poor, of any improvement in the
deplorably backward state of education, of any attempt to raise the low
moral tone which prevails in their own class, little or nothing is ever
heard. There is, however, an increasing number of educated young men who
are doing what lies in their power to promote a better state of things.
They have to contend, not only against the active hostility of the
clericals, but against the dead weight of middle-class apathy and
ignorance, and in consequence their labour is as that of Sisyphus. Yet
they patiently struggle on against all discouragements, and their circle
of influence is widening every year.
But while the upper-class Spaniard is intent on the pursuit of wealth
and indifferent to higher things, the peasant has an ideal which he has
set before him, and for which he makes every effort in his power,
against obstacles which anywhere but in Spain would be inconceivable.
And that ideal, as has been said, is some sort of education for his
children, whom he does not wish to be handicapped, as he has been, by
inability to read and write. If he can only pay for the schooling of one
child, that child has to share his knowledge with the rest of the
family, reading to them all he can get to read, and sometimes even
passing on the little instruction he has received, and teaching his
parents and brothers and sisters their letters at night, after the
day's work is done.[4] And his chosen reading is not the republican, or
socialist, or anarchical stuff against which the Church inveighs with
theological fervour as the mental pabulum beloved of the masses, but
certain papers with moderate Liberal views, which preach education and
loyalty to existing institutions as the best hope for the country. These
papers point out that any upheaval of the social order, with its
necessarily attendant paralysis of trade and agriculture, can only
result in making the hard lot of the labourer harder still; and the
peasant, whom his masters take to be indifferent and half brutal, has
the sense to see the wisdom of this teaching and to be guided by it.
That the Ultramontane party should maintain, as they do, that every
disturbance that may occur in Spain is the fruit of the working man's
attachment to seditious and anti-religious literature, is only another
proof of their determination to misrepresent or slander him. Had this
been the case, no measures of repression would have saved Spain, in July
1909, from an outburst of rage against the Religious Orders all over the
country. That the fires lighted in Barcelona did not spread was not due
to the suspension of the Constitution or to any terrorism exercised by
the priest-ridden Government of Señor Maura. The people were kept in
bounds by the influence of their chosen organ, the _Liberal_, which
costs less than a farthing and has the largest circulation of any paper
in Spain. And this paper, like the others of its party and all the best
of the Radical and Republican Press, throughout all the turmoil of the
three months before the Maura Ministry fell, steadily urged the people
to have patience, keep the peace, and show by their actions that they
were worthy of liberty.
THE RELIGION OF THE PEOPLE
[Illustration: PEASANT WOMEN.
[To face page 39.]
CHAPTER II
THE RELIGION OF THE PEOPLE
If you ask upper-class Spaniards, priestly or lay, about the religion of
the people of Spain, you will be told that half the nation are bigots
and the other half free-thinkers and atheists, or at best indifferent
Laodiceans: a sweeping assertion that has so often been made that it has
become a commonplace with foreign journalists and magazine writers.
To accuse the nation at large of bigotry, atheism, or indifferentism, is
nevertheless as unjust as to accuse the army of cowardice. Small though
is the attendance of the working classes at Mass, and hostile though
they are to the practice of confession, they are none the less deeply
religious--firm believers in the efficacy of prayer, and loyal to the
fundamental tenets of their faith, such as dependence on the will of
God, gratitude for small mercies vouchsafed by a good Providence, and
devotion to the Virgin and the saints.
In the middle class there is, no doubt, a good deal of rather shallow
free-thinking, although it usually goes little beyond a scoff at
superstition and contempt for miracles and images, and is confined to
the men. The women usually follow in their mothers' footsteps, attend
Mass, run through the rosary, and thoroughly enjoy the processions which
enliven so many Church festivals. Confession, however, is perfunctory
even among middle-class women, and the poor avoid it altogether. For
strict observance of the ordinances and for material support of the
Church you must go to women of higher social position, ladies of title
and the wives of rich men, whose political relations keep them hand in
hand with the priests and the Religious Orders. They are the bulwark of
the Church in Spain. Indeed, it is often said that if all the ladies of
the aristocracy could be locked up for a few years, the Church of Spain
would go to pieces, so little real hold has it on any other element in
the national life.
These ladies attend Mass every day and confess with great regularity.
They consider it the highest privilege to be "wardrobe keepers" for the
_santos_ (saints-images) in their favourite churches; they dress and
undress the image of the Virgin with their own hands for festivals, and
they keep in their own houses the jewels and other treasures belonging
to her. In some cases they also look after the vestments of the priests
and take charge of the altar linen. And they give or bequeath large
fortunes to different monasteries and convents, and to religious houses
built to receive orphans and old people, repentant Magdalens, and girls
in training for domestic service. But no one is admitted to institutions
supported by ladies of devout life save on condition of daily attendance
at Mass and regular confession and communion. And therefore the people
say that such charity is not dictated by love for their poorer brethren,
but is merely given in order to prop up a decadent Church, and many will
starve rather than ask for it.
The people have a word of contempt for the religious principles of women
such as these. They call them _beata_, which according to the dictionary
means "devout," but which the poor translate as "canting." There is a
world of difference to them between the lady who is _religiosa_
(religious) and the one who is _beata_. _Religiosa_ is applied to a
woman who devotes her life to God and works for the sake of doing good;
_beata_ means one who lives, moves, and has her being under the thumb of
the priests and the Religious Orders.
The poor say that unless they are prepared to attend Mass and confess
regularly, they can expect nothing from women, however rich, who are
known to be _beatas_. For alms given unquestioningly and without
insistence on previous compliance with the rules of the Church, the sick
and needy turn by preference either to persons recognised as "religious"
or to those who "have nothing to do with those follies." That is how the
practice of confession is characterised by the democracy in the privacy
of their own homes. They dread and distrust the confessors, and no poor
man or woman will speak freely in the presence of one of their own class
who is in the habit of confessing.
Yet notwithstanding their antagonism to this primary dogma of their
religion, the working classes, and especially the peasantry, are, as
already stated, deeply and sincerely devout, and firmly uphold the
Christian faith as they understand it.
One of the most remarkable features in the spiritual life of the nation
is the clear comprehension of even the least educated among them that
the sins of the priests and the Religious Orders stand apart from and
leave unsmirched the national religion.
"What have I to do with those people?" said a young fisherman to the
writer. "Confess to a priest? Never! I confess to God and my mother, and
I want no priest to come between me and my God."
"I? Confess to a priest? What for? Every night when I go to bed I
confess my sins to the Virgin, and I can die as well after that as if I
had received the holy oils," said an old woman of deep and sincere
faith.
"I do not allow my wife to go to confession," said a master mason. "If
she insisted I should refuse to provide for her. I will have no traffic
with the gentry of the long skirts in _my_ family."
"No, I did not call in a priest when my husband was dying. He would have
died all the sooner if I had, he hated them so. We poor people never
call the priest if we can help it. We say 'death gave us no time.' The
priests pretend to believe it; they are glad enough to be saved the
trouble of coming to our houses because, if we send for them, they have
to give the holy oils gratis. And we get buried all the same," said a
young widow who had lost husband and child within three months of each
other.
"And you are not afraid the dead will stay longer in purgatory if they
die without the holy oils?" I have frequently asked on hearing such
statements.
"Why should they? My brother, may his soul rest in peace! was a good
man. God will look after him without any priest putting in his oar. Yes,
it is true that the priests talk of purgatory, but for my part I have
never well understood what it is, and I do not care. I say a prayer for
my dead on All Souls' Day, and there is an end of it. There may be
purgatory, God knows. But certainly I will not pay money to a priest on
that account. I want it more than he does."
The popular idea of purgatory is very confused, and many declare that
they do not believe in it, while betraying in every word that they pray
heartily for the souls that they assume to be there.
"Do you think I could believe that my brother or my mother are in
purgatory, or that _I_ shall go there, I, who would give the clothes off
my back to the poor?"
You cannot pay a greater compliment to a sceptic of this kind than to
say: "By your good deed of this or that kind you have certainly taken a
soul (or two souls) out of purgatory to-day." "Taking souls out of
purgatory" is a favourite occupation. It is effected by prayer or by
good works, not necessarily, so far as the popular belief can be
understood, by both practices together.
"There are always seven souls clinging to the cloak of the Virgin--not
the Virgin on any altar, but the real Virgin in heaven. They are all
climbing up, one above the other, and by prayers or good works you can
help the uppermost to get out and make room for the next."
"And where do they go then?"
"I don't know. To heaven, I suppose. Purgatory is not a very bad place
to be in, it is pretty fair. The wicked people go to the _Tinieblas_
[tenebræ]. I do not know what that is, but it is very bad. It is always
well to say a prayer for those in _Tinieblas_."
"But do you suppose that any of _your_ friends are there?"
"No, indeed; but you never know who may be clinging to the robe of the
Virgin, and some one belonging to you might just be climbing up. At
least, nothing is lost by saying a prayer."
"If the souls in the _Tinieblas_ are allowed to cling to the Virgin, I
suppose she also is there?"
"How do I know? Perhaps these are all lies--things of priests
[_mentiras, cosas de sacerdotes_]. What does it matter? What is needful
is to share your _puchero_[5] with any poor man who is hungrier than
you, and God knows I do that."
The custom of attending a Mass for the dead on All Souls' Day is very
general. There are thousands of men and women who never set foot in a
church during the rest of the year, yet rise an hour earlier than usual
to go to Mass before beginning their work on November 2nd. But the
proportion of communicants even on this occasion is very small. I have
counted the congregations present at churches attended by the working
and the lower middle classes on All Souls' Day. At one early Mass, out
of forty present, four communicated; at another, two out of thirteen;
and so on. Communion involves previous confession, and the poor will not
confess. Nevertheless, their faces show that this Mass is not a mere
empty form to them. They do not, of course, understand a syllable of the
words the priest mutters at the altar, but they are absorbed in earnest
intercession for the dead whom they are commemorating. Then they go
their way to take up the round of work, and probably do not attend
another Mass until All Souls' Day comes round again, while the rich
celebrate the "Day of the Dead" by paying for and attending frequent
Masses, and by taking or sending wreaths of flowers to adorn the graves
of relatives in the distant cemetery.
Curiously enough, infant baptism bulks far larger in the religion of the
poor than any other office of the Church, and the parents, and
especially the mother, will make heavy sacrifices to obtain the fee
demanded for the performance of this rite. The ceremony itself has some
singular features, for the mother must on no account be present, and
even the father remains in the background. But the social function which
follows the ceremony in the Church is almost as important an event in
the family life as a wedding, and the festivities are kept up far into
the night. It may seem fanciful to trace these baptismal customs back to
the time of Islam, but it is a fact that the accounts of the
birth-feasts (_buenas fadas_) among the Moslems of Spain offer certain
resemblances to those of to-day, while the term used to describe an
unbaptized child among the peasantry links us directly to the time when
to be a follower of the Prophet was to be an object of contumely. The
explanation of the efforts made by the family and friends of a child of
poor parents to scrape together the 7.50 pesetas demanded by the priest
for the performance of the baptismal office is: "I could not leave him a
Moor" (_No podia dejarle Moro_).
Burial often takes place without the offices of the Church, for there
are few among the working classes who can afford to pay for a funeral
Mass, and very many are unaware that they can insist upon the attendance
of a priest even without a fee. And since the charge for a marriage in
church amounts in many parishes to as much as 25 pesetas--the average
weekly wage of the agricultural labourer certainly not exceeding half
that sum--it is only to be expected that the civil ceremony, which costs
one peseta, or the stolen "blessing" snatched from an unwilling priest
by the pair proclaiming themselves man and wife at the close of any
Mass, should be more frequently resorted to than the orthodox function.
Many couples, moreover, live all their lives as husband and wife, as
faithfully as if married by the Church or the mayor, without any
religious or legal tie at all.
"The women don't like it," said a working man to the writer, "but what
is one to do? How can we pay twenty-five pesetas to get married? And the
women are only now beginning to understand that the civil marriage is
quite as good as the other, if there is any question of money to be
left to the children. I could show you plenty among my neighbours who
live as if married, and no one takes notice that they are not. The
priests only say such couples are living in sin because they have not
got the marriage fee out of them."
"It is true that my daughter-in-law could leave my son if she liked,"
said an old woman when discussing a quarrel between her hot-tempered son
and his hotter-tempered "wife." "There was no money for the marriage, so
I consented to their marrying without going to church. They will never
separate: it does not occur to them that it would be possible. It is not
as if they were not faithful to each other. My son does not look at
other women, and as for my daughter-in-law (_mi nuera_), he would kill
her if she set her eyes on another man, and well she knows it. There is
no sin in marriages like that, whatever the priests may say about it. Of
course I would have preferred that they should be married in church, and
so would my daughter-in-law, but what are you to do when there is no
money?"
The use of the term _nuera_ here is significant. No social stigma
attaches to these "wives" who are no wives at all, unless they leave one
man to go to another. Then they are branded as "women of bad repute" by
their neighbours, and shunned accordingly.
Thus the religion of the people seems to be entirely dissociated from
the forms imposed by the Church upon its members, save only that of
baptism, which is respected mainly owing to an unconscious traditional
antipathy to the unbaptized;--the "Moor" or Moslem, of bygone days--and
an almost complete indifference to the rites of marriage and death has
sprung up as a consequence of inability to pay the fees demanded for
their performance. In the towns perhaps few really care if their dead
are buried without a prayer, but in the villages there still remains
enough feeling about it to arouse an occasional growl of indignation
when a coffin is borne through the streets attended only by the
mourners, without the priest, the acolytes, and the censer-bearers, who
lend distinction to the last journey of those who possess a few pesetas.
As for the children, who are born and die like flies, the poor have
become so accustomed to see the little coffins carried by on the
shoulders of small elder brothers or school friends, led by the father
or uncle of the dead child, that the piteous sight no longer calls forth
a comment. It is often only one out of half a dozen of the same family
who have gone the same way, and notwithstanding the heartbroken
lamentations of the mother when the breath leaves the little body, every
one knows she will soon be consoled. The lot of the poor is too hard for
indulgence in sorrow, and it is not uncommon to hear a woman who is
approaching childbirth say with the utmost unconcern, "We shall see what
happens. Perhaps it will please God to allow the infant to be born
dead." It is not heartlessness or want of love for her offspring, for
Spanish parents of both sexes and of every class are very affectionate,
and indulge their children to excess. It is simply that every extra
mouth to feed means so much less to fill the stomachs of the rest, while
the national custom of suckling the child for at least a year, and often
for two renders the mother meanwhile weak and unfit for the washing,
sewing, or charing which helps out the family resources. That a great
effort should be made to baptize the new baby when it comes shows the
strength of the feeling in regard to this religious duty, and would make
the general indifference to the intervention of the Church in marriage
and death doubly remarkable, were it not that the tradition connected
with the first ceremony does not extend to the other two.
Among the upper classes more attention seems to be paid to the
religious funeral ceremony than to the actual committal to the grave.
When a death takes place in a family of social position, all the friends
and relatives are invited to attend the funeral Mass, which takes place
in the parish church of the defunct, and it is expected of the guests
that they shall accompany the funeral procession on foot as far as the
outskirts of the town or village, the cemetery generally being some
little distance beyond. There, however, the party disperse; few attend
the coffin to the grave itself, and very often it is shuffled into its
last resting-place with what to English eyes seems indecent haste and
carelessness.
Indeed, whatever be the reason, small respect is shown for the empty
shell, once the spirit of life has fled. The rich buy a freehold grave
for themselves and their family, but the poor can seldom afford to pay
for more than a six years' concession, if that; and if they do not renew
payment the bones of their dead are disinterred and thrown on a heap in
the _osario_ or bone-house, a building with a locked door built for the
purpose within the walls of the cemetery.
The mental attitude of the people towards images is intricate and
difficult to disentangle. Even persons who have had what ought to be a
liberal education in many cases believe in the miraculous virtues of
the images, scapulars, and medals of their particular devotion.
"If you would only wear this medal," a devout lady said to the writer,
"I know you would be converted to the true faith, for it is very
miraculous, and has converted many. But you would not wear it, so it is
useless to give it to you."
The speaker was a woman of culture, artistic, and fairly well read for a
Spanish lady, yet she was obviously sincere in her belief in the virtues
of the little cast-lead medal washed over with silver.
Nor is this singular simplicity confined to women. Every year men of the
upper classes (never, I think, of the lower) may be seen during Holy
Week walking barefoot before the images carried in procession through
the streets; and since their faces are covered and there is nothing to
reveal their identity to the world at large, it cannot be supposed that
the act of penance is performed for political reasons, as,
unfortunately, is too often the case with public demonstrations of
adherence to the Church. Moreover, these processions are attended by men
of Liberal as well as Conservative opinions. That the particular image
plays an important part is shown by the fact that the act of penitence
is never performed save in connection with its appearance in the
streets. The penitent walks barefoot before or after the platform on
which is carried the Virgin of his adoration, and although it may be one
among fifty representations of the Virgin in his city, it is understood
that no other would have the same efficacy in cleansing his particular
sin. It must be a genuinely penitential experience for a man used to
luxury to tramp barefoot over badly paved streets at a rate of progress
which makes the two or three miles of distance occupy twice as many
hours of time, and sometimes these aristocratic penitents reach the end
of their journey in a state of complete exhaustion. But there does not
seem to be any sentiment of shame or disgrace attached to the act, as
though there were some great sin to purge, for it is not unusual to hear
a young man of orthodox proclivities say to a girl whom he meets in
society shortly before Holy Week: "You must look out for me at such a
place in the route of the procession; I am going in penitence and I will
lift my hood there for you to know it is I." Yet, frivolous as such
penitence may appear, the rich man shares with his poor and ignorant
brother a personal feeling for the image of his devotion, which leads
him to disregard even danger to life in connection with it, should the
need arise. This is quite dramatically shown in the case of the fires
which frequently occur in churches and chapels, where lights burn
continually before images adorned with lace and other combustible
fabrics. The _santos_ are always the first thought of the crowd on these
occasions, and even men who scoff aloud at "all those fooleries" in
daily life will be seen risking personal injury to save "the Virgin of
Hope" or "Our Lady of Miracles" from destruction.
One very puzzling question in connection with this worship of the images
is how far even the better educated Spaniards recognise the fact that
the different images, _e.g._, of the Virgin--the Virgin of Sorrows, of
Miracles, of the Pillar, of the Kings, and hundreds more--are all
representations of one and the same Virgin Mary, and how far they
consider them to be distinct individuals. Probably the worshippers
themselves are not at all clear on the point: that the prayers offered
before these images are in most cases addressed, not to the Person
represented, but to the image itself, there seems little doubt. In the
case of the populace the images certainly seem to be distinct
individuals; indeed, I have been pitied more than once by kindly
peasants for having "only one Christ." "_We_ have many: there is the
Christ of the Descent from the Cross, and the Christ of the Waters,
besides the Christ of the Flagellation in the Parish Church, all very
miraculous."
An intelligent man of middle age, better educated than most of his
class, said to me in reference to the affection of the Spanish peasants
for their images of the Virgin.
"You would be shocked if you could hear what we say to the Virgin in our
houses and when we see her in the streets. But it is not irreverence or
disrespect, as you would consider it. It is that we feel towards her as
one of the family and talk to her as we should to one of ourselves."
The return of certain confraternities after carrying their images
through the streets in Holy Week presents an extraordinary spectacle.
This is especially the case with images belonging to the poorer
quarters. In one town the procession of one of these images returns
early on the morning of Easter Eve, after moving slowly through the
streets, from its church to the distant cathedral and back, all through
the night. The bearers of the platform, which is a great weight, the
members of the confraternity, the soldiers--for the Army always has a
place in these functions--and the band in attendance, are all worn out
with fatigue, but when they reach the threshold of the church they
revive, the band strikes up an animated march, and the whole crowd
assembled to do honour to "Our Lady" seem to go crazy with joy at having
brought her safely back to her "home" (_á su casa_). The richly dressed
life-sized image is lifted down from the platform by many eager hands,
and swayed to and fro in time to the music almost as if dancing, and the
whole atmosphere of the scene is that of a rejoicing welcome to a
beloved being who has returned to her family after a long absence
fraught with danger. Nothing brings home to the observer the intense
reality of the people's feeling for their _santos_ like such a scene as
this. It is, however, seldom witnessed by foreigners or even by the
well-to-do of their own nation. It is so much a matter of course in
Spain that no one goes out of his way to see it, and I was present on
such an occasion only by the merest chance.
A bright, clever woman of the working classes, with a strong sense of
humour, told me that she could only pray to a certain Christ. "All the
others are only sticks (_palos_) to me. I can never pass our Lord of
Pity without kneeling down, and I know by the look in his eyes if he is
going to grant my prayer, but I cannot pray to any of the others."
"Then when you pray to that image of Our Lord, it really is the Christ
to you?"
"No; the Christ is in heaven with His Mother, but I pray to our Lord of
Pity, and he always answers me. No other is the same. When I pass Our
Lord of the Miracles, for instance, in the Church of San José, I have to
say: 'Excuse me, Lord, but you are only a stick to me, and I cannot pray
to you. I do not know why this should be so, Lord, but that is how I
find it.'" All this was said quite gravely, and the prayer addressed to
"Our Lord of Pity" was recited with sincere piety.
A good old widow of my acquaintance finds St. Anthony of Padua
particularly sympathetic, and feels constrained to pray for the soul of
her husband at 7 a.m. on All Souls' Day before one particular St.
Anthony in one particular chapel at a quite inconvenient distance from
her home. On any other occasion the first St. Anthony of Padua she comes
across serves her purpose, and I once saw her stop short and break into
a fervent prayer under her breath at the sight of an abominable penny
chromo of the saint which suddenly attracted her attention in a shop
window.
MORALITY AND CEREMONIAL
[Illustration: NEWSPAPER SELLERS IN MADRID.
(At the offices of the _Nuevo Mundo_.)
[To face page 61.]
CHAPTER III
MORALITY AND CEREMONIAL
That it is a duty to speak the truth is a proposition practically
unrecognised in Spain. This is chiefly, if not entirely, due to the
influence of the Church, for, as a great historian says in reference to
this question, "when credulity is inculcated as a virtue, falsehood will
not long be stigmatised as a vice."[6] I have heard the peasant's creed
on this point put into a nutshell, thus:
"Very often it is necessary to lie, either for your own or for some one
else's benefit. There is nothing wrong in that. But to tell an
unnecessary lie is a sin."
This sophism, which I have translated word for word, seems altogether
too subtle to be instinctive, and we trace in it some echo of the
Church's teaching, instilled into the mind of the uneducated, who have
come to adopt it as an axiom of common morality.
The honesty of the Spaniard is, according to our views, relative. It is
very rare for a working man or woman to take cash which does not belong
to him. But the same people--_e.g._, servants--who would consider it a
disgrace to steal a peseta in coin, will have no hesitation in
falsifying their accounts and cheating their employer out of ten or
twenty times that amount.
In certain matters there is extreme sensitiveness to any suspicion of
dishonesty, but it is not clear that any conscious religious principle
underlies this feeling. It seems rather an instinct of self-protection;
for when we learn that it is a common practice for employers to examine
their servants' boxes when they leave a situation, even although their
good conduct has not been called into question, we see that the
friendliest relations between master and man do not necessarily imply
confidence in the honour of the latter. The result of assuming evil
where there is none is to encourage its genesis. I have heard
working-class Spaniards say bitterly: "The rich people believe that we
are all thieves, so what is the use of being honest? Yet most of us are
honest, even though we go hungry for being so." Stealing is considered
by the poor as a sin, but I am inclined to think that the degree of
sinfulness depends in the criminal's eyes upon the nature of the theft.
Thus, while no respectable peasant will steal money or clothes, servants
have no hesitation in appropriating sweets and wine, to which, indeed,
they think they have a right, very possibly dating from the time when
domesticated aliens were fed on the leavings of their masters' table. It
would be difficult to convince them that to drink their employer's wine
without permission is just as immoral as to steal his cash.
The instruction given by the Church on these points is hardly ambiguous,
if one may judge by a parish leaflet in my possession. It contains the
following questions of conscience resolved under the head of
"Consultations."
"May a servant give to the poor the food which remains over,
without asking permission of her master?"
"She may do so when her master does not make use of or dispose of
it."
"And may she give it to her poor relations?"
"Without any doubt; but _it is better_ to consult her master"
(italics mine).
Such moral teaching as this would quite account for the conduct of a
pious cook once in my employ, who fed her entire family for some time at
my expense. She, it is perhaps needless to say, did not "consult her
master" on the point. She may have consulted her priest, for all I know,
and if she did was probably told that it was a meritorious act to rob a
heretic.
To turn to another branch of the subject, it will probably be news to
many people that a "Bull of the Crusade" is still largely sold in Spain.
This indulgence was first instituted in the days of the Moorish wars, to
permit those who were fighting the infidel to keep up their strength by
eating meat whenever they could get it. Few or none of the poor purchase
this or any other indulgence nowadays, but it is still freely sold to
people of means, and the day of its issue is kept as a minor feast day.
It now costs the modest sum of pesetas 1.75, having been gradually
reduced from pesetas 7.50, and is a source of income to the Government,
producing, according to the Budget for 1909, 2,670,000 pesetas, or, say,
£106,800. Any one can obtain it, as no questions are asked as to the
religion of the purchaser.
An interesting survival is the penitential purple dress, with yellow
cord and tassels round the neck and waist, which is worn on occasion by
women of all classes in the rural districts, and by the poor in many
cities. It is not, generally speaking, a penance imposed by the priest,
but a free-will offering to the Virgin, made on behalf of some one dear
to the wearer. A woman will promise to wear the _hábito_ or penitential
robe for a specified number of months, or a year, or sometimes even for
life, if the Virgin will intercede for her invalid husband; or a girl
will undertake to wear nothing else until the dress is torn or worn past
repair, when the sacrifice is completed. A girl of seventeen explained
that she had volunteered to assume the _hábito_ she was wearing because
her only sister was very ill.
"But my sister got better and persuaded me to put it out of my head.
Then she suddenly became very ill again; all one night she seemed to be
dying, so I knew I must keep my promise to the Virgin, and after that I
would not let any of them put it out of my head."
The Spaniards have two distinct ways of crossing themselves. One,
described by the verb _santiguar_, consists in making the sign of the
cross with the first and middle finger from the forehead to the breast
and from the left to the right shoulder, invoking the Trinity. The
other, called _signar_, consists in making, with the thumb and first
finger crossed, or with the thumb alone, the sign of the cross on the
forehead, mouth, and breast, praying God by the sign of our Redemption
to deliver us from our enemies. In some parts a third method is often
employed, which peasants will tell you "is from the times of the Moors."
In this the nose is touched as well as the forehead and mouth, with the
thumb-nail, which is kissed at the end. The two forms are usually
combined (_persignarse_), and the invocation is divided as follows:
By the sign of the Holy Cross from our
(forehead) (nose) (L. cheek) (R. cheek) (nose) (chin)
enemies deliver us Lord.
(L. cheek) (R. cheek) (L. shoulder) (R. shoulder)
In the name of the Father, of the Son, and of the Holy Ghost.
(breast) (L. shoulder) (R. shoulder)
Amen.
(thumb kissed).
A good Catholic, say the peasants, must _persignarse_ thirty-three times
in the course of the Mass, "and that would be very well if we understood
the language and knew why we were doing it."
In the south people always take up a handful of water and cross
themselves before bathing in the sea or in a river, some even before
taking an ordinary bath at home. It will be remembered that the Moslem,
when preparing for prayer, washes his nose, mouth, and ears, as well as
his hands and feet, and possibly this elaborate mode of making the sign
of the cross may be a survival of the Moslem ceremonial of purification,
especially when combined with the water. One distinctly Islamic
tradition is seen in the custom of touching anything unclean, if it has
to be touched, with the left hand, the right being put behind the back.
A woman of Andalusia when washing the dead for burial always begins
operations with the left hand, just as the Moslem does, and will not use
the right until it becomes necessary. Thus it is not impossible that the
curious sign of the cross described, like the traditional reason for
insistence on infant baptism, even when the other offices of the Church
are viewed with indifference, may be connected more or less closely, as
the peasants say, with Mohammedan practices.
In the south and west the peasants never put on clean underlinen without
the _persignar_, and previous to the crossing they recite the following
prayer:
"Blessed and washed be the most holy Sacrament of the Altar, pure and
clean, of the always Virgin Mary, Our Lady, conceived without spot of
original sin from the first instant of her most pure human nature.
Amen."
No matter how great their aversion from the Confessional and
indifference to the offices of the Church, the most careless never omit
this invocation when they change their underclothes.
Another prayer, which is universal, reminds one of the--
"Matthew, Mark, Luke, and John,
Bless the bed that I lie on"
of our own peasantry in bygone days. It runs thus:
"_Con Dios me acuesto,
Con Dios me levanto,
Con la Virgen Maria,
Y el Espíritu Santo_."
("With God I lie down, with God I arise, with the Virgin
Mary, and the Holy Ghost.")
As will be observed, the Virgin here takes the place of Christ in the
Trinity. I have inquired of a number of people how the verse goes, and
find it does not vary. They say that evil would befall them if they
failed to recite the lines every morning and night.
The Radicals, Republicans, and Socialists, who are all branded alike as
atheists by the Ultramontanes, understand the people's faith better than
their priests do. The cry of the Church is that the nation is
indifferent to all things holy. But men like Melquiades Alvarez, the
novelist Galdos, Sol y Ortega, and many other leaders of the Lefts,
continually explain that the national quarrel is only with the priests
and the Religious Orders, not with the Church as an institution, for
they recognise and proclaim that religion is an essential part of
Spanish national life.
There is, indeed, no room for doubt that the mass of the people love
God, Christ, the Virgin, and the saints with a warmth and sincerity rare
in these materialistic days. His God is a living God to the peasant of
Spain, his Virgin a mother always prepared to protect him, her image and
those of the saints the most beautiful things in the world to his
unsophisticated eyes. This may seem to the enfranchised intellect a
degrading superstition. But the fact remains that the religion of the
working classes of Spain in the mass does what many a more advanced
creed cannot do, for it carries conviction and comfort to its
possessors.
THE CONFESSIONAL AND CHURCH ABUSES
CHAPTER IV
THE CONFESSIONAL AND CHURCH ABUSES
Something must now be said about the way in which the people refer to
the confessional, and this I will endeavour to do in their own words,
premising that I offer no opinion as to the truth or falsehood of their
stories, most of which have been told me by women. The abuse of the
confessional is such a heinous sin that Catholics of other nations will
not believe what is currently said as to its prevalence in Spain; they
hold that such things are impossible, and it is to be hoped, for the
sake of the Church, that prejudice distorts the popular view, and that
what the working classes in town and country assert to be of frequent
occurrence does not in fact take place. But whatever be the actual
truth, it is impossible to doubt that the people are convinced that the
confessional is habitually abused, and this conviction--which nothing
can shake--constitutes a peril which must ultimately endanger the very
existence of the Church in this country.
When first I was told, several years ago, that the secrets of the
confessional were betrayed, as a matter of course, in the interests of
the rich as against the poor, I flatly said that I did not believe it.
The thing was unthinkable to one brought up in the belief that such
secrets were inviolate. I was given actual instances of domestic
servants sent to confession "so that the mistress might learn from the
priest what the maid had been doing wrong." As my informant was a young
foreigner, born and bred a Catholic, in the employment of a family of
title, and with a somewhat limited knowledge of Spanish, I found it
easier to assume that he had misunderstood what was said in his presence
than to believe that he had accurately repeated his employer's words,
although he declared that the above remark had been made in his hearing
on several occasions. It should be said that he was in a house noted for
its clerical leanings.
A similar assertion was made to a member of my family by the daughter of
a professional man, better educated than most of her class, and touched
with the superficial scepticism prevalent among clever young Spaniards,
whose hatred of the priests and the Religious Orders tends to alienate
them altogether from the religion in which they were brought up. In this
instance I attributed the accusation to prejudice, and attached no more
importance to it than to the young man's story. And as it is not a
matter that can be discussed with practising Catholics, the two
mentioned, and one other, who confirmed their statements, are the only
educated persons whose opinions I can quote.
But among the poor this offence is spoken of freely, and they accuse the
priests, not only of betraying their trust by repeating what is told
them under the seal of the confessional, but also of using the
opportunities it offers to ruin young and foolish women who obey the
Church's order to confess with frequency. Indeed, many working men have
gravely assured me that such is their distrust of the priests that
nothing would induce them to allow their women-folk to go to confession
on any pretence whatever.
The following are some among many stories of this kind, which I give as
they were told me, only omitting the expressions of anger with which
some of them were punctuated:
"I was laundress in a priest's house for several years. His sister
lived with him, and she really was his sister, for a wonder; not
the sort they generally call their 'sisters.' They also kept a
young girl to help in the house, for the priest was well off. One
day my fellow-servant committed a sin, for the devil tempted her to
steal a ring belonging to the Señora. But she could not rest happy
with it, and at last she went to a priest and confessed that she
had stolen it, and asked what she should do. He told her to put it
back, and gave her a penance. So she put it back. And the priest
went and told her mistress, and she sent the girl to prison."
"There are several maidservants in the house of Doña Dolores, and
one of them goes to confession frequently. The others all have to
be very careful what they say before her, for the priest repeats it
all to Doña Dolores, and then it is 'into the street' with those
who have done anything silly or wrong in the kitchen or elsewhere."
A friend of mine--a foreigner--was begged by her servants not to engage
an attractive-looking housemaid from one of the convent training schools
who applied for a situation. "She will repeat everything that is done in
the house to her priest, and he will make unpleasantness for you and us
too. That is done every day here. We who have not had the misfortune to
be brought up in a convent never, if we can help it, take a situation
where a convent-trained girl lives."
"Juan Cabrito was hung through the priest telling the authorities
that he had confessed that he was a murderer. The priest went
straight to the Governor and told him everything Cabrito had said.
He well deserved hanging, and no one thought anything of the priest
betraying his confession. We are quite used to that in Spain."
"I often used to be called in to help to wait in the evening in the
house of a priest who had a _tertulia_ for priests every week. His
niece kept house for him. I have often heard the priests laughing
and joking about the confessions of the nuns. They would imitate
their voices, speaking high up and whining: 'Father, I lost my
temper and spoke harshly to the dog or the cat to-day.' 'How
tedious they are with their dogs and their cats and their tempers!'
the priest who confessed the nuns would say, and then they all
laughed together, very much amused. But it was wrong, for the
priest is forbidden to repeat a confession. I am not very fond of
the nuns myself, but I did not like to hear those coarse men
[_nombres brutos_] making jokes about their penitence."
"It is many years since I have confessed. When I went to confess
before my wedding the priest asked me a question which no man
should put to a decent woman, so I never went again."
"In my last situation my mistress made me go to Mass with her every
Sunday. I had to get up at five in the morning, so as to be back in
time to do my work in the house. Every Sunday she asked me if I had
confessed so that I might take the Communion, but I always told her
I had not had time and would confess next time. I will not go to
confession. I would rather lose my place."
"It is true that I am over seventy, and it is very hard to earn my
bread and pay a penny a day for rent by picking up rags and rubbish
for sale. I am ill too: I have never been well since my daughter
ran away from me to live with a priest. But I do not wish to go
into the Asylum of the ---- Sisters. They not only make the poor
people there confess and communicate every day, but they make them
work quite as hard as I am working now. And in my own place, though
I may be hungry, at least I am not obliged to get up at six in the
morning to go to Mass, and then carry firewood for the convent, as
my poor old brother was."
In another town in the diocese where the rag-picker lived, an old
acquaintance of mine thankfully accepted an opportunity I was able to
obtain for her, through friends, of entering an asylum for aged paupers,
managed by nuns under the supervision of the municipality. That town has
long been markedly Liberal in its politics, and possibly this may have
something to do with the more humane administration of the asylum. With
this instance in my mind I was surprised at the rag-picker's rejection
of a similar refuge for her old age, but further inquiries convinced me
that the rule of the one convent was in truth very different from that
of the other.
"Every one knows that Higuero was the son of the Bishop, and that
was why they didn't hang him. There was no doubt at all that he
murdered his paramour: he was caught almost in the act. How upset
the Bishop was! His son and his daughter married a brother and
sister, and both turned out badly, very badly. The son--Higuero was
his nickname--and the daughter's husband--Pepita her name was--fell
in love with the same woman, and that was the cause of the murder.
If the Bishop had not used all his influence with the Government
Higuero would not have escaped hanging. He was taken away to ----
Prison, and no one ever heard of him again. Of course he was not
really taken to prison, he was allowed to escape. How did we know
he was the son of the Bishop? Very simply. His mother had been _ama
de gobierno_ [housekeeper] in the Bishop's house before he was made
bishop. No, she was never married. She was well provided for, and
the children had some education, but they were bad from the
beginning. I lived for some years in the same tenement house with
them. Many of the priests' children turn out ill. What can be
expected of the children of such bad men?"
These are a few out of hundreds of such stories told. _And the people
believe they are true._
Certain scandals, relating to the disappearance of valuable paintings
from one Spanish cathedral or another, are familiar to all who travel in
Spain. One such incident has always been a mystery to the outside world,
owing to the seeming impossibility of a thief getting access to the
picture in question, which was in a chapel in the cathedral, protected
by a heavy grille extending from floor to ceiling, the door of which was
always kept locked.
The following explanation was given by the widow of a former cathedral
servant:
"I know quite well how it was done. The assistant-keeper of the
keys was on duty that night, his superior having leave of absence
because his daughter was ill. The priest in charge of the chapel
made some excuse to take the keys from him that afternoon. Next day
he and several others were sent to prison, accused of having been
concerned in the theft. They were released in a week, for there was
no evidence against them, and the proof is that not one of them
lost his place. The priest soon after left the city. It was said
that he had been promoted, but no one ever heard of him again." The
husband of the speaker was one of those accused.
A scandal which gave rise to a question in the Cortes was the
disappearance of two valuable pictures from the Cathedral of Toledo. It
appears that these pictures were in a chapel which had been built and
endowed in the seventeenth century by a certain family. Two or three
years ago their descendants claimed these pictures as their private
property, and entered into treaty to sell them to a "foreigner." The
State intervened, declaring the whole contents of the cathedral to be
inviolate. Soon afterwards "it was found necessary to repair" the chapel
in question, and the pictures were taken down "for safe custody"
meanwhile. What happened after that has never been cleared up, but a
"foreigner" and a motor figure in the story, and the chapel is now
without the pictures. No steps were ever taken, so far as the public
could learn, to bring the matter home to any one.
That quantities of valuable old laces and embroideries have disappeared
from the cathedrals and parish churches of Spain there is no doubt. I
know of one case myself in which an antique chasuble was exchanged for
one of cheap jute imitating brocade. The explanation given was that the
old one was worn out, but as it now figures in a private museum it is
difficult not to believe, as the people say, that some money changed
hands with the chasuble.
In the cathedrals each canon had, until quite lately, entire control of
the chapel he served, and was responsible to no one for its contents.
The temptation to sell old lace and vestments and altar fittings, and to
replace them by new, was no doubt great, especially if there is any
truth in the popular belief that the priests in many cases maintain a
home and bring up families like men to whom marriage is not forbidden.
And no one could bring him to book for any change made in the
appointments of his chapel or (in the case of a parish priest) his
church, because, as a rule, no one in authority over him knew what it
contained when he took possession. Even after his death it would
generally be impossible to prove peculation, did the superior officers
of the Church desire to do so, for it is a rare thing for any cathedral
or church to keep an inventory of the valuables it is supposed to
possess.
It is said that the priests in many cathedrals and parish churches allow
their linen vestments and altar fittings to be taken away from the
precincts for laundry purposes. The facility with which valuable old
laces can be exchanged for modern machine-made stuff in these cases need
not be dwelt on.
Another opportunity for those who wished to profit by the sale of church
treasures was said to be afforded by the fact that fabrics, sometimes
several centuries old, stand in occasional need of repair. I have heard
the "store-room" or "workshop" laughed at by employees of the church.
"Once anything worth money goes into the store-room for repairs we never
expect to see it again."
"And where is this store-room?"
"Don't you know? The dealers in antiques can tell you."
The hostility of the people towards the priests doubtless colours their
views in these as in all other matters relating to them. But it is a
fact that a distinguished Spanish archæologist a few years ago was
refused further access to the archives of a certain cathedral after he
had asked the Chapter to permit him to publish an inventory of the
treasures under their charge. Now, I am glad to say there are at any
rate some dioceses in which all this has been changed. The archbishops
have had the contents of the churches examined and catalogued, to the
annoyance of certain persons, but to the satisfaction of the
parishioners, who obtained no benefit from the sale of the Church
treasures under the old system.
The following incident was reported to the Press at the end of the year
1909. I have not seen any contradiction published, and I give the story
for what it is worth.
