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

Question: What was the best performing model for the Spanish language in Track-1?

Read the following text and answer briefly.

Context 1: Paper Info

Title: Two-stage Pipeline for Multilingual Dialect Detection
Publish Date: Unkown
Author List: Ankit Vaidya (from Pune Institute of Computer Technology), Aditya Kane (from Pune Institute of Computer Technology)

Figure

Figure 1: Class distribution of dialects
Figure 2: System diagram for dialect classification The LID classifies the input into one of 3 languages The sample is then further classified into dialects by language specific models 
Figure 3: Confusion matrix of 9-way classification Note that rows are normalized according to the number of samples is that class 
Our complete results for Track-1 using the two-stage dialect detection pipeline Model-* denotes the language of the models used for the experiments 
Performance on Track-1 validation dataset of individual models used in the two-stage pipeline "Lg" stands for language of the model used 
Comparative results of two-way classification using the finetuned (F T ) predictions and predictions adapted from three-way classification models 

abstract

Dialect Identification is a crucial task for localizing various Large Language Models  This paper outlines our approach to the VarDial 2023 DSL-TL shared task  Here we have to identify three or two dialects from three languages each which results in a 9-way classification for Track-1 and 6-way classification for Track-2 respectively 
Our proposed approach consists of a two-stage system and outperforms other participants' systems and previous works in this domain  We achieve a score of 58 54% for Track-1 and 85 61% for Track-2  Our codebase is available publicly 1  

Introduction

Language has been the primary mode of communication for humans since the pre-historic ages  Studies have explored the evolution of language and outlined mathematical models that govern the intricacies of natural language   Inevitably, as humans established civilization in various parts of the world, this language was modified by, and for the group of people occupied by that particular geographical region 
This gave rise to multiple national dialects of the same language  The VarDial workshop (colocated with EACL 2023) explores various dialects and variations of the same language  We participated in the Discriminating Between Similar Languages -True Labels (DSL-TL) shared task  In this task, the participants were provided with data from three languages, with each language having three varieties 
This shared task consisted of two tracks -Track-1 featuring nine-way classification and Track-2 featuring six-way classification  The second track included two particular national dialects of each language (eg  American English and British English), and the first track had one general  We ranked 1 st in both of the tracks 
Moreover, we beat the next best submission by a margin of 4 5% in the first task and 5 6% in the second task We were the only team to surpass the organizer baseline scores  We present our winning solution in this paper  We used an end-to-end deep learning pipeline which consisted of a language identification model and three language-specific models, one for each language 
We converged upon the best combination by doing an elaborate analysis of various models available  Furthermore, in this work we also analyze the performance of the pipeline as a whole and also provide an ablation study  Lastly, we provide some future directions in this area of research 

Related Work

The present literature encompasses various aspects of dialect identification  We study this from three perspectives: large language models, language identification and dialect classification problems 

Large Language Models

The success of transformers and BERT based models was inevitable since the initial boom of the transformer  2017) model  In recent years, many other architectures like RoBERTa and ELECTRA have further pushed the state-of-the-art in this domain  Moreover, autoregressive models like GPT and GPT-2 have also shown their prowess 
Multilingual versions of RoBERTA, namely XLM-RoBERTa are also available  Lastly, language specific models like Spanish BERT (la Rosa y Eduardo G  Ponferrada y Manu Romero y Paulo Villegas y Pablo González de Prado Salas y María Grandury, 2022) and Portuguese BERT are available as well  Our winning solution makes use of these large language models trained on specific languages 

Language Identification Models

Many multilingual language identification models have been developed in order to classify the language of the input sentence beforehand  Even though the initial works used n-gram models and generative mixture models or even conditional random fields and other classical machine learning methods like naive bayes , modern methods have shifted to the use of deep learning for language identification  
Recent works have mainly focused on deep learning based language identification, where handling codemixed data is a big challenge in the domain  For our experiments, we use a version of XLM-RoBERTa finetuned on a language identification dataset 2   This model has near-perfect test accuracy of 99 6% 

Dialect Classification

Dialect classification has been previously solved using statistical methods like Gaussian Mixture Models and Frame Selection Decoding or Support Vector Machines (SVM)   It has been explored relatively sparsely, mostly in the case for local languages   Deep learning approaches have been explored in previous editions of the VarDial workshop shared tasks and otherwise  
Dialect classification was also explored previously as a part of other shared tasks   We want to stress that given the multilingual nature of the dataset, using the present methods directly was not an option  In our work, although we take inspiration from the previous works, we propose a novel system that surpasses the performance of the previous systems by a large margin 

Data

The dataset  We observed that the class PT-BR had the most number of samples (2,724) and the class EN had the least number of samples (349), and thus the imbalance ratio was almost 1:8  We have illustrated the data distribution in Figure   We tried to mitigate this imbalance using over-sampling and weighted sampling methods 
However, the improved data sampling method did not affect the performance 

System Description

This was a problem of multi-class classification having 9 classes for Track-1 and 6 classes for Track-2  The samples were belonging to 3 languages having 3 varieties each, so the classification pipeline was made in 2 stages  The Language Identification (LID) model which is the first stage classifies the sentence into 3 languages: English (EN), Spanish (ES) and Portuguese (PT) 
The LID is a pretrained XLM-RoBERTa that is fine-tuned for the task of language identification  It is able to classify the input sentence into 20 languages  We classify and separate the samples according to their language  The samples corresponding to the specific languages are then fed into the language specific models for dialect identification 
For dialect identification we have used models like BERT and RoBERTa with a linear layer connected to the pooler output of the models  Then fine-tuning is done on the models for dialect identification   using the samples corresponding to the specific languages  For the task of dialect identification we experimented with several pretrained models like XLM-RoBERTa, BERT, ELECTRA, GPT-2 and RoBERTa 
All models were fine-tuned for 20 epochs with a learning rate of 1e-6 and weight decay 1e-6 with a batch size of 8  The best performing model checkpoint was chosen according to the epoch-wise validation macro-F1 score  5 Experiments and Results

Experiments using Large Language Models

For the task of Dialect Identification we have tried various language specific models like XLM-RoBERTa, BERT, ELECTRA, RoBERTa and GPT- 2  The base variant of all these models were used and all the models were used through the Hugging-Face library  The pooler output of these models was passed through a linear layer and the models were fine-tuned 
First, we experimented with different models for Track-1  All the models were trained for 20 epochs with learning rate 1e-6, weight decay 1e-6 and a batch size of 8  We used XLM-RoBERTa as the baseline for all 3 languages  The best performing models for the English language were RoBERTa and BERT whereas GPT-2 was the worst performing 
Similarly the language specific versions of RoBERTa and BERT performed well for the Spanish and Portuguese respectively  Overall the worst performing model was GPT-2 across all 3 languages  The validation F1 scores are present in Table   The two best-performing models for every language were chosen for Track-2 
The same procedure as specified above was used and the F1 scores are present in Table   The train and validation F1 scores for 2-class classification are higher for all models as compared to the F1 score of the same models for 3-class classification  This was mainly due to the poor representation and accuracy of classification of the third class 
We observed symptoms of overfitting in all models after 12-15 epochs and the best validation F1 score was obtained in the range of 4-8 epochs 

