Input: What is the future direction mentioned in the conclusion?

Context:
Paper Info

Title: Is In-hospital Meta-information Useful for Abstractive Discharge Summary Generation?
Publish Date: 10 Mar 2023
Author List: Mamoru Komachi (from Tokyo Metropolitan University), Takashi Okumura (from Kitami Institute of Technology), Hiromasa Horiguchi (from National Hospital Organization), Yuji Matsumoto

Figure

Fig. 1.Example of part of a discharge summary which is a dummy we created.
Fig. 2. Overview of our proposed method.A new feature embedding layer encoding hospital, physician, disease, and length of stay is added to the standard transformer architecture.The figure shows an example of hospital embedding.
Statistics of our data for experiment.
of summarization models with different meta-information.The best results are highlighted in bold.Each score is the average of three models with different seeds.The BS and BR indicate BERTScore and BLEURT, respectively.
Statistics on the number of cases handled by physicians.C/P denotes Cases/Physician, which indicates how many cases an individual physician has.Method of Grouping Physician IDs A most naive method of mapping physician IDs to features is without any grouping process.The data contains 4,846 physicians, so |M | was set to 4,846.However it caused our model's training to be unstable.This might be due to the many physician IDs appearing for the first time in the test time.Table

abstract

During the patient's hospitalization, the physician must record daily observations of the patient and summarize them into a brief document called "discharge summary" when the patient is discharged. Automated generation of discharge summary can greatly relieve the physicians' burden, and has been addressed recently in the research community.
Most previous studies of discharge summary generation using the sequenceto-sequence architecture focus on only inpatient notes for input. However, electric health records (EHR) also have rich structured metadata (e.g., hospital, physician, disease, length of stay, etc.) that might be useful. This paper investigates the effectiveness of medical meta-information for summarization tasks.
We obtain four types of meta-information from the EHR systems and encode each meta-information into a sequence-to-sequence model. Using Japanese EHRs, meta-information encoded models increased ROUGE-1 by up to 4.45 points and BERTScore by 3.77 points over the vanilla Longformer. Also, we found that the encoded meta-information improves the precisions of its related terms in the outputs.
Our results showed the benefit of the use of medical meta-information.

INTRODUCTION

Clinical notes are written daily by physicians from their consults and are used for their own decision-making or coordination of treatment. They contain a large amount of important data for machine learning, such as conditions, laboratory tests, diagnoses, procedures, and treatments. While invaluable to physicians and researchers, the paperwork is burdensome for physicians , .
Discharge summaries, a subset of these, also play a crucial role in patient care, and are used to share information between hospitals and physicians (see an example in Figure ). It is created by the physician as a summary of notes during hospitalization at the time of the patient's discharge, which is known to be very time-consuming.
Researchers have begun to apply automatic summarization techniques to address this problem - . Previous studies used extractive or abstractive summarization methods, but most of them focused on only progress notes for inputs. Properly summarizing an admission of a patient is a quite complex task, and requires various meta-information such as the patient's age, gender, vital signs, laboratory values and background to specific diseases.
Therefore, discharge summary generation needs more medical meta-information, than similar but narrower tasks such as radiology report generation. However, what kind of meta-information is important for summarization has not been investigated, even though it is critical not only for future research on medical summarization but also for the policy of data collection infrastructure.
In this paper, we first reveal the effects of meta-information on neural abstractive summarization on admissions. Our model is based on an encoder-decoder transformer with an additional feature embedding layer in the encoder (Figure ). Hospital, physician, disease, and length of stay are used as meta-information, and each feature is embedded in the vector space.
For experiments, we collect progress notes, discharge summaries and coded information from the electronic health record system, which are managed by a largest multi-hospital organization in Japan. Our main contributions are as follows: • We found that a transformer encoding meta-information generates higher quality summaries than the vanilla one, and clarified the benefit of using meta-information for medical summarization tasks.
• We found that a model encoding disease information can produce proper disease and symptom words following the source. In addition, we found that the model using physician and hospital information can generate symbols that are commonly written in the summary. • We are the first to apply the abstractive summarization method to generate Japanese discharge summaries.
In the studies of summarization of medical documents, it is common to retrieve key information such as disease, examination result, or medication from EHRs - . Other researchs more similar to our study targeted to help physicians get the point of medical documents quickly by generating a few key sentences - .
Studies generating contextualized summaries can be categorized by the type of model inputs and architectures. Some studies produced a whole discharge summary using structured data for input - The sensitivity of the gram stain for bacterial meningitis is about 60%, and the sensitivity of the culture is not high either.
Also, the glucose in the cerebrospinal fluid would have been slightly lower. Although no definitive diagnosis could be made, bacterial meningitis was the most suspicious disease. The causative organism was assumed to be MRSA, and vancomycin and meropenem (meningitis dose) were used to cover a wide range of enteric bacteria.
a whole discharge summary from free-form inpatient records - . The free-form data is more challenging since it is noisier than structured data. In inputting of the free-form data, extractive summarization methods, which extract sentences from the source, are commonly used , - . On the other hands, an encoder-decoder model was used for abstractive summarization , , with a limited number of studies.
The various issues in the abstractive generation of discharge summary would be studied in the future. Studies using medical meta-information have long been conducted on a lot of tasks - . In abstractive summarization on discharge summary,  developed a model incorporating similarity of progress notes and information of the record author.
They presented an idea of integrating meta-information into the abstractive summarization model on medical documents, but did not reveal how meta-information would affect the quality of the summaries. Our method is based on the encoder-decoder transformer model. The transformer model is known for its high performance and has been widely used in recent studies, thus it is suitable for our purpose.
As shown in Figure , the standard input to a transformer's encoder is created by a token sequence T = [t 0 , t 1 , ..., t i ] and position sequence P = [p 0 , p 1 , ..., p i ], where i is the maximum input length. The token and position sequences are converted into token embeddings E T and positional embeddings E P by looking up the vocabulary tables.
The sum of E T and E P is input into the model. In this paper, we attempt to encode meta-information to feature embeddings. We follow the segment embeddings of BERT and the language embeddings of XLM , which provide additional information to the model. It is not a new idea but is suitable for our validation.
Our method is formulated as follows: Let M be feature type, M ∈ {Vanilla, Hospital, Physician, Disease, Length of stay}, since we set five types of features. Feature embeddings E M is created by looking up the feature table where m j is featue value (e.g., pysician ID, disease code, etc.) and |M | is the maximum number of differences in a feature.
In our study, |M | is set to four different values depending on features. Specifically, they are as follows. a) Hospital: As shown in Table , the data includes five hospital records. They were obtained mechanically from the EHR system. b) Physician: Physicians are also managed by IDs in the EHR systems. We hashed the physician IDs into 485 groups containing 10 people each.
Specifically, as a naive strategy, we shuffled and listed the cases within each hospital, and hashed them into groups in the order of appearance of the physician IDs. So each group has the information about the relevance of the hospitals. The reason for employing a grouping strategy is described in Appendix A.
c) Disease: Two types of disease information exist in our EHRs: disease names and disease codes called ICD-10 . We did not use any disease names in the inputs for our experiment. Instead, we encoded diseases with the first three letters of the ICD-10 code, because they represent well the higher level concept.
The initial three letters of the ICD-10 codes are arranged in the order of an alphabetic letter, a digit, and a digit, so there are a total of 2,600 ways to encode a disease. In our data, some ICD-10 codes were missing, although all disease names were systematically obtained from the EHR system. For such cases, we converted the disease names into ICD-10 codes using MeCab with the J-MeDic (MANBYO 201905) dictionary.
Also, diseases can be divided into primary and secondary diseases, but we only deal with the primary diseases. d) Length of stay: The length of stay can be obtained mechanically from the EHR system and the maximum value was set to 1,000 days. We set |M | for vanilla, hospital, physician, disease, and length of stay to 1, 5, 485, 2,600, and 1,000, respectively .
The vanilla embedding is prepared for the baseline in our experiment and to equalize the total number of parameters with the other models. The input to our model is the sum of E T , E P and E M . We also prepare an extra model with all features for our experiments. This takes all four feature embeddings (hospital, physician, disease, and length of stay) added to the encoder.

Datasets and Metrics

We evaluated our proposed method on a subset of data from National Hospital Organization (NHO), the largest multiinstitutional organization in Japan. The statistics of our data are shown in Table  , which includes 24,630 cases collected from five hospitals. Each case includes a discharge summary and progress notes for the days of stay.
The data are randomly split into 22,630, 1,000, and 1,000 for train, validation, and test, respectively. Summarization performances are reported in ROUGE-1, ROUGE-2, ROUGE-L and BERTScore in terms of F1. In addition, we also employed BLEURT , which models human judgment.