In one of the great cities a certain church was condemned as unsafe, and
the congregation were told that ere long they would have to attend other
churches in the neighbourhood. One of the Religious Orders entered into
treaty for the purchase of the condemned building, in order to build on
the site. But nothing was settled, and as the danger of collapse was not
immediate the services continued to be conducted as usual. When the time
came to collect money for Masses to be said on All Souls' Day, the
parish priest found his usual request for alms refused, on the ground
that the ---- Brothers had already been round to say that the church was
given up and its congregation attached to the Brothers' Church in such a
street, and this being so they had come to collect the payment for the
All Souls' Masses, which was usually given to the parish priest. He
indignantly reported the affair to his superiors, and so it got into the
papers.
It was added that the priest declined to say the Masses for the dead, as
he had not been paid for them, and the ---- Brothers, although they got
the money, provided no special service for the congregation who had paid
for it. So that the souls for whom these poor folk had given their alms
will--in their belief--remain so much longer in purgatory. That the alms
were given by the poor, not by the rich of the parish, is evident from
the donors not knowing that their parish church still existed. The whole
affair throws an instructive light on the relations of the poor with
their Church or their parish priest. Had he been in the habit of
visiting them, or did they make a practice of attending Mass even
occasionally, the mistake could never have arisen. But, as the story
shows, the priest had no intercourse with his people save when he went
to beg from them. The incident, even allowing a wide margin for
journalistic exaggeration, goes a long way to support the assertion of
the woman who gave as her reason for not going to confession, that "the
priest would only ask her for money, which she wanted more than he did."
One more case, and I have done with this unattractive subject.
Some twenty years ago a large dole to the poor, which had been given
annually for about four centuries in a certain chapel, was suddenly cut
off, and has never been renewed. It came out that the priest in charge
had sold the bonds in which the capital was invested, with the
connivance of a Government official in the Finance Department, and the
two between them spent the money. The priest was convicted and
imprisoned for a twelvemonth. Then he was released and appointed to
another church in the same diocese. My informant said he had been a
witness at the trial. "And to-day," said he, "that bad man holds the
sacred Elements in his hands, and gives the people his blessing. Such
things ought not to be allowed."
THE POOR AND THE RELIGIOUS ORDERS
CHAPTER V
THE POOR AND THE RELIGIOUS ORDERS
My readers may be inclined to think that the Religious Orders are a kind
of King Charles' head, which I, a twentieth century Mr. Dick, am unable
to keep out of this book. The truth is that in an attempt such as this
to make intelligible the views and aspirations of the working classes of
Spain, the Religious Orders are the central and dominating fact which
overshadows everything else. Whether we discuss the material condition
of the poor, their education, their political disabilities, or whatever
it may be, and make any attempt to analyse the matter and discover the
reasons of their deplorably backward condition, we always get back to
the Religious Orders as the cause--if not in actual fact, at any rate in
the firm and unshakeable conviction of the people--of all their
misfortunes.
It must be remembered, in connection with the Religious Orders, that the
position of nearly all of them in Spain is illegal. According to the
Concordat, made before the expulsion of Isabel II., the only Orders
allowed in Spain are those of St. Vincent of Paul, St. Philip Neri, and
one other, to be nominated by the Pope by agreement with the Government,
while all closed Orders of nuns are prohibited. The Pope has never yet
named the third Order, and apparently no steps are ever taken to oblige
him to carry out his part of the bargain.
I will now give--generally in the words of the narrators--typical
instances of the way in which the Religious Orders are said to interfere
with the livelihood of the working classes, and of the manner in which
once wealthy families have been brought to ruin through their
machinations.
The porter of a Jesuit college--for the servants of these institutions
love their employers no better than do their friends and relatives
outside--told his brother, who told me, that every night during the
first two or three weeks in August, 1909, after the Barcelona riots,
refugees were admitted to the college. At least eighty, he said, came in
all. They slipped in secretly, after the lights were out, disguised in
lay dress, often of the poorest description, having travelled half dead
with fear [_muertos de miedo_] from Cataluña. That the porter's story
was true was proved by the large purchases of provisions made by the
college during that month. A baker told me that the _frailes_ were more
insistent than ever that all the waste bread should be given to them
"for the poor." And, he added, the "good Fathers" were already buying
twice their usual supply of him. "The _frailes_ always demand all the
bread we put by for the poor," said my friend. "We would prefer to give
it direct to the poor ourselves, for we do not feel sure how much of it
they get from the _frailes_, whose house-keepers are great hands at
making _pasteles_ and _dulces_[7] for sale to good Catholic families.
These good Catholic families prefer to buy their _pasteles_ cheap from
the friars, who say that they are sold for the good of the Church. We do
not care to give our stale bread to be used in injuring the trade of our
companions the confectioners; for the friars, having no taxes to pay,
can naturally undersell ordinary tradesmen, and all the more when they
get the bread for their confectionery free. But if we said that we
wished to give our bread to our own acquaintances among the poor, the
Jesuits would ruin us. They would tell all their clients that we were
bad men and enemies of the Church, and we should lose all our trade. We
know this by experience. So we give our stale bread to the _frailes_
and they let us live. But the poor are getting no bread from the
_frailes_ since the Barcelona business."
During the disturbances in Cataluña it was said that "shiploads" of
monks and nuns were being landed in the middle of the night at sundry
ports along the coast, and that they so effectually betrayed themselves
by their nervousness of manner that the country people had not the
slightest doubt as to who and what they were. But as the people had no
desire to injure them personally--notwithstanding a certain amount of
talk about cutting throats and hanging--they were permitted to pass
unmolested, though it is true that there were occasional scowls at
ill-clad individuals who wore their trousers "with a difference," as
though they missed the flowing skirts of their cloth.
And it must be remembered that at the very time that these frightened
men and women were travelling the country in disguise, numbers of
families were sorrowfully bidding farewell to sons, brothers, and
husbands, on their departure to the war which, as the people will always
believe, was begun in the interest of Jesuit capitalists, sheltering
their ownership of the Morocco mines and the great steamer companies
behind the names of lay millionaires.
The popular suspicion of Jesuit interference in these, as in almost all
the other big commercial concerns in the Peninsula, may or may not be
justified, but its effect on the attitude of the people towards the
Religious Orders cannot be over-rated. Not the least extraordinary
feature in the situation is that the Religious Orders profess to
disregard the feeling that exists against them, although it is apparent
on every hand to any one who goes about with eyes and ears open.
For years past I have noticed that no member of the working classes
salutes a priest or friar in the streets. Day after day one summer I saw
the same priests taking their afternoon walk along the same by-way,
where the same artisans, to the number of twenty or thirty, watched the
"long skirts" from the doors of their workshops. I never saw an artisan
greet a priest or friar, or vice versa. The flowing robes of the
ecclesiastics swept against the patched garments of the workmen, but no
glance was exchanged. The priests kept their eyes bent on the ground,
one hand grasping the skirts and the other pressed on the breast, a
typical attitude, which is jeered at by the poor as "canting." The
workmen kept their eyes fixed on the work on which they were engaged. It
is impossible to imagine anything more hostile than the silent defiance
of the men, as they turned to watch the "long skirts" out of sight. I
have seen single instances of the same thing elsewhere in many places,
but here I had special facilities for observing the daily exhibition of
armed neutrality, owing to the accident of my room at the back of the
little country hotel looking out on the by-path. "I hate to see them,"
one of the men said to me; "they are the ruin of us and our country."
What made it the more significant was that these same workmen had a
pleasant word of greeting for every lay person, man or woman,
acquaintance or stranger, who passed by them.
The economic question bulks largely among the causes of the popular
hostility to the Religious Orders, and if only half the complaints
generally made are based on fact, the people have reason on their side.
Formerly, say the women, it was easy to obtain a day's wage by washing
in well-to-do houses, and a laundress could make a decent living. Now in
every town of any importance there are one or more convents called
"Domestic Colleges," where orphans or servants out of place are
received, and these girls repay the nuns for their board and lodging by
doing laundry-work for rich Catholic families. If the girls were allowed
to keep even a portion of what they earn, the women say that they would
not feel the system to be so unjust. But they declare that this is not
the case. Whatever is paid goes to the nuns, and as they, having no
taxes or wages to pay, can undersell the laundresses, who are called
upon to provide both charges, the lay laundry trade is steadily
declining, although the quality of the work is on a par with that done
in the convents.
The nuns teach their protégées every class of needlework, lace-making,
and a kind of embroidery or net-work which is largely used for priests'
vestments, altar-cloths, &c. This competition, which was one of the
reasons given for the presence of women (if, indeed, they were present)
in the attack on the convents in Barcelona, is felt in every part of
Spain, but perhaps especially in the south and south-west, where skilled
needlework, which is almost the only employment of women above the
domestic servant class, is exceedingly common and badly paid at best.
Nowadays, say such women, it is increasingly difficult to obtain
employment of this kind at any price, owing to the quantity done in the
convents and the reduced prices at which the nuns undertake it.
And finally, although this does not so directly affect women, the nuns
do a large trade in the sweetmeats and _patisserie_ already referred
to.
The grievances against the nuns, then, are chiefly related to their
interference with the industrial market, and this, although a very real
source of hardship, has not yet, except in a few isolated instances,
given birth to anything like the active hostility that is expressed
against the male Religious Orders. The closed Orders of nuns are
regarded with aversion and contempt, as living at the expense of the
nation and leading lives which, to the working classes, seem purely idle
and self-indulgent--"doing nothing all day but pray for their own souls,
or worse," as they describe the life of contemplation. I have never
heard working women express any desire to injure the nuns, much though
they dislike them as a class. But when we come to the Jesuits, Maristas,
Carmelites, Franciscans, Dominicans, and all the long list of Orders
lumped together in one condemnation by both men and women under the
derisive name of "long skirts" (_sayones_), we find a much worse state
of affairs.
The people declare that in many places the leading industries have been
completely ruined by the competition of persons in the employ of the
Jesuits--for they call all the friars indiscriminately Jesuits, although
they are perfectly aware of the distinctions between the various Orders.
And they will point out to you one family after another who have been
reduced to penury by the "good Fathers," and will relate innumerable
instances of the methods they believe to have been employed to this end.
Perhaps the best way to explain these will be to give a literal
translation of one or two of the stories I have heard from the people,
which, told as they are with an absolute conviction of their truth, show
what ground the working classes have for distrusting and detesting those
who ought to set an example of virtue and self-abnegation.
"A man I know saved 5,000 _duros_ [about £1,000], and he lent it all to
the Jesuits for a building they were putting up, a building attached to
the monastery, so that he looked upon it as a work for God. Some years
went by, and my friend was growing old and wished to retire from his
little business and live on his capital. So he asked the Jesuits to
repay his loan. 'Oh, no!' they said, 'do you not know, my son, that he
who lends to God must expect no return? A loan to God is a gift for the
salvation of thy soul.' As I say, he was an old man, and he found
himself ruined, without hope of earning more money. He left the good
Fathers and went and cut his throat."
"Why is the old Marquesa de Fulano starving? I will tell you. When
her father, the last Marquis, was dying, the Jesuits never left him
for a moment, and at last they persuaded him that his soul was of
more consequence than his daughter's livelihood, and he made a will
by which he left all his money to found a college for boys in ----.
When he was dead his daughter discovered that all she had was the
family land, and not a farthing of the capital her father had
invested. Soon afterwards a famine came, and there was no rain for
nine months. The Marquesa gave food to all the labourers on the
estate, although there was no work for them, for she is a very
charitable lady. She spent all the money she had, and then sold all
her jewels and other valuables to buy them food. You see, she is a
widow without family to advise and help her. Of course she was too
proud to betray her poverty, but even if she had told her friends
they could have done nothing, for many landowners were ruined that
year. Now the estate is mortgaged to the last acre, and she has
sold everything she has and is almost without food for herself. If
you wish to hear about the Jesuits, ask the Marquesa de Fulano! And
you will understand that all the people employed on the estate lost
their livelihood too, for it is now long since she has been able to
afford to have it cultivated."
"Yes, it is a pity to see that fine old oil-mill falling to ruin.
It used to belong to a very rich man, but when he died the Jesuits
got hold of his widow and induced her to build a large new chapel
in the monastery of ----. Millions of pesetas they squeezed out of
her for the work, and when it was finished, there was nothing left
of the business. One of the sons meanwhile became a Jesuit, and as
they have a big business in oil over there he naturally took the
olive-groves for his share of the property. This happened twenty
years ago: the younger brothers are married and have children to
bring up. They have to earn their bread as they can. One of them
rents ten acres and cultivates them himself, so he does not starve,
but the other poor fellow has taken to drink, and he and his family
mostly go hungry. It is all the work of the Jesuits."
There are many such stories of gifts made to the Church _in articulo
mortis_. The priests are said to urge the dying penitent to save his
soul by benefiting the Religious Orders instead of providing for his
family, on the ground that if he acts as his duty and instincts dictate
he will lengthen his stay in purgatory. There seems no room for doubt
that many once wealthy families have been reduced to poverty in
consequence of such legacies to the Church. Indeed, it almost seems as
if a new class of society is gradually arising among the very people who
formerly were the strongest supporters of the Church--people of good
birth and gentle breeding, with a family tradition of injury at the
hands of the Jesuits, which has alienated them for ever from a Church to
which they owe their worldly misfortunes, and is converting them into
earnest recruits to the cause of Free Thought. From these men of gentle
breeding will eventually come the leaders of the people in their final
struggle against the Ultramontanes.
The ways which the Jesuits are reputed to employ in order to ruin those
who defy them are many. The following story shows how easily it can be
done, if it is true that the Company of Jesus condescends to such
contemptible action against the industrial and working classes.
"Francisco Mengano used to have a very good business. He employed
nine men to work for him. But he hated the friars, and he used to
talk against them to a man who pretended to think as he did and
came to sit with him every day, and encouraged him to say he would
never let his daughters go to confession because he was afraid the
priest would make love to them, and many other things. That vile
man always talked in the same sort of way himself, and poor
Francisco looked upon him as a friend. But when his eldest girl was
old enough for her first Communion and Francisco refused to let her
go to confession, he discovered that the man he trusted was himself
a Jesuit, and had told the Jesuits everything Francisco had said.
They waited to be sure his daughter was not going to confession,
and then set to work to ruin him. It was quite simple. He was a
cart-builder and wheelwright, and depended on the landowners in the
neighbourhood for most of his work. The Jesuits merely sent word
round that he was charging too much and doing bad work, and his
trade was ruined and he became what you see--a poor old jobbing
carpenter, who cannot even afford to employ a boy to do his heavy
work."
When I heard this story I recollected that about a year earlier, when
passing Francisco's workshop in company with a gentleman reputed to be
friendly with the Jesuits, he had remarked, _apropos des bottes_, "Don't
employ Francisco if you should want any carpentering done; his work is
bad and he overcharges abominably." It naturally did not occur to me
that this observation could have any other object than to save me, as a
foreigner, from being cheated, and, all unconscious of what I afterwards
discovered to be its injustice, I gave my work elsewhere.
Not only do the people accuse the Religious Orders of depriving them of
employment by underselling them and destroying their trade by slanders,
but they also bring grave charges of indifference, if not actual
brutality, to the poor who ask them for help of any kind.
It seems to be a fact that no assistance was volunteered by any
Religious House during the epidemic of typhus in Madrid in 1909. In
another town, where a seminary for priests was temporarily converted
into a hospital for the sick and wounded from Melilla, the cisterns ran
dry one night owing to the unusual quantity of water used for the
invalids. Not a man or a boy among the seminarists would take the
trouble to pump more water, though a quarter of an hour's work would
have done all that was needed for the time, so workmen had to be fetched
in the middle of the night to supply what was immediately required for
the sufferers. This I heard from one of the men who did the work.
The working classes have as yet no plan of campaign against the
Religious Orders. They are waiting in the hope that at no distant day
they will have the suffrage in fact, not, as at present, in name only.
But the bitterness of their hostility may be judged from the following
incident, related to me by an eye-witness.
Three country people, dealers in charcoal, were sitting in a tramcar. My
informant was sitting immediately behind them, and at his side was a
priest. One of the charcoal-merchants, pretending to be unaware of the
priest's presence, related how he had been overtaken by night on the
mountains, where he was buying wood in pursuit of his trade, and how he
had gone to a large Jesuit college standing alone on the hillside, to
ask permission to sleep under the portico, the season being mid-winter
and the weather bitterly cold. The "good Father" who opened the door at
his knock refused to admit him, telling him that "the college was not a
house of call for tramps, and he could go and sleep under a tree by the
roadside." The narrator had no option but to do this, for the door was
shut in his face, and "he thought he would have died of cold before
morning." "I wish," he concluded, "that all the _frailes_ in Spain would
come to my house some cold night and ask for shelter. Before morning I
would leave every one of them under my trees with his throat cut."
I have no doubt that the charcoal merchant uttered his theatrical threat
on purpose to frighten the priest, and if that was his object he
certainly succeeded, for the poor man turned white and trembled with
alarm; but it is certain that no one of his class would have dared to
express such sentiments before a priest or a monk previous to the
Barcelona affair.
I have heard a gentle-looking old woman say deliberately: "I wish all
the _frailes_ were going out to be shot this morning! How I should enjoy
seeing them killed!"
And I have heard an artisan remark as a couple of "long skirts" went by:
"How I hate those vermin! It makes me sick to see them near me."
The people who say these things are not Socialists nor Anarchists, nor
even Republicans. They are decent, quiet, industrious working people,
who know and care little about current politics, and simply judge of the
priests and the Religious Orders by what they see. Once the confidence
of such people is won, you will hear similar remarks by the score
wherever two or three of the working class are gathered together,
whether in town or country. Nor are their wives and daughters one whit
behind the men in their expressions of hostility.
Here is an outburst which I took down word for word from a clever but
quite illiterate working woman. The reference to the "ovens," as will be
seen, tallies with what I have quoted about the bakers, although the
speakers were in different provinces, far apart.
"While we have that lot here we cannot live. The alms which ought
to be given to the poor are given to them. I don't believe they
give to the poor the bread which they beg from the ovens. I believe
they use it all to make _alforjillas_ and _piñonates_ for sale.[8]
They cannot make them without bread. The Jesuits do not make them
themselves. All the monasteries keep _amas de gobierno_[9] to cook
and wash and mend and do everything required. They are quite
independent, answerable to nobody. They eat us up as if they were
ants. They let no one live, meddling with everything that doesn't
concern them. They tack themselves to a lady, an acquaintance, and
she has linen to launder, and they order her to send it at once to
be washed in one of the [religious] houses where poor unfortunates
are taken in. Many of them are the children of the friars
themselves and of the priests. There is a family in ---- Street whom
they call 'Curitas,' five brothers and sisters, all children of one
parish priest [_cura_]. Their 'Uncle Cura,' as they called him,
brought them up and educated them and left them all he possessed.
They were all the children of one mother; it was the same as if the
priest had been married to her. He lived with her and maintained
her all his life. But that is a great sin for the priest! They say
that in your country the priests are allowed to marry. If it were
the same here the Church would be purged of many sins, for all the
priests live with women. If they are faithful to one woman I do not
see what sin there is in it. It is natural. But it is more usual
for them to have many women. I know that one old priest had at
least ten or twelve children in ---- [a low quarter of the town], all
by different women. He brought them all up, and gave them every
year enough for their food. He was very rich. They called him 'the
Prior.' The mothers said to the children: 'Run along; the Prior
will give thee money to buy food.' I know this because my father
used to work for him. The priest often said to him: 'Look, what
pretty children mine are!' He was not in the least ashamed of
owning that they were his."
And here is another statement made by the father of growing lads whom he
was educating as best he could to try for appointments in the Civil
Service.
"You should be careful not to say anything about the Jesuits in
those letters you are always writing," he said. "In Madrid or
Barcelona it may be all very well, but in a little country town
like this you can never be sure how much the 'good Fathers' will
find out. It is well known that Paco, who attends to the registered
letters here, is the son of a Jesuit. Many of the clerks in the
post-offices are the sons of priests or _frailes_, and that is why
honest lads like my sons have no chance of getting a place there.
The Jesuits have a plan by which their sons slip in without the
examination imposed on others. Do you think that fool Paco would be
where he is if he had had to pass a competitive examination? But be
warned! He has been clever enough to learn how to open letters and
seal them up again. The 'good Fathers' have taken care of that. And
if they suspect that you write stories about them, they will take
care to read your letters before they leave the post-office."
In this connection I may mention a curious incident. A book sent to me
from England went astray, and some six months later, after inquiry made
by the English post-office, reached me minus its wrapper, with a note of
apology from the local post-office, explaining that it had been
recovered from a Jesuit college at least fifty miles from the place to
which it had been addressed and registered by the publisher. Why it was
delivered at the college instead of to me was not explained. I thought
at the time it was merely a piece of characteristic Spanish
carelessness, but I was reminded of the occurrence by my friend's
remarks about "Paco" and the post-office.
THE MONARCHY AND THE PEOPLE
[Illustration: THE QUEEN AND THE QUEEN-MOTHER OF SPAIN.
[To face page 111.]
CHAPTER VI
THE MONARCHY AND THE PEOPLE
If Spain at large had attributed the misfortunes of 1909--the war in
Melilla, the outbreak in Cataluña, the suspension of the Constitution,
the attacks on the country made by the foreign Press--to the influence
of Don Alfonso, the throne would have been in greater danger than at any
time since the expulsion of Isabel II., for the whole nation was roused
to indignation by the general conduct of the Clericalist Ministry then
in power.
But happily for Spain, and indeed for Europe, since civil war in the
Peninsula would be an European disaster, not even the most violent of
the Republicans or Socialists taxed the King, the Queen, or any member
of the Royal Family with indifference to the feelings of the people or a
disregard of the sufferings of the poor.
A fact not recognised in England is the extent to which conscription
tends to consolidate the Monarchy in a country where the King, the head
of the Army, enjoys personal popularity among his working-class
subjects.
Under an unpopular ruler conscription would probably lend itself to the
speedy establishment of a Republic. But every year that King Alfonso
lives he binds the Army, which is the very flesh and blood of the
nation, more firmly to himself by ties of personal affection. And
personal affection is a stronger force than political conviction alone
ever has been or ever can be.
In the seventies the Army stood for liberty and the Republic against
Carlism and Ultramontanism, until Alfonso XII. was brought from his
English college and offered to the nation which had seen his mother
dethroned, as the mass of the nation always will believe, at the
instigation of the Church. It was his mother's personal popularity with
the masses which made her son's path comparatively smooth,
notwithstanding the chaos of conflicting interests among which his lot
was cast. His own honesty of purpose, his devotion to his people's
welfare, the Spartan simplicity of his private life, and the personal
charm which he, in common with all his race, possessed, gave him a
higher place in the affections of the nation than is at all realised
outside of Spain, and the greatest hope expressed for King Alfonso XIII.
by the poor is that he may take after his father. "He was a _man_,"
they say. It was for the father's sake that all parties agreed to call a
truce during the anxious months that followed on his premature death,
until his son was born, and it would be difficult to say how many times,
while Queen Maria Cristina held the reins of Regency, the memory of her
dead husband may have turned the tide in favour of their child, when the
Ultramontanes would have used the national unrest to the profit of the
proscribed branch.
From the day that Alfonso XII. breathed his last, the Ultramontanes have
consistently tried to represent the Queen Mother as closely attached to
their party. The accusation is manifestly absurd. No mother would
support a policy directed in the interests of a Pretender before that
which maintains the rights of her own son. It is, indeed, a matter of
history that in order to give the Opposition no excuse for agitation,
Canovas, with true patriotism, recommended the grief-stricken Regent in
the early days of her widowhood to entrust the Government to his
opponents, the Liberals under Sagasta, in order to avoid a contest, so
that to the latter fell the duty of proclaiming to the waiting nation
the birth of Alfonso XIII., on May 17, 1886.[10] Canovas would not have
acted thus had there been any real doubts of the loyalty of the Liberal
party.
The _Imparcial_, in an article dealing with Señor Maura's assertion,
immediately after his fall in 1909, that the Conservatives are the only
bulwark against revolution and the only support of the Throne, recalled
this fact, and added that "without the Liberals the Throne would not
exist now, because the Liberals rescued it from revolution after it had
been shaken by the bloody attacks of Carlism. Without Sagasta, without
Castelar, the Spanish monarchy would not be."
Yet so persistently has the story of the Queen Mother's clericalist
leanings been repeated by those interested in its acceptance during the
twenty-three years that King Alfonso XIII. has been on the throne, that
the mass of the people still believe that she defers to the Jesuits even
in matters in which their interference cannot fail to injure the King in
the eyes of his people--a preposterous misconception, which cannot be
corrected too soon. Quite lately I heard a working woman say:
"She cannot be a Jesuit, as they say. A Jesuit mother could not
have borne such children as hers. Look at the King! He has none too
much love for the _curas_ (priests). Yet we have always been told
that Queen Cristina is a Jesuit! Why should that be said? These are
_cosas de los frailes_ (doings of the friars) 'said to make us
dislike her.'"
In one town where I had some acquaintances among the clergy, I was
struck by the malicious things that were said by them about the young
Queen, and especially about her relations with the Queen Mother. Not
long before I went there I happened to have heard a very pleasant
account of the private life of the Royal Family from a foreigner,
entirely outside of politics, who was for a short time employed in one
of the palaces while the Royal Family were in residence. His description
left no doubt at all as to the happiness of their home life.
With this in my mind I did not feel greatly concerned at being informed
by various Ultramontanes that "Queen Victoria was on the worst terms
with the Queen Mother, who had never forgiven her for having been
brought up a Protestant," and that "Maura had refused to let her go to
England after the Barcelona affair, because she was so miserable in
Madrid that she had declared she would never return to Spain if once she
got back to her own country."
No one who has seen the young King and Queen together believes this kind
of thing, although it has been repeated in clericalist circles ever
since the marriage. But, unfortunately, comparatively few of their
subjects have the opportunity of seeing them, and during the last
half-year of the Maura administration photographs and picture postcards
of the Royal Family, which formerly were on sale everywhere, became
noticeably absent. Throughout the three months that the press was
censured it was almost impossible to find an illustrated paper
containing any picture of the King, the Queen, the Queen Mother, the
Infanta Maria Teresa, or the Royal children. During that time everything
that could tend to recall the King and Queen to the minds of the people
and increase their popularity was suppressed. My attention was first
called to this state of affairs by finding that in one large town not a
single picture postcard of King Alfonso could be bought. The shops had
sold out their last year's stock, and no new photographs of any kind had
been issued since the war broke out.
It would have been natural that portraits of the Queen should appear in
connection with the War Fund initiated by her Majesty and taken up with
enthusiasm all over the country. But no. A portrait and several pictures
of the Marquesa de Squilache, who acted as honorary secretary, were
published, showing that lady at work in her office, distributing money
to applicants, &c. But I have not been able to discover that any such
pictures appeared with the young Queen as the central figure. The
Marquesa de Squilache is a philanthropist whose fame deservedly extends
all over Spain, and the admirable organisation of the fund was certainly
due in a great measure to her clear-headed and business-like methods.
But she would be the first to acknowledge that the Queen, and not
herself, should have been represented in the picture-papers as the head
and front of this effort to alleviate the misery caused by the war. It
is difficult to believe that the marked omission of her Majesty's
portrait in the illustrated papers during the clericalist
Press-censorship was accidental, while at the same time a series of
thirty-six postcards of Don Jaime of Bourbon in the Castle of Frohsdorf
was being freely advertised in Madrid.
The War Fund, initiated and presided over by the young Queen, was
perhaps the first charitable appeal ever issued direct from the Court to
the nation, without the intervention of the Church. At first it was
stated that applicants for relief from this Fund must bring certificates
of birth, baptism, marriage, &c., from their parish priests.[11] But
the _Heraldo_, one of the leading Liberal-Monarchist papers, pointed out
that such a condition would deprive all those who had been married by
the civil authority of participation in the Fund, and put in a further
plea for the children of soldiers not born in wedlock. The Queen and her
committee of ladies decided on the widest interpretation of the family
limitation, and at an early stage in the war relief was given to a child
whose father was at the front, although the mother did not bear his
name. This broadly charitable decision commended the Fund warmly to the
mass of the people, for, as already shown, the prohibitive cost of the
marriage licence in many, if not all, the Spanish dioceses compels
numbers of decent couples to use the civil rite or none. Thus the
decided action taken by, the Queen and her committee, notwithstanding
the recommendations of the Church, endeared Queen Victoria Eugénie to
thousands of mothers who, if the first conditions proposed had been made
obligatory, would have been without the pale.
The interminable lists of subscribers, appearing day after day and week
after week, and the innumerable small subscriptions, often not exceeding
ten centimes, and sometimes falling as low as five, proved how
whole-heartedly the poor gave of their penury, and various incidents
which occurred showed a real spirit of self-sacrifice in the
wage-earners. Such was the action of the cigarette-makers of Seville,
the two thousand women of all ages whose fame has been so often sung in
the opera of "Carmen." They were ordered to make up several thousand
boxes of cigarettes with the legend "For the Army at Melilla." They
immediately asked to be allowed to do the whole work gratis as a tribute
to the Army; and on being informed that this could not be permitted,
because the consignment was a gift to the troops from the Company which
rents the tobacco rights from the Crown, the _cigarreras_ volunteered to
forfeit a whole day's pay, to be given to the Queen's Fund for the
Wounded. Numbers of these women are mothers of families, and many of
them have only three or four days' work weekly, at a wage ranging from
75 centimes to pesetas 1.50, so that a whole day's pay was a serious
consideration to them. Nor were they by any means alone in their
generosity, for many industrial guilds, companies, trade unions, and
civil servants, such as, _e.g._, the minor post-office officials and
telegraph operators, also gave a day's wage.
Judging from the results of previous appeals to the public for
charitable purposes, it is safe to say that the enthusiastic response to
the Queen's Fund was due in a great measure to the national confidence
that the money would be well and wisely administered under her Majesty's
auspices, for it is a melancholy fact that similar confidence is not
felt by the poor in the case of subscriptions raised under the patronage
of the Church.
I have quoted at random a few observations from among many betraying
animus against her Majesty on the part of the priests. Here is another,
which shows why they dislike the young Queen so much. I met one day in a
mountain village a Franciscan friar who had come from a neighbouring
city to deliver a course of sermons. He mistook me for a Frenchman, and
therefore had the less hesitation in enlarging upon the evils that the
King's marriage would bring upon the country. One remark particularly
impressed me, as expressing in a few words the attitude of the Church
towards education.
"She will do untold harm by trying to introduce her English ideas about
the education of women. The women of Spain have quite as much education
as is good for them. More would only do them harm."
In this connection it seems worth while to mention that what most
appealed to the working women (who certainly are not over-burdened with
education) in relation to the birth of the Prince of Asturias was the
announcement that the Queen intended to nurse her baby herself, instead
of following the old-fashioned custom, universal among the upper
classes, of employing a wet-nurse. This is not the place to discuss the
unhappy, results of the system on the general health and morale of the
nation. But the announcement was seized upon by the poor as bringing the
royal mother into close contact with themselves.
"Have you heard that she is suckling her child, just as we do?"
And when soon after it was stated that "owing to the Queen's state of
health, and having regard to the duties of her position" the infant
Prince had been handed over to a wet-nurse like any other rich man's
child, a sigh of disappointment went up.
"You see, the doctors would not let her do as she wished. Health?
Rubbish! Any one can see that she is the picture of health. But what
would become of the commissions the doctors get from the wet-nurses for
recommending them if the Queen put wet-nursing out of fashion?"
The Queen was not blamed for relinquishing her maternal duties. Every
poor mother believed that she would have nursed her baby, had the
decision rested with her. This is characteristic of the attitude of the
mass of the people towards both the King and the Queen. Whatever they do
that is worthy of respect and admiration is taken as fresh evidence of
their intrinsic virtues. But whatever happens in regard to them that
does not please the country people is attributed to the malign influence
of those who stand between them and their subjects.
I was struck by the popular comments on the announcement that Don
Alfonso was not going to Melilla, among which this was one:
"Do we not know that he is dying to go? He is young and brave, and he
loves our soldiers. It is Maura who forbids him to go to the war."
A suggestive remark was made by a journeyman plumber with whom I had a
long conversation while the war was at its height.
"No doubt he would have liked to lead the Army. He is brave enough. But
kings are too expensive to be risked in that way. If we have a king he
may as well be taken care of."
"You do not seem a very enthusiastic Monarchist," I said.
"I? Monarchist! I am republican to the bones."
"Ah! Then I suppose you would like to turn Don Alfonso out of the
country?"
"I? Why? What harm has that boy done me? Everybody likes him."
And he seemed quite puzzled by the smile I found it impossible to
repress at this exposition of "republicanism to the bones."
For fully a year before the fall of the Maura Ministry anecdotes of the
charity and generosity shown by the King, the Queen, and the Royal
Family were growing rarer in the papers which had formerly supplied
these little pieces of information to the many people who like thus to
be brought into contact with the home life of their rulers. The omission
was introduced so gradually that at first no one noticed it. But when
soldiers returning from the war talked of gifts sent out by the Queen,
and other evidences of active sympathy shown by the Royal Family, it was
realised that no steps had been taken to make these things known to the
public at home. The King's gift of thirty thousand solar topees out of
his private purse was one instance. The Queen's present of thousands of
warm vests to wear under the uniform was another. Queen Maria-Cristina,
and the Infanta Maria Teresa (who by her gentleness and unassuming
manner has won for herself an affectionate nickname among the poor of
Madrid), as well as the Infantas Doña Isabel, Doña Paz (Princess Louis
of Bavaria) and Doña Eulalia, the King's aunts, all devoted themselves
during the war to working with their own hands for the soldiers, besides
giving generously to the Queen's fund, but not a word of this appeared
in any of the papers. I heard something of their work from private
sources: the public heard nothing.
It may be suggested that the ladies of the Royal Family, who are
instinct with patriotism and love of their fellow-countrymen, may have
preferred that their charities should pass unpraised by the nation. But
even were that so, one would expect that the expenditure of some £11,000
out of the King's private purse would have been reported far and wide,
especially since it had been impossible to conceal that the troops were
suffering severely from want of proper headgear in the tropical summer
of North Africa. But beyond the bare announcement that the King had
ordered this immense number of sun-helmets to be procured for the troops
in urgent haste, from abroad, because they could not be purchased at
home, no comment was made on an act of truly royal generosity. A Liberal
paper said that information on the subject was held back by ministerial
instructions "until a suitable time for publication arrived," but
beyond the bare fact of the number given and the price said to have been
paid, no further details were ever published. The Conservative organs
confined themselves to commenting unfavourably on the size, shape, and
colour of the new headgear, and one of their correspondents turned the
whole affair into ridicule, describing the soldiers in the new helmets
as "having the appearance of walking mushrooms, which destroyed all that
had hitherto been picturesque in the campaign." But when the illustrated
papers brought out one picture after another in which the men were seen
wearing these solar topees, and the soldiers began to write home to
their families that "the King's helmets" not only protected them from
the sun by day, but kept their heads dry and warm while sleeping on the
damp ground by night, the people scored another black mark against Señor
Maura, crediting him with a deliberate intention to conceal evidences of
the King's care for the soldiers from the people at home.
The vests sent out by the Queen were never mentioned at all by the
Press. Yet my informant, a returned soldier, told me they must have
numbered thousands, for, said he, "there seemed to be enough for all of
us; at any rate, all I knew had them."
It was thanks to these, he said, that there were not many more fever
patients when the torrential rains of October fell on an Army destitute
of winter clothing and even of sufficient sleeping accommodation, so
that for nights at a stretch "men lay on soaked mattresses or blankets
only, sunk in a bed of mud." "The Queen's vests kept us warm in the
middle, and that helped us to bear the wet and cold," he said.
Why was the Queen's gift, equally with the King's, treated with such
discourteous silence under the Press censorship of the Clericalist
Ministry? It was not for want of space in the papers, nor for want of
goodwill on the part of the editors, for full particulars were given of
innumerable generous offerings by commercial houses and private
individuals, and column after column was daily filled with names of
subscribers to the War Fund, which was designated "The Patriotic Fund
presided over by H.M. the Queen," or "The Patriotic Fund under the
Committee of Ladies," according to the political bias of the paper
publishing the lists.
If anything had been wanting to arouse national enthusiasm for the
Queen, her prompt action in initiating this fund would have provided it.
To English people it seems natural that the Queen should undertake the
work, for the Queen of England has been for many a long day regarded as
the head and front of charity organised on behalf of the nation. But
Spanish women, accustomed for centuries to bow to the dictates of the
Church, had come to believe that what the Church looked on coldly could
not be carried out at all, and least of all by a woman. The Church, with
certain exceptions, stood aloof from the Queen's Fund on the pretext
that men of peace might not aid in any matter connected with war. The
nation translated this into a protest on the part of the Ultramontanes
against a national work of charity headed by a Queen who is not popular
with the priesthood. And the response to the Queen's appeal for the sick
and wounded is not only a testimony of the love of the nation for the
Army, but also evidence of its confidence in the Monarchy as opposed to
the Ultramontanes.
A pretty incident in regard to another royal gift made on the first
visit of the young King and Queen to a certain large provincial town may
be worth relating. The usual largesse of so many thousand pesetas to the
municipality, for the poor, was announced in the newspapers when they
left. But by chance I heard how much farther their unannounced charity
had extended. They had given a considerable sum to a convent in each
district of the city to buy bread for the poor, and of this no notice
was taken by the papers. I heard of it from a journeyman painter, whose
sick wife had received two loaves.
"Her aunt is portress at the Convent of ----, so she was able to get her
share. Everybody in our parish was very pleased. The only thing we
should have liked better would be to receive the bread from the King's
and Queen's own hands, so that we might have thanked them as they
deserve. But such a crowd of people would have gone to the palace that
the Queen would have got very tired, which was no doubt the reason why
they did not give us the bread themselves."
Strangely enough, the Queen's Protestant upbringing, which prejudices
the Ultramontanes so strongly against her, has just the opposite effect
upon the people. They look upon her as being, like themselves, a victim
of clericalist injustice, and so deep-rooted is the conviction that
whatever the Jesuits object to must be good for the people, that the
knowledge of their oppositions to the marriage would have been
sufficient in itself to secure her a welcome from the proletariat.
But her hold upon the masses goes deeper than this. The peasants
appreciate, far more than many of the upper classes seem to do, the
vital importance to the nation of a settled Dynasty and Constitution.
They know that for many years the Monarchy hung on a thread, while the
frail life of a little child was all that preserved Spain from the chaos
that another conflict between Republicans, Carlists, and Monarchists
would have produced. Therefore when King Alfonso grew up, married, and
became the father of an heir to the throne, the rejoicing of the nation
was heartfelt and sincere. The discussions which arose in 1905 on the
death of the poor young Infanta Mercedes, the King's eldest sister, as
to whether her son was or was not entitled to be Prince of Asturias in
the absence of a direct heir, had aroused all serious-minded Spaniards
to the ever-present dangers that would take shape in action should King
Alfonso die unmarried or childless. So that when the birth of the little
Prince of Asturias--the first son born to a reigning King of Spain for
over a century--was speedily followed by that of a second, the poor,
always the worst sufferers from civil discord and changes of Government,
learnt to look upon the young Queen who has given these hostages for
peace to the nation, with a feeling compounded of admiration and
affection. And each fresh child that comes to fill the royal nurseries
seem a fresh bulwark to the State in the eyes of the working classes,
who remember how their own flesh and blood were thrown to the dogs of
war time after time by opposing forces during the century when Spain had
either no King or no Crown Prince.
THE REVIVAL OF CARLISM
CHAPTER VII
THE REVIVAL OF CARLISM
For a long time past it has been assumed abroad that Carlism is dead in
Spain, and probably few even among diplomatists in other countries could
say off-book what the proscribed branch of the Spanish Bourbons now
consists of, where the different members of the family live, and what
relations they maintain with each other and with the country from which
they have been exiled since 1876. Even so careful an observer as Major
Martin Hume wrote in 1899 that Carlism as a political system was dead in
Spain, and the absolutism upon which that party pin their faith, past
revival.[12]
It is, therefore, hardly surprising that the Carlists have been left out
of account by those who observe Spanish political troubles from outside.
Indeed, the very existence of the old Pretender seemed to have been
forgotten by the generation which has grown up in England in the
thirty-four years which have elapsed since the close of the last civil
war. But the Spanish working classes have not forgotten Don Carlos, nor
have they for a moment lost sight of the continued existence of this
party in their own country.
The root of their long memory lies in their antipathy to the Religious
Orders. To the people the Carlists are indissolubly associated with the
Ultramontanes, and who says Jesuit says Carlist in their vocabulary of
distrust. And that the people have reason on their side has been proved
by the words and actions of the Ultramontanes themselves since the
events that took place in the summer of 1909.