LID experiments

The pipeline for dialect identification is divided into two parts as the sentences in the dataset belong to different languages  The stages are described in Section 4  The XLM-RoBERTa we have used for language classification has a test accuracy of 99 6% meaning it correctly classifies all input sentences and hence, can be considered as a perfect classifier 
For the final pipeline we experimented using the two best performing models for each language in Track-1 and Track-2  For both the tracks we experimented with all 8 (2 3 ) possible combinations of models and calculated the validation F1 score for the combined validation dataset which had sentences belonging to all languages 
The validation scores for Track-1 and Track-2 are shown in Table and Table respectively  For both the tracks, the three pipelines with the best validation F1 scores were chosen for submission 

Using 3-way classifier as a 2-way classifier

In Track-1, participants are expected to train a classifier which classifies amongst 9 classes, and in Track-2, participants are expected to train a classifier which classifies amongst 6 classes  These 6 classes are a proper subset of the 9 classes from Track-1  Thus, an intuitive baseline for Track-2 is to use the model finetuned for Track-1, whilst considering only the relevant classes for the latter task 
The classes EN , ES and P T , i e  the classes without any national dialect associated with them are not included in Track-2 as compared to Track-1  Thus, we calculate the predictions for the Track-2 validation dataset using the models for Track-1 and exclude the metrics for Track-1 specific classes to get the metrics for this "adapted" 2-way classification 
We show the results of this experiment in Table and observe that, as expected, the adapted 2-way classification performs worse compared to the explicitly finetuned variant 

Results for Track-1 and Track-2

We now present our experiments and their performance for both tracks  Our experiments for Track-1 are described in Table and our experiments for Track-2 are described in Table   The participants were allowed three submissions for evaluation on the test set, so we submitted predictions using the three systems which performed the best on the validation set 
As mentioned in Section 5 2, we performed 2 3 , i e  a total of 8 experiments using the two best models for each language  We observed that RoBERTa base on English, Spanish BERT base on Spanish and Portuguese BERT base performed the best on the testing set for Track-1  The same combination, with RoBERTa base for English, worked best for Track-2 
All of our submissions were the top submissions for each track, which surpassed the next best competitors by a margin of 4 5% and 5 6% for Track-1 and Track-2 respectively 

Ablation of best submissions

We hereby make some observations of our submissions and other experiments  To assist this, we plot the confusion matrices of our best submissions for Track-1 and Track-2 in Figures respectively  Note that these confusion matrices have their rows (i e  true labels axes) normalized according to the number of samples in the class 
Here are observations from our experiments: 1  BERT-based models outperform other models across all languages: We observe that BERT-based models outperform ELECTRA-based and GPT-2-based models, as shown in Table   We speculate this is because of the inherent architecture of BERT, which combines semantic learning with knowledge retention 
This combination of traits is particularly useful for this task  2  Common labels perform the worst across all languages: We observe that the common labels EN , ES and P T perform the worst, both in the individual as well as the two-stage setup  We hypothesize this is because of the absence of dialect specific words, or words that are specific to the geographical origin of the national dialect (for example, "Yankees" for EN-US and "Oxford" for EN-GB) 
3  English models work better than models of other languages: It can be noted from Figures 4 and 3 that the English models have the best performance across all classes  This can be attributed to two reasons: absence of national dialect specific words and lesser pretraining data in the case of Portuguese 
4  British English is most correctly classified class: We can observe that the Spanish or Portuguese models make equal number of mistakes in the case of either national dialect, in the case of Track-2 (see Figure )  However, in the case of English, the label EN − GB is correctly classified for more than 95% of the cases 
We speculate this is because British English involves slightly distinctive grammar and semantics, which help the model separate it from other classes  5  The proposed 2-step method is scalable for multiple language dialect classification: We can strongly assert that the novel 2-step deep learning method for multilingual dialect classification is a scalable method for the task due to two specific reasons: firstly, the multilingual models (like XLM-RoBERTa) might not have the vocabulary as well as the learning capabilities to learn the minute differences between individual dialects 
Secondly, this system can be quickly expanded for a new language by simply adding a language specific dialect classifier, provided the language identification model supports that particular language 

Conclusion

In this paper we propose a two-stage classification pipeline for dialect identification for multilingual corpora  We conduct thorough ablations on this setup and provide valuable insights  We foresee multiple future directions for this work  The first is to expand this work to many languages and dialects 
Secondly, it is a worthwhile research direction to distill this multi-model setup into a single model with multiple prediction heads  The obvious limitation of this system is the excessive memory consumption due to the usage of language specific models  For low resource languages this system is difficult to train and scale 
We hope that these problems will be addressed by researchers in future works 



ground_truth: The best performing model for the Spanish language in Track-1 was Spanish BERT.

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

Question: What was the Buckeyes' record in their first game of the season?

Read the following text and answer briefly.

Context 2: The 1951 Ohio State Buckeyes baseball team represented the Ohio State University in the 1951 NCAA baseball season  The head coach was Marty Karow, serving his 1st year 

The Buckeyes lost in the College World Series, defeated by the Texas A&M Aggies 

Roster

Schedule 

! style="" | Regular Season
|- valign="top" 

|- align="center" bgcolor="#ccffcc"
| 1 || March 16 || at  || Unknown • San Antonio, Texas || 15–3 || 1–0 || 0–0
|- align="center" bgcolor="#ffcccc"
| 2 || March 17 || at B  A  M  C  || Unknown • San Antonio, Texas || 7–8 || 1–1 || 0–0
|- align="center" bgcolor="#ffcccc"
| 3 || March 19 || at  || Clark Field • Austin, Texas || 0–8 || 1–2 || 0–0
|- align="center" bgcolor="#ffcccc"
| 4 || March 20 || at Texas || Clark Field • Austin, Texas || 3–4 || 1–3 || 0–0
|- align="center" bgcolor="#ccffcc"
| 5 || March 21 || at  || Unknown • Houston, Texas || 14–6 || 2–3 || 0–0
|- align="center" bgcolor="#ffcccc"
| 6 || March 22 || at Rice || Unknown • Houston, Texas || 2–3 || 2–4 || 0–0
|- align="center" bgcolor="#ccffcc"
| 7 || March 23 || at  || Unknown • Fort Worth, Texas || 4–2 || 3–4 || 0–0
|- align="center" bgcolor="#ccffcc"
| 8 || March 24 || at TCU || Unknown • Fort Worth, Texas || 7–3 || 4–4 || 0–0
|- align="center" bgcolor="#ccffcc"
| 9 || March 24 || at  || Unknown • St  Louis, Missouri || 10–4 || 5–4 || 0–0
|-