Architectures and Hyperparameters

Due to our hardware constraints we need a model that is computationally efficient, so we employed the Longformer instead of the conventional transformer. Longformer can  . In our model, number of layers, window size, dilation, input sequence length, output sequence length, batch size, learning rate and number of warmup steps are 8, 256, 1, 1024, 256, 4, 3e-5 and 1K, respectively.
Other hyperparameters are the same as in the original Longformer, except for the maximum number of epochs is not fixed and the best epoch. It is selected for each training using the validation data based on ROUGE-1. Also, the original Longformer imports pretrained-BART parameters to initial values, but we do not use pre-trained Japanese BART in this study.
We used three GeForce RTX 2080 TI for our experiments. Our vocabulary for preparing input to Longformer is taken from UTH-BERT , which is pre-trained on the Japanese clinical records. Since the vocabulary of UTH-BERT is trained by WordPiece , we also tokenize our data with WordPiece. However, the vocabulary does not include white space and line breaks, which cannot be handled, so we add those two tokens to the vocabulary, resulting in a total size of 25,002.
The vocabulary has all tokens in full characters, so we normalized full-wdith characters by converting all alphanumeric and symbolic characters to half-width for byte fallback. , we found that all the models with encoded medical meta-information perform better in ROUGE-1, ROUGE-L and BLEURT than the vanilla Longformer.
However, in BERTScore, only hospital and disease models outperform the vanilla. Specifically, disease information is most effective, improving ROUGE-1, ROUGE-2, ROUGE-L, BERTScore and BLEURT by 4.45, 0.73, 3.12, 3.77 and 0.21 points over the vanilla model, respectively. This seems to be because disease information and the ICD-10 ontology efficiently cluster groups with similar representations.
In contrast, in ROUGE-2 and ROUGE-L, the model with physician embedding is inferior to the vanilla model. This seems to be a negative effect of grouping physicians without any consideration of their relevance. It would be better to cluster them by department, physician attributes, similarity of progress notes, etc. Regarding low ROUGE-2 scores in all models, a previous study using the English data set also reported a low ROUGE-2 score of about 5%, which may indicate an inherent difficulty in discharge summary generation.
In BERTScore, the models with the physician and the length of stay did not reach the performance of the vanilla model, suggesting that the system's outputs are semantically inferior. The model with all features performed the lowest of all models in BERTScore. The reason for the low score of the model with all features seems to be that its number of parameters in feature embedding was four times larger than that of the model with the individual feature, and the amount of training data was insufficient.
In BLEURT, all models with meta-information outperform vanilla, which suggests that they are more natural to humans. To analyze the influence of encoded meta-information on the outputs, we evaluate the precisions of the generated text. Specifically, we measure the probability that the generated words are included in the gold summary to investigate if the proper words are generated.
Some previous studies on faithfulness, which also analyze the output of summarization, have employed words or entities - . In this study, we focused on words, not entities, because we wanted to visualize expressions that are not only nouns. The words were segmented by MeCab with the J-MeDic. For each segmented word, the numeral and symbol labels were assigned as parts of speech by MeCab, the morphological analyzer, while the disease and symptom were assigned by the J-Medic dictionary.
The results, shown in Figure , indicate that the encoded disease information leads to generate more proper disease and symptom words. This indicates that the meta-information successfully learns disease-related expressions. The encoded hospital or physician information also improved the precision of symbols generation.
This suggests that different hospitals and physicians have different description habits (e.g., bullet points such as "•", "*" and "-", punctuation such as "。" and ".", etc.), which can be grouped by meta-information. In this paper, we conducted a discharge summary generation experiment by adding four types of information to Longformer and verified the impact of the meta-information.
The results showed that all four types of information exceeded the performance of the vanilla Longformer model, with the highest performance achieved by encoding disease information. We found that meta-information is useful for abstractive summarization on discharge summaries. Our limitations are that we used Japanese EHR, the limited number of tested features and not performing human evaluations.
As for the efficacy of the meta-information, we believe that our results are applicable to non-Japanese, but it is left as Fig. . The precisions of words in the generated summaries. The vertical axis shows the probability that the words exist in the gold summary. a future work. Other meta-information may be worth verifying such as the patient's gender, age, race, religion and used EHR system, etc.
It is hard to collect a large amount of medical information and process it into meta-information, so we may need to develop a robust and flexible research infrastructure to conduct a more large scale cross-sectional study in the future. In the discharge summary generation task, which demands a high level of expertise, the human evaluation requires a lot of physicians' efforts and it is a very high cost which is unrealistic.
This is a general issue in tasks dealing with medical documents, and this study also could not perform human evaluations. On this research, informed consent and patient privacy are ensured in the following manner. Notices about their policy and the EHR data usage are posted at the hospitals. The patients who disagree with the policies can request opt-out and are excluded from the archive.
In case of minors and their parents, followed the same manner. In the case of minors and their parents are same. To conduct a research on the archive, researchers must submit their research proposals to the institutional review board. After the proposal is approved, the data is anonymized to build a dataset for analysis.
The data is accessible only in a secured room at the NHO headquarters, and only statistics are brought out of the secured room, for protection of patients' privacy. In the present research, the analysis was conducted under the IRB approval (IRB Approval No.: Wako3 2019-22) of the Institute of Physical and Chemical Research (RIKEN), Japan, which has a collaboration agreement with the National Hospital Organization.
This data is not publicly available due to privacy restrictions. shows the detailed number of cases handled by physicians. In all hospitals, there is a large difference between the median and the maximum of cases/physician. This indicates that a few physicians handle a large number of cases and many physicians handle fewer cases.
It is impossible to avoid physician IDs first seen at test time without some process that averages the number of cases a physician holds.

Answer: Verifying other meta-information such as patient's gender, age, race, etc.


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

Context:
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)).

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


Input: What are the stability conditions for a solution of $-\Delta u = f(u)$?

Context:
\section{Introduction and main results}

In this note we are interested in the existence versus non-existence of stable sub- and super-solutions of equations of the form
\begin{equation} \label{eq1}
-div( \omega_1(x) \nabla u ) = \omega_2(x) f(u) \qquad \mbox{in $ {\mathbb{R}}^N$,}
\end{equation} where $f(u)$ is one of the following non-linearities: $e^u$,  $ u^p$ where $ p>1$ and $ -u^{-p}$ where $ p>0$.  We assume that $ \omega_1(x)$ and $ \omega_2(x)$, which we call \emph{weights},  are smooth positive functions (we allow $ \omega_2$ to be zero at say a point) and which satisfy various growth conditions at $ \infty$.    Recall that we say that a solution $ u $ of $ -\Delta u = f(u)$ in $ {\mathbb{R}}^N$ is stable provided
\[ \int f'(u) \psi^2 \le \int | \nabla \psi|^2, \qquad \forall \psi \in C_c^2,\] where $ C_c^2$ is the set of $ C^2$ functions defined on $ {\mathbb{R}}^N$ with compact support.  Note that the stability of $u$ is just saying that the second variation at $u$ of the energy associated with the equation is non-negative.     In our setting this becomes:  We say a $C^2$ sub/super-solution $u$ of (\ref{eq1}) is \emph{stable} provided
\begin{equation} \label{stable}
\int \omega_2 f'(u) \psi^2 \le \int \omega_1 | \nabla \psi|^2 \qquad \forall \psi \in C_c^2.
\end{equation}
One should note that (\ref{eq1}) can be re-written as
\begin{equation*}
- \Delta u  + \nabla \gamma(x) \cdot \nabla u ={ \omega_2}/{\omega_1}\ f(u) \qquad \text{ in $ \mathbb{R}^N$},
\end{equation*}
where
$\gamma = - \log( \omega_1)$ and on occasion we shall take this point of view.

\begin{remark} \label{triv} Note that  if $ \omega_1$ has enough integrability then it is immediate that if $u$ is a stable solution  of (\ref{eq1}) we have $ \int \omega_2 f'(u) =0 $  (provided $f$ is increasing).   To see this let $ 0 \le \psi \le 1$ be supported in a ball of radius $2R$ centered at the origin ($B_{2R}$) with $ \psi =1$ on $ B_R$ and such that $  | \nabla \psi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$.   Putting this $ \psi$ into $ (\ref{stable})$ one obtains
\[ \int_{B_R} \omega_2 f'(u) \le \frac{C}{R^2} \int_{R < |x| <2R} \omega_1,\] and so if the right hand side goes to zero as $ R \rightarrow \infty$ we have the desired result.

\end{remark}

The existence versus non-existence of stable solutions of  $ -\Delta u =  f(u)$ in $ {\mathbb{R}}^N$ or $ -\Delta u = g(x) f(u)$ in $ {\mathbb{R}}^N$ is now quite well understood, see  \cite{dancer1, farina1, egg, zz, f2, f3, wei, ces, e1, e2}.  We remark that some of these results are examining the case where $ \Delta $ is replaced with $ \Delta_p$ (the $p$-Laplacian) and also in many cases the authors are interested in finite Morse index solutions or solutions which are stable outside a compact set.
  Much of the interest in these Liouville type theorems  stems from the fact that the non-existence of a stable solution is related to the existence of a priori estimates for stable solutions of a related equation on a bounded domain.