The Catalonian question has been discussed at such length and with so
much confidence by writers living in other countries that I may be
forgiven if I add to their pronouncements on the causes and effects of
the "Red Week" certain information which is not common knowledge beyond
the Pyrenees, unless possibly at the Castle of Frohsdorf or in the
palaces of the proscribed branch in Venice or Trieste. The censorship
exercised over the press, and over telegrams and even letters addressed
to newspaper offices, was so severe while the country was under martial
law, and, indeed, right up to the fall of the Maura Cabinet in October,
1909, that representatives of foreign journals who tried to put the
facts before their leaders found it impossible to do so. Any man who had
once tried--and failed--to get off even the most cryptic telegram
relating to the part played by the Ultramontanes in the riots, was
thenceforth marked by the Intelligence Department of the Society of
Jesus, and if he continued his efforts to communicate what he knew, he
not only found his telegrams suppressed after they had been accepted and
paid for, but stood a good chance of having his personal liberty
interfered with. There was plenty of excitement about the work of a
foreign correspondent in Spain in the summer of 1909, wherever he
happened to be stationed. But it was not precisely the form of danger
suggested by the reports of revolution and anarchy which were supplied
to the foreign Press. Notices of that class could be procured and sent
through without the slightest difficulty. The newspaper correspondent
who was in danger was the man who crossed the frontier to telegraph the
facts as he saw them, and who was not unlikely to meet with an
"accident" as he made his way back to the scene of his labours.
The result of this regime of espionage was that all Europe was
hoodwinked as to the real crisis in Spain, for naturally as soon as
affairs in Cataluña ceased to be sensational, the foreign Press relapsed
into its usual indifference to what was going on in the Peninsula.
Foreign residents, living as quietly and comfortably as usual, were
considerably astonished when their home newspapers reached them, packed
with sensational tales of revolution, incendiarism, military sedition,
and wholesale executions, and asked what on earth the Spanish Government
was about that it allowed these slanders to be propagated all over the
world? Who was responsible for these perversions of the truth? What
advantage was hoped for by those who fostered so colossal a
misrepresentation of the conduct of the inarticulate proletariat of
Spain?[13]
The Censor handled the national Press even more sternly than the
foreign, for the suspension of the Constitution gave the Government a
perfectly free hand, and although the Constitutional rights were
nominally restored everywhere except in Cataluña a few weeks after the
rising, the Press was in reality gagged as long as Señor Maura remained
in office. The Opposition indeed was placed on the horns of an
impossible dilemma. So long as the party kept silence as to what they
knew, Spain would continue to be held up to foreign contumely for a
condition of affairs which did not exist. Yet if any Liberal dared to
criticise the Government, he was clapped into prison until such time as
it might suit the Government to release him.
The position was recognised by Señor Moret, the veteran
Liberal-Monarchist leader. He possesses the invaluable quality of
knowing when to speak and when to keep silence. And throughout the time
when the fair fame of his nation was being dragged in the dust, he urged
patience and submission upon his followers, pointing out that when the
time came to call the Ultramontanes to account for their conduct of the
Government, the strong men of his party must not be found in prison, for
it would be their business to speak: and the Liberal-Monarchist
statesmen, without exception, supported their leader in his patriotic
policy.
Only a very strong man could have controlled the rising tide of wrath
against the Religious Orders, whom the people hold responsible for
everything that goes wrong with the nation. Señor Maura, the leader of
the Ultramontane party, is supposed, by those who do not know the facts,
to be the only really strong man in Spain, and it appears to be honestly
believed abroad that he holds the Conservative party together by sheer
force of statesmanship. The truth is that Maura is a weak man who owes
his position as the leader of the party he is supposed to control only
to the unflagging energy and intrigue of the Ultramontanes--the richest
men and the subtlest intellects in the Peninsula; while Moret's power,
on the other hand, is based upon unswerving political rectitude,
maintained against the onslaughts of corrupt politicians, and upon his
capacity for silence among men who spend half their lives in talking.
This is why he has obtained such a hold upon the people that the whole
forces of political immorality have laboured to bring about his
overthrow each time that he has taken office, lest he end by leading the
nation into paths where corruption will have no standing ground. Maura's
policy of repression gave a great impetus to the revolutionary spirit
against which it professed to be directed. And yet Moret's influence was
strong enough to keep the nation quiet, because the nation trusts him.
For once the low level of popular education, which Moret and his
followers are working hard to raise, was on the side of the Liberal
leader. Only some twenty-five per cent. or so of the nation can read,
and of that number few indeed know any language but their own. Had the
working classes realised that the Army, of which all Spain is so proud,
was being traduced by the foreigner, neither Moret nor Maura could have
controlled the storm of wrath that would have overwhelmed those held
responsible for the lie.
Happily for Spain, the syndicate of newspapers known as the _Sociedad
Editorial de España_, which is edited under the direction of the
Liberal-Monarchist party in Madrid, and read by thousands, as against
hundreds of readers of the journals which support a different policy,
never wavered for a day in upholding Moret's recommendation of patience
and submission to the law, and refrained from increasing their sales by
pandering to popular excitement with allusions to what was going on
outside of Spain, notwithstanding the grossest insults from the
Ultramontane press. Those who control the organs of the party knew well
enough that if they had raised the cry of "Down with the Jesuits!" they
would have called up a tempest not easily or speedily to be allayed. But
they knew that their adversaries wished for nothing better, and they
kept on their own course and saved Spain from a violent revolution
against the Church. The syndicate of Liberal-Monarchist papers is
continually accused by the Ultramontane press of being responsible for
the attack on the religious houses in Cataluña, and is held up to
reprobation for "encouraging the destruction of the country by
maintaining the right of the people to have lay schools." But the truth
is, and the Ultramontanes know it, that to the Liberal-Monarchist press
is due the present security of innumerable buildings belonging to the
Church, which, but for the influence of the Liberal party, would be in
smoke-blackened ruins to-day.
Many members of the educated middle classes of Barcelona assert that the
disturbances in Cataluña in July, 1909, were deliberately instigated by
the Jesuits. The object was, they say, by hook or by crook, to close the
lay schools, and that it had long been an open secret that the
Ultramontane party were determined to take the first excuse they could
find to destroy an educational movement which they find inimical to
their interests. And the link connecting the Jesuits of Barcelona with
Don Jaime, the son of the recently deceased Pretender, Don Carlos, was
provided by an indignant disclaimer of Carlist participation in the
affair, published by a Carlist organ edited in Paris, long before any
suggestion had been made that such participation was suspected. This was
so clearly a case of _qui s'excuse s'accuse_ that no thoughtful
observer, unbiassed by political passion, could fail to put two and two
together.
The peasants had no doubt whatever as to the origin of the disturbances.
One of them gave me his view of the situation, as follows:
"The Carlists and the Jesuits plotted to turn out Isabel II., and now
they are trying to overthrow King Alfonso. They ruined Queen Isabel
because she loved the people and hated the priests, and now they are
trying to do the same with _los Reyes_ because they are popular in the
country. But let them try! There are still many of us who remember what
we suffered in the last Carlist war, and we do not intend to have
another. Let them try to touch _los Reyes_! We will kill every priest in
the country before they shall put a hand to that work!"
"Well," I objected, "this is very fine talk now that Don Carlos is dead
and buried, but if you did not want him for your king, why did so many
of you fight for him?"
"If you were ordered to fight and knew that you and your children would
be put in the street if you refused, would you not fight rather than let
your family starve? The men who paid our wages said: 'You will go with
us to fight for Don Carlos or you will never have another day's work
from us.' What help had we against our masters? Do you think we wished
to take arms against Queen Isabel? If the Jesuits had not been in the
affair, no one would have taken notice of her little faults. The Jesuits
intended her to commit faults when they married her to a man who was no
man. If she had not been good to us they would have let her stay."
The popular songs expressing loyalty to Queen Isabella have not been
forgotten in the forty odd years that have elapsed since she was
dethroned, for in 1909 they were revived, with such slight alterations
as were needed to bring them up to date. Here is a specimen:
"_Si la Reina de España moriera
Y Don Carlos quisiera reinar,
Los arroyos de sangre correrian
Por el campo de la libertad._"
("If the Queen of Spain were to die and Don Carlos wanted
to reign, the streams would run with blood on the field of
liberty.")
The intricacies of succession being little understood by the people,
this song was modernised by substituting "Victoria" for "de España" and
"Don Jaime" for "Don Carlos." During the suspension of the Constitution
the song was not sung aloud: the people said that "Maura had forbidden
it." Directly the Liberals came in it was heard again everywhere.
Here is another:
"_No reinará Don Jaime,
No reinará, no, no!
Mientras España tenga
Bayoneta y cañon._"
("Don Jaime shall not reign while Spain has a bayonet and
a cannon.")
And another, perhaps in some ways the most interesting of the three:
"_Dicen de Barcelona
De un mitin clerical,
Que Don Jaime asistió
Provisto de un disfraz.
Al ver la bronca de palos y morral
Creyó que le tiraban bomba encima de nocedal!_"
The literal translation is as follows: "They say from Barcelona, of a
clerical meeting, that Don Jaime attended it provided with a disguise.
When he saw the row with sticks and a nosebag, he thought they were
throwing a bomb on the top of a walnut-copse." Taken in its literal
meaning it is of course nonsense, and the popularity of the song was
evidently due to the puns on "_morral_" and "_nocedal_." By writing
these words with capitals we get the names of the man who threw the bomb
at the King's wedding (Morral), and of a well-known Carlist leader
(Nocedal). This song was sung in Barcelona when the colonies were lost,
with another name in place of Morral's, and was revived in 1909, brought
up to date as above, until with the suspension of the Constitution it
was severely repressed. It expresses the popular opinion as to the
authorship of the bomb of 1906 and the troubles in Cataluña.
It was curious to observe how constantly the Carlist and Cuban wars
seemed to be in the minds of the people during the regime of repression.
Frequent comparisons were drawn between the reign of Queen Isabella and
the rule of Señor Maura, all highly unfavourable to the latter. It was
always Maura and the Jesuits: never was the King blamed in Spain for the
sins of his ministers. The Carlist war seemed as fresh in the minds of
the unlettered masses as Mellila itself. Tales were raked up of shocking
cruelty to the rank and file, and of a callous disregard of their
sufferings in Cuba and the Philippines, and it seemed to be believed by
many of the speakers that similar abuses were being repeated in Morocco.
A nation which has been prevented from developing its intellectual life
is inevitably thrown back upon its recollections, and traditions of
class injury cannot fail to be more permanent among a people who have no
other occupation for their thoughts.
For nearly forty years the uneducated Spanish peasantry, and the artisan
classes, have nursed their wrath against the body whom their parents
believed to have dethroned, for their own ends and at the cost of a
bloody civil war, a Queen desirous of ameliorating the lot of her
people; and for ten years of that time their resentment has been
increased by the conviction that the same body plotted to sell the last
of the once world-wide Spanish colonies, and strewed the road to that
sale with the corpses of Spanish peasants, set to fight, without arms or
equipment, against the overwhelming forces of the enemy, while the
Jesuits appropriated to their own purposes the money wrung from the
nation for the expenses of the war.
How much of the indictment against the Jesuits is justified I do not
pretend to decide; but all the world knows that the Spanish Army again
and again went into action in Cuba and the Philippines so destitute of
munitions as to be practically unarmed, while the tragic loss of the
Spanish Navy at Santiago will never be forgotten by those whose friends
and relatives were sent to an inevitable death "by order of the
Government in Madrid."
Had the people been allowed to educate themselves during the years that
have passed since those fatal adventures, the wound, though it will long
remain unhealed, would have been skinned over by consoling comparisons
drawn between these and the great disasters of other nations. But the
Religious Orders have always opposed the spread of popular education.
The people have been driven back upon tradition, old and new, for their
mental nourishment, while other nations have been forging ahead. Thus
the Religious Orders have sharpened a sword for their own undoing. The
longer the Spanish peasant is left to nurse the memory of his
grievances, the more bitter grows his resentment against those whom he
holds responsible for them.
To the Jesuits the people attribute the downfall of Isabel II. and the
years of internecine strife which followed; to the Jesuits they
attribute the fiasco of the Spanish-American War, with all the suffering
it entailed upon the poor; to the Jesuits they attribute the war in
Morocco, with its heavy account of bloodshed, sickness, and money-cost;
and to the Jesuits they impute the chronic unrest in Cataluña, which
they believe to be fostered in the interests of Don Jaime of Bourbon.
They are convinced that all these things were and are engineered by the
Carlist party, being well aware that, as the Pretender himself stated,
in a published document, the "Monarch" of the Ultramontanes has no hope
of entering Spain again, save on the waves of a national revolution,
which would bring misery and desolation to thousands of homes.
Let us now see what evidence may be found in incidents that have
occurred and statements that have been published since the regime of
repression was abolished, to support the popular theory of
Jesuit-Carlist intervention in the events of 1909.
The belief, industriously promulgated in Spain and abroad, that Ferrer
engineered and conducted the July outbreak, fell flat, generally
speaking, among the Spanish working people; always excepting, of course,
the more educated elements in the larger industrial cities.
"They are saying that this Ferrer, whoever he may be, paid Morral to
throw the bomb at the King and Queen. If that is true he deserved to be
shot. But others say that the Jesuits themselves paid Morral, and
others again say that Don Carlos employed him.[14] What do _I_ know
about it? The only thing certain is that the Jesuits had a hand in this
Barcelona business, for they have a hand in everything that is bad for
the country. Where the Jesuits are, there are the Carlists also. As for
this Ferrer, who is he? We never heard of him till Maura shot him."
This commentary on "the execution of an anarchist" or "the martyrdom of
an enthusiast" is, of course, that of the peasantry alone, not that of
the Republicans and Socialists, to whom he was well known long before
the Barcelona riots took place. But let it be remembered that the
unlettered peasant forms the great majority of the working classes in
Spain.
The calm with which the mass of the nation regarded the affair was,
however, shaken when a report got about that Ferrer was denounced by or
at the instigation of a Dominican monk--even a name being given--who,
having quarrelled with him some years before, had determined to contrive
his destruction. I do not say that there was any foundation for this
vague story. But its ready acceptance as exculpating
[Illustration: SEÑOR MAURA.
Leader of the Ultramontanes.
[To face page 149.]]
Ferrer, by those who had previously been indifferent or hostile to him,
shows how the people twist everything to the prejudice of the Religious
Orders, and believe all evil possible to them. Had the Liberal papers
lent themselves to agitation then, the result might have been serious.
No better incitement to riot could have been found than the story of the
Dominican. The death of Ferrer in itself left the mass of the people
unmoved, but the ease with which churches and monasteries were destroyed
in Barcelona had already set many aggrieved people thinking how easy it
would be to follow Barcelona's example in other towns where the Jesuits
are numerous, and only a leader and a party-cry were needed to raise the
working classes against the Religious Orders.
The whole of the Opposition Press, however, in spite of great
provocation, as usual stuck to its guns and steadily, continued to
condemn violence and to point out that the duty of the nation, unjustly
deprived of its constitutional rights, was to prove by its
self-restraint and moderation how entirely it was worthy to be trusted.
"If we could only kill Maura without hurting the King," a working man
said to me, "he would have been dead long ago, for he is the cause of
all our troubles. But the Jesuits would make out that any act of
violence on our part was directed, not against Maura himself, but
against the party which is supposed to support the King; they would
never admit that it was only their friend Maura whom we were attacking,
and it would be made to appear that we were trying to overthrow the
Monarchy. That is why Maura is still alive." The conviction, and the
rancour expressed in these words, cannot be rendered in print.
The speaker could not read or write. He and some ten or twelve of his
friends were in the habit of meeting quietly together after nightfall,
when no priest or Jesuit was likely to see them. One of the better
instructed, generally a reservist who had "got education while serving
the King," would read aloud to the rest, and all would discuss the
pronouncements of their chosen newspaper and form a collective opinion
on them. I have sat with many such groups, in small towns and country
villages, and have taken care to notice what newspaper they read. It was
invariably the _Libéral_. It often struck me, during the three months of
"repression," that Señor Maura literally owed his life to the organs of
the so-called "Trust," which he and his party accuse of working hand in
hand with the anarchists; for the sentiment recorded above was
expressed in my presence many times by members of the working classes in
connection with Barcelona, the war, the want of education, and all their
other grievances. Maura, the Jesuits, and the Carlists, are regarded as
one by the mass of the nation, and the three-fold hostility is
concentrated on each member of the trinity in turn.
When you have some twelve or thirteen million people brooding over their
grievances, and cherishing the conviction that a certain party in the
State refuses to recognise or redress those grievances because by
preserving the _status quo_ they put money in their own pockets, the
situation becomes serious for the party to which such action, or
inaction, is attributed. And it must be remembered that though the
Spanish peasant can read but little of the literature disseminated by
revolutionaries--anarchists and such-like--across the Pyrenees, an echo
of their campaign can hardly fail to reach him sooner or later.
Several little incidents occurred about this time which, though trivial
in themselves, lend support to the popular view that the Carlists were
at the bottom of the trouble.
Thus we have the disclaimer of participation in the outbreak, by the
_Correo Catalan_, the official organ of the party, synchronising with
the publication of extracts from a "forthcoming" manifesto of Don
Jaime, suggesting the possible expulsion of the King by a revolution, to
which I have already alluded; while early in August the Ultramontane
journals said that a quantity of weapons, which they allege to have been
taken from the hands of the mob, were stored in the Carlist club in
Barcelona. Don Jaime's full manifesto was not published till November,
when the equivocal passage did not appear. But it is worth observing
that some time before the expected death of the old Pretender his
supporters in the Press had been hinting that those who believed Carlism
to be dead in Spain "would presently see things that would surprise
them."
Then we have the inexplicable favours accorded by the Government to
Señor Llorens, a Carlist Deputy to Cortes and one of the most prominent
of the "Court" of Don Jaime. This gentleman paid more than one visit to
the Army at Melilla, and was allowed privileges at the front which were
granted to no other civilian. The favours shown to him and his own
proceedings were so marked as to call forth outspoken comment from the
_Ejército Español_, a military paper professedly without political bias,
which, after recalling the fact that he is a well-known supporter of the
anti-dynastic party, and had
[Illustration: DON JAIME OF BOURBON IN MOROCCO.
[To face page 153.]
taken part in the last Carlist war, plainly warned him that any attempt
to tamper with the loyalty of the Army would be in vain, and asked what
was the meaning of the exceptional privileges he enjoyed.
On July 24th a great meeting of the Carlists was held at Trieste on the
occasion of the funeral of Don Carlos, the most noteworthy feature of
which was that on the evening of the same day Don Jaime left his
followers to be entertained by his mother and sisters, and went himself,
it was said, to Frohsdorf. Why did he, on the very day of his father's
funeral, abandon the delegates of his party when they travelled long
distances to see him and discuss the situation? The rioting at Barcelona
was just then at its height, having begun, it was said, some days sooner
than was intended.
About this time all sorts of reports were spread, calculated to alarm
the country and prejudice it against the Monarchy. The story of the
mutiny and execution of soldiers on their arrival at the front I have
already mentioned. The details of this varied with each telling:
sometimes two men were shot, sometimes nine, often a whole battalion,
once several of them. The immediate preparation of accommodation was
called for for the thousands of sick and wounded, who could not be
received in the already overflowing hospitals. Real sacrifices were made
by the poor to help in these preparations, for every one wished to do
his share for the sufferers; and when at length it became clear that no
wounded were coming, at any rate at that time, and that the demands made
on the public sympathy--for the moment at least--were a sham, much
indignation was naturally aroused. These alarmist reports circulated
with great rapidity, even in remote villages where no one received
newspapers. The people had no hesitation in attributing them to the
parish priests, "who have their own ways of spreading what Jesuits wish
them to make known," and tales of all sorts of horrors for which they
had been held responsible in the past began to be raked up and repeated
as happenings of the moment. One such tale was of a walled-up nun found
in a convent in Barcelona during the July riots. I took some trouble to
track this story, and finally convinced myself that it was merely an
echo from the past--a tale of the Inquisition or of some monastic crime.
But it formed another instance of the hold that tradition has on the
Spanish peasant, and of the way in which it is combined with the events
of the day to pile up the indictment against the Religious Orders.
I asked some of my middle-class acquaintances on one occasion where was
to be found the "army" of Don Jaime, which I had seen mentioned in the
report of a Carlist meeting. One of them laughed and said it existed
only in the region of comic opera, but another proceeded to explain with
conviction that the Pretender had a strong following in Cataluña and the
Basque Provinces and a good many adherents in Andalusia, and expatiated
at length on the benefits the nation would derive from the autocracy and
the abolition of popular rights, which he seemed to think would bring
about a social millennium.
And while he was speaking I mentally recalled the commentary of one of
the people, to whom I had read aloud Don Jaime's manifesto, asking
whether he would welcome the advent of the "Legitimist Monarch" in
Spain.
"Don Jaime? I? I would like to burn him and then blow his ashes to the
winds, and so would all my friends, both men and women. What dealings do
we desire with the seed of Don Carlos? There is no poor man in Spain who
liked Don Carlos or wishes ever to see his son."
A Basque friend of mine, a highly educated man, whose position as a
large employer of labour enables him to judge fairly of the political
leanings of the people, made the following remark to me one day:
"The Carlists," he said, "may think they have the Basque Provinces with
them, but they are completely mistaken. The working classes of my
country have no more desire for civil war than those of any other part
of Spain."
This gentleman is not a man who would use illicit means to influence the
votes of his dependants, and his opinion may be taken as representing
the true state of public opinion in his district. On the other hand, he
said that among his wealthy clients there was no attempt to disguise the
desire for a dynastic change: the portrait of Don Jaime hangs in a place
of honour in many of the great houses which he visits in the course of
his business, and the general devotion of this class to Carlism is open
and avowed.
"But," he said, "what can be done by a party which is all head and no
body? The army of Don Jaime may be well supplied with would-be officers
and with all the munitions of war, but they have no troops behind them,
in the Basque Provinces or anywhere else."
His description of the Carlist "army" reminded me of the famous raid of
the Fhairshon:
"For he did resolve to extirpate the vipers
With four and twenty men and five and thirty pipers."
THE CHURCH MILITANT
CHAPTER VIII
THE CHURCH MILITANT
The Church of Spain asserts that its mission is peace, and as has been
said, supported the assertion, when Queen Victoria initiated the
patriotic fund for the sick and wounded at Melilla, by declining as a
body to contribute, on the ground that men of peace would be stultifying
their office if they supported a war fund. When it was pointed out that
the healing of the sick and the binding up of wounds, however incurred,
was as much the Church's mission as the preaching of peace, the reply
was given that the priests as a class were poor men who could not afford
to give away money. It may be remarked that what they call their poverty
is so well recognised by the working classes that they never dream of
applying to their parish priests when they are in distress; they say it
would be useless to do so, because "the priests do nothing without
securing their fee in advance."
But for a body whose first duty, on their own showing, is the preaching
of peace, it cannot be denied that the Religious Orders, if not the
secular clergy, are distinctly militant.
While Maura was in office nothing about priests and firearms would have
been allowed to appear in the papers, but about a fortnight after his
fall the _Pais_ asserted that previous to the Barcelona outbreak the
Carlists and Jesuits had accumulated a great quantity of arms in some of
the small towns in Cataluña, which were subsequently conveyed at night
from one religious house to another in Barcelona. And, said the _Pais_,
while the local authorities were imprisoning luckless working men who
neither possessed weapons nor made any sort of revolutionary movement,
the contraband purchase of arms was still going on, "the priests and the
Religious Orders shamelessly lending their aid to it, and the people
keeping silence because they believe that every Government, whatever it
may be called, is either friendly to Carlism or is afraid of it and
cannot or will not interfere with it."
On reading this article (which the Ultramontane organs did not
contradict) I was reminded of a story which had been told me three
months before by one of my working class friends. A pious old woman, the
wife of a small shopkeeper in a town where there are many religious
houses, went one night to a service at a church attached to a
monastery.[15] The weather was hot and the old woman tired. She fell
sound asleep in a dark corner and woke at midnight to find the church
empty and the doors locked.
Recognising at once that she had no choice but to stay where she was
until morning, she was looking about for the most comfortable bench on
which to pass the night, when she saw a light in the sacristy
communicating with the monastery and heard steps approaching. Fearing
lest the fathers should accuse her of being there with intent to rob the
church, she crawled under a bench and lay trembling. From this position
she saw a number of monks and priests file into the nave, form up in
ranks, and go through various military exercises under the command of
one of the number, who looked and spoke like an officer. The drill
continued for some time, and after it was over the unwilling witness had
to stay where she was until the doors were opened for early Mass, when
she made her escape, ran home as fast as her poor old legs would carry
her, and related what she had seen to her husband and neighbours. This
was told me by a lad who sold fruit to the husband, who declared that he
had heard it from the old woman herself.
At the time I paid no attention to the story, knowing how the dramatic
instinct of the Spaniard lends itself to exaggeration in repeating
anything that appeals to the imagination, and thinking that the whole
thing might have been a dream. But later on I found reason to think
there might be some basis of fact in what was related by my young
fruit-seller. When the _Pais_ article appeared I was told, in the course
of a conversation about it, that a priest in a neighbouring town had
said in the hearing of my interlocutor--of course unaware that he was
listening--that his party were all armed and prepared to shoot "on
sight" every one whom they knew to be inimical to them, directly the
opportunity offered. And thenceforward for some weeks constant reports
of the arming of friars and their lay allies--the "Young Catholics,"
"Luises," and other such associations--were published by the one party
and denied by the other with equal frequency.
In this connection the following passages from an article in the _Correo
Español_ are rather significant.
"In Barcelona ten Carlists sufficed to prevent the burning of a church,
and put the mob to flight, so that they left in the hands of our friends
the weapons they were carrying in pursuit of their vandalic designs" [an
incident already referred to]. "And there are 100,000 brave men such as
these in Spain.... We are prepared for all! all!! all!!!" (in crescendo
capitals). "The fight, which inevitably had to come sooner or later, has
now begun between Catholics and sectarians, between civilisation and
barbarism, and we must not stop till we have destroyed them."
It all reads like transpontine melodrama, and as such I at first
regarded it. But when day after day announcements appeared that new
Carlist clubs were being opened in one small town after another, when
Señor Llorens returned from his second sojourn with the troops, loaded
with plans, sketches, reports, and what not, relating to the campaign
and the general condition of the Army there, and openly announced that
he had obtained them for Don Jaime, and when, although the people were
shouting songs of defiance to the Carlists and their "King," the
militant "Catholic Association of Social Defence" announced that it had
increased its working class membership from 31,000 to 200,000, one began
to wonder whether the Carlist "army" might be something more than comic
opera.[16]
The stories related of secret arming and drilling in the churches at
night are obviously not capable of verification by a layman and a
foreigner,[17] but that the Jesuits in Barcelona were armed before the
revolt began, and used their arms with skill, seems certain. A near
relative of one of these warlike men of religion told me that they had
twice driven back the mob by firing from their balconies, so it seems
fair to assume that when the newspapers talked of the shooting down of
the crowd by the Jesuits they had some ground for their statement.
Civilians in Barcelona found in the possession of arms were arrested,
even though they had not used them, but it does not appear that the
Jesuits incurred any penalty for using their weapons on the mob.
One mysterious feature in the events of that week has never been cleared
up, and possibly never will be.
On the first two days of the rioting there was fighting about the
barricades which had been raised in many of the central streets, but the
scarcity of firearms among the rioters was noticeable, a large number of
them being without arms of any kind. Mainly, no doubt, in consequence of
this, the struggle was practically over by the third day, after which
there was no more street fighting, the troops occupied the city, and the
attack on the Religious Orders, which might so easily have spread all
over Spain, was at an end.
Yet, notwithstanding that the fighting was over, shooting from the roofs
of the houses went on for two days more. No one ever saw those who
fired: the shots came from invisible persons concealed behind the
parapets and other sheltered positions. And, what was the more
remarkable, whether the shooting was in working class districts, or, as
was frequently the case, from houses in those quarters of the city where
rich men live, the noise of the report and the bullets which were found
were always the same. The "man on the roof" invariably used a Browning
pistol, a weapon not easily procured by a poor artisan. Thirty, forty,
fifty such shots would be fired in succession, the troops would hurry up
to the roof from which the bullets came, find no one there, and see
nothing suspicious, yet hear the rattle of the shots again as they
returned to their duty in the street below. A civilian who ran up the
stairs from the ground floor in one of the "haunted" houses told me that
although several shots were fired as he ran, no one was to be seen
above, except a young priest professedly on the same errand as his own.
It was said that among the many people arrested there was at least one
priest. But nothing more was heard of him, and whether he was released
as innocent, or allowed to disappear, was not revealed to the public.
No one has yet explained who organised the expensively-armed
sharpshooters who displayed such remarkable skill in firing from an
elevation without being caught in the act. The people believe that they
were members of the clerical party whose object was to exasperate the
troops against the rioters who were supposed to be firing at them, and
thus to bring about a fight in which the whole town should be involved.
Meanwhile Don Jaime was to convert the mêlée into an organised
revolution against the established order of things, which should spread
from Barcelona all over Cataluña, and from Cataluña throughout Spain.
This, for what it is worth, is the popular explanation of one of the
most mysterious features in the "anarchist" rising of July, 1909.
But the people go farther still. They attribute not only the incidents
of July, but the whole of the political unrest in Cataluña to the
underground activities of the Carlists and their allies the
Ultramontanes. It is firmly believed by the unlettered peasantry, who
read or listened to the accounts of the beginnings and endings of the
"Red Week," that the emissaries of the Pretender planned and carried out
every incident that led up to the general strike with which the rioting
began.
The protest against the calling out of the reservists--the greatest
error of the many committed by the Government at that time--was said to
have been engineered by the Carlists. It was not spontaneous and found
no real echo in the feeling of the nation.
The next step was to proclaim a general strike, but even then there was
so little idea among the working classes that anything like violence was
intended, that women and children strolled out to the meeting-place as
for an outing, with the men who were unconsciously being led into
action which was to brand them as revolutionaries and assassins.
To this day no one has been able to say how or why the rioting began.
The only thing clear is that the great majority of the strikers expected
and intended to proceed peaceably to formulate their demands, although
no one knows exactly what these were to be, for no formal report of the
strikers' complaints, or even of the factories they worked in, has ever
been published.
The Civil Governor, Señor Osorio, objected to the calling out of the
troops, and fell into permanent disgrace with Maura and his Cabinet for
saying that but for the undue harshness employed by the military
authorities, the rising would never have attained serious proportions.
He was dismissed from his post--perhaps inevitably, since he had not
foreseen events. It is worth noting that the week before the riots the
Government had expressed themselves as perfectly satisfied with the
tranquil condition of Barcelona under Señor Osorio, and had withdrawn
most of the troops in garrison in the province.
Meanwhile the Ultramontane Press never wearied of repeating
blood-curdling tales of the awful scenes of carnage, rapine, and
sacrilege, brought about by the teaching given in the lay schools, a
hundred of which, they said, Maura had been compelled to close in order
to put an end to a system of education which produced such horrors: and
since the Opposition newspapers were not allowed to publish a line
without the sanction of Señor La Cierva, the Minister of the Interior,
the nation, had it read the Ultramontane papers, would have supped its
fill of uncontradicted libels upon the working people of Cataluña. But
the nation does not read the Ultramontane papers. The Press of that
party, indeed, admits the exiguousness of its circulation by pathetic
appeals to the faithful to furnish money for the propaganda which in
Ultramontane opinion constitutes the only hope of arresting the crimes
born of the instruction given in the lay schools, and fostered by the
seditious labours of the _Liberal_. But although the people closed their
ears to the fulminations of the Church papers, the hand of the Church
lay heavy on all Spain in 1909, for the continual reports of bombs and
arrests, and the whispered tales of the secret drilling and arming of
"good Catholics," kept everybody on the rack, fearing they knew not
what. The slow progress of the campaign in Melilla, the constant arrival
of shiploads of sick and wounded, and the impossibility of obtaining
trustworthy news of what was really going on, filled the cup of anxiety,
and every one was in low spirits, for every family had friends or
relatives in the war.
Meanwhile Don Jaime, in his castle of Frohsdorf, was occupied in editing
a verbose document which he published later on, addressed "to those
loyal to me." The gist of this was that as long as Spain was engaged in
war he would make no move, but that when the flag waved victorious he
would remember that he had to fulfil unavoidable duties imposed by his
birth. "And," said he, "social order, shaken by the revolution, is
tottering to its foundations. And this not so much from the attack of
anarchical crowds as from the cowardice of the powers who make compact
with them, delivering themselves as hostages in order to save their life
and property. In the violent struggle which is approaching between
civilisation and barbarism I yield to no one the first place in the
vanguard in the fight for society and the country."
Curiously enough, an incident in which the nation at large took very
little interest nearly proved the last straw. This was the execution of
Ferrer.
Everything had been done beforehand to excite the public over the
affair. Columns upon columns of matter prejudging the case had filled
the Ultramontane Press for weeks, while the _Sociedad Editorial_ and the
republican _Pais_ were accused of complicity with the prisoner because
they pointed out that the publication of incriminating documents alleged
to have been found in his house, before the Court had pronounced them
genuine, was contrary to all the principles of justice. In Republican
and Socialist circles this action on the part of the Government--for
copies of the documents in question were sent to the Press by persons in
Government employ--produced the indignation that might be
expected--indignation that probably was counted upon to bring about an
outbreak of violence. But the mass of the people, thanks to their lack
of education, knew and cared very little about Ferrer and his alleged
offences against society.
While all Europe was excited about the fate of the founder of the lay
schools, the Spanish people, believed abroad to be seething with anarchy
and sedition, were peaceably if dispiritedly pursuing their usual
avocations, only interested in Ferrer, if they took any interest in him
at all, as another victim of the tyranny of the Church, whose "tool," as
they call Maura, had brought Spain so low.
This was because the _Sociedad Editorial_, and especially the
_Liberal_, laboured as indefatigably to keep the temper of the people
within bounds as their opponents on the Ultramontane press laboured to
produce irritation. At one period in the protracted controversy I
wondered whether the editors or staff of the _Sociedad Editorial_ could
actually be unaware of the lies spread broadcast concerning the
political party for which they stand, so temperate in quality and so
limited in quantity were their comments on the foreign campaign against
the honour of the Spanish nation. But I soon came to understand that it
was not ignorance of what was going on, although the Censorship used all
its wits to keep foreign newspapers out of the Liberal-Monarchist
newspaper offices. It was the deliberate policy of the wise and
far-sighted Liberal-Monarchist party to keep their working-class readers
in the dark about the Ferrer incident, because they knew that if the
mass of the people became aware of the attack upon their honour, a civil
war between the Ultramontanes and the people would have broken out
within a week.
It seems impossible to doubt that the desire of those who pull the
strings that work the Ultramontane party leaders was to provoke such a
war. The declaration of the _Correo Catalan_ that a hundred thousand
good Catholics were ready to follow the example of the Jesuits who
fired on the crowd in Barcelona and to "go all lengths" against the
forces of "anarchy" bears no other interpretation. The
Liberal-Monarchists, who know that in any such war the people would
stand as one man for the King and the Constitution against the
Ultramontanes with the hated Pretender at their head, might have been
excused had they dallied with the idea of sweeping out the Religious
Orders by force, and thus settling once for all the eternal quarrel
between the State and the Church of Rome. But no such course of action
would have been admitted as possible by Moret, who is, and will remain
while he lives, the spiritual if not the ostensible leader of his party.
Well aware that he was offending the more advanced and impetuous among
his followers, and that he was being accused of lukewarmness in
defending the Liberal party from attacks both at home and abroad, Moret
firmly pursued his lifelong policy of conciliation instead of
provocation, and it was thanks to his firmness alone, during the last
three months of Maura's rule, that Spain was not once more thrust into
the horrors of internecine strife.
The week before Maura's Government fell the Radical and Republican party
in Madrid demanded permission to hold a meeting on the following
Sunday, to protest against what they considered the illegality of a
trial in which witnesses for the defence were not summoned. The
organising committee frankly stated that whether Maura gave leave or
not, the demonstration would equally take place.
What might have happened had the Ultramontane Government still been in
office on the day of the demonstration, no one can pretend to say. But
in the meantime the climax came and the Maura Government fell, amid
general rejoicings. The demonstrations took place, not only in Madrid
but in all the large towns, and were in every case conducted with the
most perfect order. Their original object seemed to be lost sight of in
the satisfaction at the change of Government. The speakers said very
little about Ferrer, because Ferrer was of so little interest to the
people; in the majority of cases the demonstrators limited themselves to
a protest against Maura's policy and a demand that he should never hold
office again.
The Religious Orders were, or professed to be, in a state of panic
terror when the demonstrations were announced. They declared that they
expected violence, incendiarism, and robbery; treasures of gold and
silver work, images, paintings, &c., were removed to private houses for
[Illustration: A DEMONSTRATION OF REJOICING AT THE FALL OF THE
ULTRAMONTANE MINISTRY, NOVEMBER, 1909.
[To face page 174.]]
safe keeping; and the general exhibition of alarm on the part of friars,
nuns, and parish priests made them a laughing-stock to the working
classes for the month during which the demonstrations continued. The
Civil Guard were sent, at the request of the ecclesiastical authorities,
to assist the friars in their projected self-defence and to instil
courage into the trembling nuns, and the garrisons were everywhere kept
in barracks in readiness for attacks which nobody dreamed of making. A
Civil Guard told me, with a twinkle in his eye, that he and his
companion had sat up all night in the portal of a convent, knowing all
the time that they might just as well have been in their beds for all
the danger the convent was in. No doubt many nuns seriously believed
their houses to be in peril, although the Jesuits must have been
perfectly aware of the truth, and it is not easy to find words in which
to characterise the folly, to say no worse, of a policy which tries to
forward its ends by permitting women cut off and completely ignorant of
the world to spend hours of misery anticipating dangers which their
leaders must know to be imaginary.
It cannot, however, be denied that the deep-seated and chronic hostility
of the people to the Religious Orders became manifest all over Spain,
as reports of panic-stricken friars spread from mouth to mouth,
converting their traditional dread of the Church into a feeling of
contempt. The working-class Spaniards fear the underground action of the
Church because they know it may mean starvation for their wives and
children. But it was something new for them to see the "long skirts"
fleeing from Cataluña in fear of their lives, and the spectacle led to
open exhibitions of scorn, which are a new feature in the history of the
Church in Spain.
There were not wanting either journalists or private persons to hint
that the alarm shown by the Religious Orders at the demonstrations
against Señor Maura was fictitious, and a renewal of the Catalonian
riots would have suited their plans. It was said that the slightest
hostile action on the part of the working classes would have been made
the signal for a Carlist rising, and that numbers of priests and monks,
as well as civilians of that party, were armed in readiness for such a
contingency.
This was why the organisers of the demonstration so urgently appealed to
their followers not to be provoked into recrimination by "persons
subsidised by the other party, who would place themselves among the
demonstrators with the intention of causing disturbances." They thought
it necessary to warn the public that what might seem the merest act of
personal aggression on the part of an ordinary loafer might really be
the initiation of an organised plan to raise a serious revolt. And they
prayed their friends to bear in mind that persons committing such acts
of aggression might be the secret agents of the Jesuits, and therefore
on no account to be induced to retaliate. These appeals were issued in
leaflets which were distributed by the thousand in all the towns where
demonstrations were to be held, and no doubt contributed largely to the
self-restraint and good conduct of the crowd everywhere.
If the organisers were justified in believing that the Jesuits wanted to
create disturbances, the angry and exceedingly untruthful comments on
these leaflets in the Ultramontane Press might be accounted for. They
were described as deliberate incentives to the usual list of
crimes--incendiarism, sacrilege, &c.--and "good Catholics" were ordered
to destroy any that fell into their hands without reading the infamies
uttered by the "anarchist canaille." Naturally the description given by
the Clericalists of their opponents' circular only excited the curiosity
of the "good Catholics." The "good" working man read the paper with the
added interest given by its prohibition, and finding nothing criminal
in it, went with the rest to the meeting to hear what it was all about.
It is quite likely that the Church's anathema of the essentially
constitutional leaflets issued in most of the industrial cities on the
first two Sundays of November, 1909, resulted in making new converts to
Liberalism among the small minority of working men who till then were
still following the dictates of the priests.
BARCELONA AND THE LAY SCHOOLS
CHAPTER IX
BARCELONA AND THE LAY SCHOOLS
I have already referred to the popular belief that the riots in
Barcelona in July, 1909, were deliberately instigated by the Jesuits and
the Carlists acting in concert, the object of the Churchmen being
primarily to provide an excuse for closing the lay schools established
by Ferrer, the hope of the Pretender and his party being that the
disturbances would spread and assume the proportions of a revolution,
"on the waves" of which he hoped to ride to the throne.
As the course of events in Barcelona which culminated in the "Red Week"
has not unnaturally perplexed foreign observers, it may be worth while,
in the absence of any proof as to who was at the bottom of the trouble,
to suggest a hypothesis which at any rate has the merit of giving a
plausible explanation of the incidents.
Throughout the three years that Señor Maura was at the head of affairs,
Barcelona had been in a state of continual unrest and anxiety. Bomb
outrages were reported every two or three weeks with monotonous
regularity, but strange to say, the explosions seldom or never took
place in public buildings or in places where people congregate. Now and
then some inoffensive passerby was killed or wounded, and once in a way
an insignificant house would be damaged more or less seriously. But the
total injuries inflicted by this long series of bombs were so few that
the object of their authors must have been to terrorise rather than to
kill. When the King and Queen went to Barcelona in the autumn of 1908,
the inevitable bomb was let off--or was reported to have been let
off--on the sea shore, where no one could possibly have been hurt by it.