|- align="center" bgcolor="#ccffcc"
| 10 || April 6 ||  || Varsity Diamond • Columbus, Ohio || 2–0 || 6–4 || 0–0
|- align="center" bgcolor="#ccffcc"
| 11 || April 7 ||  || Varsity Diamond • Columbus, Ohio || 15–1 || 7–4 || 0–0
|- align="center" bgcolor="#ffcccc"
| 12 || April 14 ||  || Varsity Diamond • Columbus, Ohio || 0–1 || 7–5 || 0–0
|- align="center" bgcolor="#ccffcc"
| 13 || April 20 ||  || Varsity Diamond • Columbus, Ohio || 10–9 || 8–5 || 1–0
|- align="center" bgcolor="#ccffcc"
| 14 || April 21 || Minnesota || Varsity Diamond • Columbus, Ohio || 7–0 || 9–5 || 2–0
|- align="center" bgcolor="#ffcccc"
| 15 || April 24 || at  || Unknown • Oxford, Ohio || 3–4 || 9–6 || 2–0
|- align="center" bgcolor="#ffcccc"
| 16 || April 27 || at  || Hyames Field • Kalamazoo, Michigan || 2–3 || 9–7 || 2–0
|- align="center" bgcolor="#ffcccc"
| 17 || April 28 || at Western Michigan || Hyames Field • Kalamazoo, Michigan || 5–7 || 9–8 || 2–0
|-

|- align="center" bgcolor="#ccffcc"
| 18 || May 1 || at  || Unknown • Athens, Ohio || 7–6 || 10–8 || 2–0
|- align="center" bgcolor="#ccffcc"
| 19 || May 4 ||  || Varsity Diamond • Columbus, Ohio || 12–6 || 11–8 || 3–0
|- align="center" bgcolor="#ccffcc"
| 20 || May 5 || Purdue || Varsity Diamond • Columbus, Ohio || 14–4 || 12–8 || 4–0
|- align="center" bgcolor="#ffcccc"
| 21 || May 8 ||  || Varsity Diamond • Columbus, Ohio || 6–8 || 12–9 || 4–0
|- align="center" bgcolor="#ccffcc"
| 22 || May 9 || at Dayton || Unknown • Dayton, Ohio || 11–2 || 13–9 || 4–0
|- align="center" bgcolor="#ccffcc"
| 23 || May 12 ||  || Varsity Diamond • Columbus, Ohio || 6–5 || 14–9 || 5–0
|- align="center" bgcolor="#ccffcc"
| 24 || May 12 || Indiana || Varsity Diamond • Columbus, Ohio || 5–2 || 15–9 || 6–0
|- align="center" bgcolor="#ccffcc"
| 25 || May 15 || Ohio || Varsity Diamond • Columbus, Ohio || 6–0 || 16–9 || 6–0
|- align="center" bgcolor="#ffcccc"
| 26 || May 18 || at  || Northwestern Park • Evanston, Illinois || 1–3 || 16–10 || 6–1
|- align="center" bgcolor="#ccffcc"
| 27 || May 19 || at Northwestern || Northwestern Park • Evanston, Illinois || 10–3 || 17–10 || 7–1
|- align="center" bgcolor="#ccffcc"
| 28 || May 22 || at Cincinnati || Carson Field • Cincinnati, Ohio || 8–4 || 18–10 || 7–1
|- align="center" bgcolor="#ccffcc"
| 29 || May 25 ||  || Varsity Diamond • Columbus, Ohio || 4–1 || 19–10 || 8–1
|- align="center" bgcolor="#ffcccc"
| 30 || May 25 || Michigan || Varsity Diamond • Columbus, Ohio || 3–6 || 19–11 || 8–2
|- align="center" bgcolor="#ffcccc"
| 31 || May 30 || Miami (OH) || Varsity Diamond • Columbus, Ohio || 3–4 || 19–12 || 8–2
|-

|- align="center" bgcolor="#ccffcc"
| 32 || June 1 || at  || Old College Field • East Lansing, Michigan || 8–0 || 20–12 || 9–2
|- align="center" bgcolor="#ccffcc"
| 33 || June 2 || at Michigan State || Old College Field • East Lansing, Michigan || 9–8 || 21–12 || 10–2
|-

|-
|-
! style="" | Postseason
|- valign="top"

|- align="center" bgcolor="#ccffcc"
| 34 || June 8 || Western Michigan || Varsity Diamond • Columbus, Ohio || 1–0 || 22–12 || 10–2
|- align="center" bgcolor="#ffcccc"
| 35 || June 8 || Western Michigan || Varsity Diamond • Columbus, Ohio || 2–4 || 22–13 || 10–2
|- align="center" bgcolor="#ccffcc"
| 36 || June 9 || Western Michigan || Varsity Diamond • Columbus, Ohio || 3–2 || 23–13 || 10–2
|-

|- align="center" bgcolor="#ffcccc"
| 37 || June 13 || Oklahoma || Omaha Municipal Stadium • Omaha, Nebraska || 8–9 || 23–14 || 10–2
|- align="center" bgcolor="#ffcccc"
| 38 || June 13 || Texas A&M || Omaha Municipal Stadium • Omaha, Nebraska || 2–3 || 23–15 || 10–2
|-

Awards and honors 
Dick Hauck
 First Team All-Big Ten

Stewart Hein
 First Team All-Big Ten

References 

Ohio State Buckeyes baseball seasons
Ohio State Buckeyes baseball
Big Ten Conference baseball champion seasons
Ohio State
College World Series seasons
Context 3: Paper Info

Title: An CUSUM Test with Observation-Adjusted Control Limits in Change Detection
Publish Date: March 9, 2023
Author List: Fuquan Tang (from Department of Statistics, Shanghai Jiao Tong University), Dong Han (from Department of Statistics, Shanghai Jiao Tong University)