       In \cite{Ni}  equations similar to  $ -\Delta u = |x|^\alpha u^p$  where examined on the unit ball in $ {\mathbb{R}}^N$ with zero Dirichlet boundary conditions.  There it was shown that for $ \alpha >0$ that  one can obtain positive solutions for $ p $ supercritical with respect to Sobolev embedding and so one can view that the term $ |x|^\alpha$ is restoring some compactness.      A similar feature happens for equations of the form
\[ -\Delta u = |x|^\alpha f(u) \qquad \mbox{in $ {\mathbb{R}}^N$};\]    the value of $ \alpha$ can vastly alter the existence versus non-existence of a stable solution,  see \cite{e1, ces, e2, zz, egg}.

We now come to our main results and for this we need to define a few quantities:

\begin{eqnarray*}
I_G&:=& R^{-4t-2} \int_{ R < |x|<2R} \frac{ \omega_1^{2t+1}}{\omega_2^{2t}}dx , \\
 J_G&:=& R^{-2t-1} \int_{R < |x| <2R} \frac{| \nabla \omega_1|^{2t+1} }{\omega_2^{2t}} dx  ,\\I_L&:=& R^\frac{-2(2t+p-1)}{p-1}  \int_{R<|x|<2R }{ \left(   \frac{w_1^{p+2t-1}}{w_2^{2t}}    \right)^{\frac{1}{p-1} }  }  dx,\\ J_L&:= &R^{-\frac{p+2t-1}{p-1} }  \int_{R<|x|<2R }{ \left(   \frac{|\nabla w_1|^{p+2t-1}}{w_2^{2t}}     \right)^{\frac{1}{p-1} }  }  dx,\\
I_M  &:=&   R^{-2\frac{p+2t+1}{p+1} }  \int_{R<|x|<2R }{ \left(   \frac{w_1^{p+2t+1}}{w_2^{2t}}    \right)^{\frac{1}{p+1} }  } \ dx, \\
J_M  &:= &  R^{-\frac{p+2t+1}{p+1} }  \int_{R<|x|<2R }{ \left(   \frac{|\nabla w_1|^{p+2t+1}}{w_2^{2t}}     \right)^{\frac{1}{p+1} }  }  dx.
\end{eqnarray*}

The three equations we examine are
\[ -div( \omega_1 \nabla u ) = \omega_2 e^u \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (G), \]
\[  -div( \omega_1 \nabla u ) = \omega_2 u^p \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (L), \]
\[ -div( \omega_1 \nabla u ) = - \omega_2 u^{-p} \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (M),\]  and where we restrict $(L)$ to the case $ p>1$ and $(M)$ to $ p>0$.    By solution we always mean a $C^2$ solution.   We now come to our main results in terms of abstract $ \omega_1 $ and $ \omega_2$.    We remark that our approach to non-existence of stable solutions is the approach due to Farina, see \cite{f2,f3,farina1}.

\begin{thm} \label{main_non_exist} \begin{enumerate}

\item   There is no  stable sub-solution of $(G)$ if $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$   for some   $0<t<2$.
 \item    There is no positive   stable sub-solution (super-solution) of $(L)$  if  $ I_L,J_L \rightarrow 0$ as $ R \rightarrow \infty$   for some   $p- \sqrt{p(p-1)}   < t<p+\sqrt{p(p-1)} $  ($0<t<\frac{1}{2}$).
\item   There is no positive   stable super-solution of (M) if  $ I_M,J_M \rightarrow 0$ as $ R \rightarrow \infty$  for some $0<t<p+\sqrt{p(p+1)}$.

\end{enumerate}

\end{thm}     If we assume that $ \omega_1$ has some monotonicity we can do  better.   We will assume that the monotonicity conditions is satisfied for big $x$  but really all ones needs is for it to be satisfied on a suitable sequence of annuli.

\begin{thm} \label{mono}  \begin{enumerate} \item There is no  stable sub-solution of $(G)$ with $  \nabla \omega_1(x) \cdot  x \le 0$ for big $x$ if  $ I_G \rightarrow 0$ as $ R \rightarrow \infty$   for some   $0<t<2$.

\item There is no positive  stable sub-solution of  $(L)$ provided  $  I_L \rightarrow 0$ as $ R \rightarrow \infty$ for either:
\begin{itemize}
\item  some $  1 \le t < p + \sqrt{p(p-1)}$ and $  \nabla \omega_1(x) \cdot x \le 0$ for big $x$,  or  \\

\item some $ p - \sqrt{p(p-1)} < t \le 1$ and $ \nabla \omega_1(x) \cdot x \ge 0$ for big $ x$.
\end{itemize}
 There is no positive super-solution of $(L)$ provided $ I_L \rightarrow 0$ as $ R \rightarrow \infty$ for some $ 0 < t < \frac{1}{2}$ and $ \nabla \omega_1(x) \cdot x \le 0$ for big $x$.

\item  There is no positive stable super-solution of $(M)$ provided  $ I_M \rightarrow 0$ as $ R \rightarrow \infty$  for some  $0<t<p+\sqrt{p(p+1)}$.

\end{enumerate}

\end{thm}

\begin{cor}  \label{thing}  Suppose $ \omega_1 \le C  \omega_2$  for big $ x$, $ \omega_2 \in L^\infty$,  $ \nabla \omega_1(x) \cdot  x \le 0$ for big $ x$.
\begin{enumerate} \item There is no stable sub-solution of $(G)$ if $ N \le 9$.

\item  There is no positive stable sub-solution of $(L)$ if  $$N<2+\frac{4}{p-1} \left( p+\sqrt{p(p-1)}  \right).$$

\item  There is no positive stable super-solution of $(M)$ if $$N<2+\frac{4}{p+1} \left( p+\sqrt{p(p+1)}  \right).$$

\end{enumerate}

\end{cor}

If one takes $ \omega_1=\omega_2=1$ in the above corollary, the results obtained for $(G)$ and  $(L)$,  and for some values of $p$ in $(M)$, are optimal, see \cite{f2,f3,zz}.

We now drop all monotonicity conditions on $ \omega_1$.

\begin{cor} \label{po} Suppose  $ \omega_1 \le C \omega_2$ for big $x$, $ \omega_2 \in L^\infty$, $ | \nabla \omega_1| \le C \omega_2$ for big $x$.
\begin{enumerate} \item  There is no stable sub-solution of $(G)$ if $ N \le 4$.

\item  There is no positive stable sub-solution of $(L)$ if $$N<1+\frac{2}{p-1} \left( p+\sqrt{p(p-1)}  \right).$$

\item There is no positive super-solution of $(M)$ if $$N<1+\frac{2}{p+1} \left( p+\sqrt{p(p+1)}  \right).$$

\end{enumerate}

\end{cor}

Some of the conditions on $ \omega_i$ in Corollary \ref{po} seem somewhat artificial.  If we shift over to the advection equation (and we take $ \omega_1=\omega_2$  for simplicity)
\[ -\Delta u + \nabla \gamma \cdot \nabla u = f(u), \] the conditions on $ \gamma$ become: $ \gamma$ is bounded from below and has a bounded gradient.

In what follows we examine the case where $ \omega_1(x) = (|x|^2 +1)^\frac{\alpha}{2}$ and $ \omega_2(x)= g(x) (|x|^2 +1)^\frac{\beta}{2}$,  where $ g(x) $ is positive except at say a point, smooth and where $ \lim_{|x| \rightarrow \infty} g(x) = C \in (0,\infty)$.     For this class of weights we can essentially obtain optimal results.

\begin{thm} \label{alpha_beta}   Take $ \omega_1 $ and $ \omega_2$ as above.
\begin{enumerate}

\item If $ N+ \alpha - 2 <0$ then there is no stable sub-solution for $(G)$, $(L)$ (here we require it to be positive) and in the case of $(M)$ there is no positive  stable  super-solution. This case is the trivial case, see Remark \ref{triv}.  \\

\textbf{Assumption:} For the remaining cases we assume that $ N + \alpha -2 > 0$.

  \item If  $N+\alpha-2<4(\beta-\alpha+2)$ then there is no  stable sub-solution for $ (G)$.

\item If $N+\alpha-2<\frac{ 2(\beta-\alpha+2)   }{p-1} \left( p+\sqrt{p(p-1)}  \right)$ then there is  no positive stable sub-solution of $(L)$.

\item If $N+\alpha-2<\frac{2(\beta-\alpha+2)   }{p+1} \left( p+\sqrt{p(p+1)}  \right)$ then there is no positive stable super-solution of $(M)$.

\item Further more 2,3,4 are optimal in the sense if $ N + \alpha -2 > 0$ and the remaining inequality is not satisfied (and in addition we assume we don't have equality in the inequality) then we can find a suitable function $ g(x)$ which satisfies the above properties and a stable sub/super-solution $u$ for the appropriate equation.