Here, by way of parenthesis, I should like to call attention to the
courage and devotion to duty shown by both the King and the Queen on
this occasion. It was considered advisable by the Ultramontane
Government that the young wife and mother should accompany her husband
to the city which has been made to bear such an evil reputation as the
home of anarchy and sedition. The nation watched the proceedings with
admiration. "What courage the Queen had, to face the chance of another
bomb being exploded in her presence so soon after that tragedy in
Madrid!" said those who appreciated the human fear which they knew must
be concealed under the smiles demanded by the exigencies of her
position. Not a word of this was permitted to appear in the Press, of
course. It was only the common talk of the common people. But one little
paragraph slipped, through some mismanagement, into a popular paper,
which revealed the Queen's realisation of the danger she might be
running. It was to the effect that "the alteration of their Majesties'
itinerary, by which they would spend two days in Madrid instead of
travelling direct to Cataluña from Vienna, was dictated by the Queen's
wish to embrace her children before going to Barcelona." The next day
the paragraph was corrected by a careful explanation that the Queen had
wished to see the royal children because they were suffering from
childish ailments. But the people were not deceived by the second
notice. They said that Doña Victoria's conduct was worthy of a Queen of
Spain.
I do not believe that the people of Barcelona would hurt a hair of Queen
Victoria's head, nor that they would have raised a hand against King
Alfonso had he appeared there during the riots of 1909: what advantage
his secret enemies might have taken of his presence during the
disturbances is another matter. And my personal belief is that the
people of Barcelona were not responsible for any of the bomb outrages
which have made their city a byword in Europe.
Two things go to show that the industrial classes in Barcelona had
nothing to do with the bombs. The first is that they are too clever to
commit stupid crimes by which their class could not possibly benefit.
The second is that during the "Red Week," when Barcelona was given over
to mob law, the mob, said to be responsible for the bomb outrages, did
not explode a single bomb. It is not likely that if letting off bombs
were the favourite occupation of the criminal classes of Barcelona, they
would have lost the opportunities afforded them during the first three
days of the riots. Yet when the rising was quelled and the whole
province was under martial law, the bombs began again, and twenty-three
were reported to have been exploded between August 15th and October
20th.
The stringent censorship exercised then and for three months afterwards
prevented Europe from hearing of either this remarkable feature of the
riots or their real object. But every one in Spain knew perfectly well
that the riots were directed solely against the Religious Orders,
whereas the bomb outrages never affected a building belonging to the
Church or a person attached to the Clericalist party so long as Maura
held office.
Is there any previous instance in history of a mob, said to be composed
of the lowest and most degraded of the community, firing monasteries,
convents, and churches, while they left public buildings, banks, and
rich men's dwellings untouched? Is there any other revolt on record in
which troops of people containing the dregs of the criminal classes
protected and brought food to orphanages supported by the objects of
their attack? And can we find a parallel, in the circumstances, to the
organisation which had the markets opened for two hours every morning
and kept its forces under such complete discipline that during those two
hours persons of either sex could walk all over the town secure from
molestation?
These things I have heard from people of unimpeachable veracity who were
in Barcelona at the time; not only Catalans and Spaniards, but also
foreigners unconnected with any political party. I do not attempt to
deny that some half-hundred or so of buildings belonging to the Church
and the Religious Orders were damaged or destroyed, nor that many evil
deeds were done by the criminal hangers-on of the movement; nor do I at
all desire to minimise the crime of destroying property to gratify
feelings of personal revenge. But I do say that the mob, as a mob,
behaved with extraordinary self-restraint, and proved by their conduct
that they had no complicity with the miscreants who for so long
terrorised the unoffending inhabitants of Barcelona by exploding bombs,
without apparent intent to injure.
No one disputes that every suspect in the province was imprisoned or
fled from the country when the iron hand of military law closed on the
insurgents. Nevertheless the bomb outrages began again after the "Red
Week" came to an end, and only ceased with the fall of Maura and his
Cabinet of repression.
I have related in the previous chapter the continued shooting from the
roofs of the town, after the riots were quelled, by persons who were
never seen, and the stories that were told of the secret arming of the
Religious Orders. When we remember that the hope of the Ultramontanes
lies in a Carlist restoration, which is only possible through a
revolution, and that a revolution cannot be brought about except by
fomenting unrest and discontent in the country, and when further we
recall that the bomb explosions ceased with the fall of Maura's
Ministry, when the officials of a Government not in sympathy with the
aspirations of the Religious Orders might have instituted inconvenient
inquiries had the bombs continued, it may at any rate be conjectured, in
the absence of any evidence as to who instigated this long series of
comparatively harmless outrages, that their authors were the only party
who expected to benefit by a subversion of the social order such as
might have ensued had the patience of the people given way under this
long series of provocations. This theory of the bombs, I may add, is
that held by the working classes.
From the moment that Moret took office in October, 1909, Barcelona began
to resume her normal aspect, although the constitutional rights were not
restored until the new Civil Governor and the new Captain-General had
taken possession of their respective offices and reported that the whole
province was quiet.
From that date a strict watch was kept upon newspaper reports of
explosions, and the _Heraldo_ got into trouble for publishing a
paragraph saying that what proved to have been merely a slight explosion
of gas was a bomb. The authorities at once explained to the Press that
the explosion was purely accidental, and that no one in Barcelona had
for a moment believed it to be otherwise, yet the report that it was a
bomb had reached the _Heraldo_ office in a form circumstantial enough to
deceive an experienced editor.
It is not surprising, therefore, that doubts are now expressed whether a
good many of the alleged bombs may not have been as fictitious as this
last. The persons who let them off, or were supposed to have let them
off, in order to maintain unrest in Barcelona, could certainly have
provided means to deceive the Press, as in the attempt upon the
_Heraldo_, frustrated by the prompt action of the Civil Governor.
Two or three bombs, if they can be given so imposing a name, were
exploded in Zaragoza in December, 1909, under the conditions which had
become so familiar to Barcelona under the Maura regime. They were made
of bits of old iron, mixed with some mild form of explosive and placed
in a meat tin, the whole being wrapped in a black cotton material, said
to be of the same make as that found on remains of bombs at Barcelona.
The tin cans on these occasions were placed in or near the porch of a
convent church, and no harm was done beyond some slight damage to the
plaster on the walls. The progressive Press, freed from censorship,
expressed the conviction that this affair was the work of the monks,
desirous of raising disturbances in Zaragoza because they were now
powerless to do so in Barcelona, with the result that the public
remained entirely indifferent to the incident. One cannot but hope,
therefore, that that may have been the expiring effort of the
bomb-throwers, whatever their real purpose was and whoever their
employers may have been.
I should like, before closing this branch of my subject, to point out
once more the wide differences that exist between the methods, objects,
and results of the Barcelona and Zaragoza bomb outrages and those of
similar attempts elsewhere on the Continent. The murderous anarchist
makes a direct attack on the personage whose death he believes to be
necessary to the furtherance of his political creed, and when he lets
off a bomb he takes care that it shall do as much damage as possible,
regardless of risk to himself. Abhorrent though the creed of the
militant anarchist is, he has at least the courage of his convictions,
since he so frequently pays the penalty of his act with his life. The
wretch who tried to murder the King and Queen of Spain on their
wedding-day was the tool of some one working on the usual anarchist
lines, and his crime bore no resemblance in detail to the work of the
mysterious party interested in terrorising, without injuring, the
inhabitants of Barcelona.
A volume of school statistics published in November, 1909, to which
further reference is made in another chapter, shows that there are in
Spain 91 protestant and 107 lay schools, 43 of which are in Barcelona.
On the other hand, there are 5,000 private Catholic schools, in addition
to some 25,000 Government schools, in which the rudiments of the
Catholic religion are supposed to be taught. These few Protestant and
lay schools are the subject of furious and unceasing abuse from the
Clericalist party and Press, who make every effort to traduce and vilify
them. It would not be edifying, nor is it necessary, to cull specimens
of their flowers of invective: the language in which the _odium
theologicum_ is habitually expressed is tolerably well known. The
schools in Barcelona, many of which were established by Ferrer, who
devoted his fortune to the work of education, are the special subject of
clerical hostility, and there is no doubt that they cost him his life.
As far as can be learnt about these schools the teaching given in them
contains absolutely nothing of the socialistic or anarchistic or other
doctrines subversive of society of which their enemies so freely accuse
them. They are more or less hostile to the form of religion taught by
the Church in Spain, which is the chief reason for the venom with which
they are attacked; but setting this on one side, there is, I am credibly
informed, nothing either in the text-books used or the teaching given to
which objection need be taken.
Nevertheless the Clericalist campaign against these schools is carried
on without intermission, and at the end of February, 1910, about the
time that Moret fell, unusual efforts were made against them. Thus in
Valencia several thousands of priests and friars, ladies of the
aristocracy, and members of the militant religious associations filled
the great open-air theatre of Jai-Alai: a telegram giving the Papal
benediction to the objects of the meeting was read, and cheers for the
Pretender were raised at intervals during the afternoon. The reactionary
papers asserted that twenty thousand people were present on this
occasion, and although this was doubtless an exaggeration, no one
attempted to deny that a very large number attended.
The number of public bodies and associations said to have sent letters
and telegrams of adherence to the objects of the meeting would be
alarming to any one unacquainted with the arithmetical methods employed
on these occasions in Spain. The grand total was given at 280,000,
"composed of 100 newspapers, 83 town councils, 135 mayors, 429 clubs,
1,714 congregations, and 272 parishes." But no names of these parishes
and congregations were given, and verification of the figures is
impossible. It was also said that "9,000" ladies who had been present at
the meeting subsequently left their cards on the Civil Governor.
Admission to the meeting was by ticket, and there were not wanting
working men who declared that whole villages had been coerced into
attending by the action of their priests and their caciques, but I give
this for what it is worth. It is, however, safe to say that the great
majority of those present were priests and friars, and members of the
upper classes. Only one speech by a working man was mentioned in the
long report published in the _Correo Español_, although the Clericalist
papers always give prominence to the smallest indication of sympathy
with their cause on the part of the people.
The really serious feature in the affair was the Papal benediction of
the speakers and the audience. There is nothing in the Constitution to
forbid the existence of the lay schools, to protest against which the
meeting was held. Thus the Pope, by his benediction, set the seal of his
approval upon an effort to subvert, in this respect, the Constitution
of the country. But, further, the introduction by the speakers of the
name of the Pretender and the reception given to references to him
turned the whole affair into a frankly seditious gathering. The Pope's
support of the meeting was the more significant because his official
reception of Don Jaime at the Vatican had been reported by the Spanish
and foreign Press a few days before.
The Valencia meeting was followed by others in many of the large towns,
and about this time Count Romanones, in his capacity of Minister of
Education, closed a lay school[18] on the pretext that it was
insanitary, but this only irritated the Liberals without conciliating
the Church party, and Romanones hastily declared that the school would
be re-opened as soon as certain structural alterations had been made.
On February 27th a Clericalist meeting was held at Bilbao, at which,
notwithstanding the efforts of the police and Civil Guards, serious
disturbances occurred. The circulars inviting people to the meeting were
so inflammatory in tone that the Civil Governor found it necessary to
suppress some of them. The following extracts from one of these will
give an idea of the kind of language employed.
"In the name of religion outraged, and society menaced with total ruin
... and in the name of our own personal independence, closely bound up
with the faith in our souls, let us go to the Catholic meeting to
protest against the ignorance of those who desire to separate us from
all other civilised nations, tearing faith and Christian morality from
the souls of the young, together with all decency, all virtue, and every
quality necessary to human dignity.... We unite our voices with those of
all Catholics, speaking through the mouths of the most eminent men of
science, to condemn this monstrous birth engendered by error and lies."
The Liberal element in Bilbao is strong, and naturally great indignation
was created among the working classes by these insults to their politics
and religion. Down to that date there had been no lay school in the
city, but now it was announced that one would be opened immediately.
The noteworthy feature of this meeting was a denunciation of the
Conservatives by a Carlist speaker, who included them with the Liberals
in his fulmination against the "cowardly incendiaries of Barcelona,"
urging the Catholics "to have done with patient endurance and enter
upon the period of action." The result of this was that the
Conservatives of Bilbao refrained from sending any representative of
their party to the banquet given after the meeting to the orators who
had spoken at it, thus definitely dissociating themselves from the
policy of the Clericalists in their city.
I have made special mention of two of these demonstrations against the
lay schools, one because of its magnitude and importance, the other
because of its results. To chronicle more of them would be tedious and
unnecessary. The campaign against these schools is unceasing: the
defence is by no means equal in vigour to the attack, and is limited to
articles now and then in the _Pais_ and an occasional meeting in their
support. Whether this apparent indifference is due to weakness on the
part of those who uphold the lay schools, or to a feeling of strength
which can afford to despise the fulminations of opponents, I am unable
to say. It is a quarrel, as the satirist says,
"_Ubi tu pulsas, ego vapulo tantum._"
THE ARMY, PAST AND PRESENT
[Illustration: A CONSCRIPT.
[To face page 199.]
CHAPTER X
THE ARMY, PAST AND PRESENT
It is allowed that great abuses were committed by those in power during
the long war in Cuba, which ended with the struggle in the United States
and the final expulsion of Spain from the last of her American colonies,
and it is common knowledge that the munitions, provisions, and all the
supplies of the Army fell lamentably short of what was required. It may
be imagined, therefore, that the survivors of these long years of
warfare brought back stories of experiences little calculated to inspire
their friends with confidence in the governing classes, who were
responsible for such shortcomings. Fully to appreciate the difference
between the sentiment of the Army to-day and what it was so late as
1901, when the defeated troops from the lost colonies came home with
their tale of suffering, it is necessary to show what convictions have
had to be changed and what prejudices overcome by Don Alfonso before he
could win the place which he now holds in the affections of his
soldiers.
I will only deal with the rank and file, whose loyalty is even of more
importance to the nation than that of the officers. My own impression is
that, after making all due allowance for differences in politics and
traditions, the great majority of the Spanish officers to-day are
staunch supporters of the Monarchy and the Constitution they have sworn
to uphold. But beyond putting on record my private opinion, formed on
the utterances of officers of all arms, I do not propose to deal with
this side of the question.
It was natural that reminiscences of the Cuban and American Wars should
be continually brought forward during the operations in Morocco, and
that the popular expectation of the treatment the troops would there
receive should be based on what took place in Cuba; and it was
inevitable that the unlettered mass of the community, agitated as they
were in the early days of the war by rumours of wholesale massacre and
tales of thousands of dead and wounded, should have imagined that their
friends and relatives were once more being sacrificed without mercy on
the altar of political corruption. Not long ago I heard the following
conversation among a party of working people who were entertaining a
soldier at a tavern on the eve of his departure for Melilla.
"Poor fellow!" said a stout elderly matron, with a tear in her eye. "So
young and so good-looking, to be killed by the Moors!"
"Don't distress yourself, Señora," said the lad, a slim, active young
fellow. "I'm going to make mincemeat of at least eight before they kill
me, and I shall be in no greater danger there than up at the mines
of ----, where I was knocked to pieces by a landslide. Three months I've
been in hospital, and it's just like my luck to be called out to Melilla
the moment I get out. I'm not afraid. If they kill me it can't hurt more
than that landslide did."
"He'll sing a different song when he gets out there," remarked an
elderly man gloomily. "I know how the soldiers are treated--not enough
to eat, and that bad, no clothes, no beds, and no cartridges to put into
their rifles when they go into action. I saw it with my own eyes in
Cuba."
I ventured to suggest that Melilla was nearer to the resources of Spain
than Cuba, and that the general condition of military affairs had
considerably improved of late years.
"Don't you believe it!" said the old soldier. "The Government sold Cuba
to put money into their own pockets, and they will do the same in
Morocco. Do you know what happened to us one day in the Cuban War? We
found ourselves attacked by the enemy, and we had nothing, _nothing_ to
fight with. There were no officers; the chiefs were in a safe place,
spending the money they had robbed us of (for we got no pay), and the
inferiors were hiding from the Cubanos wherever they could, behind us,
to be out of the fighting. I assure you this is true. When the Cubanos
came upon us we tried to load the guns, but many of the balls did not
fit, and we had no wadding.[19] We tore up our white drawers and our
shirts to make wadding, but what was the good? It was hopeless for us to
fight. And seeing the enemy upon us and we helpless to defend ourselves,
we went mad with rage and despair and turned on each other, not knowing
what we were doing. It was all the fault of the Jesuits at home, who
stole the money which the nation gave for the Army. And it will be the
same thing now with this Maura and his Jesuits, you will see!"
"It is all quite true," said another old man. "My son has often told me
the same. He said they tied their officers to the gun-carriages in
[Illustration: A FORT ON MOUNT GURUGÚ.
The War in Melilla.]
his company more than once to prevent them from running away. They said:
'If we, the common soldiers, are to be killed like flies, at least you,
the officers, shall take your share.'"
With such traditions firmly embedded in the popular belief, it would not
have been surprising had a real spirit of mutiny been shown on the
calling out the reservists in July, 1909. But this was not the case.
In an interview given to a representative of _Le Journal_ of Paris, in
November, 1909, by General Primo de Rivera, who was Minister of War
previous to the disasters of July, that officer threw some light on
Señor Maura's conduct of military affairs, and explained why he had no
alternative but to retire from office, to be abused by the Clericalists
in power as "unpatriotic" for so doing. Here is a brief résumé of his
statement:
"From the moment I took office, foreseeing what was brewing at Melilla,
I began to fortify our positions in the Riff. Expecting that General
Marina would need reinforcements, I brought the regiments of the
Cazadores del Campo de Gibraltar up to their full strength, and put the
Orozco Division, in all three arms of the service, on a war footing. In
order to secure rapidity of transport, I contracted with the
Transatlantica Company to make the voyage in twenty-four hours, on only
four hours' notice. When General Linares replaced me in the Ministry, he
thought fit to improvise all that was required, and this caused complete
disorganisation in the Army. He refused to call out the divisions which
I had held in readiness, and by drawing the troops from Cataluña not
only gave rise to the melancholy events of the "Red Week," but rendered
it necessary to incorporate many reservists who had married and set up
homes in the belief that they were free from service, thus bringing
misery on thousands of previously contented families. And after all this
mismanagement it was necessary in the end to send the Orozco Division
which I had prepared so long before."
At the time one heard on all sides the question: "Why does the
Government call out the reservists while the Orozco Division stands idle
at home?" to which there has never been any reply but that of the
people, who said: "The Government wants the war to go on because it
suits the Jesuits, who are making a fortune out of it."
But notwithstanding the acute distress throughout the country, the
reports of an organised and widespread protest against the calling out
of the reserves, which flooded the foreign Press at the time, were
entirely unjustified and incorrect. Parents in Madrid wrote, full of
anxiety, to their children in provincial towns, saying: "What is all
this we hear about disturbances in your city? What is happening? What
have the reservists been doing?" While the children were writing with
equal urgency to ask what was amiss in the capital, that "such bad
things" were being said of the soldiers in Madrid. I know these reports
were spread, for I was asked to read aloud more than one such letter by
working people who could not read for themselves.
It was not long before the people discovered that they had been deceived
and vilified by some persons unknown, who were making it their business
to represent Spain as in the throes of a revolution, and it was then
that they became convinced that the rising in Cataluña, represented by
the Government as springing from a protest against the calling out of
the reserves, was in fact a Carlist plot, gone wrong so far as the
Carlists were concerned.
As one travelled about the country in 1909 it seemed as if every village
had sent one or more of its sons to Melilla. Yet, although their
families made sure that they were going straight to destruction, few
endeavours were made to evade the call to arms.
I heard one man, an artisan, say with a shrug of his shoulders that he
was going because he might as well be shot in action as shot for a
deserter at home, and I saw another fling himself flat on the platform
when the train came in, howling that "he was afraid of being killed and
didn't want to go to the war." The first was a professed republican; the
second, as the bystanders promptly informed me, was "drunk, as usual."
Very likely there were other cases of the same kind, but they were
certainly exceptional. I made it my business to travel as much as I
could at that time, on purpose to observe the people, for, knowing the
Spanish peasant, I did not believe the tales current in the foreign
Press of his cowardly and mutinous conduct, and I wished to see for
myself how he behaved. I saw no such disgraceful exhibitions as were
described by English and French journalists.
The conversations that I overheard were very naïve: not at all the talk
of a rebellious people, notwithstanding the tales of suffering in Cuba
and in the Carlist wars which balked so large in the popular
imagination.
"My son! my son!" wailed one woman. "They will kill thee! I shall never
see thee again!"
"Hush, mother!" answered the young man. "Rest assured that if they do
kill me I shall have killed plenty of them first."
"Why will they not let us women go too!" cried another mother. "We could
kill all the _Moras_ [female Moors] and then they would bring no more
little Moors into the world to be the ruin of Spain."
It was curious to observe how the eternal race-hatred came out at the
very name of Moor--the tradition of the long contest between Christian
and Moslem. The Moors of Morocco cannot be held to have inflicted any
serious injury on the nation for many centuries past, yet such is the
force of ancient tradition among the peasantry that the very name of
_Moro_ calls forth the cry, "They are the ruin of Spain," and if you ask
for an explanation you will be told that "The Moors are always pressing
upon us and trying to take our country from us."
One pathetic yet humorous incident was related by the Infanta Doña Paz
(aunt to Don Alfonso) in a letter which she wrote to the Press about
this time, exhorting her fellow-countrywomen to have patience and be of
good courage. Describing her experiences of the patriotism of the men
and the devotion of the women, she told how a poor mother, learning that
her son was ordered to report himself for service, followed him from
village to village, as he pursued his avocation of pedlar, carrying his
regimental trousers, which had been put away in the family
clothes-chest. When she found him at last, there was barely time for him
to catch the appointed train, and the two hurried together to the
station with the trousers flapping like a flag as they ran.
The sons of mothers like these do not shirk their duty when called upon
to fight for their country. I believe that if the whole truth were told,
we should find that no one was more indignant at the protest supposed to
have been initiated by the working classes of Barcelona than the
reservists whose grievances were its ostensible object.
Fresh from an exceptionally rough crossing, weak with sea-sickness,
rusty in their drill after three years of home life, the reservists who
sailed from Barcelona found themselves led straight from the ship on to
the field of battle. This I had from a naval officer on the man-of-war
that took them out.
"If they had been veterans," he said, "such a situation would have been
trying to them, and they were only raw fellows who hardly remembered the
words of command. And yet I tell you they behaved with such courage and
[Illustration: A RESERVIST At THE FRONT.
[To face page 208.]
discipline that I felt proud to be among them. I was sorry and ashamed
to see those sea-sick boys ordered into action, but now I am glad to
remember what I saw my compatriots do that day."
The naval officer spoke of an incident in the early days of the war,
before the foreign correspondents had reached the scene of action. But
for some time the censors, both at the front and in Madrid, had made it
impossible for the truth about the campaign to be told; and England, at
any rate, was for several weeks allowed to remain under the impression
that the Spanish rank and file were a cowardly lot, driven into action
at the point of their officers' swords. That impression was corrected as
time went on, and it is, I believe, now generally admitted that the
Spanish troops do not lack courage.
In Spain the conscripts join at the age of eighteen, and serve three
years with the colours, when they are drafted into the first reserve.
But those who can afford it may buy exemption from service for 1,500
pesetas (say £60), and from this source the Government makes an income
estimated in the Budget for 1909 at pesetas 12,800,000 (about £512,000).
Naturally the well-to-do always buy themselves out, as do also a certain
number of the more prosperous of the working classes in the industrial
towns. Señor Maura's Government, not long before they went out, suddenly
made an order calling on all those who had already bought their
exemption to pay another 500 pesetas or join the colours at once, a
proceeding which, differing as it does in no respect from highway
robbery, naturally caused a good deal of indignation. No one likes to be
called on to pay a second time over for what he has already bought; and
in the case of the workmen, who generally secure themselves against
service by means of one of the numerous insurance companies formed _ad
hoc_, the premiums they had already paid were of course thrown away, and
few indeed of them could produce 500 pesetas at a moment's notice.
A scheme is on foot for doing away with the present unjust system, and
making service compulsory on all alike. It provides for six months'
instead of three years' service with the colours, the term to be
extended, in the case of the illiterate, until they can read and write.
This scheme obtained from the first the support of the whole Liberal,
Radical, and Republican Press, but the opposition of the Clericalists
must always be counted on in Spain, and the proposals most obviously
beneficial to the nation are usually those which meet with the
strongest opposition. Another popular clause in the scheme affects the
officers, whose pay is small. At present the officers live where they
can: they have no mess, and their quarters in barracks are so much the
reverse of luxurious that a lieutenant in a smart regiment apologised
for not asking me to visit him there, as he, knowing our English
customs, would have liked to do, because, said he, "it is not fit for an
Englishman to see."
It is now proposed, in order to reduce the cost of living in the Army,
that quarters and a mess shall be provided for the officers in barracks.
Most Spanish officers have to live on their pay, and even a captain in
the cavalry only gets about £140 a year. On the other hand, their social
expenses are very small, subscription dances, dinners, sports, and the
numerous calls on the purse of a British officer being unknown in
Spain.
THE POLICE
CHAPTER XI
THE POLICE
The visitor to Spain is frequently struck with the number of persons
whom he meets on all sides clad in various uniforms and armed, some with
cutlasses alone, others with revolvers in addition. If he asks who they
are, he is told that they are the police, and then he is perplexed to
find such a large number of distinct bodies, all apparently performing
much the same duties. A few words of explanation as to the various
police-forces of the country and their different functions may not be
out of place.
In the first place, every town has its body of municipal police, under
the orders of the Alcalde. Their chief duty is to regulate the traffic,
to maintain order in the streets, and to report to the Town Council any
infraction of the municipal by-laws, and to another body of police
anything or any person whom they may regard as suspicious or a possible
danger to the public security. They do not themselves, as a rule,
arrest malefactors, though no doubt they are empowered to do so on
emergency.
These policemen are well-intentioned, but on the whole ineffective, not
from any fault of their own so much as from the conditions of their
appointment and tenure of place. In Spain anything can be done by
influence, and it is practically impossible to enforce the by-laws
against a person in high place who chooses to break them. Not long ago I
was at an exhibition, which a very great number of people had gone to
see on that particular day. The municipal police were doing their best
to make the crowd "pass along," but at one point there was a block,
caused by one or two well-dressed men who refused to move. I asked the
policeman why he did not make them, and he replied that one of them was
So-and-so (a person of local importance) and that if he said anything to
him he would find himself dismissed the next day![20] In a certain town
not long ago a body of police inspectors was established, whose duty it
was to supervise the municipal police and report derelictions of duty,
and as far as I could learn they were doing useful work. After about
three months they all disappeared. On inquiry I was told that the
reason of their suppression was that one of them had reported the
carriage of some duke or marquis for obstructing the traffic, and that
the indignant nobleman had insisted on and obtained the abolition of the
force.
The municipal police go off duty about 8 p.m., and are replaced by the
_serenos_ or night-watchmen, who patrol the streets all night carrying a
pike and a lantern, and in some towns still cry the hours. Hence their
name, from their not unmusical cry, "_Las doce han dado y sereno_"
("twelve o'clock"--or whatever the hour is--"and a fine night.")
Alongside of the municipal police is what is known as the _Vigilancia_.
It is they who have to deal with criminals of all sorts within their own
districts, arrest pickpockets and other offenders, investigate thefts,
murders, &c., and catch the guilty. To them the hotels report the
arrival and departure of guests, and it is their business to find any
persons who are wanted on extradition warrants. In short, they perform
most of the ordinary police duties except those assigned to the
municipal police.
There is also a body of rural police, whose duty it is to patrol the
country districts; they are few in number and not particularly
effective. It is not often that one runs across any of them, even in
their own districts.
The most important and by far the finest body of men in Spain is the
Civil Guard, popularly known as the _Benemerita_ (well-deserving). This
force is one which, both in physique and morale, would do credit to any
country in the world. They are under very strict discipline, and are
prevented as far as possible from associating with any one outside of
their own body--for instance, with the ordinary police-forces. Even the
officers are under stricter regulations than those of the regular Army.
I was told of one case where a junior officer, after due warning, was
broke for gambling. The force is officered from the regular Army, and so
highly is the service esteemed that an officer who obtains a commission
in the Civil Guard _ipso facto_ loses a step. Very great care is
exercised both in their selection and in recruiting the rank and file.
They are something in the nature of a military police, and may be
generally compared to the Irish Constabulary; they do not perform
ordinary police duties, but in case of anything serious, such as a riot,
they would act, and they are expected to hunt down and catch malefactors
who are escaping from justice--which, indeed, they usually succeed in
doing. They practically have power of life and death, as if, in the
execution of their duty, they think it necessary to shoot, no questions
are asked. They always go about in couples, a young man accompanying an
older one, sometimes on foot, generally on horseback. They are the
terror of evildoers, and some years ago entirely stamped out the
brigandage which was then rife in the South of Spain, by the simple
expedient of shooting down the brigands wherever they caught them. But I
have never heard it suggested that they abuse their powers, and every
one, foreigners as well as Spaniards, speaks well of them. Moreover--and
this is rare in Spain--they are said, I believe with truth, to be
incorruptible, and everybody has the utmost confidence in them.
I have already referred to this force being called on to protect the
nervous nuns and the ostensibly non-militant clergy during the
anticlericalist demonstrations in November, 1909. It may be interesting
to show how they regarded the political situation at that time,
premising that as they are in daily contact with the people, no body in
the kingdom has its finger more closely pressed on the public pulse.
"You seem to have had your work for nothing," I remarked to a couple of
my friends at their barracks on the evening after one of the
demonstrations. "I never saw a more orderly crowd."
"What did you expect?" they replied. "These are political matters in
which we take no part beyond going where we are ordered. It seems to be
the fashion to talk about the prevalence of anarchical ideas in our
country, so presumably it suits some persons that the public should
think our people are anarchists. But _we_ see no symptoms of it. No
doubt it is right for the authorities to take precautions if they
believe there are fellows of that sort about. It is not our place to
inquire why they believe in a condition of affairs which we know does
not exist. The Civil Governor naturally does not ask for our opinion on
matters connected with politics. If he did we could tell him that he
need not be nervous, for anarchy is a disease which does not progress in
our nation, at any rate in any part of the districts _we_ have to travel
over."
Remarks of this tenour have been made to me by members of the force in
many times and in many places. A couple of Civil Guards accompany every
train, and detachments of them are stationed in every town and village,
in addition to mounted men charged with the care of the rural districts.
They are continually changed from place to place, to prevent any danger
of becoming too friendly with those they are intended to control, and
the result is that they have an exhaustive knowledge of the feeling of
the people. A Government genuinely desirous of gauging the popular point
of view at any crisis need only apply to the _Benemerita_ for
information. But so long as the Civil Governors who command the Civil
Guard are appointed for party purposes and changed with every change in
the Government, this means of contact between the Government and the
governed will be neglected in the interests of party.
It must not be supposed that the Civil Guard talk in public about the
duty with which they are entrusted. On the contrary, their non-committal
attitude is always honourably maintained before their fellow-countrymen.
But when I travel alone with them--for I frequently take a second-class
ticket merely for the sake of their company--they are not unwilling to
express an opinion on affairs in general, feeling secure that I, as an
Englishman, can be trusted not to turn anything they may say to their
disadvantage. Before Spaniards they are extremely reserved, but when the
compartment is empty save for myself and them conversation runs easily.
I was struck by this one day when a hot-headed individual shouted his
vehemently Radical views to a friend at the opposite end of the
carriage. The second-class carriages in many parts of the country are
only divided half-way up, so that it is not unusual to talk from one end
to the other on country lines where simple manners prevail. The Radical
knelt on his seat and his opponent stood up on his, and the passengers
sitting between them chimed in at intervals. The Constitution was
suspended at the time, and Señor Maura would certainly have had the
whole company clapped into prison had he heard what was said. But the
Civil Guards turned a deaf ear, affecting to be entirely absorbed in
their cigarettes. Later on I took an opportunity of asking what they
thought of the oratorical exhibition we had been favoured with.
"We think nothing at all, and that is just how much it is worth," they
said. "We know that gentleman very well, and he would no more commit an
act of violence or an offence against the law than we would ourselves.
But all Spaniards love talking, and if he could not relieve himself by
that sort of gabble he might become a danger to the public peace. There
are a fair number of his kidney scattered about the country, though they
are chiefly to be found in the big northern cities. They revel in the
nonsense spouted at Republican meetings and love to read out violent
articles from the _Pais_ and the _Motin_, but they are quite satisfied
with talking. Very few indeed of them would fire a shot for what they
call their principles. That is why we never take any notice of what they
say before us in the trains or elsewhere. We know it means nothing and
is an excellent safety-valve. If Maura had done as we always do--let
them talk and take no notice--there would have been no riots in
Barcelona. But Spaniards have hot tempers, and if you make them angry,
trouble begins. What harm does their talk do to any one? You have only
to reflect that they are pretty nearly all fathers of families, who know
very well that any such revolution as they romance about would only make
it ten times harder for them to earn a living for their children, and
God knows it is hard enough already to live in our country. We have to
eat beans and bread, and often don't get enough even of that. Do you
imagine that any working man wants civil war in the country to make his
food dearer still? _Ca!_ Let them talk! It amuses them and it makes no
difference to the Government. Whichever party is in power the poor have
the same difficulty in bringing up their families."
The Civil Guards had to shut their ears to a good deal of conversation
which the Ultramontane Government would have found it desirable to
suppress, during the three months after the "Red Week" in Cataluña; for
the attitude of the people towards the priests and Religious Orders, not
only in the North of Spain, but all over the country, became daily more
aggressive, and I have frequently admired the tact and good temper with
which members of that force contrived to do their duty and yet avoid
fanning the embers of discontent into a blaze of passion.
It has been sometimes remarked to me that it is the Civil Guard who
really govern Spain, and that without them anarchy would shortly ensue.
So far as the maintenance of public order is concerned, there is a good
deal of truth in this remark. They go quietly about their business,
never interfering with any one unless there is need, but if there is,
their intervention is immediately and conclusively efficacious. They are
at once feared and respected, and it is only in extreme cases that
resistance to them is ever attempted.
POLITICS
[Illustration: DON SEGISMUNDO MORET.
Leader of the Liberal-Monarchists.
[To face page 227.]
CHAPTER XII
POLITICS
The apparently purposeless and kaleidoscopic changes in Spanish politics
are very apt to puzzle foreign observers, who cannot understand what has
happened to bring about the resignation of a Minister or an entire
Cabinet, for which the cause, if any, alleged in the papers seems wholly
inadequate. Internal and external affairs appear to be pursuing a
tranquil course: no disputed question is agitating the country or the
Cortes, when suddenly comes a bolt from the blue in the shape of an
announcement of a Ministerial crisis, and the Government is changed.
Thus, early in the year 1910, Señor Moret, who after overthrowing the
Government of Maura in the previous October, seemed to be pretty firmly
seated in the saddle, suddenly resigned, in spite of the fact that at
the municipal elections a month or so before his policy had been
endorsed by overwhelming majorities all over the country. One of the
English newspapers, in commenting on this seemingly inexplicable change
of Ministry, frankly confessed that it was useless for foreigners to
attempt to understand Spanish politics.
Generally speaking, Ministerial changes in Spain are the outcome of a
tacit arrangement made some thirty years ago between Canovas and
Sagasta, the then leaders of the two main parties, the Liberals and the
Conservatives, and continued by their successors, that each side should
have its fair share of the loaves and fishes. After one party had been
in office three or four years it was agreed by common consent that the
time had come for the other side to have a turn. Thus, as Major Martin
Hume says:[21] "Dishonest Governments are faced in sham battle by
dishonest Oppositions, and parliamentary institutions, instead of being
a public check upon abuses, are simply a mask behind which a large
number of politicians may carry on their nefarious trade with impunity."
But sometimes, though more rarely, another cause operates to upset
Governments, and that is the underground intrigues of disappointed
place-hunters. If the Premier in his distribution of appointments
happens to omit any important person or section of people who think
themselves entitled to a share in the plums of office, they will not
hesitate to join with political opponents and turn out their own nominal
leader, if circumstances happen to make this possible.
It is often said by foreign critics that the people--the mass of the
nation--are to blame for the sins of their Governments. They have the
franchise: if they are not satisfied, why do they not elect better men?
This criticism proceeds from ignorance of an important factor in Spanish
politics--one of the tentacles of the octopus of corruption which holds
the whole country in its grip.
The simple fact is that the great mass of the people have no voice at
all in the election of their representatives. Nominally voting is free:
actually it is not.[22]
The whole administrative system is centralised in Madrid, and the
various Government offices interfere in local affairs to an extent
inconceivable to an Englishman, accustomed for generations to manage his
own affairs his own way. One result of this is that the elections to the
Cortes are, in fact if not in theory, conducted from Madrid. In every
small town and rural district there is a person known as the _Cacique_,
usually a large employer of labour or a moneylender, to whom most of the
working population of the district look for employment, or in whose debt
they are. So enormous is the usury that once a loan has been raised,
many a borrower has been unable to free himself from debt for the rest
of his life. I have known cases where as much as 75 per cent. per annum
has been paid for a trifling loan. Thus the _Cacique_, whether as
employer or moneylender, or both, has the majority of the constituency
under his thumb. He receives his instructions from Madrid, and issues
his orders accordingly. If by chance the voting goes wrong, the returns
are falsified; but this does not often happen, for the voters are so
convinced that the exercise of their legal right of choice, if in
opposition to the wish of the authorities, will result in loss of
employment, that either they abstain or they vote as they are ordered.
The existence of the _Cacique_ is one of the great obstacles to any
effective decentralisation. If the villages and rural districts were
given the management of their own affairs, the _Cacique_ would be more
absolute than ever. One can hardly open a paper without finding a report
of some case of his arbitrary interference with local matters. If he is,
as he usually is, the friend or creature of the Civil Governor of the
Province, who is the nominee of the Ministry, he does what he likes and
there is no redress against his illegal and oppressive action.
The following stories illustrate the method of conducting elections in
Spain.
* * * * *
One man complained that a Conservative had given him a dollar for his
vote, and after he had voted he found that the dollar was bad. "Had I
not already voted, how gladly would I have given you gentlemen the
advantage!" he said to a group of Liberals. "But you see I am left
without my vote in exchange for a bad dollar. Never again will I sell my
vote to the Conservatives!"
* * * * *
Another rascal went to the office of a Liberal paper to complain that "a
thief" had contracted with him to engage some twenty fellow-rogues to
vote to order. He fulfilled his part of the contract and took his twenty
to the poll, but when he went to claim his pay the contractor had
disappeared.
"And here I am many pesetas out of pocket," he lamented; "for not only
have I lost the large profit the thief offered me, but I had to pay my
friends two reals apiece before they would stir out of the wine-shop."
* * * * *
In one district the Liberals boasted that for years they had never
bought a vote. "Partly," as my informant ingenuously said, "because we
have always had a safe majority, but partly also because we prefer to be
honest. But," he continued, "we learnt this time that a party of
Conservatives intended to interfere with us, so we prepared a party of
the same kind to receive them. 'Do not begin to fight,' said my father,
'but if they begin, hit hard.' They did begin, and our leader obeyed
orders. He hit the leader of the other side so hard that he knocked out
four of his front teeth, and that was the end of the fighting in our
district."
* * * * *
All these incidents are said to have occurred in the municipal elections
of 1909. One more is worth mentioning.
In a town of some twenty thousand inhabitants, where for many years past
an Ultramontane _Cacique_ has been supreme, that gentleman rose early on
the polling-day and personally roused the dwellers in the gipsy
quarters--mostly the biggest ruffians in the place--out of their beds.
"Get up, my sons," he said, "and go and vote, and there will be a dollar
apiece for you when you leave the polling-booth."
"They said they would go and vote," said my informant, "and they got
their dollars. But the Republicans came out at the head of the poll, and
the Liberals next, and the _Cacique_ and his Conservatives were
nowhere."
* * * * *
I happen to be aware that the _Cacique_ in this instance is a man of
great wealth and high social position, whose clericalist leanings are
well known. If, indeed, it be the fact that the working classes have
gained courage to defy men like him, the rising in Cataluña, the Maura
regime of repression, and the campaign led against Spain by
Ultramontanes and Socialists abroad will have borne fruit.
There is, however, one political leader in Spain who stands for purity
of election and is the lifelong foe of the "caciquism" and corruption
which paralyse any and every effort at political regeneration. Don
Segismundo Moret has thrice been Premier of Spain. Each time he could
have retained office had he consented to purchase the favour of the
place-hunters by giving posts in the Ministry, not to those best
qualified for the work, but to those who could command the largest
following among the "Liberal mercenaries" who, as long as the system of
"caciquism" continues, can make or mar electoral majorities. This he has
never consented to do. So it has happened that each time that he has
been in office he has had to sacrifice place and power rather than
pander to an evil system.
The story of his late short tenure of the Premiership, and of the
intrigues by which he was ousted is worth telling at some little length,
because it throws light on the workings of the political machine, and on
some of the difficulties with which a reformer has to contend in Spain.