Figure

exp{−cg(µ)(θ − x Hv (θ) + o(1))} for 1 ≤ k ≤ ac − 1, bc ≤ n ≤ m, where Zi = −g ′ (µ)(Z i − µ)/a and Hv (θ) = ln hv (θ) + ( ac k − 1) ln ĥv (θ), ĥv (θ) = E v (e θ Zi ) 
i < cg(µ)(1 + o(1))) exp{−cg(µ)θ * v (1 + o(1))} (A  5) for ac ≤ k ≤ bc − 1, bc ≤ n ≤ m,andP v (
i + g ′ (µ)a −1 Tc(g)−1 i=Tc(g)−ac (Z i − µ)] −→ µas c → ∞ By the uniform integrability of {T c (g)/c} and using Theorem A 1 1 in Gut's book(1988), we haveE v (T c (g)) = (1 + o(1)) cg(µ) µfor a large c This completes the proof of Theorem 2 Proof of Theorem 4  Since g(x) < 0 for x > a * , a * ≤ µ * and µ * ≥ 0, it follows thatP v m Ẑm < cg( Ẑm ), Ẑm > a * ≤ P v ( Ẑm < µ * )andP v (T c (g) > m) = P v n i=n−k+1 Z i < cg( Ẑn ), 1 ≤ k ≤ n, 1 ≤ n ≤ m ≤ P v m Ẑm < cg( Ẑm ) = P v m Ẑm < cg( Ẑm ), Ẑm ≤ a * + P v m Ẑm < cg( Ẑm ), Ẑm > a * ≤ 2P v ( Ẑm < µ * ) Furthermore,P v ( Ẑm < µ * ) = P v ( m i −Z i > −mµ * ) = P v ( m i (µ − Z i ) > m(µ − µ * )) = P v (e θ m i (µ−Z i ) > e θm(µ−µ * ) ) ≤ e −m[θ(µ−µ * )−ln M (θ)] ,whereM(θ) = E v (e θ(µ−Z 1 )) and the last inequality follows from Chebychev's inequality Note thath(θ) = θ(µ − µ * ) − ln M(θ) attains its maximum value h(θ * ) = θ * (µ − µ * ) − ln M(θ * ) > 0 at θ = θ * > 0, where h ′ (θ * ) = 0  So, E v (T c (g)) = 1 + ∞ m=1 P v (T c (g) > m) ≤ 1 + m=1 −m[θ * (µ−µ * )−ln M (θ * )] = e θ * (µ−µ * )−ln M (θ * ) + 1 e θ * (µ−µ * )−ln M (θ * ) − Let k > 1 It follows that E vk (T c (g) − k + 1) + = ∞ m=1 P vk (T c (g) > m + k − 1, T c (g) > k − 1) ≤ (a 0 + 1)(k − 1)P 0 (T c (g) > k − 1) + ∞ m≥(a 0 +1)(k−1) P vk (T c (g) > m + k − 1) Similarly, we haveP vk (T c (g) > m + k − 1) = P vk n i=n−k+1 Z i < cg( Ẑn ), 1 ≤ k ≤ n, 1 ≤ n ≤ m + k − 1 ≤ 2P vk ( Ẑm+k−1 < µ * ) − Z i ) > m(µ − µ * ) + (k − 1)(µ 0 − µ * ) ≤ 2 exp{−m θ * (µ − µ * ) − ln M(θ * ) + k − 1 m [µ 0 − µ * − ln M 0 (θ * )] } ≤ e −mb for m ≥ (a 0 + 1)(k − 1), since θ * (µ − µ * ) − ln M(θ * ) + k − 1 m [µ 0 − µ * − ln M 0 (θ * )] ≥ b for m ≥ (a 0 + 1)(k−1) Thus, E vk (T c (g) − k + 1) + ≤ (a 0 + 1)(k − 1)P 0 (T c (g) ≥ k) + m≥(a 0 +1)(k−1) e −mb ≤ (a 0 + 1)(k − 1)P 0 (T c (g) >≥ k) + 2e −(a 0 +1)(k−1)b 1 − e −b  
Simulation of E τ i ,v and J ACE for detecting two mean shifts v = 0 1, v = 1 The parameters for T * M are k1=1, k2=150, r 1 = 5 2 * 10 −5 , r 2 = 1 1 * 10 −5 , and the expectation and standard deviation in both cases are 1717 06with 13459 80 and 3918 33 with 16893 25,respectively 

abstract

In this paper, we not only propose an new optimal sequential test of sum of logarithmic likelihood ratio (SLR) but also present the CUSUM sequential test (control chart, stopping time) with the observation-adjusted control limits (CUSUM-OAL) for monitoring quickly and adaptively the change in distribution of a sequential observations 
Two limiting relationships between the optimal test and a series of the CUSUM-OAL tests are established  Moreover, we give the estimation of the in-control and the out-of-control average run lengths (ARLs) of the CUSUM-OAL test  The theoretical results are illustrated by numerical simulations in detecting mean shifts of the observations sequence 

INTRODUCTION

In order to quickly detect a change in distribution of observations sequence without exceeding a certain false alarm rate, a great variety of sequential tests have been proposed, developed and applied to various fields since proposed a control chart method, see, for example, , , One of popular used sequential tests is the following upper-sided CUSUM test which was proposed by  
where c > 0 is a constant control limit, Z i = log[p v 1 (X i )/p v 0 (X i )], p v 0 (x) and p v 1 (x) are prechange and post-change probability density functions respectively for a sequence of mutually independent observations {X i , i ≥ 1}, that is, there is a unknown change-point τ ≥ 1 such that X 1 ,    , X τ −1 have the probability density function p v 0 , whereas, X τ , X τ +1 ,     have the probability density function p v 1  
By the renewal property of the CUSUM test T C we have , where E 1 (T C ) is the out-of-control average run length (ARL 1 ), P k and E k denote the probability and expectation respectively when the change from p v 0 to p v 1 occurs at the change-point τ = k for k ≥ 1  Though we know that the CUSUM test is optimal under Lorden's measure (see Moustakides 1986 and Ritov 1990), the out-of-control ARL 1 of the CUSUM test is not small, especially in detecting small mean shifts ( see Table in Section 4) 
In other words, the CUSUM test is insensitive in detecting small mean shifts  Then, how to increase the sensitivity of the CUSUM test ? Note that the control limit in the CUSUM test is a constant c which does not depend on the observation samples  Intuitively, if the control limit of the CUSUM test can become low as the samples mean of the observation sequence increases, then the alarm time of detecting the increasing mean shifts will be greatly shortened 
Based on this idea, by selecting a decreasing function g(x) we may define the ( upper-sided ) CUSUM chart T C (cg) with the observation-adjusted control limits cg( Ẑn ) ( abbreviated to the CUSUM-OAL chart ) in the following where c > 0 is a constant and Ẑn = n i=1 Z i /n  In other words, the control limits cg( Ẑn ) of the CUSUM-OAL test can be adjusted adaptively according to the observation information { Ẑn } 
Note that the control limits cg( Ẑn ) may be negative  In the special case, the CUSUM-OAL chart T C (cg) becomes into the conventional CUSUM chart T C (c) in (1) when g ≡ 1  Similarly, we can define a down-sided CUSUM-OAL test  In this paper, we consider only the upper-sided CUSUM-OAL test since the properties of the down-sided CUSUM-OAL test can be obtained by the similar method 
The main purpose of the present paper is to show the good detection performance of the CUSUM-OAL test and to give the estimation of its the in-control and out-of-control ARLs  The paper is organized as follows  In Section 2, we first present an optimal SLR sequential test, then define two sequences of the CUSUM-OAL tests and prove that one of the two sequences of CUSUM-OAL tests converges to the optimal test, another sequences of CUSUM-OAL tests converges to a combination of the optimal test and the CUSUM test 
The estimation of the in-control and out-of-control ARLs of the CUSUM-OAL tests and their comparison are given in Section 3  The detection performances of the three CUSUM-OAL tests and the conventional CUSUM test are illustrated in Section 4 by comparing their numerical out-ofcontrol ARLs  Section 5 provides some concluding remarks 
Proofs of the theorems are given in the Appendix 