\end{enumerate}

\end{thm}

\begin{remark}  Many of the above results can be extended to the case of equality in either the $ N + \alpha - 2 \ge 0$ and also the other inequality which depends on the equation we are  examining.  We omit the details because one cannot prove the results in a unified way.

\end{remark}

In showing that an explicit solution is stable we will need the   weighted Hardy inequality given in \cite{craig}.
\begin{lemma} \label{Har}
Suppose $ E>0$ is a smooth function.  Then one has
\[ (\tau-\frac{1}{2})^2 \int E^{2\tau-2} | \nabla E|^2 \phi^2 + (\frac{1}{2}-\tau) \int (-\Delta E) E^{2\tau-1} \phi^2 \le \int E^{2\tau} | \nabla \phi|^2,\] for all $ \phi \in C_c^\infty({\mathbb{R}}^N)$ and $ \tau \in {\mathbb{R}}$.
\end{lemma}  By picking an appropriate function $E$ this gives,

\begin{cor} \label{Hardy}
For all $ \phi \in C_c^\infty$ and $ t , \alpha \in {\mathbb{R}}$. We have
  \begin{eqnarray*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2 &\ge& (t+\frac{\alpha}{2})^2 \int |x|^2 (1+|x|^2)^{-2+\frac{\alpha}{2}}\phi^2\\
&&+(t+\frac{\alpha}{2})\int  (N-2(t+1)   \frac{|x|^2}{1+|x|^2}) (1+|x|^2)^{-1+\frac{\alpha} {2}} \phi^2.
\end{eqnarray*}
 \end{cor}

\section{Proof of main results}

\textbf{ Proof of Theorem \ref{main_non_exist}.}   (1). Suppose $ u$ is a stable sub-solution of $(G)$ with $ I_G,J_G \rightarrow 0$ as $ R \rightarrow \infty$ and  let $ 0 \le \phi \le 1$ denote a smooth compactly supported function.  Put $ \psi:= e^{tu} \phi$ into (\ref{stable}), where $ 0 <t<2$, to arrive at
\begin{eqnarray*}
\int \omega_2 e^{(2t+1)u} \phi^2 &\le & t^2 \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \\
&& +\int \omega_1 e^{2tu}|\nabla \phi|^2 + 2 t \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi.
\end{eqnarray*}  Now multiply $(G)$ by $ e^{2tu} \phi^2$ and integrate by parts to arrive at
\[ 2t \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \le \int \omega_2 e^{(2t+1) u} \phi^2 - 2 \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi,\]
and now if one equates like terms they arrive at
\begin{eqnarray} \label{start}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^2  & \le & \int \omega_1 e^{2tu} \left( | \nabla \phi |^2 - \frac{ \Delta \phi}{2} \right) dx \nonumber \\
&& - \frac{1}{2} \int e^{2tu} \phi \nabla \omega_1 \cdot \nabla \phi.
\end{eqnarray}   Now substitute $ \phi^m$ into this inequality for $ \phi$ where $ m $ is a big integer to obtain
\begin{eqnarray} \label{start_1}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m}  & \le & C_m \int \omega_1 e^{2tu} \phi^{2m-2}  \left( | \nabla \phi |^2 + \phi |\Delta \phi|  \right) dx \nonumber \\
&& - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi
\end{eqnarray} where $ C_m$ and $ D_m$ are positive constants just depending on $m$.   We now estimate the  terms on the right but we mention that when ones assume the appropriate monotonicity on $ \omega_1$ it is the last integral on the right which one is able to drop.

\begin{eqnarray*}
\int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2 & = & \int \omega_2^\frac{2t}{2t+1} e^{2tu} \phi^{2m-2}  \frac{ \omega_1 }{\omega_2^\frac{2t}{2t+1}} | \nabla \phi|^2  \\
& \le &  \left( \int \omega_2 e^{(2t+1) u} \phi^{(2m-2) \frac{(2t+1)}{2t}} dx \right)^\frac{2t}{2t+1}\\ &&\left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1}.
\end{eqnarray*}
Now, for fixed $ 0 <t<2$ we can take $ m $ big enough so $ (2m-2) \frac{(2t+1)}{2t} \ge 2m $ and since $ 0 \le \phi \le 1$ this allows us to replace the power on $ \phi$ in the first term on the right with $2m$   and hence we obtain
 \begin{equation} \label{three}
 \int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2  \le \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} \left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1}.
 \end{equation}    We now take the test functions $ \phi$ to be such that $ 0 \le  \phi \le 1$ with $ \phi $ supported in the ball $ B_{2R}$ with $ \phi = 1 $ on $ B_R$ and $ | \nabla \phi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$.   Putting this choice of $ \phi$ we obtain
 \begin{equation} \label{four}
 \int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi |^2 \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}.
 \end{equation}  One similarly shows that
 \[ \int \omega_1 e^{2tu} \phi^{2m-1} | \Delta \phi| \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}.\]
 So, combining the results we obtain

 \begin{eqnarray} \label{last} \nonumber \frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m} &\le& C_m \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}\\
 &&- D_m \int e^{2tu} \phi^{2m-1}  \nabla \omega_1 \cdot \nabla \phi.
 \end{eqnarray}
 We now estimate this last term.  A similar argument using H\"{o}lder's inequality shows that
 \[ \int e^{2tu} \phi^{2m-1} | \nabla \omega_1| | \nabla \phi| \le \left(  \int \omega_2 \phi^{2m} e^{(2t+1) u} dx \right)^\frac{2t}{2t+1} J_G^\frac{1}{2t+1}. \] Combining the results gives that
\begin{equation} \label{last}
(2-t) \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{1}{2t+1} \le I_G^\frac{1}{2t+1} + J_G^\frac{1}{2t+1},
\end{equation} and now we send $ R \rightarrow \infty$ and use the fact that $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$ to see that
\[ \int \omega_2 e^{(2t+1) u} =0, \] which is clearly a contradiction.   Hence there is no stable sub-solution of $(G)$.

(2).   Suppose that $u >0$ is a stable sub-solution (super-solution) of $(L)$.  Then a similar calculation as in (1) shows that for  $ p - \sqrt{p(p-1)} <t < p + \sqrt{p(p-1)}$,  $( 0 <t<\frac{1}{2})$ one has

\begin{eqnarray}   \label{shit}
(p  -\frac{t^2}{2t-1}   )\int \omega_2 u^{2t+p-1} \phi^{2m} & \le & D_m \int \omega_1 u^{2t} \phi^{2(m-1)} (|\nabla\phi|^2  +\phi |\Delta \phi |) \nonumber \\
&& +C_m \frac{(1-t)}{2(2t-1)} \int u^{2t} \phi^{2m-1}\nabla \omega_1 \cdot  \nabla \phi.
 \end{eqnarray}    One now applies H\"{o}lder's argument as in (1) but the terms $ I_L$ and $J_L$ will appear on the right hand side of the resulting
 equation.      This shift from a sub-solution to a super-solution depending on whether $ t >\frac{1}{2}$ or $ t < \frac{1}{2}$ is a result from the sign change of $ 2t-1$ at $ t = \frac{1}{2}$.   We leave the details for the reader.

(3).  This case is also similar to (1) and (2).

\hfill $ \Box$

 \textbf{Proof of Theorem \ref{mono}.}   (1).  Again we suppose there is a stable sub-solution $u$ of $(G)$.  Our starting point  is (\ref{start_1}) and we wish to be able to drop the term
 \[ - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi, \] from (\ref{start_1}).    We can choose $ \phi$ as in the proof of Theorem \ref{main_non_exist} but also such that $ \nabla \phi(x) = - C(x) x$ where $ C(x) \ge 0$.     So if we assume that $ \nabla \omega_1 \cdot x \le 0$ for big $x$ then we see that this last term is non-positive and hence we can drop the term.  The the proof is as before but now we only require that $ \lim_{R \rightarrow \infty} I_G=0$.

 (2).     Suppose that $ u >0$ is a stable sub-solution of $(L)$  and so (\ref{shit}) holds for all $  p - \sqrt{p(p-1)} <t< p + \sqrt{p(p-1)}$.   Now we wish to use monotonicity to drop the term from (\ref{shit}) involving the term $ \nabla \omega_1 \cdot \nabla \phi$.      $ \phi$ is chosen the same as in (1)  but here one notes that the co-efficient for this term changes sign at $ t=1$ and hence by restriction $t$ to the appropriate side of 1 (along with the above condition on $t$ and $\omega_1$) we can drop the last term depending on which monotonicity we have and hence to obtain a contraction we only require that $ \lim_{R \rightarrow \infty} I_L =0$.   The result for the non-existence of a stable super-solution is similar be here one restricts $ 0 < t < \frac{1}{2}$.