Moret took office in 1909 against his own better judgment, for he would
have preferred that the Conservatives should bear the responsibility of
their own misdeeds, and solve the many difficulties resulting from
Maura's "policy of repression." But the country had been brought to such
a pitch of irritation and unrest by the reactionaries that the
situation was becoming dangerous. The Riff question was attracting the
unfriendly attention of foreign diplomatists; Barcelona was impatient
under a rigid application of martial law, and the Ferrer incident had
called forth a storm of condemnation from all the countries where the
assumption that a prisoner is innocent until he has been proved guilty
is an axiom of criminal law, while the advanced parties in the State
were getting out of hand and had begun to defy the Government, as,
_e.g._, in the matter of the demonstrations already referred to.
From the moment that Moret accepted office he was assailed by a stream
of the most virulent abuse, not only by the Carlist but also by the
Conservative and Ultramontane newspapers. He was "the destruction of
Spain," "the ruin of the nation," "the arch-priest of irreligion and
immorality," and not only was his policy attacked in terms of unmeasured
vilification, but the editors of these papers, which are owned and
supported by some of the best born and wealthiest men in the country,
did not hesitate to descend to vulgar personal abuse. His "grey hair,"
for instance, was a favourite subject of their ridicule, and his
"vacillation," "infirmity of purpose," and "inability to keep his party
together" were accounted for by jeers at his "senile decay," his
"failing intellect," his "body bent double by the weight of years," and
so forth, while the party led by him are usually spoken of in the
clericalist organs as _canaille_.
But on his acceptance of the Premiership the aspect of affairs underwent
a complete and immediate change. The political horizon began to clear.
Terms of peace were arrived at in Morocco. Foreign susceptibilities were
soothed. Cataluña was immediately relieved from the burden of martial
law, and the constitutional rights were restored in Barcelona. The
troops began to return from the war and were received with the greatest
enthusiasm; the trials of persons arrested in connection with the
disorders in Cataluña, who had been kept in prison on suspicion for four
or five months, were pushed forward, and numbers of them were released
for want of any evidence against them. Most of the lay schools were
reopened, on showing that nothing seditious had been taught in them. The
depleted treasury was replenished, and means were found to provide three
months' pay for the Melilla forces, which the outgoing Ministry had left
out of account. A great project of irrigation was vigorously promoted by
Moret's Minister of Public Works, Gasset, who has devoted practically
the whole of his political life to this subject, and has produced a
scheme which would convert vast tracts, now arid waste, into fertile
land. And the municipal elections, which took place about six weeks
after the change of Government, were conducted, so far as time had
permitted any modification of existing conditions, according to law,
with the result that the Liberal-Monarchists swept the board all over
the country. The official figures were as follows: Liberal-Monarchists,
2,961; Conservatives, 1,213; Carlists, 185; Republicans, 193;
Socialists, 4. Thus Moret's party nearly doubled the Conservatives,
Carlists, and Republicans put together. The smallness of the Socialist
vote should be noticed.
In any other country it would have been certain that a leader who could
so well and so quickly convert popular indignation into contentment and
hope was in for a long term of office. Not so in Spain.
During his four months of office, from October, 1909, to February, 1910,
Moret tried hard to obtain the decree of dissolution of the Conservative
Cortes, in order that the nation might have an opportunity of expressing
its opinion on recent events. At first it almost seemed as if he would
obtain the King's consent to dissolve. But the place-hunters were
afraid, and the Ultramontanes were more afraid. They played so
successfully into each other's hands that the decree of dissolution was
postponed day after day, while all his enemies proclaimed the incapacity
of a Premier who was "afraid" to go to the country.
The first attempt to upset him was a so-called "military demonstration"
in front of the offices of the _Ejercito Español_, a military paper
which had been confiscated for publishing an article written by a
Carlist, accusing the Premier of unjust favouritism in the distribution
of rewards for good service in Melilla. The demonstration was described
by the Conservative papers as of "overwhelming importance," and the
number of demonstrators was placed by some of them at two thousand. The
truth is that it was confined to a few officers well known for their
Carlist leanings, and the rank and file of their regiments stood
resolutely aloof. Moret and his Minister of War, General Luque, retired
the Captain-General of Madrid and the colonels of the regiments in
question for failure to maintain discipline, and ordered the actual
participants a couple of months' arrest--a proceeding which called forth
general applause from all except the reactionaries. The small
significance of the affair was made manifest when it came out that
these arrests did not exceed half a dozen, including the editor (also an
officer) responsible for the publication of the seditious article.
The result of this fiasco was still more to strengthen Moret's influence
with the nation, and it became evident that he would sweep the country
should he obtain the long-deferred decree of dissolution. All the
ingenuity of the Church was therefore exercised to secure his fall
before this could take place, and the cupidity of a cabal of
disappointed candidates for place was skilfully used to bring about--the
catastrophe, I was going to say, but the triumph of morality would be a
truer expression.
At the municipal elections in December, 1909, an endeavour had been made
by Moret to secure something in the direction of freedom of voting for
the working classes, and the result, as I have shown, was a triumph for
the Liberal-Monarchists. The Republicans--to their honour be it said,
for they did not do as well in these elections as they had
expected--worked harder than ever after this to secure to the electors
the free exercise of their legal privileges, and Moret accepted their
programme, so far as it was designed to help in cleansing the Augean
stable of corruption by limiting the powers of the local _Caciques_.
This gave an opportunity to those who live by political immorality, and
the intrigue which followed is typical of Spanish politics.
In the December elections Madrid returned a Republican majority to the
Town Council. The Alcalde, Señor Aguilera, an old and staunch ally of
Señor Moret's, although himself a Monarchist, ranged himself on the side
of the Republicans by supporting their demand for the limitation of the
Alcalde's power to appoint and thus control the votes of the very
numerous municipal employees. It was proposed that the Alcalde, instead
of being, as now, nominated by the Government, should be elected by the
Councillors, who in their turn have been elected by the popular vote,
and that the posts under the Council should be filled by open
competition.
Most of the Alcaldes, even in the small towns, enter office poor and
leave rich. But it is admitted even by his opponents that Señor
Aguilera, a man with but small private means, who has twice been Alcalde
of Madrid under Moret, has each time gone out of office as poor as when
he came in.
A crisis was deliberately provoked by the President of the so-called
"Liberal" election committees of Madrid, Count Romanones, a man who held
office under Moret in a former Cabinet, and has long been suspected of
aspiring to the Premiership of the party to which he belongs. The
election committees, represented by Count Romanones, although nominally
Liberal, objected to the proposed limitation of the power of the
Alcalde, and finding Moret firm on the point, went so far as to hand him
an ultimatum. Briefly, their terms were, "Leave to the Alcaldes" (often
the _Caciques_) "throughout Spain the appointment of the municipal
employees, or we will refuse to act, and leave you without any electoral
organisation at all when the Cortes are dissolved." It is not denied
that this resolution was handed to the Premier by Count Romanones a day
or two before his resignation. Meanwhile other opponents took advantage
of Aguilera's temporary alliance with the Republicans, and represented
that if a programme of electoral reform supported by that party were
carried out, the Throne would be endangered by a Republican majority in
the new Cortes. This danger was imaginary, for there is no doubt that
both the numerical strength of the Republicans and their hostility to
the reigning House have always been greatly exaggerated by all the
various factions desirous of clogging the wheels of reform.
Señor Moret of course declined to compromise with Count Romanones on
any terms, which in a man of his recognised probity was a certainty,
doubtless counted upon by the "Liberal" cabal and by the Ultramontanes.
He then once more asked the King for the decree of dissolution, that he
might place his programme of reform before those whom it most concerned.
Exactly what passed at this interview was not divulged, but at its
conclusion he placed his resignation in Don Alfonso's hands. It was
accepted, and the veteran Liberal-Monarchist, after forty years' service
to the Throne and the country, found himself dismissed at a moment's
notice, through the machinations of the opponents of electoral reform.
No plausible reason was given for the dismissal of Moret. It was
reported that "the representative men of the party," when applied to by
the King for advice, recommended the appointment of Canalejas, on the
ground that Moret had lost their confidence. But it was not stated who
these representative men were. The _Daily Mail_ gave half a dozen names,
which had been telegraphed by its correspondent in Madrid, but that list
was obviously untrustworthy because Montero Rios figured in it, and it
is well known that the leader of the Radical group sets the unity of the
party above every other consideration, and has always urged loyalty to
Moret upon Liberals of all shades.
The circumstances were calculated to embitter the most even temper;
nevertheless Moret's first thought was for the welfare of the nation,
whose whole governmental machinery was thrown out of gear. Some of his
followers wanted to make a complete split with Canalejas, and one or two
articles were published in the heat of the moment, expressed in terms
tending to a final division in the Liberal camp. But in his own
utterances for the Press Moret showed himself true to his ideal--the
good of the country before personal ambition.
"The most serious feature in this crisis," he said, "is that both the
event and its solution were foretold by the reactionary newspapers,
proving the intervention of the reactionary party in the intrigue. They
interfered because they wish to prevent my conducting the elections in
accordance with my programme of electoral reform."
Moret's assertion that the intrigue which brought about his fall was
engineered by the Ultramontanes received confirmation from the _Correo
Catalan_, a Carlist paper, which committed itself to the following
prophecy:
"Canalejas will govern without altering the Cabinet until the autumn.
Before the re-opening of the Cortes there will be ministerial changes.
And in order to make compensation to Señor Moret a couple of
unconditional friends of his will enter the Government. In the autumn
Maura will have become tired of acting as guardian to Canalejas, who
will fall irremediably. The Maurist restoration will be inaugurated next
year."
Working-class opinion on the situation was quite definite. For a day or
two satisfaction was expressed, because Canalejas was reputed to be
devoted to the interests of the people. But no sooner was suspicion
aroused that his elevation to the Premiership had been engineered by the
Ultramontanes than the poor were up in arms: the mere suggestion that
the Jesuits were at work being sufficient to revive all the irritation
and anxiety that Moret had succeeded in allaying.
"Canalejas talks a great deal, and we have long looked upon him as our
friend," a journeyman mason remarked to me. "But here we are again with
everything in a state of confusion, and work in every direction waiting
while our employers are busy with their politics. We shall get nothing
done now till things have quieted down, so I don't see what advantage it
is to us to have Canalejas in power."
[Illustration:
GENERAL MARINA. SEÑOR CANALEJAS.
Commander-in-Chief at Melilla. Leader of the Liberal Democrats.
[To face page 244.]
"If it is true that Canalejas is in league with the Jesuits to bring
Maura back, there will be trouble," said another man. "We will not have
Maura ruining the country again just when it was beginning to pick up. I
would rather shoot him myself. The poor can't live under Maura, so I
should lose nothing by killing him, even if I paid for it with my life."
A woman burst out crying when she heard her husband talking about Maura.
"Why does the good God let that man live?" she sobbed. "If it is true
that he is coming into power again, all our sons will be sent to Melilla
to be killed. And we have been so contented because we thought we had
got rid of him!"
The hope of the Ultramontanes was that the downfall of Moret would bring
about a final and irremediable split in the Liberal party, which would
facilitate the overthrow of Canalejas when the time came. And at first
it seemed probable that this hope would be realised, for practically the
whole of Moret's Cabinet resigned with him, and refused to take office
under Canalejas, while Canalejas himself at first acted as though he
desired a permanent breach, by claiming that his appointment as Premier
necessarily carried with it the leadership of the party--a proposition
to which the party was by no means disposed to agree. But in time better
counsels prevailed, and an interview between the Premier and his
predecessor has lately been reported in the Press, in the course of
which Canalejas frankly admitted the obligations of the party to Moret
and the need that exists for his co-operation and advice--which Moret
for his part professed himself quite ready to give, as indeed he had
done ever since his resignation. So that it looks as though the danger
of a breach had been avoided, at any rate for the present.
It is worth noting that the Government of Spain can be carried on for an
indefinite time without the sessions of the Cortes. The Cortes adjourned
for the summer recess in June, 1909, before the troubles began in
Barcelona, and never met again. Throughout his tenure of office Moret
tried without success to obtain the Royal decree for a dissolution.
Canalejas was in office two months before he could get the decree
signed, but at length, in April, 1910, it was announced that the General
Election would definitely be held in May. The outgoing Cortes has a
Conservative majority: what the next will be no man can say, although,
having regard to the fact that a Liberal Ministry is in power, the
presumption is that a Liberal majority will be returned. There is,
however, no shadow of doubt that if the elections were conducted fairly
and freely and the people could vote in accordance with their
convictions, a Cortes would be returned with an overwhelming majority in
favour of the Constitutional Monarchy, reform of abuses, and the
destruction of the political influence and privileges of the Church.
POLITICAL PARTIES
CHAPTER XIII
POLITICAL PARTIES
It must not be supposed that the whole of the Conservative party shares
the Carlist and Ultramontane views of the majority. The old school of
Conservatives, led by Canovas, supported the Constitutional Monarchy as
strongly as do the Liberals, and even now a contingent of strong
constitutional Conservatives exists, although it is not easy to detect
their influence on the general policy of the party with which they act.
Their existence, however, was proved in October, 1909, when some of the
leading men of Señor Maura's party withdrew their adhesion to his
leadership upon his declaration of "implacable hostility" towards the
whole of the Liberal party. They saw, as did every one else, that the
reactionary policy of the Ultramontane Premier was imperilling the
existence of the Constitutional Monarchy.
To appreciate the disinterestedness of men who thus cut themselves off
from the acknowledged leader of their party, whether in office or in
opposition, the unwritten law of an alternate share of the spoils must
be borne in mind. Thus a politician who deliberately deserts his party,
from whatever motive, loses all chances of a salaried appointment when
that party again has power to confer these political plums.
I make no apology for putting the facts thus plainly. They are spoken of
with cynical frankness by all Spaniards, and it is considered a matter
of course that any statesman who refuses to sell his favours to the
highest bidder will be removed from place at the earliest opportunity by
the intrigues of the many who live by political corruption.
It will be seen that once the rule of alternating office by mutual
arrangement be broken, the end of the whole system of gerrymandering the
elections would be brought within sight, for any Government which
enjoyed the confidence of the nation would remain in office year after
year, once the people were permitted to make their voice heard in the
elections.
Señor Maura, or those who inspired him, of course foresaw this after the
fall of his Cabinet in October, 1909, and his address to his party on
that occasion marks an epoch in the history of political reform in
Spain, although perhaps not precisely on the lines he intended.
His party, he said, must fight without truce against the Liberals,
"ex-Ministers and ex-Presidents of Council who reach office in
collaboration with anarchists." Nor must his own party stand alone, he
said, for it would be necessary to seek the alliance of all, no matter
how different their political ideals, who desired to check the advance
of revolution. From that moment all relations between Liberals and
Conservatives must cease. Any other course would be treachery, for only
thus could the nation be saved from the reproach and the ruin with which
it was threatened.
In the opinion of Señor Maura, the party against which he thus
proclaimed war to the knife includes every shade of Liberalism in Spain,
from the most loyal Monarchists down to the rioters in Cataluña, while
his right-hand man, Señor La Cierva, Minister of the Interior in the
Ultramontane Cabinet, went so far as to accuse Señor Moret of being at
"one end of a chain which linked Liberals, Radicals, Democrats,
Republicans, and Socialists with the persons who fired the Religious
Houses in Barcelona and scattered anarchy broadcast by encouraging the
abominable sedition taught in the lay schools."
When Señor Maura declared that all relations between the two parties
must cease, he no doubt expected that the party to obtain power and
keep it would be his own. But the dissatisfaction caused in his own
party by his violent speech showed that they did not all share his
views, and for a time it looked as though there might be a split in the
camp. The Clericalists foresaw days of leanness, for if Maura's
calculations went wrong and Moret was able to carry out his project of
electoral reform, their occupation and their livelihood would be gone.
How the dissentients were brought into line we need not inquire. But the
occasion forced one of the most statesmanlike of the Conservatives to
make a public confession of his faith, and it then became manifest that
in Señor Sanchez Toca, ex-Minister and ex-Alcalde of Madrid, the spirit
of Canovas and Silvela still survives, although he with one or two more
seem to stand almost alone among their party against the tide of
reaction.
After three years of loyal support, said Sanchez Toca, in an interview
with the representative of the Conservative _Correspondencia de
España_--the organ of all that is left of the old Conservative
policy--Señor Maura should have consulted his Cabinet before taking a
resolution which completely alters the normal course of national
politics. "I stood aghast," said he, "at the thought of the incalculable
results that must spring from those furious voices convoking the whole
Christian world to a holy war against Ministers holding office under the
Crown, to whom even the name of Liberals was denied, and shouting with
anathemas that he was no true Conservative who held other relations than
those of implacable hostility with men appointed to office by the King."
And he concluded by pointing out that the true Conservative faith
irresistibly impels those who hold it towards conciliation instead of
provocation, and that if this reason for its existence failed, the
Conservative party must disappear.
Whether Sanchez Toca could have formed a party under his own leadership
on the old Conservative lines it is not possible to guess; for after
making the protest of which the above is a brief abstract, he left
Madrid, and it was announced that he intended to make a protracted
sojourn abroad, so that no one was able to accuse him of self-seeking in
his secession from the Ultramontane party. The dignity of his position,
as compared with that taken up by the supporters of the "implacable
hostility" which has become a byword among the scoffers, needs no
emphasis.
I have dwelt at perhaps undue length upon his part in the affair,
because it is assumed abroad that Señor Maura represents an united
Conservative party, which, as the above declaration proves, is by no
means the case.
How completely this was misunderstood by many of the English journalists
who wrote about Spain in 1909 was shown by their comments upon the
strength displayed by Maura in holding the whole "Conservative" forces
together, and their complete misapprehension of the real causes of the
dissensions which have always prevailed in the Liberal camp.
The modern Conservatives and the modern Liberals are so nearly alike in
their policy as regards the Crown and the Constitution, that they might
almost be classed as one party, under the general name of Monarchists.
In the matter of electoral reform there seems hardly anything to choose
between them, although on the question of the Religious Orders Moret's
views are perhaps rather more advanced than those of Sanchez Toca, Dato,
or Gonzalez Besada, the three most prominent Conservative-Monarchists of
to-day. Unfortunately, the popular distrust of the very name of
Conservative is so great that it would be difficult for any one thus
labelled to convince the people that he meant fairly by them. Even
Moret's policy of conciliation is taken by the masses to indicate fear
of the Jesuits, rather than as a calculated avoidance of action which
might lead to disturbances.
The constant commendation given by the Conservative and even by the
Liberal Press of England to the strength and unity of the "Conservative"
party under Señor Maura, and their adverse comments on the dissensions
in the Liberal camp, have materially added to the difficulties, already
serious enough, which block the path of Moret and those of his creed,
and have strengthened the party of clerical reaction and absolutism.
The _Heraldo_, in an article on the benefits to the nation to be
expected from Moret's support of Canalejas' Government, spoke as follows
of the influence of England upon Spanish affairs:
"It now appears probable that the democratic Government will be
consolidated by the disappearance of the danger to which we have
referred [the split with Moret's party]. If this proves the case, all
Europe will recognise with satisfaction how the personal convictions of
the monarch, strengthened perhaps by the healthy influence of his
illustrious connections by marriage, are leading Spain along the paths
of prosperity and gradually relieving us of the nightmare of reaction
which has weighed so heavily upon our nation during the minority of Don
Alfonso and the early years of his manhood."
Centuries of government by the rich for the rich, and by the Church for
the Church, have contributed to make reform exceedingly difficult, but
at length the issues between political morality and the maintenance of
the old abuses have been clearly set before the nation, in the struggle
which ended with the dismissal of Señor Moret. He determined to have the
country freed from the tyranny of the _Cacique_. His opponents desired
to maintain the system. That was the whole point at issue.
At the moment it seemed as if those interested in the maintenance of a
corrupt system had won a signal victory, and the men who are working for
the moralising of political life would have been more than human had
they spoken no word of the bitterness they felt at seeing, as it seemed,
their work undone and their hopes frustrated. But there are apparent
defeats which mark a stage on the road to final victory, and such a
stage was marked, for the people of Spain, by the fall of Moret in
February, 1910.
Turning to the other main body of political opinion, the Liberal party,
with its offshoot, the Republicans, it is worth noting that many of
these, including several of their most prominent and influential
leaders, although professing republican opinions, are in reality
staunch upholders of the constitutional Monarchy, their republicanism
being more in the nature of a political counsel of perfection than a
policy that they are actively forwarding. Thus Montero Rios, the leader
of the Radical wing of the Liberals, who, if not avowedly a Republican,
is closely allied to that party, recently said, _à propos_ of the split
in the Liberal camp which seemed imminent after the resignation of
Moret, "I have always urged that our group should submit to the
leadership of Moret, because he alone can hold the party together."
Melquiades Alvarez, one of the acknowledged leaders of the Republicans,
made in October, 1909, an important speech in which he offered "a final
truce" with the Monarchy, and Republican support to a programme of
liberty of worship, restriction of the power of the Religious Orders,
neutral schools, and social reform. "With the adoption of this programme
permanent stability would be afforded to the Throne on the model of the
English dynasty--a crowned Republic." And Soriano, another prominent man
in that party, said about the same time "The Republican revolution
should be spiritual, not material. _We do not desire to overthrow the
Monarchy_, but to implant education and progress" (italics mine). The
term "Republican," as used by the men of this school of thought, seems
to connote a social and political Utopia rather than a particular form
of government, and "republican" principles are quite compatible with an
undeviating support of the Constitutional Monarchy.
These "idealist" Republicans would not thus group their party with the
supporters of the Monarchy if they believed that the existence of the
Throne were prejudicial to the nation. Nor would the Liberal-Monarchists
accept without protest such an association with themselves, did they
believe that these men were working to overthrow the Monarchy.
The truth is that all Spanish politicians who have the good of their
country at heart recognise, even though they may disapprove, the
traditional respect for the kingly office which is implanted in the mind
of most peoples who have lived from childhood under the Monarchical
system. In Spain, where the King who united Castile with Leon and
expelled the Moslems from nearly the whole of the South is venerated as
a saint, the tradition exists more strongly and has greater weight in
determining the action of the masses at any given moment than in any
other country except perhaps Russia.
EDUCATION
[Illustration: A STREET HAWKER DESCRIBING BATTLE SCENES TO AN ILLITERATE
AUDIENCE.
[To face page 263.]]
CHAPTER XIV
EDUCATION
Of the many evils that afflict Spain, one of the gravest, for it lies at
the root of most of the others, is the deplorably backward state of
education. It is commonly said that 75 per cent. of the population
cannot read or write. This figure may or may not be exaggerated, but it
is certainly the exception to find a member of the working classes who
can do either. And this ignorance is not confined to the working
classes, but extends, in a relative degree, throughout all social ranks.
People of good position, presumably educated, frequently cannot write
and spell their own language correctly. I have even been told as a fact
that there are, or were until quite recently, grandees of Spain who
could not sign their names. And ignorance of the commonest facts of
geography and history is astonishingly prevalent even in the middle
classes. It would not in the least surprise any one who knows Spanish
society to be asked whether Germany lies to the south of Switzerland,
or if Berlin is the capital of London. Even in the universities things
are no better. The course of study in any subject consists in the
scholar getting up a textbook written _ad hoc_ by the professor of that
subject, in which alone he is examined for his certificate or diploma,
and outside of which he is not expected to travel. Indeed, in some of
the universities the students are actively discouraged from reading
anything except the prescribed textbook of the subject they are
studying, and the natural consequence is that a young man who has passed
through the University with credit may be, and often is in fact, quite
illiterate.
The administrative educational system in Spain is as follows:
At the head is the Minister of Public Instruction, assisted by a
consultative Council, which includes the Rectors of the universities.
Certain of the functions of the Minister are delegated to the
sub-secretary, who is the second authority in the department.
The local administration is complicated. The whole country is divided
into ten university districts, at the head of each of which is the
Rector of the university. He, _inter alia_, exercises a general
supervision over all the schools in his district, appoints teachers
whose salary is below 1,000 pesetas per annum, and proposes to the
Minister of Education the appointment of those of higher grade or
salary. He is assisted by a consultative Council. The teachers are
appointed after some sort of competitive examination.
The Civil Governor of each province is responsible for the fulfilment of
all the obligations imposed by the law, but has no voice in the internal
management, teaching, &c., of the schools and colleges. He, too, is
assisted by a provincial consultative Council. Lastly, in the
municipality, the Alcalde, the President of the _Ayuntamiento_, has the
same functions in his district as those of the Civil Governor in the
province.[23] The Alcalde also has his local consultative committee,
whose functions are not unlike those of school managers in England.
The inspection of all schools except the elementary is the duty of the
Rector of the university. The inspection of elementary schools is
committed to an Inspector-General, under the immediate orders of the
sub-Secretary of State, and forty-nine inspectors, one for each
province.
Elementary education in Spain has been compulsory since 1857, and free,
since 1901, to children whose parents "are unable to pay." The
compulsory school age is from 6 to 12. The provision of schools and the
upkeep of the buildings is the duty of the _Ayuntamiento_, and a small
sum is set aside in the annual Budget of the kingdom for grants in aid
to poor districts.
The system under which the teachers are paid is peculiar. The locality
finds the money and hands it to the Ministry of Education, which pays
the salaries. The reason for this arrangement is characteristic. It was
made because of the irregularities in the payment of the teachers which
frequently occurred when the local authority administered the funds. The
salaries of the teachers in the Elementary schools are from 500 to 3,000
pesetas--say £20 to £120 a year, with a house, and they are entitled
after twenty years' service or upwards, to a pension of from 50 to 80
per cent. of their salary. They also have a right to the fees of
"children who can pay them."
It is easy to see that this system of overlapping authorities and
divided responsibility must necessarily lead to waste of time and
general inefficiency, even assuming that every one concerned is
genuinely anxious to do his duty and to work for the good of education,
an assumption which it would be very rash to make.
The teachers in the Government schools have to hold a Government
certificate, which is obtained after a two years' course in a normal
school. In the higher-grade schools a superior certificate is required,
involving an additional two years' training. Private schools are
reckoned as part of the school supply, in a proportion to the total
which varies with the population of the school district. The school
supply is calculated, not on the basis of school places but of teachers;
roughly speaking, one master and one mistress are allowed to each
thousand of population. The total number of qualified teachers of both
sexes is eventually to be brought up to 30,000. The private schools have
to satisfy the inspector as to sanitary requirements only: no control
seems to be exercised over the instruction, nor is any certificate of
competence demanded of the teacher.
In addition to the elementary schools, the law provides for the
establishment of infant and night schools, schools for the deaf and
dumb, and all the machinery of a complete and comprehensive system of
secondary and higher education.
So much for the theory: the practice is another matter.
In the universities all the elements of university education are
lacking. The professors in many cases are ignorant and incompetent, and
those who are properly qualified for their work--and some of them are
men of scholastic and scientific attainments of no mean order, often
acquired abroad--are isolated, working in unsympathetic and even hostile
surroundings, and their knowledge is almost useless in the lecture-room,
owing to the fact that the students are absolutely destitute of the
grounding necessary in order to benefit by proper teaching. When by
chance a group of professors with real knowledge and enthusiasm happens
to be formed in an university, the results are surprising. This has
occurred at Oviedo, where, thanks to the existence of such a teaching
staff, an educational atmosphere was formed, the students were educated
and not merely crammed with a few useless facts, and extension lectures
were established with extraordinary success, especially among the
working classes.
In addition to a body of competent professors, the universities, if they
are to fulfil their function, need (_a_) a certain amount of
autonomy--at present they are merely bureaus for conducting
examinations and issuing diplomas; (_b_) reorganisation of studies, with
some liberty of choice to the student; (_c_) reform of the examination
system in the direction of substituting for the present examination by
subjects either a certificate based on attendance and general
proficiency or a final examination on leaving; (_d_) in scientific
subjects, a good deal more practical work; and in all branches of study
the abolition of the textbook, the committing of which to memory is now
the sole demand made on the candidate for examination.
A professor at one of the universities has summed up to me the present
state of these centres of learning in the following words:
"There is an absolute lack of any educational spirit, of any contact
between teachers and pupils, of any feeling of solidarity among the
students, of any organisation of games, excursions, &c., of any artistic
refinement, and of any organised effort to raise the moral standard,
to-day perhaps the most degraded in the world."
When we turn to the administration of the elementary schools, the part
of the educational system which more directly and immediately affects
the working classes, we find the same general state of inefficiency and
neglect.
A volume of school statistics was officially issued not long ago, of
which a useful summary was published in the _Heraldo de Madrid_ in
November, 1909.
From this it appears that while four provinces have the full complement
of Elementary schools required by the law, the supply in all the
remaining 45 is deficient, the shortage per province being from 772
schools downwards, and the total deficiency amounting to 9,505 schools.
The total increase of school supply between 1870 and 1908 is 2,150
schools, or an average of about 56 schools per year. At this rate it
would take over 150 years to catch up even to the school provision
required by the school law of 1857, without allowing for any increase of
population.
But in another way, about two-thirds of the school districts of Spain,
or some thirty thousand towns and villages, have no Government school.
In Madrid about half the schools required by law are wanting. Barcelona
has a somewhat similar deficiency. And be it remembered that the school
supply is calculated in accordance with the law of 1857, the
requirements of which are far below those which obtain in any other
country in Europe, so that even in the very few districts where there is
nominally sufficient school accommodation there is actually a serious
deficiency according to modern standards of what is necessary. The
consequence, as the _Heraldo_ observes, is that some 12,000,000 of the
population do not know their letters.
But the towns where there is a school are not really much better off,
educationally, than those that have none. Save in very exceptional
cases, no attempt is made to enforce school attendance, and though some
of the parents send their children to school, the careless and
indifferent do not. The Alcalde, whose business it is to see that the
law is carried out, probably is--except in the larger towns--entirely
uneducated himself, and is not going to stir up possible ill-feeling by
enforcing a law which does not benefit him personally, and of which he
does not see the necessity. The Civil Governor, who is over the Alcalde,
and whose duty it is to see that the education laws are carried out,
probably is equally indifferent, and in any case has not the time to
supervise all the Alcaldes of the scores of towns and villages in his
province. The schoolmaster naturally does not trouble himself; his
salary does not depend on the number of his pupils, but on the
population of the school district. And, lastly, any attempt to enforce
the attendance required by law, were it made, must necessarily fail,
simply for lack of school accommodation, for already most if not all of
the elementary schools have many more children on the register than
there is room for. Thus school attendance, although nominally
compulsory, is in fact purely voluntary, with the usual results.
The supply of school material is the duty of the Central Government, as
already stated. This duty the Government delegates, not to the local
authority, but to the schoolmaster, who receives, in addition to his
salary, an allowance to scale for providing books, &c. The natural
result is that he looks on this allowance as an augmentation of salary,
and reduces the supply of books to the barest minimum, or to zero. It is
not to be expected that the schools should be liberally or even decently
supplied with these necessaries when every penny that can be saved on
them is so much clear profit to the teacher. The consequence--seeing
that books cannot be altogether dispensed with--is that the children
have to pay for them, and the intention of the law, that schooling
should be free to the poor, is frustrated.
The working classes, who, as has been said, honestly desire that their
children may receive some rudiments of education, do not as a rule like
the Government schools, because, they say, nothing is taught in them. It
is not at all uncommon for the teacher to absent himself altogether
from school during school hours. He may or may not set the children some
lesson--for instance, a passage to repeat over and over again--and he
may or may not lock the door after him when he goes away, but very often
the children are left entirely alone during the hours when the school is
open and they are supposed to be receiving instruction. Parents say, and
no doubt with truth, that the moral consequences of this lack of
supervision are exceedingly bad, and that a great deal of harm is done
to the majority by uncontrolled association with a few demoralised
children. A working man in a small provincial town complained to me that
a whole school had been corrupted by the evil influence of one boy older
than the rest.
It will naturally be asked why such a state of things is tolerated. The
answer is easy. It is the duty of the Alcalde to see that the
schoolmaster does his work and does not absent himself without leave.
But the Alcalde may be a friend or relative of the schoolmaster, or may
have other reasons for not worrying him by pedantically insisting that
he do his duty. Besides which, it is quite probable that the
schoolmaster is not being paid. His salary may not have been sent to the
_Ayuntamiento_, or if sent may not have reached him. According to the
article in the _Heraldo_ above referred to, the _Ayuntamientos_ are now
in debt to the school teachers for arrears of salary amounting to
7,000,000 pesetas--say £280,000. Therefore the negligent schoolmaster is
not unlikely to have a conclusive answer to any remonstrance that the
Alcalde might be inclined to make: "Pay me my wages and I'll do my
work."
The parents dare not complain. The Alcalde or the teacher, or both of
them, would make things unpleasant for the audacious parent who hinted
that either of them was not doing all he should, and there is further
the tradition of hopeless submission to misrule of all kinds, from long
experience of the uselessness and danger of protesting, which in itself
makes the working man reluctant to take any steps against those in
authority.
As far as can be gathered, the working classes seem on the whole to
prefer the private schools, in spite of the fee charged, on the ground
that in these schools the children are under some sort of supervision
and do learn something, if only, a little. But the fee, even where quite
low, is a serious obstacle to a labourer with a large family, who is
only earning some ten or twelve pesetas a week, and, as has been already
said, it often happens that only one child can be sent, who in the
evenings passes on what he has learnt to the rest of the family. But in
these schools it is the custom for the children to bring presents to the
master on certain occasions, and it is said that a child who does not
bring his present is neglected. "Only the children of those who have
money get any teaching," the parents say.
In many towns most of the private schools are kept by nuns. These
schools, generally speaking, have not a good name. It is said that the
children who attend them are taught nothing but catechism and
needlework, and that the punishments given are often cruel and sometimes
disgusting. There was a great scandal lately in a town in the south, on
account of a punishment of a nature impossible to describe, inflicted on
a little girl in one of these schools. The father took steps to bring an
action against the convent in question, but the Civil Governor
interfered, and compensation was paid in order to have the matter hushed
up.
There are schools, both public and private, where a better state of
things prevails, where the master is more or less of an enthusiast, and
where in consequence the children get decent elementary teaching;
indeed, in one village I was told that "the master of the Government
school taught very well, when he was sober." Nevertheless it is a fact
that most of those few of the working classes who can read and write do
so badly; indeed, to decipher a letter, say from a domestic servant, or
a workman or small tradesman, is a labour of great difficulty, not only
owing to the bad writing, but to the extraordinary spelling, although
Spanish is the easiest of languages to spell correctly.
Still, in spite of all the obstacles created by administrative
incompetence, neglect, and corruption, some progress is being
made--enough at least to prove that the people would take immediate
advantage of a decently efficient school system. If any members of a
family can read, it is usually the children. I have often seen little
groups of older people seated round some child who reads the paper to
them.
The desire of the working classes for education has brought about a
remarkable change in their attitude towards the conscription. This
change is the growth of recent years, and is the result of personal
efforts on the part of certain distinguished officers. Their feeling in
the matter may be summed up in the words of Colonel Ibañez Marin, who
met his death in the early days of the Melilla War: "My ambition is that
no conscript shall return to his home, after serving his first three
years, without being able at least to read and write."
I am continually asked about the education of the working classes in
England.
"They say that in your country every one is taught to read. Is it
possible that that is true? But you have a different kind of Government
from ours. Over there it seems that they attend to the interests of the
poor. Here you must be able to pay if you wish to learn anything.
"England must do something for us now that our King has married your
King's daughter. If things here were conducted as they are over there,
Spain would be the happiest country in the world. But England will take
our part in everything now, so matters will improve."
If the connection between King Edward and the Queen of Spain is
explained, and it is observed that one nation cannot interfere with the
internal affairs of another on the sole ground of relationship between
their respective rulers, the speaker will reply:
"You say that because Governments talk in that way, but we know better.
Are not kings human beings like ourselves? And if the Spanish Government
knows that Don Alfonso asks the King of England for advice, will they
not have to respect the advice he gives? It is not possible that England
should take no interest in our affairs when our Queen is your King's
niece."
Perhaps one ought to explain that the only influence that England could
exert in their favour is that of public opinion, and that England is too
busy with her own affairs to have time to form an opinion about those of
Spain, far less to express it in a convincing manner. But it would be
cruel to deprive these people of the gleam of hope which has come to
them through the King's marriage, so perhaps I say, "In the meantime
here are a few little reals for teaching," and get the reply: "May God
repay you! My second boy can go to the night school for a month for
seven reals."
A movement which has in it great promise for the future was started a
few years ago by certain able young university professors, who fully
realise how much of the backwardness of their country is due to lack of
education, with its resultant narrowness of mind and outlook, and
ignorance of the modes of life and thought of other nations.
The fundamental idea of this movement, as described to me by its
originator, is to create an organic body, independent of political
changes, which shall endeavour little by little to promote contact
between the teachers of all grades in Spain and their foreign
colleagues, and to form within the country small nuclei of workers to
diffuse in Spain the ideas brought from abroad, and to create an
atmosphere of sympathy and enthusiasm, without which scientific work
cannot flourish. On these lines two Committees were formed by Royal
Decree in January, 1907. One of these was charged with reforms in
elementary teaching, to be carried out by the establishment of classes
for teachers, by school inspection on modern lines, by sending selected
teachers abroad (the Government gives a grant for this purpose, the
administration of which was entrusted to the committee) and by grouping
and grading the schools, and encouraging and supervising holiday resorts
for teachers, school games, &c.--the whole of the reforms to be
introduced gradually, as circumstances might permit. To the other body
then created, a "Committee for the development of studies and scientific
research," was entrusted the gradual formation of a staff of competent
teachers and professors for higher education generally and for
scientific studies.
But the work of these Committees, which, if steadily pursued, offers the
best hope for the intellectual regeneration of Spain, was paralysed in
the first year by Government interference. Between the formation of the
Committees and the issue of their first Annual Report a change of
Government took place, and the Ministry of Señor Maura, true to the
traditions of Clericalism, did their best to bring all effective work to
an end. They suppressed altogether the Committee charged with reforms in
elementary education, and set up in its place a mere official bureau,
powerless and useless. And though they did not actually abolish the
Committee for Higher Education, they succeeded in putting an end to all
effective work, by overriding its statutory constitution and curtailing
its freedom of action, by stopping supplies, and by delaying or refusing
the necessary official consent to measures proposed by the Committee.
For instance, whereas the Committee had made arrangements with the
French Ministry of Public Instruction for the disposal of the teachers
who held grants for study abroad, the Government refused to recognise
these arrangements unless they were made officially through the
ambassadors, with the result that in the year 1908 none of these
teachers were sent abroad. In the Budget for that year the sum set aside
for foreign study was reduced by 110,000 pesetas (about £4,400),
although in the previous year a quite exceptional number of applicants
for these grants had come forward. These instances are sufficient to
show the attitude of the Clericalists towards education, but the whole
Report of the Committee shows how at every turn their work was checked
and hampered after Señor Maura took office. With the return of a Liberal
Ministry to power it is hoped that the work will be once more
effectively taken up.
* * * * *
Since the above chapter was written a Report on the present condition of
education, addressed to the Cortes, has been prepared by the Minister of
Education, and summarised in the Spanish Press. The following quotations
from this summary throw a lurid light on the actual state of affairs.
"More than 10,000 schools are on hired premises, and many of these are
absolutely destitute of hygienic conditions. There are schools mixed up
with hospitals, with cemeteries, with slaughterhouses, with stables. One
school forms the entrance to a cemetery, and the corpses are placed on
the master's table while the last responses are being said. There is a
school into which the children cannot enter until the animals have been
taken out and sent to pasture. Some are so small that as soon as the
warm weather begins the boys faint for want of air and ventilation. One
school is a manure-heap in process of fermentation, (_sic_) and one of
the local authorities has said that in this way the children are warmer
in the winter. One school in Cataluña adjoins the prison. Another, in
Andalusia, is turned into an enclosure for the bulls when there is a
bullfight in the town.
"The school premises are bad, but most of the Town Councils do not pay
the rent, for which reason the proprietors refuse to let their houses.
Ninety per cent. of the buildings in which the schools are held are the
worst dwellings in the town.
"In Lucena the salary of a mistress is held back because she guaranteed
the school rent. The Municipality did not pay it, the school was going
to be evicted and the teaching to be interrupted, and that mistress, in
order to prevent this, pledged her miserable pay."
Comment is needless: the facts, vouched for by the Minister of
Education, speak for themselves.
TAXATION
[Illustration: SAFFRON PICKERS SORTING THEIR CROP.
To face page 285.]
CHAPTER XV
TAXATION
Among the sources of the national revenue of Spain there are several
which more especially affect the poorer portion of the community, or, by
hampering trade and manufactures, put obstacles in the way of the
national prosperity. Among these may be especially mentioned the Customs
duties, the tax on trades, business, and professions (_contribucion
industrial_) the _octroi_, or _consumo_, the creation and sale of
monopolies, and the national lottery. The total taxation of the country
is absolutely crushing, and makes Spain one of the dearest places of
residence in Europe.