AN OPTIMAL SLR TEST, TWO CUSUM-OAL TESTS AND THEIR LIMITING RELATIONSHIPS

Let P 0 and E 0 denote the probability and the expectation respectively with the probability density p v 0 when there is no change for all the time  It is known that It follows from Proposition 2 38 in and (5 8)-(5 9) in Chow et al, P 108) that the following sequence test of sum of logarithmic likelihood ratio (SLR)
for B > 1, is optimal in the following sense min for P 0 (T SLR < ∞) = α, where c = log B and 0 < α < 1  In particular, if P 0 is the standard normal distribution with mean shift µ > 0 after changepoint, we have Z j − µ 0 = µX j , where µ 0 = −µ 2 /2  It follows from proposition 4 in that the SLR test T SLR in (4) is also optimal (minimal ARL 1 ) with the same false alarm probability P 0 (T < τ ) 
It can be seen that the in-control average run length of T SLR is infinite, that is, ARL 0 = E 0 (T SLR ) = ∞  However, the minimal ARL 1 with finite ARL 0 is a widely used optimality criterion in statistical quality control (see ) and detection of abrupt changes (see   In order to get finite ARL 0 for T SLR , we replace the constant control limit c of T SLR in (3) or (4) with the dynamic control limit n(µ 0 − r) and obtain a modified SLR test T SLR (r) in the following
for r ≥ 0  For comparison, the in-control ARL 0 of all candidate sequential tests are constrained to be equal to the same desired level of type I error, the test with the lowest out-of-control ARL v has the highest power or the fastest monitoring (detection) speed  In the following example 1, the numerical simulations of the out-of-control ARLs of the CUSUM-OAL tests T C (cg u,0 ) in detecting the mean shifts of observations with normal distribution will be compared with that of the SLR tests T * (r) and T * (0), and that of the CUSUM-SLR test T C (c) ∧ T * (0) := min{T C (c), T * (0)} in the following Table  
These comparisons lead us to guess that there are some limiting relationships between T C (cg u,r ) and T * (r), and T C (c g u ) and T C (c) ∧ T * (0), respectively  Example 1  Let X 1 , X 2 ,      be mutually independent following the normal distribution N(0, 1) if there is no change  After the change-point τ = 1, the mean E µ (X k ) ( k ≥ 1 ) will change from v 0 = 0 to v = 0 1, 0 25, 0 5, 0 75, 1, 1 25, 1 5, 3  Here, we let
, where v 1 = 1 is a given reference value which for the CUSUM test is the magnitude of a shift in the process mean to be detected quickly  We conducted the numerical simulation based on 1,000,000 repetitions  The following Table lists the simulation results of the ARLs of the tests T C (c), T C (c g u ) for u = 1, 10, 10 2 , 10 3 , 10 4 , T * (0 0007), T C (c) ∧ T * (0) and T * (0) for detecting the mean shifts, where the mean shift 0 0 means that there is no change which corresponds to the in-control ARL 0 and all tests have the common ARL 0 ≈ 1000 except the test T * (0) which has ARL 0 = ∞ 
The values in the parameters are the standard deviations of the tests  From the last row in Table , it's a little surprising that though the ARL 0 of T * (0) is infinite, that is, E 0 (T * (0)) = ∞, the detection speed of T * (0) is faster than that of the CUSUM chart T C for all mean shifts, in particular, for detecting the small mean shift 0 1, the speed of T * (0) is only 7 47 which is very faster than the speed, 439, of the CUSUM test 
Moreover, both control charts T * (0 0007) and T C (11 9271) ∧ T * (0) not only have the nearly same detection performance as T * (0) but also can have the finite in-control ARL 0   Note particularly that when the number u in g u is taken from 0 to 1, 10, 10 2 , 10 3 , 10 4 , the detection speed of T C (c g u ) is getting faster and faster, approaching to that of T C (c) ∧ T * (0) 
This inspires us to prove the following theoretic results  Let τ = 1 and {X k , k ≥ 1} be an i i d  observations sequence with Theorem 2 shows that when the constant control limit c of the CUSUM test T C (c) is replaced with the observation-adjusted control limits {cg u,r ( Ẑn )} and {c g u ( Ẑn )} respectively, the corresponding two CUSUM-OAL tests {T C (cg u,r )} and {T C (c g u )} will converge to the optimal SLR test T * (r) and the CUSUM-SLR test T C (c) ∧ T * (0) as u → ∞, respectively 
In other words, the fastest alarm times that {T C (cg u,r )} and {T C (c g u )} can be reached are T * (r) and T C (c) ∧ T * (0), respectively  u ≥ 0} can be seen as two "long bridges" connecting T C (c) and T * (r), and T C (c) and T C (c) ∧ T * (0), respectively 

ESTIMATION AND COMPARISON OF ARL OF THE CUSUM-OAL TEST

In this section we will give an estimation of the ARLs of the following CUSUM-OAL test that can be written as where g( ) is a decreasing function, Ẑn (ac x] denotes the smallest integer greater than or equal to x  Here Ẑn (ac) is a sliding average of the statistics, Next we discuss on the the post-change probability distribution in order to estimate the ARLs of T C (cg) 
Usually we rarely know the post-change probability distribution P v of the observation process before it is detected  But the possible change domain and its boundary (including the size and form of the boundary) about v may be determined by engineering knowledge, practical experience or statistical data 
So we may assume that the region of parameter space V and a probability distribution Q on V are known  If we have no prior knowledge of the possible value of v after the change time τ , we may assume that v occurs equally on V , that is, the probability distribution Q is an equal probability distribution (or uniform distribution ) on V  
For example, let P v be the normal distribution and v = (µ, σ), where µ and σ denote the mean and standard deviation respectively, we can take the set V = {(µ, σ) : and Q is subject to the uniform distribution U(V ) on V if v occurs equally on V , where the numbers µ 1 , µ 2 , σ 1 and σ 2 are known  It means that we know the domain of the possible post-change distributions, P v , v ∈ V , i e , the boundary ∂V of the parameter space V is known 
Next we shall divide the parameter space V into three subsets V + , V 0 and V − by the Kullback-Leibler information distance  Let and are two Kullblak-Leibler information distances between P v , P v 0 and P v , P v 1   Since I(p|q) = 0 if and only if p = q, where p and q are two probability measures, it follows that
, it means that P v is closer to P v 0 than to P v 1 according to the Kullblak-Leibler information distance  There is a similar explanation for v ∈ V + or ∈ V 0   Suppose the post-change distribution P v and the function g(x) satisfy the following conditions: (I) The probability P v is not a point mass at E v (Z 1 ) and P v (Z 1 > 0) > 0 
(II) The moment-generating function h v (θ) = E v (e θZ 1 ) satisfies h v (θ) < ∞ for some θ > 0  (III) The function g(x) is decreasing, its second order derivative function g ′′ (x) is continuous and bounded, and there is a positive number x * such that g(x * ) = 0  ) and and therefore, Θ ′ (θ(u)) = −H(θ(u)) = −H(θ * v ) = 0, Θ ′ (θ(1/x)) > 0 for x > 1/u and Θ ′ (θ(1/x)) < 0 for x > 1/u 
Hence, there exists a positive number b defined in (??)  It can be seen, the main part of ARL v (T c (g)) will be an exponential function, square function, and linear function of c when the process {Z k : k ≥ 0} has no change or a "small change", a "medium change" and a "large change" from P v 0 to P v , respectively 
Here, the "small change" (v ∈ V − ) means that P v is closer to P v 0 than to P v 1 , i e , I(P v |P v 0 ) < I(P v |P v 1 ), and the "large change" is just the opposite  The "medium change" (v ∈ V 0 ) corresponds to In this paper, we will use another method to prove Theorem 3 since Wald's identity and the martingale method do not hold or can not work for showing the ARLs estimation of the test T c (g) when g is not constant 
Next we compare the detection performance of the CUSUM-OAL test (ARL v (T c ′ (g))) with that of the CUSUM test (ARL v (T C (c))) by using (??) in Theorem 4 1  ) when µ 0 < µ < 0 and for θ * v 0 > g(µ)/g(µ 0 ) when µ ≥ 0  This means that ARL v (T c (g)) can be smaller than ARL v (T C (c)) as long as g(µ)/g(µ 0 ) is small for all µ > µ 0  