(3).  The proof here is similar to (1) and (2) and we omit the details.

 \hfill $\Box$

\textbf{Proof of Corollary \ref{thing}.}   We suppose  that $ \omega_1 \le C  \omega_2$  for big $ x$, $ \omega_2 \in L^\infty$,  $ \nabla \omega_1(x) \cdot  x \le 0$ for big $ x$.     \\
(1).    Since $ \nabla \omega_1 \cdot x \le 0$ for big $x$ we can apply Theorem \ref{mono} to show the non-existence of a stable solution to $(G)$.   Note that  with the above assumptions on $ \omega_i$ we have that
\[ I_G \le \frac{C R^N}{R^{4t+2}}.\]  For $ N \le 9$  we can take $ 0 <t<2$  but close enough to $2$ so the right hand side goes to zero as $ R \rightarrow \infty$.

Both (2) and (3) also follow directly from applying Theorem \ref{mono}.   Note that one can say more about (2) by taking the multiple cases as listed in Theorem \ref{mono} but we have choice to leave this to the reader.

\hfill $ \Box$

\textbf{Proof of Corollary \ref{po}.}   Since we have no monotonicity conditions now we will need both $I$ and $J$ to go to zero to show the non-existence of a stable solution.   Again the results are obtained immediately by applying Theorem \ref{main_non_exist}  and we prefer to omit the details.

\hfill $\Box$

\textbf{Proof of Theorem \ref{alpha_beta}.}  (1).  If $ N + \alpha -2 <0$ then using Remark \ref{triv}  one easily sees there is no stable sub-solution of $(G)$ and $(L)$ (positive for $(L)$) or a positive stable super-solution of $(M)$.   So we now assume that $ N + \alpha -2 > 0$.     Note that the monotonicity of $ \omega_1$ changes when $ \alpha $ changes sign and hence one would think that we need to consider separate cases if we hope to utilize the monotonicity results.   But a computation shows that in fact $ I$ and $J$ are just multiples of each other in all three cases so it suffices to show, say, that $ \lim_{R \rightarrow \infty} I =0$. \\
(2).   Note that for $ R >1$ one has
\begin{eqnarray*}
I_G & \le  & \frac{C}{R^{4t+2}} \int_{R <|x| < 2R} |x|^{ \alpha (2t+1) - 2t \beta} \\
& \le  &  \frac{C}{R^{4t+2}}  R^{N + \alpha (2t+1) - 2t \beta},
\end{eqnarray*} and so to show the non-existence we want to find some $ 0 <t<2$ such that
$  4t+2 > N  + \alpha(2t+1) - 2 t \beta$,   which is equivalent to  $ 2t ( \beta - \alpha +2) > (N + \alpha -2)$.    Now recall that we are assuming that $ 0 < N + \alpha -2 < 4 ( \beta - \alpha +2) $ and hence we have the desired result by taking $  t <2$ but sufficiently close.
The proof of the non-existence results for
(3) and (4) are similar and we omit the details.   \\
(5).  We now assume that $N+\alpha-2>0$.  In showing the existence of stable sub/super-solutions we need to consider  $ \beta - \alpha + 2 <0$ and $ \beta - \alpha +2 >0$ separately.

\begin{itemize} \item $(\beta - \alpha + 2 <0)$  Here we take $ u(x)=0$ in the case of $(G)$ and $ u=1$ in the case of $(L)$ and $(M)$. In addition we take $ g(x)=\E$.  It is clear that in all cases $u$ is the appropriate sub or super-solution.  The only thing one needs to check is the stability.    In all cases this reduces to trying to show that we have
\[ \sigma \int       (1+|x|^2)^{\frac{\alpha}{2}  -1}    \phi^2 \le \int      (1+|x|^2)^{\frac{\alpha}{2}}   | \nabla\phi  |^2,\]  for all $ \phi \in C_c^\infty$ where  $ \sigma $ is some small positive constant; its either $ \E$ or $ p \E$ depending on which equation were are examining.
To show this we use the result from Corollary \ref{Hardy} and we drop a few positive terms to arrive at
\begin{equation*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2\ge (t+\frac{\alpha}{2})\int \left (N-2(t+1)   \frac{|x|^2}{1+|x|^2}\right) (1+|x|^2)^{-1+\frac{\alpha} {2}}
\end{equation*} which holds for all $ \phi \in C_c^\infty$ and $ t,\alpha \in {\mathbb{R}}$.
  Now, since $N+\alpha-2>0$, we can choose $t$ such that $-\frac{\alpha}{2}<t<\frac{n-2}{2}$.  So, the integrand function in the right hand side is positive and since for small enough $\sigma$ we have
  \begin{equation*}
\sigma \le  (t+\frac{\alpha}{2})(N-2(t+1)   \frac{|x|^2}{1+|x|^2})  \ \ \ \text {for all} \ \ x\in \mathbb{R}^N
\end{equation*}
 we get stability.

\item ($\beta-\alpha+2>0$) In the case of $(G)$ we take   $u(x)=-\frac{\beta-\alpha+2}{2} \ln(1+|x|^2)$ and $g(x):= (\beta-\alpha+2)(N+(\alpha-2)\frac{|x|^2}{1+|x|^2})$. By a computation one sees that $u$ is a sub-solution of $(G)$ and hence we need now to only show the stability, which amounts to showing that
\begin{equation*}
\int \frac{g(x)\psi^2}{(1+|x|^{2   })^{-\frac{\alpha}{2}+1}}\le \int\frac{|\nabla\psi|^2}{    (1+|x|^2)^{-\frac{\alpha}{2}}     },
\end{equation*} for all $ \psi \in C_c^\infty$.  To show this we use  Corollary \ref{Hardy}.  So  we  need to choose an appropriate $t$ in   $-\frac{\alpha}{2}\le t\le\frac{N-2}{2}$  such that for all $x\in {\mathbb{R}}^N$ we have
 \begin{eqnarray*}
 (\beta-\alpha+2)\left(    N+  (\alpha-2)\frac{|x|^2}{1+|x|^2}\right)         &\le& (t+\frac{\alpha}{2})^2 \frac{ |x|^2 }{(1+|x|^2}\\
&&+(t+\frac{\alpha}{2}) \left(N-2(t+1)   \frac{|x|^2}{1+|x|^2}\right).
\end{eqnarray*}
With a  simple calculation one sees we need just to have
   \begin{eqnarray*}
 (\beta-\alpha+2)&\le& (t+\frac{\alpha}{2}) \\
  (\beta-\alpha+2)     \left(    N+  \alpha-2\right)      &   \le&  (t+\frac{\alpha}{2}) \left(N-t-2+\frac{\alpha}{2}) \right).
 \end{eqnarray*}     If one takes $ t= \frac{N-2}{2}$ in the case where $ N \neq 2$ and $ t $ close to zero in the case for $ N=2$ one easily sees the above inequalities both hold, after considering all the constraints on $ \alpha,\beta$ and $N$.

 We now consider the case of $(L)$.  Here one takes $g(x):=\frac {\beta-\alpha+2}{p-1}(    N+  (\alpha-2-\frac{\beta-\alpha+2}{p-1})
\frac{|x|^2}{1+|x|^2})$ and $ u(x)=(1+|x|^2)^{ -\frac  {\beta-\alpha+2}{2(p-1)} }$.   Using essentially the same approach as in $(G)$ one shows that $u$ is a stable sub-solution of $(L)$ with this choice of $g$.   \\
For the case of $(M)$ we take   $u(x)=(1+|x|^2)^{ \frac  {\beta-\alpha+2}{2(p+1)}   }$ and $g(x):=\frac {\beta-\alpha+2}{p+1}(    N+  (\alpha-2+\frac{\beta-\alpha+2}{p+1})
\frac{|x|^2}{1+|x|^2})$.

\end{itemize}

\hfill  $  \Box$

Answer: $\int f'(u) \psi^2 \le \int | \nabla \psi|^2, \forall \psi \in C_c^2$.


Input: How many massive star-forming regions were studied?

Context:
\section{Introduction}

Spectral line surveys have revealed that high-mass star-forming
regions are rich reservoirs of molecules from simple diatomic species
to complex and larger molecules (e.g.,
\citealt{schilke1997b,hatchell1998b,comito2005,bisschop2007}).
However, there have been rarely studies undertaken to investigate the
chemical evolution during massive star formation from the earliest
evolutionary stages, i.e., from High-Mass Starless Cores (HMSCs) and
High-Mass Cores with embedded low- to intermediate-mass protostars
destined to become massive stars, via High-Mass Protostellar Objects
(HMPOs) to the final stars that are able to produce Ultracompact H{\sc
  ii} regions (UCH{\sc ii}s, see \citealt{beuther2006b} for a recent
description of the evolutionary sequence). The first two evolutionary
stages are found within so-called Infrared Dark Clouds (IRDCs).  While
for low-mass stars the chemical evolution from early molecular
freeze-out to more evolved protostellar cores is well studied (e.g.,
\citealt{bergin1997,dutrey1997,pavlyuchenkov2006,joergensen2007}),
it is far from clear whether similar evolutionary patterns are present
during massive star formation.