The Customs are excessively high, especially for a country that has
comparatively few manufactures of its own to protect. To take a few
instances at random from the Tariff: Straw hats pay about 20 pesetas per
kilogramme; preserves, 3 pesetas per kilogramme; typewriters, 8 pesetas
per kilogramme; timber in boards, 6 pesetas per cubic metre; materials
of silk and velvet, from 30 pesetas per kilogramme; woollen materials,
from 13 pesetas per kilogramme; drugs at various rates from 36 pesetas
per kilogramme downwards, and so forth. The consequence is seen in the
prices charged in the shops for ordinary commodities. Thus Danish tinned
butter costs pesetas 2.75 per lb.; tapioca, 1 peseta per packet of about
¼ lb.; coffee, pesetas 2.50 to 3 per lb.; Keiller's marmalade, 3 to 4
pesetas the 1 lb. pot; Burroughs & Welcome's tabloids, pesetas 2.50 the
small bottle of 25; and most other things in about the same
proportion.[24] The Customs receipts form, with the exception of the tax
on real estate, the largest single item in the Budget. For the year 1909
they were estimated at pesetas 153,600,000 (£6,144,000).
The _Contribucion industrial_ is a tax upon every imaginable trade,
profession, and industry that can be exercised: on merchants,
manufacturers, shopkeepers, professional men of all sorts, on means of
transit, on public entertainments, on schools, newspapers--in short, it
is almost impossible to find any kind of business or occupation that is
not taxed.
The amount of the tax varies with the population, a scale of ten
different rates being drawn up according to the size of the town.
Bankers pay from about £300 a year downwards, according to the place of
residence; merchants from £200; shopkeepers from £70; hotels from £60;
contractors 6 per mil. on their contracts; railways £10 per kilometre
constructed, and a tax on their receipts; brokers from £80; newspapers
from £35; publishers from £25; private schools from £10; and so on. It
is tedious and unnecessary to accumulate instances.[25]
That such a burdensome tax as this necessarily hampers trade and goes to
prevent commercial and industrial development needs no demonstration:
the thing is self-evident. The compiler of the manual of this law says
in his Introduction, with perfect truth, that it has the radical defect
of making heavier demands as the trader's profits fall, and that the
framers of the law, so far from attempting to harmonise the interests of
the Treasury with those of the taxpayer, thought only how to squeeze
him, so that the tax nips in the bud whatever might aid in the increase
of prosperity or open new fields for the productiveness of the nation.
This tax immediately affects the professional and commercial classes:
the poor, such as street hawkers, journeymen labourers, fishermen, &c.,
are exempt; but indirectly they too suffer, as naturally it helps to
increase the price of necessaries.
The greatest burden on the working classes--and it is a very grievous
one--is the _octroi_, or _consumo_, a heavy tax on nearly every kind of
food, drink, and fuel, and on timber, stone, lime, &c.; in short, on
nearly everything that is consumed in use. The fisherman has to pay
_consumo_ on his catch before he can sell it; the farmer on his dead
meat, poultry, and eggs brought to market; the charcoal-burner on his
charcoal; and so on. The tax, varying in details, is levied in every
town and village, and thus may be, and often is, paid twice over by the
same goods, if they happen to be conveyed from one town to another.
It is obvious that such a tax on the necessaries of life presses with
exceptional severity on the poor, and it is, moreover, steadily rising,
while wages remain stationary. It is usually farmed out to syndicates
which are said, and no doubt with truth, to be making enormous profits
out of it. These syndicates are believed by the people to consist in
many cases of persons who
[Illustration:
A SELLER OF PALM-LEAF BRUSHES AND FANS.
[To face page 289.]
represent the Jesuits, and the oppressiveness of the tax and the steady
rise in its amount form another count in the heavy indictment of the
poor against the Religious Orders. The estimated receipts from this tax
in the Budget for 1909 were pesetas 58,000,000 (about £3,520,000).
"Everything in the country is dying of the _consumos_," said a working
woman of about sixty years of age, who remembers with regret how much
easier the life of the poor was in the days of Isabel II. "Every four
years the contract for the _consumos_ in our province is put up to
auction, and every time they are sold the price is raised four or five
thousand duros,[26] and _we_ have to pay the difference. Yesterday
Manolo paid four duros _consumo_ for the fish he sold in the market, and
all he had for himself after twenty-four hours' work was ten reals. The
man who rents the _consumos_ from the Government is rotten with money
(_podrido de dinero_): millions and millions of pesetas he has, all
wrung from the necessities of the poor. Don Alfonso does not like it;
every one knows that. If he had his own way there would be no _consumo_
for the poor. Already since he came into power we have been relieved of
the _consumo_ on wine, green vegetables, and potatoes, and they say that
two years hence, when the contract runs out, he wishes that it shall not
be renewed. But that would not suit the Government nor the Jesuits, who
are mixed up in this business. They would lose too much which they now
are able to put into their own pockets. So they would like to make
another revolution to get rid of Don Alfonso, as they got rid of his
grandmother, before their contract comes to an end. In her time bread
cost just half what it does now, twenty-five eggs cost five reals
(pesetas 1.25) instead of two pesetas a dozen, and for four cuartos we
could buy a piece of pork as big as we get now for two reals.[27] Salt
was free of _consumo_, so was oil, so was cheese, and shell-fish and
chestnuts sold in the street were not taxed, so that they could be
bought for much less than now, and the whole reason is because the
Government lets the taxes instead of taking the trouble to collect them
as was done in the time of Queen Isabella."
Whether this good lady, whose words I have translated literally from
notes made at the time, was right or wrong in her supposition as to the
interest of the Jesuits in this tax, and as to the quadrennial increase
in the amount paid for it by the syndicate of farmers who exploit it, I
cannot say. I quote her words as an instance of what is said on all
sides by her class whenever the subject is mentioned, and as far as I
can learn she is quite correct in her comparison of the prices of food
now with those of forty years or so ago.
Tobacco and sugar are Government monopolies, farmed out to companies,
which also are popularly believed to be under the control of the
Jesuits. I have never seen any accounts of the profits of the Tobacco
Company, but their shares are quoted at about 390 to 400, which speaks
for itself. The tobacco they supply is very bad, and outrageously dear.
The estimated receipts from this source were pesetas 140,400,000
(£5,616,000).
The sugar trust was created comparatively recently. A short account of
the last annual meeting of the shareholders was published by the Press
in November, 1909, from which it appears that the trust made a profit in
the year, in round figures, of pesetas 8,400,000 on a gross income of
pesetas 14,600,000 (say £336,000 on £584,000), and pays a dividend of 8
per cent. And during the past twelve months the price of sugar has been
rising, and now stands at about 7d. per lb. Figures like these,
relating to a necessary of life which is the one luxury of the poor, do
not require comment, especially in view of the fact that enough cane and
beetroot sugar for the entire needs of the population could be produced
in the country, where both soil and climate are suitable in a great part
of the southern provinces. But one company after another has been
crushed out of existence, and only ruined factories remain to remind the
traveller of what ought to be prosperous undertakings, beneficial to the
whole nation. From this source the State gets pesetas 31,600,000
(£1,264,000).
Matches are another monopoly, also farmed out. They are of course bad
and very dear--½d. or 1d. for fifty matches, according to quality. The
conditions under which the operatives work are, I am told on good
authority, simply deplorable, and growing worse instead of better. The
estimated receipts are pesetas 10,000,000 (£400,000).
A tax which combines a maximum of irritation with a minimum of profit is
one which is levied on the business books of persons engaged in
commerce. Every page of the ledger, cash book, press copy book, &c., has
to be officially stamped at a charge of so much per page: the total
charge for a complete set of commercial books sometimes amounting to 500
pesetas (£20), and not only so, but the Government--presumably in order
to get more out of the tax--prescribes the method by which the merchant
must keep his books. I was told by the manager of a large foreign
industrial concern that he has to employ twice as many clerks as he
needs, solely because the authorities insist on a cumbrous and obsolete
system of book-keeping.
The law enacts that pious foundations which offer their manufactures for
public sale are liable to taxation. It is currently said that this
obligation is evaded. Whether this is the case or not I cannot say from
personal knowledge, but certainly any visitor can purchase sweets or
needlework made in the convents. Indeed, some of them are celebrated for
their confectionery, which is always sold a trifle under the cost of
similar goods made by a lay tradesman.
If the taxes were fairly and honestly collected, their amount could be
materially reduced. But as a matter of fact many are not collected at
all from the persons most able to pay. The tax-collector is usually
willing, for a consideration, to play the part of the unjust steward,
and take less than the proper amount. It is sometimes said that only
fools and foreigners pay the taxes, and cases have occurred in my own
knowledge where bribery in the proper quarter has effected a
substantial reduction in the amount accepted. Every resident in Spain
knows of such instances: the thing is notorious, is talked of quite
openly, and is done with hardly any attempt at concealment. It is
impossible to conjecture what proportion of the total taxes due is thus
informally remitted, but it must be something considerable.
Complaints about evasions of taxation frequently appear in the papers:
thus it was stated as a fact in the _Liberal_ in February, 1910, that
about 45 per cent, of those liable for _Contribution industrial_ evade
payment. In the same newspaper, in the same month, appeared a long
statement, signed by the officials of the Guild of Cab Proprietors in
one of the large towns, accusing certain owners of livery stables, who
let smart carriages for hire, of defrauding the municipality of some
50,000 pesetas (about £2,000) a year by falsifying the declarations on
which they take out their licences, and no attempt was made to show that
the accusation was unfounded. Complaints about evasion of taxation by
large landowners also are of frequent occurrence.
Quite recently the Government has seriously taken up this question of
falsified returns, especially in the case of real estate, and is making
a systematic inspection of the properties liable for taxation. An
immense amount of fraud has already been discovered in the towns, and
the case of the rural estates is probably worse. I was lately told of an
instance where, to my informant's knowledge, an estate which adjoins his
own has been paying 60 pesetas a year, whereas it should have paid about
2,000. In some parts of the country the large landowners are doing their
utmost to oppose the carrying on of the Ordnance Survey, because the
effect of it would be to define and make public the extent of their
property.
An ingenious mode of defrauding the exchequer of succession duties is
practised on a gigantic scale. This consists in depositing personal
property in the banks in the joint names of all concerned, actual
holders and heirs apparent, to the order of any one of them. Thus on the
death of the father, the owner of the personal estate, it passes to his
son without any legal intervention, and the Treasury is powerless to
collect the succession duties. Under the Spanish law as it now stands,
if one of the owners of such a joint deposit dies, the deposit pays a
proportion of the duties corresponding to the number of names in which
it stands: a half if there are two, a third if there are three, and so
on. In January, 1910, there were "undefined deposits" (_depósitos
indistintos_) as they are called amounting to nearly 519,700,000
pesetas (about £20,788,000) in the Bank of Spain alone, and Alvarado,
Moret's Minister of Finance, obtained a Royal Decree dealing with these
deposits. His plan was simple: merely to make the joint deposit liable
for the whole duty on the death of any one of those interested. As this
would oblige the owner to pay if the heir died first, it is obvious that
the practice of depositing in joint names would at once come to an end.
But Cobian, Alvarado's successor in Canalejas' Ministry, suspended the
decree, a proceeding inexplicable in a Minister whose Chief loudly
proclaims his democratic principles. Meanwhile the depositors took
immediate advantage of the respite afforded them by the suspension of
the decree to transfer some 200,000,000 pesetas (about £8,000,000) to
banks abroad, and most probably a good deal more will go the same way.
The Religious Orders are fighting the decree tooth and nail, because
while legally formed associations, who do not desire to conceal their
capital, do not object to the decree, illegal associations, who have
reasons for secrecy as to their affairs, find in the system of joint
deposits an easy way of escaping their liabilities. It must be
remembered that most of the Religious Orders now established in Spain
are illegal, the Concordat only allowing of two, together with a third
not yet named.
It is always assumed, as a matter of course, that the whole
administration of the country is corrupt. When an unexpected deficit
appears in public accounts, governmental or municipal, when a sum of
money voted for a certain purpose has evidently not been spent as
intended, when, as frequently happens, money owing by the State or the
Municipality is not paid--in short, whenever there is anything in the
national or local administration of the public funds which calls for
explanation, it is taken for granted that some one in office has been
stealing. Whether this assumption is justified or not I do not pretend
to say. All I know is that it is universally made.
I asked a Spaniard on one occasion why a certain public building had
never been finished. "No doubt the Alcalde uses the money to keep up his
carriage," was the reply. The man certainly did not know the facts, but
this was to him the most plausible explanation. When a few years ago
Admiral Cervera was ordered to fight the United States with ships armed
with obsolete guns and shells that did not fit them, every one said, and
still says, if the subject is spoken of, that officials in the
Government stole the money that ought to have been spent on the Navy.
The system extends, or is said to extend, from the highest ranks of
officialdom downwards, and if this is true, it must necessarily operate
in substantially reducing the total funds available for the Treasury.
A minor matter, which I only mention because, it goes to illustrate once
more the system of over-taxation with no adequate result, is the postal
service. A letter in Spain does not cost a penny, as it does everywhere
else; it costs twopence: of this three-halfpence are paid by the sender
and a halfpenny by the receiver. In exchange for this, the Government
gives a service which is indifferent in the large towns, and infamously
bad in the smaller towns and the rural districts, where there is no
security whatever that any given letter will reach sender or receiver,
and where, to my own knowledge, a very large number are lost.
Gambling in the national lottery, which is drawn about three times a
month, is almost universal, and an immense amount of money must be
wasted on it. I remember seeing a man in a second-class railway
carriage, after borrowing my newspaper to see the result of a drawing,
throw away at least a dozen tickets, representing a cost of either three
or five pesetas each. The lottery is conducted with absolute fairness,
and it might be argued that, as people will gamble, it is better that
they should do so on a straightforward lottery than, _e.g._, on
horse-racing or some other sport of doubtful honesty. On the other hand,
there is no doubt that the fact of these lottery tickets being thrust
under the noses of the public all day long, coupled with the reports
current in conversation and the particulars given in the Press of the
sudden wealth which has accrued to this and the other working man
through a lucky number, must foment the gambling spirit, which is
sufficiently rife in Spain without any such official encouragement. The
estimated net receipts from the lottery for 1909 were pesetas 35,250,000
(£1,410,000).
It must be borne in mind, in connection with the universal venality of
the lower grades of the bureaucracy, that a certain amount of excuse is
to be found in the salaries they receive, which are miserably small in
amount and often in arrears. When a man has to keep himself and his
family on two pesetas a day, it is not surprising that he takes
advantage of the opportunities which his official position gives him to
increase by illicit means a wage on which it is quite impossible that he
should live decently and honestly.
THE PROCESS OF REGENERATION
CHAPTER XVI
THE PROCESS OF REGENERATION
The regeneration of Spain must necessarily be a slow process, for the
causes of her degradation are deep-seated, and are not to be removed by
mere legislative enactments or alteration of the machinery of
government. One of the principal difficulties with which the country has
to contend is the dishonesty of the bureaucracy, which paralyses any
reform that may be attempted. Of what use is legislation, when the laws
are not honestly administered? If what is the common talk of all classes
has any foundation whatever in fact, the whole of the bureaucracy, from
top to bottom, not excluding the inferior judiciary, is venal and
corrupt, and until a tradition of honest administration is established
amendment will be difficult, if not impossible.
The history of Spain for the last three hundred years affords an
illustration of the proposition established by Lecky[28] that "the
period of Catholic ascendancy was on the whole one of the most
deplorable in the history of the human mind." In no country in Western
Europe has the Church of Rome been so entirely absolute and dominant,
since the Reformation, as in Spain, where the Inquisition instantly and
finally crushed out all freedom of thought and all opposition to
theological orthodoxy. The Church in Spain to-day enjoys the unique
position of holding a monopoly of the spiritual direction of the nation.
Although other creeds and forms of worship are tolerated, there is no
religious liberty. Everywhere else, even in Catholic countries, there is
a vigilant and hostile body of opinion, of more or less weight, which
necessarily contributes by its very existence to moralise the Church and
to enforce on the priesthood a certain standard of duty. In Spain this
check is absent. There is no rival Church, for the Spanish Protestants
are too few in number and too insignificant in position to make their
influence felt, and the working classes, who, as has been shown, are
bitterly hostile to the priesthood, are inarticulate, and powerless as
an influence corrective of abuses, while the middle classes, who might
do something towards enforcing a higher standard, are generally
speaking, indifferent.
To what extent the corruption of the spiritual power in Spain is
responsible for the low moral standard of the laity is an exceedingly
difficult question, on which I am not capable of pronouncing an opinion.
There is no doubt that Spain for the last three hundred years has
suffered from a succession of some of the worst, the most incompetent,
and the most corrupt rulers known to history. During all this time,
except perhaps during the thirty years when Charles III. was on the
throne, the Church was supreme. If the clergy, the directors of the
conscience of the nation, armed with the power of the confessional and
supported if necessary by the secular arm, had deliberately set their
faces against the system of public venality and corruption instituted by
Lerma and Olivares and continued by their subordinates and successors,
it is difficult to believe that the upas-tree would have grown so tall
and struck its roots so deeply as it has.
The excessive centralisation of the whole administration in Madrid,
coupled with the Spanish habit of writing long letters and reports about
every trivial question, which reports are referred for further
information from one official to another before the Minister or other
authority gives his final decision, paralyses all initiative and causes
infinite delays and annoyances over the simplest matters. On the other
hand, if effective local self-government were given under existing
conditions, the _Cacique_ would be even more powerful than he now is,
and Spain would be ruled, not by a single bureaucracy, but by a number
of irresponsible autocrats.
Thus before Spain can effectually reform herself there is needed a
change of heart, a vital conviction that only through honest and
fearless administration is redemption possible. An educated Spaniard
once observed to me, when discussing this matter: "In England you act on
the supposition that a person in office is an honest man, and if you
find that he is not, you punish him severely. In Spain we presuppose
dishonesty, and do not chastise the rogue when he is found out." This is
perfectly true. There are swarms of official inspectors who are supposed
to inspect everything connected with the public administration. But the
inspectors themselves are venal, and for a sufficient consideration will
report that all is well when it is far from well. _Quis custodiet ipsos
custodes?_ It is the rarest thing to hear of any official being punished
for peculation or receiving bribes.
Every educated Spaniard is fully aware of this canker, which is rotting
the whole body politic: they talk to each other and to foreigners about
it with the utmost frankness, entirely recognising the greatness of the
evil and usually despairing of any amendment.
To turn to another side of the same question. In a different way the
bullfight is responsible for an amount of moral degradation that no one
but a Spaniard can adequately estimate. It is not only that the
spectacle of broken-down horses gored to death and a wild beast worried
for half an hour at a stretch is in itself debasing, but the whole
atmosphere created by the amusement is thoroughly vicious and degrading.
This is not merely my private opinion: I repeat what has been told me by
cultured and thoughtful Spaniards, who see in its popularity one of the
many obstacles to the growth of a higher standard of morality. Happily
there are indications that the taste for the sport is on the wane. I
know numerous members of the upper middle class and many working people,
both men and women, who object strongly to the institution, and never
attend a bullfight, and bullrings have been closed in many of the
smaller towns during the last ten years or so, for want of support. But
the vested interests--the cattle-breeders who make their living by
breeding the bulls, the impresarios who get up the shows, the companies
who have invested millions of pesetas in building bullring's, the
thousands of men employed in them in various capacities, and the
bull-fighters themselves--form together a very powerful combination with
a good deal of political influence, and it will be many years yet before
this blot on civilisation disappears.[29]
One deplorable fact connected with the bullfights is the extent to which
they are patronised by foreign visitors, and of these the English are
among the worst offenders. I have been told, though I cannot vouch for
the truth of the statement, that one bullring close to Gibraltar is
practically kept going by the English spectators, and that but for their
support it would be closed. I know that Englishmen and English women, in
scores and hundreds, every year, some of them ardent supporters at home
of the Society for the Prevention of Cruelty to Animals, make a point,
when they come to Spain, of going to see the show. "No, I daresay I
shan't like it," they will say, "but when one is in Spain it is one of
the things one ought to see." Let us hope that they do not realise that
their example goes to make the task of the Spanish social reformers
even more up-hill and heart-breaking than it need be.
I may instance an University professor who was wearing himself out in
the endeavour to raise the moral and intellectual standard of his
pupils. He himself was educated in England, and had the highest respect
for English customs and institutions and for the general code of English
honour. He told me that he had lain awake all one night trying to find a
reply to his lads when they said: "If the English, whom you hold up to
us as an example in so many ways, support the bullfight, there can be no
reason why we should condemn it."
"And meanwhile," said the professor bitterly, "your English ladies come
out of the bullring and tell me that what they have seen there proves us
to be a nation of barbarians."
In this connection it should be remembered that Spaniards of all classes
have a great admiration for England and English institutions, which has
been recently increased thanks to the popularity of the Queen. One sees
this in all directions. English is beginning to replace French as the
first foreign language a young Spaniard learns; English games and
English fashions are rapidly being introduced; and one of the leaders
of the Republican party has proclaimed a democratic Monarchy on English
lines to be the best compromise possible under existing conditions in
Spain. So that the support which English visitors give to the bullring
is probably more influential for harm than that of other foreigners.
Materially, moreover, the bullring operates in a manner prejudicial to
the country. All the best land has to be given up to the bulls, which
require immense space to keep them from fighting each other. Thus great
estates, which, if cultivated, would employ numerous labourers and
produce a rich return, are lost to the nation.
In this matter, too, the Church might exercise a good influence and does
not. On the contrary, the Clericalist newspapers give at least as much
space to reports of the bullfights as do any others, and one of the
reproaches levelled against the clergy by the working classes is that
they attend these shows disguised in lay dress, and associate with
bullfighters, regardless of the prohibition of the Church. In the parish
leaflet already quoted, one of the cases of conscience put is "whether
it is a sin to attend a bullfight": to which the answer returned is,
"No, it is not."
* * * * *
Setting aside the question of a moral reform, without which legislative
and administrative changes can produce little or no fruit, it may be
useful to consider what measures are urgently needed to contribute to
the intellectual and material development of the country.
First and foremost the Church should be confined to its spiritual
functions, and restrained from active interference in politics,
education, and business.
In a circular issued by the Bishop of Madrid in December, 1909, on the
duties of Catholics in the elections, it is laid down that the Catholic
voter must not vote for a Liberal as against a Catholic, and that a
Liberal is, _inter alia_, "one who refuses adhesion to the propositions
and doctrines laid down by the Apostolic See, _principally in reference
to the relations of the Church to the State_" (italics mine).
The attitude of Rome to what it calls "liberalism" is so well known that
there is no need to dilate upon it here. It is quite certain that unless
and until the Church can be excluded from intervention in the State, no
progress will be possible. The struggle will, no doubt, be severe, for
Spain is now the last stronghold of the Roman Church; but once the
democracy can make its voice effectively heard, the end will not be
doubtful.
In education the dominance of the Church is, if possible, more
prejudicial, more of an obstacle to progress of the best kind, than it
is in other branches of the work of the State, and the clergy in Spain,
as elsewhere, are resisting with might and main every attempt to set up
schools which are not under their control. A sufficiency of good and
well-conducted schools is one of the crying needs of the country; the
Clericalists say they are unable to finance even the Catholic schools
which already exist, yet the lay schools supported by the party of
progress, although trivial in number, are not only virulently attacked,
but are made the basis of a campaign against the Crown and the
Constitution, and every nerve is strained to rally "good Catholics" to
the fight against the spread of education among the poor.
Decentralisation of the machinery of administration is badly needed,
because under the present system vexatious and unnecessary delays must
occur, even were there every desire for progress on the part of all
concerned. But local government cannot be effective, as has already been
said, until the _Cacique_ is abolished.
Among what may be called the material elements of progress may be
briefly mentioned the need of improved means of communication,
especially good roads. Most of the roads which do exist in Spain are
very bad, and there are officially stated to be five thousand villages
to which there is no road at all--nothing but a track or path,
impassable for wheeled vehicles.
Much needs to be done to encourage agriculture, and to introduce
improved methods. Systematic irrigation would render fertile hundreds of
square miles of land, now sterile for lack of water. Phylloxera is
ravaging the vineyards, and a contagious blight is devastating the
orange plantations all over the southern provinces. Neither of these
plagues can be effectively combated by private enterprise: public aid
and public organisation are essential.
The postal service requires to be overhauled, and security taken, which
now does not exist, that postal matter shall reach its destination, and
that the contents be not stolen _en route_, as not infrequently happens.
Last, but not least, the conduct of elections must be reformed, so that
the working classes may have an effectual, instead of, as now, a merely
nominal vote. With a few notable exceptions they distrust their rulers,
of whatever party; it should be made possible for them to return to the
local councils and to the Cortes men in whom they have confidence, who
know what they want, and who will devote themselves with singleness of
mind to getting it. The hope for the future of Spain lies in the
democracy. The peasantry, from whose ranks the whole of the working
classes are more or less directly recruited, are sober, honest, and
industrious. They work long hours for low wages without complaint, and
employers--English, American, and so on--who come into contact with
large numbers of them in the numerous industries established by foreign
enterprise in the Peninsula, all speak in the highest terms of them as
labourers. In America, too, they are highly valued, and it is said that
the men who in the long run prove the most satisfactory and the best
able to bear the trying conditions of work on the Panama Canal are the
Spanish emigrants, of whom thousands cross the Atlantic every year.
As yet practically no member of this class, no matter what his natural
gifts may have been, has ever risen to a position in which he could make
his voice heard in the counsels of his nation. Many Spanish peasants
have, no doubt, succeeded in Spanish South America, and some of them
have come home again to spend their money and their declining years in
their native land. I am not aware that such men have been encouraged to
play a part in the politics of Spain, although their experience of the
outside world would be of the greatest value. But the frequent
instances of Spanish peasants rising to affluence abroad show that it is
not their own incapacity, but the crushing burdens imposed on them by
those in power, which are the cause of the miserable condition of the
peasantry at home. When a Spanish peasant gets a chance, he is well able
to profit by it.
Spain always seems to me like a great tree which for centuries has been
allowed to go unpruned. It is half smothered with branches which bear no
fruit, and the top is a mass of decay. Yet the trunk and the roots are
sound and strong, so that once the barren wood which saps the life of
the tree is cut away, a new and healthy growth will soon replace it. But
the longer the difficult and painful process of pruning away the dead
wood is delayed, the greater must grow the danger of a storm which will
tear up the tree, roots and all.
Still, in spite of all the drags on the wheels of progress, in spite of
ignorance, incapacity, and corruption, in spite of all the forces of
reaction and all their efforts to keep Spain in the Slough of Despond
from which she is struggling to emerge, one may say with Galileo, "_e
pur si muove_." Some little advance is being made, slight and slow
though it be, and among the more thoughtful members of the younger
generation one sees signs of a new spirit--an intelligent appreciation
of the needs of the country and an honest and sincere resolve to work
for their attainment, which cannot fail to spread and to bear fruit in
due season. From the older generation nothing is to be hoped, but ere
long they will have yielded their places to the young men--university
professors, officers in the Army, journalists, and so forth, many of
whom have ideas and ideals, and only lack power and opportunity to put
them in practice. The little leaven is working, and though as yet it is
small in amount and the lump is large, those who wish Spain well need
not despair.
"For while the tired waves, vainly breaking,
Seem here no painful inch to gain,
Far back, through creeks and inlets making,
Comes silent, flooding in, the main.
And not by eastern windows only,
When daylight comes, comes in the light,
In front, the sun climbs slow, how slowly!
But westward, look! the land is bright."
POSTSCRIPT
While this book was in the press, the Spanish Government took a step,
the ultimate consequences of which may be of the utmost moment for the
country. In June, 1910, Señor Canalejas resolved to take definite action
in the matter of the Religious Orders.
The immediate cause of his determination appears to have been the
general discontent created by the numerous cases of clerical corruption
and intimidation alleged to have occurred in the recent elections to the
Cortes. A great meeting of protest was held at Madrid, in which both
Republicans and Socialists took part, and Señor Melquiades Alvarez, the
Republican leader, who not many months before had expressed his
willingness to compromise with the Monarchists on the lines of a
democratic Monarchy like that of England, deliberately went over to the
Socialists. This important _volte face_, coupled with the fact that at
the elections Madrid returned an overwhelming majority of Republicans,
seems to have spurred Canalejas to action.
This action consisted of a modest Royal Order requiring the fulfilment
of an edict of 1902, which compels the registration of all Religious
Orders established in the country since that date, and the payment of
the industrial tax on the trades they carry on. This was followed by a
decree permitting members of other than the State religion to display
emblems and notices outside their places of worship, and to hold funeral
processions, in accordance with the provision made by the law of the
land for liberty of conscience.
These two decrees hardly strike one as revolutionary; but they have been
enough to set the whole of the Church party in an uproar; and the
Primate, the Archbishop of Toledo, has thrown down the gauntlet, defying
the Government to put the decrees in force, on the ground that the
Church owes obedience to Rome alone, and that the State has no power to
interfere with it. At the moment of writing the Vatican is trying to
bully the Government by threatening to break off relations, the Catholic
Associations have telegraphed their grief and distress at the outrage
inflicted upon the Pope by these Royal Orders, the ladies of the
Ultramontane aristocracy have petitioned the Premier to reconsider his
determination to destroy the national Church and drag Spain's religion
in the dust, and the whole clerical party are preparing a furious
campaign against the Government, which, needless to say, is warmly
supported by all the Liberal elements in the country. The working
classes are naturally delighted, and several of them have congratulated
themselves in my hearing on this excellent result of the King's
marriage.
"He has seen what religious liberty means in England, and that has given
him courage to defy the Jesuits. Viva Alfonsito!"
APPENDIX
NOTES ON POLITICIANS AND PERIODICALS
LIBERAL-MONARCHISTS
When Sagasta died three men were proposed as leaders of the Liberal
party, Moret, Montero Rios, and Canalejas, Montero Rios gave way in
favour of Moret, in order to secure the unity of the party, but
Canalejas preferred to lead a group of his own.
=Moret.=--Was a Republican until Alfonso XII. was proclaimed. He then
joined the Monarchical forces, the road being opened to him and many
others by the broadly liberal policy of Sagasta. He is English on the
mother's side.
=Montero Rios.=--Also was a Republican until the Monarchy was
re-established. Then he also adhered to Sagasta, bringing in with him
his own group, thenceforth to be known as the Radical wing of the
Liberal-Monarchists.
=Rafael Gasset.=--A staunch supporter of Moret's policy. He is the author
of the great irrigation scheme which is one of the most popular features
in Moret's programme. His enthusiasm for this improvement in the
conditions of agriculture is so strong that his opponents have nicknamed
him "The Duke of the Reservoirs." He is one of the strongest of the
younger Liberals, and his sincerity and devotion to the interest of the
working-classes have won him their confidence and respect.
The policy of the Liberal-Monarchist party is supported by the _Sociedad
Editorial de España_, which publishes three daily papers, all sold at 5
cmes. per copy, or 1 peseta per month:
=El Liberal.=--This paper has by far the largest circulation of any in
Spain. Its political news is edited in Madrid, and telegraphed thence
twice daily, for the morning and evening editions, to branch offices at
Bilbao, Murcia, Barcelona, and Seville, where the local notes and news
are added. Although conducted on Liberal-Monarchical lines, it is tinged
with democratic feeling. The Reactionists profess to consider it a
dangerous enemy to religion, and label its readers atheists and
anarchists. It is universally popular with the working classes.
=El Heraldo de Madrid.=--Edited and published in the capital on
Radical-Monarchical lines. On sale all over the country, but with a
comparatively small circulation among the working men outside of Madrid.
=El Imparcial.=--Edited on Liberal-Monarchical lines in the interest of
the working classes, with full reports and articles on public works of
every description, trades unions, schemes for social and industrial
reform, &c. It is on sale everywhere, and probably has the largest
circulation of any Madrid paper among the working classes in the
provinces, but does not come near _El Liberal_ in popularity.
The literary style of the writers employed by the _Sociedad Editorial_
is cultivated and refined, the flying of political kites is discouraged,
and personal abuse of opponents in politics finds no favour with the
directors. The Society is abusively called a "Trust" by the Opposition,
and reactionary journals daily publish headlines proclaiming that they
do not belong to the "Trust." As a matter of fact the _Sociedad_ is an
ordinary limited liability company, well managed, and paying a good
dividend, and partaking in no respect of the evils of the Trust system.
LIBERAL DEMOCRATIC GROUP
=Canalejas.=--Was a Republican, but maintained his independence, although
adhering to Sagasta's party, by proclaiming himself chief of a group of
progressive Liberals with Republican sympathies. The main plank in his
programme has always been a direct attack upon the Church and Religious
Orders. His policy is supported by the _Diario Universal_, but it has a
small sale and is hardly known by working men outside of Madrid.
THE REPUBLICAN PARTY
The three most distinguished men in this party--=Melquiades Alvarez=,
=Blasco Ibañez= and =Rodrigo Soriano=,--are all celebrated for their
literary and oratorical gifts, and enjoy the respect and confidence of
the veteran Liberal leaders, Moret and Montero Rios. Their policy may be
described as Republican in idea, but democratically Monarchical in
practice, and their demands for vigorous measures of reform have
materially strengthened the hands of the Liberal-Monarchists.
The organ of this party is _El Pais_, which, although its sale is very
much smaller, has the largest circulation among the working classes
after _El Liberal_. The paper, as might be expected from the literary
renown of the leaders who direct it, is extremely well written, the
staff including some of the most highly educated Progressives in Spain.
It is possible, however, that the standard of intellectuality maintained
in its leading articles militates against its success with the people.
The numerical strength of the Republicans is small. Thus, the
circulation of _El Pais_ being comparatively limited, the Reactionists
are not nearly so much afraid of its influence on the country as of that
of _El Liberal_, and indeed seem to treat it almost with indifference.
It is sold at the same price as the papers of the _Sociedad Editorial_.
THE SOCIALIST PARTY
=Lerroux, Pablo Iglesias, Nakens.=--The Socialists in Spain have a very
small following, and that confined to a few of the industrial cities,
chiefly in the north. They formed a coalition with the Republicans to
secure the rout of the Clericalists at the Municipal Elections of 1909,
but the party is disunited, Iglesias and Lerroux seldom coming into line
with each other, while neither of them goes so far as Nakens, editor of
the Socialist organ _El Motin_ and a violent revolutionary. _El Motin_
has a very small circulation, and the programme of the Socialists has no
serious influence in Spanish politics.
The Separatist, Regionalist, and other groups of Catalans exist solely
for the political purposes of that province, and play no part in the
programme of either of the national parties.
The so-called Anarchist party, of which so much has been heard abroad,
is practically non-existent. Their sporadic publications have no genuine
circulation and seldom live for over a month.[30]
THE REACTIONARY, CLERICALIST, OR ULTRAMONTANE PARTY
The leader of this party, =Maura=--for many years a Liberal and the
intimate friend of Moret--adopted Conservative principles under Silvela,
and on his death was chosen to be leader of the Conservative party. His
Liberal proclivities at first influenced him in the direction of reform,
and gave him a strong and united following among the true Conservatives.
But as time passed he developed so much religious fervour that he has
now become recognised as the protagonist of the Religious Orders and the
hope of the Church in the rapidly approaching final struggle with the
State. Down to July, 1909, Maura was able to hold the Conservative party
together, notwithstanding the marked development of reaction in his
policy. But after the events at Barcelona the Conservatives proper
withdrew their support on his programme of repression, and since his
Cabinet fell in October of that year, he has been universally regarded
more as the tool of the Ultramontanes than the leader of the
Conservative party.
The organ of Maura is _La Epoca_. It is sold in Madrid at 10 cmes., but
is never seen on the bookstalls at any distance from the capital, and
can only be obtained in provincial towns by paying three months'
subscription to the Madrid office in advance. Its circulation is
exclusively confined to the Clericalist aristocracy and plutocracy, by
whom it is subsidised.
THE CARLIST, JAIMIST, OR TRADITIONALIST PARTY
This party, which numbers many of the richest men in Spain among its
adherents, besides all the Religious Orders, with their enormous wealth
and influence, is directed from the Castle of Frohsdorf by =Don Jaime,
Duke of Madrid=, through persons whom he appoints in every province of
Spain. The name brought most frequently before the public in connection
with the party, after the Pretender's own, is that of =Llorens=, whose
work in the Melilla campaign is referred to in Chapter VII. The
Pretender has a complete organisation all over Spain, with _Caciques_ in
a large number of provincial towns and villages, and is supported by
numerous religious associations, clubs, colleges, &c., of a confessedly
militant character, but confined to the upper classes.
The leading organs of the Carlists are the _Correo Español_ and the
_Correo Catalan_, with offices in Madrid, Paris, and Barcelona; but
practically all the reactionary Press supports the claims of the
Pretender more or less openly. The Carlist papers have no sale among the
working classes, and can only be obtained outside of Madrid (like _La
Epoca_) by paying three months' subscriptions in advance.
* * * * *
Among military politicians much in the public eye may be mentioned
Generals =Luque=, =Weyler=, and =Lopez Dominguez=, all on the Liberal side,
and all strong men, in whom the people feel confidence. =Aguilera=, twice
Alcalde of Madrid under Moret, who has been referred to in Chapter
XIII., is highly popular with the poor of Madrid, owing to his
consistent kindness to the children, whom he takes under his special
protection.
Count =Romanones=, who engineered the crisis of February, 1910, is
credited by the working classes with having large interests in the mines
of Beni-bu-Ifrur, and with having schemed to bring about the war in
Morocco, in order to put money into his own pockets. This impression,
whether well or ill founded, is sufficient to make him cordially hated
by them. He is credited with aspiring to the leadership of the Liberal
party, but it is hardly probable that his following would prove strong
enough to give him that position.
=La Cierva=, Minister of the Interior in Maura's Cabinet, obtained an
unenviable reputation in 1909, through his share in administering
Maura's policy of repression. Since his leader went out of office La
Cierva's name has hardly been mentioned among the working classes.
THE CONSERVATIVE-MONARCHIST PARTY
=Dato=, =Sanchez Toca=, and _Gonzalez Besada_ are the three leading
dissentients from Maura's policy of reaction, and now stand for the old
Conservative-Monarchical programme of peace and conciliation without
sensational reforms. Their organ is the _Correspondencia de España_, an
eight-paged paper, well printed and got up, containing the fullest
military intelligence and the best foreign news to be found in the
Spanish Press. It has a far larger circulation than any other
Conservative or Clericalist paper, and is to be seen on most of the
kiosks in large towns. If it were not believed by the people to be
subsidised by the party opposed to electoral and social reforms, its
influence in the country would doubtless be considerably stronger than
it is. At present the working classes do not read it, although no other
paper gives nearly as much matter for the price, which is 5 cmes.