NUMERICAL SIMULATION AND A REAL EX-AMPLE ILLUSTRATION

4 1 Numerical Simulation of ARLs for τ ≥ 1 By the simulation results of ARLs in Table , we see that the detection performance of T * (r), T C (c)∧T * (0), T * (0) and T C (c g u ) for large u is much better than that of the conventional CUSUM test T C for τ = 1  The following Table illustrates the simulation values of E τ i ,v and J ACE of nine tests in detecting two mean shifts v = 0 1 and v = 1 after six change-points, τ i , 1 ≤ i ≤ 6 with ARL 0 (T ) = E 0 (T ) ≈ 500 
Note that H v (θ) is a convex function and H ′ v (0) = µ < 0  This means that there is a unique positive number   It follows from (A 9) that for a large c  Taking θ ց θ * v and u ′ ց u, we have for a large c  Thus, by (A 11) we have as c → ∞  By the properties of exponential distribution, we have for a large c 
To prove the downward inequality of (A 10), let where b is defined in (??) and without loss of generality, we assume that b > a  Obviously, Let k = xcg(µ)  By Chebyshev's inequality, we have Since Hv (θ) and H v (θ) are two convex functions and Let m = tcg(µ)θ * v /bc for t > 0  By (A 13), (A 14), (A 15) and Theorem 5 1 in Esary, Proschan and Walkup (1967) we have
Finally, as c → +∞, where θ 0 > 0 satisfies h v (θ 0 ) = 1  Thus as c → ∞  This implies that for a large c  This completes the proof of (A 10)  Let v ∈ V 0   Let m 1 = (cg(0)) 2 /σ 2   It follows that Note that for a large c, where A = |g ′ (0)|/a, and , where Φ( ) is the standard normal distribution  Let m 2 = (cg(0)) 2 /(8σ 2 ln c) 
Note that as c → ∞, where the third inequality comes from Theorem 5 1 in Esary, Proschan and Walkup (1967)  Thus, we have Let v ∈ V + and let The uniform integrability of {T c (g)/c} for c ≥ 1, follows from the well-known uniform integrability of {T 0 /c} (see Gut (1988)) 



ground_truth: They won their first game with a score of 15-3.

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

Question: What are the three subsets into which the parameter space V is divided?

Read the following text and answer briefly.

Context 4: 51$ ($b=1 50$) for $U'=0$ ($U'=U/10$),
correspondingly; see \figs{G-T}(b-c) 
Moreover, as follows from \eq{Jeff}, $U'$ reduces CAR exchange, and therefore diminishes $T^*$ 
For the same values of $J^{\mathrm{eff}}$, the dependence of $G(T)$ for $t=0$ and $\GS{}>0$ is hardly different 
from the one for $\GS{}=0$ and $t>0$ for $T\geq T^*$ (results not shown) 
However, $G(T)$ saturates at residual value $G_{\mathrm{min}}$ as $T\to 0$ only for finite
$\GS{}$, which at particle-hole symmetry makes $G_{\mathrm{min}}$
the hallmark of SC proximity and the corresponding CAR exchange processes 
From numerical results, one can estimate it as
\begin{equation}
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{}^2}{U^2} 
	\qquad {\scriptstyle (\GS{1}=\GS{2}=\GS{})} ,
\label{Gmin}
\end{equation}
with $c\approx 2 25$, barely depending on $U'$ and getting smaller for $t>0$  
This is illustrated in \figs{G-T}(d-e), where the dotted line corresponds to \eq{Gmin} with $c=2 25$  

Lastly, in \fig{G-T}(a) we also present the curves obtained for $t=\GS{}$ chosen such, 
that the quantity $\xi=\sqrt{t^2+\GS{}^2}$ remains the same 
in all the cases 
This is to illustrate what happens when \emph{both} (direct and CAR) exchange interactions are
present  \fig{G-T}(c) clearly shows that $T^*$ remains practically unaltered for $U'=0$ 
The comparison with \fig{G-T}(b) proves that in this case it practically does not depend 
on $U'$  The enhancement of direct exchange is compensated by the decrease of the CAR one  
On the contrary, $G_{\mathrm{min}}$ decreases for larger $t$ below the estimation given by Eq ~(\ref{Gmin}), 
as can be seen in \figs{G-T}(d-e)  

While analyzing the results concerning $G_{\mathrm{min}}(\GS{})$ plotted in \figs{G-T}(d-e) 
one needs to keep in mind that $G_{\mathrm{min}}$ is obtained at deeply cryogenic conditions  To illustrate
this better, $G(\GS{})$ obtained for $t=0$ and $T=10^{-6}U$ is plotted with solid line 
in \fig{3}  Clearly, for weak $\GS{}$ the system exhibits rather conventional (single-stage)
Kondo effect with $G=G_{\mathrm{max}}\approx 2e^2/h$, while QD2 is effectively decoupled ($G_{\mathrm{max}}<2e^2/h$
in the proximity of SC lead \cite{KWIW})  Only for larger values of $\GS{}$
the CAR exchange is strong enough, such that $T^*>T$ and the dependence $G(\GS{})$ continuously 
approaches the $T=0$ limit estimated by \eq{Gmin} and presented in \figs{G-T}(d-e) 

\section{CAR-RKKY competition}
\label{sec:RKKY}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig3 pdf}
\caption{Linear conductance $G$ vs  $\GS{}$ calculated
		 for $t=0$, $\Gamma=U/5$, $U'=U/10$, finite $T=10^{-6}U$
		 and different values of RKKY coupling $J$, as indicated  
		 Inset shows QD1 spectral function $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0 1U$ curve, indicated with corresponding symbols 
		}
\label{fig:3}
\end{figure}

Let us now discuss the effects introduced by the conventional RKKY interaction 
We choose $t=0$ for the sake of simplicity and
analyze a wide range of $\GS{}$, starting from the case of anti-ferromagnetic 
RKKY interaction ($J>0$)  Large $J>0$ leads to the formation of a molecular singlet in the 
nanostructure  This suppresses the conductance, unless $\GS{}$ becomes of the order of $U/2$, 
when the excited states of DQD are all close to the ground state  This is illustrated 
by double-dotted line in \fig{3} 
Smaller value of $J>0$ causes less dramatic consequences, namely it just increases $J^{\mathrm{eff}}$ according
to \eq{Jeff}, leading to enhancement of $T^*$, cf  \eq{Tstar}  This is presented with
dot-dashed line in \fig{3} 