To better understand the chemical evolution of high-mass star-forming
regions we initiated a program to investigate the chemical properties
from IRDCs to UCH{\sc ii}s from an observational and theoretical
perspective. We start with single-dish line surveys toward a large
sample obtaining their basic characteristics, and then perform
detailed studies of selected sources using interferometers on smaller
scales. These observations are accompanied by theoretical modeling of
the chemical processes.  Long-term goals are the chemical
characterization of the evolutionary sequence in massive star
formation, the development of chemical clocks, and the identification
of molecules as astrophysical tools to study the physical processes
during different evolutionary stages. Here, we present an initial
study of the reactive radical ethynyl (C$_2$H) combining single-dish
and interferometer observations with chemical modeling.  Although
C$_2$H was previously observed in low-mass cores and Photon Dominated
Regions (e.g., \citealt{millar1984,jansen1995}), so far it was not
systematically investigated in the framework of high-mass star
formation.

\section{Observations}
\label{obs}

The 21 massive star-forming regions were observed with the Atacama
Pathfinder Experiment (APEX) in the 875\,$\mu$m window in fall 2006.
We observed 1\,GHz from 338 to 339\,GHz and 1\,GHz in the image
sideband from 349 to 350\,GHz.  The spectral resolution was
0.1\,km\,s$^{-1}$, but we smoothed the data to
$\sim$0.9\,km\,s$^{-1}$. The average system temperatures were around
200\,K, each source had on-source integration times between 5 and 16
min. The data were converted to main-beam temperatures with forward
and beam efficiencies of 0.97 and 0.73, respectively
\citep{belloche2006}. The average $1\sigma$ rms was 0.4\,K.  The main
spectral features of interest are the C$_2$H lines around 349.4\,GHz
with upper level excitation energies $E_u/k$ of 42\,K (line blends of
C$_2$H$(4_{5,5}-3_{4,4})$ \& C$_2$H$(4_{5,4}-3_{4,3})$ at
349.338\,GHz, and C$_2$H$(4_{4,4}-3_{3,3})$ \&
C$_2$H$(4_{4,3}-3_{3,2})$ at 349.399\,GHz). The beam size was $\sim
18''$.

The original Submillimeter Array (SMA) C$_2$H data toward the
HMPO\,18089-1732 were first presented in \citet{beuther2005c}. There
we used the compact and extended configurations resulting in good
images for all spectral lines except of C$_2$H. For this project, we
re-worked on these data only using the compact configuration. Because
the C$_2$H emission is distributed on larger scales (see
\S\ref{results}), we were now able to derive a C$_2$H image. The
integration range was from 32 to 35\,km\,s$^{-1}$, and the achieved
$1\sigma$ rms of the C$_2$H image was 450\,mJy\,beam$^{-1}$.  For more
details on these observations see \citet{beuther2005c}.

\section{Results}
\label{results}

The sources were selected to cover all evolutionary stages from IRDCs
via HMPOs to UCH{\sc ii}s. We derived our target list from the samples
of \citet{klein2005,fontani2005,hill2005,beltran2006}.  Table
\ref{sample} lists the observed sources, their coordinates, distances,
luminosities and a first order classification into the evolutionary
sub-groups IRDCs, HMPOs and UCH{\sc ii}s based on the previously
available data. Although this classification is only based on a
limited set of data, here we are just interested in general
evolutionary trends. Hence, the division into the three main classes
is sufficient.

Figure \ref{spectra} presents sample spectra toward one source of each
evolutionary group. While we see several CH$_3$OH lines as well as
SO$_2$ and H$_2$CS toward some of the HMPOs and UCH{\sc ii}s but not
toward the IRDCs, the surprising result of this comparison is the
presence of the C$_2$H lines around 349.4\,GHz toward all source types
from young IRDCs via the HMPOs to evolved UCH{\sc ii}s.  Table
\ref{sample} lists the peak brightness temperatures, the integrated
intensities and the FWHM line-widths of the C$_2$H line blend at
349.399\,GHz. The separation of the two lines of 1.375\,MHz already
corresponds to a line-width of 1.2\,km\,s$^{-1}$. We have three C$_2$H
non-detections (2 IRDCs and 1 HMPO), however, with no clear trend with
respect to the distances or the luminosities (the latter comparison is
only possible for the HMPOs). While IRDCs are on average colder than
more evolved sources, and have lower brightness temperatures, the
non-detections are more probable due to the relatively low sensitivity
of the short observations (\S\ref{obs}). Hence, the data indicate
that the C$_2$H lines are detected independent of the evolutionary
stage of the sources in contrast to the situation with other
molecules.  When comparing the line-widths between the different
sub-groups, one finds only a marginal difference between the IRDCs and
the HMPOs (the average $\Delta v$ of the two groups are 2.8 and
3.1\,km\,s$^{-1}$).  However, the UCH{\sc ii}s exhibit significantly
broader line-widths with an average value of 5.5\,km\,s$^{-1}$.

Intrigued by this finding, we wanted to understand the C$_2$H spatial
structure during the different evolutionary stages.  Therefore, we
went back to a dataset obtained with the Submillimeter Array toward
the hypercompact H{\sc ii} region IRAS\,18089-1732 with a much higher
spatial resolution of $\sim 1''$ \citep{beuther2005c}.  Albeit this
hypercompact H{\sc ii} region belongs to the class of HMPOs, it is
already in a relatively evolved stage and has formed a hot core with a
rich molecular spectrum.  \citet{beuther2005c} showed the spectral
detection of the C$_2$H lines toward this source, but they did not
present any spatially resolved images. To recover large-scale
structure, we restricted the data to those from the compact SMA
configuration (\S\ref{obs}). With this refinement, we were able to
produce a spatially resolved C$_2$H map of the line blend at
349.338\,GHz with an angular resolution of $2.9''\times 1.4''$
(corresponding to an average linear resolution of 7700\,AU at the
given distance of 3.6\,kpc). Figure \ref{18089} presents the
integrated C$_2$H emission with a contour overlay of the 860\,$\mu$m
continuum source outlining the position of the massive protostar. In
contrast to almost all other molecular lines that peak along with the
dust continuum \citep{beuther2005c}, the C$_2$H emission surrounds the
continuum peak in a shell-like fashion.

\section{Discussion and Conclusions}

To understand the observations, we conducted a simple chemical
modeling of massive star-forming regions. A 1D cloud model with a mass
of 1200\,M$_\sun$, an outer radius of 0.36\,pc and a power-law density
profile ($\rho\propto r^p$ with $p=-1.5$) is the initially assumed
configuration. Three cases are studied: (1) a cold isothermal cloud
with $T=10$\,K, (2) $T=50$\,K, and (3) a warm model with a temperature
profile $T\propto r^q$ with $q=-0.4$ and a temperature at the outer
radius of 44\,K. The cloud is illuminated by the interstellar UV
radiation field (IRSF, \citealt{draine1978}) and by cosmic ray
particles (CRP). The ISRF attenuation by single-sized $0.1\mu$m
silicate grains at a given radius is calculated in a plane-parallel
geometry following \citet{vandishoeck1988}. The CRP ionization rate is
assumed to be $1.3\times 10^{-17}$~s$^{-1}$ \citep{spitzer1968}. The
gas-grain chemical model by \citet{vasyunin2008} with the desorption
energies and surface reactions from \citet{garrod2006} is used.
Gas-phase reaction rates are taken from RATE\,06 \citep{woodall2007},
initial abundances, were adopted from the ``low metal'' set of
\citet{lee1998}.

Figure \ref{model} presents the C$_2$H abundances for the three models
at two different time steps: (a) 100\,yr, and (b) in a more evolved
stage after $5\times10^4$\,yr. The C$_2$H abundance is high toward the
core center right from the beginning of the evolution, similar to
previous models (e.g., \citealt{millar1985,herbst1986,turner1999}).
During the evolution, the C$_2$H abundance stays approximately
constant at the outer core edges, whereas it decreases by more than
three orders of magnitude in the center, except for the cold $T=10$~K
model.  The C$_2$H abundance profiles for all three models show
similar behavior.