INDEX
Aguilera, Señor, 240
_Alcalde_, the, 265 note
Alfonso XII., 112
Alfonso XIII., 122-4, 182, 318
All Souls' Day, observation of, 46;
scandal connected with Masses on, 84-6
_Ayuntamiento_, the, 265 note; 274
Baptism, 47
Barcelona, effect of riots, 17;
refugees from, 90, 92;
Carlist activities in, 134 ff.;
stories of riots, 165-6;
bombs in, theory of, 181 ff
_Beatas_, 21
"Bull of the Crusade," 64
Bullfight, the, 307 ff
Burial, 48, 50-1
_Cacique_,the 230-3
Canalejas, Señor, 243-6, 317-9
Carlists, alleged plots of, 167, 176;
army, 155-6;
party, 321
"Catholic Associations," 163
Church, attitude towards people, 31-2;
illegal disposal of property, 80 ff;
unique position of,in Spain, 304
Civil Guard, the, 175, 218 ff
Clergy, children of, 79-80, 106-7;
arming and drilling of, 161, 164
Clerical Press, the, 28, 169, 235, 320-1
Confessional, the, 43, 73 ff
Conscription, consolidates Monarchy, 111;
conditions of exemption, 209;
proposals for alteration, 210
Conservatives, the, 251, 256, 322
_Consumo_, the, 15, 288
_Contribucion industrial_, 286
Convent schools, 275
_Correo Catalan_, the, 151, 172, 243, 321
_Correo Español_, the, 163, 321
_Correspondencia de España_, the, 254, 322
Crossing, modes of, 65-7
Cuban War, stories of, 200-3
Customs' duties, 285
Demonstrations, clerical, 191-4
" popular, 174
Education, desire for, 15, 33
_Ejercito Español_, the, 152, 238
Electoral system, 229-33
Employers and employed, relations of, 23 ff
England, misunderstanding of Spanish politics in, 227, 257;
hopes of people from, 277, 318;
admiration of, in Spain, 309
Ferrer, 147-9, 170, 325
Gasset, Señor, 236, 317
Governments, distrusted by working classes, 30
_Heraldo de Madrid_, the, 187, 229 note, 257, 270, 318
Honesty, 62
Hume, Major Martin, quoted, 133, 228
Illiteracy, 263, 271
Images, belief in, 52-3, 55-8
_Imparcial_, the, 114, 318
Irrigation scheme, 237
Jaime, Don, of Bourbon, 117, 153-4, 166, 170
Lay schools, the, 169;
clerical campaign against, 190-1, 193
_Liberal_, the, 35, 172, 294, 317
Llorens, Señor, 152-3, 163, 321
Lottery, national, 298
Luque, General, 238
Madrid, attitude of, towards the South, 26-8
Marriage among working classes, 48-9
Matches, monopoly of, 292
Maura, Señor, 35, 115, 137-8, 144, 150, 203, 234, 245, 251-3, 256, 280, 320
Melquiades Alvarez, Señor, 259, 317, 319
Monopolies, Government, 291-2
Montero Rios, Señor, 242, 317
Moret, Señor, 137, 173, 228, 234, 237, 239, 241-3, 253, 258, 317
Morocco, war in, 200 ff
Morral, 144, 148
Moslems, mixed with Spaniards, 28-9;
traditional feeling against, 207
Municipal elections, 1909, 237
_Nuevo Mundo_, the, 7
_Pais_, the, 171, 223, 319
Paz, Infanta Doña, 207
Penitential dress, 64
Penitents, 53
Police, various bodies of, 215 ff
Politics, difficulties of understanding, 228
Popular songs, 142-3
Postal service, 298
Prayers quoted, 67-8
Primo de Rivera, General, 203
Public instruction, system of,
264 ff
Purgatory, popular view of, 44
Queen, the, animus of clergy against, 120;
feeling of working people towards, 121-2, 128;
courage shown by, 182
Queen-Mother, the, 113, 115
Religious Orders, the, change of people's attitude towards, 17;
positions in Spain illegal, 90;
relations to working classes, 93;
underselling of workpeople by, 94-5, 105;
people ruined by, 97-102;
refusal to help at time of distress, 102;
evasion of taxation by, 295;
measures adopted by Government, 317-8
Republicans, the, 239, 258-60, 317, 319
Reservists, supposed protest against calling out of, 204
Romanones, Count, 240, 322
Royal Family, suppression of news about, during the Maura _régime_, 116, 123-6
Sanchez Toca, Señor, 254
School supply, facts about, 270
_Serenos_, the, 217
Socialists, the, 237
_Sociedad editorial_, the, 139, 171-2, 317
Squilache, Marquesa de, 116-7
Sugar monopoly, 291
Taxation, evasions of, 294
Tobacco monopoly, 291
Tradition, influence of, 16, 145
Truth-telling, 61
Universities, the, 268
Upper classes, general character of, 32; religion of, 40
_Vigilancia_, the, 217
War Fund, initiated by the Queen, 117 ff, 127;
contributions of workpeople to, 119
Working classes, general character of, 14, 30;
what they read, 34;
religion of, 39 ff
Zaragoza, explosion of bombs at, 188
The Gresham Press
UNWIN BROTHERS, LIMITED
WOKING AND LONDON
FOOTNOTES:
[1] The last edict of expulsion was issued in 1712.
[2] Isabella the Catholic made an order for the expulsion of the
unconverted "Moors" in 1501, but a very large number of them, whether
nominally Christian or not, remained until driven out by Philip III.
After the massacres commanded by Philip II. in Granada, the Moriscos
who were expelled from that kingdom did not apparently leave Spain, for
two years later an edict was issued for their registration.
[3] What Lecky says about the seventh and following centuries might be
applied to the religion of the upper classes in Spain to-day: "It is no
exaggeration to say that to give money to the priests was for several
centuries the first article of the moral code" ("History of European
Morals," ii. 216).
[4] I know several cases of lads of fourteen or fifteen who return
after working from 6 a.m. to 6 p.m. in the fields, to sit over their A
B C and pot-hooks until they can keep their eyes open no longer, while
the rest of the family look on and encourage the student.
[5] One of the various local terms for what the guide-books call _olla
podrida_--a universal dish in Spain.
[6] Lecky, "History of European Morals," ii. 213.
[7] Sweet cakes and _patisserie_, the foundation of which is generally
finely grated stale bread.
[8] Two favourite sweetmeats.
[9] From the Basque _ama_, a mother; applied to the head servant in the
house of a priest or other man living alone.
[10] Hume, "Modern Spain," p. 550.
[11] It is said--although I repeat the statement with all reserve--that
there are "parish" doctors employed by the Municipality of Madrid who
refuse to prescribe for a dying child unless the mother can show her
marriage certificate.
[12] "Modern Spain," p. 563.
[13] It was stated as a fact that nineteen men in one regiment had
been shot for refusing to go into action, and an Ultramontane of my
acquaintance, who never reads anything but the newspapers of his own
party and never travels ten miles from his own village, solemnly
assured me that the tale was true!
[14] I was told at the time that many people in Madrid thought the bomb
was thrown on behalf of the Pretender.
[15] The names of the monastery and of all the people concerned were
given me, but I refrain for obvious reasons from publishing them.
[16] It is said that the Association of Social Defence promises its
working men members a retaining fee of 3 pesetas a day should political
exigencies compel them to leave their work at any time, the average
labourer's wages all over the country being from 1.50 to 2 pesetas. It
has not been possible to obtain trustworthy information, either as to
terms of membership or the actual numbers who have joined the league
during the last twelve months, but there is evidence that it has no
influence among the working classes generally.
[17] I have been told by an English friend that a Spanish acquaintance
of his has, to his knowledge, lately made a substantial sum by selling
arms to the Religious Orders.
[18] Most of them had been re-opened after Moret took office in
October, 1909, as already mentioned.
[19] This story evidently relates to the early days of the Cuban war.
[20] This is not the only statement of the kind that I have heard.
[21] "Modern Spain," p. 531.
[22] So clearly is this recognised on all sides, and so impossible does
political honesty on the part of the rich appear to Spaniards, that the
_Heraldo_, the leading moderate-Liberal paper, in the course of its
comments on the rejection by the English House of Lords of the Budget
of 1909, said that if the Lords permitted the people to vote as they
pleased, this action on their part would have been justifiable, but
that naturally they would take the usual means to secure the suffrages
of those over whom they had control, and with the immense wealth at
their command would easily influence the elections in the direction
they desired.
[23] In Spain not only every city, but every town and nearly every
village, has its _Ayuntamiento_, more or less equivalent to our town or
village Council, and its Alcalde, who has a good deal more power than
the Mayor of a Corporation.
[24] The peseta is the same as the franc.
[25] The sums set down in the schedules are less than those named. The
tax has been increased at different times, and the additions amount in
all to about 66 per cent.
[26] The Spanish dollar, value five pesetas, and counted by the poor as
twenty reals.
[27] The cuarto was a little over two centimes.
[28] "History of European Morals," Chap. IV.
[29] In a decaying town of some 15,000 inhabitants, once wealthy and
prosperous, two large new buildings have been erected during the last
half-century, while on all sides dwelling-houses, great and small, are
falling into ruin. These are the Jesuit College and the bullring; and
the people say that the one is the parent of the other.
[30] For a full account of the political parties in Spain see "The
Backwardness of Spain," by John Chamberlain. The author has an
exhaustive knowledge of the country, and of many phases of society in
Spain, but in my opinion he has not informed himself of the mind of
the provincial and rural population. This class, if only from their
numbers, cannot fail to exercise a strong influence over politics, when
once they obtain the right to vote which the Constitution gives them.
End of the Project Gutenberg EBook of Spain from within, by Rafael Shaw
***
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Subject: Standards for Assessing the Diversity Policies and Practices of Regulated Entities
To: Chief Executive Officers of National Banks, Federal Savings Associations, and Federal Branches and Agencies; Department and Division Heads; All Examining Personnel; and Other Interested Parties
Description: Final Interagency Policy Statement
Section 342 of the Dodd-Frank Wall Street Reform and Consumer Protection Act of 2010 requires the directors of the Offices of Minority and Women Inclusion at the Office of the Comptroller of the Currency (OCC) and other federal financial regulators1 (collectively, the agencies) to develop standards for assessing their regulated entities' diversity policies and practices. The agencies published a proposed policy statement establishing joint standards in the Federal Register on October 25, 2013. On June 10, 2015, the agencies published a final policy statement establishing joint standards, which provides a framework for assessing a regulated entity's diversity policies and practices. The final policy statement is effective upon publication.
The final policy statement reflects the collective efforts of the agencies, which held extensive discussions with depository institutions, holding companies, and industry trade groups and consulted with financial professionals, consumer advocates, and community representatives. The agencies also considered more than 200 public comments submitted on the proposed policy statement.
In addition, the agencies, as required by the Paperwork Reduction Act, are asking for public comment on the voluntary collection of information from regulated entities identified in the final policy statement. Comments are due within 60 days after the final policy statement is published in the Federal Register. The voluntary collection of information will be effective upon announcement of Office of Management and Budget approval in the Federal Register.
When drafting this final policy statement, the agencies focused primarily on regulated entities with 100 or more employees. Given that small or remotely located regulated entities face different challenges and have different options available to them compared with larger regulated entities or those located in more populated areas, the agencies encourage each regulated entity to use this final policy statement in a manner appropriate to its specific characteristics.
The final policy statement does the following:
Provides a framework for assessing a regulated entity's organizational commitment to diversity, its workforce profile and employment practices, its procurement and business practices, and its efforts to promote transparency in its organizational diversity and inclusion.
Includes a framework for an entity to conduct a self-assessment of its policies and practices and explains how the agencies will use the assessment information voluntarily provided by a regulated entity.
Please contact Betty Washington, Program Manager-Regulated Entities, Office of Minority and Women Inclusion, at (202) 649-6460; or Karen McSweeney, Counsel, Legislative and Regulatory Activities Division, (202) 649-5490.
Joyce B. Cofield
Executive Director, Office of Minority and Women Inclusion
"Final Interagency Policy Statement Establishing Joint Standards for Assessing the Diversity Policies and Practices of Regulated Entities" (PDF)
1 These other agencies are the Board of Governors of the Federal Reserve System, Federal Deposit Insurance Corporation, National Credit Union Administration, Consumer Financial Protection Bureau, and U.S. Securities and Exchange Commission.
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New wearable tech from Western may hold big benefits for people with Parkinson's
Michael Naish from Mechanical and Materials Engineering at Western and Ana Luisa Trejos from Electrical and Computer Engineering at Western examine a prototype of a wearable tremor suppression glove modeled by Western doctoral student Yue Zhou, who 3D-printed its key components.
A new prototype for wearable tremor suppression gloves has a team of Western University researchers believing real change is on the way for the more than 6 million people in the world afflicted by Parkinson's disease.
Ana Luisa Trejos, an Electrical and Computer Engineering professor at Western, and members of her Wearable Biomechatronics Laboratory Group have developed a novel approach for designing wearable technology that allows those with Parkinson's to exhibit improved motor control while reducing or even restricting involuntary muscle contractions commonly associated with the long-term and degenerative neurological disorder. More than 25 per cent of people with Parkinson's disease have an associated action tremor.
Previous studies from Trejos and her team show suppression devices targeting elbows or wrists often produce exaggerated tremors in the fingers, which causes even more difficulty for those with Parkinson's.
" If you have seen anybody with Parkinson's that has tremors, they have them in their entire body but it's the ones in their fingers that really prevent them from performing the activities of daily living," explains Trejos, also a key investigator at Western's Bone and Joint Institute.
Instead of suppressing tremors, which is what most other tremor suppression devices do, these new personalized gloves actually track voluntary movement so if a person is trying to accomplish a particular task, the glove allows the action to happen while minimizing the tremor.
"Our gloves don't suppress all movements, which is what most other wearable tech systems do," says Trejos. "They are either suppressing or not suppressing movement so when a person is trying to perform a specific task, the devices actually prevent them from performing the action they are trying to perform. They have to act against it. Our gloves actually allow the voluntary movement to happen and at the same time, prevent the tremor from occurring."
The new gloves will be custom designed for both hands of each patient to maximize the benefits of the wearable technology. The prototype was created specifically for the left hand of Western doctoral student Yue Zhou, who 3D-printed its key components. Mary Jenkins from Western's Schulich School of Medicine & Dentistry and Michael Naish from Mechanical and Materials Engineering at Western also collaborated on the project.
"While collecting data, we have seen first-hand that people with Parkinson's get really frustrated when they can't do something on their own and I feel our glove will allow them to get back to their daily living," says Trejos. "It can be very frustrating to not be able to eat or button a shirt on your own. Or even draw. Things we take for granted. By creating a glove that allows people to perform these actions while suppressing the tremors, I think they could go back to being much more independent in their own homes for a far longer period of time."
This project was supported by the Peter C. Maurice Fellowship in Biomedical Engineering.
MEDIA CONTACT: Jeff Renaud, Senior Media Relations Officer, 519-661-2111, ext. 85165, 519-520-7281 (mobile), jrenaud9@uwo.ca, @jeffrenaud99
Western delivers an academic experience second to none. Since 1878, The Western Experience has combined academic excellence with life-long opportunities for intellectual, social and cultural growth in order to better serve our communities. Our research excellence expands knowledge and drives discovery with real-world application. Western attracts individuals with a broad worldview, seeking to study, influence and lead in the international community.
Western University's Yue Zhou models new prototype for wearable tremor suppression glove. The doctoral student from the Wearable Biomechatronics Laboratory Group 3D-printed its key components.
Tags: Ana Luisa Trejos, Bone and Joint Institute, Mary Jenkins, Michael Naish, Schulich School of Medicine & Dentistry, Wearable Biomechatronics Laboratory Group, wearable tech, wearable tremor suppression gloves, Western Engineering
jrenaud9@uwo.ca
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Q: make multiple, composite query in oracle How can i make multiple, composite query in oracle?
for example this several queries in one step?
1
CREATE TABLE test (id NUMBER PRIMARY KEY, name VARCHAR2(30));
2
CREATE SEQUENCE test_sequence START WITH 1 INCREMENT BY 1;
3
CREATE OR REPLACE TRIGGER test_trigger
BEFORE INSERT
ON test
REFERENCING NEW AS NEW
FOR EACH ROW
BEGIN
SELECT test_sequence.nextval INTO :NEW.ID FROM dual;
END;
4
INSERT INTO test (name) VALUES ('Jon');
5
INSERT INTO test (name) VALUES ('Meloun');
A: We solved it by wrapping each statement in an EXECUTE IMMEDIATE command inside a PL/SQL script:
BEGIN
EXECUTE IMMEDIATE 'CREATE TABLE test (id NUMBER PRIMARY KEY, name VARCHAR2(30))';
EXECUTE IMMEDIATE 'CREATE SEQUENCE test_sequence START WITH 1 INCREMENT BY 1';
-- etc
END;
A: By and large DDL statements have to executed one at a time. It is true that Oracle supports the CREATE SCHEMA syntax, but this only permits the creation of multiple tables and/or views plus grants on them. It doesn't include ancilliary objects such as sequences or triggers.
You could turn the two inserts into a single statement like this:
INSERT INTO test (name)
select 'Jon' from dual
union all
select 'Meloun' from dual
/
This might be more trouble than its worth, but I suppose it does give you a simple transactionality: it inserts either all the rows or none of them.
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List of landmark African-American legislation
Ordinance of 1787: The Northwest Territorial Government (Northwest Ordinance)
Fugitive Slave Law of 1850 - Made any federal marshal or other official who did not arrest an alleged runaway slave liable to a fine of $1,000
Missouri Compromise (1850) - Series of Congressional legislative measures addressing slavery and the boundaries of territories acquired during the Mexican-American War (1846–1848)
Enrollment Act (Conscription) - Resulted in Draft Riots in several American cities. Noted for the devastating loss of life and property among African-Americans in New York City
Civil Rights Act of 1866 - Declared that all persons born in the United States were now citizens, without regard to race, color, or previous condition
Reconstruction Act - A series of four acts provided for the division of all former Confederate states into five military districts; Each district would be headed by a military commander, who was charged with ensuring that the states would create new constitutions and ratify the Fourteenth Amendment
Naturalization Act of 1870 - Allowed persons of African descent to become citizens of the United States
Enforcement Act of 1870 - enacted 31 May 1870
Enforcement Act of 1871 - enacted February 1871
Enforcement Act of 1871 Also known as the Ku Klux Klan Force Act. It was the third enforcement act passed by Congress. The act gave the United States President the power to suspend the writ of habeas corpus to combat the Ku Klux Klan and other white terrorist organizations during the Reconstruction Era.
Morrill Land Grant Colleges Act (1890) - Required each state to show that race was not an admissions criterion, or else to designate a separate land-grant institution for persons of color. Among the seventy colleges and universities which eventually evolved from the Morrill Acts are several of todays Historically Black colleges and universities
McLaurin v. Oklahoma State Regents (overturned low court decision by same name) (1950)
Jones v. Mayer (1968) - A United States Supreme Court case which held that
Arts Facts
Disney updates classic films with stronger racism content warnings - Voice Online
Four arrests in clampdown on illegal electricity vendors
Darren Lewis: Chadwick Boseman was hero who gave us template for better future - African American News Today - EIN News
The Betsy's Overture to Overtown Festival Celebrates its 9th Year (Virtually) - African American News Today - EIN News
U.S. to require all arriving passengers to get COVID-19 test | L.A. Focus News
Growth & Jobs | Embracing the new norm - Small and medium tourism enterprises hopeful about reopening of sector
Viola Davis Buys Childhood Home on Former Slave Plantation
Brandy and Monica 'Verzuz' Battle Breaks Records with 1.2 Million Viewers
United States Facts
Morehouse School of Medicine [Atlanta] (1975- )
Abolitionism
Lee, Herbert (1912–1961)
Benjamin L. Hooks
Adu, Freddy (1989-- )
(1974) Congresswoman Barbara Jordan's Statement: The Richard Nixon Impeachment Hearings
Jones, Stephanie Tubbs (1949-2008)
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: d3 v6 force simulation: center larger nodes / circles I am using d3 force simulation (adapted mostly from Yan Holtz's example). I would like bigger nodes to be drawn to the centre of the plot. When clustering all colours together I'd like to prioritise size over colour - bigger nodes of any cluster/colour in the centre.
This question is the closest I found to mine: D3 Force Layout where larger nodes cluster in center
But I can't figure out how to adapt the d3.v3 force layout code to my d3.v6 force simulation. Ideas?
My code:
<!doctype html>
<html>
<head>
<script src="https://d3js.org/d3.v6.js"></script>
</head>
<body>
<!-- checkbox to toggle cluster grouping -->
<input type="checkbox" id="clusterCheck">
<label for="clusterCheck">Cluster by colour</label>
<script>
// clustering checkbox
var clustering = false;
const clusterCheck = document.getElementById("clusterCheck");
clusterCheck.setAttribute("onclick", "toggleClustering()");
const svgWidth = 500;
const svgHeight = 400;
// append the svg object to the body of the page
const targetSvg = d3.select("#my_dataviz")
.append("svg")
.attr("width", svgWidth)
.attr("height", svgHeight)
.style('background-color', "lightgray")
// two-dimentional dataset
const wordFreq_data = [ { name: "Word1", cluster: "a", occurance: 4 },
{ name: "Word2", cluster: "a" , occurance: 8},
{ name: "Word3", cluster: "a" , occurance: 1},
{ name: "Word4", cluster: "a" , occurance: 10},
{ name: "Word5", cluster: "a" , occurance: 3},
{ name: "Word6", cluster: "a" , occurance: 2},
{ name: "Word1", cluster: "b" , occurance: 5},
{ name: "Word2", cluster: "b" , occurance: 4},
{ name: "Word7", cluster: "b" , occurance: 6},
{ name: "Word8", cluster: "b" , occurance: 9},
{ name: "Word9", cluster: "b" , occurance: 2},
{ name: "Word10", cluster: "b" , occurance: 7},
{ name: "Word1", cluster: "c" , occurance: 4},
{ name: "Word2", cluster: "c" , occurance: 1},
{ name: "Word3", cluster: "c" , occurance: 9}
]
// A scale that gives an X target position for each group
const groupX = d3.scaleOrdinal()
.domain(["a", "b", "c"])
.range([100, 250, 400])
// A color scale
const colour = d3.scaleOrdinal()
.domain(["a", "b", "c"])
.range([ "gold", "blue", "green"])
// Size scale
const circleSize = d3.scaleLinear()
.domain([0, 10])
.range([7,40]) // circle radius will be between 7 and 55 px wide
const nodePadding = 1.5;
// Initialize the circle: all located at the center of the svg area
const node = targetSvg.append("g")
.selectAll("circle")
.data(wordFreq_data)
.join("circle")
.attr("r", d => circleSize(d.occurance)) // radius depends on occurance
.attr("cx", svgWidth/2)
.attr("cy", svgHeight/2)
.style("fill", d => colour(d.cluster)) // colour depends on cluster
.style("fill-opacity", 0.8)
.call(d3.drag() // call handlers when circle is dragged
.on("start", dragstarted)
.on("drag", dragged)
.on("end", dragended));
// Features of the forces applied to the nodes:
var simulation = d3.forceSimulation()
.force("x", d3.forceX().strength(.1).x( svgHeight/2))
.force("y", d3.forceY().strength(.1).y( svgHeight/2 ))
.force("center", d3.forceCenter().x(svgWidth / 2).y(svgHeight / 2)) // Attraction to the center of the svg area
.force("charge", d3.forceManyBody().strength(.1)) // Nodes are attracted one each other if value is > 0
// Apply these forces to the nodes and update their positions.
simulation
.nodes(wordFreq_data)
.force("collide", d3.forceCollide().strength(.3).radius(d => circleSize(d.occurance) + nodePadding).iterations(1))
.on("tick", function(d){
node
.attr("cx", d => d.x)
.attr("cy", d => d.y)
})
;
// handling drag
function dragstarted(event, d) {
if (!event.active) simulation.alphaTarget(.03).restart();
d.fx = d.x;
d.fy = d.y;
}
function dragged(event, d) {
d.fx = event.x;
d.fy = event.y;
}
function dragended(event, d) {
if (!event.active) simulation.alphaTarget(.03);
d.fx = null;
d.fy = null;
}
// toggle clustring based on checkbox input
function toggleClustering() {
if(!clustering) {
// change force to form colour clusters
simulation
.force("x", d3.forceX().strength(.2).x(d => groupX(d.cluster)))
.force("y", d3.forceY().strength(.1).y( svgHeight/2 ))
simulation.alpha(1).restart();
clustering = true;
}
else {
simulation
.force("x", d3.forceX().strength(.1).x( svgHeight/2))
.force("y", d3.forceY().strength(.1).y( svgHeight/2 ))
simulation.alpha(1).restart();
clustering = false;
}
}
</script>
</body>
</html>
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{
"redpajama_set_name": "RedPajamaStackExchange"
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| 5,081
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\section{Introduction}
The potential of gravitationally bound structures in the Universe, ranging in size from dwarf galaxies to galaxy clusters, is sourced by a composite mass distribution of dark matter, baryonic matter in gas form, and collapsed objects such as stars in galaxies and galaxies in clusters. The investigation of these mass distributions entails a number of questions: what is the shape of the distributions? Is it universal across ten magnitudes of mass and at all redshifts? Does it depend on cosmology or on the merger history of the individual halos? Since the dominant component of relaxed structures is dark matter, much focus has been aimed at dark matter-only halos.
There is little theoretical understanding of the distribution of matter in a dark matter halo. The main developments have been found through numerical simulations of the formation of structure in the universe within a given cosmological model. Advances have been achieved through the improvement of numerical codes as well as the increase of raw computing power on one hand and a more refined understanding of which questions that need to be answered on the other. Perhaps the most fundamental idea that has come out of the numerical approach is that relaxed halos are (nearly) universal in many respects, including the distribution of matter \citep{1997ApJ...490..493N,2001ApJ...563..483T} and the dynamical structure \citep{2001ApJ...555..240B,2006NewA...11..333H}. However, the simulations have not been able to reach agreement about the exact behavior of the profiles in the innermost regions, where the limited force resolution of simulations sets a lower limit to the radial range that can be probed. Various authors claim that the logarithmic slope of the density profile reaches a value between $-1$ and $-1.5$, perhaps dependent on mass or merger history, and there is also discussion whether the inner slope is actually universal or not \citep{1998ApJ...499L...5M,2001ApJ...554..903K,2004MNRAS.349.1039N,2004ApJ...606..625F,2006AJ....132.2685M,2006AJ....132.2701G,2008MNRAS.387..536G}. A further complication arises when the simulations are compared with observations since the gravitational potential of the baryonic component, which is very time consuming to model in the simulations, cannot be neglected in the center. This complication can in principle both change the slope of the dark matter profile as well as alter the total mass profile \citep{1986ApJ...301...27B,2001ApJ...560..636E,2004ApJ...616...16G,2009arXiv0906.0573S}.
Theoretical efforts are hampered by the fact that, even under the strongest simplifying assumptions, there are not enough constraints to obtain unique solutions to the collisionless Boltzmann equation \citep{1987gady.book.....B} which governs a dark matter structure. Instead, one can take phenomenological input from numerical simulations such as the density profile itself, the pseudo-phase space density \citep{2001ApJ...563..483T,2005MNRAS.363.1057D}, or the density slope-velocity anisotropy relation (from which \citet{2006JCAP...05..014H} predict an inner slope of $0.8$), and implement this into a Jeans equation analysis to predict the consequences of the `inspired guess' (see also \citet{2008ApJ...682..835Z} and references therein). Alternatively one can attempt to model the formation history of the halo including major mergers and steady accretion (e.g., \citet{1987ApJ...318...15R,2004MNRAS.352.1109A,2007ApJ...666..181S,2009ApJ...698.2093D}, and references therein). While these approaches typically yield results in rough agreement with simulations, the modeling can also explore the physical connection between the static and dynamic properties of the halo as well as offer constrained extrapolations which are not accessible in simulations.
Observationally, there is a strong discrepancy between the numerical results and the inferred mass distributions in dwarf and low surface brightness galaxies, which are much shallower than predicted, the so-called cusp/core-problem (see, e.g., \citet{2003A&A...409...53S,2005AJ....129.2119S,2007ApJ...663..948G}). At the opposite end of the mass spectrum, galaxy clusters are typically found to be in rough agreement with the cuspy numerical simulations, but with even greater scatter for the inferred inner slope. There is also significant discussion about the type of model and number of parameters that are necessary in order to obtain an acceptable description of the data. One common method is based on mass modeling through weak or strong gravitational lensing, which can yield results which are in good agreement with numerical simulations \citep{2005ApJ...619L.143B,2006ApJ...642...39C,2008A&A...489...23L,2009A&A...498...37R}, but also profiles that are significantly shallower \citep{2004ApJ...604...88S,2008ApJ...674..711S}. Another method is based on X-ray observations of the intracluster medium (ICM) which is supported against gravitational collapse by its own pressure. Again, authors find a range of inner slopes \citep{2002MNRAS.331..635E,2003ApJ...586..135L,2006ApJ...650..777Z,2009ApJ...690..154S}. For both lensing and X-ray studies most authors focus on only one or a few clusters, which of course makes it more difficult to assess the universality of the profiles on an observational foundation.
In the present work we take a sample of 11 highly relaxed clusters and use the measurements of the X-ray emitting gas to infer model-independent mass profiles. We then compare with a number of different models that have been applied as mass profiles in the literature, focusing on three key questions: Which parameterized model is the most successful? How many free parameters are needed to describe the data adequately? Is there evidence for a universal inner slope/shape-type parameter? We answer these questions using a detailed statistical analysis based on Bayesian inference where we use the Bayesian evidence (or marginal likelihood) to make judgments about which model is preferred by the data.
\section{Density profile models}
Most models that are used for modeling the mass distribution in halos have been proposed or introduced as fitting formulae applied to the halos found in the numerical simulations. Hence these models are not theoretically well-founded but rather form a basis on which the predictions of numerical simulations can be compared with observations. Almost all of these models have two free parameters which determine the mass scale and the spatial extent of the halo, and these two parameters are specific to each halo. Some models have one or more additional parameters which determines the shape of the profile, and which may or may not be universal. Here we consider a number of two- and three-parameter models.
\begin{deluxetable*}{lcccc}
\tablewidth{0pt}
\tablecaption{Density profile models\label{tb:models}}
\tablehead{\colhead{Model} & \colhead{$(\alpha,\beta,\gamma)$} &\colhead{$r_{-2}/r_s$} &\colhead{$\rho_{-2}/\rho_0$} & \colhead{$\mu(x=r/r_s)$}}
\startdata
NFW & $(1,3,1)$ & 1 & $\frac{1}{4}$ & $\ln (1+x)-x/(1+x)$ \\[1mm]
D\&M & $(\frac{7}{9},\frac{31}{9},\frac{4}{9})$ & $\frac{121}{169}$ & $0.0338$ & $\frac{9}{20}(1+x^{4/9})^{-5} $\\[1mm]
Hernquist & $(1,4,1)$ & $ \frac{1}{2} $ & $\frac{16}{27} $ & $x^2/[2(1+x)^2] $\\[1mm]
Moore & $(\frac{3}{2},3,1)$ & $\frac{1}{2}$ & $\frac{8}{3\sqrt{3}}$ & $2\sinh^{-1}(\sqrt{x})-2\sqrt{x/(1+x)} $ \\[1mm] \tableline
slopeNFW & $(\alpha,3,1)$ & $2-\alpha$ & $(2-\alpha)^{-\alpha}(3-\alpha)^{\alpha-3}$ & \nodata\\[1mm]
transNFW & $(1,3,\gamma)$ & $1$ & $\frac{1}{4}$ & \nodata \\[1mm]
S\'ersic & \nodata & 1 & 1 & $8^{-n}e^{2n}n^{1-3n}\gamma(3n,2 n x^{1/n})$
\enddata
\tablecomments{Properties of the density profiles that we consider, including the $(\alpha,\beta,\gamma)$ specification, the relations between $(r_{-2},\rho_{-2})$ and $(r_s,\rho_0)$, and the shape $\mu (r)$ of the mass profile $M(r)=4\pi r_s^3\rho_0\mu (r)$, if analytical. $\gamma(a,x)$ is the lower incomplete gamma function, $\gamma(a,x)=\int_0^x t^{a-1}e^{-t}dt$.}
\end{deluxetable*}
A whole class of models are `double power-laws' which asymptote to power laws at very small and very large radii. These models can conveniently be summarized in Hernquist's $(\alpha,\beta,\gamma)$ parametrization \citep{1990ApJ...356..359H,1996MNRAS.278..488Z},
\begin{equation}
\rho(r)=\rho_0\left(\frac{r}{r_s}\right)^{-\alpha} \left[1+\left(\frac{r}{r_s}\right)^\gamma\right]^{-\frac{\beta-\alpha}{\gamma}},
\end{equation}
where $\rho_0$ and $r_s$ are scaling constants to be determined for each halo individually. The inner power-law slope is $\alpha$ and the outer slope is $\beta$, while the width of the transition region is controlled by $\gamma$. We consider four such two-parameter profiles: the NFW \citep{1997ApJ...490..493N}, the Dehnen-McLaughlin \citep{2005MNRAS.363.1057D}, the Hernquist, and the Moore profile \citep{1998ApJ...499L...5M}. The properties of these models are summarized in Table \ref{tb:models}.
We also consider three three-parameter models: Two are simply generalized NFW profiles where, in the first case, we allow the inner slope $\alpha$ to vary. The motivation for this slopeNFW model was already apparent from the introduction. The second case, transNFW, is also a generalization of the NFW where now the transition parameter $\gamma$ is free. Such a profile can mimic a steeper inner slope by pushing the inner power law behavior closer to the center. The third profile is the S\'ersic (or Einasto) profile \citep{1963BAAA....6...41S,1969Ap......5...67E},
\begin{equation}
\rho(r)=\rho_s\exp\left(-2n\left[\left(\frac{r}{r_s}\right)^{1/n}-1\right]\right),
\end{equation}
where the parameter $n$ determines the shape of the profile. For $n=4$ the de Vaucouleurs' law describing the surface brightness of elliptical galaxies is recovered. The shape parameter is sometimes given as $\alpha_s=n^{-1}$. Recently, the S\'ersic profile has been claimed to provide a better fit than the NFW to Milky Way-sized haloes formed in numerical simulations, and, interestingly, with a shape parameter that varies significantly from halo to halo \citep{2007ApJ...666..181S,2008arXiv0810.1522N}.
We map the scale radius $r_s$ and scale densities $\rho_s$ or $\rho_0$ of each model to the model-independent parameters $r_{-2}$ and $\rho_{-2}$, which are the radius at which the slope of the density profile is $-2$ and the density at that radius, respectively. This mapping makes comparison of the models easier and enables identical priors to be used in the statistical analysis in all models.
\section{Data analysis}
We revisit the sample of 11 highly relaxed, low redshift ($z<0.1$) galaxy clusters observed with {\it XMM-Newton} which we already used in \citet{2009ApJ...690..358H} to measure the dark matter velocity anisotropy profile for the first time (see also \citet{2007A&A...476L..37H}). The members of this sample were selected to appear close to round on the sky and not have strong features in the temperature and density profiles. The spectral analysis and deprojection of the X-ray data was carried out in \citet{2004A&A...413..415K} and \citet{2005A&A...433..101P}. The deprojection method was non-parametric, i.e.~without any parametric modeling of the radial temperature or density profiles. The outcome, and the starting point for the present analysis, was estimates of the ICM temperature $T_i$ and electron number density $n_{e,i}$ with associated uncertainties in six or seven radial bins, for each of the clusters.
Assuming hydrostatic equilibrium, the ICM gas traces the gravitational potential according to \citep{1978A&A....70..677C}
\begin{equation}\label{eq:he}
\frac{k_BT}{\mu m_H}\left(\frac{d\ln n_e}{d\ln r}+\frac{d\ln T}{d\ln r}\right)=-\frac{GM_{\mathrm{tot}}(r)}{r},
\end{equation}
where $\mu=0.6$ is the mean molecular weight of the ICM. Almost all of the cluster mass resides in dark matter and the ICM, and therefore the dark matter mass distribution can be determined through $M_\mathrm{DM}(r)=M_\mathrm{tot}(r)-M_\mathrm{ICM}(r)$. The ICM mass profile is given straight-forwardly by the density $\rho_\mathrm{ICM}=\mu m_H n_e$.
We calculate $M_\mathrm{DM}(r_i)$ of each radial bin through a Monte Carlo (MC) analysis in order to propagate uncertainties accurately. In detail, the prescription for each MC realization is as follows: In each bin $i$ the best estimates of $T_i$ and $n_{e,i}$ are added to random numbers drawn from Gaussian distributions representative of the uncertainties $\delta T_i$ and $\delta n_{e,i}$. In order to apply Equation \eqref{eq:he}, we estimate the logarithmic derivative of, e.g., $T$ at the bin-radius $r_i$ by the slope of the unique parabola that passes through $(\ln r_{i-1},\ln T_{i-1})$, $(\ln r_i,\ln T_i)$, and $(\ln r_{i+1},\ln T_{i+1})$. In this way we can calculate the total mass interior to $r_i$ for that data realization. We subtract the gas mass, estimated through a five-point Newton-Cotes integration formula applied to the same realization of the density data, and we arrive at the dark matter mass $M_{\mathrm{DM},i}$. We impose a number of checks to determine if the derived data realization is physically sensible: the ICM temperature and density must be greater than zero in all bins, the total mass profile must be increasing with radius, and the dark matter mass profile and derived density profile must also be everywhere positive. If these conditions are not met the entire realization is discarded. This process is repeated until $N=5000$ realizations have been accepted. From these the sample mean of $\ln M_i$ in each bin is determined, as well as the sample covariance matrix with elements
\begin{equation}\label{eq:cov}
C_{ij}=\frac{1}{1-N}\sum_k^N(\ln M_{ik}-\langle{\ln M_i}\rangle)
(\ln M_{jk}-\langle{\ln M_j}\rangle),
\end{equation}
where $N$ is the number of Monte Carlo realizations. Even though we sample the ICM temperature and density in each bin independently, the covariance matrix is not diagonal since the derivatives and physical consistency checks induce bin-to-bin correlations in the accepted sample. We use the mean and covariance of $\ln M$ rather than $M$ since, by inspection, the former is closer to being Gaussian distributed.
\section{Statistical analysis}
We take a Bayesian approach to the statistical analysis and the usual starting point is the likelihood function, which we calculate in the following manner.
It requires less manipulation of the data to calculate the mass profile from the observations than to calculate the density profile. Therefore we integrate the density profile analytically or numerically for each model to obtain the mass distribution and compare with the data in mass space, not density space. Further, as mentioned above, we have found in the MC analysis that the mass samplings in each bin are close to being log-normally distributed. Therefore we construct the likelihood $\mathcal{L}(M_i)=\exp(-\chi^2/2)$ from the $\chi^2$ function,
\begin{equation}
\chi^2=\sum_{i,j}(\ln M_i-\ln M(r_i)) C_{ij}^{-1}(\ln M_j-\ln M(r_j)),
\end{equation}
where $M(r_i)$ is the model mass profile at the radial centre $r_i$ of bin $i$, and $\ln M_i$ and $C_{ij}$ are determined by the MC analysis.
The main goal is to decide which model is the better representation of the data. We do this by calculating the Bayesian evidence of each model, which is a quantitative measure of the agreement between model and data \citep{2008ConPh..49...71T}. First we calculate the likelihoods of each model on a grid in the parameter space $\theta=(\log r_{-2},\log\rho_{-2})$. Next, we construct the posterior probability distribution by combining the likelihood function with a prior probability distribution $\pi(\theta)$ resembling our knowledge of $\log r_{-2}$ and $\log \rho_{-2}$ before taking the data into account. We discuss the choice of prior below. We then integrate the posterior to obtain the Bayesian evidence,
\begin{equation}
E=\int_\mathrm{all}d\theta \pi(\theta)\mathcal{L}(\theta,\ln M_i),
\end{equation}
which is essentially the weighted average of the likelihood over the prior volume. The evidence of a model, given the data and a prior, quantifies how well that model explains the data. It is important to stress that the comparison is made over all of the prior volume, not just at the best fitting set of parameters. When comparing models the Bayes factor $B_{12}=E_1/E_2$ shows how much more (or less) probable model 1 is than model 2, in light of the data. Traditionally, this is gauged on Jeffrey's scale where a Bayes factor of $\ln B_{12}<1$ is labeled `inconclusive' evidence for model 1 over model 2 while `weak', `moderate', and `strong' evidence corresponds to $\ln B_{12}$ values $<2.5$, $<5$, and $>5$, respectively.
We choose priors which are constant in the logarithms of $r_{-2}$ and $\rho_{-2}$. The flat logarithmic prior is the uninformative prior for scaling parameters \citep{2008ConPh..49...71T} since it reflects ignorance about the magnitude of the parameter. However we restrict the range of the priors, so that we end up with top-hat priors in $\log r_{-2}$ and $\log \rho_{-2}$. As a reference point we first assume a top-hat prior relative to the best estimate of $r_{2500}$ as determined in the MC analysis. (The scale radius $r_{2500}$ is defined as the radius within which the mean density is $2500$ times the critical density of the universe.) The top-hat prior in $\log r_{-2}$ ranges from $1.5$ magnitudes below $r_{2500}$ to 0.5 above. The basic idea behind this prior is that the transition or `roll' of a model should occur close to $r_{2500}$, as it does in haloes in numerical simulations, and also to prevent the model from behaving as a simple power-law by pushing the transition from the inner to the outer power law far away from the range of the data. We emphasize that this is still a conservative prior, as current simulations typically resolve 2--3 radial magnitudes with $r_{-2}$ located about one order of magnitude below the virial radius \citep{2001MNRAS.321..559B}. The prior in $\rho_{-2}$ is also a top-hat in the logarithm and a range of $10^{-26}$--$10^{-21}\,$kg$\,$m$^{-3}$, which in practice means that the likelihood is vanishingly small at the boundaries of the prior.
\begin{figure*}[htbp]
\includegraphics[width=\textwidth]{fig1.pdf}
\caption{Mass profile of each cluster with 68\% uncertainties and best-fit models. The radial axis has been scaled to the best estimate of $r_{2500}$ from the MC analysis, and the mass axis has been scaled by $r^{-1}$ so that the fitted models are approximately horizontal at $r_{-2}$.}\label{fi:mass}
\end{figure*}
\begin{deluxetable}{lrrrr}
\tablewidth{\columnwidth}
\tablecaption{Bayes Factor $\ln B$ for the two-parameter models, relative to the NFW profile\label{tb:bayes2}}
\tablehead{\colhead{Cluster} & \colhead{$z$} &\colhead{D\&M} &\colhead{Hernq.}&\colhead{Moore}}
\startdata
A262 & 0.015 & -2.0 & 0.9 & -3.0 \\
NGC533 & 0.018 & -1.7 & 1.2 & -3.0 \\
A496 & 0.032 & -1.4 & 0.6 & -1.2 \\
2A0335+096 & 0.034 & 0.5 & -0.1 & 13.2 \\
A2052 & 0.036 & 1.9 & -0.3 & 5.8\\
MKW9 & 0.040 & 0.5 & -0.1 & 1.4 \\
MKW3s & 0.046 & 1.8 & -0.3 & 6.2 \\
A4059 & 0.047 & 1.5 & -0.4 & 9.5 \\
S\'ersic 159--3 & 0.057 & -0.5 & 1.6 & 2.7 \\
A1795 & 0.064 & 2.5 & -0.5 & 17.9 \\
A1837 & 0.071 & 0.5 & -0.2 & 1.2 \\
\tableline
Total & \nodata & 3.6 & 2.4 & 51\phd\phn
\enddata
\tablecomments{A positive value of $\ln B$ indicates that the NFW profile is preferred over the considered model. Note that this does not imply any bias towards the NFW as the Bayes factor of any two other models is just the difference between the respective Bayes factors given here.}
\end{deluxetable}
\subsection{Two-parameter model results}
The result of the model comparison is summarized in Table \ref{tb:bayes2}, where the NFW model is compared against each of the other two-parameter models. A positive Bayes factor indicates that the NFW model is preferred. This does not imply any bias on the NFW since any two models can be compared by subtracting the Bayes factors we give for them from one another. We find that, individually, the clusters yield strong constraints only against the Moore model, while the evidences for or against the D\&M and Hernquist models are either weak or inconclusive on Jeffrey's scale. If instead we consider the cumulative Bayes factor summed over the full sample, the NFW is found to be the preferred model overall, i.e., as a universal two-parameter profile our sample favors the NFW model. The Hernquist profile and the D\&M profile are weakly and moderately disfavored, respectively, with cumulative Bayes factors of 2.4 and 3.6 while the Moore profile is convincingly ruled out with a factor of 51. The weak constraint on the Hernquist profile is not surprising as data extending out to the virial radius would likely be needed to properly distinguish this model from the NFW.