The situation changes qualitatively for ferromagnetic RKKY coupling, $J<0$ 
Then, RKKY exchange and CAR exchange have opposite signs and compete with each other 
Depending on their magnitudes and temperature, one
of the following scenarios may happen 

For $J^{\mathrm{eff}} > 0$, {\it i e } large enough $\GS{}$, and $T<T^*$, the system is in the 
singlet state due to the two-stage Kondo screening of DQD spins  $G(T\!=\!0)$ is reduced 
to $G_{\mathrm{min}}$, which tends to increase for large negative $J$; see dashed lines in \fig{3}  
In the inset to \fig{3}, the spectral density of QD1 representative for this regime is plotted 
as curve indicated by triangle  It corresponds to a point on the $J=-0 1U$ curve in the main 
plot, also indicated by triangle  The dip in $\mathcal{A}(\omega)$ has width of order of $T^*$ 

For finite $T$, there is always a range of sufficiently small $|J^{\mathrm{eff}}|$, where QD2 becomes effectively
decoupled, and, provided $T<T_K$, $G$ reaches $G_{\mathrm{max}}$ due to conventional Kondo effect 
at QD1  This is the case for sufficiently small $\GS{}$ for $J=0$ or $J=-0 01U$, and in the narrow
range of $\GS{}$ around the point indicated by a circle in \fig{3} for $J=-0 1U$ (for $J=0 01U$, 
the considered $T$ is close to $T^*$ and $G$ does not reach $G_{\rm max}$)  The conventional Kondo effect manifests itself with 
a characteristic peak in $\mathcal{A}(\omega)$, as illustrated in the inset in \fig{3} with line denoted by circle 

Finally, large enough $J^{\mathrm{eff}} < 0$ and low $T$, give rise to an effective ferromagnetic coupling of DQDs 
spins into triplet state  Consequently, the underscreened Kondo effect occurs 
\cite{Mattis,NozieresBlandin} for weak $\GS{}$ and, {\it e g }, $J=-0 1U$; 
see the point indicated by square in \fig{3} 
This leads to $G=G_{\mathrm{max}}$ and a peak in $\mathcal{A}(\omega)$, whose shape is significantly different from the
Kondo peak, cf  the curve denoted by square in the inset in \fig{3}  



\section{Effects of detuning from the particle-hole symmetry point}
\label{sec:asym}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig4 pdf}
\caption{
         (a) Linear conductance between the normal leads $G$ as a function of temperature $T$
         for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, and additional curves for finite 
         detuning from particle-hole symmetry point, $\delta_1=-\delta_2$, 
         and two values of $\xi=\sqrt{t^2+\GS{}^2}$, as indicated in the figure 
         (b) $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of QD1 detuning $\delta_1$ for different
         exchange mechanisms, $\xi=U/10$ and $\delta_2=\pm\delta_1$ (as indicated) 
		}
\label{fig:asym}
\end{figure}

At PHS $G_{\mathrm{min}}=G(T \!=\! 0)=0$ in the absence of superconducting lead, making $G_{\mathrm{min}} > 0$ a hallmark
of SC-induced two-stage Kondo effect  However, outside of PHS point $G_{\mathrm{min}} > 0$ even in the case of 
the two-stage Kondo effect caused by the direct exchange  
Exact PHS conditions are hardly possible in real systems, and the fine-tuning of the QD energy
levels to PHS point is limited to some finite accuracy 
Therefore, there may appear a question, if the results obtained at PHS are of any importance for the
realistic setups  As we show below --- they are,
in a reasonable range of detunings $\delta_i=\varepsilon_i +U/2$ 

In \fig{asym}(a) we present the $G(T)$ dependence in and outside the PHS, corresponding to 
parameters of \fig{G-T}(a)  
Clearly, for considered small values of $\delta_1=\delta_2=\delta$, 
$G_{\mathrm{min}}<10^{-3}e^2/h$ for direct exchange only, while $G_{\mathrm{min}}$ in the presence of a superconductor is 
significantly increased and close to the PHS value  Furthermore, for $|\delta_1| \sim |\delta_2| 
\sim \delta$, the residual conductance caused by the lack of PHS, $G_{\mathrm{min}} \approx e^2/h \cdot (\delta/U)^2$,
which is a rapidly decreasing function in the vicinity of PHS point, as illustrated in \fig{asym}(b)
with lines denoted by a square  Evidently, in the regime $|\delta_i| < 0 01U$ the residual conductance
caused by SC is orders of magnitude larger, leading to the plateau in $G_{\mathrm{min}}(\delta_1)$ dependence,
visible in \fig{asym}(b) 
Taking into account that the realistic values of $U$ in the semiconductor quantum dots are rather 
large, this condition seems to be realizable by fine-tuning of QD gate voltages 

Lastly, let us point out that while in the presence of only one exchange mechanism, \emph{CAR} or
\emph{direct}, $G_{\mathrm{min}}(\delta_1)$ dependencies depicted in \fig{asym}(b) are symmetrical with respect
to sign change of $\delta_1$, for \emph{both} exchange mechanisms the dependence is non-symmetric  

\section{Effects of asymmetry of couplings to superconductor}
\label{sec:x}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig5 pdf}
\caption{
		 (a) Linear conductance between the normal leads, $G$, as a function of temperature, $T$,
		 for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, for different values 
		 of asymmetry coefficient $x$ [see \eq{xGS}], in the presence of \emph{CAR} exchange only 
		 %
		 (b) The second-stage Kondo temperature $T^*$ normalized by $T_K$ as a function of $x$, 
		 calculated with the aid of NRG (points) and a fit to \eq{Tstar} (lines) 
		 with $J^{\mathrm{eff}}$ from \eq{Jeff} 
		 %
		 (c) The zero-temperature conductance $G_{\mathrm{min}}$ as a function of QD1 coupling to SC lead, $\GS{1}$,
		 compiled from data obtained at different circumstances (as indicated in the legend)
		 for different $x$  Dotted line corresponds to \eq{Gmin2} with $c=2 25$ 
		}
\label{fig:x}
\end{figure}