The chemical evolution of ethynyl is determined by relative removal
rates of carbon and oxygen atoms or ions into molecules like CO, OH,
H$_2$O. Light ionized hydrocarbons CH$^+_{\rm n}$ (n=2..5) are quickly
formed by radiative association of C$^+$ with H$_2$ and hydrogen
addition reactions: C$^+$ $\rightarrow$ CH$_2^+$ $\rightarrow$
CH$_3^+$ $\rightarrow$ CH$_5^+$.  The protonated methane reacts with
electrons, CO, C, OH, and more complex species at later stage and
forms methane.  The CH$_4$ molecules undergo reactive collisions with
C$^+$, producing C$_2$H$_2^+$ and C$_2$H$_3^+$. An alternative way to
produce C$_2$H$_2^+$ is the dissociative recombination of CH$_5^+$
into CH$_3$ followed by reactions with C$^+$.  Finally, C$_2$H$_2^+$
and C$_2$H$_3^+$ dissociatively recombine into CH, C$_2$H, and
C$_2$H$_2$. The major removal for C$_2$H is either the direct
neutral-neutral reaction with O that forms CO, or the same reaction
but with heavier carbon chain ions that are formed from C$_2$H by
subsequent insertion of carbon. At later times, depletion and
gas-phase reactions with more complex species may enter into this
cycle.  At the cloud edge the interstellar UV radiation
instantaneously dissociates CO despite its self-shielding,
re-enriching the gas with elemental carbon.

The transformation of C$_2$H into CO and other species proceeds
efficiently in dense regions, in particular in the ``warm'' model
where endothermic reactions result in rich molecular complexity of the
gas (see Fig.~\ref{model}).  In contrast, in the ``cold'' 10\,K model
gas-grain interactions and surface reactions become important.  As a
result, a large fraction of oxygen is locked in water ice that is hard
to desorb ($E_{\rm des} \sim 5500$~K), while half of the elemental
carbon goes to volatile methane ice ($E_{\rm des} \sim 1300$~K). Upon
CRP heating of dust grains, this leads to much higher gas-phase
abundance of C$_2$H in the cloud core for the cold model compared to
the warm model. The effect is not that strong for less dense regions
at larger radii from the center.

Since the C$_2$H emission is anti-correlated with the dust continuum
emission in the case of IRAS\,18089-1732 (Fig.\,\ref{18089}), we do
not have the H$_2$ column densities to quantitatively compare the
abundance profiles of IRAS\,18089-1732 with our model. However, data
and model allow a qualitative comparison of the spatial structures.
Estimating an exact evolutionary time for IRAS\,18089-1732 is hardly
possible, but based on the strong molecular line emission, its high
central gas temperatures and the observed outflow-disk system
\citep{beuther2004a,beuther2004b,beuther2005c}, an approximate age of
$5\times10^4$\,yr appears reasonable.  Although dynamical and chemical
times are not necessarily exactly the same, in high-mass star
formation they should not differ to much: Following the models by
\citet{mckee2003} or \citet{krumholz2006b}, the luminosity rises
strongly right from the onset of collapse which can be considered as a
starting point for the chemical evolution. At the same time disks and
outflows evolve, which should hence have similar time-scales.  The
diameter of the shell-like C$_2$H structure in IRAS\,18089-1732 is
$\sim 5''$ (Fig.\,\ref{18089}), or $\sim$9000\,AU in radius at the
given distance of 3.6\,kpc.  This value is well matched by the modeled
region with decreased C$_2$H abundance (Fig.\,\ref{model}).  Although
in principle optical depths and/or excitation effects could mimic the
C$_2$H morphology, we consider this as unlikely because the other
observed molecules with many different transitions all peak toward the
central submm continuum emission in IRAS\,18089-1732
\citep{beuther2005c}. Since C$_2$H is the only exception in that rich
dataset, chemical effects appear the more plausible explanation.

The fact that we see C$_2$H at the earliest and the later evolutionary
stages can be explained by the reactive nature of C$_2$H: it is
produced quickly early on and gets replenished at the core edges by
the UV photodissociation of CO. The inner ``chemical'' hole observed
toward IRAS\,18089-1732 can be explained by C$_2$H being consumed in
the chemical network forming CO and more complex molecules like larger
carbon-hydrogen complexes and/or depletion.

The data show that C$_2$H is not suited to investigate the central gas
cores in more evolved sources, however, our analysis indicates that
C$_2$H may be a suitable tracer of the earliest stages of (massive)
star formation, like N$_2$H$^+$ or NH$_3$ (e.g.,
\citealt{bergin2002,tafalla2004,beuther2005a,pillai2006}). While a
spatial analysis of the line emission will give insights into the
kinematics of the gas and also the evolutionary stage from chemical
models, multiple C$_2$H lines will even allow a temperature
characterization. With its lowest $J=1-0$ transitions around 87\,GHz,
C$_2$H has easily accessible spectral lines in several bands between
the 3\,mm and 850\,$\mu$m.  Furthermore, even the 349\,GHz lines
presented here have still relatively low upper level excitation
energies ($E_u/k\sim42$\,K), hence allowing to study cold cores even
at sub-millimeter wavelengths.  This prediction can further be proved
via high spectral and spatial resolution observations of different
C$_2$H lines toward young IRDCs.

\acknowledgments{H.B. acknowledges financial support
  by the Emmy-Noether-Programm of the Deutsche Forschungsgemeinschaft
  (DFG, grant BE2578). }

Answer: 21.