In Table \ref{tb:priors} we present the effects of varying the priors. The evidence against the D\&M profile increases to the level of strong when we limit the range of the prior in $\log r_{-2}$ to the smaller interval $(-0.75,0.25)$, while the Bayes factor is reduced slightly on the larger range $(-3,3)$. The evidence also becomes strong if we choose top-hat priors in $(r_{-2},\rho_{-2})$ instead of the logarithmic priors. Finally, the D\&M model is disfavored slightly more if we apply a `soft' Gaussian prior in $\log r_{-2}$. The Bayes factor for the Hernquist model is robust under the same variations, while the Moore profile is very strongly ruled out in all cases. We conclude that our two-parameter model selection results are stable against variation amongst reasonable choices of priors, which means that the data are of sufficient quality to make robust conclusions.
A more interesting issue to consider than the priors is that the preference for the NFW profile over the Hernquist and D\&M profiles is somewhat susceptible to `jackknife' resampling: if we recompute the cumulative Bayes factor eleven times systematically leaving a single cluster out each time, there are a few cases where the strength of the evidence is reduced to inconclusive but also cases where it is increased to strong (against the D\&M). This is largely due to the fact that our data sample is somewhat inhomogeneous in terms of the relative statistical uncertainty on the mass profile. For example, a comparison of the error bars of MKW9 with those of A1795 or S\'ersic 159-3 (see Figure \ref{fi:mass}) immediately shows that the former is much less constraining than the latter two. This means that our sample is a mixture of strongly and weakly constraining clusters and this is reflected in Figure \ref{fi:chart} where the contributions from individual clusters clearly varies. There appears to be a trend that the clusters A262, NGC533, and A496, which are the lowest redshift and some of the least massive in our sample, stand out by preferring the D\&M and the Moore profile. However, such trends are just as likely spurious selection effects caused by the relatively small sample but could be investigated with a larger sample. The D\&M profile can easily be preferred by clusters that also prefer the Moore profile since, by extending the transition region, the D\&M profile can push the inner asymptotic power law well inside the radial range of the data.
Finally we compare with a standard goodness--of--fit test: the minimum $\chi^2$ values for the models support our more detailed analysis: for a total of 53 degrees of freedom we get minimum $\chi^2$'s of 81 for the NFW, 93 for the D\&M, 82 for the Hernquist, and 190 for the Moore profile. Major contributions to these $\chi^2$ values come from the two clusters MKW3s with $\chi^2=14.2$ and A4059 with $\chi^2=13.2$ for the NFW model and similar or larger values for the other models. The corresponding $p$-values imply that the D\&M $\chi^2$ is about 20 times less likely to have occurred by chance (if the D\&M model is correct) than the NFW model is (if the NFW model is correct). Compare this with the Bayesian odds that the NFW is $\sim40$ times more probable than the D\&M. Note that the actual best-fits are slightly smaller since we evaluate the $\chi^2$ on a grid instead of minimizing it with a dedicated search. The $\chi^2$ values show that, also in terms of goodness--o-f-fit, our sample is rather inhomogeneous. The rather poor total fit should not be judged too harshly since the halos in numerical simulations also show halo--to--halo scatter, which is not accounted for by the fitting profiles.
\begin{deluxetable}{lrrrr}
\tablewidth{\columnwidth}
\tablecaption{Total Bayes factor $\ln B$ for the two-parameter models assuming various priors, relative to the NFW profile\label{tb:priors}}
\tablehead{\colhead{Prior} &\colhead{Range $\log\rho_{-2}$}&\colhead{D\&M} &\colhead{Hernq.}&\colhead{Moore}}
\startdata
Top-hat in $\log r_{-2}$ & (-1.5,0.5) & 3.6 & 2.4 & 51\\
Top-hat in $\log r_{-2}$ & (-3,3) & 2.8 & 2.5 & 36\\
Top-hat in $\log r_{-2}$ & (-0.75,0.25) & 6.8 & 2.3 & 63\\
Top-hat in $(r_{-2},\rho_{-2})$ & (-1.5,0.5) & 8.8 & 1.3 & 60\\
Gaussian in $\log r_{-2}$ & \nodata & 4.8 & 2.3 & 50
\enddata
\tablecomments{The top line is the fiducial prior used in Table \ref{tb:bayes2}. In the next two cases the range of the prior in $\log r_{-2}$ (in units of $r_{2500}$, see text) is varied, and in the following two cases a top-hat prior in $r_{-2}$ and both $r_{-2}$ and $\rho_{-2}$ is applied. The final case assumes a Gaussian prior in $\log r_{-2}$ with mean -0.25 and width 0.5.}
\end{deluxetable}
\subsection{Three-parameter model results}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=.9\textwidth]{fig2.pdf}
\caption{Bar chart of the Bayes factors $\ln B$ for the various models considered, relative to the NFW, as given in Table \ref{tb:bayes2} and \ref{tb:bayes3}. The Bayes factors are additive so the contribution of individual clusters to the total Bayes factor is easily assessed. The values shown are based on the fiducial priors discussed in the text.}\label{fi:chart}
\end{center}
\end{figure*}
For the three-parameter models we again want to test whether the models represent the data better than the NFW. In this case the comparison is slightly more involved since there is the freedom of an additional parameter to take into account. This naturally yields a lower value of the evidence if the extra parameter does not provide a better description of the data, or, more specifically, the third parameter must improve the fit over a significant volume of parameter space in order to be preferred over the NFW.
It is important to stress that there is no assumption about the third parameter being universal. On the contrary, we ask whether the data require the additional freedom of an extra parameter which is determined individually for each cluster.
The model comparison proceeds as before: we calculate the evidence for each of the three-parameter models with the same priors in $\log r_{-2}$ and $\log \rho_{-2}$ as in the fiducial two-parameter analysis for all models. For the slopeNFW we choose a top-hat prior for $\alpha$ which ranges from 0 to 1.75, i.e.~from a cored profile to a profile slightly steeper than the Moore profile. We do not want to go all the way to $-2$ since $r_{-2}$ tends to zero and eventually becomes undefined as $\alpha$ approaches $-2$. For the transNFW, we choose a logarithmic prior with $\gamma$ in the range $(0.1,4)$ which allows this profile to mimic a steeper inner profile by pushing the asymptotic inner power law inside the radial range of the data. Finally, we take a logarithmic prior for $n$ in the range (2,15) for the S\'ersic model, motivated by numerical simulations which have best fits S\'ersic profiles with $n$=5--9. The logarithmic prior has the advantage that it is invariant whether one prefers $n$ or $\alpha_s=1/n$ as the parameterization.
The resulting Bayes factors relative to the NFW are given in Table \ref{tb:bayes3} and summarized in the chart in Figure \ref{fi:chart}. The individual clusters provide only weak evidence for or against any of the models. Based on the whole sample, the model selection is inconclusive for the transNFW and S\'ersic models but there is `moderate' evidence for the slopeNFW model over the NFW with a Bayes factor of $-2.6$. This corresponds to odds of 13 to 1 in favor of the slopeNFW model and shows that, overall, the slopeNFW has the highest evidence $E$ of all models considered. Hence the data show a moderate need for a free inner slope despite the penalty against the extra freedom built into the Bayesian analysis. It must be mentioned that most of the discriminatory power is carried by a few clusters such as NGC533, A4059, and A1795 and removal of any of these clusters from the sample would change the Bayes factor significantly. Therefore we caution that the moderate preference for the slopeNFW model is somewhat susceptible to selection effects since, as noted above, the constraints from individual clusters vary in quality. We also find some sensitivity to the choice of prior: if the upper bound of $\alpha$ is extended from 1.75 up to 1.9, the Bayes factor for the slopeNFW model changes to $-1.9$, while if it is set to the Moore profile at $1.5$ the Bayes factor becomes $-3.1$. If the lower bound of $\alpha$ is increased to $0.5$, the Bayes factor remains relatively unchanged at $-2.9$. While we believe that the fiducial priors used above are reasonable descriptions of the `state of knowledge' based on numerical simulations, the sensitivity to the choice of prior indicates that the data do not necessarily confine the posterior to a sufficiently small region of the prior volume to provide unambiguous conclusions.
\begin{deluxetable}{lrrr}
\tablewidth{\columnwidth}
\tablecaption{Bayes Factor $\ln B$ for the three-parameter models, relative to the NFW profile.\label{tb:bayes3}}
\tablehead{\colhead{Cluster} & \colhead{slopeNFW}&\colhead{transNFW}&\colhead{S\'ersic}}
\startdata
A262 & -2.1 & -1.6 & -2.0 \\
NGC533 & -1.8 & -1.9 & -1.8 \\
A496 & -1.1 & -0.7 & -0.9 \\
2A0335+096 & 1.1 & 0.6 & 1.0 \\
A2052 & 0.2 & 1.2 & 0.6 \\
MKW9 & 0.2 & 0.4 & 0.3 \\
MKW3s & 1.4 & 1.7 & 1.5 \\
A4059 & -2.5 & -1.4 & -1.8 \\
S\'ersic 159--3 & 0.3 & 0.9 & 0.5 \\
A1795 & 1.8 & 1.3 & 2.4 \\
A1837 & -0.1 & 0.2 & 0.1 \\ \tableline
Total & -2.6 & 0.7 & -0.1
\enddata
\tablecomments{A positive value of $\ln B$ indicates that the NFW profile is preferred over the considered model. A top-hat prior in $\log r_{-2}$ of $(-1.5,0.5)$ around the best estimate of $r_{2500}$ for each cluster is assumed.}
\end{deluxetable}
\subsection{Constraints on the third parameters}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=.32\textwidth]{fig3a.pdf}
\includegraphics[width=.32\textwidth]{fig3b.pdf}
\includegraphics[width=.32\textwidth]{fig3c.pdf}
\caption{Probability distributions for the third parameter in each of the three-parameter models: slopeNFW $\alpha$ (left), transNFW $\gamma$ (middle), and S\'ersic $n$ (right). In each panel, the full line shows the joint posterior for all clusters combined while the dot--dashed line shows the joint posterior obtained using the method of hyper-parameters (see text). The dashed lines are the pdf's of individual clusters. Note that each posterior is normalized to unity so it is not possible to draw conclusions about the quality of fit of the individual clusters from this plot. The standard 95\% credible intervals are $(1.00,1.21)$ for $\alpha$, $(0.68,1.06)$ for $\gamma$, and $(4.3,6.2)$ for $n$. With the hyper-parameters, the intervals are instead $(0.85,1.31)$ for $\alpha$, $(0.50,1.28)$ for $\gamma$, and $(3.5,7.4)$ for $n$. We assume top-hat priors in $\alpha$, $\ln \gamma$, and $\ln n$.}\label{fi:third}
\end{center}
\end{figure*}
Finally, for the three-parameter models we also want to constrain the preferred value of the third parameter. Unlike above, this analysis assumes that there is a universal value for the third parameter and attempts to identify that value. We use the same priors as in the previous analysis for each model, but now we marginalize over the nuisance parameters $(\log r_{-2}, \log \rho_{-2})$ to find the one-dimensional posterior probability distribution for the third parameter for each cluster. Then we combine the results from the individual clusters into a joint posterior which is simply the product of the the individual ones. We calculate 95\% credible intervals for both the individual and the joint posterior. However, we know from the previous analysis that each three-parameter model is preferred by some clusters but not by others. Therefore we also use the method of hyper-parameters \citep{2000MNRAS.315L..45L} which allows the contribution from individual data-sets to the joint posterior to be weighted. These weights are marginalized over assuming logarithmic priors with the result that in the joint likelihood one replaces
\begin{equation}
\sum_i\chi_i^2\rightarrow \sum_i N_i\ln\chi^2_i,
\end{equation}
where $N_i$ is the number of data points in data-set $i$. The upshot of all this is that clusters that are not described well by the model do not constrain the parameters as strongly as clusters that are well described. The price to pay is that the effective sample size is reduced which, all other things being equal, will lead to wider and more conservative credible intervals.
The results are shown in Figure \ref{fi:third}, where in each panel the fully drawn line is the joint posterior, the dotted line is the hyper-parameters posterior, and the dashed lines are the posteriors of the individual clusters. The generalized NFW models are drawn slightly away from, but not in disagreement with, the NFW with 95\% credible intervals of $(1.00,1.21)$ for $\alpha$ and $(0.68,1.06)$ for $\gamma$. The interval for the S\'ersic $n$ parameter is $(4.3,6.2)$, in good agreement with the values reported by the Aquarius numerical simulations for Milky Way-sized halos \citep{2008arXiv0810.1522N}. The intervals derived using the method of hyper-parameters are wider as expected: $(0.85,1.31)$ for $\alpha$, $(0.50,1.28)$ for $\gamma$, and $(3.5,7.4)$ for $n$. The difference between the hyper-parameters method and the conventional calculation illustrates the need for a cautious approach to in-homogeneous data-sets. We believe the hyper-parameters yields the more trustworthy results in the case at hand, while on the other hand we acknowledge that they are not very constraining.
An inspection of the contribution from individual clusters reveals some issues: It is clear that for each model a number of clusters provide very little information about the third parameter, i.e.~the model describes the mass profile almost equally well regardless of the third parameter value. This is actually expected, given the varying size of the Bayes factors in table \ref{tb:bayes3}. There are also a few cases, particularly for the transNFW model, where the posterior peaks very close to or on the bounds of the prior. In such cases the results, e.g.~the individual credible intervals, are of course very prior-dependent which again indicates that the data are not very discriminatory with respect to the prior. On the other hand, rather drastic priors or small sub-samples must be used in order to significantly affect the credible intervals of the joint posterior, especially for the hyper-parameters method.
\begin{figure}[bt]
\includegraphics[width=\columnwidth]{fig4.pdf}
\caption{The individual clusters' constraints on the third parameter in each of the three-parameter models. In this case we show the 68\% credible intervals, and the horizontal lines indicate the 68\% range of the joint posterior calculated using the method of hyper-parameters. Refer to Table \ref{tb:bayes2} for the redshifts of each cluster.}\label{fi:zdep}
\end{figure}
Figure \ref{fi:zdep} shows the individual clusters' constraints on $\alpha$, $\gamma$, and $n$. As could be expected given the varying nature of our results, there is perhaps the slightest of hints of a redshift--dependence in the constraints but our sample size does not allow us to probe such an issue in detail. It should again be noted that any hint of a redshift--dependence could actually be caused by a mass--dependence instead, since the two lowest redshift clusters in our sample are also the least massive.
A different picture emerges when we consider the overlap of the individual clusters' credible intervals for the slopeNFW model. For example, no value of $\alpha$ is contained in all 11 95\% credible intervals, and only the very short range $(1.08,1.10)$ is contained in all but two intervals. Likewise the NFW $\alpha=1$ case is excluded from four of the eleven intervals. These results, as well as a visual inspection of Figure \ref{fi:third}, puts strong doubts about the concept of a universal shape parameter. The situation is not quite as compelling for the transNFW and S\'ersic models which is likely the reason that they do not stand out from the NFW in the model selection. In fact, we believe it is a reasonable statement that the success of the slopeNFW model is precisely due to the very different preferred values of $\alpha$ from cluster to cluster. This puts a strong question mark against the idea of universal third parameter.
We conclude that there is moderate evidence for the slopeNFW model to be preferred over the simple NFW, while the transNFW and S\'ersic models do not stand out against the two-parameter NFW profile. If the inner slope of the slopeNFW model is universal, we constrain it to be close to $-1$ but preferably slightly steeper. However, the spread of the individual clusters' preferred ranges suggests that the inner slope is not universal.
We also comment that the method of hyper-parameters method could in principle be extended to the model selection analysis. As a matter of fact, since the slopeNFW `contains' both the NFW and the Moore models as subsets, we can derive the corresponding Bayes factor for the Moore profile which is only 10. This is of course a drastic reduction numerically but it does not alter the conclusion and anyway corresponds to rather convincing odds of about $20\,000:1$.
\section{Biases}
So far we have discussed the interpretation of our results with respect to the statistical evidence. However, a number of biases, or systematic uncertainties, can be thought of that may affect our results. Loosely, these can be grouped into biases that affect both the individual cluster mass modeling and the combined analysis, and selection effects that only influence the latter.
The analysis rests on the ability to produce deprojected temperature and density profiles with uncertainties that correctly mirror the uncertainties in the spectral analysis of the X-ray data. This has been discussed extensively in \citet{2004A&A...413..415K}.
The basic assumption in determining the mass distribution of a galaxy cluster is that the cluster is relaxed, and obeys the equation of hydrostatic equilibrium. Numerical simulations indicate that the additional pressure associated with turbulence and bulk motion in the ICM yields an underestimate of the mass in the region of $5-20$\% with the larger values corresponding to large radii, $r_{500}$ and greater \citep{2007ApJ...655...98N,2008A&A...491...71P,2009arXiv0903.4895L}. We do not expect this bias to exceed 10\% in the present case since we do not model further out than to $\sim r_{2500}$. On the other hand, the same numerical simulations indicate that if the turbulent pressure is accounted for, an accurate mass reconstruction is possible. This point demonstrates that deviations from spherical symmetry are not a major concern in the error budget.
A related question is whether the parameterized profiles should be tested against the total mass distribution or the dark matter mass profile only. While the predictions of numerical simulations are founded in dark matter-only simulations, it is not clear how much a simulated dark matter-only mass profile is modified by the presence of baryons. Observationally, the ICM contributes about $5-15$\% of the total density in a cluster, again increasing with radius in the range of interest here, so formally there is a difference between the total and the dark matter profile's radial dependence. To test the impact of this, we have rerun the statistical analyses described above without subtracting the ICM mass from the mass estimate of Equation \eqref{eq:he}. We find only minor differences: For the two-parameter models, the total Bayes factors relative to the NFW profile, assuming the fiducial prior as in Table \ref{tb:bayes2}, are 2.8 (D\&M), 2.7 (Hernquist), and 54 (Moore), i.e.~there is no significant change in the interpretation of the results. For the three-parameter models, the total Bayes factors become $-3.1$ (slopeNFW), $0.6$ (transNFW), and $-0.2$ (S\'ersic), which are in good agreement with the results in Table \ref{tb:bayes3}. Finally, the constraints on the third parameters for the three-parameter models are unchanged.
The fact that our results are stable whether we test the mass models against the total or dark matter-only mass profiles allows us to gauge how important the mass bias caused by turbulent pressure is. The point is that the turbulent pressure is expected to contribute the same amount (or less) to the total mass estimate as the ICM mass: both contributions are at the $5-15$\% level and radially increasing, and at the maximum radius we consider $\sim r_{2500}$ the gas fraction ($\sim10\%$) is likely larger than the pressure bias. Since our results are the same whether we account for the ICM mass or not, we conclude that the systematic uncertainty is likely much smaller than the statistical uncertainty.
\begin{figure}[bt]
\includegraphics[width=\columnwidth]{fig5.pdf}
\caption{The mass--concentration relation of our sample, calculated within the NFW model. The contours contain 95\% of the posterior PDF and are based on the fiducial prior. We show two contours for each cluster: $(M_{2500},c_{2500})$ (red, full lines) which are derived within the radial range of the data, and $(M_{200},c_{200})$ (blue, dot-dashed) which is based on an NFW model-dependent extrapolation to $r_{200}$. The dashed lines show the mean relations for the two values of $\Delta$ from the N-body simulations of \citet{2008MNRAS.391.1940M}, based on the WMAP5 cosmology. The relations are $\log c_{200}=0.83-0.094\log(M_{200}/10^{12}M_\sun)$ and $\log c_{2500}=0.35-0.130\log(M_{2500}/10^{12}M_\sun)$. Given the low redshift of our sample, we have not made any correction for a redshift evolution of $c_\Delta$.}\label{fi:mc}
\end{figure}
\section{Mass--concentration relation}
An important consequence of the `bottom-up' scenario of structure formation is that smaller halos are denser in the center, since they formed earlier when the density of the Universe was higher. This effect is seen in numerical simulations and it can be expressed as a relation between the halo mass and the concentration parameter. The concentration parameter is defined for a given overdensity as $c_\Delta=r_\Delta/r_{-2}$ (often $r_s$ is used instead of $r_{-2}$ but for the NFW this is unimportant). Simulations usually consider the mass--concentration relation at the virial radius $r_{200}$, but we can only reach that radius by model-dependent extrapolation. Therefore, in Figure \ref{fi:mc}, we show the mass--concentration relation of our sample calculated within the NFW model at both $r_{2500}$ and extrapolated to $r_{200}$. We suggest that authors provide relations at both of these $\Delta$'s as they complement each other in physical significance and observational accessibility.
As can be seen in Figure \ref{fi:mc}, our sample is not ideally suited to derive a relation from given that six sample members cluster at almost identical values of $M_\Delta$. Instead we compare with the mass--concentration relation of the simulations presented in \citet{2008MNRAS.391.1940M}, which are in reasonable agreement with our sample except for the low mass NGC533. We emphasize that the orientation of the uncertainty ellipses is related only to the parameter degeneracies present in the combination of model and mass profile data and has nothing to do with the slope of the mass--concentration relation. The agreement between our observed mass--concentration relation and the predictions of numerical simulations resembles the recent X-ray analysis of \citet{2007ApJ...664..123B}, but stands out from the significant discrepancy of the lensing study in \citet{2008ApJ...685L...9B}.
\section{Summary \& discussion}
We have conducted a careful statistical analysis of the constraints on mass distribution models of galaxy clusters which can be derived from X-ray observations. We find that the NFW model is the preferred two-parameter model and that the Moore model is decisively ruled out. There is moderate evidence that the data require an additional free parameter that alters the shape of the mass profile, and the best choice is a model similar to the NFW but with a freely varying inner slope. If we assume this slope to be universal, we can constrain it to be close to or slightly steeper than the NFW, but our data suggest that the shape parameter must be determined individually.
Significantly, the clusters in our sample prefer very different values for the inner slope, some prefer flat cores while others prefer steep cusps. The shape-parameters of the other two three-parameter models we consider, the S\'ersic and transNFW, also show considerable scatter across our sample. We conclude that there is a strong indication in our data that the mass profile is not universal but suffers considerable halo-to-halo scatter. The limited size of our sample means that we cannot assess whether this is in disagreement with the results of numerical simulations. Of course, we can force universality of the inner slope, in which case we find that it is preferred to be slightly steeper than the NFW value of $-1$. However, when the goodness--of--fit of each cluster is taken into account using the method of Bayesian hyper-parameters, the credible interval becomes significantly larger, partly because of the smaller effective sample size, but also because of the lack of universality.
This analysis stands out from the numerous observational results that claim significant discrepancies from simulations based on only one or a few observed clusters. We acknowledge that our sample size is still limited, but it allows us to discuss the issue of universality. Given that halos in numerical simulations which include baryons are still not readily mass produced with sufficient resolution, which makes the question of halo to halo scatter difficult to assess, it is not possible to decide if the strong indication of a non-universal model that we see is at odds with the numerical predictions.
Our results are largely insensitive to whether we compare the models with the dark matter mass profile or the total mass profile, and so we cannot judge whether one type of model is more appropriate for the dark matter halo or the total gravitational potential of a halo. There are two reasons for this: firstly, the uncertainties of primarily the ICM temperature profiles are too large, and secondly, the angular resolution in the center is not good enough. Of course the ICM is rather smoothly distributed and very good statistics would be needed for a model to fit either the total or dark matter distribution significantly better. For nine of the 11 clusters of our sample we readily found 2MASS \citep{2006AJ....131.1163S} cD or BCG galaxies very close to the X-ray center but including these in the mass budget does not make a difference, unless we assumed extreme mass-to-light ratios. This is again due to the limited angular resolution of the observations which implies that a large amount of dark matter is contained even within the radial center of the innermost bin.
The robustness of our results whether we use the dark matter or total mass profile gives us reasonable confidence that deviations from hydrostatic equilibrium in the ICM are not a major problem. Such a systematic uncertainty would yield a bias of at most 10--15\% according to numerical simulations, which is similar to the difference between the total and dark matter mass profiles.
\acknowledgments{We thank Rocco Piffaretti for sharing data with us and for comments on the manuscript, and Andrea Macci\`o for providing the mass-concentration relation of his numerical simulations. The Dark Cosmology Centre is funded by the Danish National Research Foundation.}
{\it Facilities:} \facility{XMM}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 5,960
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{"url":"http:\/\/encyclopedia.kids.net.au\/page\/lu\/Lust","text":"## Encyclopedia > Lust\n\nArticle Content\n\n# Lust\n\nLust may designate intense sexual or other desires. It is considered a vice by many religions.\n\nSome moralists consider lust to be a corruption of temperance, in the sense that when temperance fails, lust is the natural result. In Christian theology, it is considered one of the Seven Deadly Sins.\n\nAll Wikipedia text is available under the terms of the GNU Free Documentation License\n\nSearch Encyclopedia\n Search over one million articles, find something about almost anything!\n\nFeatured Article\n Quadratic formula ... impossible to divide by it.) The symbol $\\pm \\sqrt {b^2-4ac}$ in the formula should be understood as \"either of those element(s) of the field ...","date":"2021-01-26 10:27:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7312380075454712, \"perplexity\": 3228.3606851742506}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610704799711.94\/warc\/CC-MAIN-20210126073722-20210126103722-00758.warc.gz\"}"}
| null | null |
Александр Анатольевич Гонченков (род. , Львов) — советский трековый и украинский и российский профессиональный шоссейный велогонщик. Заслуженный мастер спорта СССР (1990). Чемпион мира в командной гонке преследования на треке 1990 года. Участник летних Олимпийских игр 1992 года. Бронзовый призёр чемпионата России 1997 года в шоссейной групповой гонке.
На Олимпийские игры 1996 года главный тренер олимпийской сборной России Виктор Капитонов Гонченкова не взял, несмотря на то, что ранее гонщику были предоставлены письменные гарантии участия. Причина была в неурегулированных отношениях с Украинской федерацией велоспорта. Из-за этого Гонченков снялся с Тур де Франс 1996.
Спортивные достижения на треке
1990
Чемпионат СССР, командная гонка преследования — 2-е место
Чемпионат мира, командная гонка преследования — 1-е место
Спортивные достижения на шоссе
Примечания
Велогонщики СССР
Велогонщики Украины
Велогонщики России
Велогонщики на летних Олимпийских играх 1992 года
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 4,382
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namespace tzw
{
class LoadWorldUI : public IMGUIObject
{
public:
// ͨ¹ý IMGUIObject ¼Ì³Ð
void drawIMGUI(bool * isOpen) override;
LoadWorldUI();
std::function<void (std::string)> m_onCreate;
void prepare();
private:
char m_worldName[64];
std::vector<std::filesystem::path> m_entryList;
};
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,208
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A Wrong Turn on the Way to the Forum?
By Dean Williams June 8, 2011 5
Comedy isn't easy. And making it look easy takes of great deal of careful preparation. It is, as they say, all in the timing.
Mounting a full-scale musical is a daunting task for any theater and the enormous cost for performance rights and music royalties this type of show demands can hamper a community theater's ability to do all they envision. Even so, the biggest challenge Riverfront's current run of "A Funny Thing Happened On The Way To The Forum" may have faced is time itself. Coordinating a large cast (19) as well as the various technical elements that make a show shine, takes a lot of time and hands. But, let's take a look first at what does shine.
At the center of this farce is the scheming slave Pseudolus, whipping the plot through its paces and driving the madness forward. Played by Jorin Antero-Towle in his Riverfront debut, his task is Herculean. Pseudolus must act as narrator and facilitator of the story, as well as conman and matchmaker, while infecting those around him with calculated hysteria. He sings well, on key and on time and zips deftly through his lines with the energy of a house full of courtesans. Oddly, at one point, Jorin comes out in tap shoes, but barely uses them.
Jon Narducci, during rehearsals for "Forum"
The music is handled perfectly by pianist Jon Narducci. He is a pro and keeps the show lively and moving along. And it's always great to have live music onstage for a musical. He sits within the set, in Roman costume, with a clear view of the actors so no one misses a beat. Yet in their second week of performance, some of the singers still seem a bit timid (smile, girls!) and others wander off key. The best number is definitely "Everybody Ought to Have a Maid," handled deftly by the four leading men: Jim Littier, Jorin Antero-Towle, Colin Peacock, and Ian Dalziel. Littier (Senex) has a good musical theater background and carries his part with ease and a strong voice.
A Roman soldier, during rehearsals
Costumer Evie Bishop does a colorful riff off of your basic toga, making the cast look bright and showy. She also toughens up those Roman soldiers with great detailing in armor and leather.
Now, there are too many characters to mention here, but I will say that small roles, even non-speaking roles, are nevertheless important. Small parts support larger ones and are written in for good reason. One wishes that more attention could have been spared for these potential character jewels because the Courtesans (ladies-for-hire) and the Proteans (who play citizens, soldiers, and slaves), look aimless, under-directed and lost. Both groups tended to drop out of character as soon as focus was off them.
The setting of course is ancient Rome. Whatever that conjures up in your mind, I'd guess it's not the grey box-like structures that seem lifted from a suburb of Glasgow. The cramped set is supposed to represent a Mediterranean street and the houses of Lycus, Senex, and Erronius, but the dreary mottled grey walls gave the play a strangely somber tone. Perhaps a quick googling for 'Roman architecture' could have provided some design ideas.
I noted there are no light cues in this production and that baffled me at first. After all, lighting is what makes theater theatrical and is so important in delineating mood and action. I also saw that the blocking, or the movement of actors, was very basic, consisting mostly of people in straight lines, facing front. But a look at the program yields up a clue to this lack of detail. Hard working Joe Cullis, who originally brought Forum to Riverfront, is listed as not only Technical Director, but Set Design, Lighting, and Sound. I know Joe and I know he did even more than what the program gives him credit for. I think that maybe he wore a few too many hats, even for his experienced and able hands. My point is, I think the show would have benefitted greatly by having a dedicated choreographer and lighting designer and taken some of the pressure off the directors. Doing a musical comedy is a huge juggling act. But as always, the show must, and did go on, is very lively and everyone appears to be having fun. "Forum" continues to play weekends through June 18.
The overall success of the original 1962 "Forum," aside from Sondheim's great tunes, was due to its style of comedy. This was the heyday, the early '60s, of the Borscht Belt comedians, the comics who worked the resorts in the Catskills in upstate New York. Guys like Shecky Greene, Jack E. Leonard, Sid Caesar, and Phil Silvers. That's the comedy of Forum — Jewish standup in togas. That's funny. Wise-cracking Jewish comedians as Romans? Say no more. It's that essence that I would have liked to see more of from Riverfront's production– that vaudevillian flavor, the yiddishness, the scheming, self-deprecating American-Jewish humor at the core of the show. The show's title is literally the set-up for a joke and the entire play is its punchline. Bada-boom, bada-bing. Perhaps with a little more time they can explore that essence. And a good joke, as any of those old timers will tell you, is all … in … the … timing. (Rimshot!)
What: Riverfront Playhouse's "A Funny Thing Happened On the Way to the Forum"
When: Friday through Sunday (June 9-12) and June 16-18. Showtimes:
Thursday – Saturday: 7:30 p.m. (doors open 6:45 p.m.)
Sunday matinees: 2 p.m. (doors open 1:15 p.m.)
Where: 1620 East Cypress Ave, Redding, CA 96002
Cost: Opening/Closing Night: $27
Regular Evenings: $22
Matinees: $20
Tickets available at the door or in advance at:
Collectibles, Etc. (formerly Graphic Emporium), 1525 Pine Street, Redding, CA 96001
For reservations by phone, call (530) 241-4278 or (530) 246-7727.
For more information, call (530) 221-1028 or visit www.riverfrontplayhouse.net.
View the Riverfront Playhouse location in a larger map.
Photos courtesy of Riverfront Playhouse.
For the Cafe Stage Manager's listings of live theater performances in the North State, please click here.
An actor, director, and artist, Dean Williams has appeared on Shasta County stages for over 25 years in nearly 100 different roles. He has collaborated with many theater groups and is co-founder of The Root Theatre Company. He has also voiced characters for Sega and Playstation video games, and acted for a number of radio, televison and independent film projects. Dean also compiles A News Cafe's Cafe Stage Manager list of live theater events. Reach him at cafestagemanager@gmail.com.
A News Cafe, founded in Shasta County by Redding, CA journalist Doni Greenberg, is the place for people craving local Northern California news, commentary, food, arts and entertainment. Views and opinions expressed here are not necessarily those of anewscafe.com.
An actor, director, and artist, Dean Williams has appeared on Shasta County stages for over 25 years in nearly 100 different roles. He has collaborated with many theatre groups and is co-founder of The Root Theatre Company. He has also voiced characters for Sega and Playstation video games, and acted for a number of radio, televison and independent film projects. Ever since the first stories were acted out around ancient fires, theatre has held the power to move audiences like no other art form. It remains Williams's focus because live theatre has the potential to tell us every human story, intimately and impactfully. It becomes a magic mirror in which we see our own stories.
Comment Policy: We welcome your comments, with some caveats: Please keep your comments positive and civilized. If your comment is critical, please make it constructive. If your comment is rude, we will delete it. If you are constantly negative or a general pest, troll, or hater, we will ban you from the site forever. The definition of terms is left solely up to us. Comments are disabled on articles older than 90 days. Thank you. Carry on.
Polly Baker says:
What a good, detailed review! I think you might be expecting a bit much from local theater, but at least people know exactly what to expect.
Jon Lewis says:
What a thoughtful review, Dean. It has a lot of the elements I like to see in a critique, including reasoned explanations to support your opinions and helpful (but not condescending) information on theatre in general.
I think we all (actors, directors, backstage crew, audience members) benefit from constructive criticism. Anybody can be catty and go for the cheap laugh but that doesn't help anything.
I agree that 'Forum' is a big undertaking. I haven't seen the show yet so I'll reserve judgment, but I did watch part of a dress rehearsal and I was favorably impressed by the energy.
Anyway, I'm glad you've started this 'Stage Manager' forum and I applaud A News Cafe for giving you the space to inform us all.
Thank you, Jon. I'd rather hold higher expectations of local theater than settle for lowered expectations.
Constructive criticism aside, our family of amateur theatre-goers really enjoyed this production last week and we've been humming "something familiar…a comedy tonight!" ever since! We were especially impressed by the energy of Jon Narducci and Jorin Antero-Towle, but we thought there was a lot of talent and experience evident in the cast. We liked the costumes….and I thought the cartoonish set was an appropriate canvas for the production. We were delighted by this comedy and recommend people check it out!
kjohnson says:
Hi – I am definitely glad to find this. cool job!
Tags: "A Funny Thing Happened on the Way to the Forum"96002California NorthstateColin PeacockDean WilliamsEvie BishopIan DalzielJim LittierJon NarducciJorin Antero-TowleNorthern CaliforniaPerformanceplayRedding CARiverfront PlayhouseShasta County
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© A News Cafe.com, LLC. All rights reserved. This material may not be published, broadcast, rewritten or redistributed without permission.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,845
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Community choirs meet at Malvern
By Gary Bills-Geddes
CELEBRATED: The Malvern Male Voice Choir. Pic. Charles Pavey
Malvern Male Voice Choir plans to step up to mark for its centenary year with a stunning concert this month, among friends.
In fact, The Midsummer Malvern Community Choir Concert, on Sunday June 23, will include The Cradley Singers, The Hill Singers and Choirs from Malvern St James School . the U3A Choir and the Salvation Army.
A spokesman for the event said: "Witness a wonderful concert from local community choirs in the Forum, Malvern Theatres.
"The standard of singing is extremely high and the programme is varied. Come along for what should be an enchanting evening."
Malvern Male Voice Choir spokesman, Gerry Taggart said: "Our music programme includes folk songs and American and European Music from choirs singing together and separately.
"This is an important concert for us as we approach our Centenary Year. We all know how important singing is to bring communities together."
Malvern Male Voice Choir started in 1922 as a male quartet and grew rapidly to become one of the most prominent choirs in the Three Counties, with members who live in and around Malvern.
Mr Taggart said: "Our members' ages span late teens to eighties and we enjoy a wide range of talents, skills and abilities both musically and organisationally.
"We currently number about thirty voices and we always welcome new members to share our enthusiasm and to join in our enjoyment. Our repertoire is large and our library contains over 300 titles."
Tickets and further details for the June 23 concert: 01684 892277 and at
https://www.malvern-theatres.co.uk/whats-on/community-choirs-concert/
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,012
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Battleground Wisconsin Green Party to sue to get on Wisconsin ballot
Green Party to sue to get on Wisconsin ballot
The Green Party says it has retained a law firm to file a lawsuit this week challenging a decision by the state Elections Commission to keep its presidential and vice presidential candidates off the November ballot.
The move comes after entertainer Kanye West filed a suit in Brown County on Friday challenging the commission's ruling that his nomination papers were late.
Both legal actions come as locals were prepared to begin printing ballots this week ahead of the Sept. 17 deadline to mail them to voters with valid absentee ballot requests on file.
The commission deadlocked 3-3 on whether to place presidential nominee Howie Hawkins and running mate Angela Walker on the ballot. A complaint challenged the party's filings, alleging some nomination papers listed an incorrect address for Walker. She moved within the same South Carolina city while the papers were being circulated, but the party failed to amend its filings with the new address or respond to the challenge.
The Green Party said it retained the law firm of von Briesen & Roper.
See the release:
https://www.wispolitics.com/2020/green-party-candidates-hawkins-and-walker-to-sue-wisconsin-for-ballot-access/
David Wise
Wisconsin's congressional Dems call for Trump's ouster following Capitol violence
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,115
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.PHONY: clean-pyc clean-build docs help
.DEFAULT_GOAL := help
define BROWSER_PYSCRIPT
import os, webbrowser, sys
try:
from urllib import pathname2url
except:
from urllib.request import pathname2url
webbrowser.open("file://" + pathname2url(os.path.abspath(sys.argv[1])))
endef
export BROWSER_PYSCRIPT
BROWSER := python -c "$$BROWSER_PYSCRIPT"
help:
@perl -nle'print $& if m{^[a-zA-Z_-]+:.*?## .*$$}' $(MAKEFILE_LIST) | sort | awk 'BEGIN {FS = ":.*?## "}; {printf "\033[36m%-25s\033[0m %s\n", $$1, $$2}'
clean: clean-build clean-pyc clean-output
clean-output:
rm -f output/*
clean-build: ## remove build artifacts
rm -fr build/
rm -fr dist/
rm -fr *.egg-info
clean-pyc: ## remove Python file artifacts
find . -name '*.pyc' -exec rm -f {} +
find . -name '*.pyo' -exec rm -f {} +
find . -name '*~' -exec rm -f {} +
lint: ## check style with flake8
flake8 django_test_tools tests
test: clean-output ## run tests quickly with the default Python
python runtests.py tests
test-all: ## run tests on every Python version with tox
tox
coverage: ## check code coverage quickly with the default Python
coverage run --source django_test_tools runtests.py tests
coverage report -m
coverage html
open htmlcov/index.html
docs: ## generate Sphinx HTML documentation, including API docs
rm -f docs/django-test-tools.rst
rm -f docs/modules.rst
sphinx-apidoc -o docs/ django_test_tools
$(MAKE) -C docs clean
$(MAKE) -C docs html
$(BROWSER) docs/_build/html/index.html
sdist: clean test
python setup.py sdist
python setup.py sdist bdist_wheel
ls -l dist
patch: clean ## package and upload a release
python ./scripts/bump.py --action=patch
minor: clean ## package and upload a release
python ./scripts/bump.py --action=minor
upload: sdist
git push origin master
git push origin develop
git push --tags
twine upload ./dist/*
travis-push: clean
python setup.py sdist
python setup.py sdist bdist_wheel
ls -l dist
git push --tags
git push origin master
git push origin develop
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,604
|
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