Similarly to PHS, the ideal symmetry in the coupling between respective QDs and SC lead is hardly possible
in experimental reality  As shown below, it does not introduce any qualitatively new features 
On the other hand, it decreases the second stage Kondo temperature, which is already small, therefore,
quantitative estimation of this decrease may be important for potential experimental approaches 
To analyze the effects of $\GS{1}\neq\GS{2}$, we introduce the asymmetry parameter $x$ and extend
the definition of $\GS{}$,
\beq
x = \frac{\GS{1}-\GS{2}}{\GS{1}+\GS{2}}, \quad \GS{} = \frac{\GS{1}+\GS{2}}{2} 
\label{xGS}
 \end{equation} 
Note, that even for a fixed $\GS{}$, the actual CAR coupling $\GS{\rm X}=\GS{}\sqrt{1-x^2}$ decreases
with increasing $|x|$, which is a main mechanism leading to a decrease of $T^*$ outside the $x=0$ point
visible in \figs{x}(a) and (b)  To illustrate this, the curves corresponding to \emph{both} exchange
mechanisms were calculated using $x$-dependent $t=\GS{\rm X}$ instead of $t=\xi/\sqrt{2}$  
Therefore, $\xi$ was generalized for $x\neq 0$ by setting $\xi=\sqrt{t^2(1-x^2)^{-1}+\GS{}^2}$ 
Clearly, in \fig{x}(b) the curves for different exchange mechanisms are very similar and differ mainly 
by a constant factor, resulting from different influence of $U'$; see \Sec{scales}  
The magnitude of $T^*$ changes is quite large, exceeding an order of magnitude for $x=\pm 0 5$ 
and $\xi=U/20$  Moreover, $T^* \to 0$ for $x\to\pm 1$  Consequently, for strongly asymmetric
devices one cannot hope to observe the second stage of Kondo screening 

A careful observer can note that the $T^*(x)$ dependency is not symmetrical; note for example different 
$T^*$ for $x=\pm 0 5$ in \fig{x}(a)  This is caused by the dependence of the first stage Kondo temperature
$T_K$ on $\GS{1}$ \cite{part1,DomanskiIW},
\beq
\widetilde{T}_K(\GS{1}) = T_K \cdot \exp\!\left( \frac{\pi}{2} \frac{\GS{1}^2}{\Gamma U}\right) 
 \end{equation} 
Here, $T_K$ is, as earlier, defined in the absence of SC, while $\widetilde{T}_K$ is a function 
of $\GS{1}$, such that $G(\widetilde{T}_K) = G_{\rm max}(\GS{1})/2$ in the absence of QD2  
As $\widetilde{T}_K$ grows for increasing $\GS{1}$ (or $x$), $T^*$ decreases according to \eq{Tstar}  
Its $\GS{}$ dependence can be accounted for by small changes in the coefficients $a$ and $b$ in \eq{Tstar}, 
as long as $x$ is kept constant  

To close the discussion of $T^*(x)$ dependence let us point out, that in \eq{A_J} 
there appears a correction to \eq{Jeff} for $x\neq 0$  However, it is very small due to additional
factor $\GS{}^2/U^2$ in the leading order  Its influence on curves plotted in \fig{x}(b) is hardly visible 

In turn, let us examine the $x$ dependence of the $T=0$ conductance $G_{\mathrm{min}}$  As can be seen 
in \fig{x}(a), it monotonically increases with $x$, as it crosses $x=0$ point  In fact, \eq{Gmin}
can be generalized to
\beq
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{1}^2}{U^2} ,
\label{Gmin2}
 \end{equation} 
with $c\approx 2 25$ (indicated by a dotted line in \fig{x}(c))  Note that $G_{\mathrm{min}}$ is proportional to 
$\GS{1}^2=(x+1)^2 \GS{}^2$, instead of simply $\GS{}$, cf  \eq{Gmin}  The values of $G_{\mathrm{min}}$ obtained
from all analyzed $G(T)$ dependencies for different $x$ have been compiled in \fig{x}(c) 
It is evident, that \eq{Gmin2} is approximately fulfilled for all the considered cases 

Finally, it seems noteworthy that the normal-lead coupling asymmetry, 
$\Gamma_{\rm L}\neq \Gamma_{\rm R}$, is irrelevant for the results except for a constant factor
diminishing the conductance $G$ \cite{KWIWJB-asym} 



\section{The role of CAR efficiency}
\label{sec:coef}

\begin{figure}[tb]
\includegraphics[width=0 98\linewidth]{Fig6 pdf}
\caption{Linear conductance between the normal leads
		 $G$ as a function of coupling to SC lead, $\GS{}$, for indicated values of RKKY exchange $J$
		 and the efficiency of CAR processes reduced by factor (a) $\mathcal{C}=0 9$ and (b) $\mathcal{C}=0 5$ 
		 Other parameters as in \fig{3} 
		 Insets: QD1 local spectral density $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0 1U$ curve, indicated with corresponding symbols 
		} 
\label{fig:C}
\end{figure}

Up to this point we assumed $\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$, which is valid when the two 
quantum dots are much closer to each other than the coherence length in the superconductor 
This does not have to be the case in real setups, yet relaxing this assumption does not 
introduce qualitative changes  Nevertheless, the model cannot be extended to inter-dot 
distances much larger than the coherence length, where $\GS{\rm X}\to 0$ 

To quantitatively analyze the consequences of less effective Andreev coupling we define the 
CAR efficiency as $\mathcal{C} \equiv \GS{\rm X} / \sqrt{\GS{1}\GS{2}}$ and analyze $\mathcal{C} < 1$
in the wide range of $\GS{1}=\GS{2}=\GS{}$ and other parameters corresponding to \fig{3}  
The results are presented in \fig{C} 

Clearly, decreasing $\mathcal{C}$ from $\mathcal{C}=1$ causes diminishing of $\GS{\rm X}$, and consequently of CAR 
exchange  For a change as small as $\mathcal{C}=0 9$, the consequences reduce to some shift of the 
conventional Kondo regime, compare \fig{C}(a) with \fig{3}  Stronger suppression of CAR may, 
however, increase the SC coupling necessary to observe the second stage of Kondo screening caused
by CAR outside the experimentally achievable range, see \fig{C}(b)  Moreover, the reduced $T^*$
leads to narrowing of the related local spectral density dip, while the
increased critical $\GS{}$ necessary for the observation of the second stage of screening leads to the
shallowing of the dip  This is visible especially in the inset in \fig{C}(b) 


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

The CAR exchange mechanism is present in any system comprising at least
two QDs or magnetic impurities coupled to the same superconducting contact
in a way allowing for crossed Andreev reflections 
In the considered setup, comprised of two quantum dots in a T-shaped geometry 
with respect to normal leads and proximized by superconductor,
it leads to the two-stage Kondo
screening even in the absence of other exchange mechanisms 
This CAR induced exchange screening is characterized by a residual 
low-temperature conductance at particle-hole symmetric case 
We have also shown that the competition between CAR exchange and RKKY
interaction may result in completely different Kondo screening scenarios 

The presented results bring further insight into the low-temperature
behavior of hybrid coupled quantum dot systems, which hopefully could be verified
with the present-day experimental techniques 
Moreover, non-local pairing is present also in bulk systems such as non-$s$-wave superconductors 
The question if an analogue of discussed CAR exchange may play a role there
seems intriguing in the context of tendencies of many strongly correlated materials
to possess superconducting and anti-ferromagnetic phases 


\begin{acknowledgments}
This work was supported by the National Science Centre in Poland through project no 
2015/19/N/ST3/01030 
We thank J  Barna\'{s} and T  Maier for valuable discussions 
\end{acknowledgments}








ground_truth: The three subsets are V+, V0, and V-, determined by the Kullback-Leibler information distance.