Input: 对于PD3.0协议，FS312BH支持的最高诱骗电压是多少？

Context:
'无锡速芯微电子有限公司是一家集芯片 研发，销售和服务于一体的国家高新技 术企业，为客户提供高性能，高集成 度，极致体验的全协议快充芯片。 无锡速芯微电子有限公司 FastSOC Microelectronics Co.,Ltd. 销售联系方式： 联系人：顾先生 手机：1800 185 3071 邮箱：gpp@fastsoc.com 网址：www.fastsoc.com 地址：无锡市新吴区菱湖大道200号中国物联网国际创新园E-503室 顾工微信号 速芯微公众号 免责声明：本文所述方法、方案均供客户参考，用于提示或者展示芯片应用的一种或者多种方式，不作为最终产品的实际方案。文中所描述的功能和性能指标在实 验室环境下测试得到，部分可以提供第三方测试报告，但是不保证客户产品上能获得相同的数据。本文信息只作为芯片使用的指导，不授权用户使用本公司或者其 他公司的知识产权。本文信息只作为芯片使用的指导，不承担因为客户自身应用不当而造成的任何损失。 **文中信息仅供参考，详情请联系我司获取最新资料” 无锡速芯微电子有限公司 FastSOC Microelectronics Co.,Ltd. 产品手册 2023年
新品快览 FS312A：PD3.0 诱骗- FS312A支持PD2.0/PD3.0最高诱骗电压：20V - FS312AE支持PD2.0/PD3.0 最高诱骗电压：20V支持Emarker模拟功能 - 封装：SOT23-5 VBUS CC1 CC2 DM DP 用电电路 4.7K 0.47uF R C C 1 V D D F U N C C C 2F S 3 1 2 B D M D P EP GND 应用图 FS8628：A+C快充协议CC2 CC1 VBUS CC2 CC1 FS312A FUNC GND VDD 4.7K GND R 用电电路 1uF GND 应用图 多口极简方案 FS8611SP*2+CCM-8611SP-A+7533B-T 双C智能降功率方案 FS8611S USB-C AC-DC 双变压器 7533B-T CCM-8611SP-A FS8611S USB-C 采用2颗FS8611SP搭配CCM-8611SP-A （MCU），7533B-T配合工作 - 支持多种协议 - 支持I2C控制 - 任意单 C 的为 35W - 双 插 降 功 率 ， 三 档 功 率 智 能 配 置：27.4W+7.4W；17.4W+17.4W； 27.4W - BOM极简，成本低 FS312B：PD3.1 诱骗FS8611K*2+CCM-8611K-A+7550B-T 双C方案 - FS312BL支持PD2.0/PD3.0/PD3.1/第三方协议最高诱骗电压：20V - FS312BLE支持PD2.0/PD3.0/PD3.1/第三方协议最高诱骗电压：20V支持Emarker模拟功能 - FS312BH支持PD2.0/PD3.0/PD3.1/第三方协议最高诱骗电压：48V - FS312BHE支持PD2.0/PD3.0/PD3.1/第三方协议最高诱骗电压：48V 支持Emarker模拟功能 - 封装：DFN2x2-6L - 兼容兼容BC1.2、Apple2.4A、 QC2.0 Class A、QC3.0 Class A/B、 FCP、SCP、AFC、低压直充等 - 兼容Type-C PD2.0、Type-C PD3.0、 Type-C PD3.0 PPS、QC4.0协议 - 支持两路DP/DM - 支持CV/CC（分段CC）功能 - 支持定制PDO - 支持A+C双口工作，电压自动回5V - 支持FB/OPTO反馈 - 封装：QFN3x3-20L VPWR FB PowerSystem 100K GND R1 GND 19 VIN 17 FB FUNC1 FUNC2 20 15 18 13 PLUGIND VFB FS8628 QFN3x3-20L AGATE 47K 7.5K 47K 7.5K 1 16 8 7 3 4 5 6 10 9 11 CGATE CVBUS CC2 CC1 CDP CDM AVBUS DM DP ISP ISN 12 应用图 2 V3P3 100Ω 1u EP GND GND CVBUS TYPE- C CC2 CC1 CDP CDM CGND TYPE-A AVBUS DM DP 10n 200 AGND 5mΩ GND FS8611K USB-C AC-DC DC-DC 7550B-T CCM-8611K-A FS8611K USB-C 采用2颗FS8611K搭配CCM-8611K-A （MCU）工作，7550B-T配合工作 - 支持PD2.0/PD3.0/QC2.0/AFC/FCP - 支持PDO定制 - 任意单 C 的为 35W(可定制) - 双插18W（可定制15W/20W） - BOM极简，成本低 FS212C+ACM-212C-A+7550B-T 双C方案 FS212C USB-C AC-DC DC-DC 7550B-T ACM-212C-A FS8623B-A+C方案 AC-DC DC-DC FS8623B USB-A USB-C USB-A 采 用 1 颗 F S 2 1 2 C 搭 配 ACM-212C-A 工 作，7550B-T配合工作 - 支持PD2.0/PD3.0 - 支持PDO定制 - 任意单 C 的为20W - 双插7.5W回5V - BOM极简，成本低 采用一颗FS8623B实现A+C方案 - 兼容兼容Apple2.4A/低压直充 QC2.0 Class A/QC3.0 Class A/B/ FCP/SCP等 - 兼 容Type -C PD2.0 / PD3.0 / PD3.0PPS/QC4.0协议 - 支持PDO定制 - 双插回5V
多口方案选型 产品选型 受电端芯片选型 速芯微现有多种多口的方案选择：A+C，C+C，C+C+A，C+C+C，C+C+A+A等方案。对于 A+C的方案，可使用1颗芯片实现，也可用多颗芯片来实现。 速芯微现有多种受电端诱骗芯片，客户可根据应用需求进行选择。 受电端诱骗芯片应用领域 筋膜枪 无线充 线材 无人机 产品型号 PD2.0 PD3.0 PD3.1 第三方协议 诱骗电压(V) 控制方式 内置Emarker 定制 封装 FS312A √ √ 5/9/12/15/20 电阻阻值 可变电压策略 SOT23-5 FS312AE √ √ 5/9/12/15/20 电阻阻值 √ (公头专用) 可变电压策略 SOT23-5 FS312BL √ √ √ √ 5/9/12/15/20 电阻阻值 可变电压策略 DFN2x2-6 FS312BLE √ √ √ √ 5/9/12/15/20 电阻阻值 √ (公头专用) 可变电压策略 DFN2x2-6 FS312BH √ √ √ √ 5/20/28/36/48 电阻阻值 可变电压策略 DFN2x2-6 FS312BHE √ √ √ √ 5/20/28/36/48 电阻阻值 √ (公头专用) 可变电压策略 DFN2x2-6 FS312LC √ √ √ 5/9/12 电阻阻值 可变第三方 协议 SSOP10 FS312HC √ √ √ 5/9/12/15/20 电阻阻值 可变第三方 协议 SSOP10 FS2711Q √ √ √ 任意设置 I2C √ QFN3x3-16 FS2711P √ √ √ 任意设置 I2C √ QFN3x3-16 FS2711PA √ √ 全协议 任意设置 I2C √ SSOP10 FS2711SW √ √ 全协议 SSOP10 FS512 √ √ 全协议 任意设置 I2C √ SSOP10 方案 类型 产品型号 单C 单A 双插 A+C方案 FS8623 20W（PPS）（可定制） A口全协议18w 5V共享3A FS8623B 20W（PPS）(可定制) A口全协议18w 5V共享3A FS8628 20W（PPS）(可定制) A口全协议18w 5V共享3A FS8611RPC+FS116DB 65W（PPS）(可定制) A口全协议18w A口：5V/2.4A C口：45W FS8628RC+FS116DB 35W(可定制) A口全协议18w A口：5V（BC1.2，Apple 2.4） C口：20W 方案类型 产品型号 单C1 单C2 C1/C2 C+C方案 FS8611RPB*2 30W(可定制) 30W(可定制) C1/C2：5V/3A(或5V/2.4A) FS8611GH*2 35W（可定制） 35W（可定制） C1/C2:18W（可定制） FS8628P*2 35W（可定制） 35W（可定制） C1/C2：17.4W可定制） FS8611KL*2 20W（可定制） 20W（可定制） C1/C2：5V/1.5 A FS8611PC*2 35W 35W C1/C2：18W FS8611BH*2 65W（可定制） 65W（可定制） C1：45W（可定制）C2：20W（可定制） FS8628RPC+FS8611RB 45W（可定制）） 36W （可定制）） C1：30W（可定制）C2：5V/1.5A（可定制） 方案类型 产品型号 单C1 单C2 单A C1+C2 C1/C2+A C1+C2+A C+C+A FS8611S*2+FS116DB 65W（可定制） 65W（ 可定制）） A口全协议18w 智能分配功率 45W+18W C1/C2：智能分配功率 A：18W（或5V1.5A） FS8612C+FS8628P 100W（可定制） 35W （可定制）） 20W C1：65W C2：20W C1+A：65W+20W C2+A：7.5W+7.5W C1：65W C2：7.5W A：7.5W 其他
Source-TYPE C协议芯片选型 Source-TYPE A协议芯片选型 速芯微现有多种TYPE-C的快充协议芯片，支持多种协议，支持客户定制，多样化，满 足客户对TYPE C的各种快充需求。 速芯微现有多种TYPE A快充协议芯片，支持全协议，支持定制，满足客户对A口协议的各种需 求。速芯微的TYPE-A快充协议芯片的协议丰富，FS112系列拥有多种的型号；FS116D 系列带插入指示，可搭配TYPE-C快充协议芯片，实现A+C，A+C+C，A+A+C+C等多口方 案，协议丰富，其中FS116A一般用于插入指示使用 Source-TYPE A协议芯片引脚封装图 D+ VSS FB 1 2 3 FS112 6 5 4 D- VDD FUNC GATE VIN FUNC FB LED/PLUG_IN 1 2 3 4 5 FS116D 10 DM 9 8 7 6 DP CSP CSN VSS速芯微的各TYPE-C快充协议芯片之间可搭配使用，实现多口方案，更多详情请咨 询我司工作人员。 多口降功率专用快充协议芯片：FS8611RB，FS8611RC，FS8611RPB，FS8611RPC， FS8612CP。 带I2C快充协议芯片：FS8611S，FS8611SP 产品型号 BC1.2 Apple 2.4 QC2.0 QC3.0 AFC FCP SCP HISCP 大电流直充 封装 FS112 √ √ √ √ √ √ √ SOT23-6 FS112H √ √ √ √ √ √ √ √ √ SOT23-6 FS113 √ v √ √ √ √ √ √ √ SOT23-6 FS116DP √ √ √ √ √ √ √ √ SSOP10 FS116DB √ √ √ √ √ √ √ √ SSOP10 FS116E √ √ √ √ √ √ √ √ √ SSOP10 FS116A √ √ SSOP10 其他 可定制 PD2.0 PD3.0 PD3.0 PPS 第三方协议 反馈方式 MOS CV/CC 定制 封装 FS212C √ √ FB √ SOT23-6 FS212CM √ √ FB PMOS(可省) √ SOT23-6 FS212D √ √ √ FB √ SOT23-6 FS212DH √ √ √ FB √ SOT23-6 FS212DP √ √ √ FB PMOS √ SOT23-6 FS212DG √ √ √ FB PMOS √ SOT23-6 FS8611G √ √ FB PMOS(可省) √ SOP-8 FS8611K √ √ QC2.0/AFC/FCP FB PMOS(可省) √ SOP8 FS8611J √ √ √ 全协议 FB PMOS(可省) √ SOP8 FS8611B √ √ √ 全协议 FB PMOS(可省) √ SSOP10 FS8611RB √ √ 全协议 FB PMOS √ SSOP10 FS8611RC √ √ 全协议 FB PMOS √ SSOP10 FS8611S √ √ √ 全协议 FB PMOS √ SSOP10 FS8611PP √ √ √ 全协议 FB PMOS √ SSOP10 FS8611BP √ √ √ 全协议 FB PMOS(可省) √ SSOP10 FS8611RPB √ √ √ 全协议 FB PMOS √ SSOP10 FS8611RPC √ √ √ 全协议 FB PMOS √ SSOP10 FS8611SP √ √ √ 全协议 FB PMOS(可省) SSOP10 FS8612 √ √ √ 全协议 OPTO PMOS √ √ SSOP16 FS8612B √ √ √ 全协议 FB PMOS √ √ SSOP16 FS8612BP √ √ √ 全协议 FB PMOS √ √ SSOP16 FS8612C √ √ √ 全协议 FB/OPTO PMOS √ √ QFN4x4-16 FS8612CP √ √ √ 全协议 FB/OPTO PMOS √ √ QFN4x4-16
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Answer: 48V.